UNITED STATES NAVAL OBSERVATORY

CIRCULAR NO. 179

U.S. Naval Observatory, Washington, D.C. 20392 2005 Oct 20

The IAU Resolutions

on Astronomical Reference Systems,

Time Scales, and Earth Rotation Models

Explanation and Implementation

by

George H. Kaplan
2
Contents

Introduction iii

Overview of the Resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

About this Circular . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Other Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

A Few Words about Constants viii

Abbreviations and Symbols Frequently Used x

1 Relativity 1

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 The BCRS and the GCRS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Computing Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Time Scales 7

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1 Different Flavors of Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Time Scales Based on the SI Second . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Time Scales Based on the Rotation of the Earth . . . . . . . . . . . . . . . . . . . . 10

2.4 Coordinated Universal Time (UTC) . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.5 To Leap or Not to Leap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.6 Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.6.1 Formulas for Time Scales Based on the SI Second . . . . . . . . . . . . . . . . 13

2.6.2 Formulas for Time Scales Based on the Rotation of the Earth . . . . . . . . . 16

3 The Fundamental Celestial Reference System 19

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1 The ICRS, the ICRF, and the HCRF . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2 Background: Reference Systems and Reference Frames . . . . . . . . . . . . . . . . . 21

3.3 Recent Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.4 ICRS Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.4.1 The Defining Extragalactic Frame . . . . . . . . . . . . . . . . . . . . . . . . 24

3.4.2 The Frame at Optical Wavelengths . . . . . . . . . . . . . . . . . . . . . . . . 24

3.4.3 Standard Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

i
ii

3.4.4 Relationship to Other Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.4.5 Data in the ICRS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.5 Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4 Ephemerides of the Major Solar System Bodies 29

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.1 The JPL Ephemerides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.2 DE405 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.3 Sizes, Shapes, and Rotational Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.4 DE405 Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5 Precession and Nutation 33

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.1 Aspects of Earth Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.2 Which Pole? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.3 The New Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.4 Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.4.1 Formulas for Precession . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.4.2 Formulas for Nutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.4.3 Alternative Combined Transformation . . . . . . . . . . . . . . . . . . . . . . 48

5.4.4 Observational Corrections to Precession-Nutation . . . . . . . . . . . . . . . . 49

6 Modeling the Earth’s Rotation 50

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

6.1 A Messy Business . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6.2 Non-Rotating Origins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

6.3 The Path of the CIO on the Sky . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

6.4 Transforming Vectors Between Reference Systems . . . . . . . . . . . . . . . . . . . . 55

6.5 Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6.5.1 Location of Cardinal Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6.5.1.1 CIO Location Relative to the Equinox . . . . . . . . . . . . . . . . . 59

6.5.1.2 CIO Location from Numerical Integration . . . . . . . . . . . . . . . 60

6.5.1.3 CIO Location from the Arc-Difference s . . . . . . . . . . . . . . . . 61

6.5.2 Geodetic Position Vectors and Polar Motion . . . . . . . . . . . . . . . . . . . 63

6.5.3 Complete Terrestrial to Celestial Transformation . . . . . . . . . . . . . . . . 64

6.5.4 Hour Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Text of IAU Resolutions of 1997 67

Text of IAU Resolutions of 2000 70

References 82

URLs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

IAU 2000A Nutation Series 88

Errata & Updates 104
Introduction

The series of resolutions passed by the International Astronomical Union at its General Assemblies

in 1997 and 2000 are the most significant set of international agreements in positional astronomy in

several decades and arguably since the Paris conference of 1896. The approval of these resolutions

culminated a process — not without controversy — that began with the formation of an intercommission Working Group on Reference Systems at the 1985 IAU General Assembly in Delhi.

The resolutions came at the end of a remarkable decade for astrometry, geodesy, and dynamical

astronomy. That decade witnessed the successes of the Hipparcos satellite and the Hubble Space

Telescope (in both cases, after apparently fatal initial problems), the completion of the Global

Positioning System, 25-year milestones in the use of very long baseline interferometry (VLBI) and

lunar laser ranging (LLR) for astrometric and geodetic measurements, the discovery of Kuiper

Belt objects and extra-solar planets, and the impact of comet Shoemaker-Levy 9 onto Jupiter.

At the end of the decade, interest in near-Earth asteroids and advances in sensor design were

motivating plans for rapid and deep all-sky surveys. Significant advances in theory also took place,

facilitated by inexpensive computer power and the Internet. Positional and dynamical astronomy

were enriched by a deeper understanding of chaos and resonances in the solar system, advances in

the theory of the rotational dynamics of the Earth, and increasingly sophisticated models of how

planetary and stellar systems form and evolve. It is not too much of an exaggeration to say that

as a result of these and similar developments, the old idea that astrometry is an essential tool of

astrophysics was rediscovered. The IAU resolutions thus came at a fortuitous time, providing a

solid framework for interpreting the modern high-precision measurements that are revitalizing so

many areas of astronomy.

This circular is an attempt to explain these resolutions and provide guidance on their implementation. This publication is the successor to USNO Circular 163 (1982), which had a similar

purpose for the IAU resolutions passed in 1976, 1979, and 1982. Both the 1976–1982 resolutions

and those of 1997–2000 provide the specification of the fundamental astronomical reference system,

the definition of time scales to be used in astronomy, and the designation of conventional models

for Earth orientation calculations (involving precession, nutation, and Universal Time). It will

certainly not go unnoticed by readers familiar with Circular 163 that the current publication is

considerably thicker. This reflects both the increased complexity of the subject matter and the

wider audience that is addressed.

Of course, the IAU resolutions of 1997–2000 did not arise in a vacuum. Many people participated

in various IAU working groups, colloquia, and symposia in the 1980s and 1990s on these topics,

and some important resolutions were in fact passed by the IAU in the early 1990s. Furthermore,

any set of international standards dealing with such fundamental matters as space and time must

to some extent be based on, and provide continuity with, existing practice. Therefore, many of the

new resolutions carry “baggage” from the past, and there is always the question of how much of this

history (some of it quite convoluted) is important for those who simply wish to implement the latest

iii
iv INTRODUCTION

recommendations. Material in this circular generally avoids detailed history in an effort to present

the most succinct and least confusing picture possible. However, many readers will be involved with

modifying existing software systems, and some mention of previous practice is necessary simply to

indicate what needs to be changed. A limited amount of background material also sometimes aids

in understanding and provides a context for the new recommendations. The reader should be

aware that the presentation of such material is selective and no attempt at historical completeness

is attempted.

It must be emphasized that the resolutions described here affect astronomical quantities only at

the level of some tens of milliarcseconds or less at the present epoch. And, despite misinformation to

the contrary, familiar concepts such as the equinox and sidereal time have not been discarded. The

largest systematic change is due to the new rate of precession, which is 0.3 arcsecond per century

less than the previous (1976) rate; the change affects some types of astronomical coordinates and

sidereal time. Astronomical software applications that work acceptably well now at the arcsecond

or 0.1-arcsecond level (which would include most telescope control systems) will continue to work

at that level, even when used with new sources of reference data, such as the Hipparcos, Tycho-2,

or 2MASS star catalogs or the VCS3 radio source catalog. Applications that are independent of

the rotation of the Earth, such as those for differential (small-field) astrometry, are largely unaffected. For these kinds of systems, changes to computer code that implement the new resolutions

are recommended as a long-term goal, to maintain standardization of algorithms throughout the

astronomical community, but are not an immediate practical necessity. (Perhaps some readers will

stop right here and file this circular on the shelf!)

Overview of the Resolutions

The IAU resolutions described in this circular cover a range of fundamental topics in positional

astronomy:

• Relativity Resolutions passed in 2000 provide the relativistic metric tensors for reference

systems with origins at the solar system barycenter and the geocenter, and the transformation

between the two systems. While these are mostly of use to theorists — for example, in the

formulation of accurate models of observations — they provide a proper relativistic framework

for current and future developments in precise astrometry, geodesy, and dynamical astronomy.

(See Chapter 1.)

• Time Scales Resolutions passed in 1991 and 2000 provide the definitions of various kinds

of astronomical time and the relationships between them. Included are time scales based on

the Syst`eme International (SI) second (“atomic” time scales) as well as those based on the

rotation of the Earth. (See Chapter 2.)

• The Fundamental Astronomical Reference System A resolution passed in 1997 established the International Celestial Reference System (ICRS), a high precision coordinate

system with its origin at the solar system barycenter and “space fixed” (kinematically nonrotating) axes. The resolution included the specification of two sets of benchmark objects

and their coordinates, one for radio observations (VLBI-measured positions of pointlike extragalactic sources) and one for optical observations (Hipparcos-measured positions of stars).

These two sets of reference objects provide the practical implementation of the system and

allow new observations to be related to it. (See Chapter 3.)
INTRODUCTION v

• Precession and Nutation Resolutions passed in 2000 provided a new precise definition

of the celestial pole and endorsed a specific theoretical development for computing its instantaneous motion. The celestial pole to which these developments refer is called the Celestial

Intermediate Pole (CIP); the instantaneous equatorial plane is orthogonal to the CIP. There

are now new precise algorithms for computing the pole’s position on the celestial sphere at

any time, in the form of new expressions for precession and nutation. (See Chapter 5.)

• Earth Rotation A resolution passed in 2000 establishes new reference points, one on the

celestial sphere and one on the surface of the Earth, for the measurement of the rotation of the

Earth about its axis. The new points are called, respectively, the Celestial Intermediate Origin

(CIO) and the Terrestrial Intermediate Origin (TIO). Both lie in the instantaneous equatorial

plane. The rotation of the Earth is simply the geocentric angle, θ, between these two points, a

linear function of Universal Time (UT1). The CIO is analogous to the equinox, the reference

point on the celestial sphere for sidereal time. Unlike the equinox, however, the CIO has no

motion along the instantaneous equator, and unlike sidereal time, θ is not “contaminated”

by precession or nutation. The new CIO-TIO-based Earth rotation paradigm thus allows a

clean separation of Earth rotation, precession, and nutation in the transformation between

terrestrial and celestial reference systems. (See Chapter 6.)

This circular also includes a brief description of the de facto standard solar system model,

produced and distributed by the Jet Propulsion Laboratory (see Chapter 4). This model, labeled

DE405/LE405, provides the positions and velocities of the nine major planets and the Moon with

respect to the solar system barycenter for any date and time between 1600 and 2200. The positions

and velocities are given in rectangular coordinates, referred to the ICRS axes. This ephemeris is

not the subject of any IAU resolutions but has become widely adopted internationally; for example,

it is the basis for the tabulations in The Astronomical Almanac and it underlies some of the other

algorithms presented in this circular.

The 1997 and 2000 IAU resolutions form an interrelated and coherent set of standards for

positional astronomy. For example, the definitions of the SI-based time scales rely on the relativity

resolutions, and the position of the Celestial Intermediate Pole and the Celestial Intermediate

Origin can only be properly computed using the new precession and nutation expressions. Many

other links between the resolutions exist. In fact, attempting to apply the resolutions selectively

can lead to quite incorrect (or impossible to interpret) results. This circular is meant to provide an

explanatory and computational framework for a holistic approach to implementing these resolutions

in various astronomical applications. The author hopes that what is presented here does justice to

the efforts of the many people who worked very hard over the last decade to take some important

scientific ideas and work out their practical implications for positional astronomy, to the benefit of

the entire scientific community.

About this Circular

The chapters in this circular reflect the six main subject areas described above. Each of the

chapters contains a list of the relevant IAU resolutions, a summary of the recommendations, an

explanatory narrative, and, in most chapters, a collection of formulas used in implementing the

recommendations. The references for all chapters are collected in one list at the end of the circular

(p. 82 ff.). The reference list is in the usual style for astronomy journal articles. At the end of the

references, a list of Uniform Resource Locators (URLs) is given (p. 87) for documents and data
vi INTRODUCTION

that are available on the World Wide Web. These URLs are numbered and they are referred to in

the text by number — for example, a PDF version of this circular can be found at URL 1.

It is assumed that readers have a basic knowledge of positional astronomy; that the terms right

ascension, declination, sidereal time, precesssion, nutation, equinox, ecliptic, and ephemeris are

familiar. Some experience in computing some type of positional astronomy data is useful, because

the ultimate purpose of the circular is to enable such computations to be carried out in accordance

with the 1997 and 2000 IAU resolutions. The explanatory narratives deal primarily with new or

unfamiliar concepts introduced by the resolutions — concepts that would not generally be described

in most introductory textbooks on positional astronomy. This circular is not a substitute for such

textbooks.

IAU resolutions are referred to in the text in the form “res. N of year”, for example, “res. B1.2

of 2000”. The year refers to the year of the IAU General Assembly that passed the resolution. The

proceedings of each General Assembly, including the text of the resolutions, are usually published

the following year. The References section of this circular lists the various proceedings volumes

under “IAU”. An online reference for the text of IAU resolutions (beginning with those passed at

the 1994 General Assembly) is the IAU Information Bulletin (IB) series, at URL 2. Resolutions

are printed in the January IB following a General Assembly, i.e., IB numbers 74, 81, 88, 94, etc.

This circular contains two appendices containing the complete text of the resolutions passed by the

1997 and 2000 General Assemblies, which are the main focus of attention here.

Errata in this circular and updates to it are provided at URL 1.

Other Resources

An increasing number of publications, data, and software related to the recent IAU resolutions are

becoming available.

A major online resource for implementing the IAU resolutions involving Earth rotation and

time (Chapters 2, 5, and 6 here) is the document of conventions used by the International Earth

Rotation and Reference Systems Service (IERS): IERS Conventions (2003), IERS Technical Note

No. 32, edited by D. D. McCarthy and G. Petit. It is available in printed form from the IERS

and also on the web at URL 3. The online document contains links to Fortran subroutines that

implement the recommended models. The document also contains algorithms specific to geodetic

applications, such as tidal and geopotential models, that have not been the subject of IAU action

and are not discussed in this circular. The IERS also maintains an online list of FAQs on the IAU

resolutions (URL 4).

The IAU Working Group on Nomenclature for Fundamental Astronomy (2003–2006) has a

website (URL 6) with many helpful documents, including a list of definitions (some of which are

used in this circular) and other educational material.

In addition to the IERS software, two other packages of computer subroutines are available for

implementing the IAU resolutions: the Standards of Fundamental Astronomy (SOFA), at URL 7,

and the Naval Observatory Vector Astrometry Subroutines (NOVAS), at URL 8. SOFA is a collection of routines managed by an international panel, the SOFA Reviewing Board, that works under

the auspices of IAU Division 1 and is chaired by P. Wallace. The board has adopted a set of Fortran

coding standards for algorithm implementations (C versions are contemplated for the future) and

is soliciting code from the astrometric and geodetic communities that implements IAU models.

Subroutines are adapted to the coding standards and validated for accuracy before being added

to the SOFA collection. NOVAS is an integrated package of subroutines, available in Fortran and
INTRODUCTION vii

C, for the computation of a wide variety of common astrometric quantities and transformations.

NOVAS dates back to the 1970s but has been continually updated to adhere to subsequent IAU

resolutions.

The Astronomical Almanac, beginning with the 2006 edition, is also a resource for implementing

the IAU resolutions. Not only does it list various algorithms arising from or consistent with the

resolutions, but its tabular data serve as numerical checks for independent developments. Both

SOFA and NOVAS subroutines are used in preparing the tabulations in The Astronomical Almanac,

and various checks have been made to ensure the consistency of the output of the two software

packages.

Acknowledgments

Many people have contributed in some way to this circular. I have had many interesting and enlightening discussions with James Hilton, Victor Slabinski, and Patrick Wallace. I wish to thank

Bill Tangren for setting me straight on some of the basic concepts of general relativity, and Sergei

Klioner and Sergei Kopeikin for patiently trying to explain to me the relativistic aspects of the 2000

IAU resolutions. Sergei Klioner provided some text that I have used verbatim. I hope that the final

product properly represents their input. The e-mail discussions within the IAU Working Group

on Nomenclature for Fundamental Astronomy have also been quite valuable and have contributed

to what is presented here. The many scientific papers written by Nicole Capitaine and her collaborators have been essential references. Victor Slabinski and Dennis McCarthy of the U.S. Naval

Observatory, Ken Seidelmann of the University of Virginia, Catherine Hohenkerk of Her Majesty’s

Nautical Almanac Office, and Myles Standish of the Jet Propulsion Laboratory carefully reviewed

drafts of this circular and made many substantive suggestions for improvement. The remaining

defects in the circular are, of course, my sole responsibility.

— George Kaplan
A Few Words about Constants

This circular does not contain a list of adopted fundamental astronomical constants, because the

IAU is no longer maintaining such a list. The last set of officially adopted constant values was the

IAU (1976) System of Astronomical Constants. That list is almost entirely obsolete. For a while,

an IAU working group maintained a list of “best estimates” of various constant values, but the IAU

General Assembly of 2003 did not renew that mandate. It can be argued that a list of fundamental

astronomical constants is no longer possible, given the complexity of the models now used and the

many free parameters that must be adjusted in each model to fit observations. That is, there are

more constants now to consider, and their values are theory dependent. In many cases, it would be

incorrect to attempt to use a constant value, obtained from the fit of one theory to observations,

with another theory.

We are left with three defining constants with IAU-sanctioned values that are intended to be

fixed:

1. The Gaussian gravitational constant: k = 0.01720209895. The dimensions of k

2 are

AU3M
·

−1d−2 where AU is the astronomical unit, M
· is the solar mass, and d is the day

of 86400 seconds.

2. The speed of light: c = 299 792 458 m s−1.

3. The fractional difference in rate between the time scales TT and TCG: LG = 6.969290134×10−10.

Specifically, the derivative dTT/dTCG = 1 − LG. (See Chapter 3.)

The IERS Conventions (2003) includes a list of constants as its Table 1.1. Several useful ones

from this list that are not highly theory dependent (for astronomical use, at least) are:

1. Equatorial radius of the Earth: aE = 6 378 136.6 m.

2. Flattening factor of the Earth: f = 1/298.25642.

3. Dynamical form factor of the Earth: J2 = 1.0826359×10−3.

4. Nominal mean angular velocity of Earth rotation: ω = 7.292115×10−5rad s−1.

5. Constant of gravitation: G = 6.673×10−11 m3kg−1s

−2

(CODATA 2002 recommended

value: 6.6742×10−11 m3kg−1s

−2

).

The first four values above were recommended by Special Commission 3 of the International

Association of Geodesy; the first three are so-called “zero tide” values. (The need to introduce

the concept of “zero tide” values indicates how theory creeps into even such basic constants as

the radius of the Earth as the precision of measurement increases. See section 1.1 of the IERS

viii
CONSTANTS ix

Conventions (2003).) Planetary masses, the length of the astronomical unit, and related constants

used in or obtained from the Jet Propulsion Laboratory DE405/LE405 ephemeris are listed with its

description in Chapter 4. The rate of general precession in longitude (the “constant of precession”)

is given in Chapter 5 on the precession and nutation theories.

The World Geodetic System 1984 (WGS 84), which is the basis for coordinates obtained from

GPS, uses an Earth ellipsoid with aE = 6378137 m and f = 1/298.257223563.

Some astronomical “constants” (along with reference data such as star positions) actually represent quantities that slowly vary, and the values given must therefore be associated with a specific

epoch. That epoch is now almost always 2000 January 1, 12h (JD 2451545.0), which can be expressed in any of the usual time scales. If, however, that epoch is considered an event at the

geocenter and given in the TT time scale, the epoch is designated J2000.0. See Chapter 2.
Abbreviations and Symbols

Frequently Used

and index to most relevant sections

α right ascension 3.2

δ declination 3.2

∆ψ nutation in [ecliptic] longitude (usually expressed in arcseconds) 5.3, 5.4.2

∆ nutation in obliquity (usually expressed in arcseconds) 5.3, 5.4.2

 mean obliquity of date 5.4.1



0

true obliquity of date (=  + ∆) 5.4.2

0 mean obliquity of J2000.0 5.3, 5.4.1

θ Earth Rotation Angle 2.3, 6.2

μas microarcecond (= 10−6 arcsecond ≈ 4.8×10−12 radian)

σ a non-rotating origin or, specifically, the CIO 6.2

σ unit vector toward a non-rotating origin or, specifically, the CIO 6.5.1

Υ the equinox 3.2, 6.2

Υ unit vector toward the equinox 6.5.1

as or 00 arcsecond (= 1/3600 degree ≈ 4.8×10−6radian)

AU astronomical unit(s)

B frame bias matrix 3.5

BCRS Barycentric1 Celestial Reference System 1.2

BIPM Bureau International des Poids et Mesures

C matrix for transformation from GCRS to Eσ 5.4.3, 6.5.3

cen century, specifically, the Julian century of 36525 days of 86400 seconds

CIO Celestial Intermediate Origin2 3 6.2, 6.5.1

CIP Celestial Intermediate Pole 5.2, 6.5.1

CIRS (See Eσ)

EΥinstantaneous (true) equator and equinox of date 6.4, 6.5.3

Eσ Celestial Intermediate Reference System (CIRS) 6.4, 6.5.3

E$ Terrestrial Intermediate Reference System (TIRS) 6.4, 6.5.3

EΥ equation of the equinoxes 2.3, 2.6.2

Eo equation of the origins 6.5.1.1, 6.5.4

ESA European Space Agency

FKn nth Fundamental Catalog (Astronomisches Rechen-Institut, Heidelberg) 3.3

GAST Greenwich apparent sidereal time 2.3, 2.6.2

GCRS Geocentric Celestial Reference System 1.2

x
ABBREVIATIONS & SYMBOLS xi

GMST Greenwich mean sidereal time 2.3, 2.6.2

GPS Global Positioning System

HCRF Hipparcos Celestial Reference Frame 3.1, 3.4

IAG International Association of Geodesy

IAU International Astronomical Union

ICRF International Celestial Reference Frame 3.1, 3.4

ICRS International Celestial Reference System 3.1, 3.4

IERS International Earth Rotation and Reference System Service

ITRF International Terrestrial Reference Frame 6.4

ITRS International Terrestrial Reference System 6.4, 6.5.2

IUGG International Union of Geodesy and Geophysics

J2000.0 the epoch 2000 January 1, 12h TT (JD 2451545.0 TT) at the geocenter 2.2

(“J2000.0 system” is shorthand for the celestial reference system defined

by the mean dynamical equator and equinox of J2000.0.)

3.2

JD Julian date (time scale used should be specified)

JPL Jet Propulsion Laboratory

mas milliarcsecond (= 10−3 arcsecond ≈ 4.8×10−9radian)

N nutation matrix (for transformation from mean to true system of date) 5.4, 5.4.2

n unit vector toward the CIP (celestial pole) 5.4,6.5.1

NOVAS Naval Observatory Vector Astrometry Subroutines (software)

P precession matrix (for transformation from J2000.0 system to mean system of date)5.4, 5.4.2

R1(φ) rotation matrix to transform column 3-vectors from one cartesian coordinate system to another. Final system is formed by rotating original

system about its own x-axis by angle φ (counterclockwise as viewed from

the +x direction):

R1(φ) =





1 0 0

0 cos φ sin φ

0 − sin φ cos φ





R2(φ) rotation matrix to transform column 3-vectors from one cartesian coordinate system to another. Final system is formed by rotating original

system about its own y-axis by angle φ (counterclockwise as viewed from

the +y direction):

R2(φ) =





cos φ 0 − sin φ

0 1 0

sin φ 0 cos φ





R3(φ) rotation matrix to transform column 3-vectors from one cartesian coordinate system to another. Final system is formed by rotating original

system about its own z-axis by angle φ (counterclockwise as viewed from

the +z direction):

R3(φ) =





cos φ sin φ 0

− sin φ cos φ 0

0 0 1




xii ABBREVIATIONS & SYMBOLS

s CIO locator: the difference between two arcs on the celestial sphere,

providing the direction toward the CIO

6.5.1.3

SI Syst`eme International d’Unit ́es (International System of Units)

SOFA Standards of Fundamental Astronomy (software)

T unless otherwise specified, time in Julian centuries (36525 days of 86400

seconds) from JD 2451545.0 (2000 Jan 1.5)

(The time scale used should be specified, otherwise TT is understood.)

Teph time argument of JPL planetary and lunar ephemerides 2.2, 4.2

TAI International Atomic Time 2.2

TCB Barycentric1 Coordinate Time 1.2, 2.2

TCG Geocentric Coordinate Time 1.2, 2.2

TDB Barycentric1 Dynamical Time 2.2

TIO Terrestrial Intermediate Origin2 6.2, 6.5.2

TIRS (See E$)

TT Terrestrial Time 2.2

UCAC USNO CCD Astrographic Catalog 3.4.5

USNO U.S. Naval Observatory

UT1 Universal Time (affected by variations in length of day) 2.3, 2.6.2

UTC Coordinated Universal Time (an atomic time scale) 2.4

VLBI very long baseline [radio] interferometry

W “wobble” (polar motion) matrix (for transformation from ITRS to E$) 6.5.2

WGS 84 World Geodetic System 1984 6.4

X

Y

Z







components of nGCRS, unit vector toward the CIP with respect to the

GCRS

5.4, 6.5.1

xp

yp

)

standard polar motion parameters, defining location of the CIP in the

ITRS

6.5.2

1 “Barycentric” always refers to the solar system barycenter, the center of mass of all bodies

in the solar system.

2 The fundamental reference points referred to here as the Celestial Intermediate Origin (CIO)

and the Terrestrial Intermediate Origin (TIO) were called, respectively, the Celestial Ephemeris

Origin (CEO) and the Terrestrial Ephemeris Origin (TEO) in the IAU resolutions of 2000. The

IAU Working Group on Nomenclature for Fundamental Astronomy (URL 6) has recommended the

change of nomenclature with no change in the definitions. The new terminology is already in use

in The Astronomical Almanac and in IERS documents, and will undoubtedly be adopted by the

IAU General Assembly in 2006. It is used throughout this circular, except in the verbatim text of

the IAU resolutions.

3 The abbreviation CIO was used throughout much of the 20th century to designate the

Conventional International Origin, the reference point for the measurement of polar motion.
Chapter 1

Relativity

Relevant IAU resolutions: A4.I, A4.II, A4.III, A4.IV of 1991; B1.3, B1.4, B1.5 of 2000

Summary In 2000, the IAU defined a system of space-time coordinates for (1) the solar system, and (2) the Earth, within the framework of General Relativity, by specifying

the form of the metric tensors for each and the 4-dimensional space-time transformation between them. The former is called the Barycentric Celestial Reference System

(BCRS) and the latter is called the Geocentric Celestial Reference System (GCRS).

The BCRS is the system appropriate for the basic ephemerides of solar system objects

and astrometric reference data on galactic and extragalactic objects. The GCRS is the

system appropriate for describing the rotation of the Earth, the orbits of Earth satellites, and geodetic quantities such as instrument locations and baselines. The analysis

of precise observations inevitably involves quantities expressed in both systems and the

transformations between them.

1.1 Background

Although the theory of relativity has been with us for a century (Einstein’s first papers on special

relativity were published in 1905), it has only been within the last few decades that it has become a

routine consideration in positional astronomy. The reason is simply that the observational effects of

both special and general relativity are small. In the solar system, deviations from Newtonian physics

did not need to be taken into account — except for the advance of the perihelion of Mercury — until

the advent of highly precise “space techniques” in the 1960s and 1970s: radar ranging, spacecraft

ranging, very long baseline interferometry (VLBI), pulsar timing, and lunar laser ranging (LLR).

More recently, even optical astrometry has joined the list, with wide-angle satellite measurements

(Hipparcos) at the milliarcsecond level. Currently, the effects of relativity are often treated as

small corrections added to basically Newtonian developments. But it has become evident that

the next generation of instrumentation and theory will require a more comprehensive approach,

one that encompasses definitions of such basic concepts as coordinate systems, time scales, and

units of measurement in a relativistically consistent way. It may remain the case that, for many

applications, relativistic effects can either be ignored or handled as second-order corrections to

Newtonian formulas. However, even in such simple cases, the establishment of a self-consistent

relativistic framework has benefits — it at least allows the physical assumptions and the errors

involved to be more clearly understood.

1
2 RELATIVITY

In 1991, the IAU made a series of recommendations concerning how the theory of relativity could

best be incorporated into positional astronomy. These recommendations and their implications were

studied by several working groups in the 1990s and some deficiencies were noted. As a result, a

series of new recommendations was proposed and discussed at IAU Colloquium 180 (Johnston et

al. 2000). The new recommendations were passed by the IAU General Assembly in 2000. It is

these recommendations that are described briefly in this chapter.

In special relativity, the Newtonian idea of absolute time in all inertial reference systems is

replaced by the concept that time runs differently in different inertial systems, in such a way that

the speed of light has the same measured value in all of them. In both Newtonian physics and

special relativity, inertial reference systems are preferred: physical laws are simple when written in

terms of inertial coordinates. In general relativity, however, time (and even space-time) is influenced

not only by velocity but also by gravitational fields, and there are no preferred reference systems.

One can use, in principle, any reference system to model physical processes. For an infinitely small

space-time region around an observer (considered to be a massless point), one can introduce socalled locally inertial reference systems where, according to the Einstein’s equivalence principle, all

physical laws have the same form as in an inertial reference system in special relativity. Such locally

inertial reference systems are used to describe observations taken by the point-like observer. In

general-relativistic reference systems of finite spatial extent, the geometry of space-time is defined

by a metric tensor, a 4×4 matrix of mathematical expressions, that serves as an operator on two 4-

vectors. In its simplest application, the metric tensor directly yields the generalized (4-dimensional)

distance between two neighboring space-time events. The metric tensor effectively determines the

equations through which physics is described in the reference system.

Time in general relativity can be understood as follows. As a particle moves through space-time,

each point (a space-time event) on the path that it follows can be characterized by a set of four

numbers. These four numbers are the values of the four coordinates in four-dimensional space-time

for a given coordinate system. For the same path in a different coordinate system, the numbers will,

in general, be different. Proper time is simply the time kept by a clock co-moving with the particle,

in whatever trajectory and gravity field it finds itself. Proper time is always measurable if a clock is

available that can travel with the particle. Coordinate time is one of the four independent variables

used to characterize a space-time event. Coordinate time is not measurable. The coordinate time of

a reference system is the independent argument of the equations of motion of bodies in that reference

system. The IAU resolutions on relativity passed in 2000 are concerned with two coordinate frames,

one barycentric and one geocentric, and the coordinate times used in each one.

1.2 The BCRS and the GCRS

In res. B1.3 of 2000, the IAU defined two coordinate frames for use in astronomy, one with its origin

at the solar system barycenter and one with its origin at the geocenter. In current astronomical

usage these are referred to as reference systems. (The astronomical distinction between reference

systems and reference frames is discussed in Chapter 3.) The two systems are the Barycentric

Celestial Reference System (BCRS) and the Geocentric Celestial Reference System (GCRS). Harmonic coordinates are recommended for both systems (i.e., the harmonic gauge is used). The

resolution provides the specific forms of the metric tensors for the two coordinate systems and the

4-dimensional transformation between them. (The latter would reduce to a Lorentz transformation

for a fictitious Earth moving with constant velocity in the absence of gravitational fields.) The general forms of the gravitational potentials, which appear in the metric tensors, are also presented. In
RELATIVITY 3

res. B1.4, specific expansions of the Earth’s gravitational potential in the GCRS are recommended.

In res. B1.5, the relationship between the coordinate time scales for the two reference systems,

Barycentric Coordinate Time (TCB), and Geocentric Coordinate Time (TCG), is given. Each of

the resolutions is mathematically detailed, and the formulas may be found in the text of the resolutions at the end of this circular. For interested readers, the paper titled “The IAU 2000 Resolutions

for Astrometry, Celestial Mechanics, and Metrology in the Relativistic Framework: Explanatory

Supplement” (Soffel et al. 2003), is highly recommended as a narrative on the background, meaning, and application of the relativity resolutions. Here we will make only very general comments on

the BCRS and GCRS, although the time scales TCB and TCG are described in a bit more detail

in Chapter 2.

The BCRS is a “global” reference system in which the positions and motions of bodies outside

the immediate environment of the Earth are to be expressed. It is the reference system appropriate

for the solution of the equations of motion of solar system bodies (that is, the development of

solar system ephemerides) and within which the positions and motions of galactic and extragalactic objects are most simply expressed. It is the system to be used for most positional-astronomy

reference data, e.g., star catalogs. The GCRS is a “local” reference system for Earth-based measurements and the solution of the equations of motion of bodies in the near-Earth environment,

e.g., artificial satellites. The time-varying position of the Earth’s celestial pole is defined within

the GCRS (res. B1.7 of 2000). Precise astronomical observations involve both systems: the instrumental coordinates, boresights, baselines, etc., may be expressed in the GCRS, but in general we

want the astronomical results expressed in the BCRS where they are easier to interpret. Thus it

is unavoidable that data analysis procedures for precise techniques will involve both GCRS and

BCRS quantities and the transformation between them. For example, the basic equation for VLBI

delay (the time difference between wavefront arrivals at two antennas) explicitly involves vectors

expressed in both systems — antenna-antenna baselines are given in the GCRS, while solar system

coordinates and velocities and quasar directions are expressed in the BCRS. Various relativistic

factors connect the two kinds of vectors.

In the 2000 resolutions, the coordinate axes of the two reference systems do not have a defined

orientation. They are described as kinematically nonrotating, which means that the axes have

no systematic rotation with respect to distant objects in the universe (and specifically the radio

sources that make up the ICRF — see Chapter 3). Since the axis directions are not specified,

one interpretation of the 2000 resolutions is that the BCRS and GCRS in effect define families

of coordinate systems, the members of which differ only in overall orientation. The IAU Working

Group on Nomenclature for Fundamental Astronomy has recommended that the directions of the

coordinate axes of the BCRS be understood to be those of the International Celestial Reference

System (ICRS) described in Chapter 3. And, since the transformation between the BCRS and

GCRS is specified in the resolutions, the directions of the GCRS axes are also implicitly defined

by this understanding. Here are the definitions of the two systems recommended by the working

group:

Barycentric Celestial Reference System (BCRS): A system of barycentric spacetime coordinates for the solar system within the framework of General Relativity with

metric tensor specified by the IAU 2000 Resolution B1.3. Formally, the metric tensor of

the BCRS does not fix the coordinates completely, leaving the final orientation of the

spatial axes undefined. However, for all practical applications, unless otherwise stated,

the BCRS is assumed to be oriented according to the ICRS axes.
4 RELATIVITY

Geocentric Celestial Reference System (GCRS): A system of geocentric spacetime coordinates within the framework of General Relativity with metric tensor specified

by the IAU 2000 Resolution B1.3. The GCRS is defined such that the transformation

between BCRS and GCRS spatial coordinates contains no rotation component, so that

GCRS is kinematically non-rotating with respect to BCRS. The equations of motion

of, for example, an Earth satellite with respect to the GCRS will contain relativistic

Coriolis forces that come mainly from geodesic precession. The spatial orientation of

the GCRS is derived from that of the BCRS, that is, unless otherwise stated, by the

orientation of the ICRS.

Because, according to the last sentence of the GCRS definition, the orientation of the GCRS

is determined by that of the BCRS, and therefore the ICRS, in this circular the GCRS will often

be described as the “geocentric ICRS”. However, this sentence does not imply that the spatial

orientation of the GCRS is the same as that of the BCRS (ICRS). The relative orientation of these

two systems is embodied in the 4-dimensional transformation given in res. B1.3 of 2000, which,

we will see in the next section, is itself embodied in the algorithms used to compute observable

quantities from BCRS (ICRS) reference data. From another perspective, the GCRS is just a

rotation (or series of rotations) of the international geodetic system (discussed in Chapter 6). The

geodetic system rotates with the crust of the Earth, while the GCRS has no systematic rotation

relative to extragalactic objects.

The above definition of the GCRS also indicates some of the subtleties involved in defining

the spatial orientation of its axes. Without the kinematically non-rotating constraint, the GCRS

would have a slow rotation with respect to the BCRS, the largest component of which is called

geodesic (or de Sitter-Fokker) precession. This rotation, approximately 1.9 arcseconds per century,

would be inherent in the GCRS if its axes had been defined as dynamically non-rotating rather than

kinematically non-rotating. By imposing the latter condition, Coriolis terms must be added (via the

inertial parts of the potentials in the metric; see notes to res. B1.3 of 2000) to the equations of motion

of bodies expressed in the GCRS. For example, as mentioned above, the motion of the celestial

pole is defined within the GCRS, and geodesic precession appears in the precession-nutation theory

rather than in the transformation between the GCRS and BCRS. Other barycentric-geocentric

transformation terms that affect the equations of motion of bodies in the GCRS because of the

axis-orientation constraint are described in Soffel et al. (2003, section 3.3) and Kopeikin & Vlasov

(2004, section 6).

1.3 Computing Observables

Ultimately, the goal of these theoretical formulations is to facilitate the accurate computation of the

values of observable astrometric quantities (transit times, zenith distances, focal plane coordinates,

interferometric delays, etc.) at the time and place of observation, that is, in the proper reference

system of the observer. There are some subtleties involved because in Newtonian physics and

special relativity, observables are directly related to some inertial coordinate, while according to

the rules of general relativity, observables must be computed in a coordinate-independent manner.

In any event, to obtain observables, there are a number of calculations that must be performed.

These begin with astrometric reference data: a precomputed solar system ephemeris and, if a star

is involved, a star catalog with positions and proper motions listed for a specified epoch. The

computations account for the space motion of the object (star or planet), parallax (for a star) or

light-time (for a planet), gravitational deflection of light, and the aberration of light due to the
RELATIVITY 5

Earth’s motions. Collectively, these calculations will be referred to in this circular as the algorithms

for proper place. For Earth-based observing systems, we must also account for precession, nutation,

Earth rotation, and polar motion. There are classical expressions for all these effects (except

gravitational deflection), and relativity explicitly enters the procedure in only a few places, usually

as added terms to the classical expressions and in the formulas that link the various time scales

used. It has become common, then, to view this ensemble of calculations as being carried out

entirely in a single reference system; or, two reference systems, barycentric and geocentric, that

have parallel axes and differ only in the origin of coordinates (that is, they are connected by a

Galilean transformation). For example, the coordinate system defined by the “equator and equinox

of J2000.0”, can be thought of as either barycentric or geocentric. The relativistic effects then are

interpreted simply as “corrections” to the classical result.

While such a viewpoint may be aesthetically tidy, it breaks down at high levels of accuracy

and for some types of observations. Relativity theory leads to a more correct, albeit more subtle,

interpretation for the same set of calculations. It is represented by the BCRS-GCRS paradigm

wherein some of the quantities are expressed relative to the BCRS and others are relative to the

GCRS. The two systems are quite different in a number of ways, as described in the previous

section. The situation is easiest to describe if we restrict the discussion to a fictitious observer

at the center of the Earth, that is, to observations referred to the geocenter. The transformation

between the two systems is not explicit in the normal algorithms, but is embodied in the relativistic

terms in the expressions used for aberration or VLBI delay. The distinction between the two

systems is most obvious in the formulation for angular variables. There, the algorithms for space

motion, parallax, light-time, and gravitational deflection1 all use vectors expressed in the BCRS

(star catalogs and solar system ephemerides are inherently BCRS), while the series of rotations for

precession, nutation, Earth rotation, and polar motion (if applied in that order) starts with vectors

expressed in the GCRS. In essence, the aberration calculation connects the two systems because

it contains the transformation between them: its input is a pair of vectors in the BCRS and its

output is a vector in the GCRS. In the VLBI case, aberration does not appear explicitly, but the

conventional algorithm for the delay observable incorporates vectors expressed in both systems,

with appropriate conversion factors obtained from the BCRS-GCRS transformation.2

For an observer on or near the Earth’s surface the calculations have to include the position and

velocity of the observer relative to the geocenter. These are naturally expressed in the GCRS but

for some of the calculations (parallax, light-time, light deflection, and aberration) they must be

added to the position and velocity of the geocenter relative to the solar system barycenter, which

are expressed in the BCRS. Thus another GCRS-BCRS transformation is indicated, although the

velocity is sufficiently small that a Galilean transformation (simple vector addition) suffices for

current observational accuracy (Klioner 2003). Correct use of the resulting vectors results in the

values of the observables expressed, not in the GCRS, but in the proper reference system of the

observer.

1

In the case of the observer at the geocenter, we neglect the gravity field of the Earth itself in computing gravitational deflection.

2Part of the expression for VLBI delay, in the time domain, accounts for what would be called aberration in the

angular domain; it is possible to compute aberration from the VLBI delay algorithm. See Kaplan (1998).
6 RELATIVITY

1.4 Concluding Remarks

The 2000 IAU resolutions on relativity define a framework for future dynamical developments

within the context of general relativity. For example, Klioner (2003) has described how to use the

framework to compute the directions of stars as they would be seen by a precise observing system

in Earth orbit. However, there is much unfinished business. The apparently familiar concept of

the ecliptic plane has not yet been defined in the context of relativity resolutions. A consistent

relativistic theory of Earth rotation is still some years away; the algorithms described in Chapter 5

are not such a theory, although they contain all the main relativistic effects and are quite adequate

for the current observational precision.

A local reference system similar to the GCRS can be easily constructed for any body of an

N-body system in exactly the same way as the GCRS, simply by changing the notation so that the

subscript E denotes a body other than the Earth. In particular, a celenocentric reference system

for the Moon plays an important role in lunar laser ranging.

It is also worth noting that the 2000 resolutions do not describe the proper reference system of

the observer — the local, or topocentric, system in which most measurements are actually taken.

(VLBI observations are unique in that they exist only after data from various individual antennas

are combined; therefore they are referred to the GCRS ab initio.) A kinematically non-rotating

version of the proper reference system of the observer is just a simplified version of the GCRS: x

i

E

should be understood to be the BCRS position of the observer (v

i

E and a

i

E are then the observer’s

velocity and acceleration) and one should neglect the internal potentials. See Klioner & Voinov

(1993); Kopeikin (1991); Kopeikin & Vlasov (2004); Klioner (2004).

One final point: the 2000 IAU resolutions as adopted apply specifically to Einstein’s theory of

gravity, i.e., the general theory of relativity. The Parameterized Post-Newtonian (PPN) formalism

(see, e.g., Will & Nordtvedt (1972)) is more general, and the 2000 resolutions have been discussed in

the PPN context by Klioner & Soffel (2000) and Kopeikin & Vlasov (2004). In the 2000 resolutions,

it is assumed that the PPN parameters β and γ are both 1.
Chapter 2

Time Scales

Relevant IAU resolutions: A4.III, A4.IV, A4.V, A4.VI of 1991; C7 of 1994; B1.3, B1.5, B1.7, B1.8,

B1.9, and B2 of 2000

Summary The IAU has not established any new time scales since 1991, but more

recent IAU resolutions have redefined or clarified those already in use, with no loss of

continuity. There are two classes of time scales used in astronomy, one based on the

SI (atomic) second, the other based on the rotation of the Earth. The SI second has

a simple definition that allows it to be used (in practice or in theory) in any reference

system. Time scales based on the SI second include TAI and TT for practical applications, and TCG and TCB for theoretical developments. The latter are to be used

for relativistically correct dynamical theories in the geocentric and barycentric reference systems, respectively. Closely related to these are two time scales, TDB and Teph,

used in the current generation of ephemerides. Time scales based on the rotation of

the Earth include mean and apparent sidereal time and UT1. Because of irregularities

in the Earth’s rotation, and its tidal deceleration, Earth-rotation-based time scales do

not advance at a uniform rate, and they increasingly lag behind the SI-second-based

time scales. UT1 is now defined to be a linear function of a quantity called the Earth

Rotation Angle, θ. In the formula for mean sidereal time, θ now constitutes the “fast

term”. The widely disseminated time scale UTC is a hybrid: it advances by SI seconds

but is subject to one-second corrections (leap seconds) to keep it within 0.s9 of UT1.

That procedure is now the subject of debate and there is a movement to eliminate leap

seconds from UTC.

2.1 Different Flavors of Time

The phrase time scale is used quite freely in astronomical contexts, but there is sufficient confusion

surrounding astronomical times scales that it is worthwhile revisiting the basic concept. A time scale

is simply a well defined way of measuring time based on a specific periodic natural phenomenon.

The definition of a time scale must provide a description of the phenomenon to be used (what

defines a period, and under what conditions), the rate of advance (how many time units correspond

to the natural period), and an initial epoch (the time reading at some identifiable event). For

example, we could define a time scale where the swing of a certain kind of pendulum, in vacuum at

sea level, defines one second, and where the time 00:00:00 corresponds to the transit of a specified

7
8 TIME SCALES

star across a certain geographic meridian on an agreed-upon date.

As used in astronomy, a time scale is an idealization, a set of specifications written on a piece

of paper. The instruments we call clocks, no matter how sophisticated or accurate, provide some

imperfect approximation to the time scale they are meant to represent. In this sense, time scales

are similar to spatial reference systems (see Chapter 3), which have precise definitions but various

imperfect realizations. The parallels are not coincidental, since for modern high-precision applications we actually use space-time reference systems (see Chapter 1). All time scales are therefore

associated with specific reference systems.

Two fundamentally different groups of time scales are used in astronomy. The first group of

time scales is based on the second that is defined as part of the the Syst`eme International (SI) —

the “atomic” second — and the second group is based on the rotation of the Earth. The SI second

is defined as 9, 192, 631, 770 cycles of the radiation corresponding to the ground state hyperfine

transition of Cesium 133 (BIPM 1998), and provides a very precise and constant rate of time

measurement, at least for observers local to the apparatus in which such seconds are counted. The

rotation of the Earth (length of day) is quite a different basis for time, since it is variable and has

unpredictable components. It must be continuously monitored through astronomical observations,

now done primarily with very long baseline [radio] interferometry (VLBI). The SI-based time scales

are relatively new in the history of timekeeping, since they rely on atomic clocks first put into

regular use in the 1950s. Before that, all time scales were tied to the rotation of the Earth.

(Crystal oscillator clocks in the 1930s were the first artificial timekeeping mechanisms to exceed

the accuracy of the Earth itself.) As we shall see, the ubiquitous use of SI-based time for modern

applications has led to a conundrum about what the relationship between the two kinds of time

should be in the future. Both kinds of time scales can be further subdivided into those that are

represented by actual clock systems and those that are simply theoretical constructs.

General reviews of astronomical time scales are given in Seidelmann & Fukushima (1992) and

Chapter 2 of the Explanatory Supplement (1992).

2.2 Time Scales Based on the SI Second

Let us first consider the times scales based on the SI second. As a simple count of cycles of

microwave radiation from a specific atomic transition, the SI second can be implemented, at least in

principle, by an observer anywhere. Thus, SI-based time scales can be constructed or hypothesized

on the surface of the Earth, on other celestial bodies, on spacecraft, or at theoretically interesting

locations in space, such as the solar system barycenter. According to relativity theory, clocks

advancing by SI seconds may not appear to advance by SI seconds by an observer on another

space-time trajectory. In general, there will be an observed difference in rate and possibly higherorder or periodic differences, depending on the relative trajectory of the clock and the observer and

the gravitational fields involved. The precise conversion formulas can be mathematically complex,

involving the positions and velocities not just of the clock and observer but also those of an ensemble

of massive bodies (Earth, Sun, Moon, planets). These considerations also apply to coordinate time

scales established for specific reference systems. The time-scale conversions are taken from the

general 4-dimensional space-time transformation between the reference systems given by relativity

theory (see Chapter 1). While the rate differences among these time scales may seem inconvenient,

the universal use of SI units, including “local” SI seconds, means that the values of fundamental

physical constants determined in one reference system can be used in another reference system

without scaling factors.
TIME SCALES 9

Two SI-second-based times have already been mentioned in Chapter 1: these are coordinate time

scales (in the terminology of General Relativity) for theoretical developments based on the Barycentric Celestial Reference System (BCRS) or the Geocentric Celestial Reference System (GCRS).

These time scales are called, respectively, Barycentric Coordinate Time (TCB) and Geocentric Coordinate Time (TCG). With respect to a time scale based on SI seconds on the surface of the Earth,

TCG advances at a rate 6.97×10−10 faster, while TCB advances at a rate 1.55×10−8faster. TCB

and TCG are not likely to come into common use for practical applications, but they are beginning

to appear as the independent argument for some theoretical developments in dynamical astronomy

(e.g., Moisson & Bretagnon (2001)). However, none of the current IAU recommended models used

in the analysis of astrometric data use TCB or TCG as a basis, and neither time scale appears in

the main pages of The Astronomical Almanac. This simply reflects the fact that there has not been

enough time or motivation for a new generation of dynamical models to be fully developed within

the IAU-recommended relativistic paradigm.

For practical applications, International Atomic Time (TAI) is a commonly used time scale

based on the SI second on the Earth’s surface at sea level (specifically, the rotating geoid). TAI

is the most precisely determined time scale that is now available for astronomical use. This scale

results from analyses by the Bureau International des Poids et Mesures (BIPM) in S`evres, France,

of data from atomic time standards of many countries, according to an agreed-upon algorithm.

Although TAI was not officially introduced until 1972, atomic time scales have been available since

1956, and TAI may be extrapolated backwards to the period 1956–1971 (See Nelson et al. (2001)

for a history of TAI). An interesting discussion of whether TAI should be considered a coordinate

time or a kind of modified proper time1in the context of general relativity has been given by Guinot

(1986). In any event, TAI is readily available as an integral number of seconds offset from UTC,

which is extensively disseminated; UTC is discussed at the end of this chapter. The TAI offset from

UTC is designated ∆AT = TAI–UTC. (For example, from 1999 through 2005, ∆AT = 32 s.) ∆AT

increases by 1 s whenever a positive leap second is introduced into UTC (see below). The history

of ∆AT values can be found on page K9 of The Astronomical Almanac and the current value can

be found at the beginning of each issue of IERS Bulletin A (URL 5).

The astronomical time scale called Terrestrial Time (TT), used widely for geocentric and

topocentric ephemerides (such as in The Astronomical Almanac), is defined to run at a rate of

(1 − LG) times that of TCG, where LG = 6.969290134×10−10. The rate factor applied to TCG to

create TT means that TT runs at the same rate as a time scale based on SI seconds on the surface

of the Earth. LG is now considered a defining constant, not subject to further revision. Since TCG

is a theoretical time scale that is not kept by any real clock, for practical purposes, TT can be considered an idealized form of TAI with an epoch offset: TT = TAI + 32.

s184. This expressssion for

TT preserves continuity with previously-used (now obsolete) “dynamical” time scales, Terrestrial

Dynamical Time (TDT) and Ephemeris Time (ET). That is, ET → TDT → TT can be considered

a single continuous time scale.

Important Note: The “standard epoch” for modern astrometric reference data, designated

J2000.0, is expressed as a TT instant: J2000.0 is 2000 January 1, 12h TT (JD 2451545.0 TT) at

the geocenter.

The fundamental solar system ephemerides from the Jet Propulsion Laboratory (JPL) that are

the basis for many of the tabulations in The Astronomical Almanac and other national almanacs

1These terms are described in Chapter 1, p. 2.
10 TIME SCALES

were computed in a barycentric reference system and are distributed with the independent argument

being a coordinate time scale called Teph (Chapter 4 describes the JPL ephemerides). Teph differs

in rate from that of TCB, the IAU recommended time scale for barycentric developments; the rate

of Teph matches that of TT, on average, over the time span of the ephemerides. One may treat

Teph as functionally equivalent to Barycentric Dynamical Time (TDB), defined by the IAU in 1976

and 1979. Both are meant to be “time scales for equations of motion referred to the barycenter of

the solar system” yet (loosely speaking) match TT in average rate. The original IAU definition of

TDB specified that “there be only periodic variations” with respect to what we now call TT (the

largest variation is 0.0016 s with a period of one year). It is now clear that this condition cannot be

rigorously fulfilled in practice; see Standish (1998a) for a discussion of the issue and the distinction

between TDB and Teph. Nevertheless, space coordinates obtained from the JPL ephemerides are

consistent with TDB, and it has been said that “Teph is what TDB was meant to be.” Therefore,

barycentric and heliocentric data derived from the JPL ephemerides are often tabulated with TDB

shown as the time argument (as in The Astronomical Almanac), and TDB is the specified time

argument for many of the equations presented in this circular.2 Because Teph (≈TDB) is not based

on the SI second, as is TCB, the values of parameters determined from or consistent with the JPL

ephemerides will, in general, require scaling to convert them to SI-based units. This includes the

length of the astronomical unit. Dimensionless quantities such as mass ratios are unaffected.

The problem of defining relativistic time scales in the solar system has been treated by Brumberg

& Kopeikin (1990); the paper is quite general but pre-dates the current terminology.

2.3 Time Scales Based on the Rotation of the Earth

Time scales that are based on the rotation of the Earth are also used in astronomical applications,

such as telescope pointing, that depend on the geographic location of the observer. Greenwich

sidereal time is the hour angle of the equinox measured with respect to the Greenwich meridian.

Local sidereal time is the local hour angle of the equinox, or the Greenwich sidereal time plus the

longitude (east positive) of the observer, expressed in time units. Sidereal time appears in two

forms, mean and apparent, depending on whether the mean or true equinox is the reference point.

The position of the mean equinox is affected only by precession while the true equinox is affected by

both precession and nutation. The difference between true and mean sidereal time is the equation

of the equinoxes, which is a complex periodic function with a maximum amplitude of about 1 s.

Of the two forms, apparent sidereal time is more relevant to actual observations, since it includes

the effect of nutation. Greenwich (or local) apparent sidereal time can be observationally obtained

from the right ascensions of celestial objects transiting the Greenwich (or local) meridian.

Universal Time (UT) is also widely used in astronomy, and now almost always refers to the

specific time scale UT1. Historically, Universal Time (formerly, Greenwich Mean Time) has been

obtained from Greenwich sidereal time using a standard expression. In 2000, the IAU redefined

UT1 to be a linear function of the Earth Rotation Angle, θ, which is the geocentric angle between

two directions in the equatorial plane called, respectively, the Celestial Intermediate Origin (CIO)

2The IAU Working Group on Nomenclature for Fundamental Astronomy is considering a recommendation to

correct the definition of TDB so that a distinction between TDB and Teph would no longer be necessary; TDB would

be a linear function of TCB with a rate as close to that of TT as possible over the time span of the ephemeris to

which it applies.
TIME SCALES 11

and the Terrestrial Intermediate Origin (TIO) (res. B1.8 of 20003). The TIO rotates with the

Earth, while the CIO has no instantaneous rotation around the Earth’s axis, so that θ is a direct

measure of the Earth’s rotational motion:  ̇θ = ω, the Earth’s average angular velocity of rotation.

See Chapter 6 for a more complete description of these new reference points. The definition of UT1

based on sidereal time is still widely used, but the definition based on θ is becoming more common

for precise applications. In fact, the two definitions are equivalent, since the expression for sidereal

time as a function of UT1 is itself now based on θ.

Since they are mathematically linked, sidereal time, θ, and UT1 are all affected by variations

in the Earth’s rate of rotation (length of day), which are unpredictable and must be routinely

measured through astronomical observations. The lengths of the sidereal and UT1 seconds, and

the value of  ̇θ, are therefore not precisely constant when expressed in a uniform time scale such as

TT. The accumulated difference in time measured by a clock keeping SI seconds on the geoid from

that measured by the rotation of the Earth is ∆T = TT–UT1. A table of observed and extrapolated

values of ∆T is given in The Astronomical Almanac on page K9. The long-term trend is for ∆T

to increase gradually because of the tidal deceleration of the Earth’s rotation, which causes UT1

to lag increasingly behind TT.

In predicting the precise times of topocentric phenomena, like solar eclipse contacts, both TT

and UT1 come into play. Therefore, assumptions have to be made about the value of ∆T at the

time of the phenomenon. Alternatively, the circumstances of such phenomena can be expressed

in terms of an imaginary system of geographic meridians that rotate uniformly about the Earth’s

axis (∆T is assumed zero, so that UT1=TT) rather than with the real Earth; the real value of ∆T

then does not need to be known when the predictions are made. The zero-longitude meridian of

the uniformly rotating system is called the ephemeris meridian. As the time of the phenomenon

approaches and the value of ∆T can be estimated with some confidence, the predictions can be

related to the real Earth: the uniformly rotating system is 1.002738 ∆T east of the real system

of geographic meridians. (The 1.002738 factor converts a UT1 interval to the equivalent Earth

Rotation Angle — i.e., the sidereal/solar time ratio.)

2.4 Coordinated Universal Time (UTC)

The worldwide system of civil time is based on Coordinated Universal Time (UTC), which is now

ubiquitous and tightly synchronized. (This is the de facto situation; most nations’ legal codes,

including that of the U.S., do not mention UTC specifically.) UTC is a hybrid time scale, using

the SI second on the geoid as its fundamental unit, but subject to occasional 1-second adjustments

to keep it within 0.s9 of UT1. Such adjustments, called leap seconds, are normally introduced at

the end of June or December, when necessary, by international agreement. Tables of the remaining

difference, UT1–UTC, for various dates are published by the International Earth Rotation and

Reference System Service (IERS), at URL 5. Both past observations and predictions are available.

DUT1, an approximation to UT1–UTC, is transmitted in code with some radio time signals, such as

those from WWV. As previously discussed in the context of TAI, the difference ∆AT = TAI–UTC

is an integral number of seconds, a number that increases by 1 whenever a (positive) leap second

is introduced into UTC. That is, UTC and TAI share the same seconds ticks, they are just labeled

differently.

3

In the resolution, these points are called the Celestial Ephemeris Origin (CEO) and the Terrestrial Ephemeris

Origin (TEO). The change in terminology has been recommended by the IAU Working Group on Nomenclature for

Fundamental Astronomy and will probably be adopted at the 2006 IAU General Assembly.
12 TIME SCALES

Clearly UT1–UTC and ∆T must be related, since they are both measures of the natural “error”

in the Earth’s angle of rotation at some date. In fact, ∆T = 32.s184 + ∆AT – (UT1–UTC).

For the user, then, UTC, which is widely available from GPS, radio broadcast services, and the

Internet, is the practical starting point for computing any of the other time scales described above.

For the SI-based time scales, we simply add the current value of ∆AT to UTC to obtain TAI. TT

is then just 32.s184 seconds ahead of TAI. The theoretical time scales TCG, TCB, TDB, and Teph

can be obtained from TT using the appropriate mathematical formulas. For the time scales based

on the rotation of the Earth, we again start with UTC and add the current value of UT1–UTC to

obtain UT1. The various kinds of sidereal time can then be computed from UT1 using standard

formulas.

Figure 2.1 Differences in readings of various time scales compared to International Atomic Time (TAI). TT and its predecessors, TDT and ET, are all shown as

TAI+32.184 s. The periodic terms of TCB and TDB are exaggerated by a factor of

100. The “stair-step” appearance of UTC is due to the leap seconds inserted into

that time scale so that it tracks UT1. TT and the “steps” of UTC are parallel to

the TAI line because they are all based on the SI second on the geoid. TDB (or

Teph) tracks TT on average over the time span of the specific ephemeris to which

it applies. Note the instant at the beginning of 1977 when TT, TCB, and TCG all

had the same value. The figure is from Seidelmann & Fukushima (1992).

2.5 To Leap or Not to Leap

Because of the widespread and increasing use of UTC for applications not considered three decades

ago — such as precisely time-tagging electronic fund transfers and other networked business transactions — the addition of leap seconds to UTC at unpredictable intervals creates technical problems
TIME SCALES 13

and legal issues for service providers. There is now a movement to relax the requirement that UTC

remain within 0.9 seconds of UT1. The issue is compounded by the unavoidable scientific fact that

the Earth’s rotation is slowing due to tidal friction, so that the rate of addition of leap seconds to

UTC must inevitably increase. Aside from monthly, annual, and decadal variations, the Earth’s

angular velocity of rotation is decreasing linearly, which means that the accumulated lag in UT1

increases quadratically; viewed over many centuries, the ∆T curve is roughly a parabola. The

formulas for sidereal time, and length of the old ephemeris second to which the SI second was originally calibrated, are based on the average (assumed fixed) rate of Earth rotation of the mid-1800s

(Nelson et al. 2001). All of our modern timekeeping systems are ultimately based on what the

Earth was doing a century and a half ago.

An IAU Working Goup on the Redefinition of Universal Time Coordinated (UTC) was established to consider the leap second problem and recommend a solution, working with the IERS, the

International Union of Radio Science (URSI), the Radiocommunication Sector of the International

Telecommunications Union (ITU-R), the International Bureau for Weights and Measures (BIPM),

and the relevant navigational agencies (res. B2 of 2000). Possibilities include: using TAI for technical applications instead of UTC; allowing UT1 and UTC to diverge by a larger amount (e.g., 10

or 100 seconds) before a multi-second correction to UTC is made; making a variable correction to

UTC at regularly scheduled dates; eliminating the corrections to UTC entirely and allowing UTC

and UT1 to drift apart; or changing the definition of the SI second. No solution is ideal (including

the status quo) and each of these possibilities has its own problems. For example, if we keep leap

seconds, or a less frequent multi-second correction, can current systems properly time-tag the date

and time of an event that occurs during the correction? Does a time scale that diverges from UT1

provide a legally acceptable representation of civil time? If corrections are made less frequently,

will the possibility of technical blunders increase? If leap seconds are eliminated, won’t natural

phenomena such as sunrise and sunset eventually fall out of sync with civil time? How do we find all

the existing computer code that assumes |UT1–UTC| ≤ 0.9 s? The matter is now being considered

by the ITU-R, where a working group has proposed eliminating leap seconds from UTC entirely.

Contact Dr. Dennis McCarthy, U.S. Naval Observatory, dmc@maia.usno.navy.mil, for a copy of a

report or if you wish to comment. In any event, it would take a number of years for any proposed

change to take place because of the many institutions and international bodies that would have to

be involved.

For scientific instrumentation, the use of TAI — which is free of leap seconds — has much

to recommend it. Its seconds can be easily synchronized to those of UTC (only the labels of the

seconds are different). It is straightforward to convert from TAI to any of the other time scales.

Use of TAI provides an internationally recognized time standard and avoids the need to establish

an instrument-specific time scale when continuity of time tags is a requirement.

2.6 Formulas

2.6.1 Formulas for Time Scales Based on the SI Second

For the SI-based time scales, the event tagged 1977 January 1, 00:00:00 TAI (JD 2443144.5 TAI) at

the geocenter is special. At that event, the time scales TT, TCG, and TCB all read 1977 January 1,

00:00:32.184 (JD 2443144.5003725). (The 32.s184 offset is the estimated difference between TAI

and the old Ephemeris Time scale.) This event will be designated t0 in the following; it can be

represented in any of the time scales, and the context will dictate which time scale is appropriate.
14 TIME SCALES

From the perspective of a user, the starting point for computing all the time scales is Coordinated

Universal Time (UTC). From UTC, we can immediately get International Atomic Time (TAI):

TAI = UTC + ∆AT (2.1)

where ∆AT, an integral number of seconds, is the accumulated number of leap seconds applied to

UTC.

The astronomical time scale Terrestrial Time (TT) is defined by the epoch t0 and its IAUspecified rate with respect to Geocentric Coordinate Time (TCG):

dTT

dTCG = 1 − LG where LG = 6.969290134×10−10 (exactly) (2.2)

from which we obtain

TT = TCG − LG (TCG − t0) (2.3)

However, TCG is a theoretical time scale, not kept by any real clock system, so in practice,

TT = TAI + 32.

s184 (2.4)

and we obtain TCG from TT.

The relationship between TCG and Barycentric Coodinate Time (TCB) is more complex. TCG

and TCB are both coordinate time scales, to be used with the geocentric and barycentric reference

systems (the GCRS and BCRS), respectively. The exact formula for the relationship between TCG

and TCB is given in res. B1.5 of 2000, recommendation 2. For a given TCB epoch, we have

TCG = TCB −

1

c

2

Z TCB

t0

v

2

e

2

+ Uext(xe)



dt −

ve

c

2

· (x − xe) + · · · (2.5)

where c is the speed of light, xe and ve are the position and velocity vectors of the Earth’s center

with respect to the solar system barycenter, and Uext is the Newtonian potential of all solar system

bodies apart from the Earth. The integral is carried out in TCB since the positions and motions of

the Earth and other solar system bodies are represented (ideally) as functions of TCB. The last term

on the right contains the barycentric position vector of the point of interest, x, and will be zero for

the geocenter, as would normally be the case. The omitted terms are of maximum order c

−4

. Note

that the transformation is ephemeris-dependent, since it is a function of the time series of xe and

ve values. The result is a “time ephemeris” associated with every spatial ephemeris of solar system

bodies expressed in TCB. It is to be expected that ephemeris developers will supply appropriate

time conversion algorithms (software) to allow the positions and motions of solar system bodies to

be retrieved for epochs in conventional time scales such as TT or TAI. It is unlikely that ordinary

ephemeris users will have to compute eq. (2.5) on their own.

The functional form of the above expressions may seem backwards for practical applications;

that is, they provide TCG from TCB and TT from TCG. These forms make sense, however, when

one considers how an ephemeris of a solar system body (or bodies) or a spacecraft is developed.

The equations of motion for the body (or bodies) of interest are expressed in either the barycentric

or geocentric system as a function of some independent coordinate time argument. For barycentric

equations of motion, expressed in SI units, we would be tempted immediately to identify this time

argument with TCB. Actually, however, the association of the time argument with TCB is not

automatic; it comes about only when the solution of the equations of motion is made to satisfy

the boundary conditions set by the ensemble of real observations of various kinds. Generally, these
TIME SCALES 15

observations will be time-tagged in UTC, TAI, or TT (all of which are based on the SI second on

the geoid) and these time tags must be associated with the time argument of the ephemeris. The

above formulas can be used to make that association, which then allows the ephemeris to be fit

to the observations. (More precisely, the space-time coordinates of the observation events must

be transformed to the BCRS.) As a consequence, the time argument of the ephemeris becomes a

realization of TCB. The fit of the computed ephemeris to observations usually proceeds iteratively,

and every iteration of the spatial ephemeris produces a new time ephemeris. With each iteration,

the spatial coordinates of the ephemeris become better grounded in reality (as represented by

the observations) and the time coordinate becomes a better approximation to TCB. Viewed from

this computational perspective, the ephemeris and its time argument are the starting point of the

process and the sequence TCB → TCG → TT makes sense.

One can compute an ephemeris and fit it to observations using other formulas for the time

scale conversions. A completely valid and precise ephemeris can be constructed in this way, but its

independent time argument could not be called TCB. The values of various constants used in, or

derived from, such an ephemeris would also not be SI-based and a conversion factor would have to

be applied to convert them to or from SI units. Such is the case with the solar system Development

Ephemeris (DE) series from the Jet Propulsion Laboratory. DE405 is now the consensus standard

for solar system ephemerides and is described in Chapter 4. The DE series dates back to the

1960s, long before TCB and TCG were defined, and its independent time argument is now called

Teph. Teph can be considered to be TCB with a rate factor applied. Or, as mentioned above,

Teph can be considered to be functionally equivalent to the time scale called TDB. Both Teph and

TDB advance, on average, at the same rate as TT. This arrangement makes accessing the DE

ephemerides straightforward, since for most purposes, TT can be used as the input argument with

little error. The total error in time in using TT as the input argument is <2 ms, which for the

geocentric position of the Moon would correspond to an angular error of <1 mas. When more

precision is required, the following formula can be used:

Teph ≈ TDB ≈ TT + 0.001657 sin (628.3076 T + 6.2401)

+ 0.000022 sin (575.3385 T + 4.2970)

+ 0.000014 sin (1256.6152 T + 6.1969)

+ 0.000005 sin (606.9777 T + 4.0212) (2.6)

+ 0.000005 sin (52.9691 T + 0.4444)

+ 0.000002 sin (21.3299 T + 5.5431)

+ 0.000010 T sin (628.3076 T + 4.2490) + · · ·

where the coefficients are in seconds, the angular arguments are in radians, and T is the number of

Julian centuries of TT from J2000.0: T = (JD(TT) − 2451545.0)/36525. The above is a truncated

form of a much longer and more precise series given by Fairhead & Bretagnon (1990). The maximum

error in using the above formula is about 10 μs from 1600 to 2200; that is, its precision is more than

two orders of magnitude better than the approximation Teph≈TDB≈TT. For even more precise

applications, the series expansion by Harada & Fukushima (2003) is recommended.

A word of caution: The idea that “Teph and TDB advance, on average, at the same rate as TT”

is problematic. The independent time argument of a barycentric ephemeris (whether considered to

be Teph, TDB, or TCB) has a large number of periodic components with respect to TT. Some of

the periods are quite long, and may extend beyond the time period of the ephemeris. Thus, the

“average rate” of the time argument of the ephemeris, with respect to TT, depends on the averaging
16 TIME SCALES

method and the time span considered. Differences in rate of some tens of microseconds per century

are possible (Fairhead & Bretagnon 1990). These rate ambiguities are probably unimportant

(amounting to fractional errors of only ∼10−14) for retrieving positions and velocities from the

ephemeris but do affect pulsar timing that is reduced to the barycentric time scale.

2.6.2 Formulas for Time Scales Based on the Rotation of the Earth

For the time scales that are based on the rotation of the Earth, we again start with UTC.

UT1 = UTC + (UT1–UTC) (2.7)

≈ UTC + DUT1 (2.8)

where DUT1 is a broadcast approximation to UT1–UTC (precision ±0.s1). We also have

UT1 = TT − ∆T (2.9)

where ∆T = 32.s184 + ∆AT – (UT1–UTC). The most recent values of UT1–UTC and ∆AT are

listed in IERS Bulletin A (URL 5). Values of ∆T are listed in The Astronomical Almanac on page

K9.

The Earth Rotation Angle, θ, is

θ = 0.7790572732640 + 1.00273781191135448 DU (2.10)

where DU is the number of UT1 days from 2000 January 1, 12h UT1: DU = JD(UT1) – 2451545.0.

The angle θ is given in terms of rotations (units of 2π radians or 360◦). The above rate coefficient

gives an Earth rotation period of 86164.0989036903511 seconds of UT1. If we consider this to be

the adopted average rotation period of the Earth in SI seconds, it is consistent with the nominal

mean angular velocity of Earth rotation, ω = 7.292115×10−5radian s−1, used by the International

Association of Geodesy. The above expression is taken directly from note 3 to res. B1.8 of 2000. An

equivalent form of this expression (if the integral number of rotations is neglected) that is usually

more numerically precise is

θ = 0.7790572732640 + 0.00273781191135448 DU + frac(JD(UT1)) (2.11)

where frac(JD(UT1)) is the fractional part of the UT1 Julian date, i.e., JD(UT1) modulus 1.0.

Then Greenwich mean sidereal time in seconds is

GMST = 86400 · θ + (0.014506 + 4612.156534 T + 1.3915817 T

2

−0.00000044 T

3 − 0.000029956 T4 − 0.0000000368 T5

)/15 (2.12)

where T is the number of centuries of TDB (equivalently for this purpose, TT) from J2000.0: T

= (JD(TDB) – 2451545.0)/36525. The polynomial in parentheses is the accumulated precession of

the equinox in right ascension, in arcseconds, as given for the P03 solution (eq. 42) in Capitaine

et al. (2003). Note that two time scales are now required to compute sidereal time: in the “fast

term”, θ is a function of UT1, while in the remaining terms, T is expressed in TDB (or TT).

To obtain Greenwich apparent sidereal time in seconds, we must add the equation of the

equinoxes:

GAST = GMST + EΥ/15 (2.13)
TIME SCALES 17

which accounts for the motion of the equinox due to nutation. An extended series is now used for

the equation of the equinoxes. The new series includes so-called complementary terms and more

fully accounts for the accumulated effect of combined precession and nutation on the position of

the equinox. The equation of the equinoxes in arcseconds is

EΥ = ∆ψ cos 

+ 0.00264096 sin (Ω)

+ 0.00006352 sin (2Ω)

+ 0.00001175 sin (2F − 2D + 3Ω)

+ 0.00001121 sin (2F − 2D + Ω)

− 0.00000455 sin (2F − 2D + 2Ω) (2.14)

+ 0.00000202 sin (2F + 3Ω)

+ 0.00000198 sin (2F + Ω)

− 0.00000172 sin (3Ω)

− 0.00000087 T sin (Ω) + · · ·

where ∆ψ is the nutation in longitude, in arcseconds;  is the mean obliquity of the ecliptic; and F,

D, and Ω are fundamental luni-solar arguments. All of these quantities are functions of TDB (or

TT); see Chapter 5 for expressions (esp. eqs. 5.12, 5.15, & 5.19). The above series is a truncated

form of a longer series given in the IERS Conventions (2003), but should be adequate for almost

all practical applications.

Local mean sidereal time (LMST) and local apparent sidereal time (LAST) in seconds can then

be computed respectively from

LMST = GMST + 

3600

15 

λ and LAST = GAST + 

3600

15 

λ (2.15)

where λ is the longitude of the place of interest, in degrees, positive for places east of Greenwich.

In the above, “Greenwich” actually refers to a plane containing the geocenter, the Celestial

Intermediate Pole (CIP), and the point called the Terrestrial Intermediate Origin (TIO). These

concepts are described in Chapters 5 and 6. Loosely, the CIP is the rotational pole, defined by

the precession and nutation theories. For astronomical purposes, the TIO can be considered to be

a point on the rotational equator (the plane orthogonal to the CIP) essentially fixed at geodetic

longitude 0. Strictly, then, the longitude λ should be measured around the axis of the CIP from

the TIO to the location of interest. Because of polar motion, the pole of the conventional system

of geodetic coordinates is not at the CIP so the longitude we need is not quite the same as the

geodetic longitude. The longitude, in degrees, to be used in eq. 2.15 is

λ = λG + (xp sin λG + yp cos λG) tan φG /3600 (2.16)

where λG and φG are the usual geodetic longitude and latitude of the place, with λG in degrees

(north latitudes and east longitudes are positive); and xp and yp are the coordinates of the pole

(CIP) with respect to the geodetic system, in arcseconds (xp and yp can be a few tenths of an

arcsecond). The geodetic system is formally the International Terrestrial Reference System (ITRS),

which matches WGS-84 (available from GPS) to several centimeters. The local meridian assumed

by the formula for LAST, using the longitude λ, passes through the local zenith (orthogonal to the

local surface of the WGS-84 ellipsoid) and the north and south celestial poles — close to but not
18 TIME SCALES

through the local geodetic north and south points. This is the meridian that all stars with apparent

topocentric right ascension equal to LAST will pass over at time UT1. More information can be

found in sections 6.4, 6.5.2, and 6.5.4.

The above formulas are entirely geometric. Not described are astronomical latitude and longitude, which are based on the local direction of gravity. Astronomical latitude and longitude are

affected by the deflection of the vertical caused by permanent gravitational anomalies and, at a

much lower level, semidiurnal tides. Astronomical latitude and longitude must be corrected for

such effects to obtain geodetic latitude and longitude.
Chapter 3

The Fundamental Celestial Reference

System

Relevant IAU resolutions: A4.VI, A4.VII of 1991; B5 of 1994; B2 of 1997; B1.2 of 2000

Summary Reference data for positional astronomy, such as the data in astrometric

star catalogs or barycentric planetary ephemerides, are now specified within the International Celestial Reference System (ICRS). The ICRS is a coordinate system whose

origin is at the solar system barycenter and whose axis directions are effectively defined

by the adopted coordinates of 212 extragalactic radio sources observed by VLBI. These

radio sources (quasars and active galactic nuclei) are assumed to have no observable

intrinsic angular motions. Thus, the ICRS is a “space-fixed” system (more precisely, a

kinematically non-rotating system) without an associated epoch. However, the ICRS

closely matches the conventional dynamical system defined by the Earth’s mean equator

and equinox of J2000.0; the alignment difference is at the 0.02 arcsecond level, negligible

for many applications.

Strictly speaking, the ICRS is somewhat of an abstraction, a coordinate system that

perfectly satisfies a list of criteria. The list of radio source positions that define it for

practical purposes is called the International Celestial Reference Frame (ICRF). In the

terminology that is now commonly used, a reference system like the ICRS is “realized”

by a reference frame like the ICRF, and there can be more than one such realization.

In the case of the ICRS, there is, in fact, a second, lower-accuracy realization for work

at optical wavelengths, called the Hipparcos Celestial Reference Frame (HCRF). The

HCRF is composed of the positions and proper motions of the astrometrically wellbehaved stars in the Hipparcos catalog.

Astrometric data referred to the ICRS is becoming more common, with new catalogs

now available in the optical, infrared, and radio.

The ICRS is itself a specific example of a Barycentric Celestial Reference System, incorporating the relativistic metric specified in res. B1.3 of 2000 for solar system barycentric coordinate systems. In other words, the ICRS provides the orientation of the BCRS

axes.

19
20 CELESTIAL REFERENCE SYSTEM

3.1 The ICRS, the ICRF, and the HCRF

The fundamental celestial reference system for astronomical applications is now the International

Celestial Reference System (ICRS), as provided in res. B2 of 1997. The ICRS is a coordinate

system with its origin at the solar system barycenter and axis directions that are fixed with respect

to distant objects in the universe; it is to be used to express the positions and motions of stars,

planets, and other celestial objects. Its relativistic basis is defined by res. B1.3 of 2000; in the

words of that resolution, it is a Barycentric Celestial Reference System (BCRS), and as such its

axes are kinematically non-rotating (see Chapter 1). To establish the ICRS as a practical system,

the IAU specified a set of distant benchmark objects, observable at radio wavelengths, whose

adopted coordinates effectively define the directions of the ICRS axes. This “realization” of the

ICRS, called the International Celestial Reference Frame (ICRF), is a set of high accuracy positions

of extragalactic radio sources measured by very long baseline interferometry (Ma & Feissel 1997;

Ma et al. 1998). The ICRS is realized at optical wavelengths — but at lower accuracy — by the

Hipparcos Celestial Reference Frame (HCRF), consisting of the Hipparcos Catalogue (ESA 1997)

of star positions and motions, with certain exclusions (res. B1.2 of 2000). The coordinates of the

ICRF radio sources and HCRF stars are given relative to the ICRS origin at the solar system

barycenter, and a number of transformations are required to obtain the coordinates that would be

observed from a given location on Earth at a specific date and time.

Although the directions of the ICRS coordinate axes are not defined by the kinematics of the

Earth, the ICRS axes (as implemented by the ICRF and HCRF) closely approximate the axes

that would be defined by the mean Earth equator and equinox of J2000.0 (to within about 0.02

arcsecond), if the latter is considered to be a barycentric system. Because the ICRS axes are

meant to be “space fixed”, i.e., kinematically non-rotating, there is no date associated with the

ICRS. Furthermore, since the defining radio sources are assumed to be so distant that their angular

motions, seen from Earth, are negligible, there is no epoch associated with the ICRF. It is technically

incorrect, then, to say that the ICRS is a “J2000.0 system”, even though for many current data

sources, the directions in space defined by the equator and equinox of J2000.0 and the ICRS axes

are the same to within the errors of the data.

The ICRS, with its origin at the solar system barycenter and “space fixed” axis directions,

is meant to represent the most appropriate coordinate system currently available for expressing

reference data on the positions and motions of celestial objects.

The IAU Working Group on Nomenclature for Fundamental Astronomy has recommended the

following definitions for the ICRS and ICRF:

International Celestial Reference System (ICRS): The idealized barycentric coordinate system to which celestial positions are referred. It is kinematically non-rotating

with respect to the ensemble of distant extragalactic objects. It has no intrinsic orientation but was aligned close to the mean equator and dynamical equinox of J2000.0

for continuity with previous fundamental reference systems. Its orientation is independent of epoch, ecliptic or equator and is realized by a list of adopted coordinates of

extragalactic sources.

International Celestial Reference Frame (ICRF): A set of extragalactic objects

whose adopted positions and uncertainties realize the ICRS axes and give the uncertainties of the axes. It is also the name of the radio catalogue whose 212 defining sources

are currently the most accurate realization of the ICRS. Note that the orientation of the

ICRF catalogue was carried over from earlier IERS radio catalogs and was within the
CELESTIAL REFERENCE SYSTEM 21

errors of the standard stellar and dynamic frames at the time of adoption. Successive

revisions of the ICRF are intended to minimize rotation from its original orientation.

Other realizations of the ICRS have specific names (e.g. Hipparcos Celestial Reference

Frame).

3.2 Background: Reference Systems and Reference Frames

The terminology that has become standard over the past decade or so distinguishes between a

reference system and a reference frame. A reference system is the complete specification of how a

celestial coordinate system is to be formed. Both the origin and the orientation of the fundamental

planes (or axes) are defined. A reference system also incorporates a specification of the fundamental

models needed to construct the system; that is, the basis for the algorithms used to transform

between observable quantities and reference data in the system. A reference frame, on the other

hand, consists of a set of identifiable fiducial points on the sky along with their coordinates, which

serves as the practical realization of a reference system.

For example, the fundamental plane of an astronomical reference system has conventionally

been the extension of the Earth’s equatorial plane, at some date, to infinity. Declination is the

angular distance north or south of this plane, and right ascension is the angular distance measured

eastward along the equator from some defined reference point. This reference point, the right

ascension origin, has traditionally been the equinox : the point at which the Sun, in its yearly

circuit of the celestial sphere, crosses the equatorial plane moving from south to north. The Sun’s

apparent yearly motion lies in the ecliptic, the plane of the Earth’s orbit. The equinox, therefore,

is a direction in space along the nodal line defined by the intersection of the ecliptic and equatorial

planes; equivalently, on the celestial sphere, the equinox is at one of the two intersections of the

great circles representing these planes. Because both of these planes are moving, the coordinate

systems that they define must have a date associated with them; such a reference system must

therefore be specified as “the equator and equinox of (some date)”.

Of course, such a reference system is an idealization, because the theories of motion of the Earth

that define how the two planes move are imperfect. In fact, the very definitions of these planes

are problematic for high precision work. Even if the fundamental planes of a reference system are

defined without any reference to the motions of the Earth, there is no way magically to paint them

on the celestial sphere at any particular time. Therefore, in practice, we use a specific reference

frame — a set of fiducial objects with assigned coordinates — as the practical representation of

an astronomical reference system. The scheme is completely analogous to how terrestrial reference

systems are established using survey control stations (geodetic reference points) on the Earth’s

surface.

Most commonly, a reference frame consists of a catalog of precise positions (and motions, if

measurable) of stars or extragalactic objects as seen from the solar system barycenter at a specific

epoch (now usually “J2000.0”, which is 12h TT on 1 January 2000). Each object’s instantaneous

position, expressed as right ascension and declination, indicates the object’s angular distance from

the catalog’s equator and origin of right ascension. Any two such objects in the catalog (if they are

not coincident or antipodal) therefore uniquely orient a spherical coordinate system on the sky —

a reference frame.

A modern astrometric catalog contains data on a large number of objects (N), so the coordinate

system is vastly overdetermined. The quality of the reference frame defined by a catalog depends

on the extent to which the coordinates of all possible pairs of objects (≈ N2/2) serve to define
22 CELESTIAL REFERENCE SYSTEM

the identical equator and right ascension origin, within the expected random errors. Typically,

every catalog contains systematic errors, that is, errors in position that are similar for objects that

are in the same area of the sky, or are of the same magnitude (flux) or color (spectral index).

Systematic errors mean that the reference frame is warped, or is effectively different for different

classes of objects. Obviously, minimizing systematic errors when a catalog is constructed is at least

as important as minimizing the random errors.

To be useful, a reference frame must be implemented at the time of actual observations, and

this requires the computation of the apparent coordinates of the catalog objects at arbitrary dates

and times. The accuracy with which we know the motions of the objects across the sky is an

essential factor in this computation. Astrometric star catalogs list proper motions, which are the

projection of each star’s space motion onto the celestial sphere, expressed as an angular rate in right

ascension and declination per unit time. Because the tabulated proper motions are never perfect,

any celestial reference frame deteriorates with time. Moreover, systematic errors in the proper

motions can produce time-dependent warpings and spurious rotations of the frame. Therefore,

the accuracy and consistency of the proper motions are critical to the overall quality, utility, and

longevity of reference frames defined by stars. Even reference frames defined by extragalactic

objects, which are usually considered to have zero proper motion, may deteriorate, because many

of these objects show small apparent motions that are artifacts of their emission mechanisms.

The positions of solar system objects can also be used to define a reference frame. For each

solar system body involved, an ephemeris (pl. ephemerides) is used, which is simply a table of the

celestial coordinates of the body as a function of time (or an algorithm that yields such a table). A

reference frame defined by the ephemerides of one or more solar system bodies is called a dynamical

reference frame. Because the ephemerides used incorporate the motion of the Earth as well as that

of the other solar system bodies, dynamical reference frames embody in a very fundamental way

the moving equator and ecliptic, hence the equinox. They have therefore been used to correct the

orientation of star catalog reference frames (the star positions were systematically adjusted) on

the basis of simultaneous observations of stars and planets. In a sense, the solar system is used

as a gyrocompass. However, dynamical reference frames are not very practical for establishing a

coordinate system for day-to-day astronomical observations.

Descriptions of reference frames and reference systems often refer to three coordinate axes, which

are simply the set of right-handed cartesian axes that correspond to the usual celestial spherical

coordinate system. The xy-plane is the equator, the z-axis points toward the north celestial pole,

and the x-axis points toward the origin of right ascension. Although in principle this allows us to

specify the position of any celestial object in rectangular coordinates, the distance scale (based on

stellar parallaxes) is not established to high precision beyond the solar system. What a reference

system actually defines is the way in which the two conventional astronomical angular coordinates,

right ascension and declination, overlay real observable points in the sky. (See eqs. 5.1 & 5.2 for

the conversion between rectangular and spherical celestial coordinates.)

3.3 Recent Developments

The establishment of celestial reference systems is coordinated by the IAU. The previous astronomical reference system was based on the equator and equinox of J2000.0 determined from observations

of planetary motions, together with the IAU (1976) System of Astronomical Constants and related

algorithms (Kaplan 1982). The reference frame that embodied this system for practical purposes

was the Fifth Fundamental Catalogue (FK5). The FK5 is a catalog of 1535 bright stars (to magni-
CELESTIAL REFERENCE SYSTEM 23

tude 7.5), supplemented by a fainter extension of 3117 additional stars (to magnitude 9.5), compiled

at the Astronomische Rechen-Institut in Heidelberg (Fricke et al. 1988; Fricke, et al. 1991). The

FK5 was the successor to the FK3 and FK4 catalogs, all of which were based on meridian observations taken in the visual band — many such observations were, in fact, taken by eye. The formal

uncertainties in the star positions of the FK5 at the time of its publication in 1988 were about

30–40 milliarcseconds over most of the sky, but the errors are considerably worse when systematic

trends are taken into account.

In recent years, the most precise wide-angle astrometry has been conducted not in the optical

regime but at radio wavelengths, involving the techniques of very long baseline interferometry

(VLBI) and pulsar timing. Uncertainties of radio source positions listed in all-sky VLBI catalogs

are now typically less than one milliarcsecond, and often a factor of ten better. Furthermore,

because these radio sources are very distant extragalactic objects (mostly quasars) that are not

expected to show measurable intrinsic motion, a reference frame defined by VLBI positions should

be “more inertial” (less subject to spurious rotation) than a reference frame defined by galactic

objects such as stars or pulsars. The VLBI catalogs do have the disadvantage that their origin of

right ascension is somewhat arbitrary; there is no real equinox in VLBI catalogs, since VLBI has

little sensitivity to the ecliptic plane. However, this problem has turned out to be more conceptual

than practical, since methods have been developed to relate the VLBI right ascension origin to the

equinox as conventionally defined.

Because of these considerations, since the mid 1980s, astronomical measurements of the Earth’s

rotation — from which astronomical time is determined — have depended heavily on VLBI, with

classical methods based on star transits being phased out. Hence the situation evolved to where the

definition of the fundamental astronomical reference frame (the FK5) became irrelevant to some of

the most precise and important astrometric measurements. VLBI revealed, in addition, that the

models of the Earth’s precession and nutation that were part of the old system were inadequate for

modern astrometric precision. In particular, the “constant of precession” — a measurement of the

long-term rate of change of the orientation of the Earth’s axis in space — had been overestimated

by about 0.3 arcseconds per century. Moreover, the success of the European Space Agency’s

Hipparcos astrometric satellite, launched in 1989, promised to provide a new, very accurate set of

star coordinates in the optical regime.

Thus, beginning in 1988, a number of IAU working groups began considering the requirements

for a new fundamental astronomical reference system (Lieske & Abalakin 1990; Hughes, Smith,

& Kaplan 1991). The resulting series of IAU resolutions, passed in 1991, 1994, 1997, and 2000

effectively form the specifications for the ICRS. The axes of the ICRS are defined by the adopted

positions of a specific set of extragalactic objects, which are assumed to have no measurable proper

motions. The ICRS axes are consistent, to about 0.02 arcsecond, with the equator and equinox of

J2000.0 defined by the dynamics of the Earth. However, the ICRS axes are meant to be regarded

as fixed directions in space that have an existence independent of the dynamics of the Earth or the

particular set of objects used to define them at any given time.

Feissel & Mignard (1998) have written a concise review of the ICRS adoption and its implications. Seidelmann & Kovalevsky (2002) published a broader review of the ICRS and the new IAU

Earth orientation models.

The promotion, maintenance, extension, and use of the ICRS are the responsibilities of IAU

Division 1 (Fundamental Astronomy).
24 CELESTIAL REFERENCE SYSTEM

3.4 ICRS Implementation

3.4.1 The Defining Extragalactic Frame

The International Celestial Reference Frame (ICRF) is a catalog of adopted positions of 608 extragalactic radio sources observed with VLBI, all strong (>0.1 Jy) at S and X bands (wavelengths

13 and 3.6 cm) (Ma & Feissel 1997; URL 10). Most have faint optical counterparts (typically

mV  18) and the majority are quasars. Of these objects, 212 are defining sources that establish

the orientation of the ICRS axes, with origin at the solar system barycenter. Typical position

uncertainties for the defining sources are of order 0.5 milliarcsecond; the orientation of the axes

is defined from the ensemble to an accuracy of about 0.02 milliarcsecond. As described in section 3.4.4, these axes correspond closely to what would conventionally be described as “the mean

equator and equinox of J2000.0”.

The International Earth Rotation and Reference Systems Service (IERS) monitors the radio

sources involved in the ICRF. This monitoring is necessary because, at some level, most of the

sources are variable in both flux and structure and the centers of emission can display spurious

motions. It is possible that, eventually, the defining list of sources will have to be amended to

maintain the fixed orientation of the overall frame.

3.4.2 The Frame at Optical Wavelengths

The ICRS is realized at optical wavelengths by stars in the Hipparcos Catalogue of 118,218 stars,

some as faint as visual magnitude 12 (ESA 1997). Only stars with uncomplicated and well-determined proper motions (e.g., no known binaries) are used for the ICRS realization. This subset,

referred to as the Hipparcos Celestial Reference Frame (HCRF), comprises 85% of the stars in

the Hipparcos catalog. Hipparcos star coordinates and proper motions are given within the ICRS

(J2000.0) coordinate system but are listed for epoch J1991.25. (That is, the catalog effectively

represents a snapshot of the positions and motions of the stars taken on 2 April 1991, a date that is

near the mean epoch of the Hipparcos observations.) At the catalog epoch, Hipparcos uncertainties

for stars brighter than 9th magnitude have median values somewhat better than 1 milliarcsecond in

position and 1 milliarcsecond/year in proper motion (ESA 1997; Mignard 1997). Thus, projected

to epoch J2000.0, typical Hipparcos star position errors are in the range 5–10 milliarcseconds.

3.4.3 Standard Algorithms

Chapters 1, 2, 5, and 6 of this circular describe IAU-sanctioned algorithms used in the construction,

maintenance, and use of the ICRS.

The 2000 IAU resolutions provide the relativistic metric tensors for what it called the Barycentric Celestial Reference System (BCRS) and the Geocentric Celestial Reference System (GCRS), as

well as expressions for the transformation between the two systems; see Chapter 1 and res. B1.3 of

2000. As noted in Chapter 1, the resolutions specify only the relativistic basis of the two reference

systems, and there is no prescription given for establishing the axis directions. The BCRS and

GCRS could therefore be considered families of coordinate systems, each member differing from

the others only in overall orientation. The construction of the ICRS (in particular, the analysis

of VLBI observations) was consistent with the definition of the BCRS in the resolutions. Thus,

the ICRS could be considered one implementation of a BCRS; i.e., a member of the BCRS family.

Recently, the IAU Working Group on Nomenclature for Fundamental Astronomy has recommended

that the orientation of the BCRS axes be understood to be that of the ICRS/ICRF.
CELESTIAL REFERENCE SYSTEM 25

In 2000, the IAU also adopted new models for the computation of the Earth’s instantaneous

orientation, which affect the analysis of VLBI observations that are used to define and maintain the

ICRS, as well as the calculation of various observable quantities from ICRS-compatible reference

data. The new models include what is referred to as the IAU 2000A precession-nutation model,

a new definition of the celestial pole, and two new reference points in the equatorial plane for

measuring the rotational angle of the Earth around its instantaneous axis. Despite the IAU action

in 2000, some aspects of the models were not finalized until late 2002 (mid 2005 for agreement

on the final precession expressions). These algorithms are described in Chapters 5 and 6 of this

circular and in the IERS Conventions (2003).

The new Earth orientation models are, of course, relevant only to fundamental observations

made from the surface of the Earth. Astrometric observations taken from space platforms, or

those that are differential in nature (based on reference objects all located within a small field),

do not use these models. There are, of course, other effects that must be taken into account in

analyzing astrometric observations — e.g., proper motion, parallax, aberration, and gravitational

light-bending — and algorithms for these may be found in Volumes 1 and 3 of the Hipparcos Catalogue documentation (ESA 1997). For analysis of very high accuracy satellite observations, see

the development by Klioner (2003).

As described in the Introduction, there are two collections of general-purpose computer subroutines that implement the new IAU-sanctioned algorithms for practical applications: the Standards

of Fundamental Astronomy (SOFA), at URL 7, and the Naval Observatory Vector Astrometry

Subroutines (NOVAS), at URL 8. NOVAS also implements many of the Hipparcos algorithms, or

the equivalent.

For ground-based applications requiring accuracies of no better than 50 milliarcseconds between

about 1990 and 2010, the algorithms described in Chapter 3 of the Explanatory Supplement (1992)

can still be used with ICRS data. (For such purposes, ICRS data can be treated as being on the

dynamical equator and equinox of J2000.0.) A major revision of the Explanatory Supplement to

reflect the adoption of the ICRS and all the new models is in progress.

3.4.4 Relationship to Other Systems

The orientation of the ICRS axes is consistent with the equator and equinox of J2000.0 represented

by the FK5, within the errors of the latter. See Feissel & Mignard (1998) for a short discussion.

Systematically, the FK5 equator is tilted by 22 mas and its origin of right ascension is offset by

23 mas with respect to the ICRS. But the uncertainties of the FK5 equator and right ascension

system with respect to the dynamical equator and equinox of J2000.0 are 50 and 80 mas, respectively. Since, at J2000.0, the errors of the FK5 are significantly worse than those of Hipparcos, the

ICRS (as realized by the HCRF) can be considered to be a refinement of the FK5 system (ESA

1997) at (or near) that epoch.

The ICRS can also be considered to be a good approximation (at least as good as the FK5) to the

conventionally defined dynamical equator and equinox of J2000.0 (Feissel & Mignard 1998), if the

latter system is considered to be barycentric. This follows from an IAU resolution passed in 1991

that provided the original specifications for the new fundamental astronomical reference system

based on extragalactic objects — what became the ICRS. In fact, the equator is well determined

fundamentally from the VLBI observations that are the basis for the ICRS, and the ICRS pole is
26 CELESTIAL REFERENCE SYSTEM

within 20 milliarcseconds of the dynamical pole.1 The zero point of VLBI-derived right ascensions

is arbitrary, but traditionally has been set by assigning to the right ascension of source 3C 273B a

value derived from lunar occultation timings — the Moon’s ephemeris thus providing an indirect

link to the dynamical equinox. The ICRS origin of right ascension was made to be consistent

with that in a group of VLBI catalogs previously used by the IERS, each of which had been

individually aligned to the lunar occultation right ascension of 3C 273B. The difference between

the ICRS origin of right ascension and the dynamical equinox has been independently measured

by two groups that used different definitions of the equinox, but in both cases the difference found

was less than 0.1 arcsecond.

Because of its consistency with previous reference systems, implementation of the ICRS will be

transparent to any applications with accuracy requirements of no better than 0.1 arcseconds near

epoch J2000.0. That is, for applications of this accuracy, the distinctions between the ICRS, FK5,

and dynamical equator and equinox of J2000.0 are not significant.

3.4.5 Data in the ICRS

Although the ICRF and HCRF are its basic radio and optical realizations, the ICRS is gradually

being extended to fainter magnitudes and other wavelengths. Thus, an increasing amount of fundamental astronomical data is being brought within the new system. A number of projects for the

densification of the ICRS have been completed or are in progress.

As described above, the ICRF consists of the adopted positions of about 600 extragalactic radio

sources, a third of which are defining sources. In its original presentation, the ICRF contained 608

extragalactic radio sources, including 212 defining sources. All observational data were part of a

common catalog reduction (Ma & Feissel 1997) and thus the adopted coordinates of all the sources

are in the ICRS. Of the 396 non-defining sources, 294 are candidate sources that do not meet all

of the accuracy and observing history requirements of the defining sources but which may at some

later time be added to the defining list. The remaining 102 other sources show excess apparent

position variation and are of lower astrometric quality. ICRF Extension 2 (ICRF-Ext.2) was issued

in 2004 (Fey et al. 2004); the positions of the candidate and other sources were refined and 109

new sources were added. The positions of the defining sources were left unchanged.

The VLBA Calibrator Survey is a list of radio sources, with positions in the ICRS, to be used

as calibrators for the Very Long Baseline Array and the Very Large Array. The original list was

prepared by Beasley et al. (2002); the list has been extended several times and the current version is

known as VCS3 (Petrov et al. 2005). The ICRS is also being established at higher radio frequencies

(24 and 43 GHz); see, e.g., Jacobs et al. (2005).

In the optical regime, the Tycho-2 Catalogue (Høg et al. 2000) (which supersedes the original

Tycho Catalogue and the ACT Reference Catalog) combines a re-analysis of the Hipparcos star

mapper observations with data from 144 ground-based star catalogs. The ground-based catalogs

include the Astrographic Catalogue (AC), a large photographic project carried out near the beginning of the 20th century involving 20 observatories worldwide. Tycho-2 contains 2,539,913 stars,

to about magnitude 12, and combines the accuracy of the recent Hipparcos position measurements

1The reason that the ICRS pole is not perfectly aligned with the dynamical pole is complex. The ICRF was

created from almost 20 years of VLBI observations from which a grand solution was made for the directions to the

extragalactic radio sources and the changing position of the celestial pole. A specific decision made in that analysis

(see Figure 5.2 and footnote on page 37) resulted in an offset of the ICRF (hence ICRS) pole at J2000.0. With

respect to the ICRS X and Y axes, the dynamical mean pole has coordinates on the unit sphere, in milliarcseconds,

of approximately (-16.6,-6.8).
CELESTIAL REFERENCE SYSTEM 27

with proper motions derived from a time baseline of almost a century. Proper motion uncertainties

are 1-3 milliarcseconds/year. At epoch J2000.0, the Tycho-2 positions of stars brighter than 9th

magnitude will typically be in error by 20 milliarcseconds. However, the positional accuracy degrades quite rapidly for magnitudes fainter than 9, so that 12th magnitude stars may be expected

to have a median J2000.0 position error exceeding 100 milliarcseconds.

Also in the optical band, the UCAC catalog is nearing completion and will provide ICRScompatible positions and proper motions for stars as faint as visual magnitude 16. See Zacharias

(2004) for information on the second release of UCAC data.

The ICRS has been extended to the near infrared through the 2MASS survey (Cutri et al.

2003). This ground-based program provides positions for 471 million point sources, most of which

are stars, observed in the J, H, and Ks infrared bands.

The Jet Propulsion Laboratory DE405/LE405 planetary and lunar ephemerides (usually just

referred to as DE405) (Standish 1998b) have been aligned to the ICRS. These ephemerides provide

the positions and velocities of the nine major planets and the Moon with respect to the solar system

barycenter, in rectangular coordinates. The data are represented in Chebyshev series form and

Fortran subroutines are provided to read and evaluate the series for any date and time. DE405

spans the years 1600 to 2200; a long version, DE406, spans the years –3000 to +3000 with lower

precision. See Chapter 4.

The barycentric data tabulated in The Astronomical Almanac are in the ICRS beginning with

the 2003 edition. Planetary and lunar ephemerides are derived from DE405/LE405. The Astronomical Almanac for 2006 is the first edition fully to support the new ICRS-related algorithms,

including the new IAU Earth rotation models. Geocentric coordinates are therefore given with

respect to the GCRS.

3.5 Formulas

A matrix B is required to convert ICRS data to the dynamical mean equator and equinox of J2000.0

(the “J2000.0 system”), the latter considered for this purpose to be a barycentric system. The same

matrix is used in the geocentric transformations described in Chapters 5 and 6 to adjust vectors in

the GCRS (the “geocentric ICRS”) so that they can be operated on by the conventional precession

and nutation matrices. The matrix B is called the frame bias matrix, and it corresponds to a fixed

set of very small rotations. In the barycentric case it is used as follows:

rmean(2000) = B rICRS (3.1)

where rICRS is a vector with respect to the ICRS and rmean(2000) is a vector with respect to the

dynamical mean equator and equinox of J2000.0. Both of the r’s are column vectors and, if they

represent a direction on the sky, are of the general form

r =





cos δ cos α

cos δ sin α

sin δ



 (3.2)

where α is the right ascension and δ is the declination, with respect to the ICRS or the dynamical

system of J2000.0, as appropriate.

In the geocentric case, rICRS is replaced by rGCRS and rmean(2000) is then a geocentric vector. This

transformation must be carried out, for example, before precession is applied to GCRS vectors, since
28 CELESTIAL REFERENCE SYSTEM

the precession algorithm assumes a dynamical coordinate system. That is, the above transformation

is a necessary step in obtaining coordinates with respect to the mean equator and equinox of date,

if one starts with ICRS reference data. See Chapter 5 for more information.

The matrix B is, to first order,

B =





1 dα0 −ξ0

−dα0 1 −η0

ξ0 η0 1



 (3.3)

where dα0 = −14.6 mas, ξ0 = −16.6170 mas, and η0 = −6.8192 mas, all converted to radians

(divide by 206 264 806.247). The values of the three small angular offsets are taken from the

IERS Conventions (2003). They can be considered adopted values; previous investigations of the

dynamical-ICRS relationship obtained results that differ at the mas level or more, depending on

the technique and assumptions. See the discussion in Hilton & Hohenkerk (2004). The angles ξ0

and η0 are the ICRS pole offsets, and dα0 is the offset in the ICRS right ascension origin with

respect to the dynamical equinox of J2000.0, as measured in an inertial (non-rotating) system.

The above matrix can also be used to transform vectors from the ICRS to the FK5 system at

J2000.0. Simply substitute dα0 = −22.9 mas, ξ0 = 9.1 mas, and η0 = −19.9 mas. However, there

is also a time-dependent rotation of the FK5 with respect to the ICRS (i.e., a slow spin), reflecting

the non-inertiality of the FK5 proper motions; see Mignard & Frœschl ́e (2000).

Although the above matrix is adequate for most applications, a more precise result can be

obtained by using the second-order version:

B =





1 −

1

2

(dα2

0 + ξ

2

0

) dα0 −ξ0

−dα0 − η0ξ0 1 −

1

2

(dα2

0 + η

2

0

) −η0

ξ0 − η0dα0 η0 + ξ0dα0 1 −

1

2

(η

2

0 + ξ

2

0

)



 (3.4)

The above matrix, from Slabinski (2005), is an excellent approximation to the set of rotations

R1(−η0)R2(ξ0)R3(dα0), where R1, R2, and R3 are standard rotations about the x, y, and z axes,

respectively (see “Abbreviations and Symbols Frequently Used” for precise definitions).
Chapter 4

Ephemerides of the Major Solar

System Bodies

Relevant IAU resolutions: (none)

Summary The de facto standard source of accurate data on the positions and motions

of the major solar system bodies is currently the ephemeris designated DE405/LE405

(or simply DE405) developed at the Jet Propulsion Laboratory. This ephemeris provides

instantaneous position and velocity vectors of the nine major planets and the Earth’s

Moon, with respect to the solar system barycenter, for any date and time between 1600

and 2201. Lunar rotation angles are also provided. The ephemeris has been the basis

for the tabulations in The Astronomical Almanac since the 2003 edition. The DE405

coordinate system has been aligned to the ICRS.

IAU-standard data on the sizes, shapes, rotational parameters, and latitude-longitude

systems for the major planets and their satellites are given in the reports of the IAU/IAG

Working Group on Cartographic Coordinates and Rotational Elements, issued every

three years.

4.1 The JPL Ephemerides

A list of positions of one or more solar system bodies as a function of time is called an ephemeris

(pl. ephemerides). An ephemeris can take many forms, including a printed tabulation, a sequential

computer file, or a piece of software that, when interrogated, computes the requested data from

series approximations or other mathematical schemes.

Ephemerides of the major solar system bodies, with respect to the solar system barycenter, have

been calculated for many years at the Jet Propulsion Laboratory (JPL) to support various spacecraft

missions. These ephemerides have been widely distributed and, because of their quality, have

become the de facto standard source of such data for applications requiring the highest accuracy.

Between the early 1980s and about 2000, the JPL ephemeris designated DE200/LE200 was most

frequently used for such applications; it was the basis for the tabulations in The Astronomical

Almanac from the 1984 to 2002 editions. A more recent JPL ephemeris, DE405/LE405, has now

come into widespread use, and has been the basis for The Astronomical Almanac since the 2003

edition. These ephemerides are usually referred to as just DE200 and DE405, respectively. Neither

29
30 EPHEMERIDES

DE200 nor DE405 have been the subject of any IAU resolution, although they have been frequently

reported on at various IAU-sponsored meetings, and DE405 is a recommended standard of the IERS

(IERS Conventions 2003). A comparison of DE405 with DE200, with an estimate of their errors,

has been given by Standish (2004).

The JPL ephemerides are computed by an N-body numerical integration, carried out in a

barycentric reference system which is consistent, except for the time scale used, with the Barycentric

Celestial Reference System (BCRS) described in Chapter 1. The equations of motion, the method of

integration, and the techniques used to adjust the starting conditions of the integration so that the

results are an optimal fit to observations are described in Chapter 5 of the Explanatory Supplement

(1992). That chapter specifically describes DE200, but the basic procedures are largely the same

for all of the JPL ephemerides.

The position and velocity data provided by the JPL ephemerides represent the centers of mass

of each planet-satellite system (although data for the Earth and Moon can be extracted separately).

Therefore, the positions, when converted to geocentric apparent places — angular coordinates as

seen from Earth — do not precisely indicate the center of the apparent planetary disk. Displacements can amount to a few tens of milliarcseconds for Jupiter and Saturn, a few milliarcseconds

for Uranus and Neptune, and about 0.1 arcsecond for Pluto.

4.2 DE405

The JPL DE405/LE405 ephemeris provides the coordinates and velocities of the major planets,

the Sun, and the Earth’s Moon for the period 1600 to 2200 (Standish 1998b). The position and

velocity 3-vectors are in equatorial rectangular coordinates referred to the solar system barycenter.

The reference frame for the DE405 is the ICRF; the alignment onto this frame, and therefore onto

the ICRS, has an estimated accuracy of a few milliarcseconds, at least for the inner-planet data.

Optical, radar, laser, and spacecraft observations were analyzed to determine starting conditions

for the numerical integration and values of fundamental constants such as the Earth/Moon mass

ratio and the length of the astronomical unit in meters. In addition to the planetary and lunar

coordinates and velocities, the ephemerides, as distributed, include the nutation angles of the Earth

and the rotation (libration) angles of the Moon. (Note, however, that the nutation angles are not

derived from the IAU 2000A theory described in Chapter 5.)

As described in Chapter 2, DE405 was developed in a barycentric reference system using Teph,

a barycentric coordinate time (Standish 1998a). Teph is rigorously equivalent to TCB in a mathematical sense, differing only in rate: the rate of Teph matches the average rate of TT, while the

rate of TCB is defined by the SI system. The IAU time scale TDB, often (but erroneously) considered to be the same as Teph, is a quantity that cannot be physically realized, due to its flawed

definition. So, in fact, the use of the name TDB actually refers to quantities based on or created

with Teph.

1 Astronomical constants obtained from ephemerides based on Teph (or TDB) are not

in the SI system of units and must therefore be scaled for use with TCB or other SI-based time

scales.

The ephemerides are distributed by JPL as plain-text (ASCII) computer files of Chebyshev

series coefficients and Fortran source code. Third-party C versions of the code are also available

and, for Unix users, the data files can be downloaded in binary form. Once the system is installed

on a given computer, a Fortran subroutine named PLEPH can be called to provide the position and

1Because of this, the IAU Working Group on Nomenclature for Fundamental Astronomy has recommended changing the definition of TDB to be consistent with that of Teph.
EPHEMERIDES 31

velocity of any requested body, at any date and time; PLEPH supervises the process of reading the

Chebyshev file and evaluating the appropriate series. Normally the position and velocity vectors

returned are expressed in units of AU and AU/day, respectively. The date/time requested must be

expressed as a Teph or TDB Julian date. (An entry named DPLEPH is provided that allows the

input Julian date to be split into two parts for greater precision.) And, since |TT−Teph| < 0.002 s,

a TT Julian date may be used for applications not requiring the highest accuracy. The data and

software files can be obtained on CD-ROM from Willmann-Bell, Inc. (URL 11a), or downloaded

from a JPL ftp server (URL 11b). A “README” file provides export information and software

documentation (available separately at URL 12).

An extended version of DE405/LE405, called DE406/LE406, is available that spans the years

–3000 to +3000, but with coordinates given to lower precision (they are represented by shorter

Chebyshev series). The nutation angles and the lunar rotation angles are also omitted from the

DE406 files. DE406 is provided only in Unix binary format. These files are about 1/3 the size of

those for DE405 for a given span of time. The additional error in the coordinates (DE406 – DE405)

may amount to 25 m for the planets and 1 m for the Moon, which may be significant for some

applications.

The NOVAS software package mentioned in the Introduction provides an interface to an existing

DE405 or DE406 installation through Fortran subroutine SOLSYS or C function ephemeris.

4.3 Sizes, Shapes, and Rotational Data

The IAU/IAG2 Working Group on Cartographic Coordinates and Rotational Elements (URL 13)

produces a report every three years (for each IAU General Assembly) giving the best estimates

for the dimensions and rotational parameters of the planets, satellites, and asteroids, as far as

is known. The working group is also responsible for establishing latitude-longitude coordinate

systems for these bodies. The rotational elements given in the report for the 2000 General Assembly

(Seidelmann et al. 2002) serve to orient these coordinate systems within the ICRS as a function

of time. (Note that the time scale in this report is TDB, not TCB as stated.) The working

group’s reports are the basis for the physical ephemerides of the planets given in The Astronomical

Almanac.

Although the rotational elements of the Earth and Moon are given in each report for completeness, the expressions given there provide only an approximation to the known motions and should

not be used for precise work (e.g., for the Earth, precession is accounted for only to first order

and nutation is neglected). Lunar rotation (libration) angles can be obtained from DE405, and

Chapters 5 and 6 of this circular describe algorithms for the precise instantaneous alignment of the

terrestrial coordinate system within the GCRS (the “geocentric ICRS”).

4.4 DE405 Constants

The constants below were used in or determined from the DE405 ephemeris. Many DE405 constants

are expressed in what have been called “TDB units”, rather than SI units. (Perhaps it is more

proper to say that the constants have TDB-compatible values.) A scaling factor, K = 1/(1 − LB),

where LB is given below, is required to convert such constants, with dimensions of length or time,

to SI units (Irwin & Fukushima 1999). Dimensionless quantities such as mass ratios do not require

2

IAG = International Association of Geodesy
32 EPHEMERIDES

scaling.

LB = 1.550 519 767 72×10−8 =⇒ K = 1.000 000 015 505 198 (4.1)

The value of LB is taken from note 3 of res. B1.5 of 2000.

These conversions also depend on assumptions about what constants are truly fundamental and

what their values are (Klioner 2005); therefore, the SI value of the astronomical unit in meters is

not given below. The planetary masses below include contributions from satellites and atmospheres.

Constant Symbol DE405 Value Equivalent SI value

astronomical unit in seconds τA 499.004 783 806 1 s 499.004 786 385 2 s

astronomical unit in meters A or cτA 149 597 870 691 m

heliocentric gravitational GS or GM
· 1.327 124 400 2×1020 1.327 124 420 8×1020

constant m3s

−2 m3

s

−2

Moon/Earth mass ratio μ 1/81.30056

Sun/planet mass ratios:

Mercury 6 023 600

Venus 408 523.71

Earth + Moon 328 900.561400

Earth 332 946.050895. . .

Mars 3 098 708

Jupiter 1 047.3486

Saturn 3 497.898

Uranus 22 902.98

Neptune 19 412.24

Pluto 135 200 000
Chapter 5

Precession and Nutation

Relevant IAU resolutions: B1.6, B1.7 of 2000

Summary Precession and nutation are really two aspects of a single phenomenon, the

overall response of the spinning, oblate, elastic Earth to external gravitational torques

from the Moon, Sun, and planets. As a result of these torques, the orientation of the

Earth’s rotation axis is constantly changing with respect to a space-fixed (locally inertial) reference system. The motion of the celestial pole among the stars is conventionally

described as consisting of a smooth long-term motion called precession upon which is

superimposed a series of small periodic components called nutation.

The algorithms for precession used generally from about 1980 through 2000 (in

The Astronomical Almanac from the 1984 through 2005 editions) were based on the

IAU (1976) value for the rate of general precession in ecliptic longitude (5029.0966

arcseconds per Julian century at J2000.0). Nutation over most of the same time period

was given by the 1980 IAU Theory of Nutation. However, not long after these algorithms

were widely adopted, it became clear that the IAU (1976) rate of precession had been

overestimated by approximately 3 milliarcseconds per year. Further observations also

revealed periodic errors of a few milliarcseconds in the 1980 IAU Theory of Nutation.

For many applications these errors are negligible, but they are significant at the level

of the best ground-based astrometry and geodesy.

As part of the 2000 IAU resolutions, the IAU 2000A precession-nutation model was

introduced, based on an updated value for the rate of precession and a completely

new nutation theory. As before, the model actually consists of two parts, a precession

algorithm describing the smooth secular motion of the celestial pole and a nutation

algorithm describing the small periodic variations in the pole’s position. The precession

algorithm consists of short polynomial series for the values of certain angles. The sines

and cosines of these angles, in combination, then define the elements of a precession

matrix, P. The nutation algorithm consists of a rather long series expansion in Fourier

terms for the angular offsets, in ecliptic longitude and latitude, of the actual celestial

pole (as modeled) from the precession-only pole (true pole – mean pole). The sines and

cosines of these offsets, in combination, then define the elements of a nutation matrix,

N. The P and N matrices are applied to the coordinates of celestial objects, expressed

as 3-vectors, to transform them from the equator and equinox of one epoch to the

equator and equinox of another.

33
34 PRECESSION & NUTATION

5.1 Aspects of Earth Rotation

The Earth is a relatively well-behaved rotating body, and illustrates the three basic elements of

classical spin dynamics: precession, nutation, and Eulerian wobble. In fact, to first order, the Earth

can be considered to be a rigid “fast top”, and very good approximations to its rotational motion

can be obtained from elementary developments. Although the effects of the Earth’s liquid core,

elastic mantle, and oceans are not negligible for modern observations, they can be considered to be

small perturbations on the rigid-body motion. Since the Earth is nearly spherical and experiences

relatively weak torques, its axis of rotation moves slowly with respect to both the stars and the

body of the Earth itself.

The orientation of any rotating rigid body can be described as a time series of three Euler angles

that relate a body-fixed coordinate system to a space-fixed coordinate system. If the body-fixed

coordinate system can be defined such that the rate of change of one of the three Euler angles

is much greater than that of the other two — as is the case for the Earth — then the rotational

kinematics are usually described in terms of the slowly changing orientation of an axis of rotation

passing through the body’s center of mass. We can equivalently speak of the kinematics of the

pole: one of the points where the axis of rotation intersects the body’s surface or, extended to

infinity, the “celestial sphere”. For this kinematic construction to work well, the angular motion of

the axis or pole should be small and nearly linear over one rotation, predictable from theory, and

observable.

However, as was pointed out by Eubanks (1993), when we use such an axis or pole, we need

five angles, not three, to fully describe the instantaneous orientation of the body: two angles to

describe the orientation of the body with respect to the axis, one to describe the angle of the

body’s rotation about the axis, and two more to describe the orientation of the axis in the fixed

external system (“inertial space”). For the Earth, these five angles correspond to the five standard

parameters of Earth orientation disseminated by organizations such as the IERS: the coordinates

of the pole, xp and yp, measured in a terrestrial coordinate system; the Universal Time difference,

UT1–UTC; and the celestial pole offsets, dψ and d, measured in a celestial coordinate system.

Phenomenologically, the parameters divide up as follows: xp and yp describe polar motion, the

variations in the position of the pole with respect to the Earth’s crust; UT1–UTC measures the

integrated variation in length of day, the departure from a constant angular rate of rotation; and

dψ and d are the errors in the computed position of the celestial pole, reflecting deficiencies in the

adopted algorithms for precession and nutation.

What we call polar motion corresponds, in rigid-body rotation, to the free Eulerian wobble of

the figure axis about the rotation axis. On the real Earth, the phenomenon is not that simple.

From an Earth-fixed (rotating) frame of reference, polar motion is a 10-meter (0.3 arcsecond) quasicircular excursion in the pole position, with principal periods of 12 and 14 months. The 14-month

component corresponds to the Eulerian wobble, as modified by the Earth’s elasticity, while the

12-month component undoubtedly is a seasonal effect. Smaller, quasi-random variations are not

well understood. None of the components is regular enough to permit reliable predictions, and

polar motion must be obtained from observations.

Variations in the Earth’s rotation rate are due to several causes. There are fortnightly, monthly,

semiannual, and annual tidal effects, and other short-term and seasonal changes are largely due to

exchange of angular momentum with the atmosphere. Longer-term variations (decade fluctuations)

are less well understood. For a discussion of time scales that are based on the variable rotation of

the Earth, see sections 2.1 and 2.3.

Precession and nutation refer to the changing orientation of the Earth’s axis, with respect to a
PRECESSION & NUTATION 35

space-fixed (kinematically non-rotating) system, in response to external torques. The torques are

due to the gravitational attraction of the Moon and Sun (and, to a much lesser extent, the planets)

on the equatorial bulge of the Earth. Precession and nutation are really different aspects of a

single physical phenomenon, and it has become more common in recent years to write “precessionnutation”. Precession is simply the secular term in the response, while nutation is the set of periodic

terms. On the celestial sphere, the celestial pole traces out a circle, about 23◦in radius, centered

on the ecliptic pole (the direction orthogonal to the ecliptic plane), taking about 26,000 years to

complete one circuit (≈20 arcseconds/year). Precession theory describes this smooth, long-term

motion, and the precessional pole is referred to as the mean pole (the orthogonal plane is the mean

equator). But the pole also undergoes a hierarchy of small epicyclic motions, the largest of which

is a 14×18 arcsecond ellipse traced out every 18.6 years (see Fig. 1). Nutation theory describes

these periodic motions. To get the path of the true pole on the celestial sphere (i.e., the direction of

the Earth’s axis in space), it is necessary to compute both precession and nutation; conventionally,

they are described by separate time-dependent rotation matrices, P(t) and N(t), which are either

multiplied together or applied sequentially.

Figure 5.1 The path of the true celestial pole on the sky, over an 18-year period,

compared to the mean pole. The mean pole moves along a smooth arc at a rate of

20 arcseconds per year due to precession only. The complex epicyclic motion of the

true pole is nutation. The inset shows the detail of one year’s motion.
36 PRECESSION & NUTATION

5.2 Which Pole?

In theoretical developments of Earth rotation, the first issue that must be confronted is the definition

of the celestial pole. If the Earth were a rigid oblate spheroid, there would be three possible axes,

and corresponding poles, to choose from: the angular momentum axis; the rotation axis, defined

by the instantaneous angular velocity vector; and the figure axis, which is the body-fixed axis

orthogonal to the geometric equator and along the unique eigenvector of the Earth’s inertia tensor.

The distinctions among the axes arise from the physics of rotation. For example, as previously

noted, in a rotating rigid body, the free Eulerian wobble describes the motion of the figure axis

with respect to the rotation axis (or vice versa). The analog of this on the real Earth is polar

motion.

In the previous section, precession-nutation was described as the changing orientation of the

Earth’s axis in response to external torques, expressed in a space-fixed (non-rotating) frame of

reference. Which axis? The principal components of the response are rather large, amounting to

many arcseconds over the course of a year, and are nearly the same for all three axes. (However,

the three axes cannot coincide in the presence of external torques.) For a rigid Earth, the forced

oscillations of the figure and rotation axes differ by about 10 milliarcseconds, and those of the

angular momentum and rotation axes differ by only about 1 milliarcsecond. Until the mid-20th

century, observations were not accurate enough to distinguish between the axes, so the choice of the

best axis for theory was academic. But with improving observational accuracy and new techniques

coming online in the 1960s and 1970s, the question of which axis should be used for the theoretical

developments became important. After considerable discussion, the consensus emerged that the

forced motion of the figure axis was the most relevant for observations, and therefore also for theory.

At about the same time, new theoretical work was being undertaken based on Earth models

that were triaxial and contained a liquid core and elastic mantle. Such theories complicate the axis

question considerably, because the inertia tensor varies with time as the Earth’s shape responds

to tidal forces, and the tidal deformation results in large daily excursions of the Earth’s axis of

figure. These excursions do not, in general, reflect the changing overall orientation of the Earth’s

crust in space, which is relevant to astronomical observations. That is, for the elastic Earth, the

figure axis as classically defined is not an astronomically useful axis. The solution is to construct a

rotating cartesian coordinate system tied to the elastic, rotating Earth in such a way that (1) the

net angular momentum of the tidal deformation, relative to this system, is always zero; and (2) for

zero tidal deformation, the axes correspond to the principal axes of the Earth’s mantle. These axes

are the “Tisserand mean axes of the body” (Munk & MacDonald 1960), and the Tisserand axis of

the maximum moment of inertia is referred to in res. B1.7 of 2000 as “the mean surface geographic

axis”. Almost all modern theories of nutation refer to the principal Tisserand axis; in the previously

used 1980 IAU Theory of Nutation it was referred to as axis B, and the corresponding pole called

the “Celestial Ephemeris Pole”.

However, even if we have chosen an axis that best reflects the overall rotation of stations (observatories) on the Earth’s surface, a further complication arises as the observations and theoretical

developments become more sensitive to short-period motions. The problem is the small but nonnegligible circular components of nutation or polar motion with periods near one day. One can

imagine the geometric confusion that arises when the pole undergoes a circular motion in one rotation period; in fact, it becomes difficult to disentangle the various effects, and our conventional

labels become nearly meaningless. For example, any prograde nearly-diurnal nutation is equivalent

to a long-period variation in polar motion, and any retrograde nearly-diurnal polar motion appears

as a long-period nutation component (Capitaine 2000). In practice, this means a potential “leak-
PRECESSION & NUTATION 37

age” or “crossover” from the Earth orientation parameters xp and yp to dψ and d or vice versa.

The only practical solution is an explicit (although somewhat arbitrary) cutoff in the periods of

what is considered precession-nutation, embodied in the definition of the celestial pole.

Therefore, the new IAU definition of the celestial pole to be used for the new precessionnutation model (res. B1.7 of 2000) is defined by the motions of Tisserand mean axis of the Earth

with periods greater than two days in the celestial reference system. This pole is called the Celestial

Intermediate Pole (CIP). The position of the CIP is given by the adopted precession-nutation model

plus observational corrections. The word intermediate reminds us that the definition of the pole is

merely a convention, serving to impose a division between what we call precession-nutation (the

Earth orientation angles measured in the celestial system) and polar motion (the Earth orientation

angles measured in the terrestrial system). The CIP is the true pole, orthogonal to the true

equator of date. Its motion is defined within the Geocentric Celestial Reference System (GCRS)

— see Chapter 1. Therefore, the geometric transformations described in this chapter (as well as

those in Chapter 6) all apply within a geocentric system. The GCRS can be described loosely as

the “geocentric ICRS”, since its axis directions are obtained from those of the ICRS.

5.3 The New Models

The variables dψ and d are the small angular offsets on the sky expressing the difference between

the position of the celestial pole that is observed and the position predicted by the conventional

precession and nutation theories. These angles are just the differential forms of the angles ∆ψ and

∆ in which nutation theories are conventionally expressed (dψ and d are sometimes labeled ∆∆ψ

and ∆∆). ∆ψ and ∆ are in turn differential forms of the ecliptic coordinates of the celestial pole

(see Fig. 5.1).

Obviously the time series of dψ and d values, if they show systematic trends, can be used to

improve the theories of precession and nutation. In fact, 20 years of dψ and d values from VLBI

show significant patterns — see Fig. 5.2. Most obvious is the overall downward slope in longitude

and an annual periodicity in both longitude and obliquity, suggesting the need for substantial

corrections to the precession rate as well to the annual nutation term. A long-period sinusoid

is also evident, and spectral analysis reveals the presence of a number of periodic components.1

Other techniques, particularly lunar laser ranging (LLR), confirm the general trends. As a result,

there has been a major multinational effort to improve the precession and nutation formulation

and obtain interesting geophysical information in the process. This project, coordinated by an

IAU/IUGG2 working group, has involved dozens of investigators in several fields, and the resulting

algorithms, taken together, are referred to as the IAU 2000A precession-nutation model.

1The figure indicates the origin of the ICRS “frame bias” discussed in Chapter 3. The pole offsets shown are taken

from the solution for the ICRF catalog. The ICRS frame biases in longitude and obliquity are essentially the values,

at J2000.0 (Time=100), of the two curves fitted to the data. The data was arbitrarily zeroed near the beginning of

the data span, which led to non-zero values at J2000.0.

2

IUGG = International Union of Geodesy and Geophysics
38 PRECESSION & NUTATION

Figure 5.2 Observed values of celestial pole offsets from VLBI data. Offsets in

longitude have been multiplied by the sine of the obliquity to allow the same scale

to be used for both components. Circled points with error bars represent the offset

of the observed pole with respect to the computed pole, and the solid line in each

plot is a curve fitted to the data. The computed pole is given by the Lieske et al.

(1977) precession expressions and the 1980 IAU Theory of Nutation. These plots

are from Ma et al. (1998).

The VLBI observations of dψ and d indicate the error in the computed position of the pole

with respect to a space-fixed system defined by the positions of extragalactic objects. However,

the conventional expressions for precession and nutation have used angles measured with respect

to the ecliptic, a plane to which VLBI is not sensitive. The ecliptic plane has a slow precessional

movement of its own due to planetary perturbations on the heliocentric orbital motion of the

Earth-Moon barycenter.3

In the theoretical developments it is necessary to distinguish between

precession of the equator and precession of the ecliptic, which were formerly called, respectively,

lunisolar precession and planetary precession. Both types of precession are measured with respect

to a space-fixed system. The algorithms for precession and nutation provide the motion of the

3The mean ecliptic is always implied. This is the smoothly moving plane that does not undergo the periodic

oscillations of the instantaneous orbital plane of the Earth.
PRECESSION & NUTATION 39

Table 5.1 Precession-Nutation: Old & New

Values in arcseconds at J2000.0

Quantity Old value New value New–Old

General precession in longitude (/cen) 5029.0966 5028.796195 −0.3004

Mean obliquity 84381.448 84381.406 −0.042

Mean obliquity rate (/cen) −46.8150 −46.836769 −0.0218

In-phase nutation amplitudes:

18.6-year longitude −17.1966 −17.2064161 −0.0098

18.6-year obliquity 9.2025 9.2052331 0.0027

9.3-year longitude 0.2062 0.2074554 0.0013

9.3-year obliquity −0.0895 −0.0897492 −0.0002

annual longitude 0.1426 0.1475877 0.0050

annual obliquity 0.0054 0.0073871 0.0020

semiannual longitude −1.3187 −1.3170906 0.0016

semiannual obliquity 0.5736 0.5730336 −0.0006

122-day longitude −0.0517 −0.0516821 0.0000

122-day obliquity 0.0224 0.0224386 0.0000

monthly longitude 0.0712 0.0711159 −0.0001

monthly obliquity −0.0007 −0.0006750 0.0000

semimonthly longitude −0.2274 −0.2276413 −0.0002

semimonthly obliquity 0.0977 0.0978459 0.0001

equator, as appropriate for most observations, but generally use a moving ecliptic as a reference

plane for at least some of the angles involved (there are different formulations of precession using

different angle sets). This allows the precession and nutation transformations to properly account

for the motion of the equinox as well as that of the equator. The precession of the ecliptic is

obtained from theory (although indirectly tied to observations through the JPL DE405 ephemeris),

as are the high-order (unobserved) components of the precession of the equator. However, because

of the mix of theory and observation that is involved in the final expressions, raw corrections to

rates of precession from VLBI observations will not in general propagate exactly to the familiar

precession quantities.

The changes in the amplitudes of the nutation components are also not directly taken from

these observations; instead, a new nutation theory is developed and fit to observations by allowing

a small number of geophysical constants to be free parameters. These parameters are constants

in a “transfer function” that modifies the amplitudes of the terms from a rigid-Earth nutation

development. Since there are fewer solved-for geophysical constants than the number of terms with

observed amplitudes, the fit cannot be perfect. For the IAU 2000A model, 7 geophysical parameters

were determined based on the observed amplitudes of 21 nutation terms (prograde and retrograde

amplitudes for each) together with the apparent change in the rate of precession in longitude. Note

that the number of observational constraints and the number of free parameters in the model are

both quite small compared to the 1365 terms in the new, full nutation series.

Table 5.1 compares the old and new values, at epoch J2000.0, of some of the primary quantities

involved in the precession and nutation algorithms. In the table, all quantities are in arcseconds,

and the rates (marked /cen) are per Julian century of TDB (or TT). The longitude components

should be multiplied by the sine of the obliquity (≈0.3978) to obtain the corresponding motion of
40 PRECESSION & NUTATION

the pole on the celestial sphere. The new mean obliquity at J2000.0 is 23◦ 260 21.

00406. The theories

from which the values are taken are:

• Old precession: Lieske et al. (1977), based on the IAU (1976) values for general precession

and the obliquity at J2000.0, shown in the table

• Old nutation: 1980 IAU Theory of Nutation: Wahr (1981), based on Kinoshita (1977); see

report of the IAU working group by Seidelmann (1982)

• New precession: P03 solution in Capitaine et al. (2003); see report of the IAU working group

by Hilton et al. (2006)

• New nutation: Mathews et al. (2002) (often referred to as MHB), based on Souchay et al.

(1999); series listed at URL 14

The new precession development will probably be formally adopted by the IAU in 2006. The

MHB nutation was adopted in res. B1.6 of 2000, even though the theory had not been finalized at

the time of the IAU General Assembly of that year. Used together, these two developments yield

the computed path of the Celestial Intermediate Pole (CIP) as well as that of the true equinox.

The formulas given below are based on these two developments.

5.4 Formulas

In the development below, precession and nutation are represented as 3×3 rotation matrices that

operate on column 3-vectors. The latter are position vectors in a specific celestial coordinate

system — which must be stated or understood — with components that are cartesian (rectangular)

coordinates. They have the general form

r =





rx

ry

rz



 =





d cos δ cos α

d cos δ sin α

d sin δ



 (5.1)

where α is the right ascension, δ is the declination, and d is the distance from the specified origin.

For stars and other objects “at infinity” (beyond the solar system), d is often simply set to 1. The

celestial coordinate system being used will be indicated by a subscript, e.g., rGCRS. If we have the

vector r in some coordinate system, then the right ascension and declination in that coordinate

system can be obtained from

α = arctan (ry/rx)

δ = arctan rz/

q

r

2

x + r

2

y



(5.2)

where rx, ry, and rz are the three components of r. A two-argument arctangent function (e.g.,

atan2) will return the correct quadrant for α if ry and rx are provided separately.

In the context of traditional equatorial celestial coordinate systems, the adjective mean is applied

to quantities (pole, equator, equinox, coordinates) affected only by precession, while true describes

quantities affected by both precession and nutation. This is a computational distinction only, since

precession and nutation are simply different aspects of the same physical phenomenon. Thus, it is

the true quantities that are directly relevant to observations; mean quantities now usually represent
PRECESSION & NUTATION 41

an intermediate step in the computations, or the final step where only very low accuracy is needed

(10 arcseconds or worse) and nutation can be ignored.

Thus, a precession transformation is applied to celestial coordinates to convert them from

the mean equator and equinox of J2000.0 to the mean equator and equinox of another date, t.

Nutation is applied to the resulting coordinates to transform them to the true equator and equinox

of t. These transformations should be understood to be inherently geocentric rotations and they

originate in dynamical theories. Generally we will be starting with celestial coordinates in the

GCRS, which are obtained from basic ICRS data by applying the usual algorithms for proper

place.4 As discussed in Chapter 3, the ICRS is not based on a dynamically defined equator and

equinox and so neither is the GCRS. Therefore, before we apply precession and nutation — and

if we require a final accuracy of better than 0.02 arcsecond — we must first apply the frame bias

correction (section 3.5) to transform the GCRS coordinates to the dynamical mean equator and

equinox of J2000.0. Schematically,

GCRS

k

frame bias

⇓

mean equator & equinox of J2000.0

k

precession

⇓

mean equator & equinox of t

k

nutation

⇓

true equator & equinox of t

Mathematically, this sequence can be represented as follows:

rtrue(t) = N(t) P(t) B rGCRS (5.3)

where rGCRS is a direction vector with respect to the GCRS and rtrue(t)is the equivalent vector

with respect to the true equator and equinox of t. N(t) and P(t) are the nutation and precession

rotation matrices, respectively. The remainder of this chapter shows how to compute the elements

of these matrices. B is the (constant) frame-bias matrix given in section 3.5.

The transformation from the mean equator and equinox of J2000.0 to the mean equator and

equinox of t is simply

rmean(t) = P(t) rmean(J2000.0) (5.4)

4Computing proper place involves adjusting the catalog place of a star or other extra-solar system object for proper

motion and parallax (where known), gravitational light deflection within the solar system, and aberration due to the

Earth’s motions. For a solar system object there are comparable adjustments to its position vector taken from a

barycentric gravitational ephemeris. See section 1.3. In conventional usage, an apparent place can be considered to

be a proper place that has been transformed to the true equator and equinox of date. The details of the proper

place computations are beyond the scope of this circular but are described in detail in many textbooks on positional

astronomy, and in Hohenkerk et al. (1992), Kaplan, et al. (1989), and Klioner (2003).
42 PRECESSION & NUTATION

and the reverse transformation is

rmean(J2000.0) = P

T(t) rmean(t)

(5.5)

where PT(t) is the transpose of P(t).

What is described above are all conventional, equinox-based transformations. The equinox is

the traditional origin of right ascension. An alternative transformation has been developed based on

another point on the celestial equator called the Celestial Intermediate Origin (CIO); see res. B1.8

of 2000. The alternative scheme has been introduced because the equinox is based on a barycentric

concept (the ecliptic, the Earth’s mean orbit) that is not relevant to a geocentric system; furthermore, the equinox suffers from ambiguity of definition below the 0.1 arcsecond level. Additionally,

the new scheme cleanly separates different aspects of Earth orientation in the overall transformation between the terrestrial and celestial coordinate systems. The conventional transformations are

described in this chapter because of widespread current usage and the fact that even the newest

theories of precession and nutation provide the angles needed for these transformations. The new

transformation, which combines frame bias, precession, and nutation into a single matrix, is given in

section 5.4.3 and is described more fully in Chapter 6, where the CIO is introduced and explained.

The true celestial pole of date t — the Celestial Intermediate Pole (CIP) — has, by definition,

unit vector coordinates (0,0,1) with respect to the true equator and equinox of date. Therefore we

can obtain the computed coordinates of the CIP with respect to the GCRS by simply reversing the

transformation of eq. 5.3:

Computed position of CIP: rGCRS =





X

Y

Z



 = B

T PT(t) NT(t)





0

0

1





=





(NPB)31

(NPB)32

(NPB)33



 (5.6)

where NPB = N(t) P(t) B

and where the superscript T’s indicate that the transpose of the matrix is used. Daily values of

the elements of the combined matrix NPB are listed in The Astronomical Almanac.

The IERS Conventions (2003) list series expansions that directly provide X and Y , the two

most rapidly changing components of the pole position unit vector. Daily values of X and Y are

also listed in The Astronomical Almanac. The values of X and Y are given in arcseconds and

are converted to dimensionless unit vector components simply by dividing them by the number of

arcseconds in one radian, 206264.806247. . . . Also, then, Z =

√

1 − X2 − Y 2. The values of X

and Y are used in the new transformation scheme discussed in section 5.4.3 and in several places

in Chapter 6.

5.4.1 Formulas for Precession

To construct the precession matrix for the transformation of coordinates from one date to another,

we must evaluate short polynomials for the angles involved. The expressions for these angles in

the IAU 2000A model, given below, have only a single time argument, since precession from or

to J2000.0 (actually, the TDB equivalent of J2000.0) is assumed. As used in this circular (and

The Astronomical Almanac), the matrix P(t) always denotes precession from J2000.0 (TDB) to
PRECESSION & NUTATION 43

another date, t. To precess in the opposite direction, the angles are the same but the transpose

of the precession matrix, PT(t), is used. To precess coordinates from one arbitrary date, t1, to

another, t2, it is necessary to precess them from t1 to J2000.0 (using PT(t1)), then from J2000.0

to t2 (using P(t2)). Where high accuracy is not required, and t1 and t2 are not more than a few

years apart, a simpler procedure for precession from t1 to t2 is available and is given at the end of

this subsection.

All expressions given in this subsection are from Section 7 (P03 solution) of Capitaine et al.

(2003) and all coefficients are expressed in arcseconds. This is the theory of precession recommended

by the IAU Working Group on Precession and the Ecliptic (Hilton et al. 2006).

For a given TDB date and time t, let T be the number of Julian centuries of TDB since

2000 Jan 1, 12h TDB. If the dates and times are expressed as Julian dates, then T = (t −

2451545.0)/36525. TT dates and times can be used equally well — the resulting error in precession is only a few ×10−9 arcseconds.

Then the mean obliquity of the ecliptic at J2000.0 (or the equivalent TDB date) is 0 =

84381.406 arcseconds and let

ψA = 5038.481507 T − 1.0790069 T

2 − 0.00114045 T3 + 0.000132851 T4 − 0.0000000951 T5

ωA = 0 − 0.025754 T + 0.0512623 T

2 − 0.00772503 T3 − 0.000000467 T4 + 0.0000003337 T5

χA = 10.556403 T − 2.3814292 T

2 − 0.00121197 T3 + 0.000170663 T4 − 0.0000000560 T5

(5.7)

Equivalently, in notation appropriate for computer programs,

ψA = (((( − 0.0000000951 T

+ 0.000132851 ) T

− 0.00114045 ) T

− 1.0790069 ) T

+ 5038.481507 ) T

ωA = (((( + 0.0000003337 T

− 0.000000467 ) T

− 0.00772503 ) T (5.8)

+ 0.0512623 ) T

− 0.025754 ) T + 0

χA = (((( − 0.0000000560 T

+ 0.000170663 ) T

− 0.00121197 ) T

− 2.3814292 ) T

+ 10.556403 ) T

The precession matrix is then simply P(t) = R3(χA) R1(−ωA) R3(−ψA) R1(0), where R1 and

R3 are standard rotations about the x and z axes, respectively (see “Abbreviations and Symbols

Frequently Used” for precise definitions). This 4-angle precession formulation is comprised of

1. A rotation from the mean equator and equinox of J2000.0 to the mean ecliptic and equinox of

J2000.0. This is simply a rotation around the x-axis (the direction toward the mean equinox of

J2000.0) by the angle 0, the mean obliquity of J2000.0. After the rotation, the fundamental

plane is the ecliptic of J2000.0.
44 PRECESSION & NUTATION

2. A rotation around the new z-axis (the direction toward the ecliptic pole of J2000.0) by the

angle −ψA, the amount of precession of the equator from J2000.0 to t.

3. A rotation around the new x-axis (the direction along the intersection of the mean equator

of t with the ecliptic of J2000.0) by the angle −ωA, the obliquity of the mean equator of t

with respect to the ecliptic of J2000.0. After the rotation, the fundamental plane is the mean

equator of t.

4. A rotation around the new z-axis (the direction toward the mean celestial pole of t) by the

angle χA, accounting for the precession of the ecliptic along the mean equator of t. After the

rotation, the new x-axis is in the direction of the mean equinox of date.

If we let

S1 = sin (0)

S2 = sin (−ψA)

S3 = sin (−ωA)

S4 = sin (χA)

C1 = cos (0)

C2 = cos (−ψA) (5.9)

C3 = cos (−ωA)

C4 = cos (χA)

then the precession matrix can also be written:

P(t) =





C4C2 − S2S4C3 C4S2C1 + S4C3C2C1 − S1S4S3 C4S2S1 + S4C3C2S1 + C1S4S3

−S4C2 − S2C4C3 −S4S2C1 + C4C3C2C1 − S1C4S3 −S4S2S1 + C4C3C2S1 + C1C4S3

S2S3 −S3C2C1 − S1C3 −S3C2S1 + C3C1





(5.10)

Existing applications that use the 3-angle precession formulation of Newcomb and Lieske can be

easily modified for the IAU 2000A precession, by replacing the current polynomials for the angles

ζA, zA, and θA with the following:

ζA = 2.650545 + 2306.083227 T + 0.2988499 T

2 + 0.01801828 T3 − 0.000005971 T4

− 0.0000003173 T

5

zA = −2.650545 + 2306.077181 T + 1.0927348 T

2 + 0.01826837 T3 − 0.000028596 T4

− 0.0000002904 T

5

(5.11)

θA = 2004.191903 T − 0.4294934 T

2 − 0.04182264 T3 − 0.000007089 T4

− 0.0000001274 T

5

The 3-angle precession matrix is P(t) = R3(−zA) R2(θA) R3(−ζA), but any existing correct construction of P using these three angles can still be used.

The expression for the mean obliquity of the ecliptic (the angle between the mean equator and

ecliptic, or, equivalently, between the ecliptic pole and mean celestial pole of date) is:

 = 0 − 46.836769 T − 0.0001831 T

2 + 0.00200340 T3 − 0.000000576 T4 − 0.0000000434 T5

(5.12)

where, as stated above, 0 = 84381.406 arcseconds. This expression arises from the precession

formulation but is actually used only for nutation. (Almost all of the obliquity rate — the term

linear in T — is due to the precession of the ecliptic.)
PRECESSION & NUTATION 45

Where high accuracy is not required, the precession between two dates, t1 and t2, not too far

apart (i.e., where |t2 − t1|  1 century), can be approximated using the rates of change of right

ascension and declination with respect to the mean equator and equinox of date. These rates are

respectively

m ≈ 4612.16 + 2.78 T

n ≈ 2004.19 − 0.86 T (5.13)

where the values are in arcseconds per century and T is the number of centuries between J2000.0

and the midpoint of t1 and t2. If the dates are expressed as Julian dates, T = ((t1 + t2)/2 −

2451545.0)/36525. Then, denoting the celestial coordinates at t1 by (α1, δ1) and those at t2 by

(α2, δ2),

α2 ≈ α1 + τ (m + n sin α1 tan δ1)

δ2 ≈ δ1 + τ (n cos α1) (5.14)

where τ = t2 − t1, expressed in centuries. These formulas deteriorate in accuracy at high (or low)

declinations and should not be used at all for coordinates close to the celestial poles (how close

depends on the accuracy requirement and the value of τ ) .

5.4.2 Formulas for Nutation

Nutation is conventionally expressed as two small angles, ∆ψ, the nutation in longitude, and ∆, the

nutation in obliquity. These angles are measured in the ecliptic system of date, which is developed

as part of the precession formulation. The angle ∆ψ is the small change in the position of the

equinox along the ecliptic due to nutation, so the effect of nutation on the ecliptic coordinates of

a fixed point in the sky is simply to add ∆ψ to its ecliptic longitude. The angle ∆ is the small

change in the obliquity of the ecliptic due to nutation. The true obliquity of date is 

0 =  + ∆.

Nutation in obliquity reflects the orientation of the equator in space and does not affect the ecliptic

coordinates of a fixed point on the sky.

The angles ∆ψ and ∆ can also be thought of as small shifts in the position of the celestial

pole (CIP) with respect to the ecliptic and mean equinox of date. In that coordinate system,

and assuming positive values for ∆ψ and ∆, the nutation in longitude shifts the celestial pole

westward on the sky by the angle ∆ψ sin , decreasing the pole’s mean ecliptic longitude by ∆ψ.

Nutation in obliquity moves the celestial pole further from the ecliptic pole, i.e., southward in

ecliptic coordinates, by ∆. (Negative values of ∆ψ and ∆ move the pole eastward and northward

in ecliptic coordinates.)

The effect of nutation on the equatorial coordinates (α,δ) of a fixed point in the sky is more

complex and is best dealt with through the action of the nutation matrix, N(t), on the equatorial

position vector, rmean(t). Where high accuracy is not required, formulas that directly give the

changes to α and δ as a function of ∆ψ and ∆ are available and are given at the end of this

subsection.

The values of ∆ψ and ∆ are obtained by evaluating rather lengthy trigonometric series, of the

general form

∆ψ =

X

N

i=1



(Si + S ̇

iT) sin Φi + C

0

i

cos Φi


46 PRECESSION & NUTATION

∆ =

X

N

i=1



(Ci + C ̇

iT) cos Φi + S

0

i

sin Φi



(5.15)

where, in each term, Φi =

X

K

j=1

Mi,j φj (T) (5.16)

For the IAU 2000A model, N=1365 and K=14. The 14 φj (T) are the fundamental arguments,

which are, except for one, orbital angles. The main time dependence of the nutation series enters

through these arguments. The expressions given below are all taken from Simon et al. (1994) and

all coefficients are in arcseconds.

The first eight fundamental arguments are the mean heliocentric ecliptic longitudes of the

planets Mercury through Neptune:

φ1 = 908103.259872 + 538101628.688982 T

φ2 = 655127.283060 + 210664136.433548 T

φ3 = 361679.244588 + 129597742.283429 T

φ4 = 1279558.798488 + 68905077.493988 T (5.17)

φ5 = 123665.467464 + 10925660.377991 T

φ6 = 180278.799480 + 4399609.855732 T

φ7 = 1130598.018396 + 1542481.193933 T

φ8 = 1095655.195728 + 786550.320744 T

In all of these expressions, T is the number of Julian centuries of TDB since 2000 Jan 1, 12h TDB (or,

with negligible error, the number of Julian centuries of TT since J2000.0). In some implementations

it may be necessary to reduce the resulting angles, which are expressed in arcseconds, to radians

in the range 0–2π. The ninth argument is an approximation to the general precession in longitude:

φ9 = 5028.8200 T + 1.112022 T

2

(5.18)

The last five arguments are the same fundamental luni-solar arguments used in previous nutation

theories, but with updated expresssions. They are, respectively, l, the mean anomaly of the Moon;

l

0

, the mean anomaly of the Sun; F, the mean argument of latitude of the Moon; D, the mean

elongation of the Moon from the Sun, and Ω, the mean longitude of the Moon’s mean ascending

node:

φ10 = l = 485868.249036 + 1717915923.2178 T + 31.8792 T

2 + 0.051635 T3 − 0.00024470 T4

φ11 = l

0 = 1287104.79305 + 129596581.0481 T − 0.5532 T2 + 0.000136 T3 − 0.00001149 T4

φ12 = F = 335779.526232 + 1739527262.8478 T − 12.7512 T

2 − 0.001037 T3 + 0.00000417 T4

φ13 = D = 1072260.70369 + 1602961601.2090 T − 6.3706 T

2 + 0.006593 T3 − 0.00003169 T4

φ14 = Ω = 450160.398036 − 6962890.5431 T + 7.4722 T

2 + 0.007702 T3 − 0.00005939 T4

(5.19)

The first step in evaluating the series for nutation for a given date is to compute the values of

all 14 fundamental arguments for the date of interest. This is done only once. Then the nutation

terms are evaluated one by one. For each term i, first compute Φi according to eq. 5.16, using the

14 integer multipliers, Mi,j , listed for the term; i.e., sum over Mi,j × φj (where j=1–14). Then the

cosine and sine components for the term can be evaluated, as per eq. 5.15, using the listed values
PRECESSION & NUTATION 47

of the coefficients Si, S ̇

i

, C

0

i

, Ci, C ̇

i

, and S

0

i

for the term. Generally it is good practice to sum the

terms from smallest to largest to preserve precision in the sums.

The entire IAU 2000A nutation series is listed at the end of this circular. About the first half

of the series consists of lunisolar terms, which depend only on l, l

0

, F, D, and Ω (= φ10 to φ14).

In all of these terms, the first nine multipliers are all zero. The generally smaller planetary terms

comprise the remainder of the series. As an example of how the individual terms are computed

according to eqs. 5.15 and 5.16, term 6 would be evaluated

∆ψ6 = (−0.0516821 + 0.0001226 T) sin Φ6 − 0.0000524 cos Φ6

∆6 = ( 0.0224386 − 0.0000667 T) cos Φ6 − 0.0000174 sin Φ6

where Φ6 = φ11 + 2φ12 − 2φ13 + 2φ14

since M6,1 through M6,10 are zero, and only φ11 through φ14 are therefore relevant for this term. It

is assumed that all the φj have been pre-computed (for all terms) using the appropriate value of T

for the date and time of interest. A printed version of a 1365-term nutation series is obviously not

the most convenient form for computation; it is given here only for the record, since the full series

has not previously appeared in print. As noted earlier, the series is available as a pair of plain-text

computer files at URL 14, and the SOFA and NOVAS software packages (URL 7, URL 8) include

subroutines for evaluating it. There are also shorter series available where the highest precision is

not required. The IERS web site provides, in addition to the full IAU 2000A series, an IAU 2000B

series, which has only 77 terms and duplicates the IAU 2000A results to within a milliarcsecond for

input times between 1995 and 2050. NOVAS also provides a subroutine that evaluates a truncated

series, with 488 terms, that duplicates the full series to 0.1 milliarcsecond accuracy between 1700

and 2300.

Once the nutation series has been evaluated and the values of ∆ψ and ∆ are available, the nutation matrix can be constructed. The nutation matrix is simply N(t) = R1(−

0

) R3(−∆ψ) R1(),

where, again, R1 and R3 are standard rotations about the x and z axes, respectively (see “Abbreviations and Symbols Frequently Used” for precise definitions), and 

0 = +∆ is the true obliquity

(compute  using eq. 5.12). This formulation is comprised of

1. A rotation from the mean equator and equinox of t to the mean ecliptic and equinox of t.

This is simply a rotation around the x-axis (the direction toward the mean equinox of t) by

the angle , the mean obliquity of t. After the rotation, the fundamental plane is the ecliptic

of t.

2. A rotation around the new z-axis (the direction toward the ecliptic pole of t) by the angle

−∆ψ, the amount of nutation in longitude at t. After the rotation, the new x-axis is in the

direction of the true equinox of t.

3. A rotation around the new x-axis (the direction toward the true equinox of t) by the angle

−

0

, the true obliquity of t. After the rotation, the fundamental plane is the true equator of

t, orthogonal to the computed position of the CIP at t.

If we let
48 PRECESSION & NUTATION

S1 = sin ()

S2 = sin (−∆ψ)

S3 = sin (− − ∆)

C1 = cos ()

C2 = cos (−∆ψ) (5.20)

C3 = cos (− − ∆)

then the nutation matrix can also be written:

N(t) =





C2 S2C1 S2S1

−S2C3 C3C2C1 − S1S3 C3C2S1 + C1S3

S2S3 −S3C2C1 − S1C3 −S3C2S1 + C3C1



 (5.21)

Where high accuracy is not required, coordinates corrected for nutation in right ascension and

declination can be obtained from

αt ≈ αm + ∆ψ (cos 

0 + sin 0

sin αm tan δm) − ∆ cos αm tan δm

δt ≈ δm + ∆ψ sin 

0

cos αm + ∆ sin αm (5.22)

where (αm, δm) are coordinates with respect to the mean equator and equinox of date (precession

only), (αt, δt) are the corresponding coordinates with respect to the true equator and equinox of

date (precession + nutation), and 

0

is the true obliquity. Note the tan δm factor in right ascension

that makes these formulas unsuitable for use close to the celestial poles.

The traditional formula for the equation of the equinoxes (the difference between apparent and

mean sidereal time) is ∆ψ cos 

0

, but in recent years this has been superceded by the more accurate

version given in eq. 2.14.

5.4.3 Alternative Combined Transformation

The following matrix, C(t), combines precession, nutation, and frame bias and is used to transform

vectors from the GCRS to the Celestial Intermediate Reference System (CIRS). The CIRS is defined

by the equator of the CIP and an origin of right ascension called the Celestial Intermediate Origin

(CIO). The CIO is discussed extensively in Chapter 6. There, the CIRS is symbolized Eσ; it is

analogous to the true equator and equinox of date, but with a different right ascension origin.

The matrix C(t) is used in the sense

rCIRS = C(t) rGCRS (5.23)

and the components of C(t), as given in the IERS Conventions (2003) and The Astronomical

Almanac, are

C(t) = R3(−s)





1 − bX2 −bXY −X

−bXY 1 − bY 2 −Y

X Y 1 − b(X2 + Y

2

)



 (5.24)

where X and Y are the dimensionless coordinates of the CIP in the GCRS (unit vector components),

b = 1/(1 + Z), Z =

√

1 − X2 − Y 2, and s is the CIO locator, a small angle described in Chapter 6.

All of these quantities are functions of time. R3 is a standard rotation around the z axis; see

“Abbreviations and Symbols Frequently Used” for a precise definition.
PRECESSION & NUTATION 49

5.4.4 Observational Corrections to Precession-Nutation

The IERS still publishes daily values of the observed celestial pole offsets, despite the vast improvement to the pole position predictions given by the IAU 2000A precession-nutation model. The

offsets now have magnitudes generally less than 1 mas. The fact that they are non-zero is due

in part to an effect of unpredictable amplitude and phase called the free core nutation (or nearly

diurnal free wobble), caused by the rotation of the fluid core of the Earth inside the ellipsoidal

cavity that it occupies. The free core nutation appears as a very small nutation component with a

period of about 430 days. Any other effects not accounted for in the adopted precession-nutation

model will also appear in the celestial pole offsets. In any event, the celestial pole offsets are now

so small that many users may now decide to ignore them. However, it is worth noting again that,

by definition, the Celestial Intermediate Pole (CIP) includes these observed offsets.

The IERS now publishes celestial pole offsets with respect to the IAU 2000A precession-nutation

model only as dX and dY — corrections to the pole’s computed unit vector components X and

Y in the GCRS (see eq. 5.6 and following notes). The IERS pole offsets are published in units

of milliarcseconds but they can be converted to dimensionless quantities by dividing them by the

number of milliarcseconds in one radian, 206264806.247... . Then, the observationally corrected

values of X and Y are

Xcor = X + dX and Ycor = Y + dY (5.25)

The corrected values, expressed as dimensionless quantities (unit vector components), are used,

e.g., in the matrix C given in sections 5.4.3 and 6.5.3. That is, in eqs. 5.24 and 6.18, assume

X = Xcor and Y = Ycor.

The ecliptic-based pole celestial offsets, dψ and d, which are used to correct the nutation

theory’s output angles ∆ψ and ∆, are no longer supplied (actually, they are supplied but only for

the old pre-2000 precession-nutation model). Software that has not been coded to use X and Y

directly — which includes all software developed prior to 2003 — will need a front-end to convert

the IERS dX and dY values to dψ and d. A derivation of a conversion algorithm and several

options for its implementation (depending on the accuracy desired) are given by Kaplan (2003).

Succinctly, given dimensionless dX and dY values for a given date t, let





dX0

dY 0

dZ0



 = P(t)





dX

dY

dZ



 (5.26)

where P(t) is the precession matrix from J2000.0 to date t, and we can set dZ= 0 in this approximation, which holds for only a few centuries around J2000.0. Then we compute the ecliptic-based

correction angles in radians using

dψ = dX0/ sin  and d = dY 0

(5.27)

where  is the mean obliquity of the ecliptic of date t, computed according to eq 5.12. The observationally corrected values of ∆ψ and ∆ are obtained simply by adding dψ and d, respectively:

∆ψcor = ∆ψ + dψ and ∆cor = ∆ + d (5.28)

where care must be taken to ensure that all angles are expressed in the same units. The corrected

values are used in forming the nutation matrix N(t) and in other nutation-related expressions. That

is, in eqs. 5.20 and 5.22, assume ∆ψ = ∆ψcor and ∆ = ∆cor. At the same time, the corrected

value of ∆ψ should be used in forming the equation of the equinoxes using eq. 2.14.
Chapter 6

Modeling the Earth’s Rotation

Relevant IAU resolutions: B1.6, B1.7, B1.8 of 2000

Summary Res. 1.8 of 2000 establishes two new reference points in the plane of the

moving (instantaneous) equator for the measurement of Earth rotation: the point on

the geocentric celestial sphere is called the Celestial Intermediate Origin (CIO) and the

point on the surface of the Earth is called the Terrestrial Intermediate Origin (TIO).

The CIO and TIO are specific examples of a concept called a non-rotating origin that

was first described by Guinot (1979, 1981).

The Earth Rotation Angle, θ, is the geocentric angle between the directions of

the CIO and TIO, and provides a new way to represent the rotation of the Earth in

the transformation from terrestrial to celestial systems or vice versa. Traditionally,

Greenwich sidereal time, which is the hour angle of the equinox with respect to the

Greenwich meridian, has served this purpose. The CIO and TIO are defined in such a

way that θ is a linear function of Universal Time (UT1) and independent of the Earth’s

precession and nutation; it is a direct measure of the rotational angle of the Earth around

the Celestial Intermediate Pole (see Chapter 5). Since none of these statements holds

for sidereal time, the scheme based on the CIO, TIO, and θ represents a simplification of

the way the rotation of the Earth is treated. In particular, the transformation between

Earth-fixed and space-fixed reference systems can now be specified by three rotation

matrices that are independent of each other: one for polar motion, one for “pure”

rotation (i.e., θ), and one for precession-nutation.

The recent IAU resolutions do not eliminate sidereal time or the use of the equinox

as a fundamental reference point. Instead, the resolutions establish an alternative way

of dealing with Earth rotation. The comparison between the two schemes can be illuminating. For example, the CIO helps to clarify the relationship between sidereal

time and the Earth’s rotation, since θ is now the “fast term” in the formula for sidereal

time as a function of UT1. The remaining terms comprise the equation of the origins

and represent the accumulated amount of precession and nutation along the equator

as a function of time. The equation of the origins is the length of the arc between the

equinox and the CIO.

50
MODELING THE EARTH’S ROTATION 51

6.1 A Messy Business

In the computation of the positions of celestial objects with respect to an Earth-fixed system — or,

equivalently, in the transformation between terrestrial and celestial coordinate systems — sidereal

time has conventionally represented the Earth’s rotation about its axis. For example, the hour

angle of an object is simply the local apparent sidereal time minus the object’s apparent right

ascension with respect to the true equator and equinox of date (see section 2.6.2). Once its hour

angle and declination are available, the object’s zenith distance and azimuth, or its coordinates

with respect to some ground-based instrumental system, can be easily obtained. The same result

can be accomplished by a direct transformation between the celestial and terrestrial coordinate systems, conventionally represented by a series of rotation matrices, one each for precession, nutation,

sidereal time, and polar motion.

Yet there is something untidy about these procedures. The computation of apparent sidereal

time mixes quantities related to Earth rotation, precession, and nutation (see eqs. 2.10–2.14).

Because sidereal time is defined as the hour angle of the equinox, the precession of the equinox

in right ascension must be a part of the expression for sidereal time (the terms in parentheses in

eq. 2.12), and the mean sidereal day is thereby shorter than the rotation period of the Earth by

about 0.s008. Nutation also appears, in the equation of the equinoxes (eqs. 2.13 & 2.14). The

result is that in the computation of hour angle, precession and nutation enter twice: once in the

sidereal time formula and again in the computation of the star’s apparent right ascension; the two

contributions cancel for stars on the equator. Similarly, in the transformation between the celestial

and terrestrial coordinate systems, precession and nutation each enter into two of the rotation

matrices, and none of the matrices represents Earth rotation alone.

A consequence of this way of doing things is that whenever improvements are made to the

theory of precession, the numerical coefficients in the expression for sidereal time must also change.

This was not an issue for most of the twentieth century, since no adjustments were made to the

standard precession algorithm, and the expression for mean sidereal time derived from Newcomb’s

developments was used without much thought given to the matter. It was the change to this

expression, necessitated by the adjustment of the precession constant in the IAU (1976) System

of Astronomical Constants, that first motivated the search for a fundamental change of procedure.

At about the same time, new high-precision observing techniques, such as VLBI and lunar laser

ranging, were being applied to the study of all components of the Earth’s rotation, and a review

of the basic algorithms seemed appropriate. In particular, there was interest in constructing a

new geometrical picture and set of expressions for the orientation of the Earth as a function of

time that would cleanly separate the effects of rotation, precession and nutation, and polar motion.

Furthermore, since VLBI is not sensitive to the equinox, a procedure that used a different reference

point seemed desirable.

To bring the Earth’s rotation period explicitly into the terrestrial–celestial transformation, we

must define an angle of rotation about the Earth’s axis. As described in Chapter 5, what we

specifically mean by “the Earth’s axis” is the line through the geocenter in the direction of the

Celestial Intermediate Pole (CIP). The angle of rotation about this axis must be measured with

respect to some agreed-upon direction in space. Since the CIP moves small amounts during one

rotation of the Earth (∼0.1 arcsecond with respect to the stars and ∼0.005 arcsecond with respect

to the Earth’s crust), the reference direction cannot be simply a fixed vector or plane in inertial
52 MODELING THE EARTH’S ROTATION

space. What we need is an appropriate azimuthal1 origin — a point in the moving equatorial plane,

which is orthogonal to the CIP.

6.2 Non-Rotating Origins

The reference point that we define must be such that the rate of change of the Earth’s rotation

angle, measured with respect to this point, is the angular velocity of the Earth about the CIP. As

the CIP moves, the point must move to remain in the equatorial plane; but the point’s motion must

be such that the measured rotation angle is not contaminated by some component of the motion

of the CIP itself.

The concept of a “non-rotating origin” (NRO) on the equator can be applied to any rotating

body. The NRO idea was first described by Bernard Guinot (Guinot 1979, 1981) and further

developed by Nicole Capitaine and collaborators (Capitaine et al. 1986; Capitaine 1990; Capitaine

& Chollet 1991; Capitaine et al. 2000; Capitaine 2000). The condition on the motion of such a

point is simple: as the equator moves, the point’s instantaneous motion must always be orthogonal

to the equator. That is, the point’s motion at some time t must be directly toward or away from the

position of the pole of rotation at t. Any other motion of the point would have a component around

the axis/pole and would thus introduce a spurious rate into the measurement of the rotation angle

of the body as a function of time. The point is not unique; any arbitrary point on the moving

equator could be made to move in the prescribed manner. For the Earth, the difference between

the motion of a non-rotating origin and that of the equinox on the geocentric celestial sphere is

illustrated in Fig. 6.1.

As illustrated in the figure, the motion of the non-rotating origin, σ, is always orthogonal to the

equator, whereas the equinox has a motion along the equator (the precession in right ascension).

How do we specify the location of a non-rotating origin? There are three possibilities, outlined

in the Formulas section of this chapter. In the most straightforward scheme, one simply uses the

GCRS right ascension of σ obtained from a numerical integration (the GCRS is the “geocentric

ICRS”). Alternatively, the position of σ can be defined by a quantity, s, that is the difference

between the lengths of two arcs on the celestial sphere. Finally, one can specify the location of σ

with respect to the equinox, Υ: the equatorial arc Υσ is called the equation of the origins. Whatever geometry is used, the position of σ ultimately depends on an integral over time, because the

defining property of σ is its motion — not a static geometrical relationship with other points or

planes. The integral involved is fairly simple and depends only on the coordinates of the pole and

their derivatives with respect to time. The initial point for the integration can be any point on the

moving equator at any time t0.

1The word “azimuthal” is used in its general sense, referring to an angle measured about the z-axis of a coordinate

system.
MODELING THE EARTH’S ROTATION 53

equator of t3

ecliptic

σ1 Υ1

equator of t2

equator of t1

Υ2

Υ3

σ2

σ3

RA

Dec

Figure 6.1 Motion of a non-rotating origin, σ, compared with that of the true

equinox, Υ. “Snapshots” of the positions of the points are shown at three successive

times, t1, t2, and t3. The positions are shown with respect to a geocentric reference

system that has no systematic rotation with respect to a set of extragalactic objects.

The ecliptic is shown in the figure as fixed, although it, too, has a small motion in

inertial space.

So far we have discussed a non-rotating origin only on the celestial sphere, required because of

the movement of the CIP in a space-fixed reference system. But there is a corresponding situation

on the surface of the Earth. The CIP has motions in both the celestial and terrestrial reference

systems. Its motion in the celestial system is precession-nutation and its motion in the terrestrial

system is polar motion, or wobble. From the point of view of a conventional geodetic coordinate

system “attached” to the surface of the Earth (i.e., defined by the specified coordinates of a group

of stations), the CIP wanders around near the geodetic pole in a quasi-circular motion with an

amplitude of about 10 meters (0.3 arcsec) and two primary periods, 12 and 14 months. Thus the

equator of the CIP has a slight quasi-annual wobble around the geodetic equator. Actually, it

is better thought of in the opposite sense: the geodetic equator has a slight wobble with respect

to the equator of the CIP. That point of view makes it is a little clearer why a simple “stake in

the ground” at the geodetic equator would not be suitable for measuring the Earth rotation angle

around the CIP. The situation is orders of magnitude less troublesome than that on the celestial

sphere, but for completeness (and very precise applications) it is appropriate to define a terrestrial

non-rotating origin, designated $. It stays on the CIP equator, and assuming that the current

amplitude of polar motion remains approximately constant, $ will bob north and south by about

10 m in geodetic latitude every year or so and will have a secular eastward motion in longitude
54 MODELING THE EARTH’S ROTATION

of about 1.5 mm/cen. The exact motion of $ depends, of course, on what polar motion, which is

unpredictable, actually turns out to be.

The two non-rotating origins, σ and $, are called the Celestial Intermediate Origin (CIO) and

the Terrestrial Intermediate Origin (TIO). Both lie in the same plane — the equator of the CIP.

The Earth Rotation Angle, θ, is defined as the geocentric angle between these two points. The

angle θ is a linear function of Universal Time (UT1). The formula, given in the note to res. B1.8 of

2000, is simply θ = 2π (0.7790572732640 + 1.00273781191135448 DU ), where DU is the number of

UT1 days from JD 2451545.0 UT1. The formula assumes a constant angular velocity of the Earth:

no attempt is made to model its secular decrease due to tidal friction, monthly tidal variations,

changes due to the exchange of angular momentum between the atmosphere and the solid Earth,

and other phenomena. These effects will be reflected in the time series of UT1–UTC or ∆T values

(see Chapter 2) derived from precise observations.

The expression given above for θ is now the “fast term” in the formula for mean sidereal time;

see eq. 2.12. It accounts for the rotation of the Earth, while the other terms account for the motion

of the equinox along the equator due to precession.

The plane defined by the geocenter, the CIP, and TIO is called the TIO meridian. For most

ordinary astronomical purposes the TIO meridian can be considered to be identical to what is often

referred to as the Greenwich meridian. The movement of this meridian with respect to a conventional geodetic system is important only for the most precise astrometric/geodetic applications.

It is worth noting that the TIO meridian, and the zero-longitude meridians of modern geodetic

systems, are about 100 m from the old transit circle at Greenwich (Gebel & Matthews 1971).

The term “Greenwich meridian” has ceased to have a technical meaning in the context of precise

geodesy — despite the nice line in the sidewalk at the old Greenwich observatory. This has become

obvious to tourists carrying GPS receivers!

6.3 The Path of the CIO on the Sky

If we take the epoch J2000.0 as the starting epoch for evaluating the integral that provides the

position of the CIO, the only mathematical requirement for the initial point is that it lie on the

instantaneous (true) equator of that date — its position along the equator is arbitrary. By convention, however, the initial position of the CIO on the instantaneous equator of J2000.0 is set so that

equinox-based and CIO-based computations of Earth rotation yield the same answers; we want the

hour angle of a given celestial object to be the same, as a function of UT1 (or UTC), no matter

how the calculation is done. For this to happen, the position of the CIO of J2000.0 must be at

GCRS right ascension 0◦ 0

0 00.00002012. This is about 12.8 arcseconds west of the true equinox of

that date.

Since the CIO rides on the instantaneous equator, its primary motion over the next few millenia

is southward at the rate of precession in declination, initially 2004 arcseconds per century. Its rate

of southward motion is modulated (but never reversed) by the nutation periodicities. Its motion

in GCRS right ascension is orders of magnitude less rapid; remember that the CIO has no motion

along the instantaneous equator, and the instantaneous equator of J2000.0 is nearly co-planar with

the GCRS equator (xy-plane). The motion of the CIO in GCRS right ascension over the next few

millenia is dominated by a term proportional to t

3

; the GCRS right ascension of the CIO at the

beginning of year 2100 is only 0.

00068; at the beginning of 2200 it is 0.00573; and at the beginning of

2300 it is 1.

00941. Nutation does produce a very slight wobble in the CIO’s right ascension, but the

influence of the nutation terms is suppressed by several orders of magnitude relative to their effect
MODELING THE EARTH’S ROTATION 55

on the position of the pole. We can say, therefore, that to within a few arcseconds error, the path

of the CIO on the celestial sphere over the next few centuries is nearly a straight line southward

along the GCRS α=0 hour circle.

ecliptic

Start at J2000.0

Figure 6.2 Locus of the CIO (solid line) and equinox (dashed line) on the celestial

sphere over 5×104 years, with respect to space-fixed coordinates. During this time

the equinox wraps around the figure twice and ends up approximately at the starting

point.

The solid line on the left side of Figure 6.2 indicates the locus of the CIO in the GCRS over

50,000 years — about two precession cycles. The ecliptic is shown as a dashed line. The initial

nearly straight southward motion from the starting point at J2000.0 is clearly shown. There are

occasional “cusps” in the CIO’s motion, where its secular motion comes to a temporary halt before

reversing. The first of these stationary points occurs in just over a quarter of a precession cycle,

as the section of the moving equator that is farthest south in ecliptic coordinates precesses to near

the GCRS α=0 hour circle. At that time, the CIO will exhibit only nutational oscillations around

a point that remains fixed on the celestial sphere to within 10 mas for almost a decade. Then its

motion resumes, this time northward and westward.2 The motion of the equinox over the same

50,000-year time period begins at nearly the same point as the CIO (on the plot scale used, the

points overlap), but smoothly follows the ecliptic westward (to the right on the plot), wrapping

around twice and ending up essentially at the starting point.

6.4 Transforming Vectors Between Reference Systems

The reference points described above allow us to define three geocentric reference systems that

share, as a common reference plane, the instantaneous, or true, equator of date. The instantaneous

equator is now defined as the plane through the geocenter orthogonal to the direction of the CIP

at a given time, t. The three reference systems are:

2There is nothing profound about the stationary points or the dates on which they occur. If we had started the

CIO at GCRS right ascension 6hor 18hat J2000.0, it would have started at a stationary point.
56 MODELING THE EARTH’S ROTATION

1. True equator and equinox of t — azimuthal origin at the true equinox (Υ) of t

2. Celestial Intermediate Reference System (CIRS) — azimuthal origin at the Celestial Intermediate Origin (CIO or σ) of t

3. Terrestrial Intermediate Reference System (TIRS) — azimuthal origin at the Terrestrial Intermediate Origin (TIO or $) of t

In this circular we will often refer to these reference systems by the symbols EΥ, Eσ, and E$,

respectively; E denotes an equatorial system, and the subscript indicates the azimuthal origin. E$

rotates with the Earth whereas the orientations of EΥand Eσ change slowly with respect to local

inertial space. The transformation between EΥand E$ is just a rotation about the z-axis (which

points toward the CIP) by GAST, the angular equivalent of Greenwich apparent sidereal time.

The transformation between Eσ and E$ is a similar rotation, but by θ, the Earth Rotation Angle.

These two transformations reflect different ways — old and new — of representing the rotation of

the Earth.

A short digression into terrestrial, i.e., geodetic, reference systems is in order here. These

systems all have their origin at the geocenter and rotate with the crust of the Earth. Terrestrial

latitude, longitude, and height are now most commonly given with respect to a reference ellipsoid,

an oblate spheroid that approximates the Earth’s overall shape (actually, that best fits the geoid,

a gravitational equipotential surface). The current standard reference elliposid for most purposes

is that of the World Geodetic System 1984 (WGS 84), which forms the basis for the coordinates

obtained from GPS. The WGS 84 ellipsoid has an equatorial radius of 6,378,137 meters and a

polar flattening of 1/298.257223563. For the precise measurement of Earth rotation, however, the

International Terrestrial Reference System (ITRS) is used, which was defined by the International

Union of Geodesy and Geophysics (IUGG) in 1991. The ITRS is realized for practical purposes by

the adopted coordinates and velocities3 of a large group of observing stations. These coordinates are

expressed as geocentric rectangular 3-vectors and thus are not dependent on a reference ellipsoid.

The list of stations and their coordinates is referred to as the International Terrestrial Reference

Frame (ITRF). The fundamental terrestrial coordinate system is therefore defined in exactly the

same way as the fundamental celestial coordinate system (see Chapter 3): a prescription is given

for an idealized coordinate system (the ITRS or the ICRS), which is realized in practice by the

adopted coordinates of a group of reference points (the ITRF stations or the ICRF quasars). The

coordinates may be refined as time goes on but the overall system is preserved. It is important to

know, however, that the ITRS/ITRF is consistent with WGS 84 to within a few centimeters; thus

for all astronomical purposes the GPS-obtained coordinates of instruments can be used with the

algorithms presented here.

Our goal is to be able to transform an arbitrary vector (representing for example, an instrumental position, axis, boresight, or baseline) from the ITRS (≈WGS 84≈GPS) to the GCRS. The three

equatorial reference systems described above — EΥ, Eσ, and E$ — are waypoints, or intermediate

stops, in that process. The complete transformations are:

3The velocities are quite small and are due to plate tectonics and post-glacial rebound.
MODELING THE EARTH’S ROTATION 57

Equinox-Based Transformation

ITRS or WGS 84

k

polar motion

⇓

E$ — Terrestrial Intermediate Ref. System

k

Greenwich apparent sidereal time

⇓

EΥ — true equator & equinox

k

equinox-based rotation for

nutation + precession + frame bias

⇓

GCRS

CIO-Based Transformation

ITRS or WGS 84

k

polar motion

⇓

E$ — Terrestrial Intermediate Ref. System

k

Earth Rotation Angle

⇓

Eσ — Celestial Intermediate Ref. System

k

CIO-based rotation for

nutation + precession + frame bias

⇓

GCRS

which are equivalent. That is, given the same input vector, the same output vector will result

from the two procedures. In the CIO-based transformation, the three sub-transformations (for

polar motion, Earth Rotation Angle, and nutation/precession/frame bias) are independent. That

is not true for the equinox-based method, because apparent sidereal time incorporates precession

and nutation. Each of the two methods could be made into a single matrix, and the two matrices

must be numerically identical. That means that the use of the CIO in the second method does not

increase the precision of the result but simply allows for a mathematical redescription of the overall

transformation — basically, a re-sorting of the effects to be taken into account. This redescription

of the transformation provides a clean separation of the three main aspects of Earth rotation, and

recognizes that the observations defining modern reference systems are not sensitive to the equinox.

It thus yields a more straightforward conceptual view and facilitates a simpler observational analysis

for Earth-rotation measurements and Earth-based astrometry.

These transformations are all rotations that pivot around a common point, the geocenter. Although developed for observations referred to the geocenter, the same set of rotations can be applied

to observations made from a specific point on the surface of the Earth. (This follows from the assumption that all points in or on the Earth are rigidly attached to all other points. The actual

non-rigidity — e.g., Earth tides — is handled as a separate correction.) In such a case, the computations for parallax, light-bending, aberration, etc., must take into account the non-geocentric

position and velocity of the observer. Then the final computed coordinates are referred, not to the

GCRS, but rather to a proper reference system (in the terminology of relativity) of the observer.

We return to the more familiar problem mentioned at the beginning of the chapter: the computation of local hour angle. In the usual equinox-based scheme, the apparent place4 of the star or

planet is expressed with respect to the true equator and equinox of date (EΥ). The local hour angle

is just h = GAST – αΥ + λ, where GAST is Greenwich apparent sidereal time, αΥis the apparent

4See footnote 1 on page 41.
58 MODELING THE EARTH’S ROTATION

right ascension of the object, measured with respect to the true equinox, and λ is the longitude

of the observer (corrected, where necessary, for polar motion). Obviously these quantities must

all be given in the same units. In the CIO-based scheme, the apparent place would be expressed

in the Celestial Intermediate Reference System (Eσ), and h = θ – ασ + λ, where θ is the Earth

Rotation Angle and ασ is the apparent right ascension of the object, measured with respect to the

CIO. (The recommended terminology is intermediate place for the object position in the Eσ and

intermediate right ascension for the quantity ασ, although some people hold that the term right

ascension should refer only to an origin at the equinox.) In the CIO-based formula, precession and

nutation come into play only once, in expressing the object’s right ascension in the Eσ system. See

section 6.5.4 for more details.

6.5 Formulas

The formulas below draw heavily on the developments presented previously. In particular, the

3×3 matrices P, N, and B represent the transformations for precession, nutation, and frame bias,

respectively, and are taken directly from Chapters 5 and 3. The matrices P and N are functions

of time, t. The time measured in Julian centuries of TDB (or TT) from J2000.0 is denoted T and

is given by T = (JD(TDB) – 2451545.0)/36525. The elementary rotation matrices R1, R2, and

R3 are defined in “Abbreviations and Symbols Frequently Used”. Formulas from Chapter 2 for

sidereal time and the Earth Rotation Angle are used. Explanations of, and formulas for the time

scales UT1, TT, and TDB are also found in Chapter 2.

The ultimate objective is to express “local” Earth-fixed vectors, representing geographic positions, baselines, instrumental axes and boresights, etc., in the GCRS, where they can be related to

the proper coordinates5 of celestial objects. As mentioned above, the “GCRS” is a reference system

that can be thought of as the “geocentric ICRS”. Celestial coordinates in the GCRS are obtained

from basic ICRS data (which are barycentric) by applying the usual algorithms for proper place;

see section 1.3. In this chapter we will be working entirely in a geocentric system and the GCRS

will be obtained from a series of rotations that start with an ordinary Earth-fixed geodetic system.

6.5.1 Location of Cardinal Points

We will start by establishing the positions of three cardinal points within the GCRS: the Celestial

Intermediate Pole (CIP), the true equinox of date (Υ), and the Celestial Intermediate Origin (CIO).

The unit vectors toward these points will be designated nGCRS, ΥGCRS, and σGCRS, respectively. As

the Earth precesses and nutates in local inertial space, these points are in continual motion.

The CIP and the equinox can easily be located in the GCRS at any time t simply by recognizing

that they are, respectively, the z- and x-axes of the true equator and equinox of date system (EΥ)

at t. The unit vectors therefore are:

CIP : nGCRS(t) = B

T PT(t) NT(t)





0

0

1





(6.1)

Equinox: ΥGCRS(t) = B

T PT(t) NT(t)





1

0

0





5See footnote 1 on page 41.
MODELING THE EARTH’S ROTATION 59

where the matrix B accounts for the GCRS frame bias (same as for the ICRS) and the matrices

P(t) and N(t) provide the transformations for precession and nutation, respectively, at time t.

These matrices were developed in sections 3.5 and 5.4; the superscript T’s above indicate that the

transpose of each of these matrices as previously developed is used (i.e., we are using the “reverse”

transformations here, from the true equator and equinox of t to the GCRS). The first equation

above is simply eq. 5.6 rewritten. Note that ΥGCRS is orthogonal to nGCRS at each time t.

The components of the unit vector in the direction of the pole, nGCRS, are denoted X, Y , and

Z, and another approach to determining nGCRS is to use the series expansions for X and Y given

in the IERS Conventions (2003). There is a table of daily values of X and Y in Section B of The

Astronomical Almanac (there labeled X and Y). Once X and Y are converted to dimensionless

values, Z =

√

1 − X2 − Y 2. (The IERS series for X and Y are part of a data analysis approach

adopted by the IERS that avoids any explicit reference to the ecliptic or the equinox, although the

underlying theories are those described in Chapter 5.)

There are three possible procedures for obtaining the location of the Celestial Intermediate

Origin on the celestial sphere at a given time: (1) following the arc on the instantaneous equator

from the equinox to the CIO; (2) directly computing the position vector of the CIO in the GCRS

by numerical integration; or (3) using the quantity s, representing the difference in two arcs on

the celestial sphere, one of which ends at the CIO. These procedures will be described in the three

subsections below. Figure 6.3 indicates the geometric relationships among the points mentioned.

RA ≈ 6h

Σ0

Σ0

RA = 0

σ 

s Eo

N

ecliptic of date

GCRS equator

north

east

of date

RA increasing

'

equator

GCRS

Figure 6.3 Relationship between various points involved in locating the CIO. The

figure approximates the relative positions at 2020.0, although the spacings are not to

scale. The point labeled σ is the CIO; Υ is the true equinox; Σ0 is the GCRS right

ascension origin; N is the ascending node of the instantaneous (true) equator of date on

the GCRS equator; and Σ0

0

is the point on the instantaneous equator that is the same

distance from N as Σ0. As shown, the quantities s and Eo are respectively positive and

negative. The motion of the instantaneous equator (which is orthogonal to the CIP) is

generally southward (down in the figure) near RA=0, which tends to move the equinox

westward (right) and the CIO very slightly eastward (left) with respect to the GCRS.

6.5.1.1 CIO Location Relative to the Equinox

The arc on the instantaneous (true) equator of date t from the CIO to the equinox is called the

equation of the origins and is the right ascension of the true equinox relative to the CIO (or, minus
60 MODELING THE EARTH’S ROTATION

the true right ascension of the CIO). The equation of the origins is also the difference θ − GAST.

It therefore equals the accumulated precession-nutation of the equinox in right ascension, given

by the sum of the terms in parentheses from eq. 2.12 and the equation of the equinoxes given in

eq. 2.14 (all times –1). The equation of the origins in arcseconds therefore is:

Eo = − 0.014506 − 4612.156534 T − 1.3915817 T

2 + 0.00000044 T3

+ 0.000029956 T

4 + 0.0000000368 T5 − ∆ψ cos 

− 0.00264096 sin (Ω)

− 0.00006352 sin (2Ω)

− 0.00001175 sin (2F − 2D + 3Ω)

− 0.00001121 sin (2F − 2D + Ω) (6.2)

+ 0.00000455 sin (2F − 2D + 2Ω)

− 0.00000202 sin (2F + 3Ω)

− 0.00000198 sin (2F + Ω)

+ 0.00000172 sin (3Ω)

+ 0.00000087 T sin (Ω) + · · ·

where T is the number of centuries of TDB (or TT) from J2000.0; ∆ψ is the nutation in longitude,

in arcseconds;  is the mean obliquity of the ecliptic; and F, D, and Ω are fundamental luni-solar

arguments. All of the angles are functions of time; see Chapter 5 for expressions (esp. eqs. 5.12,

5.15, & 5.19). There is a table of daily values of Eo in Section B of The Astronomical Almanac.

To transform an object’s celestial coordinates from the true equator and equinox of t to the

Celestial Intermediate System (i.e., from EΥto Eσ), simply add Eo to the object’s true right

ascension. To similarly transform the components of a position vector, apply the rotation R3(−Eo).

Since many existing software systems are set up to produce positions with respect to the equator and

equinox of date, this is a relatively easy way to convert those positions to the Celestial Intermediate

Reference System if desired. Note that in such a case there is no computational difference in using

either the equinox-based or CIO-based methods for computing hour angle: Eo is computed in both

methods and is just applied to different quantities. In the equinox-based method, Eo is subtracted

from θ to form sidereal time; in the CIO-based method, Eo is added to the object’s true right

ascension so that θ can be used in place of sidereal time.

The position of the CIO in the GCRS, σGCRS, can be established by taking the position vector of

the equinox in the GCRS, ΥGCRS, and rotating it counterclockwise by the angle −Eo (i.e., clockwise

by Eo) about the axis nGCRS. Equivalently, establish the orthonormal basis triad of the equatorand-equinox system within the GCRS: ΥGCRS, (nGCRS× ΥGCRS), and nGCRS. Then

σGCRS = ΥGCRS cos Eo − (nGCRS × ΥGCRS) sin Eo (6.3)

6.5.1.2 CIO Location from Numerical Integration

As described above, a non-rotating origin can be described as a point on the moving equator whose

instantaneous motion is always orthogonal to the equator. A simple geometric construction based

on this definition yields the following differential equation for the motion of a non-rotating origin:

σ ̇ (t) = −



σ(t) · n ̇(t)



n(t) (6.4)
MODELING THE EARTH’S ROTATION 61

That is, if we have a model for the motion of the pole, n(t), the path of the non-rotating origin

is described by σ(t), once an initial point on the equator, σ(t0), is chosen. Conceptually and

practically, it is simple to integrate this equation, using, for example, a standard 4th-order RungeKutta integrator. For the motions of the real Earth, fixed step sizes of order 0.5 day work quite

well, and the integration is quite robust. This is actually a one-dimensional problem carried out

in three dimensions, since we know the non-rotating origin remains on the equator; we really need

only to know where along the equator it is. Therefore, two constraints can be applied at each step:

|σ| = 1 and σ · n = 0. See Kaplan (2005).

The above equation is quite general, and to get the specific motion of the CIO, each of the

vectors in the above equation is expressed with respect to the GCRS, i.e., σ(t) → σGCRS(t),

n(t) → nGCRS(t), and n ̇(t) → n ̇ GCRS(t). The pole’s position, nGCRS(t), is given by the first expression

in eq. 6.1. The pole’s motion, n ̇ GCRS(t), can be obtained by numerical differentiation of the pole’s

position. By numerically integrating the above equation, we obtain a time series of unit vectors,

σGCRS(ti), where each i is an integration step. The fact that this is actually just a one-dimensional

problem means that it is sufficient to store as output the CIO right ascensions (with respect to the

GCRS), using eq. 5.2 to decompose the σGCRS(ti) vectors. In this way, the integration results in

a tabulation of CIO right ascensions at discrete times. For example, see URL 17, where the times

are expressed as TDB Julian dates and the right ascensions are in arcseconds. This file runs from

years 1700 to 2300 at 1.2-day intervals.

When we need to obtain the position of the CIO for some specific time, the file of CIO right

ascensions can be interpolated to that time. The CIO’s unit vector, if required, can be readily

computed: generally, given a fixed coordinate system within which a pole and its equator move, a

point of interest on the equator has a position vector in the fixed system given by

r =





Z cos α

Z sin α

−X cos α − Y sin α



 (6.5)

where α is the right ascension of the point, and X, Y , and Z are the components of the pole’s

instantaneous unit vector, and all of these quantities are measured relative to the fixed coordinate

system. The vector r is not in general of unit length but it can be readily normalized. This formula

allows us to reconstruct the unit vector toward the CIO from just its GCRS right ascension value

at the time of interest, since we already know how to obtain the pole’s position vector in the GCRS

for that time.

Eq. 6.4 can be also made to yield the locus of the Terrestrial Intermediate Origin (TIO), simply

by referring all the vectors to the ITRS — a rotating geodetic system — rather than the GCRS.

In this case, therefore, σ(t) → $ITRS(t), n(t) → nITRS(t), and n ̇(t) → n ̇

ITRS(t). The path of the

CIP within the ITRS ( nITRS(t) ) is what we call polar motion (usually specified by the parameters

xp and yp), and is fundamentally unpredictable. The integration can therefore only be accurately

done for past times, using observed pole positions. A computed future path of the TIO on the

surface of the Earth depends on the assumption that the two major periodicities observed in polar

motion will continue at approximately the current amplitude.

6.5.1.3 CIO Location from the Arc-Difference s

On the celestial sphere, the Earth’s instantaneous (moving) equator intersects the GCRS equator

at two nodes. Let N be the ascending node of the instantaneous equator on the GCRS equator. We

can define a scalar quantity s(t) that represents the difference between the length of the arc from
62 MODELING THE EARTH’S ROTATION

N westward to the CIO (on the instantaneous equator) and the length of the arc from N westward

to the GCRS origin of right ascension (on the GCRS equator). The quantity s is called the CIO

locator. If σ represents the CIO and Σ0 represents the right ascension origin of the GCRS (the

direction of the GCRS x-axis), then

s = σN − Σ0N (6.6)

See Fig. 6.3, where the points Σ0 and Σ0

0

are equidistant from the node N. The quantity s is seen

to be the “extra” length of the arc on the instantaneous equator from N to σ, the position of the

CIO. The value of s is fundamentally obtained from an integral,

s(t) = −

Z t

t0

X(t)Y ̇ (t) − Y (t)X ̇ (t)

1 + Z(t)

dt + s0 (6.7)

where X(t), Y (t), and Z(t) are the three components of the unit vector, nGCRS(t), toward the

celestial pole (CIP). See, e.g., Capitaine et al. (2000) or the IERS Conventions (2003). The constant

of integration, s0, has been set to ensure that the equinox-based and CIO-based computations of

Earth rotation yield the same answers; s0=94 μas (IERS Conventions 2003). Effectively, the

constant adjusts the position of the CIO on the equator and is thus part of the arc σN. For

practical purposes, the value of s at any given time is provided by a series expansion, given in

Table 5.2c of the IERS Conventions (2003). Software to evaluate this series is available at the

IERS Conventions web site and is also part of the SOFA package. There is a table of daily values

of s in Section B of The Astronomical Almanac.

At any time, t, the the unit vector toward the node N is simply NGCRS = (−Y, X, 0)/

√

X2 + Y 2

(where we are no longer explicitly indicating the time dependence of X and Y ). To locate the CIO,

we rearrange eq. 6.6 to yield the arc length σN:

σN = s + Σ0N = s + arctan(X/(−Y )) (6.8)

The location of the CIO is then obtained by starting at the node N and moving along the instantaneous equator of t through the arc σN. That is, σGCRS can be constructed by taking the position

vector of the node N in the GCRS, NGCRS, and rotating it counterclockwise by the angle −σN (i.e.,

clockwise by σN) about the axis nGCRS. Equivalently,

σGCRS = NGCRS cos(σN) − (nGCRS × NGCRS) sin(σN) (6.9)

The three methods for determining the position of the CIO in the GCRS are numerically the

same to within several microarcseconds (μas) over six centuries centered on J2000.0. We now

have formulas in hand for obtaining the positions of the three cardinal points on the sky — the

CIP, the CIO, and the equinox — that are involved in the ITRS-to-GCRS (terrestrial-to-celestial)

transformations. In the following, it is assumed that nGCRS, σGCRS, and ΥGCRS are known vectors

for some time t of interest.
MODELING THE EARTH’S ROTATION 63

6.5.2 Geodetic Position Vectors and Polar Motion

Vectors representing the geocentric positions of points on or near the surface of the Earth are of

the general form

r =





(aC + h) cos φG cos λG

(aC + h) cos φG sin λG

(aS + h) sin φG



 (6.10)

where λG is the geodetic longitude, φG is the geodetic latitude, and h is the height. These coordinates are measured with respect to a reference ellipsoid, fit to the equipotential surface that

effectively defines mean sea level. The ellipsoid has an equatorial radius of a and a flattening factor

f. The quantities C and S depend on the flattening:

C = 1/

q

cos2 φG + (1 − f)

2 sin2 φG S = (1 − f)

2 C (6.11)

A complete description of geodetic concepts, reference ellipsoids, and computations is beyond the

scope of this circular, but a brief summary can be found in Section K of The Astronomical Almanac

and a more thorough account is given in Chapter 4 of the Explanatory Supplement (1992). More

information can be found in any introductory textbook on geodesy. Suffice it here to say that

the reference ellipsoid for GPS is WGS 84, with a = 6378137 m and f = 1/298.257223563. For

astronomical purposes it can be assumed that WGS 84 is a good approximation to the International

Terrestrial Reference System (ITRS) described previously in this chapter. That is, GPS provides

a realization of the ITRS.

It is worth noting that modern space techniques often measure geocentric positions in rectangular coordinates directly, without using a reference ellipsoid. Also, not all vectors of interest

represent geographic locations. Vectors representing instrumental axes, baselines, and boresights

are often of more interest to astronomers and these can usually be easily expressed in the same

geodetic system as the instrument location. All Earth-fixed vectors, regardless of what they represent, are subject to the same transformations described below. The ITRS is the assumed starting

point for these transformations, even though in most cases astronomers will be using vectors in

some system that approximates the ITRS.

For astronomical applications, we must correct ITRS vectors for polar motion (also called wobble). In current terminology, this is a transformation from the ITRS to the Terrestrial Intermediate

Reference System (TIRS; in this circular it is designated E$), and is the first transformation shown

in the flowcharts on page 56. Polar motion is the small quasi-periodic excursion of the geodetic pole

from the pole of rotation, or, more precisely stated, the excursion of the ITRS z-axis from the CIP.

It is described by the parameters xp and yp, which are the coordinates of the CIP in the ITRS, and

which generally amount to a few tenths of an arcsecond. Daily values of x and y, the observed pole

coordinates, are published by the IERS (see, e.g., IERS Bulletin A). These published values should,

for the most precise applications (<1 mas), be augmented by very small predictable components

to polar motion, with periods ≤1.2 days. These extra components are evaluated and added after

interpolation of the published x and y values — see section 5.4.2 of the IERS Conventions (2003).

The sum (total polar motion) is xp and yp. However, most users will be able to assume xp = x and

yp = y.

The transformation we seek not only must reorient the pole from the ITRS z-axis to the CIP,

it also must move the origin of longitude very slightly from the ITRS x-axis to the Terrestrial

Intermediate Origin (TIO). The latter shift is so tiny that its magnitude can be given by an

approximate formula, linear in time, based on the two main circular components of polar motion
64 MODELING THE EARTH’S ROTATION

as observed over the last few decades. That shift is s

0 = −47 μas T, where T is the time (either TT

or TDB) in centuries from J2000.0 (Lambert & Bizouard 2002). Since 47 μas amounts to 1.5 mm

on the surface of the Earth, this correction is entirely negligible for most purposes. Nevertheless,

we include it here for completeness.

The ITRS to E$ transformation then is accomplished using the polar motion (wobble) matrix

W(t):

rE$ = W(t) rITRS (6.12)

where

W(t) = R3(−s

0

) R2(xp) R1(yp) (6.13)

If we let

Sx = sin (xp)

Sy = sin (yp)

Ss = sin (−s

0

)

Cx = cos (xp)

Cy = cos (yp) (6.14)

Cs = cos (−s

0

)

then the wobble matrix can also be written:

W(t) =





CxCs SxSyCs + CySs −SxCyCs + SySs

−CxSs −SxSySs + CyCs SxCySs + SyCs

Sx −CxSy CxCy





≈





1 −s

0 −xp

s

0 1 yp

xp −yp 1



 (6.15)

where the form on the right is a first-order approximation. Due to the smallness of the angles

involved, the first-order matrix is quite adequate for most applications — further, s

0

can be set to

zero.

6.5.3 Complete Terrestrial to Celestial Transformation

The transformations corresponding to the two flowcharts on page 56 are

Equinox-based transformation: rGCRS = B

T PT NT R3(−GAST) W rITRS

CIO-based transformation: rGCRS = CT R3(−θ) W rITRS (6.16)

where all of the matrices except B are time-dependent. GAST is Greenwich apparent sidereal

time and θ is the Earth Rotation Angle; formulas are given in section 2.6.2 (eqs. 2.10–2.14). The

matrices, working from right to left, perform the following sub-transformations:

W ITRS to E$

R3(−GAST) E$ to EΥ

BT PT NT EΥto GCRS

R3(−θ) E$ to Eσ

CT Eσ to GCRS

The three “E” reference systems were described on page 55.

The matrix C has not yet been developed in this chapter. One form of the matrix C was

introduced in section 5.4.3 for the transformation from the GCRS to Eσ. We use the transpose

here because we are interested in the opposite transformation. C or CT is easy to construct because

we already have the three basis vectors of Eσ expressed in the GCRS: the z-axis is toward nGCRS,
MODELING THE EARTH’S ROTATION 65

the CIP; the x-axis is toward σGCRS, the CIO; and the y-axis is toward nGCRS × σGCRS. Call the

latter vector yGCRS. Then:

CT =



σGCRS yGCRS nGCRS =





σ1 y1 n1

σ2 y2 n2

σ3 y3 n3



 =





σ1 y1 X

σ2 y2 Y

σ3 y3 Z



 (6.17)

where X, Y , and Z are the CIP coordinates, expressed as dimensionless quantities. As in section 5.4.3, the matrix can also be constructed using only X and Y , together with the CIO locator, s:

CT =





1 − bX2 −bXY X

−bXY 1 − bY 2 Y

−X −Y 1 − b(X2 + Y

2

)





R3(s) (6.18)

where b = 1/(1 + Z) and Z =

√

1 − X2 − Y 2. The latter form is taken from the IERS Conventions

(2003), Chapter 5, where CT is called Q(t). The two constructions of CT are numerically the same.

6.5.4 Hour Angle

The local hour angle of a celestial object is given by

Equinox-based formula: h = GAST − αΥ + λ

CIO-based formula: h = θ − ασ + λ (6.19)

where GAST is Greenwich apparent sidereal time, θ is the Earth Rotation Angle, and λ is the

longitude of the observer. The quantities involved can be expressed in either angle or time units

as long as they are consistent. The right ascension in the two cases is expressed with respect to

different origins: αΥis the apparent right ascension of the object, measured with respect to the true

equinox, and ασ is the apparent right ascension of the object, measured with respect to the CIO.

That is, the coordinates of the object are expressed in system EΥin the equinox-based formula

and in system Eσ in the CIO-based formula. Since both systems share the same equator — the

instantaneous equator of date, orthogonal to the CIP — the apparent declination of the object is

the same in the two cases.

The two formulas in 6.19 are equivalent, which can be seen by substituting, in the equinox-based

formula, GAST = θ − Eo and αΥ = ασ − Eo, where Eo is the equation of the origins.

The longitude of the observer, λ, is expressed in the E$ system, that is, it is corrected for polar

motion. Using the first-order form of the matrix W, given in eq. 6.15 (and assuming s

0=0), it is

straightforward to derive eq. 2.16 for λ. Using notation consistent with that used in this chapter,

this equation is

λ ≡ λE$ = λITRS +



xp sin λITRS + yp cos λITRStan φITRS /3600 (6.20)

where λITRS and φITRS are the ITRS (geodetic) longitude and latitude of the observer, with λITRS

in degrees; and xp and yp are the coordinates of the pole (CIP), in arcseconds. This formula is

approximate and should not be used for places at polar latitudes.

The corresponding equation for the latitude, φ, corrected for polar motion is

φ ≡ φE$ = φITRS +



xp cos λITRS − yp sin λITRS/3600 (6.21)
66 MODELING THE EARTH’S ROTATION

although this equation is not needed in hour angle computations; it is given here only for completeness.

The common notion of hour angle can be expressed more precisely using concepts introduced

in Chapters 5 and 6. The local hour angle of an object is the angle between two planes: the plane

containing the geocenter, the CIP, and the observer; and the plane containing the geocenter, the

CIP, and the object. Hour angle increases with time and is positive when the object is west of

the observer as viewed from the geocenter. The two planes define meridians on the celestial sphere

that meet at the CIP. From the point of view of the observer, the CIP is not, in general, exactly

at the geodetic north point, which is the direction toward the ITRS z-axis. The azimuths of the

two directions differ by as much as qx

2

p + y

2

p/ cos φITRS, depending on time of day. This difference

is small (usually <1 arcsecond) and often negligible for practical applications. The plane defining

the astronomical Greenwich meridian (from which Greenwich hour angles are measured) can be

understood to contain the geocenter, the CIP, and TIO; there, λ ≡ λE$ = 0. This plane is now

called the TIO meridian.

The CIO-based formula for hour angle is quite simple to use (since θ is linear with time) if we

have the coordinates of the object expressed in system Eσ. Fortunately, this is straightforward if

we have the object’s coordinates expressed in the GCRS, because we also have the basis vectors of

Eσ expressed in the GCRS. If the object’s vector in the GCRS is rGCRS, then the object’s vector in

Eσ is simply

rEσ = C rGCRS =





rGCRS · σGCRS

rGCRS · yGCRS

rGCRS · nGCRS





where yGCRS = nGCRS × σGCRS (6.22)

Then

ασ = arctan 

rGCRS · yGCRS

rGCRS · σGCRS



(6.23)

As a specific case, we know the position vector of the equinox, ΥGCRS. Applying eq. 6.23 to ΥGCRS

and using it in the second formula of 6.19, with λ=0, we obtain the hour angle of the equinox at

the Greenwich (or TIO) meridian. But this is the definition of Greenwich Apparent Sidereal Time.

Therefore,

GAST = θ − arctan  ΥGCRS · yGCRS

ΥGCRS · σGCRS



(6.24)

Evidently, then,

Eo = arctan  ΥGCRS · yGCRS

ΥGCRS · σGCRS



(6.25)

which merely restates the definition of Eo — the equatorial angle from the CIO to the equinox, i.e.,

the right ascension of the equinox in system Eσ.
Text of IAU Resolutions of 1997

Adopted at the XXIIIrd General Assembly, Kyoto

Resolution B2 On the International Celestial Reference System (ICRS)

The XXIIIrd International Astronomical Union General Assembly

Considering

(a) That Recommendation VII of Resolution A4 of the 21st General Assembly specifies the coordinate system for the new celestial reference frame and, in particular, its continuity with the FK5

system at J2000.0;

(b) That Resolution B5 of the 22nd General Assembly specifies a list of extragalactic sources for

consideration as candidates for the realization of the new celestial reference frame;

(c) That the IAU Working Group on Reference Frames has in 1995 finalized the positions of these

candidate extragalactic sources in a coordinate frame aligned to that of the FK5 to within the

tolerance of the errors in the latter (see note 1);

(d) That the Hipparcos Catalogue was finalized in 1996 and that its coordinate frame is aligned to

that of the frame of the extragalactic sources in (c) with one sigma uncertainties of ± 0.6 milliarcseconds (mas) at epoch J1991.25 and ± 0.25 mas per year in rotation rate;

Noting

That all the conditions in the IAU Resolutions have now been met;

Resolves

(a) That, as from 1 January 1998, the IAU celestial reference system shall be the International

Celestial Reference System (ICRS) as specified in the 1991 IAU Resolution on reference frames and

as defined by the International Earth Rotation Service (IERS) (see note 2);

(b) That the corresponding fundamental reference frame shall be the International Celestial Reference Frame (ICRF) constructed by the IAU Working Group on Reference Frames;

(c) That the Hipparcos Catalogue shall be the primary realization of the ICRS at optical wavelengths;

(d) That IERS should take appropriate measures, in conjunction with the IAU Working Group on

67
68 IAU RESOLUTIONS 1997

reference frames, to maintain the ICRF and its ties to the reference frames at other wavelengths.

Note 1: IERS 1995 Report, Observatoire de Paris, p. II-19 (1996).

Note 2: “The extragalactic reference system of the International Earth Rotation Service (ICRS)”,

Arias, E.F. et al. A & A 303, 604 (1995).

Resolution B4 On Non-Rigid Earth Nutation Theory

The XXIIIrd International Astronomical Union General Assembly

Recognizing

that the International Astronomical Union and the International Union of Geodesy and Geophysics

Working Group (IAU-IUGG WG) on Non-rigid Earth Nutation Theory has met its goal by identifying the remaining geophysical and astronomical phenomena that must be modeled before an

accurate theory of nutation for a non-rigid Earth can be adopted, and

that, as instructed by IAU Recommendation C1 in 1994, the International Earth Rotation Service

(IERS) has published in the IERS Conventions (1996) an interim precession-nutation model that

matches the observations with an uncertainty of ± 1 milliarcsecond (mas),

endorses

the conclusions of the IAU-IUGG WG on Non-rigid Earth Nutation Theory given in the appendix,

requests

the IAU-IUGG WG on Non-rigid Earth Nutation Theory to present a detailed report to the next

IUGG General Assembly (August 1999), at which time the WG will be discontinued,

and urges

the scientific community to address the following questions in the future:

- completion of a new rigid Earth nutation series with the additional terms necessary for the theory

to be complete to within ± 5 microarcseconds, and

- completion of a new non-rigid Earth transfer function for an Earth initially in non-hydrostatic

equilibrium, incorporating mantle inelasticity and a Free Core Nutation period in agreement with

the observations, and taking into account better modeling of the fluid parts of the planet, including

dissipation.

APPENDIX

The WG on Non-rigid Earth Nutation Theory has quantified the problems in the nutation series

adopted by the IAU in 1980 by noting:

(1) that there is a difference in the precession rate of about −3.0 milliarcseconds per year (mas/year)

between the value observed by Very Long Baseline Interferometry (VLBI) and Lunar Laser Ranging

(LLR) and the adopted value,
IAU RESOLUTIONS 1997 69

(2) that the obliquity has been observed (by VLBI and LLR) to change at a rate of about

−0.24 mas/year, although there is no such change implied by the 1980 precession-nutation theory,

(3) that, in addition to these trends, there are observable peak-to-peak differences of up to 20

milliarcseconds (mas) between the nutation observed by VLBI and LLR and the nutation adopted

by the IAU in 1980,

(4) that these differences correspond to spectral amplitudes of up to several mas, and

(5) that the differences between observation and theory are well beyond the present observational

accuracy.

The WG has recognized the improvements made in the modeling of these quantities, and recommends, in order to derive a more precise nutation model, at the mas level in spectral amplitudes

and at a few mas level in the peak to peak analysis, the use of models:

(1) based on a new non-rigid Earth transfer function for an Earth initially in non-hydrostatic equilibrium, incorporating mantle inelasticity, a core-mantle-boundary flattening giving a Free Core

Nutation (FCN) period in agreement with the observed value, and a global Earth dynamical flattening in agreement with the observed precession, and

(2) based on a new rigid Earth nutation series which takes into account the following perturbing

effects:

1. in lunisolar ephemerides: indirect planetary effects, lunar inequality, J2-tilt, planetarytilt, secular variations of the amplitudes, effects of precession and nutation,

2. in the perturbing bodies to be considered: in addition to the Moon and the Sun, the

direct planetary effects of Venus, Jupiter, Mars, and Saturn, should be included,

3. in the order of the external potential to be considered: J3 and J4 effects for the

Moon, and

4. in the theory itself: effects of the tri-axiality of the Earth, relativistic effects and

second order effects.

The WG recognizes that this new generation of models still has some imperfections, the principal

one being poor modeling of the dissipation in the core and of certain effects of the ocean and the

atmosphere, and urges the scientific community to address these questions in the future.

The WG recognizes that, due to the remaining imperfections of the present theoretical nutation

models, the nutation series published in the IERS Conventions (1996), following 1994 IAU recommendation C1, still provides the users with the best nutation series. This IERS model being based

on observations of the celestial pole offset, the WG supports the recommendation that the scientific community continue VLBI and LLR observations to provide accurate estimations of nutation,

precession and rate of change in obliquity.
Text of IAU Resolutions of 2000

Adopted at the XXIVth General Assembly, Manchester

Resolution B1.1 Maintenance and Establishment of Reference Frames and

Systems

The XXIVth International Astronomical Union

Noting

1. that Resolution B2 of the XXIIIrd General Assembly (1997) specifies that “the fundamental

reference frame shall be the International Celestial Reference Frame (ICRF) constructed by the

IAU Working Group on Reference Frames,”

2. that Resolution B2 of the XXIIIrd General Assembly (1997) specifies “That the Hipparcos

Catalogue shall be the primary realization of the ICRS at optical wavelengths”, and

3. the need for accurate definition of reference systems brought about by unprecedented precision,

and

Recognizing

1. the importance of continuing operational observations made with Very Long Baseline Interferometry (VLBI) to maintain the ICRF,

2. the importance of VLBI observations to the operational determination of the parameters needed

to specify the time-variable transformation between the International Celestial and Terrestrial

Reference Frames,

3. the progressive shift between the Hipparcos frame and the ICRF, and

4. the need to maintain the optical realization as close as possible to the ICRF

Recommends

1. that IAU Division I maintain the Working Group on Celestial Reference Systems formed from

Division I members to consult with the International Earth Rotation Service (IERS) regarding the

maintenance of the ICRS,

2. that the IAU recognize the International VLBI service (IVS) for Geodesy and Astrometry as an

70
IAU RESOLUTIONS 2000 71

IAU Service Organization,

3. that an official representative of the IVS be invited to participate in the IAU Working Group

on Celestial Reference Systems,

4. that the IAU continue to provide an official representative to the IVS Directing Board,

5. that the astrometric and geodetic VLBI observing programs consider the requirements for

maintenance of the ICRF and linking to the Hipparcos optical frame in the selection of sources to

be observed (with emphasis on the Southern Hemisphere), design of observing networks, and the

distribution of data, and

6. that the scientific community continue with high priority ground- and space-based observations

(a) for the maintenance of the optical Hipparcos frame and frames at other wavelengths and (b)

for the links of the frames to the ICRF.

Resolution B1.2 Hipparcos Celestial Reference Frame

The XXIVth International Astronomical Union

Noting

1. that Resolution B2 of the XXIIIrd General Assembly (1997) specifies, “That the Hipparcos

Catalogue shall be the primary realization of the International Celestial Reference System (ICRS)

at optical wavelengths,”

2. the need for this realization to be of the highest precision,

3. that the proper motions of many of the Hipparcos stars known, or suspected, to be multiple are

adversely affected by uncorrected orbital motion,

4. the extensive use of the Hipparcos Catalogue as reference for the ICRS in extension to fainter

stars,

5. the need to avoid confusion between the International Celestial Reference Frame (ICRF) and

the Hipparcos frame, and

6. the progressive shift between the Hipparcos frame and the ICRF,

Recommends

1. that Resolution B2 of the XXIIIrd IAU General Assembly (1997) be amended by excluding from

the optical realization of the ICRS all stars flagged C, G, O, V and X in the Hipparcos Catalogue,

and

2. that this modified Hipparcos frame be labeled the Hipparcos Celestial Reference Frame (HCRF).
72 IAU RESOLUTIONS 2000

Resolution B1.3 Definition of Barycentric Celestial Reference System and

Geocentric Celestial Reference System

The XXIVth International Astronomical Union

Considering

1. that the Resolution A4 of the XXIst General Assembly (1991) has defined a system of spacetime coordinates for (a) the solar system (now called the Barycentric Celestial Reference System,

(BCRS)) and (b) the Earth (now called the Geocentric Celestial Reference System (GCRS)), within

the framework of General Relativity,

2. the desire to write the metric tensors both in the BCRS and in the GCRS in a compact and

self-consistent form, and

3. the fact that considerable work in General Relativity has been done using the harmonic gauge

that was found to be a useful and simplifying gauge for many kinds of applications,

Recommends

1. the choice of harmonic coordinates both for the barycentric and for the geocentric reference

systems.

2. writing the time-time component and the space-space component of the barycentric metric gμν

with barycentric coordinates (t, x) (t = Barycentric Coordinate Time (TCB)) with a single scalar

potential w(t,x) that generalizes the Newtonian potential, and the space-time component with a

vector potential wi(t, x); as a boundary condition it is assumed that these two potentials vanish

far from the solar system,

explicitly,

g00 = −1 + 2w

c

2 −

2w2

c

4 ,

g0i = −

4

c

3 w

i

,

gij = δij

1 + 2

c

2 w



,

with

w(t, x) = G

R

d

3x0 σ(t,x

0

)

|x−x0| +

1

2c

2G

∂

2

∂t2

R

d

3x0σ(t, x0

)|x − x

0

|

w

i

(t, x) = G

R

d

3x0 σ

i

(t,x

0

)

|x−x0|

.

Here, σ and σ

i are the gravitational mass and current densities, respectively.

3. writing the geocentric metric tensor Gαβ with geocentric coordinates (T, X) (T= Geocentric

Coordinate Time (TCG)) in the same form as the barycentric one but with potentials W(T, X)

and Wa(T, X); these geocentric potentials should be split into two parts — potentials W and Wa

arising from the gravitational action of the Earth and external parts Wext and Wa

ext due to tidal and

inertial effects; the external parts of the metric potentials are assumed to vanish at the geocenter
IAU RESOLUTIONS 2000 73

and admit an expansion into positive powers of X,

explicitly,

G00 = −1 + 2W

c

2 −

2W2

c

4 ,

G0a = −

4

c

3Wa

,

Gab = δab1 + 2

c

2W



.

The potentials W and Washould be split according to

W(T, X) = WE(T, X) + Wext(T, X),

Wa(T, X) = Wa

E(T, X) + Wa

ext(T, X).

The Earth’s potentials WE and Wa

E are defined in the same way as w and wi but with quantities

calculated in the GCRS with integrals taken over the whole Earth.

4. using, if accuracy requires, the full post-Newtonian coordinate transformation between the BCRS

and the GCRS as induced by the form of the corresponding metric tensors,

explicitly, for the kinematically non-rotating GCRS (T=TCG, t=TCB, r

i

E ≡ x

i − xi

E(t) and a

summation from 1 to 3 over equal indices is implied),

T = t −

1

c

2 [A(t) + v

i

Er

i

E] + 1

c

4 [B(t) + Bi

(t)r

i

E + Bij (t)r

i

Er

j

E + C(t, x)] + O(c

−5

),

Xa = δai

r

i

E +

1

c

2



1

2

v

i

Ev

j

E

r

j

E + wext(xE)r

i

E + r

i

Ea

j

E

r

j

E −

1

2

a

i

Er

2

E

 + O(c

−4

),

where

d

dtA(t) = 12

v

2

E + wext(xE),

d

dtB(t) = −

1

8

v

4

E −

3

2

v

2

Ewext(xE) + 4v

i

Ew

i

ext(xE) + 1

2w

2

ext(xE),

Bi(t) = −

1

2

v

2

Ev

i

E + 4w

i

ext(xE) − 3v

i

Ewext(xE),

Bij (t) = −v

i

EδajQa + 2 ∂

∂xj w

i

ext(xE) − v

i

E

∂

∂xj wext(xE) + 1

2

δ

ijw ̇ ext(xE),

C(t, x) = −

1

10 r

2

E( ̇a

i

Er

i

E).

Here x

i

E, v

i

E, and a

i

E are the barycentric position, velocity and acceleration vectors of the Earth,

the dot stands for the total derivative with respect to t, and

Qa = δai

∂

∂xi

wext(xE) − a

i

E



.

The external potentials, wext and w

i

ext, are given by

wext =

P

A6=E wA, wi

ext =

P

A6=E w

i

A,

where E stands for the Earth and wA and w

i

A are determined by the expressions for w and wi with
74 IAU RESOLUTIONS 2000

integrals taken over body A only.

Notes

It is to be understood that these expressions for w and wi give g00 correct up to O(c−5), g0i up

to O(c−5), and gij up to O(c−4). The densities σ and σ

i are determined by the components of

the energy momentum tensor of the matter composing the solar system bodies as given in the

references. Accuracies for Gαβ in terms of c−ncorrespond to those of gμν.

The external potentials Wext and Wa

ext can be written in the form

Wext = Wtidal + Winer,

Wa

ext = Wa

tidal + Wa

iner.

Wtidal generalizes the Newtonian expression for the tidal potential. Post-Newtonian expressions

for Wtidal and Wa

tidal can be found in the references. The potentials Winer, Wa

iner are inertial

contributions that are linear in Xa. The former is determined mainly by the coupling of the Earth’s

nonsphericity to the external potential. In the kinematically non-rotating Geocentric Celestial

Reference System, Wa

iner describes the Coriolis force induced mainly by geodetic precession.

Finally, the local gravitational potentials WE and Wa

E of the Earth are related to the barycentric

gravitational potentials wE and wi

E by

WE(T, X) = we(t, x)



1 + 2

c

2 v

2

E



−

4

c

2 v

i

Ew

i

E(t, x) + O(c

−4

),

Wa

E(T, X) = δai(w

i

E(t, x) − v

i

EwE(t, x)) + O(c

−2

).

References

Brumberg, V. A., Kopeikin, S. M., 1989, Nuovo Cimento, B103, 63.

Brumberg, V. A., 1991, Essential Relativistic Celestial Mechanics, Hilger, Bristol.

Damour, T., Soffel, M., Xu, C., Phys. Rev. D, 43, 3273 (1991); 45, 1017 (1992); 47, 3124 (1993);

49, 618 (1994).

Klioner, S. A., Voinov, A. V., 1993, Phys Rev. D, 48, 1451.

Kopeikin, S. M., 1988, Celest. Mech., 44, 87.

Resolution B1.4 Post-Newtonian Potential Coefficients

The XXIVth International Astronomical Union

Considering

1. that for many applications in the fields of celestial mechanics and astrometry a suitable

parametrization of the metric potentials (or multipole moments) outside the massive solar-system

bodies in the form of expansions in terms of potential coefficients are extremely useful, and
IAU RESOLUTIONS 2000 75

2. that physically meaningful post-Newtonian potential coefficients can be derived from the literature,

Recommends

1. expansion of the post-Newtonian potential of the Earth in the Geocentric Celestial Reference

System (GCRS) outside the Earth in the form

WE(T, X) = GME

R



1 + P∞

l=2

P+l

m=0(

RE

R

)

l

Plm(cos θ)(C

E

lm(T) cos mφ + S

E

lm(T) sin mφ)



,

where C

E

lm and S

E

lm are, to sufficient accuracy, equivalent to the post-Newtonian multipole moments

introduced in (Damour et al., Phys. Rev. D, 43, 3273, 1991), θ and φ are the polar angles

corresponding to the spatial coordinates Xa of the GCRS and R = |X|, and

2. expression of the vector potential outside the Earth, leading to the well-known Lense-Thirring

effect, in terms of the Earth’s total angular momentum vector SE in the form

Wa

E(T, X) = −

G

2

(X×SE)

a

R3 .

Resolution B1.5 Extended relativistic framework for time transformations

and realization of coordinate times in the solar system

The XXIVth International Astronomical Union

Considering

1. that the Resolution A4 of the XXIst General Assembly (1991) has defined systems of space-time

coordinates for the solar system (Barycentric Reference System) and for the Earth (Geocentric

Reference System), within the framework of General Relativity,

2. that Resolution B1.3 entitled “Definition of Barycentric Celestial Reference System and Geocentric Celestial Reference System” has renamed these systems the Barycentric Celestial Reference

System (BCRS) and the Geocentric Celestial Reference System (GCRS), respectively, and has specified a general framework for expressing their metric tensor and defining coordinate transformations

at the first post-Newtonian level,

3. that, based on the anticipated performance of atomic clocks, future time and frequency measurements will require practical application of this framework in the BCRS, and

4. that theoretical work requiring such expansions has already been performed,

Recommends

that for applications that concern time transformations and realization of coordinate times within

the solar system, Resolution B1.3 be applied as follows:

1. the metric tensor be expressed as

g00 = −



1 −

2

c

2 (w0(t, x) + wL(t, x)) + 2

c

4 (w

2

0

(t, x) + ∆(t, x)),
76 IAU RESOLUTIONS 2000

g0i = −

4

c

3 w

i

(t, x),

gij =



1 + 2w0(t,x)

c

2



δij ,

where (t ≡ Barycentric Coordinate Time (TCB),x) are the barycentric coordinates, w0 = G

P

A MA/rA

with the summation carried out over all solar system bodies A, rA = x−xA, xA are the coordinates

of the center of mass of body A, rA = |rA|, and where wL contains the expansion in terms of

multipole moments [see their definition in the Resolution B1.4 entitled “Post-Newtonian Potential

Coefficients”] required for each body. The vector potential w

i

(t, x) = P

A w

i

A(t, x) and the function

∆(t, x) = P

A ∆A(t, x) are given in note 2.

2. the relation between TCB and Geocentric Coordinate Time (TCG) can be expressed to sufficient

accuracy by

T CB−T CG = c

−2

"

R t

t0



v

2

E

2 +w0ext(xE)



dt+v

i

Er

i

E

#

−c

−4

"

R t

t0



−

1

8

v

4

E−

3

2

v

2

Ew0ext(xE)+4v

i

Ew

i

ext(xE)+

1

2w

2

0ext(xE)



dt −



3w0ext(xE) + v

2

E

2



v

i

Er

i

E

#

,

where vE is the barycentric velocity of the Earth and where the index ext refers to summation over

all bodies except the Earth.

Notes

1. This formulation will provide an uncertainty not larger than 5 × 10−18 in rate and, for quasiperiodic terms, not larger than 5 × 10−18 in rate amplitude and 0.2 ps in phase amplitude, for

locations farther than a few solar radii from the Sun. The same uncertainty also applies to the

transformation between TCB and TCG for locations within 50000 km of the Earth. Uncertainties

in the values of astronomical quantities may induce larger errors in the formulas.

2. Within the above mentioned uncertainties, it is sufficient to express the vector potential w

i

A(t, x)

of body A as

w

i

A(t, x) = G

"

−(rA×SA)

i

2r

3

A

+

MAv

i

A

rA

#

,

where SA is the total angular momentum of body A and v

i

A is the barycentric coordinate velocity

of body A. As for the function ∆A(t, x) it is sufficient to express it as

∆A(t, x) = GMA

rA

"

−2v

2

a +

P

B6=A

GMB

rBA

+

1

2



(r

k

A

v

k

A

)

2

r

2

A

+ r

k

Aa

k

A

#

+

2Gvk

A

(rA×SA)

k

r

3

A

,

where rBA = |xB − xA| and a

k

A is the barycentric coordinate acceleration of body A. In these

formulas, the terms in SA are needed only for Jupiter (S ≈ 6.9 × 1038m2s

−1kg) and Saturn (S ≈

1.4 × 1038m2s

−1kg), in the immediate vicinity of these planets.

3. Because the present recommendation provides an extension of the IAU 1991 recommendations

valid at the full first post-Newtonian level, the constants LC and LB that were introduced in the

IAU 1991 recommendations should be defined as < T CG/T CB > = 1 - LC and < T T /T CB > =
IAU RESOLUTIONS 2000 77

1 - LB, where TT refers to Terrestrial Time and <> refers to a sufficiently long average taken at

the geocenter. The most recent estimate of LC is (Irwin, A. and Fukushima, T., Astron. Astroph.,

348, 642–652, 1999)

LC = 1.48082686741 × 10−8 ± 2 × 10−17.

From Resolution B1.9 on “Redefinition of Terrestrial Time TT”, one infers LB = 1.55051976772 ×

10−8 ± 2 × 10−17 by using the relation 1 − LB = (1 − LC)(1 − LG). LG is defined in Resolution

B1.9.

Because no unambiguous definition may be provided for LB and LC, these constants should not

be used in formulating time transformations when it would require knowing their value with an

uncertainty of order 1 × 10−16 or less.

4. If TCB−TCG is computed using planetary ephemerides which are expressed in terms of a time

argument (noted Teph) which is close to Barycentric Dynamical Time (TDB), rather than in terms

of TCB, the first integral in Recommendation 2 above may be computed as

R t

t0



v

2

E

2 + w0ext(xE)



dt =

"

R Teph

Teph0



v

2

E

2 + w0ext(xE)



dt#/(1 − LB).

Resolution B1.6 IAU 2000 Precession-Nutation Model

The XXIVth International Astronomical Union

Recognizing

1. that the International Astronomical Union and the International Union of Geodesy and Geophysics Working Group (IAU-IUGG WG) on ‘Non-rigid Earth Nutation Theory’ has met its goals

by

a. establishing new high precision rigid Earth nutation series, such as (1) SMART97

of Bretagnon et al., 1998, Astron. Astroph., 329, 329–338; (2) REN2000 of Souchay et

al., 1999, Astron. Astroph. Supl. Ser., 135, 111–131; (3) RDAN97 of Roosbeek and

Dehant 1999, Celest. Mech., 70, 215–253;

b. completing the comparison of new non-rigid Earth transfer functions for an

Earth initially in non-hydrostatic equilibrium, incorporating mantle anelasticity and a

Free Core Nutation period in agreement with observations,

c. noting that numerical integration models are not yet ready to incorporate dissipation in the core,

d. noting the effects of other geophysical and astronomical phenomena that must

be modelled, such as ocean and atmospheric tides, that need further development;

2. that, as instructed by IAU Recommendation C1 in 1994, the International Earth Rotation Service

(IERS) will publish in the IERS Conventions (2000) a precession-nutation model that matches the

observations with a weighted rms of 0.2 milliarcsecond (mas);
78 IAU RESOLUTIONS 2000

3. that semi-analytical geophysical theories of forced nutation are available which incorporate some

or all of the following — anelasticity and electromagnetic couplings at the core-mantle and inner

core-outer core boundaries, annual atmospheric tide, geodesic nutation, and ocean tide effects;

4. that ocean tide corrections are necessary at all nutation frequencies; and

5. that empirical models based on a resonance formula without further corrections do also exist;

Accepts

the conclusions of the IAU-IUGG WG on Non-rigid Earth Nutation Theory published by Dehant

et al., 1999, Celest. Mech. 72(4), 245–310 and the recent comparisons between the various possibilities, and

Recommends

that, beginning on 1 January 2003, the IAU 1976 Precession Model and IAU 1980 Theory of

Nutation, be replaced by the precession-nutation model IAU 2000A (MHB2000, based on the

transfer functions of Mathews, Herring and Buffett, 2000 — submitted to the Journal of Geophysical

Research) for those who need a model at the 0.2 mas level, or its shorter version IAU 2000B for

those who need a model only at the 1 mas level, together with their associated precession and

obliquity rates, and their associated celestial pole offsets at J2000.0, to be published in the IERS

Conventions 2000, and

Encourages

1. the continuation of theoretical developments of non-rigid Earth nutation series,

2. the continuation of VLBI observations to increase the accuracy of the nutation series and the

nutation model, and to monitor the unpredictable free core nutation, and

3. the development of new expressions for precession consistent with the IAU 2000A model.

Resolution B1.7 Definition of Celestial Intermediate Pole

The XXIVth International Astronomical Union

Noting

the need for accurate definition of reference systems brought about by unprecedented observational

precision, and

Recognizing

1. the need to specify an axis with respect to which the Earth’s angle of rotation is defined,

2. that the Celestial Ephemeris Pole (CEP) does not take account of diurnal and higher frequency

variations in the Earth’s orientation,

Recommends
IAU RESOLUTIONS 2000 79

1. that the Celestial Intermediate Pole (CIP) be the pole, the motion of which is specified in the

Geocentric Celestial Reference System (GCRS, see Resolution B1.3) by motion of the Tisserand

mean axis of the Earth with periods greater than two days,

2. that the direction of the CIP at J2000.0 be offset from the direction of the pole of the GCRS in

a manner consistent with the IAU 2000A (see Resolution B1.6) precession-nutation model,

3. that the motion of the CIP in the GCRS be realized by the IAU 2000A model for precession

and forced nutation for periods greater than two days plus additional time-dependent corrections

provided by the International Earth Rotation Service (IERS) through appropriate astro-geodetic

observations,

4. that the motion of the CIP in the International Terrestrial Reference System (ITRS) be provided

by the IERS through appropriate astro-geodetic observations and models including high-frequency

variations,

5. that for highest precision, corrections to the models for the motion of the CIP in the ITRS may

be estimated using procedures specified by the IERS, and

6. that implementation of the CIP be on 1 January 2003.

Notes

1. The forced nutations with periods less than two days are included in the model for the motion

of the CIP in the ITRS.

2. The Tisserand mean axis of the Earth corresponds to the mean surface geographic axis, quoted

B axis, in Seidelmann, 1982, Celest. Mech., 27, 79–106.

3. As a consequence of this resolution, the Celestial Ephemeris Pole is no longer necessary.

Resolution B1.8 Definition and use of Celestial and Terrestrial Ephemeris

Origin

The XXIVth International Astronomical Union

Recognizing

1. the need for reference system definitions suitable for modern realizations of the conventional

reference systems and consistent with observational precision,

2. the need for a rigorous definition of sidereal rotation of the Earth,

3. the desirability of describing the rotation of the Earth independently from its orbital motion,

and

Noting

that the use of the “non-rotating origin” (Guinot, 1979) on the moving equator fulfills the above

conditions and allows for a definition of UT1 which is insensitive to changes in models for precession

and nutation at the microarcsecond level,
80 IAU RESOLUTIONS 2000

Recommends

1. the use of the “non-rotating origin” in the Geocentric Celestial Reference System (GCRS)

and that this point be designated as the Celestial Ephemeris Origin (CEO) on the equator of the

Celestial Intermediate Pole (CIP),

2. the use of the “non-rotating origin” in the International Terrestrial Reference System (ITRS)

and that this point be designated as the Terrestrial Ephemeris Origin (TEO) on the equator of the

CIP,

3. that UT1 be linearly proportional to the Earth Rotation Angle defined as the angle measured

along the equator of the CIP between the unit vectors directed toward the CEO and the TEO,

4. that the transformation between the ITRS and GCRS be specified by the position of the CIP

in the GCRS, the position of the CIP in the ITRS, and the Earth Rotation Angle,

5. that the International Earth Rotation Service (IERS) take steps to implement this by 1 January

2003, and

6. that the IERS will continue to provide users with data and algorithms for the conventional

transformations.

Note

1. The position of the CEO can be computed from the IAU 2000A model for precession and

nutation of the CIP and from the current values of the offset of the CIP from the pole of the ICRF

at J2000.0 using the development provided by Capitaine et al. (2000).

2. The position of the TEO is only slightly dependent on polar motion and can be extrapolated as

done by Capitaine et al. (2000) using the IERS data.

3. The linear relationship between the Earth’s rotation angle θ and UT1 should ensure the continuity in phase and rate of UT1 with the value obtained by the conventional relationship between

Greenwich Mean Sidereal Time (GMST) and UT1. This is accomplished by the following relationship:

θ(UT1) = 2π(0.7790572732640 + 1.00273781191135448 × (Julian UT1 date − 2451545.0))

References

Guinot, B., 1979, in D.D. McCarthy and J.D. Pilkington (eds.), Time and the Earth’s Rotation,

D. Reidel Publ., 7–18.

Capitaine, N., Guinot, B., McCarthy, D.D., 2000, “Definition of the Celestial Ephemeris Origin

and of UT1 in the International Celestial Reference Frame”, Astron. Astrophys., 355, 398–405.
IAU RESOLUTIONS 2000 81

Resolution B1.9 Re-definition of Terrestrial Time TT

The XXIVth International Astronomical Union

Considering

1. that IAU Resolution A4 (1991) has defined Terrestrial Time (TT) in its Recommendation 4,

and

2. that the intricacy and temporal changes inherent to the definition and realization of the geoid

are a source of uncertainty in the definition and realization of TT, which may become, in the near

future, the dominant source of uncertainty in realizing TT from atomic clocks,

Recommends

that TT be a time scale differing from TCG by a constant rate: dTT/dTCG = 1–LG, where LG =

6.969290134×10−10 is a defining constant,

Note

LG was defined by the IAU Resolution A4 (1991) in its Recommendation 4 as equal to UG/c2 where

UG is the geopotential at the geoid. LG is now used as a defining constant.

Resolution B2 Coordinated Universal Time

The XXIVth International Astronomical Union

Recognizing

1. that the definition of Coordinated Universal Time (UTC) relies on the astronomical observation

of the UT1 time scale in order to introduce leap seconds,

2. that the unpredictable leap seconds affects modern communication and navigation systems,

3. that astronomical observations provide an accurate estimate of the secular deceleration of the

Earth’s rate of rotation

Recommends

1. that the IAU establish a working group reporting to Division I at the General Assembly in 2003

to consider the redefinition of UTC,

2. that this study discuss whether there is a requirement for leap seconds, the possibility of inserting

leap seconds at pre-determined intervals, and the tolerance limits for UT1−UTC, and

3. that this study be undertaken in cooperation with the appropriate groups of the International

Union of Radio Science (URSI), the International Telecommunications Union (ITU-R), the International Bureau for Weights and Measures (BIPM), the International Earth Rotation Service (IERS)

and relevant navigational agencies.
References

Journal abbreviations:

A&A Astronomy and Astrophysics

A&A Supp Astronomy and Astrophysics Supplement Series

AJ The Astronomical Journal

ApJ Supp The Astrophysical Journal Supplement Series

Celest. Mech. Celestial Mechanics

Celest. Mech. Dyn. Astr. Celestial Mechanics and Dynamical Astronomy

Geophys. J. Royal Astron. Soc. Geophysical Journal of the Royal Astronomical Society

J. Geophys. Res. Journal of Geophysical Research

Phys. Rev. D Physical Review D

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Web URLs Note: URLs frequently change. This listing is correct as of October 2005.

[1] PDF version of this circular: http://aa.usno.navy.mil/publications/docs/Circular_179.html

. . . vi, 104

[2] IAU Information Bulletin: http://www.iau.org/Activities/publications/bulletin/

. . . vi

[3] IERS Conventions (2003): http://maia.usno.navy.mil/conv2003.html or

http://tai.bipm.org/iers/conv2003/conv2003.html

. . . vi, 85

[4] IERS FAQs: http://www.iers.org/iers/earth/faqs/iau2000.html . . . vi

[5] IERS Earth rotation data (Bulletins A & B): http://www.iers.org/iers/products/eop/

. . . 9, 11, 16, 84

[6] IAU Working Group on Nomenclature: http://syrte.obspm.fr/iauWGnfa/ . . . vi, xii

[7] SOFA: http://www.iau-sofa.rl.ac.uk/ . . . vi, 25, 47

[8] NOVAS: http://aa.usno.navy.mil/software/novas/ . . . vi, 25, 47

[9] SI brochure: http://www1.bipm.org/utils/en/pdf/si-brochure.pdf . . . 82

[10] ICRF source coordinates: http://hpiers.obspm.fr/webiers/results/icrf/icrfrsc.html

. . . 24

[11] JPL ephemerides: http://www.willbell.com/software/jpl.htm or

ftp://ssd.jpl.nasa.gov/pub/eph/export/ . . . 31

[12] JPL “README” file: http://ssd.jpl.nasa.gov/iau-comm4/README . . . 31

[13] IAU Working Group on Cartographic Coordinates and Rotational Elements:

http://astrogeology.wr.usgs.gov/Projects/WGCCRE . . . 31

[14] Nutation series: http://maia.usno.navy.mil/conv2003/chapter5/tab5.3a.txt and

http://maia.usno.navy.mil/conv2003/chapter5/tab5.3b.txt

. . . 40, 47, 88

[15] IERS nutation subroutine: http://maia.usno.navy.mil/conv2003/chapter5/NU2000A.f

. . . 88

[16] Note on (dX,dY) conversion: http://aa.usno.navy.mil/kaplan/dXdY_to_dpsideps.pdf

. . . 84

[17] File of CIO RA: http://aa.usno.navy.mil/software/novas/novas_f/CIO_RA.txt

. . . 61
IAU 2000A Nutation Series

The nutation series adopted by the IAU, developed by Mathews et al. (2002) (MHB), is listed below

in its entirety. It is also available from the IERS as a pair of plain-text computer files at URL 14,

although the arrangement of the columns differs from what is presented here. The IERS also

provides a Fortran subroutine for evaluating the nutation series, written by P. Wallace, at URL 15.

The NOVAS software package includes this subroutine, and the SOFA package contains the same

code in a subroutine of a different name. There are also subroutines available that evaluate only a

subset of the series terms for applications that do not require the highest accuracy.

There are 1365 terms in the series. The term numbers are arbitrary and are not involved in

the computation. As listed below, the first 678 are lunisolar terms and the remaining 687 are

planetary terms. In the lunisolar terms, the only fundamental argument multipliers that are nonzero are Mi,10 through Mi,14, corresponding to the arguments l, l

0

, F, D, and Ω, respectively. In

the planetary terms, there are no rates of change of the coefficients, i.e., S ̇

i and C ̇i are zero.

The formulas for evaluating the series are given in section 5.4.2; see eqs. 5.15–5.16 and the

following text.

Term Fundamental Argument Multipliers Mi,j ∆ψ Coefficients ∆ Coefficients

i j= 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Si S ̇

i C0

i Ci C ̇

i S

0

i

00 00 00 00 00 00

1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -17.2064161 -0.0174666 0.0033386 9.2052331 0.0009086 0.0015377

2 0 0 0 0 0 0 0 0 0 0 0 2 -2 2 -1.3170906 -0.0001675 -0.0013696 0.5730336 -0.0003015 -0.0004587

3 0 0 0 0 0 0 0 0 0 0 0 2 0 2 -0.2276413 -0.0000234 0.0002796 0.0978459 -0.0000485 0.0001374

4 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0.2074554 0.0000207 -0.0000698 -0.0897492 0.0000470 -0.0000291

5 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0.1475877 -0.0003633 0.0011817 0.0073871 -0.0000184 -0.0001924

6 0 0 0 0 0 0 0 0 0 0 1 2 -2 2 -0.0516821 0.0001226 -0.0000524 0.0224386 -0.0000677 -0.0000174

7 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0.0711159 0.0000073 -0.0000872 -0.0006750 0.0000000 0.0000358

8 0 0 0 0 0 0 0 0 0 0 0 2 0 1 -0.0387298 -0.0000367 0.0000380 0.0200728 0.0000018 0.0000318

9 0 0 0 0 0 0 0 0 0 1 0 2 0 2 -0.0301461 -0.0000036 0.0000816 0.0129025 -0.0000063 0.0000367

10 0 0 0 0 0 0 0 0 0 0 -1 2 -2 2 0.0215829 -0.0000494 0.0000111 -0.0095929 0.0000299 0.0000132

11 0 0 0 0 0 0 0 0 0 0 0 2 -2 1 0.0128227 0.0000137 0.0000181 -0.0068982 -0.0000009 0.0000039

12 0 0 0 0 0 0 0 0 0 -1 0 2 0 2 0.0123457 0.0000011 0.0000019 -0.0053311 0.0000032 -0.0000004

13 0 0 0 0 0 0 0 0 0 -1 0 0 2 0 0.0156994 0.0000010 -0.0000168 -0.0001235 0.0000000 0.0000082

14 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0.0063110 0.0000063 0.0000027 -0.0033228 0.0000000 -0.0000009

15 0 0 0 0 0 0 0 0 0 -1 0 0 0 1 -0.0057976 -0.0000063 -0.0000189 0.0031429 0.0000000 -0.0000075

16 0 0 0 0 0 0 0 0 0 -1 0 2 2 2 -0.0059641 -0.0000011 0.0000149 0.0025543 -0.0000011 0.0000066

17 0 0 0 0 0 0 0 0 0 1 0 2 0 1 -0.0051613 -0.0000042 0.0000129 0.0026366 0.0000000 0.0000078

18 0 0 0 0 0 0 0 0 0 -2 0 2 0 1 0.0045893 0.0000050 0.0000031 -0.0024236 -0.0000010 0.0000020

19 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0.0063384 0.0000011 -0.0000150 -0.0001220 0.0000000 0.0000029

20 0 0 0 0 0 0 0 0 0 0 0 2 2 2 -0.0038571 -0.0000001 0.0000158 0.0016452 -0.0000011 0.0000068

21 0 0 0 0 0 0 0 0 0 0 -2 2 -2 2 0.0032481 0.0000000 0.0000000 -0.0013870 0.0000000 0.0000000

22 0 0 0 0 0 0 0 0 0 -2 0 0 2 0 -0.0047722 0.0000000 -0.0000018 0.0000477 0.0000000 -0.0000025

23 0 0 0 0 0 0 0 0 0 2 0 2 0 2 -0.0031046 -0.0000001 0.0000131 0.0013238 -0.0000011 0.0000059

24 0 0 0 0 0 0 0 0 0 1 0 2 -2 2 0.0028593 0.0000000 -0.0000001 -0.0012338 0.0000010 -0.0000003

25 0 0 0 0 0 0 0 0 0 -1 0 2 0 1 0.0020441 0.0000021 0.0000010 -0.0010758 0.0000000 -0.0000003

26 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0.0029243 0.0000000 -0.0000074 -0.0000609 0.0000000 0.0000013

27 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0.0025887 0.0000000 -0.0000066 -0.0000550 0.0000000 0.0000011

28 0 0 0 0 0 0 0 0 0 0 1 0 0 1 -0.0014053 -0.0000025 0.0000079 0.0008551 -0.0000002 -0.0000045

29 0 0 0 0 0 0 0 0 0 -1 0 0 2 1 0.0015164 0.0000010 0.0000011 -0.0008001 0.0000000 -0.0000001

30 0 0 0 0 0 0 0 0 0 0 2 2 -2 2 -0.0015794 0.0000072 -0.0000016 0.0006850 -0.0000042 -0.0000005

88
NUTATION SERIES 89

Term Fundamental Argument Multipliers Mi,j ∆ψ Coefficients ∆ Coefficients

i j= 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Si S ̇

i C0

i Ci C ̇

i S

0

i

00 00 00 00 00 00

31 0 0 0 0 0 0 0 0 0 0 0 -2 2 0 0.0021783 0.0000000 0.0000013 -0.0000167 0.0000000 0.0000013

32 0 0 0 0 0 0 0 0 0 1 0 0 -2 1 -0.0012873 -0.0000010 -0.0000037 0.0006953 0.0000000 -0.0000014

33 0 0 0 0 0 0 0 0 0 0 -1 0 0 1 -0.0012654 0.0000011 0.0000063 0.0006415 0.0000000 0.0000026

34 0 0 0 0 0 0 0 0 0 -1 0 2 2 1 -0.0010204 0.0000000 0.0000025 0.0005222 0.0000000 0.0000015

35 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0.0016707 -0.0000085 -0.0000010 0.0000168 -0.0000001 0.0000010

36 0 0 0 0 0 0 0 0 0 1 0 2 2 2 -0.0007691 0.0000000 0.0000044 0.0003268 0.0000000 0.0000019

37 0 0 0 0 0 0 0 0 0 -2 0 2 0 0 -0.0011024 0.0000000 -0.0000014 0.0000104 0.0000000 0.0000002

38 0 0 0 0 0 0 0 0 0 0 1 2 0 2 0.0007566 -0.0000021 -0.0000011 -0.0003250 0.0000000 -0.0000005

39 0 0 0 0 0 0 0 0 0 0 0 2 2 1 -0.0006637 -0.0000011 0.0000025 0.0003353 0.0000000 0.0000014

40 0 0 0 0 0 0 0 0 0 0 -1 2 0 2 -0.0007141 0.0000021 0.0000008 0.0003070 0.0000000 0.0000004

41 0 0 0 0 0 0 0 0 0 0 0 0 2 1 -0.0006302 -0.0000011 0.0000002 0.0003272 0.0000000 0.0000004

42 0 0 0 0 0 0 0 0 0 1 0 2 -2 1 0.0005800 0.0000010 0.0000002 -0.0003045 0.0000000 -0.0000001

43 0 0 0 0 0 0 0 0 0 2 0 2 -2 2 0.0006443 0.0000000 -0.0000007 -0.0002768 0.0000000 -0.0000004

44 0 0 0 0 0 0 0 0 0 -2 0 0 2 1 -0.0005774 -0.0000011 -0.0000015 0.0003041 0.0000000 -0.0000005

45 0 0 0 0 0 0 0 0 0 2 0 2 0 1 -0.0005350 0.0000000 0.0000021 0.0002695 0.0000000 0.0000012

46 0 0 0 0 0 0 0 0 0 0 -1 2 -2 1 -0.0004752 -0.0000011 -0.0000003 0.0002719 0.0000000 -0.0000003

47 0 0 0 0 0 0 0 0 0 0 0 0 -2 1 -0.0004940 -0.0000011 -0.0000021 0.0002720 0.0000000 -0.0000009

48 0 0 0 0 0 0 0 0 0 -1 -1 0 2 0 0.0007350 0.0000000 -0.0000008 -0.0000051 0.0000000 0.0000004

49 0 0 0 0 0 0 0 0 0 2 0 0 -2 1 0.0004065 0.0000000 0.0000006 -0.0002206 0.0000000 0.0000001

50 0 0 0 0 0 0 0 0 0 1 0 0 2 0 0.0006579 0.0000000 -0.0000024 -0.0000199 0.0000000 0.0000002

51 0 0 0 0 0 0 0 0 0 0 1 2 -2 1 0.0003579 0.0000000 0.0000005 -0.0001900 0.0000000 0.0000001

52 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0.0004725 0.0000000 -0.0000006 -0.0000041 0.0000000 0.0000003

53 0 0 0 0 0 0 0 0 0 -2 0 2 0 2 -0.0003075 0.0000000 -0.0000002 0.0001313 0.0000000 -0.0000001

54 0 0 0 0 0 0 0 0 0 3 0 2 0 2 -0.0002904 0.0000000 0.0000015 0.0001233 0.0000000 0.0000007

55 0 0 0 0 0 0 0 0 0 0 -1 0 2 0 0.0004348 0.0000000 -0.0000010 -0.0000081 0.0000000 0.0000002

56 0 0 0 0 0 0 0 0 0 1 -1 2 0 2 -0.0002878 0.0000000 0.0000008 0.0001232 0.0000000 0.0000004

57 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -0.0004230 0.0000000 0.0000005 -0.0000020 0.0000000 -0.0000002

58 0 0 0 0 0 0 0 0 0 -1 -1 2 2 2 -0.0002819 0.0000000 0.0000007 0.0001207 0.0000000 0.0000003

59 0 0 0 0 0 0 0 0 0 -1 0 2 0 0 -0.0004056 0.0000000 0.0000005 0.0000040 0.0000000 -0.0000002

60 0 0 0 0 0 0 0 0 0 0 -1 2 2 2 -0.0002647 0.0000000 0.0000011 0.0001129 0.0000000 0.0000005

61 0 0 0 0 0 0 0 0 0 -2 0 0 0 1 -0.0002294 0.0000000 -0.0000010 0.0001266 0.0000000 -0.0000004

62 0 0 0 0 0 0 0 0 0 1 1 2 0 2 0.0002481 0.0000000 -0.0000007 -0.0001062 0.0000000 -0.0000003

63 0 0 0 0 0 0 0 0 0 2 0 0 0 1 0.0002179 0.0000000 -0.0000002 -0.0001129 0.0000000 -0.0000002

64 0 0 0 0 0 0 0 0 0 -1 1 0 1 0 0.0003276 0.0000000 0.0000001 -0.0000009 0.0000000 0.0000000

65 0 0 0 0 0 0 0 0 0 1 1 0 0 0 -0.0003389 0.0000000 0.0000005 0.0000035 0.0000000 -0.0000002

66 0 0 0 0 0 0 0 0 0 1 0 2 0 0 0.0003339 0.0000000 -0.0000013 -0.0000107 0.0000000 0.0000001

67 0 0 0 0 0 0 0 0 0 -1 0 2 -2 1 -0.0001987 0.0000000 -0.0000006 0.0001073 0.0000000 -0.0000002

68 0 0 0 0 0 0 0 0 0 1 0 0 0 2 -0.0001981 0.0000000 0.0000000 0.0000854 0.0000000 0.0000000

69 0 0 0 0 0 0 0 0 0 -1 0 0 1 0 0.0004026 0.0000000 -0.0000353 -0.0000553 0.0000000 -0.0000139

70 0 0 0 0 0 0 0 0 0 0 0 2 1 2 0.0001660 0.0000000 -0.0000005 -0.0000710 0.0000000 -0.0000002

71 0 0 0 0 0 0 0 0 0 -1 0 2 4 2 -0.0001521 0.0000000 0.0000009 0.0000647 0.0000000 0.0000004

72 0 0 0 0 0 0 0 0 0 -1 1 0 1 1 0.0001314 0.0000000 0.0000000 -0.0000700 0.0000000 0.0000000

73 0 0 0 0 0 0 0 0 0 0 -2 2 -2 1 -0.0001283 0.0000000 0.0000000 0.0000672 0.0000000 0.0000000

74 0 0 0 0 0 0 0 0 0 1 0 2 2 1 -0.0001331 0.0000000 0.0000008 0.0000663 0.0000000 0.0000004

75 0 0 0 0 0 0 0 0 0 -2 0 2 2 2 0.0001383 0.0000000 -0.0000002 -0.0000594 0.0000000 -0.0000002

76 0 0 0 0 0 0 0 0 0 -1 0 0 0 2 0.0001405 0.0000000 0.0000004 -0.0000610 0.0000000 0.0000002

77 0 0 0 0 0 0 0 0 0 1 1 2 -2 2 0.0001290 0.0000000 0.0000000 -0.0000556 0.0000000 0.0000000

78 0 0 0 0 0 0 0 0 0 -2 0 2 4 2 -0.0001214 0.0000000 0.0000005 0.0000518 0.0000000 0.0000002

79 0 0 0 0 0 0 0 0 0 -1 0 4 0 2 0.0001146 0.0000000 -0.0000003 -0.0000490 0.0000000 -0.0000001

80 0 0 0 0 0 0 0 0 0 2 0 2 -2 1 0.0001019 0.0000000 -0.0000001 -0.0000527 0.0000000 -0.0000001

81 0 0 0 0 0 0 0 0 0 2 0 2 2 2 -0.0001100 0.0000000 0.0000009 0.0000465 0.0000000 0.0000004

82 0 0 0 0 0 0 0 0 0 1 0 0 2 1 -0.0000970 0.0000000 0.0000002 0.0000496 0.0000000 0.0000001

83 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0.0001575 0.0000000 -0.0000006 -0.0000050 0.0000000 0.0000000

84 0 0 0 0 0 0 0 0 0 3 0 2 -2 2 0.0000934 0.0000000 -0.0000003 -0.0000399 0.0000000 -0.0000001

85 0 0 0 0 0 0 0 0 0 0 0 4 -2 2 0.0000922 0.0000000 -0.0000001 -0.0000395 0.0000000 -0.0000001

86 0 0 0 0 0 0 0 0 0 0 1 2 0 1 0.0000815 0.0000000 -0.0000001 -0.0000422 0.0000000 -0.0000001

87 0 0 0 0 0 0 0 0 0 0 0 -2 2 1 0.0000834 0.0000000 0.0000002 -0.0000440 0.0000000 0.0000001

88 0 0 0 0 0 0 0 0 0 0 0 2 -2 3 0.0001248 0.0000000 0.0000000 -0.0000170 0.0000000 0.0000001

89 0 0 0 0 0 0 0 0 0 -1 0 0 4 0 0.0001338 0.0000000 -0.0000005 -0.0000039 0.0000000 0.0000000

90 0 0 0 0 0 0 0 0 0 2 0 -2 0 1 0.0000716 0.0000000 -0.0000002 -0.0000389 0.0000000 -0.0000001

91 0 0 0 0 0 0 0 0 0 -2 0 0 4 0 0.0001282 0.0000000 -0.0000003 -0.0000023 0.0000000 0.0000001

92 0 0 0 0 0 0 0 0 0 -1 -1 0 2 1 0.0000742 0.0000000 0.0000001 -0.0000391 0.0000000 0.0000000

93 0 0 0 0 0 0 0 0 0 -1 0 0 1 1 0.0001020 0.0000000 -0.0000025 -0.0000495 0.0000000 -0.0000010

94 0 0 0 0 0 0 0 0 0 0 1 0 0 2 0.0000715 0.0000000 -0.0000004 -0.0000326 0.0000000 0.0000002

95 0 0 0 0 0 0 0 0 0 0 0 -2 0 1 -0.0000666 0.0000000 -0.0000003 0.0000369 0.0000000 -0.0000001

96 0 0 0 0 0 0 0 0 0 0 -1 2 0 1 -0.0000667 0.0000000 0.0000001 0.0000346 0.0000000 0.0000001

97 0 0 0 0 0 0 0 0 0 0 0 2 -1 2 -0.0000704 0.0000000 0.0000000 0.0000304 0.0000000 0.0000000

98 0 0 0 0 0 0 0 0 0 0 0 2 4 2 -0.0000694 0.0000000 0.0000005 0.0000294 0.0000000 0.0000002

99 0 0 0 0 0 0 0 0 0 -2 -1 0 2 0 -0.0001014 0.0000000 -0.0000001 0.0000004 0.0000000 -0.0000001

100 0 0 0 0 0 0 0 0 0 1 1 0 -2 1 -0.0000585 0.0000000 -0.0000002 0.0000316 0.0000000 -0.0000001

101 0 0 0 0 0 0 0 0 0 -1 1 0 2 0 -0.0000949 0.0000000 0.0000001 0.0000008 0.0000000 -0.0000001

102 0 0 0 0 0 0 0 0 0 -1 1 0 1 2 -0.0000595 0.0000000 0.0000000 0.0000258 0.0000000 0.0000000

103 0 0 0 0 0 0 0 0 0 1 -1 0 0 1 0.0000528 0.0000000 0.0000000 -0.0000279 0.0000000 0.0000000

104 0 0 0 0 0 0 0 0 0 1 -1 2 2 2 -0.0000590 0.0000000 0.0000004 0.0000252 0.0000000 0.0000002

105 0 0 0 0 0 0 0 0 0 -1 1 2 2 2 0.0000570 0.0000000 -0.0000002 -0.0000244 0.0000000 -0.0000001

106 0 0 0 0 0 0 0 0 0 3 0 2 0 1 -0.0000502 0.0000000 0.0000003 0.0000250 0.0000000 0.0000002

107 0 0 0 0 0 0 0 0 0 0 1 -2 2 0 -0.0000875 0.0000000 0.0000001 0.0000029 0.0000000 0.0000000

108 0 0 0 0 0 0 0 0 0 -1 0 0 -2 1 -0.0000492 0.0000000 -0.0000003 0.0000275 0.0000000 -0.0000001

109 0 0 0 0 0 0 0 0 0 0 1 2 2 2 0.0000535 0.0000000 -0.0000002 -0.0000228 0.0000000 -0.0000001

110 0 0 0 0 0 0 0 0 0 -1 -1 2 2 1 -0.0000467 0.0000000 0.0000001 0.0000240 0.0000000 0.0000001

111 0 0 0 0 0 0 0 0 0 0 -1 0 0 2 0.0000591 0.0000000 0.0000000 -0.0000253 0.0000000 0.0000000

112 0 0 0 0 0 0 0 0 0 1 0 2 -4 1 -0.0000453 0.0000000 -0.0000001 0.0000244 0.0000000 -0.0000001

113 0 0 0 0 0 0 0 0 0 -1 0 -2 2 0 0.0000766 0.0000000 0.0000001 0.0000009 0.0000000 0.0000000

114 0 0 0 0 0 0 0 0 0 0 -1 2 2 1 -0.0000446 0.0000000 0.0000002 0.0000225 0.0000000 0.0000001

115 0 0 0 0 0 0 0 0 0 2 -1 2 0 2 -0.0000488 0.0000000 0.0000002 0.0000207 0.0000000 0.0000001

116 0 0 0 0 0 0 0 0 0 0 0 0 2 2 -0.0000468 0.0000000 0.0000000 0.0000201 0.0000000 0.0000000

117 0 0 0 0 0 0 0 0 0 1 -1 2 0 1 -0.0000421 0.0000000 0.0000001 0.0000216 0.0000000 0.0000001

118 0 0 0 0 0 0 0 0 0 -1 1 2 0 2 0.0000463 0.0000000 0.0000000 -0.0000200 0.0000000 0.0000000

119 0 0 0 0 0 0 0 0 0 0 1 0 2 0 -0.0000673 0.0000000 0.0000002 0.0000014 0.0000000 0.0000000

120 0 0 0 0 0 0 0 0 0 0 -1 -2 2 0 0.0000658 0.0000000 0.0000000 -0.0000002 0.0000000 0.0000000
90 NUTATION SERIES

Term Fundamental Argument Multipliers Mi,j ∆ψ Coefficients ∆ Coefficients

i j= 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Si S ̇

i C0

i Ci C ̇

i S

0

i

00 00 00 00 00 00

121 0 0 0 0 0 0 0 0 0 0 3 2 -2 2 -0.0000438 0.0000000 0.0000000 0.0000188 0.0000000 0.0000000

122 0 0 0 0 0 0 0 0 0 0 0 0 1 1 -0.0000390 0.0000000 0.0000000 0.0000205 0.0000000 0.0000000

123 0 0 0 0 0 0 0 0 0 -1 0 2 2 0 0.0000639 -0.0000011 -0.0000002 -0.0000019 0.0000000 0.0000000

124 0 0 0 0 0 0 0 0 0 2 1 2 0 2 0.0000412 0.0000000 -0.0000002 -0.0000176 0.0000000 -0.0000001

125 0 0 0 0 0 0 0 0 0 1 1 0 0 1 -0.0000361 0.0000000 0.0000000 0.0000189 0.0000000 0.0000000

126 0 0 0 0 0 0 0 0 0 1 1 2 0 1 0.0000360 0.0000000 -0.0000001 -0.0000185 0.0000000 -0.0000001

127 0 0 0 0 0 0 0 0 0 2 0 0 2 0 0.0000588 0.0000000 -0.0000003 -0.0000024 0.0000000 0.0000000

128 0 0 0 0 0 0 0 0 0 1 0 -2 2 0 -0.0000578 0.0000000 0.0000001 0.0000005 0.0000000 0.0000000

129 0 0 0 0 0 0 0 0 0 -1 0 0 2 2 -0.0000396 0.0000000 0.0000000 0.0000171 0.0000000 0.0000000

130 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0.0000565 0.0000000 -0.0000001 -0.0000006 0.0000000 0.0000000

131 0 0 0 0 0 0 0 0 0 0 1 0 -2 1 -0.0000335 0.0000000 -0.0000001 0.0000184 0.0000000 -0.0000001

132 0 0 0 0 0 0 0 0 0 -1 0 2 -2 2 0.0000357 0.0000000 0.0000001 -0.0000154 0.0000000 0.0000000

133 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 0.0000321 0.0000000 0.0000001 -0.0000174 0.0000000 0.0000000

134 0 0 0 0 0 0 0 0 0 -1 1 0 0 1 -0.0000301 0.0000000 -0.0000001 0.0000162 0.0000000 0.0000000

135 0 0 0 0 0 0 0 0 0 1 0 2 -1 2 -0.0000334 0.0000000 0.0000000 0.0000144 0.0000000 0.0000000

136 0 0 0 0 0 0 0 0 0 1 -1 0 2 0 0.0000493 0.0000000 -0.0000002 -0.0000015 0.0000000 0.0000000

137 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0.0000494 0.0000000 -0.0000002 -0.0000019 0.0000000 0.0000000

138 0 0 0 0 0 0 0 0 0 1 0 2 1 2 0.0000337 0.0000000 -0.0000001 -0.0000143 0.0000000 -0.0000001

139 0 0 0 0 0 0 0 0 0 0 0 2 1 1 0.0000280 0.0000000 -0.0000001 -0.0000144 0.0000000 0.0000000

140 0 0 0 0 0 0 0 0 0 1 0 0 -2 2 0.0000309 0.0000000 0.0000001 -0.0000134 0.0000000 0.0000000

141 0 0 0 0 0 0 0 0 0 -1 0 2 4 1 -0.0000263 0.0000000 0.0000002 0.0000131 0.0000000 0.0000001

142 0 0 0 0 0 0 0 0 0 1 0 -2 0 1 0.0000253 0.0000000 0.0000001 -0.0000138 0.0000000 0.0000000

143 0 0 0 0 0 0 0 0 0 1 1 2 -2 1 0.0000245 0.0000000 0.0000000 -0.0000128 0.0000000 0.0000000

144 0 0 0 0 0 0 0 0 0 0 0 2 2 0 0.0000416 0.0000000 -0.0000002 -0.0000017 0.0000000 0.0000000

145 0 0 0 0 0 0 0 0 0 -1 0 2 -1 1 -0.0000229 0.0000000 0.0000000 0.0000128 0.0000000 0.0000000

146 0 0 0 0 0 0 0 0 0 -2 0 2 2 1 0.0000231 0.0000000 0.0000000 -0.0000120 0.0000000 0.0000000

147 0 0 0 0 0 0 0 0 0 4 0 2 0 2 -0.0000259 0.0000000 0.0000002 0.0000109 0.0000000 0.0000001

148 0 0 0 0 0 0 0 0 0 2 -1 0 0 0 0.0000375 0.0000000 -0.0000001 -0.0000008 0.0000000 0.0000000

149 0 0 0 0 0 0 0 0 0 2 1 2 -2 2 0.0000252 0.0000000 0.0000000 -0.0000108 0.0000000 0.0000000

150 0 0 0 0 0 0 0 0 0 0 1 2 1 2 -0.0000245 0.0000000 0.0000001 0.0000104 0.0000000 0.0000000

151 0 0 0 0 0 0 0 0 0 1 0 4 -2 2 0.0000243 0.0000000 -0.0000001 -0.0000104 0.0000000 0.0000000

152 0 0 0 0 0 0 0 0 0 -1 -1 0 0 1 0.0000208 0.0000000 0.0000001 -0.0000112 0.0000000 0.0000000

153 0 0 0 0 0 0 0 0 0 0 1 0 2 1 0.0000199 0.0000000 0.0000000 -0.0000102 0.0000000 0.0000000

154 0 0 0 0 0 0 0 0 0 -2 0 2 4 1 -0.0000208 0.0000000 0.0000001 0.0000105 0.0000000 0.0000000

155 0 0 0 0 0 0 0 0 0 2 0 2 0 0 0.0000335 0.0000000 -0.0000002 -0.0000014 0.0000000 0.0000000

156 0 0 0 0 0 0 0 0 0 1 0 0 1 0 -0.0000325 0.0000000 0.0000001 0.0000007 0.0000000 0.0000000

157 0 0 0 0 0 0 0 0 0 -1 0 0 4 1 -0.0000187 0.0000000 0.0000000 0.0000096 0.0000000 0.0000000

158 0 0 0 0 0 0 0 0 0 -1 0 4 0 1 0.0000197 0.0000000 -0.0000001 -0.0000100 0.0000000 0.0000000

159 0 0 0 0 0 0 0 0 0 2 0 2 2 1 -0.0000192 0.0000000 0.0000002 0.0000094 0.0000000 0.0000001

160 0 0 0 0 0 0 0 0 0 0 0 2 -3 2 -0.0000188 0.0000000 0.0000000 0.0000083 0.0000000 0.0000000

161 0 0 0 0 0 0 0 0 0 -1 -2 0 2 0 0.0000276 0.0000000 0.0000000 -0.0000002 0.0000000 0.0000000

162 0 0 0 0 0 0 0 0 0 2 1 0 0 0 -0.0000286 0.0000000 0.0000001 0.0000006 0.0000000 0.0000000

163 0 0 0 0 0 0 0 0 0 0 0 4 0 2 0.0000186 0.0000000 -0.0000001 -0.0000079 0.0000000 0.0000000

164 0 0 0 0 0 0 0 0 0 0 0 0 0 3 -0.0000219 0.0000000 0.0000000 0.0000043 0.0000000 0.0000000

165 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0.0000276 0.0000000 0.0000000 0.0000002 0.0000000 0.0000000

166 0 0 0 0 0 0 0 0 0 0 0 2 -4 1 -0.0000153 0.0000000 -0.0000001 0.0000084 0.0000000 0.0000000

167 0 0 0 0 0 0 0 0 0 0 -1 0 2 1 -0.0000156 0.0000000 0.0000000 0.0000081 0.0000000 0.0000000

168 0 0 0 0 0 0 0 0 0 0 0 0 4 1 -0.0000154 0.0000000 0.0000001 0.0000078 0.0000000 0.0000000

169 0 0 0 0 0 0 0 0 0 -1 -1 2 4 2 -0.0000174 0.0000000 0.0000001 0.0000075 0.0000000 0.0000000

170 0 0 0 0 0 0 0 0 0 1 0 2 4 2 -0.0000163 0.0000000 0.0000002 0.0000069 0.0000000 0.0000001

171 0 0 0 0 0 0 0 0 0 -2 2 0 2 0 -0.0000228 0.0000000 0.0000000 0.0000001 0.0000000 0.0000000

172 0 0 0 0 0 0 0 0 0 -2 -1 2 0 1 0.0000091 0.0000000 -0.0000004 -0.0000054 0.0000000 -0.0000002

173 0 0 0 0 0 0 0 0 0 -2 0 0 2 2 0.0000175 0.0000000 0.0000000 -0.0000075 0.0000000 0.0000000

174 0 0 0 0 0 0 0 0 0 -1 -1 2 0 2 -0.0000159 0.0000000 0.0000000 0.0000069 0.0000000 0.0000000

175 0 0 0 0 0 0 0 0 0 0 0 4 -2 1 0.0000141 0.0000000 0.0000000 -0.0000072 0.0000000 0.0000000

176 0 0 0 0 0 0 0 0 0 3 0 2 -2 1 0.0000147 0.0000000 0.0000000 -0.0000075 0.0000000 0.0000000

177 0 0 0 0 0 0 0 0 0 -2 -1 0 2 1 -0.0000132 0.0000000 0.0000000 0.0000069 0.0000000 0.0000000

178 0 0 0 0 0 0 0 0 0 1 0 0 -1 1 0.0000159 0.0000000 -0.0000028 -0.0000054 0.0000000 0.0000011

179 0 0 0 0 0 0 0 0 0 0 -2 0 2 0 0.0000213 0.0000000 0.0000000 -0.0000004 0.0000000 0.0000000

180 0 0 0 0 0 0 0 0 0 -2 0 0 4 1 0.0000123 0.0000000 0.0000000 -0.0000064 0.0000000 0.0000000

181 0 0 0 0 0 0 0 0 0 -3 0 0 0 1 -0.0000118 0.0000000 -0.0000001 0.0000066 0.0000000 0.0000000

182 0 0 0 0 0 0 0 0 0 1 1 2 2 2 0.0000144 0.0000000 -0.0000001 -0.0000061 0.0000000 0.0000000

183 0 0 0 0 0 0 0 0 0 0 0 2 4 1 -0.0000121 0.0000000 0.0000001 0.0000060 0.0000000 0.0000000

184 0 0 0 0 0 0 0 0 0 3 0 2 2 2 -0.0000134 0.0000000 0.0000001 0.0000056 0.0000000 0.0000001

185 0 0 0 0 0 0 0 0 0 -1 1 2 -2 1 -0.0000105 0.0000000 0.0000000 0.0000057 0.0000000 0.0000000

186 0 0 0 0 0 0 0 0 0 2 0 0 -4 1 -0.0000102 0.0000000 0.0000000 0.0000056 0.0000000 0.0000000

187 0 0 0 0 0 0 0 0 0 0 0 0 -2 2 0.0000120 0.0000000 0.0000000 -0.0000052 0.0000000 0.0000000

188 0 0 0 0 0 0 0 0 0 2 0 2 -4 1 0.0000101 0.0000000 0.0000000 -0.0000054 0.0000000 0.0000000

189 0 0 0 0 0 0 0 0 0 -1 1 0 2 1 -0.0000113 0.0000000 0.0000000 0.0000059 0.0000000 0.0000000

190 0 0 0 0 0 0 0 0 0 0 0 2 -1 1 -0.0000106 0.0000000 0.0000000 0.0000061 0.0000000 0.0000000

191 0 0 0 0 0 0 0 0 0 0 -2 2 2 2 -0.0000129 0.0000000 0.0000001 0.0000055 0.0000000 0.0000000

192 0 0 0 0 0 0 0 0 0 2 0 0 2 1 -0.0000114 0.0000000 0.0000000 0.0000057 0.0000000 0.0000000

193 0 0 0 0 0 0 0 0 0 4 0 2 -2 2 0.0000113 0.0000000 -0.0000001 -0.0000049 0.0000000 0.0000000

194 0 0 0 0 0 0 0 0 0 2 0 0 -2 2 -0.0000102 0.0000000 0.0000000 0.0000044 0.0000000 0.0000000

195 0 0 0 0 0 0 0 0 0 0 2 0 0 1 -0.0000094 0.0000000 0.0000000 0.0000051 0.0000000 0.0000000

196 0 0 0 0 0 0 0 0 0 1 0 0 -4 1 -0.0000100 0.0000000 -0.0000001 0.0000056 0.0000000 0.0000000

197 0 0 0 0 0 0 0 0 0 0 2 2 -2 1 0.0000087 0.0000000 0.0000000 -0.0000047 0.0000000 0.0000000

198 0 0 0 0 0 0 0 0 0 -3 0 0 4 0 0.0000161 0.0000000 0.0000000 -0.0000001 0.0000000 0.0000000

199 0 0 0 0 0 0 0 0 0 -1 1 2 0 1 0.0000096 0.0000000 0.0000000 -0.0000050 0.0000000 0.0000000

200 0 0 0 0 0 0 0 0 0 -1 -1 0 4 0 0.0000151 0.0000000 -0.0000001 -0.0000005 0.0000000 0.0000000

201 0 0 0 0 0 0 0 0 0 -1 -2 2 2 2 -0.0000104 0.0000000 0.0000000 0.0000044 0.0000000 0.0000000

202 0 0 0 0 0 0 0 0 0 -2 -1 2 4 2 -0.0000110 0.0000000 0.0000000 0.0000048 0.0000000 0.0000000

203 0 0 0 0 0 0 0 0 0 1 -1 2 2 1 -0.0000100 0.0000000 0.0000001 0.0000050 0.0000000 0.0000000

204 0 0 0 0 0 0 0 0 0 -2 1 0 2 0 0.0000092 0.0000000 -0.0000005 0.0000012 0.0000000 -0.0000002

205 0 0 0 0 0 0 0 0 0 -2 1 2 0 1 0.0000082 0.0000000 0.0000000 -0.0000045 0.0000000 0.0000000

206 0 0 0 0 0 0 0 0 0 2 1 0 -2 1 0.0000082 0.0000000 0.0000000 -0.0000045 0.0000000 0.0000000

207 0 0 0 0 0 0 0 0 0 -3 0 2 0 1 -0.0000078 0.0000000 0.0000000 0.0000041 0.0000000 0.0000000

208 0 0 0 0 0 0 0 0 0 -2 0 2 -2 1 -0.0000077 0.0000000 0.0000000 0.0000043 0.0000000 0.0000000

209 0 0 0 0 0 0 0 0 0 -1 1 0 2 2 0.0000002 0.0000000 0.0000000 0.0000054 0.0000000 0.0000000

210 0 0 0 0 0 0 0 0 0 0 -1 2 -1 2 0.0000094 0.0000000 0.0000000 -0.0000040 0.0000000 0.0000000
NUTATION SERIES 91

Term Fundamental Argument Multipliers Mi,j ∆ψ Coefficients ∆ Coefficients

i j= 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Si S ̇

i C0

i Ci C ̇

i S

0

i

00 00 00 00 00 00

211 0 0 0 0 0 0 0 0 0 -1 0 4 -2 2 -0.0000093 0.0000000 0.0000000 0.0000040 0.0000000 0.0000000

212 0 0 0 0 0 0 0 0 0 0 -2 2 0 2 -0.0000083 0.0000000 0.0000010 0.0000040 0.0000000 -0.0000002

213 0 0 0 0 0 0 0 0 0 -1 0 2 1 2 0.0000083 0.0000000 0.0000000 -0.0000036 0.0000000 0.0000000

214 0 0 0 0 0 0 0 0 0 2 0 0 0 2 -0.0000091 0.0000000 0.0000000 0.0000039 0.0000000 0.0000000

215 0 0 0 0 0 0 0 0 0 0 0 2 0 3 0.0000128 0.0000000 0.0000000 -0.0000001 0.0000000 0.0000000

216 0 0 0 0 0 0 0 0 0 -2 0 4 0 2 -0.0000079 0.0000000 0.0000000 0.0000034 0.0000000 0.0000000

217 0 0 0 0 0 0 0 0 0 -1 0 -2 0 1 -0.0000083 0.0000000 0.0000000 0.0000047 0.0000000 0.0000000

218 0 0 0 0 0 0 0 0 0 -1 1 2 2 1 0.0000084 0.0000000 0.0000000 -0.0000044 0.0000000 0.0000000

219 0 0 0 0 0 0 0 0 0 3 0 0 0 1 0.0000083 0.0000000 0.0000000 -0.0000043 0.0000000 0.0000000

220 0 0 0 0 0 0 0 0 0 -1 0 2 3 2 0.0000091 0.0000000 0.0000000 -0.0000039 0.0000000 0.0000000

221 0 0 0 0 0 0 0 0 0 2 -1 2 0 1 -0.0000077 0.0000000 0.0000000 0.0000039 0.0000000 0.0000000

222 0 0 0 0 0 0 0 0 0 0 1 2 2 1 0.0000084 0.0000000 0.0000000 -0.0000043 0.0000000 0.0000000

223 0 0 0 0 0 0 0 0 0 0 -1 2 4 2 -0.0000092 0.0000000 0.0000001 0.0000039 0.0000000 0.0000000

224 0 0 0 0 0 0 0 0 0 2 -1 2 2 2 -0.0000092 0.0000000 0.0000001 0.0000039 0.0000000 0.0000000

225 0 0 0 0 0 0 0 0 0 0 2 -2 2 0 -0.0000094 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

226 0 0 0 0 0 0 0 0 0 -1 -1 2 -1 1 0.0000068 0.0000000 0.0000000 -0.0000036 0.0000000 0.0000000

227 0 0 0 0 0 0 0 0 0 0 -2 0 0 1 -0.0000061 0.0000000 0.0000000 0.0000032 0.0000000 0.0000000

228 0 0 0 0 0 0 0 0 0 1 0 2 -4 2 0.0000071 0.0000000 0.0000000 -0.0000031 0.0000000 0.0000000

229 0 0 0 0 0 0 0 0 0 1 -1 0 -2 1 0.0000062 0.0000000 0.0000000 -0.0000034 0.0000000 0.0000000

230 0 0 0 0 0 0 0 0 0 -1 -1 2 0 1 -0.0000063 0.0000000 0.0000000 0.0000033 0.0000000 0.0000000

231 0 0 0 0 0 0 0 0 0 1 -1 2 -2 2 -0.0000073 0.0000000 0.0000000 0.0000032 0.0000000 0.0000000

232 0 0 0 0 0 0 0 0 0 -2 -1 0 4 0 0.0000115 0.0000000 0.0000000 -0.0000002 0.0000000 0.0000000

233 0 0 0 0 0 0 0 0 0 -1 0 0 3 0 -0.0000103 0.0000000 0.0000000 0.0000002 0.0000000 0.0000000

234 0 0 0 0 0 0 0 0 0 -2 -1 2 2 2 0.0000063 0.0000000 0.0000000 -0.0000028 0.0000000 0.0000000

235 0 0 0 0 0 0 0 0 0 0 2 2 0 2 0.0000074 0.0000000 0.0000000 -0.0000032 0.0000000 0.0000000

236 0 0 0 0 0 0 0 0 0 1 1 0 2 0 -0.0000103 0.0000000 -0.0000003 0.0000003 0.0000000 -0.0000001

237 0 0 0 0 0 0 0 0 0 2 0 2 -1 2 -0.0000069 0.0000000 0.0000000 0.0000030 0.0000000 0.0000000

238 0 0 0 0 0 0 0 0 0 1 0 2 1 1 0.0000057 0.0000000 0.0000000 -0.0000029 0.0000000 0.0000000

239 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0.0000094 0.0000000 0.0000000 -0.0000004 0.0000000 0.0000000

240 0 0 0 0 0 0 0 0 0 2 1 2 0 1 0.0000064 0.0000000 0.0000000 -0.0000033 0.0000000 0.0000000

241 0 0 0 0 0 0 0 0 0 3 -1 2 0 2 -0.0000063 0.0000000 0.0000000 0.0000026 0.0000000 0.0000000

242 0 0 0 0 0 0 0 0 0 -2 2 0 2 1 -0.0000038 0.0000000 0.0000000 0.0000020 0.0000000 0.0000000

243 0 0 0 0 0 0 0 0 0 1 0 2 -3 1 -0.0000043 0.0000000 0.0000000 0.0000024 0.0000000 0.0000000

244 0 0 0 0 0 0 0 0 0 1 1 2 -4 1 -0.0000045 0.0000000 0.0000000 0.0000023 0.0000000 0.0000000

245 0 0 0 0 0 0 0 0 0 -1 -1 2 -2 1 0.0000047 0.0000000 0.0000000 -0.0000024 0.0000000 0.0000000

246 0 0 0 0 0 0 0 0 0 0 -1 0 -1 1 -0.0000048 0.0000000 0.0000000 0.0000025 0.0000000 0.0000000

247 0 0 0 0 0 0 0 0 0 0 -1 0 -2 1 0.0000045 0.0000000 0.0000000 -0.0000026 0.0000000 0.0000000

248 0 0 0 0 0 0 0 0 0 -2 0 0 0 2 0.0000056 0.0000000 0.0000000 -0.0000025 0.0000000 0.0000000

249 0 0 0 0 0 0 0 0 0 -2 0 -2 2 0 0.0000088 0.0000000 0.0000000 0.0000002 0.0000000 0.0000000

250 0 0 0 0 0 0 0 0 0 -1 0 -2 4 0 -0.0000075 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

251 0 0 0 0 0 0 0 0 0 1 -2 0 0 0 0.0000085 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

252 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0.0000049 0.0000000 0.0000000 -0.0000026 0.0000000 0.0000000

253 0 0 0 0 0 0 0 0 0 -1 2 0 2 0 -0.0000074 0.0000000 -0.0000003 -0.0000001 0.0000000 -0.0000001

254 0 0 0 0 0 0 0 0 0 1 -1 2 -2 1 -0.0000039 0.0000000 0.0000000 0.0000021 0.0000000 0.0000000

255 0 0 0 0 0 0 0 0 0 1 2 2 -2 2 0.0000045 0.0000000 0.0000000 -0.0000020 0.0000000 0.0000000

256 0 0 0 0 0 0 0 0 0 2 -1 2 -2 2 0.0000051 0.0000000 0.0000000 -0.0000022 0.0000000 0.0000000

257 0 0 0 0 0 0 0 0 0 1 0 2 -1 1 -0.0000040 0.0000000 0.0000000 0.0000021 0.0000000 0.0000000

258 0 0 0 0 0 0 0 0 0 2 1 2 -2 1 0.0000041 0.0000000 0.0000000 -0.0000021 0.0000000 0.0000000

259 0 0 0 0 0 0 0 0 0 -2 0 0 -2 1 -0.0000042 0.0000000 0.0000000 0.0000024 0.0000000 0.0000000

260 0 0 0 0 0 0 0 0 0 1 -2 2 0 2 -0.0000051 0.0000000 0.0000000 0.0000022 0.0000000 0.0000000

261 0 0 0 0 0 0 0 0 0 0 1 2 1 1 -0.0000042 0.0000000 0.0000000 0.0000022 0.0000000 0.0000000

262 0 0 0 0 0 0 0 0 0 1 0 4 -2 1 0.0000039 0.0000000 0.0000000 -0.0000021 0.0000000 0.0000000

263 0 0 0 0 0 0 0 0 0 -2 0 4 2 2 0.0000046 0.0000000 0.0000000 -0.0000018 0.0000000 0.0000000

264 0 0 0 0 0 0 0 0 0 1 1 2 1 2 -0.0000053 0.0000000 0.0000000 0.0000022 0.0000000 0.0000000

265 0 0 0 0 0 0 0 0 0 1 0 0 4 0 0.0000082 0.0000000 0.0000000 -0.0000004 0.0000000 0.0000000

266 0 0 0 0 0 0 0 0 0 1 0 2 2 0 0.0000081 0.0000000 -0.0000001 -0.0000004 0.0000000 0.0000000

267 0 0 0 0 0 0 0 0 0 2 0 2 1 2 0.0000047 0.0000000 0.0000000 -0.0000019 0.0000000 0.0000000

268 0 0 0 0 0 0 0 0 0 3 1 2 0 2 0.0000053 0.0000000 0.0000000 -0.0000023 0.0000000 0.0000000

269 0 0 0 0 0 0 0 0 0 4 0 2 0 1 -0.0000045 0.0000000 0.0000000 0.0000022 0.0000000 0.0000000

270 0 0 0 0 0 0 0 0 0 -2 -1 2 0 0 -0.0000044 0.0000000 0.0000000 -0.0000002 0.0000000 0.0000000

271 0 0 0 0 0 0 0 0 0 0 1 -2 2 1 -0.0000033 0.0000000 0.0000000 0.0000016 0.0000000 0.0000000

272 0 0 0 0 0 0 0 0 0 1 0 -2 1 0 -0.0000061 0.0000000 0.0000000 0.0000001 0.0000000 0.0000000

273 0 0 0 0 0 0 0 0 0 0 -1 -2 2 1 0.0000028 0.0000000 0.0000000 -0.0000015 0.0000000 0.0000000

274 0 0 0 0 0 0 0 0 0 2 -1 0 -2 1 -0.0000038 0.0000000 0.0000000 0.0000019 0.0000000 0.0000000

275 0 0 0 0 0 0 0 0 0 -1 0 2 -1 2 -0.0000033 0.0000000 0.0000000 0.0000021 0.0000000 0.0000000

276 0 0 0 0 0 0 0 0 0 1 0 2 -3 2 -0.0000060 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

277 0 0 0 0 0 0 0 0 0 0 1 2 -2 3 0.0000048 0.0000000 0.0000000 -0.0000010 0.0000000 0.0000000

278 0 0 0 0 0 0 0 0 0 0 0 2 -3 1 0.0000027 0.0000000 0.0000000 -0.0000014 0.0000000 0.0000000

279 0 0 0 0 0 0 0 0 0 -1 0 -2 2 1 0.0000038 0.0000000 0.0000000 -0.0000020 0.0000000 0.0000000

280 0 0 0 0 0 0 0 0 0 0 0 2 -4 2 0.0000031 0.0000000 0.0000000 -0.0000013 0.0000000 0.0000000

281 0 0 0 0 0 0 0 0 0 -2 1 0 0 1 -0.0000029 0.0000000 0.0000000 0.0000015 0.0000000 0.0000000

282 0 0 0 0 0 0 0 0 0 -1 0 0 -1 1 0.0000028 0.0000000 0.0000000 -0.0000015 0.0000000 0.0000000

283 0 0 0 0 0 0 0 0 0 2 0 2 -4 2 -0.0000032 0.0000000 0.0000000 0.0000015 0.0000000 0.0000000

284 0 0 0 0 0 0 0 0 0 0 0 4 -4 4 0.0000045 0.0000000 0.0000000 -0.0000008 0.0000000 0.0000000

285 0 0 0 0 0 0 0 0 0 0 0 4 -4 2 -0.0000044 0.0000000 0.0000000 0.0000019 0.0000000 0.0000000

286 0 0 0 0 0 0 0 0 0 -1 -2 0 2 1 0.0000028 0.0000000 0.0000000 -0.0000015 0.0000000 0.0000000

287 0 0 0 0 0 0 0 0 0 -2 0 0 3 0 -0.0000051 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

288 0 0 0 0 0 0 0 0 0 1 0 -2 2 1 -0.0000036 0.0000000 0.0000000 0.0000020 0.0000000 0.0000000

289 0 0 0 0 0 0 0 0 0 -3 0 2 2 2 0.0000044 0.0000000 0.0000000 -0.0000019 0.0000000 0.0000000

290 0 0 0 0 0 0 0 0 0 -3 0 2 2 1 0.0000026 0.0000000 0.0000000 -0.0000014 0.0000000 0.0000000

291 0 0 0 0 0 0 0 0 0 -2 0 2 2 0 -0.0000060 0.0000000 0.0000000 0.0000002 0.0000000 0.0000000

292 0 0 0 0 0 0 0 0 0 2 -1 0 0 1 0.0000035 0.0000000 0.0000000 -0.0000018 0.0000000 0.0000000

293 0 0 0 0 0 0 0 0 0 -2 1 2 2 2 -0.0000027 0.0000000 0.0000000 0.0000011 0.0000000 0.0000000

294 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0.0000047 0.0000000 0.0000000 -0.0000001 0.0000000 0.0000000

295 0 0 0 0 0 0 0 0 0 0 1 4 -2 2 0.0000036 0.0000000 0.0000000 -0.0000015 0.0000000 0.0000000

296 0 0 0 0 0 0 0 0 0 -1 1 0 -2 1 -0.0000036 0.0000000 0.0000000 0.0000020 0.0000000 0.0000000

297 0 0 0 0 0 0 0 0 0 0 0 0 -4 1 -0.0000035 0.0000000 0.0000000 0.0000019 0.0000000 0.0000000

298 0 0 0 0 0 0 0 0 0 1 -1 0 2 1 -0.0000037 0.0000000 0.0000000 0.0000019 0.0000000 0.0000000

299 0 0 0 0 0 0 0 0 0 1 1 0 2 1 0.0000032 0.0000000 0.0000000 -0.0000016 0.0000000 0.0000000

300 0 0 0 0 0 0 0 0 0 -1 2 2 2 2 0.0000035 0.0000000 0.0000000 -0.0000014 0.0000000 0.0000000
92 NUTATION SERIES

Term Fundamental Argument Multipliers Mi,j ∆ψ Coefficients ∆ Coefficients

i j= 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Si S ̇

i C0

i Ci C ̇

i S

0

i

00 00 00 00 00 00

301 0 0 0 0 0 0 0 0 0 3 1 2 -2 2 0.0000032 0.0000000 0.0000000 -0.0000013 0.0000000 0.0000000

302 0 0 0 0 0 0 0 0 0 0 -1 0 4 0 0.0000065 0.0000000 0.0000000 -0.0000002 0.0000000 0.0000000

303 0 0 0 0 0 0 0 0 0 2 -1 0 2 0 0.0000047 0.0000000 0.0000000 -0.0000001 0.0000000 0.0000000

304 0 0 0 0 0 0 0 0 0 0 0 4 0 1 0.0000032 0.0000000 0.0000000 -0.0000016 0.0000000 0.0000000

305 0 0 0 0 0 0 0 0 0 2 0 4 -2 2 0.0000037 0.0000000 0.0000000 -0.0000016 0.0000000 0.0000000

306 0 0 0 0 0 0 0 0 0 -1 -1 2 4 1 -0.0000030 0.0000000 0.0000000 0.0000015 0.0000000 0.0000000

307 0 0 0 0 0 0 0 0 0 1 0 0 4 1 -0.0000032 0.0000000 0.0000000 0.0000016 0.0000000 0.0000000

308 0 0 0 0 0 0 0 0 0 1 -2 2 2 2 -0.0000031 0.0000000 0.0000000 0.0000013 0.0000000 0.0000000

309 0 0 0 0 0 0 0 0 0 0 0 2 3 2 0.0000037 0.0000000 0.0000000 -0.0000016 0.0000000 0.0000000

310 0 0 0 0 0 0 0 0 0 -1 1 2 4 2 0.0000031 0.0000000 0.0000000 -0.0000013 0.0000000 0.0000000

311 0 0 0 0 0 0 0 0 0 3 0 0 2 0 0.0000049 0.0000000 0.0000000 -0.0000002 0.0000000 0.0000000

312 0 0 0 0 0 0 0 0 0 -1 0 4 2 2 0.0000032 0.0000000 0.0000000 -0.0000013 0.0000000 0.0000000

313 0 0 0 0 0 0 0 0 0 1 1 2 2 1 0.0000023 0.0000000 0.0000000 -0.0000012 0.0000000 0.0000000

314 0 0 0 0 0 0 0 0 0 -2 0 2 6 2 -0.0000043 0.0000000 0.0000000 0.0000018 0.0000000 0.0000000

315 0 0 0 0 0 0 0 0 0 2 1 2 2 2 0.0000026 0.0000000 0.0000000 -0.0000011 0.0000000 0.0000000

316 0 0 0 0 0 0 0 0 0 -1 0 2 6 2 -0.0000032 0.0000000 0.0000000 0.0000014 0.0000000 0.0000000

317 0 0 0 0 0 0 0 0 0 1 0 2 4 1 -0.0000029 0.0000000 0.0000000 0.0000014 0.0000000 0.0000000

318 0 0 0 0 0 0 0 0 0 2 0 2 4 2 -0.0000027 0.0000000 0.0000000 0.0000012 0.0000000 0.0000000

319 0 0 0 0 0 0 0 0 0 1 1 -2 1 0 0.0000030 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

320 0 0 0 0 0 0 0 0 0 -3 1 2 1 2 -0.0000011 0.0000000 0.0000000 0.0000005 0.0000000 0.0000000

321 0 0 0 0 0 0 0 0 0 2 0 -2 0 2 -0.0000021 0.0000000 0.0000000 0.0000010 0.0000000 0.0000000

322 0 0 0 0 0 0 0 0 0 -1 0 0 1 2 -0.0000034 0.0000000 0.0000000 0.0000015 0.0000000 0.0000000

323 0 0 0 0 0 0 0 0 0 -4 0 2 2 1 -0.0000010 0.0000000 0.0000000 0.0000006 0.0000000 0.0000000

324 0 0 0 0 0 0 0 0 0 -1 -1 0 1 0 -0.0000036 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

325 0 0 0 0 0 0 0 0 0 0 0 -2 2 2 -0.0000009 0.0000000 0.0000000 0.0000004 0.0000000 0.0000000

326 0 0 0 0 0 0 0 0 0 1 0 0 -1 2 -0.0000012 0.0000000 0.0000000 0.0000005 0.0000000 0.0000000

327 0 0 0 0 0 0 0 0 0 0 -1 2 -2 3 -0.0000021 0.0000000 0.0000000 0.0000005 0.0000000 0.0000000

328 0 0 0 0 0 0 0 0 0 -2 1 2 0 0 -0.0000029 0.0000000 0.0000000 -0.0000001 0.0000000 0.0000000

329 0 0 0 0 0 0 0 0 0 0 0 2 -2 4 -0.0000015 0.0000000 0.0000000 0.0000003 0.0000000 0.0000000

330 0 0 0 0 0 0 0 0 0 -2 -2 0 2 0 -0.0000020 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

331 0 0 0 0 0 0 0 0 0 -2 0 -2 4 0 0.0000028 0.0000000 0.0000000 0.0000000 0.0000000 -0.0000002

332 0 0 0 0 0 0 0 0 0 0 -2 -2 2 0 0.0000017 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

333 0 0 0 0 0 0 0 0 0 1 2 0 -2 1 -0.0000022 0.0000000 0.0000000 0.0000012 0.0000000 0.0000000

334 0 0 0 0 0 0 0 0 0 3 0 0 -4 1 -0.0000014 0.0000000 0.0000000 0.0000007 0.0000000 0.0000000

335 0 0 0 0 0 0 0 0 0 -1 1 2 -2 2 0.0000024 0.0000000 0.0000000 -0.0000011 0.0000000 0.0000000

336 0 0 0 0 0 0 0 0 0 1 -1 2 -4 1 0.0000011 0.0000000 0.0000000 -0.0000006 0.0000000 0.0000000

337 0 0 0 0 0 0 0 0 0 1 1 0 -2 2 0.0000014 0.0000000 0.0000000 -0.0000006 0.0000000 0.0000000

338 0 0 0 0 0 0 0 0 0 -3 0 2 0 0 0.0000024 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

339 0 0 0 0 0 0 0 0 0 -3 0 2 0 2 0.0000018 0.0000000 0.0000000 -0.0000008 0.0000000 0.0000000

340 0 0 0 0 0 0 0 0 0 -2 0 0 1 0 -0.0000038 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

341 0 0 0 0 0 0 0 0 0 0 0 -2 1 0 -0.0000031 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

342 0 0 0 0 0 0 0 0 0 -3 0 0 2 1 -0.0000016 0.0000000 0.0000000 0.0000008 0.0000000 0.0000000

343 0 0 0 0 0 0 0 0 0 -1 -1 -2 2 0 0.0000029 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

344 0 0 0 0 0 0 0 0 0 0 1 2 -4 1 -0.0000018 0.0000000 0.0000000 0.0000010 0.0000000 0.0000000

345 0 0 0 0 0 0 0 0 0 2 1 0 -4 1 -0.0000010 0.0000000 0.0000000 0.0000005 0.0000000 0.0000000

346 0 0 0 0 0 0 0 0 0 0 2 0 -2 1 -0.0000017 0.0000000 0.0000000 0.0000010 0.0000000 0.0000000

347 0 0 0 0 0 0 0 0 0 1 0 0 -3 1 0.0000009 0.0000000 0.0000000 -0.0000004 0.0000000 0.0000000

348 0 0 0 0 0 0 0 0 0 -2 0 2 -2 2 0.0000016 0.0000000 0.0000000 -0.0000006 0.0000000 0.0000000

349 0 0 0 0 0 0 0 0 0 -2 -1 0 0 1 0.0000022 0.0000000 0.0000000 -0.0000012 0.0000000 0.0000000

350 0 0 0 0 0 0 0 0 0 -4 0 0 2 0 0.0000020 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

351 0 0 0 0 0 0 0 0 0 1 1 0 -4 1 -0.0000013 0.0000000 0.0000000 0.0000006 0.0000000 0.0000000

352 0 0 0 0 0 0 0 0 0 -1 0 2 -4 1 -0.0000017 0.0000000 0.0000000 0.0000009 0.0000000 0.0000000

353 0 0 0 0 0 0 0 0 0 0 0 4 -4 1 -0.0000014 0.0000000 0.0000000 0.0000008 0.0000000 0.0000000

354 0 0 0 0 0 0 0 0 0 0 3 2 -2 2 0.0000000 0.0000000 0.0000000 -0.0000007 0.0000000 0.0000000

355 0 0 0 0 0 0 0 0 0 -3 -1 0 4 0 0.0000014 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

356 0 0 0 0 0 0 0 0 0 -3 0 0 4 1 0.0000019 0.0000000 0.0000000 -0.0000010 0.0000000 0.0000000

357 0 0 0 0 0 0 0 0 0 1 -1 -2 2 0 -0.0000034 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

358 0 0 0 0 0 0 0 0 0 -1 -1 0 2 2 -0.0000020 0.0000000 0.0000000 0.0000008 0.0000000 0.0000000

359 0 0 0 0 0 0 0 0 0 1 -2 0 0 1 0.0000009 0.0000000 0.0000000 -0.0000005 0.0000000 0.0000000

360 0 0 0 0 0 0 0 0 0 1 -1 0 0 2 -0.0000018 0.0000000 0.0000000 0.0000007 0.0000000 0.0000000

361 0 0 0 0 0 0 0 0 0 0 0 0 1 2 0.0000013 0.0000000 0.0000000 -0.0000006 0.0000000 0.0000000

362 0 0 0 0 0 0 0 0 0 -1 -1 2 0 0 0.0000017 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

363 0 0 0 0 0 0 0 0 0 1 -2 2 -2 2 -0.0000012 0.0000000 0.0000000 0.0000005 0.0000000 0.0000000

364 0 0 0 0 0 0 0 0 0 0 -1 2 -1 1 0.0000015 0.0000000 0.0000000 -0.0000008 0.0000000 0.0000000

365 0 0 0 0 0 0 0 0 0 -1 0 2 0 3 -0.0000011 0.0000000 0.0000000 0.0000003 0.0000000 0.0000000

366 0 0 0 0 0 0 0 0 0 1 1 0 0 2 0.0000013 0.0000000 0.0000000 -0.0000005 0.0000000 0.0000000

367 0 0 0 0 0 0 0 0 0 -1 1 2 0 0 -0.0000018 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

368 0 0 0 0 0 0 0 0 0 1 2 0 0 0 -0.0000035 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

369 0 0 0 0 0 0 0 0 0 -1 2 2 0 2 0.0000009 0.0000000 0.0000000 -0.0000004 0.0000000 0.0000000

370 0 0 0 0 0 0 0 0 0 -1 0 4 -2 1 -0.0000019 0.0000000 0.0000000 0.0000010 0.0000000 0.0000000

371 0 0 0 0 0 0 0 0 0 3 0 2 -4 2 -0.0000026 0.0000000 0.0000000 0.0000011 0.0000000 0.0000000

372 0 0 0 0 0 0 0 0 0 1 2 2 -2 1 0.0000008 0.0000000 0.0000000 -0.0000004 0.0000000 0.0000000

373 0 0 0 0 0 0 0 0 0 1 0 4 -4 2 -0.0000010 0.0000000 0.0000000 0.0000004 0.0000000 0.0000000

374 0 0 0 0 0 0 0 0 0 -2 -1 0 4 1 0.0000010 0.0000000 0.0000000 -0.0000006 0.0000000 0.0000000

375 0 0 0 0 0 0 0 0 0 0 -1 0 2 2 -0.0000021 0.0000000 0.0000000 0.0000009 0.0000000 0.0000000

376 0 0 0 0 0 0 0 0 0 -2 1 0 4 0 -0.0000015 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

377 0 0 0 0 0 0 0 0 0 -2 -1 2 2 1 0.0000009 0.0000000 0.0000000 -0.0000005 0.0000000 0.0000000

378 0 0 0 0 0 0 0 0 0 2 0 -2 2 0 -0.0000029 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

379 0 0 0 0 0 0 0 0 0 1 0 0 1 1 -0.0000019 0.0000000 0.0000000 0.0000010 0.0000000 0.0000000

380 0 0 0 0 0 0 0 0 0 0 1 0 2 2 0.0000012 0.0000000 0.0000000 -0.0000005 0.0000000 0.0000000

381 0 0 0 0 0 0 0 0 0 1 -1 2 -1 2 0.0000022 0.0000000 0.0000000 -0.0000009 0.0000000 0.0000000

382 0 0 0 0 0 0 0 0 0 -2 0 4 0 1 -0.0000010 0.0000000 0.0000000 0.0000005 0.0000000 0.0000000

383 0 0 0 0 0 0 0 0 0 2 1 0 0 1 -0.0000020 0.0000000 0.0000000 0.0000011 0.0000000 0.0000000

384 0 0 0 0 0 0 0 0 0 0 1 2 0 0 -0.0000020 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

385 0 0 0 0 0 0 0 0 0 0 -1 4 -2 2 -0.0000017 0.0000000 0.0000000 0.0000007 0.0000000 0.0000000

386 0 0 0 0 0 0 0 0 0 0 0 4 -2 4 0.0000015 0.0000000 0.0000000 -0.0000003 0.0000000 0.0000000

387 0 0 0 0 0 0 0 0 0 0 2 2 0 1 0.0000008 0.0000000 0.0000000 -0.0000004 0.0000000 0.0000000

388 0 0 0 0 0 0 0 0 0 -3 0 0 6 0 0.0000014 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

389 0 0 0 0 0 0 0 0 0 -1 -1 0 4 1 -0.0000012 0.0000000 0.0000000 0.0000006 0.0000000 0.0000000

390 0 0 0 0 0 0 0 0 0 1 -2 0 2 0 0.0000025 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
NUTATION SERIES 93

Term Fundamental Argument Multipliers Mi,j ∆ψ Coefficients ∆ Coefficients

i j= 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Si S ̇

i C0

i Ci C ̇

i S

0

i

00 00 00 00 00 00

391 0 0 0 0 0 0 0 0 0 -1 0 0 4 2 -0.0000013 0.0000000 0.0000000 0.0000006 0.0000000 0.0000000

392 0 0 0 0 0 0 0 0 0 -1 -2 2 2 1 -0.0000014 0.0000000 0.0000000 0.0000008 0.0000000 0.0000000

393 0 0 0 0 0 0 0 0 0 -1 0 0 -2 2 0.0000013 0.0000000 0.0000000 -0.0000005 0.0000000 0.0000000

394 0 0 0 0 0 0 0 0 0 1 0 -2 -2 1 -0.0000017 0.0000000 0.0000000 0.0000009 0.0000000 0.0000000

395 0 0 0 0 0 0 0 0 0 0 0 -2 -2 1 -0.0000012 0.0000000 0.0000000 0.0000006 0.0000000 0.0000000

396 0 0 0 0 0 0 0 0 0 -2 0 -2 0 1 -0.0000010 0.0000000 0.0000000 0.0000005 0.0000000 0.0000000

397 0 0 0 0 0 0 0 0 0 0 0 0 3 1 0.0000010 0.0000000 0.0000000 -0.0000006 0.0000000 0.0000000

398 0 0 0 0 0 0 0 0 0 0 0 0 3 0 -0.0000015 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

399 0 0 0 0 0 0 0 0 0 -1 1 0 4 0 -0.0000022 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

400 0 0 0 0 0 0 0 0 0 -1 -1 2 2 0 0.0000028 0.0000000 0.0000000 -0.0000001 0.0000000 0.0000000

401 0 0 0 0 0 0 0 0 0 -2 0 2 3 2 0.0000015 0.0000000 0.0000000 -0.0000007 0.0000000 0.0000000

402 0 0 0 0 0 0 0 0 0 1 0 0 2 2 0.0000023 0.0000000 0.0000000 -0.0000010 0.0000000 0.0000000

403 0 0 0 0 0 0 0 0 0 0 -1 2 1 2 0.0000012 0.0000000 0.0000000 -0.0000005 0.0000000 0.0000000

404 0 0 0 0 0 0 0 0 0 3 -1 0 0 0 0.0000029 0.0000000 0.0000000 -0.0000001 0.0000000 0.0000000

405 0 0 0 0 0 0 0 0 0 2 0 0 1 0 -0.0000025 0.0000000 0.0000000 0.0000001 0.0000000 0.0000000

406 0 0 0 0 0 0 0 0 0 1 -1 2 0 0 0.0000022 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

407 0 0 0 0 0 0 0 0 0 0 0 2 1 0 -0.0000018 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

408 0 0 0 0 0 0 0 0 0 1 0 2 0 3 0.0000015 0.0000000 0.0000000 0.0000003 0.0000000 0.0000000

409 0 0 0 0 0 0 0 0 0 3 1 0 0 0 -0.0000023 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

410 0 0 0 0 0 0 0 0 0 3 -1 2 -2 2 0.0000012 0.0000000 0.0000000 -0.0000005 0.0000000 0.0000000

411 0 0 0 0 0 0 0 0 0 2 0 2 -1 1 -0.0000008 0.0000000 0.0000000 0.0000004 0.0000000 0.0000000

412 0 0 0 0 0 0 0 0 0 1 1 2 0 0 -0.0000019 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

413 0 0 0 0 0 0 0 0 0 0 0 4 -1 2 -0.0000010 0.0000000 0.0000000 0.0000004 0.0000000 0.0000000

414 0 0 0 0 0 0 0 0 0 1 2 2 0 2 0.0000021 0.0000000 0.0000000 -0.0000009 0.0000000 0.0000000

415 0 0 0 0 0 0 0 0 0 -2 0 0 6 0 0.0000023 0.0000000 0.0000000 -0.0000001 0.0000000 0.0000000

416 0 0 0 0 0 0 0 0 0 0 -1 0 4 1 -0.0000016 0.0000000 0.0000000 0.0000008 0.0000000 0.0000000

417 0 0 0 0 0 0 0 0 0 -2 -1 2 4 1 -0.0000019 0.0000000 0.0000000 0.0000009 0.0000000 0.0000000

418 0 0 0 0 0 0 0 0 0 0 -2 2 2 1 -0.0000022 0.0000000 0.0000000 0.0000010 0.0000000 0.0000000

419 0 0 0 0 0 0 0 0 0 0 -1 2 2 0 0.0000027 0.0000000 0.0000000 -0.0000001 0.0000000 0.0000000

420 0 0 0 0 0 0 0 0 0 -1 0 2 3 1 0.0000016 0.0000000 0.0000000 -0.0000008 0.0000000 0.0000000

421 0 0 0 0 0 0 0 0 0 -2 1 2 4 2 0.0000019 0.0000000 0.0000000 -0.0000008 0.0000000 0.0000000

422 0 0 0 0 0 0 0 0 0 2 0 0 2 2 0.0000009 0.0000000 0.0000000 -0.0000004 0.0000000 0.0000000

423 0 0 0 0 0 0 0 0 0 2 -2 2 0 2 -0.0000009 0.0000000 0.0000000 0.0000004 0.0000000 0.0000000

424 0 0 0 0 0 0 0 0 0 -1 1 2 3 2 -0.0000009 0.0000000 0.0000000 0.0000004 0.0000000 0.0000000

425 0 0 0 0 0 0 0 0 0 3 0 2 -1 2 -0.0000008 0.0000000 0.0000000 0.0000004 0.0000000 0.0000000

426 0 0 0 0 0 0 0 0 0 4 0 2 -2 1 0.0000018 0.0000000 0.0000000 -0.0000009 0.0000000 0.0000000

427 0 0 0 0 0 0 0 0 0 -1 0 0 6 0 0.0000016 0.0000000 0.0000000 -0.0000001 0.0000000 0.0000000

428 0 0 0 0 0 0 0 0 0 -1 -2 2 4 2 -0.0000010 0.0000000 0.0000000 0.0000004 0.0000000 0.0000000

429 0 0 0 0 0 0 0 0 0 -3 0 2 6 2 -0.0000023 0.0000000 0.0000000 0.0000009 0.0000000 0.0000000

430 0 0 0 0 0 0 0 0 0 -1 0 2 4 0 0.0000016 0.0000000 0.0000000 -0.0000001 0.0000000 0.0000000

431 0 0 0 0 0 0 0 0 0 3 0 0 2 1 -0.0000012 0.0000000 0.0000000 0.0000006 0.0000000 0.0000000

432 0 0 0 0 0 0 0 0 0 3 -1 2 0 1 -0.0000008 0.0000000 0.0000000 0.0000004 0.0000000 0.0000000

433 0 0 0 0 0 0 0 0 0 3 0 2 0 0 0.0000030 0.0000000 0.0000000 -0.0000002 0.0000000 0.0000000

434 0 0 0 0 0 0 0 0 0 1 0 4 0 2 0.0000024 0.0000000 0.0000000 -0.0000010 0.0000000 0.0000000

435 0 0 0 0 0 0 0 0 0 5 0 2 -2 2 0.0000010 0.0000000 0.0000000 -0.0000004 0.0000000 0.0000000

436 0 0 0 0 0 0 0 0 0 0 -1 2 4 1 -0.0000016 0.0000000 0.0000000 0.0000007 0.0000000 0.0000000

437 0 0 0 0 0 0 0 0 0 2 -1 2 2 1 -0.0000016 0.0000000 0.0000000 0.0000007 0.0000000 0.0000000

438 0 0 0 0 0 0 0 0 0 0 1 2 4 2 0.0000017 0.0000000 0.0000000 -0.0000007 0.0000000 0.0000000

439 0 0 0 0 0 0 0 0 0 1 -1 2 4 2 -0.0000024 0.0000000 0.0000000 0.0000010 0.0000000 0.0000000

440 0 0 0 0 0 0 0 0 0 3 -1 2 2 2 -0.0000012 0.0000000 0.0000000 0.0000005 0.0000000 0.0000000

441 0 0 0 0 0 0 0 0 0 3 0 2 2 1 -0.0000024 0.0000000 0.0000000 0.0000011 0.0000000 0.0000000

442 0 0 0 0 0 0 0 0 0 5 0 2 0 2 -0.0000023 0.0000000 0.0000000 0.0000009 0.0000000 0.0000000

443 0 0 0 0 0 0 0 0 0 0 0 2 6 2 -0.0000013 0.0000000 0.0000000 0.0000005 0.0000000 0.0000000

444 0 0 0 0 0 0 0 0 0 4 0 2 2 2 -0.0000015 0.0000000 0.0000000 0.0000007 0.0000000 0.0000000

445 0 0 0 0 0 0 0 0 0 0 -1 1 -1 1 0.0000000 0.0000000 -0.0001988 0.0000000 0.0000000 -0.0001679

446 0 0 0 0 0 0 0 0 0 -1 0 1 0 3 0.0000000 0.0000000 -0.0000063 0.0000000 0.0000000 -0.0000027

447 0 0 0 0 0 0 0 0 0 0 -2 2 -2 3 -0.0000004 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

448 0 0 0 0 0 0 0 0 0 1 0 -1 0 1 0.0000000 0.0000000 0.0000005 0.0000000 0.0000000 0.0000004

449 0 0 0 0 0 0 0 0 0 2 -2 0 -2 1 0.0000005 0.0000000 0.0000000 -0.0000003 0.0000000 0.0000000

450 0 0 0 0 0 0 0 0 0 -1 0 1 0 2 0.0000000 0.0000000 0.0000364 0.0000000 0.0000000 0.0000176

451 0 0 0 0 0 0 0 0 0 -1 0 1 0 1 0.0000000 0.0000000 -0.0001044 0.0000000 0.0000000 -0.0000891

452 0 0 0 0 0 0 0 0 0 -1 -1 2 -1 2 -0.0000003 0.0000000 0.0000000 0.0000001 0.0000000 0.0000000

453 0 0 0 0 0 0 0 0 0 -2 2 0 2 2 0.0000004 0.0000000 0.0000000 -0.0000002 0.0000000 0.0000000

454 0 0 0 0 0 0 0 0 0 -1 0 1 0 0 0.0000000 0.0000000 0.0000330 0.0000000 0.0000000 0.0000000

455 0 0 0 0 0 0 0 0 0 -4 1 2 2 2 0.0000005 0.0000000 0.0000000 -0.0000002 0.0000000 0.0000000

456 0 0 0 0 0 0 0 0 0 -3 0 2 1 1 0.0000003 0.0000000 0.0000000 -0.0000002 0.0000000 0.0000000

457 0 0 0 0 0 0 0 0 0 -2 -1 2 0 2 -0.0000003 0.0000000 0.0000000 0.0000001 0.0000000 0.0000000

458 0 0 0 0 0 0 0 0 0 1 0 -2 1 1 -0.0000005 0.0000000 0.0000000 0.0000002 0.0000000 0.0000000

459 0 0 0 0 0 0 0 0 0 2 -1 -2 0 1 0.0000003 0.0000000 0.0000000 -0.0000001 0.0000000 0.0000000

460 0 0 0 0 0 0 0 0 0 -4 0 2 2 0 0.0000003 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

461 0 0 0 0 0 0 0 0 0 -3 1 0 3 0 0.0000003 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

462 0 0 0 0 0 0 0 0 0 -1 0 -1 2 0 0.0000000 0.0000000 0.0000005 0.0000000 0.0000000 0.0000000

463 0 0 0 0 0 0 0 0 0 0 -2 0 0 2 0.0000000 0.0000000 0.0000000 0.0000001 0.0000000 0.0000000

464 0 0 0 0 0 0 0 0 0 0 -2 0 0 2 0.0000004 0.0000000 0.0000000 -0.0000002 0.0000000 0.0000000

465 0 0 0 0 0 0 0 0 0 -3 0 0 3 0 0.0000006 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

466 0 0 0 0 0 0 0 0 0 -2 -1 0 2 2 0.0000005 0.0000000 0.0000000 -0.0000002 0.0000000 0.0000000

467 0 0 0 0 0 0 0 0 0 -1 0 -2 3 0 -0.0000007 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

468 0 0 0 0 0 0 0 0 0 -4 0 0 4 0 -0.0000012 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

469 0 0 0 0 0 0 0 0 0 2 1 -2 0 1 0.0000005 0.0000000 0.0000000 -0.0000003 0.0000000 0.0000000

470 0 0 0 0 0 0 0 0 0 2 -1 0 -2 2 0.0000003 0.0000000 0.0000000 -0.0000001 0.0000000 0.0000000

471 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 -0.0000005 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

472 0 0 0 0 0 0 0 0 0 -1 2 0 1 0 0.0000003 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

473 0 0 0 0 0 0 0 0 0 -2 1 2 0 2 -0.0000007 0.0000000 0.0000000 0.0000003 0.0000000 0.0000000

474 0 0 0 0 0 0 0 0 0 1 1 0 -1 1 0.0000007 0.0000000 0.0000000 -0.0000004 0.0000000 0.0000000

475 0 0 0 0 0 0 0 0 0 1 0 1 -2 1 0.0000000 0.0000000 -0.0000012 0.0000000 0.0000000 -0.0000010

476 0 0 0 0 0 0 0 0 0 0 2 0 0 2 0.0000004 0.0000000 0.0000000 -0.0000002 0.0000000 0.0000000

477 0 0 0 0 0 0 0 0 0 1 -1 2 -3 1 0.0000003 0.0000000 0.0000000 -0.0000002 0.0000000 0.0000000

478 0 0 0 0 0 0 0 0 0 -1 1 2 -1 1 -0.0000003 0.0000000 0.0000000 0.0000002 0.0000000 0.0000000

479 0 0 0 0 0 0 0 0 0 -2 0 4 -2 2 -0.0000007 0.0000000 0.0000000 0.0000003 0.0000000 0.0000000

480 0 0 0 0 0 0 0 0 0 -2 0 4 -2 1 -0.0000004 0.0000000 0.0000000 0.0000002 0.0000000 0.0000000
94 NUTATION SERIES

Term Fundamental Argument Multipliers Mi,j ∆ψ Coefficients ∆ Coefficients

i j= 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Si S ̇

i C0

i Ci C ̇

i S

0

i

00 00 00 00 00 00

481 0 0 0 0 0 0 0 0 0 -2 -2 0 2 1 -0.0000003 0.0000000 0.0000000 0.0000001 0.0000000 0.0000000

482 0 0 0 0 0 0 0 0 0 -2 0 -2 4 0 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

483 0 0 0 0 0 0 0 0 0 1 2 2 -4 1 -0.0000003 0.0000000 0.0000000 0.0000001 0.0000000 0.0000000

484 0 0 0 0 0 0 0 0 0 1 1 2 -4 2 0.0000007 0.0000000 0.0000000 -0.0000003 0.0000000 0.0000000

485 0 0 0 0 0 0 0 0 0 -1 2 2 -2 1 -0.0000004 0.0000000 0.0000000 0.0000002 0.0000000 0.0000000

486 0 0 0 0 0 0 0 0 0 2 0 0 -3 1 0.0000004 0.0000000 0.0000000 -0.0000002 0.0000000 0.0000000

487 0 0 0 0 0 0 0 0 0 -1 2 0 0 1 -0.0000005 0.0000000 0.0000000 0.0000003 0.0000000 0.0000000

488 0 0 0 0 0 0 0 0 0 0 0 0 -2 0 0.0000005 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

489 0 0 0 0 0 0 0 0 0 -1 -1 2 -2 2 -0.0000005 0.0000000 0.0000000 0.0000002 0.0000000 0.0000000

490 0 0 0 0 0 0 0 0 0 -1 1 0 0 2 0.0000005 0.0000000 0.0000000 -0.0000002 0.0000000 0.0000000

491 0 0 0 0 0 0 0 0 0 0 0 0 -1 2 -0.0000008 0.0000000 0.0000000 0.0000003 0.0000000 0.0000000

492 0 0 0 0 0 0 0 0 0 -2 1 0 1 0 0.0000009 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

493 0 0 0 0 0 0 0 0 0 1 -2 0 -2 1 0.0000006 0.0000000 0.0000000 -0.0000003 0.0000000 0.0000000

494 0 0 0 0 0 0 0 0 0 1 0 -2 0 2 -0.0000005 0.0000000 0.0000000 0.0000002 0.0000000 0.0000000

495 0 0 0 0 0 0 0 0 0 -3 1 0 2 0 0.0000003 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

496 0 0 0 0 0 0 0 0 0 -1 1 -2 2 0 -0.0000007 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

497 0 0 0 0 0 0 0 0 0 -1 -1 0 0 2 -0.0000003 0.0000000 0.0000000 0.0000001 0.0000000 0.0000000

498 0 0 0 0 0 0 0 0 0 -3 0 0 2 0 0.0000005 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

499 0 0 0 0 0 0 0 0 0 -3 -1 0 2 0 0.0000003 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

500 0 0 0 0 0 0 0 0 0 2 0 2 -6 1 -0.0000003 0.0000000 0.0000000 0.0000002 0.0000000 0.0000000

501 0 0 0 0 0 0 0 0 0 0 1 2 -4 2 0.0000004 0.0000000 0.0000000 -0.0000002 0.0000000 0.0000000

502 0 0 0 0 0 0 0 0 0 2 0 0 -4 2 0.0000003 0.0000000 0.0000000 -0.0000001 0.0000000 0.0000000

503 0 0 0 0 0 0 0 0 0 -2 1 2 -2 1 -0.0000005 0.0000000 0.0000000 0.0000002 0.0000000 0.0000000

504 0 0 0 0 0 0 0 0 0 0 -1 2 -4 1 0.0000004 0.0000000 0.0000000 -0.0000002 0.0000000 0.0000000

505 0 0 0 0 0 0 0 0 0 0 1 0 -2 2 0.0000009 0.0000000 0.0000000 -0.0000003 0.0000000 0.0000000

506 0 0 0 0 0 0 0 0 0 -1 0 0 -2 0 0.0000004 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

507 0 0 0 0 0 0 0 0 0 2 0 -2 -2 1 0.0000004 0.0000000 0.0000000 -0.0000002 0.0000000 0.0000000

508 0 0 0 0 0 0 0 0 0 -4 0 2 0 1 -0.0000003 0.0000000 0.0000000 0.0000002 0.0000000 0.0000000

509 0 0 0 0 0 0 0 0 0 -1 -1 0 -1 1 -0.0000004 0.0000000 0.0000000 0.0000002 0.0000000 0.0000000

510 0 0 0 0 0 0 0 0 0 0 0 -2 0 2 0.0000009 0.0000000 0.0000000 -0.0000003 0.0000000 0.0000000

511 0 0 0 0 0 0 0 0 0 -3 0 0 1 0 -0.0000004 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

512 0 0 0 0 0 0 0 0 0 -1 0 -2 1 0 -0.0000004 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

513 0 0 0 0 0 0 0 0 0 -2 0 -2 2 1 0.0000003 0.0000000 0.0000000 -0.0000002 0.0000000 0.0000000

514 0 0 0 0 0 0 0 0 0 0 0 -4 2 0 0.0000008 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

515 0 0 0 0 0 0 0 0 0 -2 -1 -2 2 0 0.0000003 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

516 0 0 0 0 0 0 0 0 0 1 0 2 -6 1 -0.0000003 0.0000000 0.0000000 0.0000002 0.0000000 0.0000000

517 0 0 0 0 0 0 0 0 0 -1 0 2 -4 2 0.0000003 0.0000000 0.0000000 -0.0000001 0.0000000 0.0000000

518 0 0 0 0 0 0 0 0 0 1 0 0 -4 2 0.0000003 0.0000000 0.0000000 -0.0000001 0.0000000 0.0000000

519 0 0 0 0 0 0 0 0 0 2 1 2 -4 2 -0.0000003 0.0000000 0.0000000 0.0000001 0.0000000 0.0000000

520 0 0 0 0 0 0 0 0 0 2 1 2 -4 1 0.0000006 0.0000000 0.0000000 -0.0000003 0.0000000 0.0000000

521 0 0 0 0 0 0 0 0 0 0 1 4 -4 4 0.0000003 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

522 0 0 0 0 0 0 0 0 0 0 1 4 -4 2 -0.0000003 0.0000000 0.0000000 0.0000001 0.0000000 0.0000000

523 0 0 0 0 0 0 0 0 0 -1 -1 -2 4 0 -0.0000007 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

524 0 0 0 0 0 0 0 0 0 -1 -3 0 2 0 0.0000009 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

525 0 0 0 0 0 0 0 0 0 -1 0 -2 4 1 -0.0000003 0.0000000 0.0000000 0.0000002 0.0000000 0.0000000

526 0 0 0 0 0 0 0 0 0 -2 -1 0 3 0 -0.0000003 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

527 0 0 0 0 0 0 0 0 0 0 0 -2 3 0 -0.0000004 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

528 0 0 0 0 0 0 0 0 0 -2 0 0 3 1 -0.0000005 0.0000000 0.0000000 0.0000003 0.0000000 0.0000000

529 0 0 0 0 0 0 0 0 0 0 -1 0 1 0 -0.0000013 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

530 0 0 0 0 0 0 0 0 0 -3 0 2 2 0 -0.0000007 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

531 0 0 0 0 0 0 0 0 0 1 1 -2 2 0 0.0000010 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

532 0 0 0 0 0 0 0 0 0 -1 1 0 2 2 0.0000003 0.0000000 0.0000000 -0.0000001 0.0000000 0.0000000

533 0 0 0 0 0 0 0 0 0 1 -2 2 -2 1 0.0000010 0.0000000 0.0000013 0.0000006 0.0000000 -0.0000005

534 0 0 0 0 0 0 0 0 0 0 0 1 0 2 0.0000000 0.0000000 0.0000030 0.0000000 0.0000000 0.0000014

535 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0.0000000 0.0000000 -0.0000162 0.0000000 0.0000000 -0.0000138

536 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0.0000000 0.0000000 0.0000075 0.0000000 0.0000000 0.0000000

537 0 0 0 0 0 0 0 0 0 -1 2 0 2 1 -0.0000007 0.0000000 0.0000000 0.0000004 0.0000000 0.0000000

538 0 0 0 0 0 0 0 0 0 0 0 2 0 2 -0.0000004 0.0000000 0.0000000 0.0000002 0.0000000 0.0000000

539 0 0 0 0 0 0 0 0 0 -2 0 2 0 2 0.0000004 0.0000000 0.0000000 -0.0000002 0.0000000 0.0000000

540 0 0 0 0 0 0 0 0 0 2 0 0 -1 1 0.0000005 0.0000000 0.0000000 -0.0000002 0.0000000 0.0000000

541 0 0 0 0 0 0 0 0 0 3 0 0 -2 1 0.0000005 0.0000000 0.0000000 -0.0000003 0.0000000 0.0000000

542 0 0 0 0 0 0 0 0 0 1 0 2 -2 3 -0.0000003 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

543 0 0 0 0 0 0 0 0 0 1 2 0 0 1 -0.0000003 0.0000000 0.0000000 0.0000002 0.0000000 0.0000000

544 0 0 0 0 0 0 0 0 0 2 0 2 -3 2 -0.0000004 0.0000000 0.0000000 0.0000002 0.0000000 0.0000000

545 0 0 0 0 0 0 0 0 0 -1 1 4 -2 2 -0.0000005 0.0000000 0.0000000 0.0000002 0.0000000 0.0000000

546 0 0 0 0 0 0 0 0 0 -2 -2 0 4 0 0.0000006 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

547 0 0 0 0 0 0 0 0 0 0 -3 0 2 0 0.0000009 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

548 0 0 0 0 0 0 0 0 0 0 0 -2 4 0 0.0000005 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

549 0 0 0 0 0 0 0 0 0 -1 -1 0 3 0 -0.0000007 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

550 0 0 0 0 0 0 0 0 0 -2 0 0 4 2 -0.0000003 0.0000000 0.0000000 0.0000001 0.0000000 0.0000000

551 0 0 0 0 0 0 0 0 0 -1 0 0 3 1 -0.0000004 0.0000000 0.0000000 0.0000002 0.0000000 0.0000000

552 0 0 0 0 0 0 0 0 0 2 -2 0 0 0 0.0000007 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

553 0 0 0 0 0 0 0 0 0 1 -1 0 1 0 -0.0000004 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

554 0 0 0 0 0 0 0 0 0 -1 0 0 2 0 0.0000004 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

555 0 0 0 0 0 0 0 0 0 0 -2 2 0 1 -0.0000006 0.0000000 -0.0000003 0.0000003 0.0000000 0.0000001

556 0 0 0 0 0 0 0 0 0 -1 0 1 2 1 0.0000000 0.0000000 -0.0000003 0.0000000 0.0000000 -0.0000002

557 0 0 0 0 0 0 0 0 0 -1 1 0 3 0 0.0000011 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

558 0 0 0 0 0 0 0 0 0 -1 -1 2 1 2 0.0000003 0.0000000 0.0000000 -0.0000001 0.0000000 0.0000000

559 0 0 0 0 0 0 0 0 0 0 -1 2 0 0 0.0000011 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

560 0 0 0 0 0 0 0 0 0 -2 1 2 2 1 -0.0000003 0.0000000 0.0000000 0.0000002 0.0000000 0.0000000

561 0 0 0 0 0 0 0 0 0 2 -2 2 -2 2 -0.0000001 0.0000000 0.0000003 0.0000003 0.0000000 -0.0000001

562 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0.0000004 0.0000000 0.0000000 -0.0000002 0.0000000 0.0000000

563 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0.0000000 0.0000000 -0.0000013 0.0000000 0.0000000 -0.0000011

564 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0.0000003 0.0000000 0.0000006 0.0000000 0.0000000 0.0000000

565 0 0 0 0 0 0 0 0 0 0 2 0 2 0 -0.0000007 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

566 0 0 0 0 0 0 0 0 0 2 -1 2 -2 1 0.0000005 0.0000000 0.0000000 -0.0000003 0.0000000 0.0000000

567 0 0 0 0 0 0 0 0 0 0 -1 4 -2 1 -0.0000003 0.0000000 0.0000000 0.0000001 0.0000000 0.0000000

568 0 0 0 0 0 0 0 0 0 0 0 4 -2 3 0.0000003 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

569 0 0 0 0 0 0 0 0 0 0 1 4 -2 1 0.0000005 0.0000000 0.0000000 -0.0000003 0.0000000 0.0000000

570 0 0 0 0 0 0 0 0 0 4 0 2 -4 2 -0.0000007 0.0000000 0.0000000 0.0000003 0.0000000 0.0000000
NUTATION SERIES 95

Term Fundamental Argument Multipliers Mi,j ∆ψ Coefficients ∆ Coefficients

i j= 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Si S ̇

i C0

i Ci C ̇

i S

0

i

00 00 00 00 00 00

571 0 0 0 0 0 0 0 0 0 2 2 2 -2 2 0.0000008 0.0000000 0.0000000 -0.0000003 0.0000000 0.0000000

572 0 0 0 0 0 0 0 0 0 2 0 4 -4 2 -0.0000004 0.0000000 0.0000000 0.0000002 0.0000000 0.0000000

573 0 0 0 0 0 0 0 0 0 -1 -2 0 4 0 0.0000011 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

574 0 0 0 0 0 0 0 0 0 -1 -3 2 2 2 -0.0000003 0.0000000 0.0000000 0.0000001 0.0000000 0.0000000

575 0 0 0 0 0 0 0 0 0 -3 0 2 4 2 0.0000003 0.0000000 0.0000000 -0.0000001 0.0000000 0.0000000

576 0 0 0 0 0 0 0 0 0 -3 0 2 -2 1 -0.0000004 0.0000000 0.0000000 0.0000002 0.0000000 0.0000000

577 0 0 0 0 0 0 0 0 0 -1 -1 0 -2 1 0.0000008 0.0000000 0.0000000 -0.0000004 0.0000000 0.0000000

578 0 0 0 0 0 0 0 0 0 -3 0 0 0 2 0.0000003 0.0000000 0.0000000 -0.0000001 0.0000000 0.0000000

579 0 0 0 0 0 0 0 0 0 -3 0 -2 2 0 0.0000011 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

580 0 0 0 0 0 0 0 0 0 0 1 0 -4 1 -0.0000006 0.0000000 0.0000000 0.0000003 0.0000000 0.0000000

581 0 0 0 0 0 0 0 0 0 -2 1 0 -2 1 -0.0000004 0.0000000 0.0000000 0.0000002 0.0000000 0.0000000

582 0 0 0 0 0 0 0 0 0 -4 0 0 0 1 -0.0000008 0.0000000 0.0000000 0.0000004 0.0000000 0.0000000

583 0 0 0 0 0 0 0 0 0 -1 0 0 -4 1 -0.0000007 0.0000000 0.0000000 0.0000003 0.0000000 0.0000000

584 0 0 0 0 0 0 0 0 0 -3 0 0 -2 1 -0.0000004 0.0000000 0.0000000 0.0000002 0.0000000 0.0000000

585 0 0 0 0 0 0 0 0 0 0 0 0 3 2 0.0000003 0.0000000 0.0000000 -0.0000001 0.0000000 0.0000000

586 0 0 0 0 0 0 0 0 0 -1 1 0 4 1 0.0000006 0.0000000 0.0000000 -0.0000003 0.0000000 0.0000000

587 0 0 0 0 0 0 0 0 0 1 -2 2 0 1 -0.0000006 0.0000000 0.0000000 0.0000003 0.0000000 0.0000000

588 0 0 0 0 0 0 0 0 0 0 1 0 3 0 0.0000006 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

589 0 0 0 0 0 0 0 0 0 -1 0 2 2 3 0.0000006 0.0000000 0.0000000 -0.0000001 0.0000000 0.0000000

590 0 0 0 0 0 0 0 0 0 0 0 2 2 2 0.0000005 0.0000000 0.0000000 -0.0000002 0.0000000 0.0000000

591 0 0 0 0 0 0 0 0 0 -2 0 2 2 2 -0.0000005 0.0000000 0.0000000 0.0000002 0.0000000 0.0000000

592 0 0 0 0 0 0 0 0 0 -1 1 2 2 0 -0.0000004 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

593 0 0 0 0 0 0 0 0 0 3 0 0 0 2 -0.0000004 0.0000000 0.0000000 0.0000002 0.0000000 0.0000000

594 0 0 0 0 0 0 0 0 0 2 1 0 1 0 0.0000004 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

595 0 0 0 0 0 0 0 0 0 2 -1 2 -1 2 0.0000006 0.0000000 0.0000000 -0.0000003 0.0000000 0.0000000

596 0 0 0 0 0 0 0 0 0 0 0 2 0 1 -0.0000004 0.0000000 0.0000000 0.0000002 0.0000000 0.0000000

597 0 0 0 0 0 0 0 0 0 0 0 3 0 3 0.0000000 0.0000000 -0.0000026 0.0000000 0.0000000 -0.0000011

598 0 0 0 0 0 0 0 0 0 0 0 3 0 2 0.0000000 0.0000000 -0.0000010 0.0000000 0.0000000 -0.0000005

599 0 0 0 0 0 0 0 0 0 -1 2 2 2 1 0.0000005 0.0000000 0.0000000 -0.0000003 0.0000000 0.0000000

600 0 0 0 0 0 0 0 0 0 -1 0 4 0 0 -0.0000013 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

601 0 0 0 0 0 0 0 0 0 1 2 2 0 1 0.0000003 0.0000000 0.0000000 -0.0000002 0.0000000 0.0000000

602 0 0 0 0 0 0 0 0 0 3 1 2 -2 1 0.0000004 0.0000000 0.0000000 -0.0000002 0.0000000 0.0000000

603 0 0 0 0 0 0 0 0 0 1 1 4 -2 2 0.0000007 0.0000000 0.0000000 -0.0000003 0.0000000 0.0000000

604 0 0 0 0 0 0 0 0 0 -2 -1 0 6 0 0.0000004 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

605 0 0 0 0 0 0 0 0 0 0 -2 0 4 0 0.0000005 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

606 0 0 0 0 0 0 0 0 0 -2 0 0 6 1 -0.0000003 0.0000000 0.0000000 0.0000002 0.0000000 0.0000000

607 0 0 0 0 0 0 0 0 0 -2 -2 2 4 2 -0.0000006 0.0000000 0.0000000 0.0000002 0.0000000 0.0000000

608 0 0 0 0 0 0 0 0 0 0 -3 2 2 2 -0.0000005 0.0000000 0.0000000 0.0000002 0.0000000 0.0000000

609 0 0 0 0 0 0 0 0 0 0 0 0 4 2 -0.0000007 0.0000000 0.0000000 0.0000003 0.0000000 0.0000000

610 0 0 0 0 0 0 0 0 0 -1 -1 2 3 2 0.0000005 0.0000000 0.0000000 -0.0000002 0.0000000 0.0000000

611 0 0 0 0 0 0 0 0 0 -2 0 2 4 0 0.0000013 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

612 0 0 0 0 0 0 0 0 0 2 -1 0 2 1 -0.0000004 0.0000000 0.0000000 0.0000002 0.0000000 0.0000000

613 0 0 0 0 0 0 0 0 0 1 0 0 3 0 -0.0000003 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

614 0 0 0 0 0 0 0 0 0 0 1 0 4 1 0.0000005 0.0000000 0.0000000 -0.0000002 0.0000000 0.0000000

615 0 0 0 0 0 0 0 0 0 0 1 0 4 0 -0.0000011 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

616 0 0 0 0 0 0 0 0 0 1 -1 2 1 2 0.0000005 0.0000000 0.0000000 -0.0000002 0.0000000 0.0000000

617 0 0 0 0 0 0 0 0 0 0 0 2 2 3 0.0000004 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

618 0 0 0 0 0 0 0 0 0 1 0 2 2 2 0.0000004 0.0000000 0.0000000 -0.0000002 0.0000000 0.0000000

619 0 0 0 0 0 0 0 0 0 -1 0 2 2 2 -0.0000004 0.0000000 0.0000000 0.0000002 0.0000000 0.0000000

620 0 0 0 0 0 0 0 0 0 -2 0 4 2 1 0.0000006 0.0000000 0.0000000 -0.0000003 0.0000000 0.0000000

621 0 0 0 0 0 0 0 0 0 2 1 0 2 1 0.0000003 0.0000000 0.0000000 -0.0000002 0.0000000 0.0000000

622 0 0 0 0 0 0 0 0 0 2 1 0 2 0 -0.0000012 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

623 0 0 0 0 0 0 0 0 0 2 -1 2 0 0 0.0000004 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

624 0 0 0 0 0 0 0 0 0 1 0 2 1 0 -0.0000003 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

625 0 0 0 0 0 0 0 0 0 0 1 2 2 0 -0.0000004 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

626 0 0 0 0 0 0 0 0 0 2 0 2 0 3 0.0000003 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

627 0 0 0 0 0 0 0 0 0 3 0 2 0 2 0.0000003 0.0000000 0.0000000 -0.0000001 0.0000000 0.0000000

628 0 0 0 0 0 0 0 0 0 1 0 2 0 2 -0.0000003 0.0000000 0.0000000 0.0000001 0.0000000 0.0000000

629 0 0 0 0 0 0 0 0 0 1 0 3 0 3 0.0000000 0.0000000 -0.0000005 0.0000000 0.0000000 -0.0000002

630 0 0 0 0 0 0 0 0 0 1 1 2 1 1 -0.0000007 0.0000000 0.0000000 0.0000004 0.0000000 0.0000000

631 0 0 0 0 0 0 0 0 0 0 2 2 2 2 0.0000006 0.0000000 0.0000000 -0.0000003 0.0000000 0.0000000

632 0 0 0 0 0 0 0 0 0 2 1 2 0 0 -0.0000003 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

633 0 0 0 0 0 0 0 0 0 2 0 4 -2 1 0.0000005 0.0000000 0.0000000 -0.0000003 0.0000000 0.0000000

634 0 0 0 0 0 0 0 0 0 4 1 2 -2 2 0.0000003 0.0000000 0.0000000 -0.0000001 0.0000000 0.0000000

635 0 0 0 0 0 0 0 0 0 -1 -1 0 6 0 0.0000003 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

636 0 0 0 0 0 0 0 0 0 -3 -1 2 6 2 -0.0000003 0.0000000 0.0000000 0.0000001 0.0000000 0.0000000

637 0 0 0 0 0 0 0 0 0 -1 0 0 6 1 -0.0000005 0.0000000 0.0000000 0.0000003 0.0000000 0.0000000

638 0 0 0 0 0 0 0 0 0 -3 0 2 6 1 -0.0000003 0.0000000 0.0000000 0.0000002 0.0000000 0.0000000

639 0 0 0 0 0 0 0 0 0 1 -1 0 4 1 -0.0000003 0.0000000 0.0000000 0.0000002 0.0000000 0.0000000

640 0 0 0 0 0 0 0 0 0 1 -1 0 4 0 0.0000012 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

641 0 0 0 0 0 0 0 0 0 -2 0 2 5 2 0.0000003 0.0000000 0.0000000 -0.0000001 0.0000000 0.0000000

642 0 0 0 0 0 0 0 0 0 1 -2 2 2 1 -0.0000004 0.0000000 0.0000000 0.0000002 0.0000000 0.0000000

643 0 0 0 0 0 0 0 0 0 3 -1 0 2 0 0.0000004 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

644 0 0 0 0 0 0 0 0 0 1 -1 2 2 0 0.0000006 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

645 0 0 0 0 0 0 0 0 0 0 0 2 3 1 0.0000005 0.0000000 0.0000000 -0.0000003 0.0000000 0.0000000

646 0 0 0 0 0 0 0 0 0 -1 1 2 4 1 0.0000004 0.0000000 0.0000000 -0.0000002 0.0000000 0.0000000

647 0 0 0 0 0 0 0 0 0 0 1 2 3 2 -0.0000006 0.0000000 0.0000000 0.0000003 0.0000000 0.0000000

648 0 0 0 0 0 0 0 0 0 -1 0 4 2 1 0.0000004 0.0000000 0.0000000 -0.0000002 0.0000000 0.0000000

649 0 0 0 0 0 0 0 0 0 2 0 2 1 1 0.0000006 0.0000000 0.0000000 -0.0000003 0.0000000 0.0000000

650 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0.0000006 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

651 0 0 0 0 0 0 0 0 0 2 1 2 1 2 -0.0000006 0.0000000 0.0000000 0.0000003 0.0000000 0.0000000

652 0 0 0 0 0 0 0 0 0 1 0 4 0 1 0.0000003 0.0000000 0.0000000 -0.0000002 0.0000000 0.0000000

653 0 0 0 0 0 0 0 0 0 3 1 2 0 1 0.0000007 0.0000000 0.0000000 -0.0000004 0.0000000 0.0000000

654 0 0 0 0 0 0 0 0 0 3 0 4 -2 2 0.0000004 0.0000000 0.0000000 -0.0000002 0.0000000 0.0000000

655 0 0 0 0 0 0 0 0 0 -2 -1 2 6 2 -0.0000005 0.0000000 0.0000000 0.0000002 0.0000000 0.0000000

656 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0.0000005 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

657 0 0 0 0 0 0 0 0 0 0 -2 2 4 2 -0.0000006 0.0000000 0.0000000 0.0000003 0.0000000 0.0000000

658 0 0 0 0 0 0 0 0 0 -2 0 2 6 1 -0.0000006 0.0000000 0.0000000 0.0000003 0.0000000 0.0000000

659 0 0 0 0 0 0 0 0 0 2 0 0 4 1 -0.0000004 0.0000000 0.0000000 0.0000002 0.0000000 0.0000000

660 0 0 0 0 0 0 0 0 0 2 0 0 4 0 0.0000010 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
96 NUTATION SERIES

Term Fundamental Argument Multipliers Mi,j ∆ψ Coefficients ∆ Coefficients

i j= 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Si S ̇

i C0

i Ci C ̇

i S

0

i

00 00 00 00 00 00

661 0 0 0 0 0 0 0 0 0 2 -2 2 2 2 -0.0000004 0.0000000 0.0000000 0.0000002 0.0000000 0.0000000

662 0 0 0 0 0 0 0 0 0 0 0 2 4 0 0.0000007 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

663 0 0 0 0 0 0 0 0 0 1 0 2 3 2 0.0000007 0.0000000 0.0000000 -0.0000003 0.0000000 0.0000000

664 0 0 0 0 0 0 0 0 0 4 0 0 2 0 0.0000004 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

665 0 0 0 0 0 0 0 0 0 2 0 2 2 0 0.0000011 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

666 0 0 0 0 0 0 0 0 0 0 0 4 2 2 0.0000005 0.0000000 0.0000000 -0.0000002 0.0000000 0.0000000

667 0 0 0 0 0 0 0 0 0 4 -1 2 0 2 -0.0000006 0.0000000 0.0000000 0.0000002 0.0000000 0.0000000

668 0 0 0 0 0 0 0 0 0 3 0 2 1 2 0.0000004 0.0000000 0.0000000 -0.0000002 0.0000000 0.0000000

669 0 0 0 0 0 0 0 0 0 2 1 2 2 1 0.0000003 0.0000000 0.0000000 -0.0000002 0.0000000 0.0000000

670 0 0 0 0 0 0 0 0 0 4 1 2 0 2 0.0000005 0.0000000 0.0000000 -0.0000002 0.0000000 0.0000000

671 0 0 0 0 0 0 0 0 0 -1 -1 2 6 2 -0.0000004 0.0000000 0.0000000 0.0000002 0.0000000 0.0000000

672 0 0 0 0 0 0 0 0 0 -1 0 2 6 1 -0.0000004 0.0000000 0.0000000 0.0000002 0.0000000 0.0000000

673 0 0 0 0 0 0 0 0 0 1 -1 2 4 1 -0.0000003 0.0000000 0.0000000 0.0000002 0.0000000 0.0000000

674 0 0 0 0 0 0 0 0 0 1 1 2 4 2 0.0000004 0.0000000 0.0000000 -0.0000002 0.0000000 0.0000000

675 0 0 0 0 0 0 0 0 0 3 1 2 2 2 0.0000003 0.0000000 0.0000000 -0.0000001 0.0000000 0.0000000

676 0 0 0 0 0 0 0 0 0 5 0 2 0 1 -0.0000003 0.0000000 0.0000000 0.0000001 0.0000000 0.0000000

677 0 0 0 0 0 0 0 0 0 2 -1 2 4 2 -0.0000003 0.0000000 0.0000000 0.0000001 0.0000000 0.0000000

678 0 0 0 0 0 0 0 0 0 2 0 2 4 1 -0.0000003 0.0000000 0.0000000 0.0000002 0.0000000 0.0000000

679 0 0 8 -16 4 5 0 0 0 0 0 0 0 0 0.0001440 0. 0.0000000 0.0000000 0. 0.0000000

680 0 0 -8 16 -4 -5 0 0 2 0 0 0 0 0 0.0000056 0. -0.0000117 -0.0000040 0. -0.0000042

681 0 0 8 -16 4 5 0 0 2 0 0 0 0 0 0.0000125 0. -0.0000043 -0.0000054 0. 0.0000000

682 0 0 0 0 0 0 -1 2 2 0 0 0 0 0 0.0000000 0. 0.0000005 0.0000000 0. 0.0000000

683 0 0 -4 8 -1 -5 0 0 2 0 0 0 0 0 0.0000003 0. -0.0000007 0.0000000 0. -0.0000003

684 0 0 4 -8 3 0 0 0 1 0 0 0 0 0 0.0000003 0. 0.0000000 -0.0000002 0. 0.0000000

685 0 0 3 -8 3 0 0 0 0 0 0 1 -1 1 -0.0000114 0. 0.0000000 0.0000061 0. 0.0000000

686 0 10 -3 0 0 0 0 0 0 -1 0 0 0 0 -0.0000219 0. 0.0000089 0.0000000 0. 0.0000000

687 0 0 0 0 -2 6 -3 0 2 0 0 0 0 0 -0.0000003 0. 0.0000000 0.0000000 0. 0.0000000

688 0 0 4 -8 3 0 0 0 0 0 0 0 0 0 -0.0000462 0. 0.0001604 0.0000000 0. 0.0000000

689 0 0 -5 8 -3 0 0 0 0 0 0 1 -1 1 0.0000099 0. 0.0000000 -0.0000053 0. 0.0000000

690 0 0 -4 8 -3 0 0 0 1 0 0 0 0 0 -0.0000003 0. 0.0000000 0.0000002 0. 0.0000000

691 0 0 4 -8 1 5 0 0 2 0 0 0 0 0 0.0000000 0. 0.0000006 0.0000000 0. 0.0000002

692 0 -5 6 4 0 0 0 0 2 0 0 0 0 0 0.0000003 0. 0.0000000 0.0000000 0. 0.0000000

693 0 0 0 0 2 -5 0 0 2 0 0 0 0 0 -0.0000012 0. 0.0000000 0.0000000 0. 0.0000000

694 0 0 0 0 2 -5 0 0 1 0 0 0 0 0 0.0000014 0. -0.0000218 0.0000008 0. 0.0000117

695 0 0 -1 0 2 -5 0 0 0 0 0 1 -1 1 0.0000031 0. -0.0000481 -0.0000017 0. -0.0000257

696 0 0 0 0 2 -5 0 0 0 0 0 0 0 0 -0.0000491 0. 0.0000128 0.0000000 0. 0.0000000

697 0 0 -1 0 -2 5 0 0 0 0 0 1 -1 1 -0.0003084 0. 0.0005123 0.0001647 0. 0.0002735

698 0 0 0 0 -2 5 0 0 1 0 0 0 0 0 -0.0001444 0. 0.0002409 -0.0000771 0. -0.0001286

699 0 0 0 0 -2 5 0 0 2 0 0 0 0 0 0.0000011 0. -0.0000024 -0.0000009 0. -0.0000011

700 0 0 3 -7 0 0 0 0 0 2 0 -1 -1 0 0.0000026 0. -0.0000009 0.0000000 0. 0.0000000

701 0 19 -21 3 0 0 0 0 0 1 0 0 -2 0 0.0000103 0. -0.0000060 0.0000000 0. 0.0000000

702 0 2 -4 0 -3 0 0 0 0 0 0 1 -1 1 0.0000000 0. -0.0000013 0.0000000 0. -0.0000007

703 0 0 -1 0 2 0 0 0 0 1 0 0 -1 1 -0.0000026 0. -0.0000029 0.0000014 0. -0.0000016

704 0 0 -1 0 -4 10 0 0 0 0 0 1 -1 1 0.0000009 0. -0.0000027 -0.0000005 0. -0.0000014

705 0 0 2 0 0 -5 0 0 0 -2 0 0 2 1 0.0000012 0. 0.0000000 -0.0000006 0. 0.0000000

706 0 3 -7 4 0 0 0 0 0 0 0 0 0 0 -0.0000007 0. 0.0000000 0.0000000 0. 0.0000000

707 0 0 1 0 1 -1 0 0 0 0 0 -1 1 0 0.0000000 0. 0.0000024 0.0000000 0. 0.0000000

708 0 0 2 0 -2 0 0 0 0 -2 0 0 2 1 0.0000284 0. 0.0000000 -0.0000151 0. 0.0000000

709 0 18 -16 0 0 0 0 0 0 -1 0 0 0 0 0.0000226 0. 0.0000101 0.0000000 0. 0.0000000

710 0 0 1 0 -2 0 0 0 0 -2 0 1 1 2 0.0000000 0. -0.0000008 0.0000000 0. -0.0000002

711 0 18 -17 0 0 0 0 0 0 -1 0 1 -1 1 0.0000000 0. -0.0000006 0.0000000 0. -0.0000003

712 0 0 2 -2 0 0 0 0 0 -1 0 0 1 1 0.0000005 0. 0.0000000 -0.0000003 0. 0.0000000

713 0 -8 13 0 0 0 0 0 2 0 0 0 0 0 -0.0000041 0. 0.0000175 0.0000017 0. 0.0000076

714 0 -8 11 0 0 0 0 0 0 0 0 2 -2 2 0.0000000 0. 0.0000015 0.0000000 0. 0.0000006

715 0 -8 13 0 0 0 0 0 1 0 0 0 0 0 0.0000425 0. 0.0000212 0.0000269 0. -0.0000133

716 0 -8 12 0 0 0 0 0 0 0 0 1 -1 1 0.0001200 0. 0.0000598 -0.0000641 0. 0.0000319

717 0 8 -13 0 0 0 0 0 0 0 0 0 0 0 0.0000235 0. 0.0000334 0.0000000 0. 0.0000000

718 0 8 -14 0 0 0 0 0 0 0 0 1 -1 1 0.0000011 0. -0.0000012 -0.0000006 0. -0.0000007

719 0 8 -13 0 0 0 0 0 1 0 0 0 0 0 0.0000005 0. -0.0000006 0.0000003 0. 0.0000003

720 0 0 2 0 -4 5 0 0 0 -2 0 0 2 1 -0.0000005 0. 0.0000000 0.0000003 0. 0.0000000

721 0 3 -3 0 0 0 0 0 0 -2 0 0 2 2 0.0000006 0. 0.0000000 -0.0000003 0. 0.0000000

722 0 0 2 0 -3 1 0 0 0 -2 0 0 2 0 0.0000015 0. 0.0000000 0.0000000 0. 0.0000000

723 0 3 -5 0 2 0 0 0 0 0 0 0 0 1 0.0000013 0. 0.0000000 -0.0000007 0. 0.0000000

724 0 0 2 0 -4 3 0 0 0 -2 0 0 2 0 -0.0000006 0. -0.0000009 0.0000000 0. 0.0000000

725 0 0 0 2 0 0 0 0 0 0 0 -1 1 0 0.0000266 0. -0.0000078 0.0000000 0. 0.0000000

726 0 0 -1 2 0 0 0 0 0 0 0 0 0 1 -0.0000460 0. -0.0000435 0.0000246 0. -0.0000232

727 0 0 -2 2 0 0 0 0 0 0 0 1 -1 2 0.0000000 0. 0.0000015 0.0000000 0. 0.0000007

728 0 3 -5 0 0 0 0 0 0 -1 0 1 0 1 -0.0000003 0. 0.0000000 0.0000002 0. 0.0000000

729 0 3 -4 0 0 0 0 0 0 -1 0 0 1 0 0.0000000 0. 0.0000131 0.0000000 0. 0.0000000

730 0 0 2 0 -2 -2 0 0 0 -2 0 0 2 0 0.0000004 0. 0.0000000 0.0000000 0. 0.0000000

731 0 0 -5 9 0 0 0 0 0 -2 0 2 0 2 0.0000000 0. 0.0000003 0.0000000 0. 0.0000000

732 0 0 -1 0 0 0 -1 0 0 0 0 1 -1 1 0.0000000 0. 0.0000004 0.0000000 0. 0.0000002

733 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0.0000000 0. 0.0000003 0.0000000 0. 0.0000000

734 0 0 -1 0 0 0 0 2 0 0 0 1 -1 1 -0.0000017 0. -0.0000019 0.0000009 0. -0.0000010

735 0 0 0 0 0 0 0 2 1 0 0 0 0 0 -0.0000009 0. -0.0000011 -0.0000005 0. 0.0000006

736 0 0 0 0 0 0 0 2 2 0 0 0 0 0 -0.0000006 0. 0.0000000 0.0000003 0. 0.0000000

737 0 0 3 -4 0 0 0 0 0 -1 0 0 1 0 -0.0000016 0. 0.0000008 0.0000000 0. 0.0000000

738 0 0 1 0 0 2 0 0 0 0 0 -1 1 0 0.0000000 0. 0.0000003 0.0000000 0. 0.0000000

739 0 0 -1 0 0 2 0 0 0 0 0 1 -1 2 0.0000011 0. 0.0000024 -0.0000005 0. 0.0000011

740 0 0 -9 17 0 0 0 0 0 0 0 0 0 1 -0.0000003 0. -0.0000004 0.0000001 0. -0.0000002

741 0 -3 5 0 0 0 0 0 0 0 0 0 0 2 0.0000003 0. 0.0000000 -0.0000001 0. 0.0000000

742 0 0 -1 0 -1 2 0 0 0 0 0 1 -1 1 0.0000000 0. -0.0000008 0.0000000 0. -0.0000004

743 0 0 0 0 1 -2 0 0 0 0 0 0 0 0 0.0000000 0. 0.0000003 0.0000000 0. 0.0000000

744 0 17 -16 0 -2 0 0 0 0 1 0 0 -2 0 0.0000000 0. 0.0000005 0.0000000 0. 0.0000000

745 0 0 -1 0 1 -3 0 0 0 0 0 1 -1 1 0.0000000 0. 0.0000003 0.0000000 0. 0.0000002

746 0 0 5 -6 0 0 0 0 0 -2 0 0 2 1 -0.0000006 0. 0.0000004 0.0000003 0. 0.0000002

747 0 0 9 -13 0 0 0 0 0 0 0 -2 2 0 -0.0000003 0. -0.0000005 0.0000000 0. 0.0000000

748 0 0 -1 0 0 1 0 0 0 0 0 1 -1 2 -0.0000005 0. 0.0000000 0.0000002 0. 0.0000000

749 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0.0000004 0. 0.0000024 -0.0000002 0. 0.0000013
NUTATION SERIES 97

Term Fundamental Argument Multipliers Mi,j ∆ψ Coefficients ∆ Coefficients

i j= 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Si S ̇

i C0

i Ci C ̇

i S

0

i

00 00 00 00 00 00

750 0 0 1 0 0 1 0 0 0 0 0 -1 1 0 -0.0000042 0. 0.0000020 0.0000000 0. 0.0000000

751 0 5 -6 0 0 0 0 0 0 0 0 -2 2 0 -0.0000010 0. 0.0000233 0.0000000 0. 0.0000000

752 0 5 -7 0 0 0 0 0 0 0 0 -1 1 1 -0.0000003 0. 0.0000000 0.0000001 0. 0.0000000

753 0 6 -8 0 0 0 0 0 0 -2 0 0 2 0 0.0000078 0. -0.0000018 0.0000000 0. 0.0000000

754 0 -6 7 0 0 0 0 0 0 2 0 1 -3 1 0.0000000 0. 0.0000003 0.0000000 0. 0.0000001

755 0 0 0 0 1 0 0 0 0 0 0 0 0 2 0.0000000 0. -0.0000003 0.0000000 0. -0.0000001

756 0 0 1 0 1 0 0 0 0 0 0 -1 1 1 0.0000000 0. -0.0000004 0.0000001 0. -0.0000002

757 0 0 -1 0 0 0 2 0 0 0 0 1 -1 1 0.0000000 0. -0.0000008 -0.0000001 0. -0.0000004

758 0 0 0 0 0 0 2 0 1 0 0 0 0 0 0.0000000 0. -0.0000005 0.0000000 0. 0.0000003

759 0 0 0 0 0 0 2 0 2 0 0 0 0 0 -0.0000007 0. 0.0000000 0.0000003 0. 0.0000000

760 0 0 -8 15 0 0 0 0 2 0 0 0 0 0 -0.0000014 0. 0.0000008 0.0000006 0. 0.0000003

761 0 0 -8 15 0 0 0 0 1 0 0 0 0 0 0.0000000 0. 0.0000008 0.0000000 0. -0.0000004

762 0 0 -9 15 0 0 0 0 0 0 0 1 -1 1 0.0000000 0. 0.0000019 0.0000000 0. 0.0000010

763 0 0 8 -15 0 0 0 0 0 0 0 0 0 0 0.0000045 0. -0.0000022 0.0000000 0. 0.0000000

764 0 0 8 -15 0 0 0 0 0 1 0 -1 -1 0 -0.0000003 0. 0.0000000 0.0000000 0. 0.0000000

765 0 2 -5 0 0 0 0 0 0 2 0 0 -2 0 0.0000000 0. -0.0000003 0.0000000 0. 0.0000000

766 0 0 2 0 -5 5 0 0 0 -2 0 0 2 0 0.0000000 0. 0.0000003 0.0000000 0. 0.0000000

767 0 0 -6 8 0 0 0 0 0 2 0 0 -2 1 0.0000003 0. 0.0000005 -0.0000002 0. 0.0000003

768 0 0 -2 0 3 0 0 0 0 2 0 0 -2 1 0.0000089 0. -0.0000016 -0.0000048 0. -0.0000009

769 0 0 1 0 -3 0 0 0 0 -2 0 1 1 0 0.0000000 0. 0.0000003 0.0000000 0. 0.0000000

770 0 0 1 0 -3 0 0 0 0 -2 0 1 1 1 -0.0000003 0. 0.0000007 0.0000002 0. 0.0000004

771 0 0 2 0 -3 0 0 0 0 -2 0 0 2 0 -0.0000349 0. -0.0000062 0.0000000 0. 0.0000000

772 0 0 6 -8 0 0 0 0 0 -2 0 0 2 0 -0.0000015 0. 0.0000022 0.0000000 0. 0.0000000

773 0 0 2 0 -1 -5 0 0 0 -2 0 0 2 0 -0.0000003 0. 0.0000000 0.0000000 0. 0.0000000

774 0 0 1 0 -1 0 0 0 0 -1 0 0 1 0 -0.0000053 0. 0.0000000 0.0000000 0. 0.0000000

775 0 -20 20 0 0 0 0 0 0 -1 0 1 1 1 0.0000005 0. 0.0000000 -0.0000003 0. 0.0000000

776 0 20 -21 0 0 0 0 0 0 1 0 0 -2 0 0.0000000 0. -0.0000008 0.0000000 0. 0.0000000

777 0 0 8 -15 0 0 0 0 0 0 0 0 0 1 0.0000015 0. -0.0000007 -0.0000008 0. -0.0000004

778 0 0 -10 15 0 0 0 0 0 0 0 2 -2 1 -0.0000003 0. 0.0000000 0.0000001 0. 0.0000000

779 0 0 1 0 1 0 0 0 0 0 0 -1 1 0 -0.0000021 0. -0.0000078 0.0000000 0. 0.0000000

780 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0.0000020 0. -0.0000070 -0.0000011 0. -0.0000037

781 0 0 -1 0 1 0 0 0 0 0 0 1 -1 2 0.0000000 0. 0.0000006 0.0000000 0. 0.0000003

782 0 0 -1 0 -2 4 0 0 0 0 0 1 -1 1 0.0000005 0. 0.0000003 -0.0000002 0. 0.0000002

783 0 -6 8 0 0 0 0 0 0 2 0 0 -2 1 -0.0000017 0. -0.0000004 0.0000009 0. -0.0000002

784 0 5 -6 0 0 0 0 0 0 0 0 -2 2 1 0.0000000 0. 0.0000006 0.0000000 0. 0.0000003

785 0 0 0 0 0 -1 0 0 1 0 0 0 0 0 0.0000032 0. 0.0000015 0.0000017 0. -0.0000008

786 0 0 -1 0 0 -1 0 0 0 0 0 1 -1 1 0.0000174 0. 0.0000084 -0.0000093 0. 0.0000045

787 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0.0000011 0. 0.0000056 0.0000000 0. 0.0000000

788 0 0 -1 0 0 1 0 0 0 0 0 1 -1 1 -0.0000066 0. -0.0000012 0.0000035 0. -0.0000006

789 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0.0000047 0. 0.0000008 -0.0000025 0. 0.0000004

790 0 0 0 0 0 1 0 0 2 0 0 0 0 0 0.0000000 0. 0.0000008 0.0000000 0. 0.0000004

791 0 0 -9 13 0 0 0 0 0 0 0 2 -2 1 0.0000010 0. -0.0000022 -0.0000005 0. -0.0000012

792 0 0 7 -13 0 0 0 0 0 0 0 0 0 1 -0.0000003 0. 0.0000000 0.0000002 0. 0.0000000

793 0 0 5 -6 0 0 0 0 0 -2 0 0 2 0 -0.0000024 0. 0.0000012 0.0000000 0. 0.0000000

794 0 0 9 -17 0 0 0 0 0 0 0 0 0 0 0.0000005 0. -0.0000006 0.0000000 0. 0.0000000

795 0 0 -9 17 0 0 0 0 2 0 0 0 0 0 0.0000003 0. 0.0000000 -0.0000002 0. 0.0000000

796 0 0 -3 4 0 0 0 0 0 1 0 0 -1 1 0.0000004 0. 0.0000003 -0.0000002 0. 0.0000001

797 0 -3 4 0 0 0 0 0 0 1 0 0 -1 1 0.0000000 0. 0.0000029 0.0000000 0. 0.0000015

798 0 0 -1 2 0 0 0 0 0 0 0 0 0 2 -0.0000005 0. -0.0000004 0.0000002 0. -0.0000002

799 0 0 0 2 0 0 0 0 0 0 0 -1 1 1 0.0000008 0. -0.0000003 -0.0000005 0. -0.0000001

800 1 0 -2 0 0 0 0 0 0 0 0 -2 2 0 0.0000000 0. -0.0000003 0.0000000 0. 0.0000000

801 0 3 -5 0 2 0 0 0 0 0 0 0 0 0 0.0000010 0. 0.0000000 0.0000000 0. 0.0000000

802 0 0 2 0 -3 1 0 0 0 -2 0 0 2 1 0.0000003 0. 0.0000000 -0.0000002 0. 0.0000000

803 0 3 -3 0 0 0 0 0 0 -2 0 0 2 1 -0.0000005 0. 0.0000000 0.0000003 0. 0.0000000

804 0 8 -13 0 0 0 0 0 0 0 0 0 0 1 0.0000046 0. 0.0000066 -0.0000025 0. 0.0000035

805 0 8 -12 0 0 0 0 0 0 0 0 -1 1 0 -0.0000014 0. 0.0000007 0.0000000 0. 0.0000000

806 0 -8 11 0 0 0 0 0 0 0 0 2 -2 1 0.0000000 0. 0.0000003 0.0000000 0. 0.0000002

807 0 0 2 -2 0 0 0 0 0 -1 0 0 1 0 -0.0000005 0. 0.0000000 0.0000000 0. 0.0000000

808 0 18 -16 0 0 0 0 0 0 -1 0 0 0 1 -0.0000068 0. -0.0000034 0.0000036 0. -0.0000018

809 0 0 -1 0 -1 1 0 0 0 0 0 1 -1 1 0.0000000 0. 0.0000014 0.0000000 0. 0.0000007

810 0 3 -7 4 0 0 0 0 0 0 0 0 0 1 0.0000010 0. -0.0000006 -0.0000005 0. -0.0000003

811 0 0 -3 7 0 0 0 0 0 -2 0 1 1 1 -0.0000005 0. -0.0000004 0.0000003 0. -0.0000002

812 0 0 -1 0 -2 5 0 0 0 0 0 1 -1 2 -0.0000003 0. 0.0000005 0.0000001 0. 0.0000002

813 0 0 0 0 -2 5 0 0 0 0 0 0 0 1 0.0000076 0. 0.0000017 -0.0000041 0. 0.0000009

814 0 0 -4 8 -3 0 0 0 0 0 0 0 0 1 0.0000084 0. 0.0000298 -0.0000045 0. 0.0000159

815 0 -10 3 0 0 0 0 0 0 1 0 0 0 1 0.0000003 0. 0.0000000 -0.0000001 0. 0.0000000

816 0 0 -2 0 0 0 0 0 0 0 0 2 -2 1 -0.0000003 0. 0.0000000 0.0000002 0. 0.0000000

817 0 10 -3 0 0 0 0 0 0 -1 0 0 0 1 -0.0000003 0. 0.0000000 0.0000001 0. 0.0000000

818 0 0 4 -8 3 0 0 0 0 0 0 0 0 1 -0.0000082 0. 0.0000292 0.0000044 0. 0.0000156

819 0 0 0 0 2 -5 0 0 0 0 0 0 0 1 -0.0000073 0. 0.0000017 0.0000039 0. 0.0000009

820 0 0 1 0 2 -5 0 0 0 0 0 -1 1 0 -0.0000009 0. -0.0000016 0.0000000 0. 0.0000000

821 0 0 3 -7 0 0 0 0 0 2 0 -1 -1 1 0.0000003 0. 0.0000000 -0.0000002 0. -0.0000001

822 0 0 2 0 0 -5 0 0 0 -2 0 0 2 0 -0.0000003 0. 0.0000000 0.0000000 0. 0.0000000

823 0 -3 7 -4 0 0 0 0 0 0 0 0 0 1 -0.0000009 0. -0.0000005 0.0000005 0. -0.0000003

824 0 0 2 0 -2 0 0 0 0 -2 0 0 2 0 -0.0000439 0. 0.0000000 0.0000000 0. 0.0000000

825 0 -18 16 0 0 0 0 0 0 1 0 0 0 1 0.0000057 0. -0.0000028 -0.0000030 0. -0.0000015

826 0 0 1 0 -2 0 0 0 0 -2 0 1 1 1 0.0000000 0. -0.0000006 0.0000000 0. -0.0000003

827 0 -8 12 0 0 0 0 0 0 0 0 1 -1 2 -0.0000004 0. 0.0000000 0.0000002 0. 0.0000000

828 0 -8 13 0 0 0 0 0 0 0 0 0 0 1 -0.0000040 0. 0.0000057 0.0000021 0. 0.0000030

829 0 0 1 -2 0 0 0 0 1 0 0 0 0 0 0.0000023 0. 0.0000007 -0.0000013 0. 0.0000003

830 0 0 0 -2 0 0 0 0 0 0 0 1 -1 1 0.0000273 0. 0.0000080 -0.0000146 0. 0.0000043

831 0 0 1 -2 0 0 0 0 0 0 0 0 0 0 -0.0000449 0. 0.0000430 0.0000000 0. 0.0000000

832 0 0 -2 2 0 0 0 0 0 0 0 1 -1 1 -0.0000008 0. -0.0000047 0.0000004 0. -0.0000025

833 0 0 -1 2 0 0 0 0 1 0 0 0 0 0 0.0000006 0. 0.0000047 -0.0000003 0. 0.0000025

834 0 3 -4 0 0 0 0 0 0 -1 0 0 1 1 0.0000000 0. 0.0000023 0.0000000 0. 0.0000013

835 0 0 3 -4 0 0 0 0 0 -1 0 0 1 1 -0.0000003 0. 0.0000000 0.0000002 0. 0.0000000

836 0 0 -1 0 0 -2 0 0 0 0 0 1 -1 1 0.0000003 0. -0.0000004 -0.0000002 0. -0.0000002

837 0 0 -1 0 0 2 0 0 0 0 0 1 -1 1 -0.0000048 0. -0.0000110 0.0000026 0. -0.0000059

838 0 0 0 0 0 2 0 0 1 0 0 0 0 0 0.0000051 0. 0.0000114 -0.0000027 0. 0.0000061

839 0 0 0 0 0 2 0 0 2 0 0 0 0 0 -0.0000133 0. 0.0000000 0.0000057 0. 0.0000000
98 NUTATION SERIES

Term Fundamental Argument Multipliers Mi,j ∆ψ Coefficients ∆ Coefficients

i j= 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Si S ̇

i C0

i Ci C ̇

i S

0

i

00 00 00 00 00 00

840 0 3 -6 0 0 0 0 0 0 0 0 1 -1 0 0.0000000 0. 0.0000004 0.0000000 0. 0.0000000

841 0 -3 5 0 0 0 0 0 0 0 0 0 0 1 -0.0000021 0. -0.0000006 0.0000011 0. -0.0000003

842 0 -3 4 0 0 0 0 0 0 0 0 1 -1 2 0.0000000 0. -0.0000003 0.0000000 0. -0.0000001

843 0 0 -2 4 0 0 0 0 0 0 0 0 0 1 -0.0000011 0. -0.0000021 0.0000006 0. -0.0000011

844 0 -5 6 0 0 0 0 0 0 0 0 2 -2 1 -0.0000018 0. -0.0000436 0.0000009 0. -0.0000233

845 0 5 -7 0 0 0 0 0 0 0 0 -1 1 0 0.0000035 0. -0.0000007 0.0000000 0. 0.0000000

846 0 5 -8 0 0 0 0 0 0 0 0 0 0 1 0.0000000 0. 0.0000005 0.0000000 0. 0.0000003

847 0 6 -8 0 0 0 0 0 0 -2 0 0 2 1 0.0000011 0. -0.0000003 -0.0000006 0. -0.0000001

848 0 0 -8 15 0 0 0 0 0 0 0 0 0 1 -0.0000005 0. -0.0000003 0.0000003 0. -0.0000001

849 0 0 2 0 -3 0 0 0 0 -2 0 0 2 1 -0.0000053 0. -0.0000009 0.0000028 0. -0.0000005

850 0 0 6 -8 0 0 0 0 0 -2 0 0 2 1 0.0000000 0. 0.0000003 0.0000001 0. 0.0000002

851 0 0 -1 0 1 0 0 0 0 1 0 0 -1 1 0.0000004 0. 0.0000000 -0.0000002 0. 0.0000000

852 0 0 0 0 3 -5 0 0 0 0 0 0 0 0 0.0000000 0. -0.0000004 0.0000000 0. 0.0000000

853 0 0 -1 0 -1 0 0 0 0 0 0 1 -1 1 -0.0000050 0. 0.0000194 0.0000027 0. 0.0000103

854 0 0 0 0 -1 0 0 0 1 0 0 0 0 0 -0.0000013 0. 0.0000052 0.0000007 0. 0.0000028

855 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -0.0000091 0. 0.0000248 0.0000000 0. 0.0000000

856 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0.0000006 0. 0.0000049 -0.0000003 0. 0.0000026

857 0 0 -1 0 1 0 0 0 0 0 0 1 -1 1 -0.0000006 0. -0.0000047 0.0000003 0. -0.0000025

858 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0.0000000 0. 0.0000005 0.0000000 0. 0.0000003

859 0 0 0 0 1 0 0 0 2 0 0 0 0 0 0.0000052 0. 0.0000023 -0.0000023 0. 0.0000010

860 0 0 -1 0 0 -1 0 0 0 0 0 1 -1 2 -0.0000003 0. 0.0000000 0.0000001 0. 0.0000000

861 0 0 0 0 0 -1 0 0 0 0 0 0 0 1 0.0000000 0. 0.0000005 0.0000000 0. 0.0000003

862 0 0 1 0 0 -1 0 0 0 0 0 -1 1 0 -0.0000004 0. 0.0000000 0.0000000 0. 0.0000000

863 0 0 -7 13 0 0 0 0 2 0 0 0 0 0 -0.0000004 0. 0.0000008 0.0000002 0. 0.0000003

864 0 0 7 -13 0 0 0 0 0 0 0 0 0 0 0.0000010 0. 0.0000000 0.0000000 0. 0.0000000

865 0 0 -5 6 0 0 0 0 0 2 0 0 -2 1 0.0000003 0. 0.0000000 -0.0000002 0. 0.0000000

866 0 0 -8 11 0 0 0 0 0 0 0 2 -2 1 0.0000000 0. 0.0000008 0.0000000 0. 0.0000004

867 -1 0 2 0 0 0 0 0 0 0 0 2 -2 1 0.0000000 0. 0.0000008 0.0000001 0. 0.0000004

868 0 0 4 -4 0 0 0 0 0 -2 0 0 2 0 -0.0000004 0. 0.0000000 0.0000000 0. 0.0000000

869 0 0 0 0 2 -2 0 0 0 0 0 0 0 0 -0.0000004 0. 0.0000000 0.0000000 0. 0.0000000

870 0 0 -1 0 0 3 0 0 0 0 0 1 -1 1 -0.0000008 0. 0.0000004 0.0000004 0. 0.0000002

871 0 0 0 0 0 3 0 0 1 0 0 0 0 0 0.0000008 0. -0.0000004 -0.0000004 0. -0.0000002

872 0 0 0 0 0 3 0 0 2 0 0 0 0 0 0.0000000 0. 0.0000015 0.0000000 0. 0.0000007

873 0 3 -3 0 0 0 0 0 0 -2 0 0 2 0 -0.0000138 0. 0.0000000 0.0000000 0. 0.0000000

874 0 0 -4 8 -3 0 0 0 0 0 0 0 0 2 0.0000000 0. -0.0000007 0.0000000 0. -0.0000003

875 0 0 4 -8 3 0 0 0 0 0 0 0 0 2 0.0000000 0. -0.0000007 0.0000000 0. -0.0000003

876 0 0 -2 0 2 0 0 0 0 2 0 0 -2 1 0.0000054 0. 0.0000000 -0.0000029 0. 0.0000000

877 0 0 -1 0 2 0 0 0 0 0 0 1 -1 2 0.0000000 0. 0.0000010 0.0000000 0. 0.0000004

878 0 0 0 -2 0 0 0 0 0 0 0 1 -1 2 -0.0000007 0. 0.0000000 0.0000003 0. 0.0000000

879 0 0 1 -2 0 0 0 0 0 0 0 0 0 1 -0.0000037 0. 0.0000035 0.0000020 0. 0.0000019

880 0 0 2 -2 0 0 0 0 0 0 0 -1 1 0 0.0000000 0. 0.0000004 0.0000000 0. 0.0000000

881 0 0 1 0 0 -2 0 0 0 0 0 -1 1 0 -0.0000004 0. 0.0000009 0.0000000 0. 0.0000000

882 0 0 -2 0 0 2 0 0 0 0 0 2 -2 1 0.0000008 0. 0.0000000 -0.0000004 0. 0.0000000

883 0 3 -6 0 0 0 0 0 0 0 0 1 -1 1 -0.0000009 0. -0.0000014 0.0000005 0. -0.0000008

884 0 3 -5 0 0 0 0 0 1 0 0 0 0 0 -0.0000003 0. -0.0000009 0.0000003 0. -0.0000005

885 0 3 -5 0 0 0 0 0 0 0 0 0 0 0 -0.0000145 0. 0.0000047 0.0000000 0. 0.0000000

886 0 -3 4 0 0 0 0 0 0 0 0 1 -1 1 -0.0000010 0. 0.0000040 0.0000005 0. 0.0000021

887 0 -3 5 0 0 0 0 0 1 0 0 0 0 0 0.0000011 0. -0.0000049 -0.0000007 0. -0.0000026

888 0 -3 5 0 0 0 0 0 2 0 0 0 0 0 -0.0002150 0. 0.0000000 0.0000932 0. 0.0000000

889 0 -3 3 0 0 0 0 0 0 0 0 2 -2 2 -0.0000012 0. 0.0000000 0.0000005 0. 0.0000000

890 0 -3 5 0 0 0 0 0 2 0 0 0 0 0 0.0000085 0. 0.0000000 -0.0000037 0. 0.0000000

891 0 0 2 -4 0 0 0 0 1 0 0 0 0 0 0.0000004 0. 0.0000000 -0.0000002 0. 0.0000000

892 0 0 1 -4 0 0 0 0 0 0 0 1 -1 1 0.0000003 0. 0.0000000 -0.0000002 0. 0.0000000

893 0 0 2 -4 0 0 0 0 0 0 0 0 0 0 -0.0000086 0. 0.0000153 0.0000000 0. 0.0000000

894 0 0 -2 4 0 0 0 0 1 0 0 0 0 0 -0.0000006 0. 0.0000009 0.0000003 0. 0.0000005

895 0 0 -3 4 0 0 0 0 0 0 0 1 -1 1 0.0000009 0. -0.0000013 -0.0000005 0. -0.0000007

896 0 0 -2 4 0 0 0 0 1 0 0 0 0 0 -0.0000008 0. 0.0000012 0.0000004 0. 0.0000006

897 0 0 -2 4 0 0 0 0 2 0 0 0 0 0 -0.0000051 0. 0.0000000 0.0000022 0. 0.0000000

898 0 -5 8 0 0 0 0 0 2 0 0 0 0 0 -0.0000011 0. -0.0000268 0.0000005 0. -0.0000116

899 0 -5 6 0 0 0 0 0 0 0 0 2 -2 2 0.0000000 0. 0.0000012 0.0000000 0. 0.0000005

900 0 -5 8 0 0 0 0 0 2 0 0 0 0 0 0.0000000 0. 0.0000007 0.0000000 0. 0.0000003

901 0 -5 8 0 0 0 0 0 1 0 0 0 0 0 0.0000031 0. 0.0000006 -0.0000017 0. 0.0000003

902 0 -5 7 0 0 0 0 0 0 0 0 1 -1 1 0.0000140 0. 0.0000027 -0.0000075 0. 0.0000014

903 0 -5 8 0 0 0 0 0 1 0 0 0 0 0 0.0000057 0. 0.0000011 -0.0000030 0. 0.0000006

904 0 5 -8 0 0 0 0 0 0 0 0 0 0 0 -0.0000014 0. -0.0000039 0.0000000 0. 0.0000000

905 0 0 -1 0 -1 0 0 0 0 0 0 1 -1 2 0.0000000 0. -0.0000006 0.0000000 0. -0.0000002

906 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 0.0000004 0. 0.0000015 -0.0000002 0. 0.0000008

907 0 0 1 0 -1 0 0 0 0 0 0 -1 1 0 0.0000000 0. 0.0000004 0.0000000 0. 0.0000000

908 0 0 -2 0 1 0 0 0 0 0 0 2 -2 1 -0.0000003 0. 0.0000000 0.0000001 0. 0.0000000

909 0 0 -6 11 0 0 0 0 2 0 0 0 0 0 0.0000000 0. 0.0000011 0.0000000 0. 0.0000005

910 0 0 6 -11 0 0 0 0 0 0 0 0 0 0 0.0000009 0. 0.0000006 0.0000000 0. 0.0000000

911 -1 0 4 0 0 0 0 0 2 0 0 0 0 0 -0.0000004 0. 0.0000010 0.0000002 0. 0.0000004

912 1 0 -4 0 0 0 0 0 0 0 0 0 0 0 0.0000005 0. 0.0000003 0.0000000 0. 0.0000000

913 0 -3 3 0 0 0 0 0 0 2 0 0 -2 1 0.0000016 0. 0.0000000 -0.0000009 0. 0.0000000

914 0 0 2 0 0 -2 0 0 0 -2 0 0 2 0 -0.0000003 0. 0.0000000 0.0000000 0. 0.0000000

915 0 0 -7 9 0 0 0 0 0 0 0 2 -2 1 0.0000000 0. 0.0000003 -0.0000001 0. 0.0000002

916 0 0 0 0 4 -5 0 0 2 0 0 0 0 0 0.0000007 0. 0.0000000 -0.0000003 0. 0.0000000

917 0 0 0 0 2 0 0 0 0 0 0 0 0 0 -0.0000025 0. 0.0000022 0.0000000 0. 0.0000000

918 0 0 0 0 2 0 0 0 1 0 0 0 0 0 0.0000042 0. 0.0000223 -0.0000022 0. 0.0000119

919 0 0 -1 0 2 0 0 0 0 0 0 1 -1 1 -0.0000027 0. -0.0000143 0.0000014 0. -0.0000077

920 0 0 0 0 2 0 0 0 1 0 0 0 0 0 0.0000009 0. 0.0000049 -0.0000005 0. 0.0000026

921 0 0 0 0 2 0 0 0 2 0 0 0 0 0 -0.0001166 0. 0.0000000 0.0000505 0. 0.0000000

922 0 0 -2 0 2 0 0 0 0 0 0 2 -2 2 -0.0000005 0. 0.0000000 0.0000002 0. 0.0000000

923 0 0 0 0 0 5 0 0 2 0 0 0 0 0 -0.0000006 0. 0.0000000 0.0000003 0. 0.0000000

924 0 3 -5 0 0 0 0 0 0 0 0 0 0 1 -0.0000008 0. 0.0000000 0.0000004 0. 0.0000001

925 0 3 -4 0 0 0 0 0 0 0 0 -1 1 0 0.0000000 0. -0.0000004 0.0000000 0. 0.0000000

926 0 -3 3 0 0 0 0 0 0 0 0 2 -2 1 0.0000117 0. 0.0000000 -0.0000063 0. 0.0000000

927 0 0 2 -4 0 0 0 0 0 0 0 0 0 1 -0.0000004 0. 0.0000008 0.0000002 0. 0.0000004

928 0 0 -4 4 0 0 0 0 0 0 0 2 -2 1 0.0000003 0. 0.0000000 -0.0000002 0. 0.0000000

929 0 -5 7 0 0 0 0 0 0 0 0 1 -1 2 -0.0000005 0. 0.0000000 0.0000002 0. 0.0000000
NUTATION SERIES 99

Term Fundamental Argument Multipliers Mi,j ∆ψ Coefficients ∆ Coefficients

i j= 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Si S ̇

i C0

i Ci C ̇

i S

0

i

00 00 00 00 00 00

930 0 0 3 -6 0 0 0 0 0 0 0 0 0 0 0.0000000 0. 0.0000031 0.0000000 0. 0.0000000

931 0 0 -3 6 0 0 0 0 1 0 0 0 0 0 -0.0000005 0. 0.0000000 0.0000003 0. 0.0000001

932 0 0 -4 6 0 0 0 0 0 0 0 1 -1 1 0.0000004 0. 0.0000000 -0.0000002 0. 0.0000000

933 0 0 -3 6 0 0 0 0 1 0 0 0 0 0 -0.0000004 0. 0.0000000 0.0000002 0. 0.0000000

934 0 0 -3 6 0 0 0 0 2 0 0 0 0 0 -0.0000024 0. -0.0000013 0.0000010 0. -0.0000006

935 0 2 -2 0 0 0 0 0 0 0 0 -1 1 0 0.0000003 0. 0.0000000 0.0000000 0. 0.0000000

936 0 2 -3 0 0 0 0 0 0 0 0 0 0 1 0.0000000 0. -0.0000032 0.0000000 0. -0.0000017

937 0 0 -5 9 0 0 0 0 2 0 0 0 0 0 0.0000008 0. 0.0000012 -0.0000003 0. 0.0000005

938 0 0 -5 9 0 0 0 0 1 0 0 0 0 0 0.0000003 0. 0.0000000 -0.0000001 0. 0.0000000

939 0 0 5 -9 0 0 0 0 0 0 0 0 0 0 0.0000007 0. 0.0000013 0.0000000 0. 0.0000000

940 0 0 1 0 -2 0 0 0 0 0 0 -1 1 0 -0.0000003 0. 0.0000016 0.0000000 0. 0.0000000

941 0 0 -2 0 2 0 0 0 0 0 0 2 -2 1 0.0000050 0. 0.0000000 -0.0000027 0. 0.0000000

942 0 0 1 0 0 0 0 0 0 -2 0 1 1 1 0.0000000 0. -0.0000005 0.0000000 0. -0.0000003

943 0 3 -3 0 0 0 0 0 0 0 0 -2 2 0 0.0000013 0. 0.0000000 0.0000000 0. 0.0000000

944 0 -6 10 0 0 0 0 0 1 0 0 0 0 0 0.0000000 0. 0.0000005 0.0000001 0. 0.0000003

945 0 -6 10 0 0 0 0 0 2 0 0 0 0 0 0.0000024 0. 0.0000005 -0.0000011 0. 0.0000002

946 0 -2 3 0 0 0 0 0 2 0 0 0 0 0 0.0000005 0. -0.0000011 -0.0000002 0. -0.0000005

947 0 -2 3 0 0 0 0 0 1 0 0 0 0 0 0.0000030 0. -0.0000003 -0.0000016 0. -0.0000002

948 0 -2 2 0 0 0 0 0 0 0 0 1 -1 1 0.0000018 0. 0.0000000 -0.0000009 0. 0.0000000

949 0 2 -3 0 0 0 0 0 0 0 0 0 0 0 0.0000008 0. 0.0000614 0.0000000 0. 0.0000000

950 0 2 -3 0 0 0 0 0 1 0 0 0 0 0 0.0000003 0. -0.0000003 -0.0000002 0. -0.0000001

951 0 0 0 0 3 0 0 0 1 0 0 0 0 0 0.0000006 0. 0.0000017 -0.0000003 0. 0.0000009

952 0 0 -1 0 3 0 0 0 0 0 0 1 -1 1 -0.0000003 0. -0.0000009 0.0000002 0. -0.0000005

953 0 0 0 0 3 0 0 0 1 0 0 0 0 0 0.0000000 0. 0.0000006 -0.0000001 0. 0.0000003

954 0 0 0 0 3 0 0 0 2 0 0 0 0 0 -0.0000127 0. 0.0000021 0.0000055 0. 0.0000009

955 0 0 4 -8 0 0 0 0 0 0 0 0 0 0 0.0000003 0. 0.0000005 0.0000000 0. 0.0000000

956 0 0 -4 8 0 0 0 0 2 0 0 0 0 0 -0.0000006 0. -0.0000010 0.0000003 0. -0.0000004

957 0 0 2 0 -2 0 0 0 0 0 0 -2 2 0 0.0000005 0. 0.0000000 0.0000000 0. 0.0000000

958 0 0 -4 7 0 0 0 0 2 0 0 0 0 0 0.0000016 0. 0.0000009 -0.0000007 0. 0.0000004

959 0 0 -4 7 0 0 0 0 1 0 0 0 0 0 0.0000003 0. 0.0000000 -0.0000002 0. 0.0000000

960 0 0 4 -7 0 0 0 0 0 0 0 0 0 0 0.0000000 0. 0.0000022 0.0000000 0. 0.0000000

961 0 -2 3 0 0 0 0 0 0 0 0 0 0 1 0.0000000 0. 0.0000019 0.0000000 0. 0.0000010

962 0 0 -2 0 3 0 0 0 0 0 0 2 -2 1 0.0000007 0. 0.0000000 -0.0000004 0. 0.0000000

963 0 0 -5 10 0 0 0 0 2 0 0 0 0 0 0.0000000 0. -0.0000005 0.0000000 0. -0.0000002

964 0 -1 2 0 0 0 0 0 0 0 0 0 0 1 0.0000000 0. 0.0000003 0.0000000 0. 0.0000001

965 0 0 0 0 4 0 0 0 2 0 0 0 0 0 -0.0000009 0. 0.0000003 0.0000004 0. 0.0000001

966 0 0 -3 5 0 0 0 0 2 0 0 0 0 0 0.0000017 0. 0.0000000 -0.0000007 0. 0.0000000

967 0 0 -3 5 0 0 0 0 1 0 0 0 0 0 0.0000000 0. -0.0000003 -0.0000001 0. -0.0000002

968 0 0 3 -5 0 0 0 0 0 0 0 0 0 0 -0.0000020 0. 0.0000034 0.0000000 0. 0.0000000

969 0 1 -2 0 0 0 0 0 1 0 0 0 0 0 -0.0000010 0. 0.0000000 0.0000005 0. 0.0000001

970 0 1 -3 0 0 0 0 0 0 0 0 1 -1 1 -0.0000004 0. 0.0000000 0.0000002 0. 0.0000000

971 0 1 -2 0 0 0 0 0 0 0 0 0 0 0 0.0000022 0. -0.0000087 0.0000000 0. 0.0000000

972 0 -1 2 0 0 0 0 0 1 0 0 0 0 0 -0.0000004 0. 0.0000000 0.0000002 0. 0.0000000

973 0 -1 2 0 0 0 0 0 2 0 0 0 0 0 -0.0000003 0. -0.0000006 0.0000001 0. -0.0000002

974 0 -7 11 0 0 0 0 0 2 0 0 0 0 0 -0.0000016 0. -0.0000003 0.0000007 0. -0.0000001

975 0 -7 11 0 0 0 0 0 1 0 0 0 0 0 0.0000000 0. -0.0000003 0.0000000 0. -0.0000002

976 0 4 -4 0 0 0 0 0 0 0 0 -2 2 0 0.0000004 0. 0.0000000 0.0000000 0. 0.0000000

977 0 0 2 -3 0 0 0 0 0 0 0 0 0 0 -0.0000068 0. 0.0000039 0.0000000 0. 0.0000000

978 0 -4 4 0 0 0 0 0 0 0 0 2 -2 1 0.0000027 0. 0.0000000 -0.0000014 0. 0.0000000

979 0 4 -5 0 0 0 0 0 0 0 0 -1 1 0 0.0000000 0. -0.0000004 0.0000000 0. 0.0000000

980 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 -0.0000025 0. 0.0000000 0.0000000 0. 0.0000000

981 0 -4 7 0 0 0 0 0 1 0 0 0 0 0 -0.0000012 0. -0.0000003 0.0000006 0. -0.0000002

982 0 -4 6 0 0 0 0 0 0 0 0 1 -1 1 0.0000003 0. 0.0000000 -0.0000001 0. 0.0000000

983 0 -4 7 0 0 0 0 0 2 0 0 0 0 0 0.0000003 0. 0.0000066 -0.0000001 0. 0.0000029

984 0 -4 6 0 0 0 0 0 2 0 0 0 0 0 0.0000490 0. 0.0000000 -0.0000213 0. 0.0000000

985 0 -4 6 0 0 0 0 0 1 0 0 0 0 0 -0.0000022 0. 0.0000093 0.0000012 0. 0.0000049

986 0 -4 5 0 0 0 0 0 0 0 0 1 -1 1 -0.0000007 0. 0.0000028 0.0000004 0. 0.0000015

987 0 -4 6 0 0 0 0 0 1 0 0 0 0 0 -0.0000003 0. 0.0000013 0.0000002 0. 0.0000007

988 0 4 -6 0 0 0 0 0 0 0 0 0 0 0 -0.0000046 0. 0.0000014 0.0000000 0. 0.0000000

989 0 2 -2 0 0 0 0 0 0 -2 0 0 2 0 -0.0000005 0. 0.0000000 0.0000000 0. 0.0000000

990 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0.0000002 0. 0.0000001 0.0000000 0. 0.0000000

991 0 1 0 0 0 0 0 0 0 0 0 -1 1 0 0.0000000 0. -0.0000003 0.0000000 0. 0.0000000

992 0 1 -1 0 0 0 0 0 0 0 0 0 0 1 -0.0000028 0. 0.0000000 0.0000015 0. 0.0000000

993 0 0 -1 0 5 0 0 0 2 0 0 0 0 0 0.0000005 0. 0.0000000 -0.0000002 0. 0.0000000

994 0 0 1 -3 0 0 0 0 0 0 0 0 0 0 0.0000000 0. 0.0000003 0.0000000 0. 0.0000000

995 0 0 -1 3 0 0 0 0 2 0 0 0 0 0 -0.0000011 0. 0.0000000 0.0000005 0. 0.0000000

996 0 0 -7 12 0 0 0 0 2 0 0 0 0 0 0.0000000 0. 0.0000003 0.0000000 0. 0.0000001

997 0 -1 1 0 0 0 0 0 2 0 0 0 0 0 -0.0000003 0. 0.0000000 0.0000001 0. 0.0000000

998 0 -1 1 0 0 0 0 0 1 0 0 0 0 0 0.0000025 0. 0.0000106 -0.0000013 0. 0.0000057

999 0 -1 0 0 0 0 0 0 0 0 0 1 -1 1 0.0000005 0. 0.0000021 -0.0000003 0. 0.0000011

1000 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0.0001485 0. 0.0000000 0.0000000 0. 0.0000000

1001 0 1 -1 0 0 0 0 0 1 0 0 0 0 0 -0.0000007 0. -0.0000032 0.0000004 0. -0.0000017

1002 0 1 -2 0 0 0 0 0 0 0 0 1 -1 1 0.0000000 0. 0.0000005 0.0000000 0. 0.0000003

1003 0 0 -2 5 0 0 0 0 2 0 0 0 0 0 -0.0000006 0. -0.0000003 0.0000003 0. -0.0000002

1004 0 0 -1 0 4 0 0 0 2 0 0 0 0 0 0.0000030 0. -0.0000006 -0.0000013 0. -0.0000002

1005 0 0 1 0 -4 0 0 0 0 0 0 0 0 0 -0.0000004 0. 0.0000004 0.0000000 0. 0.0000000

1006 0 -1 1 0 0 0 0 0 0 0 0 0 0 1 -0.0000019 0. 0.0000000 0.0000010 0. 0.0000000

1007 0 0 -6 10 0 0 0 0 2 0 0 0 0 0 0.0000000 0. 0.0000004 -0.0000001 0. 0.0000002

1008 0 0 -6 10 0 0 0 0 0 0 0 0 0 0 0.0000000 0. 0.0000003 0.0000000 0. 0.0000000

1009 0 0 -3 0 3 0 0 0 0 0 0 2 -2 1 0.0000004 0. 0.0000000 -0.0000002 0. 0.0000000

1010 0 0 -3 7 0 0 0 0 2 0 0 0 0 0 0.0000000 0. -0.0000003 0.0000000 0. -0.0000001

1011 0 4 -4 0 0 0 0 0 0 -2 0 0 2 0 -0.0000003 0. 0.0000000 0.0000000 0. 0.0000000

1012 0 0 -5 8 0 0 0 0 2 0 0 0 0 0 0.0000005 0. 0.0000003 -0.0000002 0. 0.0000001

1013 0 0 5 -8 0 0 0 0 0 0 0 0 0 0 0.0000000 0. 0.0000011 0.0000000 0. 0.0000000

1014 0 0 -1 0 3 0 0 0 2 0 0 0 0 0 0.0000118 0. 0.0000000 -0.0000052 0. 0.0000000

1015 0 0 -1 0 3 0 0 0 1 0 0 0 0 0 0.0000000 0. -0.0000005 0.0000000 0. -0.0000003

1016 0 0 1 0 -3 0 0 0 0 0 0 0 0 0 -0.0000028 0. 0.0000036 0.0000000 0. 0.0000000

1017 0 2 -4 0 0 0 0 0 0 0 0 0 0 0 0.0000005 0. -0.0000005 0.0000000 0. 0.0000000

1018 0 -2 4 0 0 0 0 0 1 0 0 0 0 0 0.0000014 0. -0.0000059 -0.0000008 0. -0.0000031

1019 0 -2 3 0 0 0 0 0 0 0 0 1 -1 1 0.0000000 0. 0.0000009 0.0000001 0. 0.0000005
100 NUTATION SERIES

Term Fundamental Argument Multipliers Mi,j ∆ψ Coefficients ∆ Coefficients

i j= 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Si S ̇

i C0

i Ci C ̇

i S

0

i

00 00 00 00 00 00

1020 0 -2 4 0 0 0 0 0 2 0 0 0 0 0 -0.0000458 0. 0.0000000 0.0000198 0. 0.0000000

1021 0 -6 9 0 0 0 0 0 2 0 0 0 0 0 0.0000000 0. -0.0000045 0.0000000 0. -0.0000020

1022 0 -6 9 0 0 0 0 0 1 0 0 0 0 0 0.0000009 0. 0.0000000 -0.0000005 0. 0.0000000

1023 0 6 -9 0 0 0 0 0 0 0 0 0 0 0 0.0000000 0. -0.0000003 0.0000000 0. 0.0000000

1024 0 0 1 0 -2 0 0 0 0 0 0 0 0 1 0.0000000 0. -0.0000004 -0.0000001 0. -0.0000002

1025 0 -2 2 0 0 0 0 0 0 0 0 2 -2 1 0.0000011 0. 0.0000000 -0.0000006 0. 0.0000000

1026 0 0 -4 6 0 0 0 0 2 0 0 0 0 0 0.0000006 0. 0.0000000 -0.0000002 0. 0.0000000

1027 0 0 4 -6 0 0 0 0 0 0 0 0 0 0 -0.0000016 0. 0.0000023 0.0000000 0. 0.0000000

1028 0 3 -4 0 0 0 0 0 0 0 0 0 0 1 0.0000000 0. -0.0000004 0.0000000 0. -0.0000002

1029 0 0 -1 0 2 0 0 0 2 0 0 0 0 0 -0.0000005 0. 0.0000000 0.0000002 0. 0.0000000

1030 0 0 1 0 -2 0 0 0 0 0 0 0 0 0 -0.0000166 0. 0.0000269 0.0000000 0. 0.0000000

1031 0 0 1 0 -1 0 0 0 0 0 0 0 0 1 0.0000015 0. 0.0000000 -0.0000008 0. 0.0000000

1032 0 -5 9 0 0 0 0 0 2 0 0 0 0 0 0.0000010 0. 0.0000000 -0.0000004 0. 0.0000000

1033 0 0 3 -4 0 0 0 0 0 0 0 0 0 0 -0.0000078 0. 0.0000045 0.0000000 0. 0.0000000

1034 0 -3 4 0 0 0 0 0 2 0 0 0 0 0 0.0000000 0. -0.0000005 0.0000000 0. -0.0000002

1035 0 -3 4 0 0 0 0 0 1 0 0 0 0 0 0.0000007 0. 0.0000000 -0.0000004 0. 0.0000000

1036 0 3 -4 0 0 0 0 0 0 0 0 0 0 0 -0.0000005 0. 0.0000328 0.0000000 0. 0.0000000

1037 0 3 -4 0 0 0 0 0 1 0 0 0 0 0 0.0000003 0. 0.0000000 -0.0000002 0. 0.0000000

1038 0 0 2 -2 0 0 0 0 0 0 0 0 0 1 0.0000005 0. 0.0000000 -0.0000002 0. 0.0000000

1039 0 0 -1 0 2 0 0 0 0 0 0 0 0 1 0.0000000 0. 0.0000003 0.0000000 0. 0.0000001

1040 0 0 1 0 0 -3 0 0 0 0 0 0 0 0 -0.0000003 0. 0.0000000 0.0000000 0. 0.0000000

1041 0 0 1 0 1 -5 0 0 0 0 0 0 0 0 -0.0000003 0. 0.0000000 0.0000000 0. 0.0000000

1042 0 0 -1 0 1 0 0 0 1 0 0 0 0 0 0.0000000 0. -0.0000004 0.0000000 0. -0.0000002

1043 0 0 1 0 -1 0 0 0 0 0 0 0 0 0 -0.0001223 0. -0.0000026 0.0000000 0. 0.0000000

1044 0 0 1 0 -1 0 0 0 1 0 0 0 0 0 0.0000000 0. 0.0000007 0.0000000 0. 0.0000003

1045 0 0 1 0 -3 5 0 0 0 0 0 0 0 0 0.0000003 0. 0.0000000 0.0000000 0. 0.0000000

1046 0 -3 4 0 0 0 0 0 0 0 0 0 0 1 0.0000000 0. 0.0000003 0.0000000 0. 0.0000002

1047 0 0 1 0 0 -2 0 0 0 0 0 0 0 0 -0.0000006 0. 0.0000020 0.0000000 0. 0.0000000

1048 0 0 2 -2 0 0 0 0 0 0 0 0 0 0 -0.0000368 0. 0.0000000 0.0000000 0. 0.0000000

1049 0 0 1 0 0 -1 0 0 0 0 0 0 0 0 -0.0000075 0. 0.0000000 0.0000000 0. 0.0000000

1050 0 0 -1 0 1 0 0 0 0 0 0 0 0 1 0.0000011 0. 0.0000000 -0.0000006 0. 0.0000000

1051 0 0 -2 2 0 0 0 0 0 0 0 0 0 1 0.0000003 0. 0.0000000 -0.0000002 0. 0.0000000

1052 0 -8 14 0 0 0 0 0 2 0 0 0 0 0 -0.0000003 0. 0.0000000 0.0000001 0. 0.0000000

1053 0 0 1 0 2 -5 0 0 0 0 0 0 0 0 -0.0000013 0. -0.0000030 0.0000000 0. 0.0000000

1054 0 0 5 -8 3 0 0 0 0 0 0 0 0 0 0.0000021 0. 0.0000003 0.0000000 0. 0.0000000

1055 0 0 5 -8 3 0 0 0 2 0 0 0 0 0 -0.0000003 0. 0.0000000 0.0000001 0. 0.0000000

1056 0 0 -1 0 0 0 0 0 1 0 0 0 0 0 -0.0000004 0. 0.0000000 0.0000002 0. 0.0000000

1057 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0.0000008 0. -0.0000027 0.0000000 0. 0.0000000

1058 0 0 3 -8 3 0 0 0 0 0 0 0 0 0 -0.0000019 0. -0.0000011 0.0000000 0. 0.0000000

1059 0 0 -3 8 -3 0 0 0 2 0 0 0 0 0 -0.0000004 0. 0.0000000 0.0000002 0. 0.0000000

1060 0 0 1 0 -2 5 0 0 2 0 0 0 0 0 0.0000000 0. 0.0000005 0.0000000 0. 0.0000002

1061 0 -8 12 0 0 0 0 0 2 0 0 0 0 0 -0.0000006 0. 0.0000000 0.0000002 0. 0.0000000

1062 0 -8 12 0 0 0 0 0 0 0 0 0 0 0 -0.0000008 0. 0.0000000 0.0000000 0. 0.0000000

1063 0 0 1 0 1 -2 0 0 0 0 0 0 0 0 -0.0000001 0. 0.0000000 0.0000000 0. 0.0000000

1064 0 0 1 0 0 1 0 0 2 0 0 0 0 0 -0.0000014 0. 0.0000000 0.0000006 0. 0.0000000

1065 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0.0000006 0. 0.0000000 0.0000000 0. 0.0000000

1066 0 0 0 2 0 0 0 0 2 0 0 0 0 0 -0.0000074 0. 0.0000000 0.0000032 0. 0.0000000

1067 0 0 1 0 0 2 0 0 2 0 0 0 0 0 0.0000000 0. -0.0000003 0.0000000 0. -0.0000001

1068 0 -5 5 0 0 0 0 0 0 0 0 2 -2 1 0.0000004 0. 0.0000000 -0.0000002 0. 0.0000000

1069 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0.0000008 0. 0.0000011 0.0000000 0. 0.0000000

1070 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0.0000000 0. 0.0000003 0.0000000 0. 0.0000002

1071 0 0 1 0 1 0 0 0 2 0 0 0 0 0 -0.0000262 0. 0.0000000 0.0000114 0. 0.0000000

1072 0 3 -6 0 0 0 0 0 0 0 0 0 0 0 0.0000000 0. -0.0000004 0.0000000 0. 0.0000000

1073 0 -3 6 0 0 0 0 0 1 0 0 0 0 0 -0.0000007 0. 0.0000000 0.0000004 0. 0.0000000

1074 0 -3 6 0 0 0 0 0 2 0 0 0 0 0 0.0000000 0. -0.0000027 0.0000000 0. -0.0000012

1075 0 0 -1 4 0 0 0 0 2 0 0 0 0 0 -0.0000019 0. -0.0000008 0.0000008 0. -0.0000004

1076 0 -5 7 0 0 0 0 0 2 0 0 0 0 0 0.0000202 0. 0.0000000 -0.0000087 0. 0.0000000

1077 0 -5 7 0 0 0 0 0 1 0 0 0 0 0 -0.0000008 0. 0.0000035 0.0000005 0. 0.0000019

1078 0 -5 6 0 0 0 0 0 0 0 0 1 -1 1 0.0000000 0. 0.0000004 0.0000000 0. 0.0000002

1079 0 5 -7 0 0 0 0 0 0 0 0 0 0 0 0.0000016 0. -0.0000005 0.0000000 0. 0.0000000

1080 0 0 -1 0 1 0 0 0 0 0 0 2 -2 1 0.0000005 0. 0.0000000 -0.0000003 0. 0.0000000

1081 0 0 -1 0 1 0 0 0 0 0 0 0 0 0 0.0000000 0. -0.0000003 0.0000000 0. 0.0000000

1082 -1 0 3 0 0 0 0 0 2 0 0 0 0 0 0.0000001 0. 0.0000000 0.0000000 0. 0.0000000

1083 0 0 1 0 2 0 0 0 2 0 0 0 0 0 -0.0000035 0. -0.0000048 0.0000015 0. -0.0000021

1084 0 0 -2 6 0 0 0 0 2 0 0 0 0 0 -0.0000003 0. -0.0000005 0.0000001 0. -0.0000002

1085 0 2 -2 0 0 0 0 0 0 0 0 0 0 1 0.0000006 0. 0.0000000 -0.0000003 0. 0.0000000

1086 0 0 -6 9 0 0 0 0 2 0 0 0 0 0 0.0000003 0. 0.0000000 -0.0000001 0. 0.0000000

1087 0 0 6 -9 0 0 0 0 0 0 0 0 0 0 0.0000000 0. -0.0000005 0.0000000 0. 0.0000000

1088 0 -2 2 0 0 0 0 0 1 0 0 0 0 0 0.0000012 0. 0.0000055 -0.0000006 0. 0.0000029

1089 0 -2 1 0 0 0 0 0 0 0 0 1 -1 1 0.0000000 0. 0.0000005 0.0000000 0. 0.0000003

1090 0 2 -2 0 0 0 0 0 0 0 0 0 0 0 -0.0000598 0. 0.0000000 0.0000000 0. 0.0000000

1091 0 2 -2 0 0 0 0 0 1 0 0 0 0 0 -0.0000003 0. -0.0000013 0.0000001 0. -0.0000007

1092 0 0 1 0 3 0 0 0 2 0 0 0 0 0 -0.0000005 0. -0.0000007 0.0000002 0. -0.0000003

1093 0 0 -5 7 0 0 0 0 2 0 0 0 0 0 0.0000003 0. 0.0000000 -0.0000001 0. 0.0000000

1094 0 0 5 -7 0 0 0 0 0 0 0 0 0 0 0.0000005 0. -0.0000007 0.0000000 0. 0.0000000

1095 0 -2 2 0 0 0 0 0 0 0 0 0 0 1 0.0000004 0. 0.0000000 -0.0000002 0. 0.0000000

1096 0 0 4 -5 0 0 0 0 0 0 0 0 0 0 0.0000016 0. -0.0000006 0.0000000 0. 0.0000000

1097 0 1 -3 0 0 0 0 0 0 0 0 0 0 0 0.0000008 0. -0.0000003 0.0000000 0. 0.0000000

1098 0 -1 3 0 0 0 0 0 1 0 0 0 0 0 0.0000008 0. -0.0000031 -0.0000004 0. -0.0000016

1099 0 -1 2 0 0 0 0 0 0 0 0 1 -1 1 0.0000000 0. 0.0000003 0.0000000 0. 0.0000001

1100 0 -1 3 0 0 0 0 0 2 0 0 0 0 0 0.0000113 0. 0.0000000 -0.0000049 0. 0.0000000

1101 0 -7 10 0 0 0 0 0 2 0 0 0 0 0 0.0000000 0. -0.0000024 0.0000000 0. -0.0000010

1102 0 -7 10 0 0 0 0 0 1 0 0 0 0 0 0.0000004 0. 0.0000000 -0.0000002 0. 0.0000000

1103 0 0 3 -3 0 0 0 0 0 0 0 0 0 0 0.0000027 0. 0.0000000 0.0000000 0. 0.0000000

1104 0 -4 8 0 0 0 0 0 2 0 0 0 0 0 -0.0000003 0. 0.0000000 0.0000001 0. 0.0000000

1105 0 -4 5 0 0 0 0 0 2 0 0 0 0 0 0.0000000 0. -0.0000004 0.0000000 0. -0.0000002

1106 0 -4 5 0 0 0 0 0 1 0 0 0 0 0 0.0000005 0. 0.0000000 -0.0000002 0. 0.0000000

1107 0 4 -5 0 0 0 0 0 0 0 0 0 0 0 0.0000000 0. -0.0000003 0.0000000 0. 0.0000000

1108 0 0 1 1 0 0 0 0 2 0 0 0 0 0 -0.0000013 0. 0.0000000 0.0000006 0. 0.0000000

1109 0 0 -2 0 5 0 0 0 2 0 0 0 0 0 0.0000005 0. 0.0000000 -0.0000002 0. 0.0000000
NUTATION SERIES 101

Term Fundamental Argument Multipliers Mi,j ∆ψ Coefficients ∆ Coefficients

i j= 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Si S ̇

i C0

i Ci C ̇

i S

0

i

00 00 00 00 00 00

1110 0 0 0 3 0 0 0 0 2 0 0 0 0 0 -0.0000018 0. -0.0000010 0.0000008 0. -0.0000004

1111 0 1 0 0 0 0 0 0 0 0 0 0 0 0 -0.0000004 0. -0.0000028 0.0000000 0. 0.0000000

1112 0 1 0 0 0 0 0 0 2 0 0 0 0 0 -0.0000005 0. 0.0000006 0.0000002 0. 0.0000003

1113 0 -9 13 0 0 0 0 0 2 0 0 0 0 0 -0.0000003 0. 0.0000000 0.0000001 0. 0.0000000

1114 0 0 -1 5 0 0 0 0 2 0 0 0 0 0 -0.0000005 0. -0.0000009 0.0000002 0. -0.0000004

1115 0 0 -2 0 4 0 0 0 2 0 0 0 0 0 0.0000017 0. 0.0000000 -0.0000007 0. 0.0000000

1116 0 0 2 0 -4 0 0 0 0 0 0 0 0 0 0.0000011 0. 0.0000004 0.0000000 0. 0.0000000

1117 0 0 -2 7 0 0 0 0 2 0 0 0 0 0 0.0000000 0. -0.0000006 0.0000000 0. -0.0000002

1118 0 0 2 0 -3 0 0 0 0 0 0 0 0 0 0.0000083 0. 0.0000015 0.0000000 0. 0.0000000

1119 0 -2 5 0 0 0 0 0 1 0 0 0 0 0 -0.0000004 0. 0.0000000 0.0000002 0. 0.0000000

1120 0 -2 5 0 0 0 0 0 2 0 0 0 0 0 0.0000000 0. -0.0000114 0.0000000 0. -0.0000049

1121 0 -6 8 0 0 0 0 0 2 0 0 0 0 0 0.0000117 0. 0.0000000 -0.0000051 0. 0.0000000

1122 0 -6 8 0 0 0 0 0 1 0 0 0 0 0 -0.0000005 0. 0.0000019 0.0000002 0. 0.0000010

1123 0 6 -8 0 0 0 0 0 0 0 0 0 0 0 -0.0000003 0. 0.0000000 0.0000000 0. 0.0000000

1124 0 0 2 0 -2 0 0 0 0 0 0 0 0 1 -0.0000003 0. 0.0000000 0.0000002 0. 0.0000000

1125 0 0 -3 9 0 0 0 0 2 0 0 0 0 0 0.0000000 0. -0.0000003 0.0000000 0. -0.0000001

1126 0 0 5 -6 0 0 0 0 0 0 0 0 0 0 0.0000003 0. 0.0000000 0.0000000 0. 0.0000000

1127 0 0 5 -6 0 0 0 0 2 0 0 0 0 0 0.0000000 0. -0.0000006 0.0000000 0. -0.0000002

1128 0 0 2 0 -2 0 0 0 0 0 0 0 0 0 0.0000393 0. 0.0000003 0.0000000 0. 0.0000000

1129 0 0 2 0 -2 0 0 0 1 0 0 0 0 0 -0.0000004 0. 0.0000021 0.0000002 0. 0.0000011

1130 0 0 2 0 -2 0 0 0 2 0 0 0 0 0 -0.0000006 0. 0.0000000 0.0000003 0. -0.0000001

1131 0 -5 10 0 0 0 0 0 2 0 0 0 0 0 -0.0000003 0. 0.0000008 0.0000001 0. 0.0000004

1132 0 0 4 -4 0 0 0 0 0 0 0 0 0 0 0.0000008 0. 0.0000000 0.0000000 0. 0.0000000

1133 0 0 4 -4 0 0 0 0 2 0 0 0 0 0 0.0000018 0. -0.0000029 -0.0000008 0. -0.0000013

1134 0 -3 3 0 0 0 0 0 1 0 0 0 0 0 0.0000008 0. 0.0000034 -0.0000004 0. 0.0000018

1135 0 3 -3 0 0 0 0 0 0 0 0 0 0 0 0.0000089 0. 0.0000000 0.0000000 0. 0.0000000

1136 0 3 -3 0 0 0 0 0 1 0 0 0 0 0 0.0000003 0. 0.0000012 -0.0000001 0. 0.0000006

1137 0 3 -3 0 0 0 0 0 2 0 0 0 0 0 0.0000054 0. -0.0000015 -0.0000024 0. -0.0000007

1138 0 0 2 0 0 -3 0 0 0 0 0 0 0 0 0.0000000 0. 0.0000003 0.0000000 0. 0.0000000

1139 0 0 -5 13 0 0 0 0 2 0 0 0 0 0 0.0000003 0. 0.0000000 -0.0000001 0. 0.0000000

1140 0 0 2 0 -1 0 0 0 0 0 0 0 0 0 0.0000000 0. 0.0000035 0.0000000 0. 0.0000000

1141 0 0 2 0 -1 0 0 0 2 0 0 0 0 0 -0.0000154 0. -0.0000030 0.0000067 0. -0.0000013

1142 0 0 2 0 0 -2 0 0 0 0 0 0 0 0 0.0000015 0. 0.0000000 0.0000000 0. 0.0000000

1143 0 0 2 0 0 -2 0 0 1 0 0 0 0 0 0.0000000 0. 0.0000004 0.0000000 0. 0.0000002

1144 0 0 3 -2 0 0 0 0 0 0 0 0 0 0 0.0000000 0. 0.0000009 0.0000000 0. 0.0000000

1145 0 0 3 -2 0 0 0 0 2 0 0 0 0 0 0.0000080 0. -0.0000071 -0.0000035 0. -0.0000031

1146 0 0 2 0 0 -1 0 0 2 0 0 0 0 0 0.0000000 0. -0.0000020 0.0000000 0. -0.0000009

1147 0 0 -6 15 0 0 0 0 2 0 0 0 0 0 0.0000011 0. 0.0000005 -0.0000005 0. 0.0000002

1148 0 -8 15 0 0 0 0 0 2 0 0 0 0 0 0.0000061 0. -0.0000096 -0.0000027 0. -0.0000042

1149 0 -3 9 -4 0 0 0 0 2 0 0 0 0 0 0.0000014 0. 0.0000009 -0.0000006 0. 0.0000004

1150 0 0 2 0 2 -5 0 0 2 0 0 0 0 0 -0.0000011 0. -0.0000006 0.0000005 0. -0.0000003

1151 0 0 -2 8 -1 -5 0 0 2 0 0 0 0 0 0.0000000 0. -0.0000003 0.0000000 0. -0.0000001

1152 0 0 6 -8 3 0 0 0 2 0 0 0 0 0 0.0000123 0. -0.0000415 -0.0000053 0. -0.0000180

1153 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0.0000000 0. 0.0000000 -0.0000035 0. 0.0000000

1154 0 0 2 0 0 0 0 0 0 0 0 0 0 0 -0.0000005 0. 0.0000000 0.0000000 0. 0.0000000

1155 0 0 2 0 0 0 0 0 1 0 0 0 0 0 0.0000007 0. -0.0000032 -0.0000004 0. -0.0000017

1156 0 0 1 0 0 0 0 0 0 0 0 1 -1 1 0.0000000 0. -0.0000009 0.0000000 0. -0.0000005

1157 0 0 2 0 0 0 0 0 1 0 0 0 0 0 0.0000000 0. -0.0000004 0.0000000 0. 0.0000002

1158 0 0 2 0 0 0 0 0 2 0 0 0 0 0 -0.0000089 0. 0.0000000 0.0000038 0. 0.0000000

1159 0 0 -6 16 -4 -5 0 0 2 0 0 0 0 0 0.0000000 0. -0.0000086 -0.0000006 0. -0.0000019

1160 0 0 -2 8 -3 0 0 0 2 0 0 0 0 0 0.0000000 0. 0.0000000 0.0000006 0. -0.0000019

1161 0 0 -2 8 -3 0 0 0 2 0 0 0 0 0 -0.0000123 0. -0.0000416 0.0000053 0. -0.0000180

1162 0 0 6 -8 1 5 0 0 2 0 0 0 0 0 0.0000000 0. -0.0000003 0.0000000 0. -0.0000001

1163 0 0 2 0 -2 5 0 0 2 0 0 0 0 0 0.0000012 0. -0.0000006 -0.0000005 0. -0.0000003

1164 0 3 -5 4 0 0 0 0 2 0 0 0 0 0 -0.0000013 0. 0.0000009 0.0000006 0. 0.0000004

1165 0 -8 11 0 0 0 0 0 2 0 0 0 0 0 0.0000000 0. -0.0000015 0.0000000 0. -0.0000007

1166 0 -8 11 0 0 0 0 0 1 0 0 0 0 0 0.0000003 0. 0.0000000 -0.0000001 0. 0.0000000

1167 0 -8 11 0 0 0 0 0 2 0 0 0 0 0 -0.0000062 0. -0.0000097 0.0000027 0. -0.0000042

1168 0 0 11 0 0 0 0 0 2 0 0 0 0 0 -0.0000011 0. 0.0000005 0.0000005 0. 0.0000002

1169 0 0 2 0 0 1 0 0 2 0 0 0 0 0 0.0000000 0. -0.0000019 0.0000000 0. -0.0000008

1170 0 3 -3 0 2 0 0 0 2 0 0 0 0 0 -0.0000003 0. 0.0000000 0.0000001 0. 0.0000000

1171 0 0 4 -8 3 0 0 0 0 0 0 2 -2 1 0.0000000 0. 0.0000004 0.0000000 0. 0.0000002

1172 0 0 1 0 0 0 0 0 0 0 0 1 -1 0 0.0000000 0. 0.0000003 0.0000000 0. 0.0000000

1173 0 0 -4 8 -3 0 0 0 0 0 0 2 -2 1 0.0000000 0. 0.0000004 0.0000000 0. 0.0000002

1174 0 0 1 2 0 0 0 0 2 0 0 0 0 0 -0.0000085 0. -0.0000070 0.0000037 0. -0.0000031

1175 0 0 2 0 1 0 0 0 2 0 0 0 0 0 0.0000163 0. -0.0000012 -0.0000072 0. -0.0000005

1176 0 -3 7 0 0 0 0 0 2 0 0 0 0 0 -0.0000063 0. -0.0000016 0.0000028 0. -0.0000007

1177 0 0 0 4 0 0 0 0 2 0 0 0 0 0 -0.0000021 0. -0.0000032 0.0000009 0. -0.0000014

1178 0 -5 6 0 0 0 0 0 2 0 0 0 0 0 0.0000000 0. -0.0000003 0.0000000 0. -0.0000001

1179 0 -5 6 0 0 0 0 0 1 0 0 0 0 0 0.0000003 0. 0.0000000 -0.0000002 0. 0.0000000

1180 0 5 -6 0 0 0 0 0 0 0 0 0 0 0 0.0000000 0. 0.0000008 0.0000000 0. 0.0000000

1181 0 5 -6 0 0 0 0 0 2 0 0 0 0 0 0.0000003 0. 0.0000010 -0.0000001 0. 0.0000004

1182 0 0 2 0 2 0 0 0 2 0 0 0 0 0 0.0000003 0. 0.0000000 -0.0000001 0. 0.0000000

1183 0 0 -1 6 0 0 0 0 2 0 0 0 0 0 0.0000000 0. -0.0000007 0.0000000 0. -0.0000003

1184 0 0 7 -9 0 0 0 0 2 0 0 0 0 0 0.0000000 0. -0.0000004 0.0000000 0. -0.0000002

1185 0 2 -1 0 0 0 0 0 0 0 0 0 0 0 0.0000006 0. 0.0000019 0.0000000 0. 0.0000000

1186 0 2 -1 0 0 0 0 0 2 0 0 0 0 0 0.0000005 0. -0.0000173 -0.0000002 0. -0.0000075

1187 0 0 6 -7 0 0 0 0 2 0 0 0 0 0 0.0000000 0. -0.0000007 0.0000000 0. -0.0000003

1188 0 0 5 -5 0 0 0 0 2 0 0 0 0 0 0.0000007 0. -0.0000012 -0.0000003 0. -0.0000005

1189 0 -1 4 0 0 0 0 0 1 0 0 0 0 0 -0.0000003 0. 0.0000000 0.0000002 0. 0.0000000

1190 0 -1 4 0 0 0 0 0 2 0 0 0 0 0 0.0000003 0. -0.0000004 -0.0000001 0. -0.0000002

1191 0 -7 9 0 0 0 0 0 2 0 0 0 0 0 0.0000074 0. 0.0000000 -0.0000032 0. 0.0000000

1192 0 -7 9 0 0 0 0 0 1 0 0 0 0 0 -0.0000003 0. 0.0000012 0.0000002 0. 0.0000006

1193 0 0 4 -3 0 0 0 0 2 0 0 0 0 0 0.0000026 0. -0.0000014 -0.0000011 0. -0.0000006

1194 0 0 3 -1 0 0 0 0 2 0 0 0 0 0 0.0000019 0. 0.0000000 -0.0000008 0. 0.0000000

1195 0 -4 4 0 0 0 0 0 1 0 0 0 0 0 0.0000006 0. 0.0000024 -0.0000003 0. 0.0000013

1196 0 4 -4 0 0 0 0 0 0 0 0 0 0 0 0.0000083 0. 0.0000000 0.0000000 0. 0.0000000

1197 0 4 -4 0 0 0 0 0 1 0 0 0 0 0 0.0000000 0. -0.0000010 0.0000000 0. -0.0000005

1198 0 4 -4 0 0 0 0 0 2 0 0 0 0 0 0.0000011 0. -0.0000003 -0.0000005 0. -0.0000001

1199 0 0 2 1 0 0 0 0 2 0 0 0 0 0 0.0000003 0. 0.0000000 -0.0000001 0. 0.0000001
102 NUTATION SERIES

Term Fundamental Argument Multipliers Mi,j ∆ψ Coefficients ∆ Coefficients

i j= 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Si S ̇

i C0

i Ci C ̇

i S

0

i

00 00 00 00 00 00

1200 0 0 -3 0 5 0 0 0 2 0 0 0 0 0 0.0000003 0. 0.0000000 -0.0000001 0. 0.0000000

1201 0 1 1 0 0 0 0 0 0 0 0 0 0 0 -0.0000004 0. 0.0000000 0.0000000 0. 0.0000000

1202 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0.0000005 0. -0.0000023 -0.0000003 0. -0.0000012

1203 0 1 1 0 0 0 0 0 2 0 0 0 0 0 -0.0000339 0. 0.0000000 0.0000147 0. 0.0000000

1204 0 -9 12 0 0 0 0 0 2 0 0 0 0 0 0.0000000 0. -0.0000010 0.0000000 0. -0.0000005

1205 0 0 3 0 -4 0 0 0 0 0 0 0 0 0 0.0000005 0. 0.0000000 0.0000000 0. 0.0000000

1206 0 1 -1 0 0 0 0 0 0 0 0 2 -2 1 0.0000003 0. 0.0000000 -0.0000001 0. 0.0000000

1207 0 0 7 -8 0 0 0 0 2 0 0 0 0 0 0.0000000 0. -0.0000004 0.0000000 0. -0.0000002

1208 0 0 3 0 -3 0 0 0 0 0 0 0 0 0 0.0000018 0. -0.0000003 0.0000000 0. 0.0000000

1209 0 0 3 0 -3 0 0 0 2 0 0 0 0 0 0.0000009 0. -0.0000011 -0.0000004 0. -0.0000005

1210 0 -2 6 0 0 0 0 0 2 0 0 0 0 0 -0.0000008 0. 0.0000000 0.0000004 0. 0.0000000

1211 0 -6 7 0 0 0 0 0 1 0 0 0 0 0 0.0000003 0. 0.0000000 -0.0000001 0. 0.0000000

1212 0 6 -7 0 0 0 0 0 0 0 0 0 0 0 0.0000000 0. 0.0000009 0.0000000 0. 0.0000000

1213 0 0 6 -6 0 0 0 0 2 0 0 0 0 0 0.0000006 0. -0.0000009 -0.0000002 0. -0.0000004

1214 0 0 3 0 -2 0 0 0 0 0 0 0 0 0 -0.0000004 0. -0.0000012 0.0000000 0. 0.0000000

1215 0 0 3 0 -2 0 0 0 2 0 0 0 0 0 0.0000067 0. -0.0000091 -0.0000029 0. -0.0000039

1216 0 0 5 -4 0 0 0 0 2 0 0 0 0 0 0.0000030 0. -0.0000018 -0.0000013 0. -0.0000008

1217 0 3 -2 0 0 0 0 0 0 0 0 0 0 0 0.0000000 0. 0.0000000 0.0000000 0. 0.0000000

1218 0 3 -2 0 0 0 0 0 2 0 0 0 0 0 0.0000000 0. -0.0000114 0.0000000 0. -0.0000050

1219 0 0 3 0 -1 0 0 0 2 0 0 0 0 0 0.0000000 0. 0.0000000 0.0000023 0. 0.0000000

1220 0 0 3 0 -1 0 0 0 2 0 0 0 0 0 0.0000517 0. 0.0000016 -0.0000224 0. 0.0000007

1221 0 0 3 0 0 -2 0 0 2 0 0 0 0 0 0.0000000 0. -0.0000007 0.0000000 0. -0.0000003

1222 0 0 4 -2 0 0 0 0 2 0 0 0 0 0 0.0000143 0. -0.0000003 -0.0000062 0. -0.0000001

1223 0 0 3 0 0 -1 0 0 2 0 0 0 0 0 0.0000029 0. 0.0000000 -0.0000013 0. 0.0000000

1224 0 0 1 0 -1 0 0 0 0 0 0 2 -2 1 -0.0000004 0. 0.0000000 0.0000002 0. 0.0000000

1225 0 -8 16 0 0 0 0 0 2 0 0 0 0 0 -0.0000006 0. 0.0000000 0.0000003 0. 0.0000000

1226 0 0 3 0 2 -5 0 0 2 0 0 0 0 0 0.0000005 0. 0.0000012 -0.0000002 0. 0.0000005

1227 0 0 7 -8 3 0 0 0 2 0 0 0 0 0 -0.0000025 0. 0.0000000 0.0000011 0. 0.0000000

1228 0 0 -5 16 -4 -5 0 0 2 0 0 0 0 0 -0.0000003 0. 0.0000000 0.0000001 0. 0.0000000

1229 0 0 3 0 0 0 0 0 2 0 0 0 0 0 0.0000000 0. 0.0000004 0.0000000 0. 0.0000002

1230 0 0 -1 8 -3 0 0 0 2 0 0 0 0 0 -0.0000022 0. 0.0000012 0.0000010 0. 0.0000005

1231 0 -8 10 0 0 0 0 0 2 0 0 0 0 0 0.0000050 0. 0.0000000 -0.0000022 0. 0.0000000

1232 0 -8 10 0 0 0 0 0 1 0 0 0 0 0 0.0000000 0. 0.0000007 0.0000000 0. 0.0000004

1233 0 -8 10 0 0 0 0 0 2 0 0 0 0 0 0.0000000 0. 0.0000003 0.0000000 0. 0.0000001

1234 0 0 2 2 0 0 0 0 2 0 0 0 0 0 -0.0000004 0. 0.0000004 0.0000002 0. 0.0000002

1235 0 0 3 0 1 0 0 0 2 0 0 0 0 0 -0.0000005 0. -0.0000011 0.0000002 0. -0.0000005

1236 0 -3 8 0 0 0 0 0 2 0 0 0 0 0 0.0000000 0. 0.0000004 0.0000000 0. 0.0000002

1237 0 -5 5 0 0 0 0 0 1 0 0 0 0 0 0.0000004 0. 0.0000017 -0.0000002 0. 0.0000009

1238 0 5 -5 0 0 0 0 0 0 0 0 0 0 0 0.0000059 0. 0.0000000 0.0000000 0. 0.0000000

1239 0 5 -5 0 0 0 0 0 1 0 0 0 0 0 0.0000000 0. -0.0000004 0.0000000 0. -0.0000002

1240 0 5 -5 0 0 0 0 0 2 0 0 0 0 0 -0.0000008 0. 0.0000000 0.0000004 0. 0.0000000

1241 0 2 0 0 0 0 0 0 0 0 0 0 0 0 -0.0000003 0. 0.0000000 0.0000000 0. 0.0000000

1242 0 2 0 0 0 0 0 0 1 0 0 0 0 0 0.0000004 0. -0.0000015 -0.0000002 0. -0.0000008

1243 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0.0000370 0. -0.0000008 -0.0000160 0. 0.0000000

1244 0 0 7 -7 0 0 0 0 2 0 0 0 0 0 0.0000000 0. 0.0000000 0.0000000 0. -0.0000003

1245 0 0 7 -7 0 0 0 0 2 0 0 0 0 0 0.0000000 0. 0.0000003 0.0000000 0. 0.0000001

1246 0 0 6 -5 0 0 0 0 2 0 0 0 0 0 -0.0000006 0. 0.0000003 0.0000003 0. 0.0000001

1247 0 7 -8 0 0 0 0 0 0 0 0 0 0 0 0.0000000 0. 0.0000006 0.0000000 0. 0.0000000

1248 0 0 5 -3 0 0 0 0 2 0 0 0 0 0 -0.0000010 0. 0.0000000 0.0000004 0. 0.0000000

1249 0 4 -3 0 0 0 0 0 2 0 0 0 0 0 0.0000000 0. 0.0000009 0.0000000 0. 0.0000004

1250 0 1 2 0 0 0 0 0 2 0 0 0 0 0 0.0000004 0. 0.0000017 -0.0000002 0. 0.0000007

1251 0 -9 11 0 0 0 0 0 2 0 0 0 0 0 0.0000034 0. 0.0000000 -0.0000015 0. 0.0000000

1252 0 -9 11 0 0 0 0 0 1 0 0 0 0 0 0.0000000 0. 0.0000005 0.0000000 0. 0.0000003

1253 0 0 4 0 -4 0 0 0 2 0 0 0 0 0 -0.0000005 0. 0.0000000 0.0000002 0. 0.0000000

1254 0 0 4 0 -3 0 0 0 2 0 0 0 0 0 -0.0000037 0. -0.0000007 0.0000016 0. -0.0000003

1255 0 -6 6 0 0 0 0 0 1 0 0 0 0 0 0.0000003 0. 0.0000013 -0.0000002 0. 0.0000007

1256 0 6 -6 0 0 0 0 0 0 0 0 0 0 0 0.0000040 0. 0.0000000 0.0000000 0. 0.0000000

1257 0 6 -6 0 0 0 0 0 1 0 0 0 0 0 0.0000000 0. -0.0000003 0.0000000 0. -0.0000002

1258 0 0 4 0 -2 0 0 0 2 0 0 0 0 0 -0.0000184 0. -0.0000003 0.0000080 0. -0.0000001

1259 0 0 6 -4 0 0 0 0 2 0 0 0 0 0 -0.0000003 0. 0.0000000 0.0000001 0. 0.0000000

1260 0 3 -1 0 0 0 0 0 0 0 0 0 0 0 -0.0000003 0. 0.0000000 0.0000000 0. 0.0000000

1261 0 3 -1 0 0 0 0 0 1 0 0 0 0 0 0.0000000 0. -0.0000010 -0.0000001 0. -0.0000006

1262 0 3 -1 0 0 0 0 0 2 0 0 0 0 0 0.0000031 0. -0.0000006 -0.0000013 0. 0.0000000

1263 0 0 4 0 -1 0 0 0 2 0 0 0 0 0 -0.0000003 0. -0.0000032 0.0000001 0. -0.0000014

1264 0 0 4 0 0 -2 0 0 2 0 0 0 0 0 -0.0000007 0. 0.0000000 0.0000003 0. 0.0000000

1265 0 0 5 -2 0 0 0 0 2 0 0 0 0 0 0.0000000 0. -0.0000008 0.0000000 0. -0.0000004

1266 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0.0000003 0. -0.0000004 0.0000000 0. 0.0000000

1267 0 8 -9 0 0 0 0 0 0 0 0 0 0 0 0.0000000 0. 0.0000004 0.0000000 0. 0.0000000

1268 0 5 -4 0 0 0 0 0 2 0 0 0 0 0 0.0000000 0. 0.0000003 0.0000000 0. 0.0000001

1269 0 2 1 0 0 0 0 0 2 0 0 0 0 0 0.0000019 0. -0.0000023 0.0000002 0. -0.0000010

1270 0 2 1 0 0 0 0 0 1 0 0 0 0 0 0.0000000 0. 0.0000000 -0.0000010 0. 0.0000000

1271 0 2 1 0 0 0 0 0 1 0 0 0 0 0 0.0000000 0. 0.0000003 0.0000000 0. 0.0000002

1272 0 -7 7 0 0 0 0 0 1 0 0 0 0 0 0.0000000 0. 0.0000009 -0.0000001 0. 0.0000005

1273 0 7 -7 0 0 0 0 0 0 0 0 0 0 0 0.0000028 0. 0.0000000 0.0000000 0. 0.0000000

1274 0 4 -2 0 0 0 0 0 1 0 0 0 0 0 0.0000000 0. -0.0000007 0.0000000 0. -0.0000004

1275 0 4 -2 0 0 0 0 0 2 0 0 0 0 0 0.0000008 0. -0.0000004 -0.0000004 0. 0.0000000

1276 0 4 -2 0 0 0 0 0 0 0 0 0 0 0 0.0000000 0. 0.0000000 0.0000000 0. -0.0000002

1277 0 4 -2 0 0 0 0 0 0 0 0 0 0 0 0.0000000 0. 0.0000003 0.0000000 0. 0.0000000

1278 0 0 5 0 -4 0 0 0 2 0 0 0 0 0 -0.0000003 0. 0.0000000 0.0000001 0. 0.0000000

1279 0 0 5 0 -3 0 0 0 2 0 0 0 0 0 -0.0000009 0. 0.0000000 0.0000004 0. 0.0000001

1280 0 0 5 0 -2 0 0 0 2 0 0 0 0 0 0.0000003 0. 0.0000012 -0.0000001 0. 0.0000005

1281 0 3 0 0 0 0 0 0 2 0 0 0 0 0 0.0000017 0. -0.0000003 0.0000000 0. -0.0000001

1282 0 -8 8 0 0 0 0 0 1 0 0 0 0 0 0.0000000 0. 0.0000007 0.0000000 0. 0.0000004

1283 0 8 -8 0 0 0 0 0 0 0 0 0 0 0 0.0000019 0. 0.0000000 0.0000000 0. 0.0000000

1284 0 5 -3 0 0 0 0 0 1 0 0 0 0 0 0.0000000 0. -0.0000005 0.0000000 0. -0.0000003

1285 0 5 -3 0 0 0 0 0 2 0 0 0 0 0 0.0000014 0. -0.0000003 -0.0000001 0. 0.0000000

1286 0 -9 9 0 0 0 0 0 1 0 0 0 0 0 0.0000000 0. 0.0000000 0.0000000 0. -0.0000001

1287 0 -9 9 0 0 0 0 0 1 0 0 0 0 0 0.0000000 0. 0.0000000 -0.0000005 0. 0.0000000

1288 0 -9 9 0 0 0 0 0 1 0 0 0 0 0 0.0000000 0. 0.0000005 0.0000000 0. 0.0000003

1289 0 9 -9 0 0 0 0 0 0 0 0 0 0 0 0.0000013 0. 0.0000000 0.0000000 0. 0.0000000
NUTATION SERIES 103

Term Fundamental Argument Multipliers Mi,j ∆ψ Coefficients ∆ Coefficients

i j= 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Si S ̇

i C0

i Ci C ̇

i S

0

i

00 00 00 00 00 00

1290 0 6 -4 0 0 0 0 0 1 0 0 0 0 0 0.0000000 0. -0.0000003 0.0000000 0. -0.0000002

1291 0 0 6 0 0 0 0 0 2 0 0 0 0 0 0.0000002 0. 0.0000009 0.0000003 0. 0.0000004

1292 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0.0000000 0. 0.0000000 -0.0000004 0. 0.0000000

1293 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0.0000008 0. 0.0000000 0.0000000 0. 0.0000000

1294 0 0 6 0 0 0 0 0 1 0 0 0 0 0 0.0000000 0. 0.0000004 0.0000000 0. 0.0000002

1295 0 0 6 0 0 0 0 0 2 0 0 0 0 0 0.0000006 0. 0.0000000 -0.0000003 0. 0.0000000

1296 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0.0000006 0. 0.0000000 0.0000000 0. 0.0000000

1297 0 0 6 0 0 0 0 0 1 0 0 0 0 0 0.0000000 0. 0.0000003 0.0000000 0. 0.0000001

1298 0 0 6 0 0 0 0 0 2 0 0 0 0 0 0.0000005 0. 0.0000000 -0.0000002 0. 0.0000000

1299 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0.0000003 0. 0.0000000 -0.0000001 0. 0.0000000

1300 0 0 2 0 -2 0 0 0 0 1 0 0 -2 0 -0.0000003 0. 0.0000000 0.0000000 0. 0.0000000

1301 0 2 -2 0 0 0 0 0 0 1 0 0 -2 0 0.0000006 0. 0.0000000 0.0000000 0. 0.0000000

1302 0 0 1 0 -1 0 0 0 0 1 0 0 -2 0 0.0000007 0. 0.0000000 0.0000000 0. 0.0000000

1303 0 1 -1 0 0 0 0 0 0 1 0 0 -2 0 -0.0000004 0. 0.0000000 0.0000000 0. 0.0000000

1304 0 3 -3 0 0 0 0 0 0 -1 0 0 0 0 0.0000004 0. 0.0000000 0.0000000 0. 0.0000000

1305 0 0 2 0 -2 0 0 0 0 -1 0 0 0 0 0.0000006 0. 0.0000000 0.0000000 0. 0.0000000

1306 0 0 4 -8 3 0 0 0 0 -1 0 0 2 0 0.0000000 0. -0.0000004 0.0000000 0. 0.0000000

1307 0 0 4 -8 3 0 0 0 0 1 0 0 -2 0 0.0000000 0. -0.0000004 0.0000000 0. 0.0000000

1308 0 0 4 -8 3 0 0 0 0 -2 0 0 2 0 0.0000005 0. 0.0000000 0.0000000 0. 0.0000000

1309 0 0 2 0 -3 0 0 0 0 -1 0 0 0 0 -0.0000003 0. 0.0000000 0.0000000 0. 0.0000000

1310 0 0 1 0 -1 0 0 0 0 -1 0 0 0 0 0.0000004 0. 0.0000000 0.0000000 0. 0.0000000

1311 0 1 -1 0 0 0 0 0 0 -1 0 0 0 0 -0.0000005 0. 0.0000000 0.0000000 0. 0.0000000

1312 0 2 -2 0 0 0 0 0 0 -1 0 0 2 0 0.0000004 0. 0.0000000 0.0000000 0. 0.0000000

1313 0 0 1 0 0 0 0 0 0 1 0 -1 1 0 0.0000000 0. 0.0000003 0.0000000 0. 0.0000000

1314 0 0 2 0 -3 0 0 0 0 -1 0 0 2 0 0.0000013 0. 0.0000000 0.0000000 0. 0.0000000

1315 0 0 2 0 -3 0 0 0 0 -2 0 0 0 0 0.0000021 0. 0.0000011 0.0000000 0. 0.0000000

1316 0 0 4 -8 3 0 0 0 0 1 0 0 0 0 0.0000000 0. -0.0000005 0.0000000 0. 0.0000000

1317 0 0 -1 0 0 0 0 0 0 -1 0 1 -1 1 0.0000000 0. -0.0000005 0.0000000 0. -0.0000002

1318 0 0 -1 0 0 0 0 0 0 1 0 1 -1 1 0.0000000 0. 0.0000005 0.0000000 0. 0.0000003

1319 0 0 4 -8 3 0 0 0 0 -1 0 0 0 0 0.0000000 0. -0.0000005 0.0000000 0. 0.0000000

1320 0 0 2 0 -2 0 0 0 0 -1 0 0 2 1 -0.0000003 0. 0.0000000 0.0000002 0. 0.0000000

1321 0 0 2 0 -2 0 0 0 0 0 0 0 0 0 0.0000020 0. 0.0000010 0.0000000 0. 0.0000000

1322 0 0 2 0 -2 0 0 0 0 -1 0 0 2 0 -0.0000034 0. 0.0000000 0.0000000 0. 0.0000000

1323 0 3 -3 0 0 0 0 0 0 -1 0 0 2 0 -0.0000019 0. 0.0000000 0.0000000 0. 0.0000000

1324 0 0 -2 0 2 0 0 0 0 1 0 0 -2 1 0.0000003 0. 0.0000000 -0.0000002 0. 0.0000000

1325 0 -3 3 0 0 0 0 0 0 1 0 2 -2 2 -0.0000003 0. 0.0000000 0.0000001 0. 0.0000000

1326 0 0 -2 0 2 0 0 0 0 1 0 2 -2 2 -0.0000006 0. 0.0000000 0.0000003 0. 0.0000000

1327 0 1 -1 0 0 0 0 0 0 1 0 0 0 0 -0.0000004 0. 0.0000000 0.0000000 0. 0.0000000

1328 0 0 1 0 -1 0 0 0 0 1 0 0 0 0 0.0000003 0. 0.0000000 0.0000000 0. 0.0000000

1329 0 2 -2 0 0 0 0 0 0 0 0 0 -2 0 0.0000003 0. 0.0000000 0.0000000 0. 0.0000000

1330 0 0 1 0 -1 0 0 0 0 0 0 0 -2 0 0.0000004 0. 0.0000000 0.0000000 0. 0.0000000

1331 0 -2 2 0 0 0 0 0 0 0 0 2 0 2 0.0000003 0. 0.0000000 -0.0000001 0. 0.0000000

1332 0 0 -1 0 1 0 0 0 0 0 0 2 0 2 0.0000006 0. 0.0000000 -0.0000003 0. 0.0000000

1333 0 -1 1 0 0 0 0 0 0 0 0 2 0 2 -0.0000008 0. 0.0000000 0.0000003 0. 0.0000000

1334 0 -2 3 0 0 0 0 0 0 0 0 2 0 2 0.0000000 0. 0.0000003 0.0000000 0. 0.0000001

1335 0 0 2 0 -2 0 0 0 0 0 0 0 2 0 -0.0000003 0. 0.0000000 0.0000000 0. 0.0000000

1336 0 0 1 0 0 0 0 0 0 0 0 1 1 2 0.0000000 0. -0.0000003 0.0000000 0. -0.0000002

1337 0 0 1 0 0 0 0 0 0 1 0 2 0 2 0.0000126 0. -0.0000063 -0.0000055 0. -0.0000027

1338 0 10 -3 0 0 0 0 0 0 -1 0 2 0 2 -0.0000005 0. 0.0000000 0.0000002 0. 0.0000001

1339 0 0 1 0 0 0 0 0 0 0 0 1 1 1 -0.0000003 0. 0.0000028 0.0000002 0. 0.0000015

1340 0 0 1 0 0 0 0 0 0 1 0 2 0 2 0.0000005 0. 0.0000000 -0.0000002 0. 0.0000001

1341 0 0 4 -8 3 0 0 0 0 0 0 2 0 2 0.0000000 0. 0.0000009 0.0000001 0. 0.0000004

1342 0 0 -4 8 -3 0 0 0 0 0 0 2 0 2 0.0000000 0. 0.0000009 -0.0000001 0. 0.0000004

1343 0 0 -4 8 -3 0 0 0 0 -1 0 2 0 2 -0.0000126 0. -0.0000063 0.0000055 0. -0.0000027

1344 0 0 -2 0 3 0 0 0 0 2 0 2 -2 2 0.0000003 0. 0.0000000 -0.0000001 0. 0.0000000

1345 0 0 -2 0 3 0 0 0 0 1 0 2 0 1 0.0000021 0. -0.0000011 -0.0000011 0. -0.0000006

1346 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0.0000000 0. -0.0000004 0.0000000 0. 0.0000000

1347 0 0 1 0 0 0 0 0 0 -1 0 2 0 1 -0.0000021 0. -0.0000011 0.0000011 0. -0.0000006

1348 0 0 2 0 -2 0 0 0 0 -2 0 2 2 2 -0.0000003 0. 0.0000000 0.0000001 0. 0.0000000

1349 0 2 -3 0 0 0 0 0 0 0 0 2 0 2 0.0000000 0. 0.0000003 0.0000000 0. 0.0000001

1350 0 1 -1 0 0 0 0 0 0 0 0 2 0 2 0.0000008 0. 0.0000000 -0.0000004 0. 0.0000000

1351 0 0 1 0 -1 0 0 0 0 0 0 2 0 2 -0.0000006 0. 0.0000000 0.0000003 0. 0.0000000

1352 0 2 -2 0 0 0 0 0 0 0 0 2 0 2 -0.0000003 0. 0.0000000 0.0000001 0. 0.0000000

1353 0 0 -1 0 1 0 0 0 0 -1 0 2 2 2 0.0000003 0. 0.0000000 -0.0000001 0. 0.0000000

1354 0 -1 1 0 0 0 0 0 0 1 0 2 0 2 -0.0000003 0. 0.0000000 0.0000001 0. 0.0000000

1355 0 0 2 0 -3 0 0 0 0 -1 0 2 2 2 -0.0000005 0. 0.0000000 0.0000002 0. 0.0000000

1356 0 0 2 0 -3 0 0 0 0 2 0 2 0 2 0.0000024 0. -0.0000012 -0.0000011 0. -0.0000005

1357 0 0 -4 8 -3 0 0 0 0 1 0 2 0 2 0.0000000 0. 0.0000003 0.0000000 0. 0.0000001

1358 0 0 4 -8 3 0 0 0 0 1 0 2 0 2 0.0000000 0. 0.0000003 0.0000000 0. 0.0000001

1359 0 0 1 0 0 0 0 0 0 1 0 1 1 1 0.0000000 0. 0.0000003 0.0000000 0. 0.0000002

1360 0 0 1 0 0 0 0 0 0 0 0 2 0 2 -0.0000024 0. -0.0000012 0.0000010 0. -0.0000005

1361 0 0 1 0 0 0 0 0 0 2 0 2 0 1 0.0000004 0. 0.0000000 -0.0000002 0. -0.0000001

1362 0 0 2 0 -2 0 0 0 0 -1 0 2 2 2 0.0000013 0. 0.0000000 -0.0000006 0. 0.0000000

1363 0 3 -3 0 0 0 0 0 0 -1 0 2 2 2 0.0000007 0. 0.0000000 -0.0000003 0. 0.0000000

1364 0 1 -1 0 0 0 0 0 0 1 0 2 0 2 0.0000003 0. 0.0000000 -0.0000001 0. 0.0000000

1365 0 0 2 0 -2 0 0 0 0 0 0 2 2 2 0.0000003 0. 0.0000000 -0.0000001 0. 0.0000000
Errata & Updates

Errata in this circular and updates to it are given at URL 1. Please consult the web page at this

URL periodically.

104