19780004170

NASA-RP- 1009

78N12113

An introduction to orbit dynamics and its application to satellite-based

earth monitoring systems

NASA Reference Publication 1009

An Introduction to Orbit

Dynamics and Its Application

to Satellite-Based Earth

Monitoring Missions

David R. Brooks

Langley Research Center

Hampton, Virginia

National Aeronautics

and Space Administration

Scientific and Technical

Informalion Office

1977

PREFACE

This report provides, by analysis and example, an appreciation of the longterm behavior of orbiting satellites at a level of complexity suitable for the

initial phases of planning Earth monitoring missions. The basic orbit dynamics

of satellite motion are covered in detail. Of particular interest are orbit

plane precession, Sun-synchronous orbits, and establishment of conditions for

repetitive surface coverage. Orbit plane precession relative to the Sun is

shown to be the driving factor in observed patterns of surface illumination,

on which are superimposed effects of the seasonal motion of the Sun relative

to the equator. Considerable attention is given to the special geometry of Sunsynchronous orbits, orbits whose precession rate matches the average apparent

precession rate of the Sun. The solar and orbital plane motions take place

within an inertial framework, and these motions are related to the longitudelatitude coordinates of the satellite ground track through appropriate spatial

and temporal coordinate systems. It is shown how orbit parameters can be chosen

to give repetitive coverage of the same set of longitude-latitude values on a

daily or longer basis. Several potential Earth monitoring missions which illustrate representative applications of the orbit dynamics are described. The

interactions between orbital properties and the resulting coverage and illumination patterns over specific sites on the Earth's surface are shown to be at the

heart of satellite mission planning. A requirement for global surface coverage

is seen to be a potentially severe constraint on satellite systems, depending

on the mission objectives. In some contexts, this requirement strongly suggests multiple-satellite systems, whereas in less restrictive situations,

"global coverage" can be redefined such that a single satellite can provide

adequate data. The approach taken in the examples stresses the need for considering mission output as a whole, treating the entire data base as an entity

rather than a series of separate measurements. This is consistent with use of

this type of data as input to large-scale, long-term activities which can best

justify the use of satellite systems.

iii

CONTENTS

PREFACE.................................. iii

I. INTRODUCTION............................ I

II. DYNAMICSOFEARTH-ORBITINGSATELLITES................ 2

Coordinate Systems in Space .................... 2

Coordinate Systems for Time and the Relationship Between

Longitude and Right Ascension .................. 5

Equations for the Propagation of Satellite Orbits ......... 9

Sun-SynchronousOrbits ...................... 15

III. ESTABLISHINGPATTERNSOFSURFACECOVERAGEFROMA SATELLITE ..... 20

Ground Tracks for Circular Orbits ................. 20

Designing Orbits for Repetitive Longitude-Latitude Coverage .... 22

Solar Illumination Considerations in Satellite Orbit Analysis . 25

Pointing Angle Geometries for Locating the Sun .......... 28

The Geometry of Sunrise and Sunset as Viewed From a Satellite . 30

IV. SOMEEXAMPLESOFGEOGRAPHICAL COVERAGEANDSOLARILLUMINATION

PATTERNSFORLONG-DURATIONEARTHMONITORINGMISSIONS ....... 33

Establishing a Needfor Repetitive CoveragePatterns ....... 33

Yearly Illumination Patterns Over a Single Site .......... 34

Yearly Variations in Illumination Angle as a Function of

Latitude ............................ 37

Solar Illumination Effects on Global Surface Coverage ....... 38

Tangent-Point Distributions For Solar Occultation Experiments 41

V. SYMBOLS............................... 47

APPENDIXA - "BASIC" LANGUAGEALGORITHMSFORCALCULATINGORBIT

PARAMETERS...............................

APPENDIXB - ORBITELEMENTSFORTHEEARTH'SMOTIONAROUNDTHESUN.....

APPENDIXC - DETERMININGTHECONDITIONSUNDERWHICHTHESUNIS

OCCULTEDBYTHEEARTHRELATIVETOA SATELLITEHAVINGFIXEDORBIT

ELEMENTS................................

REFERENCES ................................

TABLES ..................................

51

55

56

59

60

v

I. INTRODUCTION

Since the beginning of the space age, satellites have been employed to

examine our own planet. There is currently an abundanceof missions - speculative, planned, and operational - to exploit the capabilities of a variety of

space-based sensing systems in order to understand, monitor, control, and utilize the Earth and its atmosphere. Oneof the initial steps in designing such

missions is, invariably, to gain an understanding of howa satellite orbit can

best be matched to the requirements for meeting one or more objectives. This

matching involves not just selecting the best initial or instantaneous orbit

parameters but also attempting to maximize the useful return of data over the

desired range of conditions for the entire life of the mission. The purpose of

this report is to provide, by analysis and example, an appreciation of the longterm behavior of orbiting satellites at a level of complexity suitable for the

initial phases of planning Earth monitoring missions.

Section II of the report, "Dynamics of Earth-Orbiting Satellites," deals

with the basic orbit dynamics required to depict the motion of satellites in

their orbits. The time scale of interest for Earth observation ranges typically from a few months to a few years. This meansthat the major effects of

the Earth's nonsymmetrical gravitational field must be accounted for but some

less significant orbital perturbations can be ignored at this stage of analysis. With these limits in mind, it is possible to achieve a sufficiently representative and self-consistent picture of satellite motion which allows an

analyst to study and comparevarious orbit options while avoiding the complexities and loose ends of higher order orbit dynamics. A subsection of Section II

deals with Sun-synchronous orbits, which have important implications becauseof

their special geometrical relationship to the Sun's apparent motion about the

Earth.

Section III, "Establishing Patterns of Surface Coverage From a Satellite,"

deals with viewing the Earth's surface and the ways in which orbits can be modified to produce particular patterns of coverage. The problems of keeping track

of the Sun's apparent motion with respect to a satellite are examined in detail.

This involves such requirements as looking at (or not looking at) the Sunfor

measurementsor calibration as well as accounting for variations in lighting conditions beneath the path of a satellite during the entire course of a mission.

Section IV, "SomeExamplesof Geographical Coverage and Solar Illumination

Patterns for Long-Duration Earth Monitoring Missions," presents sometypical

applications of the orbit analyses of the previous sections. Several different types of missions are considered, each with its own sampling and coverage

requirements. The limitations of a single satellite in satisfying someglobal

coverage requirements are illustrated, and the alternative of multiple-satellite

systems is briefly discussed.

The general aim of the analytic sections of this report is to present a

tutorial approach to orbit dynamics around which a mission analyst, starting
with no previous knowledge of orbital mechanics, can construct a set of selfcontained and self-consistent computing tools. The requirements for the exact

form of such tools will vary with circumstance, so that no attempt has been made

here to formulate specific computer programs, although some BASIC language algorithms have been given in appendix A for a few standard calculations. In the

applications section (Section IV) the aim has been not to exhaust the possibilities of analysis for each topic of interest, but to provide, through typical

situations, a feeling for the potential and limitations of space-based Earth

monitoring missions. In particular, Section IV is intended to provide some

insight into several types of coverage patterns and how they develop over

seasonal and yearly time periods.

II. DYNAMICS OF EARTH-ORBITING SATELLITES

In this section, the equations required to propagate a satellite along its

orbit are considered. First, the necessary coordinate systems are derived for

locating a satellite in time and space relative to the Earth and Sun. Then,

first-order perturbations are introduced to take into account the well-known

oblateness of the Earth, the result of which is that orbital paths are not fixed

in inertial space but precess about the Earth.

Coordinate Systems in Space

As outlined in the Introduction, analysis for Earth observations involves

locations on the Earth's surface (the satellite ground track, for example) and

also the Sun's position relative to a satellite in space or its viewing point on

the Earth's surface or elsewhere. Thus, it is natural to assume that two basic

coordinate systems for locating points will be required. One is a system of

locating points on the Earth's surface - longitudes and latitudes - and the

other describes the apparent motion of the Sun through space relative to the

Earth.

The longitude-latitude L-k system is described in figure I. The equator

forms the x-y plane, and the x-axis is arbitrarily defined by a vector in the

Z

f/

i \

Greenwich /

meridian/ / / i

Figure I.- Definition of a longitude-latitude coordinate system.
equatorial plane which passes through the samemeridian that passes through

Greenwich, England. This meridian has a longitude of 0° by definition. The

latitude of a point P is defined as the angle between the equator and P, measured along the meridian passing through P. Longitude is defined as the angle

between the Greenwich meridian and the meridian passing through P. Latitude is

usually defined as having values between 90° (North Pole) and -90° (South Pole).

Longitude usually takes positive values between 0° and 180° for points east of

Greenwich and between 0o and -180° for points west of Greenwich, although it is

often convenient in computer applications to define longitude between 0° and

360° measuredeast from Greenwich.

The Earth-Sun geometry is illustrated in figure 2. The ecliptic plane is

defined by the Earth's motion around the Sun. The x-axis of an Earth-centered

/ Ecliptic _S_-'-- /

/ n°rth A p°le //_ -_'_''_ /

o . ox /

Equatorial plane

Figure 2.- Earth's motion around the Sun.

Cartesian coordinate system is defined by the vector pointing from the Earth to

the Sun at the instant of the vernal equinox, that is, the instant at which the

subsolar point crosses the equator moving from south to north. This Earthcentered geometry is shown in figure 3. The celestial sphere is a fictitious

globe of apparently infinite radius which contains the "fixed "I stars against

which the Sun, the planets, and the satellite appear to move as viewed from the

Earth. The projection of the Earth's equator onto the celestial sphere is

called the equinoctial and the ecliptic is defined by the Sun's trace as it

IAs used herein, "fixed "means that the motion of stars against the celestial sphere is too small to be noticed on the time scales of importance in this

analysis.
North

Pole

Figure 3.- Celestial sphere and right-ascensionmdeclination

coordinate system.

appears to move among the stars. The vernal equinox represents a particular

location on the celestial sphere, determined by the intersection of the equinoctial and the ecliptic, as the Sun moves from south to north. From the signs of

the zodiac, this intersection is called the first point of Aries and is represented by the symbol for the ram's head T. This definite point in the heavens

is now located about 15 ° west of the constellation Pleiades. (The directions

east and west on the celestial sphere are defined in the same way as an observer

would identify these directions on a globe model of the Earth.)

The reference planes and directions thus established depend on the motion

of the Earth about the Sun and about its own axis. The Earth can be thought of

as an orbiting gyroscope which maintains the reference systems and their interrelationships. For many purposes, this is a sufficiently accurate mental picture, but in fact the gyroscope analogy suffers from some irregularities.

Chiefly because of the small unsymmetrical forces of the Sun and the Moon on

the Earth's equatorial bulge and secondarily because of the perturbing effects

of the other planets, the Earth's axis precesses in space at the rate of about

50.25 seconds of arc in a year, advancing the time of the vernal equinox by

about 20 minutes per year. This results in a slow but steady westward motion

in the location of the first point of Aries, with a period of about 25 725

years. An additional secondary effect is the variation of the obliquity (inclination) of the Earth's equator to the ecliptic plane. The obliquity is currently about 23°.44 and is decreasing at a rate of about 47 seconds of arc per

century. Detailed discussions of these effects are given in reference I.

4
It is easy to see from figure 3 that the position of a point P on the

celestial sphere can be defined by two angles relative to the vernal equinox

vector and the equinoctial. The right ascension _ is measured along the equinoctial positive to the east from T, and the declination _ is measured north

from the equinoctial. The dizzying change in perspective from figure 2 to figure 3 is actually simply accomplished by vector transformations. Starting with

an Earth ephemeris (a table of or an analytic expression for Earth positions

relative to the Sun as a function of time), the negative of the Earth's position

vector in heliocentric space is the Sun's position in a geocentric ecliptic system. Since the ecliptic and equatorial systems share the same x-axis (that is,

T), the transformation into the equatorial (equinoctial) system involves only a

vector rotation about T. Note now that the L-k and _-6 coordinate systems both

have the equatorial plane as their basis, with only the difference in x-axis to

distinguish them. Note too, that values of right ascension and declination,

although defined against the celestial sphere, are as suitable for defining

points on the Earth's surface as are longitudes and latitudes. For this analysis, declination and latitude are, in fact, equivalent and can be used interchangeably. Finally, note that the right ascension-declination system is an

inertial coordinate system in space as perceived by an observer having the vantage point implied by figure 2. That is, although the origin of the coordinate

system rotates around the Sun with the Earth, the directions of the three axes

remain fixed in space. From the vantage point of an observer on Earth, an inertial system is one whose axes are pointed at fixed positions on the celestial

sphere. The discussion of the previous paragraph shows that this inertial framework is only an approximation, as could eventually be determined (and has been)

by patient observation of the Earth-Sun system as shown in figure 2. However,

for present purposes, the inertial framework is a useful and very good approximation for analysis and will henceforth be accepted as valid. The relationship

between the inertial (right-ascension--declination) and rotating (longitudelatitude) systems depends on the location of the Greenwich meridian relative to

T, the specification of which requires definition of a coordinate system for

time. This problem is dealt with in the next section, which digresses long

enough to present the necessary background information and ends with the equations which allow transformation back and forth between longitude and right

ascensi on.

Coordinate Systems for Time and the Relationship

Between Longitude and Right Ascension

The definition of a time-measuring system requires an agreed-upon starting

point and a set of periodic events which can be used to measure intervals of

time. Fundamental to an understanding of the physical sciences is an acceptance

of the existence of a uniform time which corresponds to the time variable in

dynamical equations. However, in practice, it is necessary to admit that measures of time exist only as matters of definition, that these definitions are

imperfect ones applied to particular situations, and that discrepancies will

always exist at some level of application.

If the Earth-Sun-celestial sphere system is used as the means of defining

a time system, there are two conventional ways to proceed - sidereal time and

5
solar time. Sidereal time is governed by the rotation of the Earth relative to

the stars. If the instant of passage of a particular star over a meridian is

noted, the star will cross the samemeridian again in exactly one sidereal day,

by definition. This definition is complicated by the motion of the vernal equinox, but for most purposes the problems are not worth considering.

Solar time is governed by the rotation of the Earth relative to the Sun.

If the instant of passage of the Sun over a meridian is noted, the Sunwill

cross the samemeridian again in exactly one solar day, by definition. This

observable solar day is the basis of what is called apparent solar time. It

is not hard to see the difficulties in applying this definition to practical

measurements. It is complicated by the slightly elliptical motion of the Earth

around the Sun and by the fact that the Sun does not movein the equatorial

plane, with the result that the observed length of solar days is only approximately constant. To circumvent this problem, "meansolar time" is defined relative to a fictitious "mean sun" which movesalong the equinoctial (the projection of the equator on the celestial sphere) at a constant rate equal to the

average apparent rate of motion of the real Sun. Although the meansolar time

system is the basis of all civil time measurements- the time by which people

set their watches - it is still full of inconsistencies because of the variable

rotation rate of the Earth, the fact that the fictitious meansun cannot be

observed, and the fact that the accepted description of the fictitious meansun

is based on another time system (ephemeris time), which dependson the internal

consistency of the gravitational theories describing the entire solar system.

However, at somepoint it is necessary to cease worrying about discrepancies

and accept meansolar time as a precise and uniform method of relating nonsimultaneous events. Hereinafter, "time" meansmeansolar time unless stated otherwise. A "day" is 86 400 meansolar seconds, by definition, and a "second" is

always a meansolar second.

The meaning of a "year" is complicated by the variety of choices available.

Thus, a sidereal year is based on the motion of the Earth relative to the stars -

the cycle time for astronomical observation. A year defined relative to the Sun

is based on the passage of the Earth from one vernal (or autumnal) equinox to

the next. This interval of time is called a tropical year - the cycle time for

Earth seasons. It is nearly constant, but not quite, because of the interaction

of the planets in the solar system. It is slowly decreasing linearly with time

at a rate of 0.53 sec/century (ref. I); a current approximate length, good for

the pertinent time frame, is 365.24220 days. This discussion makesclear that

calendar systems are bound to be complicated, as they seek to maintain seasonal

relationships by using units (days) which are not exact subdivisions of the

tropical (seasonal) year.

Within the framework of meansolar days, there are two time systems of

interest in the bookkeeping sense. The first is universal time (UT), a unit of

which is, for all practical purposes, equal to the corresponding unit of mean

solar time. Universal time is the practical basis of civil time. It is

expressed in hours, minutes, and seconds or in fractions of a meansolar day.

The zero reference for a universal time day is Greenwich midnight, at the start

of a calendar day. Hence, Jan. 1.0 is the start of calendar day Jan. I, at

Greenwichmidnight.
The other system of interest, and an important one for analysis, is the

Julian date (J.D.) system, which is a method of counting meansolar days consecutively. Its origin was set long enough ago that all recorded astronomical

events can, in principle, be assigned an unambiguousJulian date. Since the

system was devised by astronomers for their own purposes, the Julian day starts

at Greenwich noon so that all astronomical observations during a night of viewing activity have the sameJulian date. Onestandard reference epoch for Julian

date tables is Jan. 0.5, 1900 (UT). This is Greenwich noon on calendar day

Dec. 31, 1899, or Julian date 241 5020.0. Julian dates are used extensively

for keeping track of days in preference to the more cumbersomecalendar system.

A Julian year is exactly 365.25 days, the average length of the calendar year

according to the Julian calendar, and, advantageously, is an exact decimal fraction. A Julian century is 36 525 days. Table I, from reference I, lists Julian

dates for the first day of each month in the last half of the 20th century. Note

that the calendar day starts at a Julian date of XXXXXXX.5, so that Jan. 1.0,

1950, is J.D. 243 3282.5.

Returning now to the relationship between right ascension and longitude,

it can be expressed in terms of a constant related to a particular epoch, the

Earth's rotational rate, and somemeasureof time. Reference I gives the following equation for the right ascension of the Greenwichmeridian at 12 midnight

(0 hours UT):

_g,o = 99°'6909833 + 36000°'7689T + 0°'00038708T2 (I)

where T is the time in Julian centuries from Jan. 0.5, 1900:

T = J.D. - 241 5020.0

36 525

(2)

The appearance of a second-order term in T is due to the accumulated effects

of small secular and periodic variations in the Earth's rotational rate and

cannot be accounted for in terms of the previous discussion; this part of equation (I) must just be accepted as fact. Note that although equation (I) appears

to give ag,o as a continuous function of T, it does not actually do so. The

value of T must correspond to a time of Oh UT, that is, to Julian dates of the

form XXX XXXX.5. A detailed discussion of equation (I) and its relationship to

the mean solar time system may be found in reference I.

At this point, it is useful to reconsider the concept of sidereal time,

that is, time relative to a "fixed" x-axis - the vernal equinox. In this sense,

the angular position of any meridian is equivalent to some fraction of a sidereal day, with 360 ° = 24 sidereal hours. Thus, sidereal time and right ascension are equivalent measurements of angular position. To determine the sidereal

time (right ascension) of any meridian at any time, it is only necessary to know

the rotational rate of the Earth @. Then

a = eg,o + (t - to)e + Lm- ag + Lm (3)

where t - to is the universal time of interest and Lm is the longitude of

the meridian. The Sun's apparent sidereal rate is 360 ° in 365.2422 mean Solar

days, so that the Earth must rotate 360 + 360/365.2422 = 360.9856473 degrees

7
per mean solar day or 0.25068447 degree per mean solar minute. An example (from

ref. 2) will make clear the necessary computations: What is the local sidereal

time !right ascension _) of a point at 298°.2213 east longitude, at 10h15m30 s UT

on Oct. 12, 1962? (The superscripts show the conventional notations for hours,

minutes, and seconds.)

Convert to Julian days:

243 7938.0

+0.5

+11.0

243 7949.5

J.D., Oct. 0.5, 1962 (from table I)

Add 0.5 day to obtain J.D., Oct. 1.0, 1962

Add 11 days to obtain J.D., Oct. 12.0, 1962

J.D., Oh UT, Oct. 12, 1962

From equations (I) and (2),

T = 243 7949.5 - 241 5020.0 = 0.62777549 Julian century

36 525

_g,o : 99°'69098 + 280°'40033 + 0°'0001525 = 20°'0915

10h15m30 s : 615.5 min

And from equation (3),

eg = 20o.0915 + (615.5 - 0)(0°.25068447) = 174°.3878

Thus,

= 174o.3878 + 298°.2213 = 112o.6091

Note the convention of expressing sidereal "time" in terms of an angle. It

could be expressed as 112o.6091 = (24)(112.6091/360) hours = 7h30m26 s, but the

time units are not the same as mean solar time unitsl Therefore, to avoid confusion, the angular notation is used.

A final topic of practical interest in a discussion of time systems is

the matter of converting back and forth between the Julian date system and a

calendar-clock system. This is a somewhat tedious procedure for hand calculations but is easily taken care of in computerized orbit analysis with the aid

of transformation subroutines. It is necessary only to keep in mind that clock

times derived from Julian dates are Greenwich times, so that the local clock

hour, if that is desired, depends on the time zone of the point of interest.

There are 24 such zones, 15 ° wide, one of which is centered about the Greenwich

meridian, ±7.5 ° • For the purposes of analysis, all time zones are considered to

follow meridians, so that local geographical and "daylight savings" fluctuations, such as are common over populated areas of the Earth, are ignored. For

example, eastern standard time is 5 hours behind Greenwich mean time, being in

the time zone centered at -75 °. Therefore, local noon EST occurs at

J.D. XXX XXXX.O + 5/24 = XXX XXXX.208333.
Equations for the Propagation of Satellite Orbits

In an unperturbed spherically symmetric gravitational field, orbiting

objects follow the classical laws of Keplerian dynamics. The position of a

satellite In a Cartesian coordinate system can be specified completely either

by a set of six orbit elements or by six vector components - a position vector

plus a velocity vector. (For a basic review of elliptic orbital motion, see

ref. 3.) Figure 4 shows the orientation of an orbit in the right ascensionz ra

(

rp

r

Equator

Y

r + r a

a= P

2

rp = a(1- e)

r a = a(1 + e)

p = a(1 - e 2)

x T

Figure 4.- Defining the position and velocity of a satellite

in orbit around the Earth.

declination system previously discussed in detail. Two of the orbit elements,

right ascension of the ascending node _ and inclination i, position the orbital plane in space. The remaining angular elements, argument of perigee m and

true anomaly f, specify the angular position of a satellite at point P along

its orbital path relative to the x-y (equatorial) plane. The size and shape of

the orbital path are determined by the semimaJor axis a and eccentricity e.

(Many orbits of interest for Earth monitoring have semimajor axes between about

6600 and 8000 km and are circular, or nearly so, with eccentricities in the

range 0.0 to 0.1.) The semimajor axis a is the average of the perigee and

apogee rp and ra (minimum and maximum radii relative to the source of the

gravitational field), as shown in figure 4. The eccentricity e can be thought

of as a dimensionless measure of the departure from a circular orbit of radius

a, re]ating a to rp and ra as shown. The semilatus rectum p is a

derived quantity which will be used in the following analysis; it is measured

through the gravitational center along a line perpendicular to a line from perigee to apogee and is equal to the semimaJor axis for a circular orbit. Knowledge of all six Keplerian elements - a, e, i, _, _, and f - is equivalent
to knowing the satellite's state vector (position and velocity), which may be

obtained in terms of orbit elements as follows (ref. 3):

rx = [cos f(cos _ cos _ - cos i sin _ sin _) + sin f(-sin _ cos

- cos i sin _ cos _ P

I + e cos f

(4a)

ry = [cos f(cos _ sin _ + cos i cos _ sin _) + sin f(-sin _ sin

+ cos i cos _ cos _ P

I + e cos f

(4b)

P

r z = Lrc°s f sin i sin _ + sin f sin i cos _71- + e COS f

(4c)

rx : \_ [(cos f + e)(-sin _ cos _ cos i sin _ cos _) - sin f(cos _ cos

Vr

- cos i sin _ sin _)] (4d)

_y : _cos f + e)(-sin _ cos _ + cos i cos _ cos _) - sin f (cos m sin

+ cos i cos _ sin _)] (4e)

rz : _p___cos f + e)(sin i cos _ - sin f sin i sin _

(4f)

The gravitational constant _ is equal to 398 601.2 km3/sec 2 for the Earth; it

is this quantity which takes into account the mass associated with a gravitational source and, hence, determines the magnitude of the velocities associated

with orbits around a particular body. The inverse of the transformation in equations (4), from state vector to orbit elements, and many other relationships

referring to Keplerian orbits and their elements are summarized in reference 3.

Because of the Earth's oblateness, not all the Keplerian elements are constant for Earth satellites. In particular, the orientation of the orbital plane

and the argument (location) of the perigee m precess as a result of the

effects of the bulge around the Earth's equator. These precessions, which can

be understood in terms of gyroscopic analogies, were thoroughly discussed in the

late 1950's. Reference 3 contains an extensive bibliography. The perturbations

in _ and _ are significant, amounting to several degrees per day, and therefore must be accounted for.

The usual treatment of gravitational potential involves expansion of the

potential in a series. The following discussion utilizes the first-order theory

of Kozai (ref. 4), involving the so-called J2 term in the gravitational representation. In this formulation

I0
= 3 J2 r@-2 M(2 - 2.5 sin 2 i)

2 p2

: _ 3 J2 r_2 _ cos i

2 p2

where

J2 = 1.6238235 × 10-3 (a dimensionless constant)

2

(5a)

(5b)

r@ Earth's mean equatorial radius, 6378.145 km

perturbed mean motion of the satellite in its orbit, rad/sec

The perturbed mean motion M (in radians per second) is based on a modification

of the unperturbed (Keplerian) value, which is

Mo : 2__ (6)

T o

where

To : 2_aV_ (7)

Then, according to Kozai,

M = MO_ + _ J2 _]_-- r_2 _ - e2_ - _ sin2 (8)

The mean anomaly at any time is, by definition, just the constant value of

multiplied by the time interval from perigee. It is equal to the true anomaly

f only for circular orbits, since according to Kepler's laws of orbital motion,

the time derivative of true anomaly f is a function of position in an orbit

when e # 0, being larger (faster) at perigee than at apogee. (If this were

not so, orbiting objects in elliptical orbits could not sweep out equal areas

in equal time, as required by Kepler's laws.) By definition, the time required

for the mean anomaly to proceed from 0 to 2w tad (0 ° to 360o), that is, from

perigee to perigee, is called the anomalistic period TA:

_A - Z. (9)

There is another equally valid definition of period brought about by the motion

of the perigee - the nodal period TN. Figure 5 illustrates the geometry.

According to this figure,

2_ (10)

TN - _ +

11
Initial __

condition: o)o,__

Anomalistic

per iod

Perigee Node

to perigee to node

Nodal period

Figure 5.- Anomalistic and nodal orbit periods.

For perigee precession in the direction shown, the anomalistic period is a little longer than the nodal period. Note that the nodal period is the one by

which equator crossings of an object in a circular orbit would be counted.

That is, if a satellite started at its ascending node on the equator, it would

cross again after one nodal period. Note, too, that TN as defined only

describes the observable behavior of objects in circular orbits, or the mean

motion for elliptical orbits. That is, the nodal crossing times of an object

in an elliptical orbit - what might be called the observed nodal periods - are

not constant but vary with the location of the perigee. Such an object does

not cross a given latitude after a length of time TN" Therefore, a distinction must in general be made between the actual position (true anomaly) andthe

"average" position described by the preceding equations for the mean anomaly.

The relation between the two involves the solution of a transcendental equation,

Kepler's equation, a discussion of which can be found in any textbook on orbital

mechanics. Briefly,

M": E - e sin E (11)

where M and E must be expressed in radians and E is called the eccentric anomaly; E can be directly related to true anomaly in a number of ways

(ref. 3). For example,

f : cos-l{ cos E- e

_I - e cos E/

(12)

There are many ways to solve equation (11) iteratively for E. For example, an

easily programed "graphical" solution is given in reference 5. In many cases, a

series expanslon to the required accuracy can be done without iteration to get

directly from M to f. (See ref. 3.)

Equations (5) can be used to propagate Earth orbit elements in the absence

of additional perturbations. Of course, a consideration of atmospheric drag

would introduce such perturbations - specifically, a secular (nonperiodic)

change in semimaJor axis. There are.other sources which could be considered

too, in addition to those due to the neglected higher order gravitational terms.

12
One example would be the effects of solar radiation, which are known to cause

substantial orbit perturbations on hollow balloon satellites. These effects

will be ignored for analysis of Earth observation mission, as they will add

little to understanding patterns of surface coverage and other pertinent

relationships.

At any time t after some initial orbit configuration,

(13a)

(13b)

A useful formulation gives the changes A_ and A_

anomaly step AM:

A_ = (1.6238235 x I0-3)r_ 2 (2 - 2.5 sin 2 i) AM

p2

corresponding to a mean-

(14a)

(-1.6238235 x 10-3)rm2.cosi

p2

(14b)

Expressing AM in degrees gives _,_ and A_ units of degrees. The time

required for a step AM is

t : TA AM

360 (15)

Note that for AM : 360 ° , the time required for a step will be one anomalistic period. Often it is desirable to use nodal periods to generate conditions

at successive crossings of a particular latitude. Then AM should be modified:

AM N : AM T--_ N (16)

TA

Thus, in a computer program for generating orbits, AM = 360 ° could he input to

produce one orbit revolution, and if desired, it could be modified in the program by an input flag to generate nodal instead of anomalistic revolutions. Of

course, any fraction or multiple of either anomalistic or nodal revolutions can

be generated with equations (14) to (16) by the appropriate choice of AM.

As a final comment on the generation and propagation of orbits with the

first-order equations, note that the concept of a perigee must be retained even

for circular orbits. It must be assigned an initial value and its motion must

be accounted for with equation (14a). This follows from figure 5, which shows

that from the point of view of an observer on the Earth, the precession of the

perigee in a circular orbit is part of, and cannot be distinguished from, the

motion of the satellite around its orbit. Thus, if an observer timed successive

equator crossings of a satellite in a circular orbit, he would measure the nodal

period, which can only be related to the orbit dynamics through both the mean

motion and the perigee precession. The distinction would become clear to the

observer only if the satellite were in an eccentric orbit, in which case he

13
would note that the altitude at each crossing and the time between crossings

undergo a periodic cycle with a period equal to the time for the perigee to

precess 360 ° .

In order to illustrate the magnitude of first-order effects on typical

Earth orbits, table 2 lists Keplerian (unperturbed) and perturbed periods for

a range of satellite altitudes h and inclinations i, and table 3 lists precession rates _ and _ as a function of inclination over the same range of

altitudes. For the usually small values of e associated with Earth observation studies, the restriction of tables 2 and 3 to circular orbits does not prevent the values from being representative, as can be seen by examining the contribution of e (through p) in equations (5). The information in tables 2 and

3 is also plotted in figures 6 and 7, respectively.

7000

6800 /"

6600 ./

6400 /

6200 /

sec°' 6000 /

/

5800 /

/

5600 /

5400

/

52OO

2O0 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500

Alti_de, km

(a) Keplerian period.

Figure 6.- Keplerian period and deviations from the Keplerian period TO

as a function of orbit altitude and inclination.

14
10

-2

h = 1500 k_ I

TN::__or- TO -10-6:2::::::_ /

TA - To, h = 200 km

sec

TN - To, h = 200 km

TN - ¢o, h = 1500 km

TA - TO, h = 200 km

_'A - To' h = 1500 krn

_14

h_-I o0 7

-18

__rh_ 200 km

-22 -- /

-26 0 10 20 30 40 50 60 70 80 90

i, deg

(b) Deviations from a Keplerian period.

Figure 6.- Concluded.

Sun-Synchronous Orbits

For many types of Earth observation it would be desirable to view the same

surface regions repeatedly under constant lighting conditions. If it were possible to establish an orbit whose nodal precession matched the solar precession,

then the Sun-orbit plane geometry would be fixed, with the lighting conditions

at a particular latitude dependent only on the north-south movement of the Sun

with the seasons. The best available way to accomplish this is to set the orbital precession rate equal tothe average solar precession rate. Recall that the

Sun precesses 360 ° in I tropical year of 365.2422 days. Therefore, from equation (5b), the orbital precession rate, in degrees per day, is

360 : 0.9856473 : _ : (-1.6238235 x 10-3)r@2_ cos i

365.2422 p2

Substituting for M from equation (8) and converting the units of M from

radians per second to degrees per day yields:

0.9856473 : -2.0645874 x 1014 cos iF1 + 6.605833 x 104 )_

a3.5(i _ e2)2 [ _(_ _ e2_3/2 (I - 1.5 sin 2 i.j (17)

15
Precession

rate,

deg/day

18

16

14

12

10

6

4

.

h, km

00

\

\

\

o\

\

0

\

\

\ \

15oo ///

/

-6 _ //

200 _/

-8 f

I

-I0

0 10 20 30 40 50 60 70 80 90 100

i , deg

Figure 7.- Precession rates for perigee and right ascension of the ascending

node for perturbed circular orbits, as a function of inclination.

16
Note that because of the minus sign appearing in the expression for _, the Sunsynchronous inclination will always be greater than 90 °. For many purposes it

may be sufficient to ignore the bracketed term in equation (17). As an example,

for a = 7000 km and e = O, the Sun-synchronous inclination is is found to

be 97°.874474 with this term ignored. Iterating once by substituting this value

for i in the bracketed term and recomputing yield i s = 97°.879528. The single iteration procedure gives a Sun-synchronous orbit precession which matches

the Sun's position after I year to better than 10 -5 deg. The variation of Sunsynchronous orbit inclination for circular orbits up to an altitude of 1600 km

is given in figure 8.

i s , deg

103

102

i01

100

99

98

97j

96

200 400 600 800 I000 1200 1400 1600

h, km

Figure 8.- Variation of Sun-synchronous orbit inclination

for circular orbits as a function of altitude h.

It is interesting to consider how well a Sun-synchronous orbit actually

maintains a constant relationship to the Sun throughout the year. To do this,

first assume a fictitious sun 2 that travels at a constant rate along the ecliptic

projection on the celestial sphere (fig. 3). This would mean that the fictitious sun travels around the celestial sphere at a constant rate of 0.9856473

deg/day. Figure 9 shows how to compare the right ascension of the ascending

node of a satellite orbit with the right ascension of the fictitious subsolar

2This "fictitious sun" is an ad hoc mathematical entity for the purposes of

this discussion, and is not the same as the fictitious mean sun which forms the

basis for the mean solar time system. (See p. 6 and ref. I.)

17
x

Celestial

/ X Fictiti_ous subsolar point

Eclip ht ascension of satellite

[_-" I-__.LL:-----'_.._-.. ascending node after time t

Equinoc tial_._._------"_

_c

T

c = 0.9856473t

tan _ = cos 23°.44 tan c

c

a c = right ascension of fictitious subsolar point after time t

Figure 9.- Relationship between a Sun-synchronous orbit and the subsolar

point for a fictitious sun which moves at a constant rate along the

ecliptic.

point. It assumes that a Sun-synchronous orbit is started at the vernal equinox, so that the subsolar point is initially in the orbital plane. For constant

motion of this fictitious sun, the right ascensions must be equal at each equinox (where the ecliptic intersects the equinoctial) and at each solstice (where

the ecliptic reaches its highest or lowest point relative to the equinoctial).

For the fictitious sun in its circular orbit, the vernal equinox, summer solstice, autumnal equinox, and winter solstice occur at exactly I/4-year

intervals.

In figure 10 the dashed line gives the resulting difference between the

right ascension of the ascending node of a Sun-synchronous orbit and that of

the fictitious subsolar point for one-fourth of the yearly cycle. The solid

line in figure 10 shows the difference between the same satellite quantity and

the right ascension of the real subsolar point throughout I year. Also shown

in the figure is the actual subsolar declination 6® as the real Sun passes

through the summer solstice, autumnal equinox, and winter solstice. The points

at which _ = 0 define the equinoxes, and the maximum and minimum values of

6_ define The summer and winter solstices, respectively. The three separate

o_dinates for _Q in figure 10 show when these events occur during the year

relative to I/4, I/2, and 3/4 of a year, as denoted by the three tick marks on

the time axis. The greater than expected lagging of the real subsolar point is

due to the eccentricity of the Earth's orbit around the Sun; the tick marks show

18
o_I

+o

+oo

oo

0

.H0

\

\

o,I _._o

oI _J

_d

00

C_J 0

P4I I I

c_o•_

0

co

L._ 0_

0

_._

r-4

co0

o_•--

om

o!

c_

•Ho

_oo

c_

_-_I

o ._

O_0

0

eJ0 ._..I

_ _-0

_0

goo

,el

19
that the autumnal equinox occurs more than 6 months after the vernal equinox,

implying that the Earth moves through its aphelion (farthest point from the Sun)

during this period. During the time between autumn and the next spring, the Sun

,catches up" as the Earth moves through its perihelion (closest point to the

Sun). Thus the satellite leads the Sun for most of the year, by as much as

about 6° . Although these data are typical for the time periods of interest,

they are, strictly speaking, a function of time, since the perihelion of the

Earth is precessing around the Sun at a rate of approximately 1.7 degrees per

century. Hence the year for which the calculations have been made (1981) is

specified.

III. ESTABLISHING PATTERNS OF SURFACE COVERAGE FROM A SATELLITE

The underlying assumption for this section is that any Earth observation

mission requires some type of repetitive surface coverage pattern to achieve its

goals. To establish these patterns, it is necessary to couple the motion of a

satellite in inertial space (that is, with respect to the inertial rightascension--declination coordinate system) with its motion over the surface of

the rotating Earth. For a randomly chosen orbit, there is no reason to expect

the satellite ground track (that is, the longitude-latitude history) of the subsatellite point to be repetitive. This is because, in general, the satellite

period bears no useful relationship to the Earth's rotational period unless it

is specifically selected for a particular purpose.

Ground Tracks for Circular Orbits

Several typical ground tracks are shown in figure 11 for circular orbits

with a semimajor axis of 7000 km and varying inclinations. One-fourth of the

I

8o I I_. ..-..__-- -85 .........__.--- -,.80

70 J'_

_ -

60 ,_r

50 50

= . ,,... ..--

71 ._.__.

2o

0 10 20 30 40 50 60 70 80 90

Longitude, deg

Figure 11 .- Ground tracks for circular orbits, through

one-fourth of a nodal period.

2O
track is shown- the track over one-fourth of a nodal period. The satellite

does not cover a longitude of 90° during this time becauseof the rotation of

the Earth and, secondarily, the orbital precession terms. The longitudelatitude history relative to an initial point on the equator can be calculated

as a function of mean-anomalystep AM (in degrees) according to the following

equations:

1 sin-IFsin i sin (AM+ _OAM)7 (18)

t J

AL = tan-1[cos i tan(AM + A_AM _ + A_AM -

AM _A(360.9856473)

(360)(86 400)

(19)

Note that these equations work only for circular orbits, or the mean motion for

elliptical orbits, for the reasons discussed earlier in reference to the definition of nodal period (eq. (10)). Since orbits for coverage of the Earth's surface are commonly circular anyhow, for a variety of reasons, this is not a particularly restrictive limitation. The longitudes and latitudes are referenced

to a fictitious spherical earth, even though the dynamical equations of the

orbit reflect the fact that the Earth is oblate. Transformation of longitudelatitude data from a sphere to points on an oblate spheroidal surface is easily

achieved (ref. 6) but is not considered necessary for the present objective of

understanding and modifying patterns of surface coverage.

The cumulative effect of the apparent westward drift of the ground track

is further illustrated in figure 12, which shows the longitude-latitude trace of

9O

7O

5O

3O

10

-10

_a -30

-5o

-7o

-9o

TN

la ='7000 km

' i 33 °

29 28 27 26 25 24 ¥_3 22 21 I 20 19 18

/

0 60 120 180 240 300

Longi_de, deg

Figure 12.- Ground track for I revolution of a circular orbit

and starting longitudes for subsequent orbits.

36O

21
a _ircular orbit with i = 63 ° and a = 7000 km. (The reason for the choice of

these particular orbit elements will be made clear in a subsequent section. It

is immediately clear, however, that the 63 ° inclination covers, in latitude at

least, most of the inhabited areas of the Earth.) The longitudes at which successive orbits cross the equator are shown in figure 12 by the numbered marks

along the axis. The westward movement of each successive orbital track is, for

any orbit, an obvious consequence of the Earth's rotation. The gaps in longitude between successive orbits constitute one of the basic restrictions in obtaining surface coverage. There is not much that can be done to alter the magnitude

of the gaps, as can be seen from table 4. This shows, as a function of altitude,

the change in longitude AL between successive orbital passes over the same latitude. To simplify construction of the table, the orbital precession effects,

which are considerably less than I degree per orbit, have been neglected. Hence

the quantity AL is the rotation of the Earth under a hypothetical unperturbed

(Keplerian) circular orbit during one Keplerian period. More exact calculations

would require the use of nodal periods and inclusion of the precession effects.

Table 4 also illustrates the range of orbital periods available for Earth observation. In principle, any orbital period between about 16 and I or less orbits

per day can be obtained; however, for reasons of sensor resolution and launch

system capability, most Earth observation orbits will remain at below about

1500 km, or a period of less than about 7000 see. The lower limit on period

is set by the lowest altitude which is practical in terms of coping with atmospheric drag - about 200 km. The last nine columns of table 4 show the longitudinal displacement in kilometers at nine different latitudes for a spherical

earth whose radius is equal to 6378.145 km (the mean equatorial radius of the

Earth).

Designing Orbits for Repetitive Longitude-Latitude Coverage

The satellite ground track and successive equator crossings presented in

figure 12 at least suggest that the longitude-latitude history will not repeat

itself in any useful way; that is, there is no discernible repetitive coverage

pattern for this particular orbit. According to table 4, the ground track cannot possibly repeat itself any more often than once a day, as it takes about I

day for the Earth's Surface to rotate once under the orbital trace. Except for

that restriction, however, it is easy to redesign the orbit to pass over the

same longitude-latitude points every day, every 2 days, and so forth. Again,

because of the general differences between mean and true anomaly, this discussion is valid only for circular orbits, or the "mean" location of satellites in

elliptical orbits.

Consider the geometry shown in figure 13. At some given time a satellite

passes over longitude L, which is shown on the equator but could be at any latitude. For repetitive coverage of L, the point L must undergo n revolutions

under the orbit trace in the time required for the satellite to undergo m nodal

revolutions (where n and m are integers!. The point L moves with respect

to the orbit plane at an angular rate e - _, and the satellite's orbital rate

is M + _. (Recall that e is the Earth's rotational rate.) Therefore,

(m)(360) = (n)(360) (20)

22
_ North

AS

Orbit trace _ I_ 1VI + I_x'_ ¢b "\ \

Figure 13.- Earth-orbit geometry for defining

the repetition factor Q.

The repetition factor Q is defined as the number of satellite revolutions

divided by the number of revolutions of the point L:

Q_m:_+_

n e-_

(21)

Note that except for the _ and _ terms, Q would be exactly equal to the

number of orbits per day; however, because of the need to maintain repeated coverage over a long period of time, it is necessary to retain these small terms

when designing for a particular repetition factor. An integer value of Q, for

example, Q = 14/I, means that the longitude and latitude points covered by a

particular ground track will be exactly duplicated 14 orbits later, the next day.

What is lost to gain this advantage is that other longitudes, those not covered

during the first day, will never be covered If Q is given the value

29 = 14_, the initial longitude coverage will be duplicated on every other

2 2

day after launch, with longitude points halfway in between being covered on the

alternate days.

Sun-synchronous orbits present a special case in defining Q. Since the

Earth's rotation rate e is 360 ° plus the average.solar precession in one day,

by definition, and the orbit is designed so that _ is the average solar precession in one day, the synchronous repetition factor is

Qs - _ + _ (22)

360

where the units of M and _ are degrees per day. That is, for Sun-synchronous

orbits, Qs is numerically identical with the number of nodal revolutions per

day.

23
The range of circular orbit altitudes required to produce specified values

of Q (convertible to semimajor axis, in the average sense discussed previously) is shown in figure 14, which also illustrates the small inclination

1700

h, km

1600 _

1500 _

1400 __= 90°

13oo _\

1200 0£_

II00

1000 9O0

800

700

6OO

500

400

3OO

15 151⁄2 16

2OO

12 121⁄2 13 13_ 14 14_

Q

Figure 14.- Circular orbit altitudes h to achieve specified

repetition factors.

24
dependence (through 6). An extensive listing of circular orbit parameters for

particular values of Q is given in table 5. Repetition rates of I to 10 days

and 2 or 3 weeks are considered. Table 6 gives the same data for Sun-synchronous

orbits, in which case both semimajor axis and inclination must have a particular

value to satisfy two independent constraints: the correct precession rate,

dependent on the satellite-Sun geometry, and repetitive ground coverage, dependent on the satellite-Earth geometry.

Appendix A presents several computing algorithms written in BASIC language

for the Wang 2200 computer system. These can be used to calculate many useful

orbit parameters based on the first-order perturbations discussed in the preceding sections. In particular, orbit parameters which produce orbits having

desired repetition factors can be generated; the necessary computations are very

tedious to do by hand to the accuracy required for internally self-consistent,

long-term orbit propagation.

Solar Illumination Considerations in Satellite Orbit Analysis

Previously it has been shown that the apparent motion of the Sun about the

Earth is not coupled in any discernible way with the Earth's rotation. As a

result, the specification of orbit parameters for a particular type of repetitive ground coverage must be made independently of what the solar illumination

conditions will be during the specified repeat cycle. Even for Sun'synchronous

orbits, which maintain a "fixed" geometry with respect to the Sun, figure 4

serves as a reminder that the resulting illumination geometry is not really

fixed at all and actually varies in a more complex way than might be expected.

As a basis for consideration of solar illumination patterns, table 7 presents a summary of solar position information, starting at the beginning of a

calendar year and continuing through the vernal equinox of the following year.

The calculations are based on equations for the Earth's motion around the Sun,

presented in appendix B. Note that figure 10 can be obtained from these data.

It should be emphasized that the large number of significant figures shown in

the position data is merely for the sake of internal consistency when working

with the data in other contexts (as in computer programs) and need not necessarily represent actual solar positions to the indicated accuracy.

As an illustration of the relationship between the Sun's apparent motion in

its nearly inertial framework and the Earth's rotation about its own axis, consider the following problem: It is well known that local clock noon does not in

general correspond to high noon - the instant at which the subsolar point crosses

an observer's meridian. For an arbitrary date, say May I, 1981, find the actual

local clock time at which high noon occurs for an observer at a longitude Lm

of -70 ° . The first step in the solution is to locate both the Sun and the

observer in a common coordinate system. The latitude (declination) does not

matter, as the location of high noon only involves lining up the observer's

meridian with the subsolar point. The right ascension of the observer is calculated from equations (I) to (3). The Julian date for Oh UT, May I, 19BI, is

244 4725.5. In the following calculations, note the practice of removing integer multiples of 360 ° from the angular values, and of converting negative angles

to their positive equivalents between 0° and 360o:

25
T = 244 4725.5 - 241 5020.0 = 0.813292 Julian century

36 525

ag,o : 990.6909833 + 36000°.7689T + 0o.00038708T 2 : 218o.838139

mobs : _g,o + Lm : 148°'838139

The right ascension of the Sun at Oh UT is given in table 7:

eQ : 38°.127349

The observer's right ascension changes at a rate equal to the sidereal rotation

of the Earth. The corresponding motion of the subsolar point is not constant

but changes with the seasons and the position of the Earth in its heliocentric

orbit. The rate can be determined to sufficient accuracy by examining the

change in solar right ascension as given in table 7. The average of the 5-day

changes prior to and following May I, 1981, is

Finally, the time at which the Sun crosses the observer's meridian is given by

148°.838139 + 360.9856473t = 38o.127349 + 0.9545191t

360.0311282t = -110°.71079 = 2490.28921

t = 0.69241016 day

Therefore, high noon occurs at J.D. 244 4726.19241016 = May I, 1981, 16h37m04 s

UT, or 11h37m04 s EST (local clock time).

Also of interest in table 7 is the variation of solar declination 6_ as

a function of time. Clearly, this seasonal variation has a dominant effect on

solar illumination during any Earth observing mission. Figure 15 illustrates

the range of Sun elevation angles available throughout the year as a function

of latitude. (A geometrical description of the elevation angle is given in the

following section.) Variation of Sun elevation angle _ with time is shown for

summer, winter, and the equinoxes in table 8. The time system used in this

table is an hour-angle (B) system; the elevation angle _ is given at any latitude X by

: 90 ° - cos-1(cos 6® cos X cos B ÷ sin 6® sin X) (23)

where B is the equatorial angle from the subsatellite meridian to the subsolar

meridian. The reference (0 hours) is local high noon (B = 0°). For example, at

a relative time of +2 hours or -2 hours (10 a.m. or 2 p.m., respectively, in a

"clock" terminology), the Sun is on a meridian +30 ° or -30 ° away from the local

meridian. The use of this relative time system serves as a warning to exercise

care when dealing with orbit timing. A "9:00 a.m." orbit could mean that a

26
90

7O

5O

30

Sun

elevation 10

angle,

deg -10

-30

-5O

-70

-90

-90 -70 -50 -30 -10 10 30 50 70 90

Subsatellite latitude, deg

Figure 15.- Range of Sun elevation angles as a function of subsatellite

latitude, with subsolar declination (time of year) as a parameter.

satellite crosses the equator at 9:00 a.m. local time, but such a description

might just as well be loosely used to mean that at the equator crossing, the

subsolar meridian is 45 ° to the east of the subsatellite meridian. These conditions are not, in general, exactly equivalentl The reason for belaboring these

distinctions will be apparent in a later example, where care is taken to separate actual Sun lighting conditions at the Earth's surface from the intuitive

feelings about sunlight that an observer from the northern temperate latitudes

tends to associate with a given local clock time.

As an example of the use of figure 15 or equation (23) to provide limiting

values for Sun lighting conditions, consider an observer at 50 ° latitude. The

range of available elevation angles is bounded by the intersection of the edges

of the rectangle appropriate to the subsolar declination at the date of observation with a vertical line at 50 ° . The subsolar declination is directly related

to the date of observation. Using table 7, for example, the observer could

relate a particular date to the corresponding subsolar declination for use with

27
figure 15, interpolating as required in the table and figure. Suppose the date

is around June 20, near the summer solstice. Then the subsolar declination is

at its maximum of 23o.44; there is an appropriate boundary indicated in figure 15 for this value which may be used without interpolation. The maximum and

minimum values of _ occur for 8 = 0° and 8 = 180 ° , respectively (0 and

12 hours). Thus, either approximately from figure 15 or more accurately from

equation (23), the maximum and minimum values of _ at the time of the summer

solstice are found to be 63o.44 and -16°.56, respectively, for an observer at

50 ° latitude.

Before examining solar illumination patterns during long-duration missions,

it is first necessary to establish the reference systems for specifying solar

position from the ground and from a satellite. Following that, some special

attention will be given to the problem of solar occultation, since the observation of sunset and sunrise relative to a satellite - at the beginning and end

of periods of darkness - has particular significance for many types of atmospheric profile measurements.

Pointing Angle Geometries for Locating the Sun

To account for the Sun's location relative to a satellite or its subpoint

on the Earth's surface, it is necessary to have three pieces of information:

(I) a reference system, (2) the solar elevation or zenith angle, and (3) the

direction from which the Sun is shining. The geometry selected for locating

the Sun from the Earth's surface is shown in figure 16. The point P is the

Z

y

x

T

Figure 16.- Pointing angles for the Sun relative

to the Earth's surface.

28
subsatellite point, _ is the unit vector to the satellite, and __ is a unit

vector in the direction of the Sun. The zenith angle _ is measured downward

from the local vertical _, and the elevation angle _ is measured upward from

the local horizon. The angle n, measured counterclockwise from the parallel

of latitude through P (that is, from a unit vector pointing east from P),

defines th_ direction from which the Sun is shining.

Note that P is a point of right ascension and declination, so that a subsatellite point given in terms of longitude and latitude must be transformed to

the proper coordinate system. Note, too, that since the Sun appears to be infinitely far away, _ has essentially constant components regardless of the

location of P. Th_s, P could Just as well be at the satellite, in which case

n and _ or _ could serve as pointing angles to the Sun from the satellite.

However, for this purpose other reference systems, such as those described in

the following paragraph, are more convenient.

Figure 17 illustrates a coordinate system for defining pointing angles

relative to a satellite. The unit vector _ is normal to the orbit, in the

z

x

F

T

yy

I_ = (sin 9sin i, -cos _sin i, cos i)

Figure 17.- Definition of a Cartesian coordinate system

for pointing relative to a satellite.

direction of the angular momentum vector. For circular orbits the heading vector H is in the same direction as the velocity vector. These two vectors,

with _, form a Cartesian coordinate system.

29
A ^

One way to find the Sun relative to H, N, and _ is to rotate first

about _ and then about N. Figure 18(a) shows the pointing angles A and

B which result. Note that A is identical with the sun elevation angle _.

Another useful way to find the Sun is to rotate first about 9, with results

as shown in figure 18(b); I is rotation in the plane of satellite motion and

0 is rotation out of plane.

i

i

i

_0

/

f

/

/

_0

.//

r

<:> I= tan-l(_

A = 90 ° - cos -I(@ • _,).

O : 90 ° - cos-I(N • _®).

Figure 18.- Pointing angles for locating the Sun relative to a satellite.

For noncircular orbits it may be useful to consider a coordinate system

which retains N, replaces H with the unit velocity vector @, and replaces

with a new unit vector @ x N to give a Cartesian system.

The Geometry of Sunrise and Sunset as Viewed From a Satellite

As previously mentioned, the times during which the Sun rises and sets on

the Earth's horizon as viewed from a satellite have a special significance for

many types of atmospheric observations. The usual astronomical definitions of

sunrise or sunset, based on the appearance of the solar image to an observer on

the ground, are not the same as employed here. It is perfectly satisfactory for

the present to define sunrise or sunset as the instant at which the center of

the solar image (as located by a unit vector _ in the right-ascension--

declination system) is coincident with the horiz'Von of a spherical earth of

radius 6378.145 km. During sunrise or sunset it is, in principle, possible to

obtain vertical profiles of many different atmospheric constituents which attenuate the Sun's radiation. Of interest during these measurements is the tangent

point on the Earth (or on a concentric spherical shell) at which the measurement

is made. The geometry is shown in sketch (a). A vector rT from the Earth's

3O
V

/_0 -

S

Satellite

Sketch (a)

center is constructed perpendicular to the line from the satellite to the Sun

(along _). The path from the satellite to the point of intersection of _T

along _(_ is given by the vector 3. The tangent latitude and longitude are

the coordinates of the point at which _T pierces the Earth's surface. Depending on the orbit parameters (and hence the apparent sunrise or sunset rate) and

the time required for measurements, the tangent point may or may not change significantly during the course of a series of measurements made as the Sun's image

passes through the atmosphere. Thus, although the problem of interpreting solar

occultation measurements is, per se, beyond the scope of this paper, the orbital

information necessary for taking these effects into account can easily be

supplied.

Since _ and __ are known, the triangular geometry for finding _ and

_T is straightforward:

(24)

_T = g + _ (25)

The apparent sunrise or sunset rate along _T is given by

R : _ • eT (26)

which can conveniently be expressed as the rate, in kilometers per second, at

which the solar image appears to move vertically with respect to the .horizon at

the tangent point, or as an angular rate by using the magnitude of _.

To provide a feeling for the magnitude of the sunrise and sunset rates,

assume that the Sun is just rising directly in front of and below a satellite in

a circular orbit with a radius of 7000 km. In terms of the angles defined in

31
figure 18, B and 0 are zero and A and I are negative. The geometry is

given in sketch (b).

v

F

(7000 k

v

F T (6378.145 km)

Sketch (b)

For this simple case:

S = \/70002 - 6378.1452 = 2884 km

W

and

I = A = -[90 ° - cos-I(s/7000)] = -24o.3

The Keplerian period of this orbit T o is 5828.5 sec, so that (to a good approximation even for the perturbed case):

v : 2_(7000/5828.5) : 7.546 km/sec

The projection of this velocity onto _T is v • rT, given by

R : v sin A : 3.11 km/sec : 0.062 deg/sec

It is easy to see from this case that the Sun's image appears to pass through

50 km or so of atmosphere in less than 20 sec. At the other extreme, however,

when the satellite is travelling along the terminator, as when the orbit is

entering a period of no occultation for several orbits, the sunrise or sunset

rate approaches zero. Then vertical profiles of the atmosphere become more

difficult to obtain because the tangent point as previously defined undergoes

large excursions across the Earth's surface.

The equations for determining the time of sunrise and sunset relative to

perigee in an inertially fixed orbit are worked out in appendix C. Strictly

speaking, this procedure must be an iterative one, since the orbit actually

moves in inertial space between the instant of perigee passage and the calculated time of entry into and exit from a period of darkness. As a practical

matter, the iteration is usually not worth the effort. With appendix C, it is

32
possible to propagate an orbit in steps of one anomalistic period, from perigee

to perigee, computing the sunrises and sunsets at each step and determining

those periods of continuous solar exposure. This information has obvious importance for the scientific objectives of any mission as well as for the engineering design of the satellite and its systems.

IV. SOMEEXAMPLESOF GEOGRAPHICAL COVERAGE AND SOLAR ILLUMINATION

PATTERNS FOR LONG-DURATION EARTH MONITORING MISSIONS

It should be clear from the preceding sections that the discussions of

orbit analysis were intended to encompass mission situations extending for a

year or so - long enough for orbits to precess completely through several

cycles, for the Sun to go through an annual illumination cycle, and for repetitive coverage of locations on the Earth's surface. The dynamic relationships

between a satellite and the targets of its sensors are often decisive factors in

assessing the worth of a mission. It is not so much a question of establishing

a particular set of viewing conditions at a particular time as it is a question

of assessing and optimizing the total output from all measurement opportunities

over the nominal course of the mission. This section presents some representative patterns of coverage based on actual orbit analysis experience with several

proposed Earth observation missions. The examples are intended to illustrate

some important types of coverage and their associated problems. They include

(I) yearly illumination patterns over a single site, (2) yearly variations in

illumination angle as a function of latitude, (3) solar illumination effects

on global surface coverage, and (4) tangent-point distributions for solar

occultation experiments.

Establishing a Need for Repetitive Coverage Patterns

Each of the following examples requires, in some form, the establishment

of repeatable patterns; therefore, it is appropriate at this point to consider

the general necessity for repetition. The quantities or features of interest

for measuring or viewing from a spacecraft have one property in common: they

all vary in at least three dimensions - longitude, latitude, and time. If they

did not, there would be little justification for viewing them from spaceborne

platforms. In order to separate the contributions of each dimension, it is

clearly necessary to have repetition - repetition of two quantities to isolate

the third. Repeated measurements over the same location (longitude and latitude) to isolate a dependence on time is one important example. Note that other

desirable repetitive pairings - longitude or latitude with time to isolate the

dependence on a space dimension - are not possible with a single satellite,

which can repeat itself in space but not in time. Thus, it must be sufficient

to match repetition in space coordinates with later but presumably identical

conditions in the time coordinate. If the time dependence is obvious or dominating, as _n a diurnal cycle, identical conditions may be easy to establish.

Otherwise, interpretation of a set of sequential measurements is one of the

major problems in utilizing spacecraft data.

33
A less obvious reason for repetitive measurements is the need to determine

error associated with measurements. Assuming that the errors resulting from

making and interpreting measurements are randomly distributed, multiple measurements under the same conditions are required to isolate error from the other

factors influencing the value of the measured quantity. The extent to which

this can be done depends on the feasibility of substituting equivalent values

in the time dimension for identical ones. Spatially distributed quantities are

not exempt from similar considerations, as it is often necessary to group nearly

identical spatial coordinates with the assumption that variations of interest

exist only over spatial scales that are large compared with the differences

among such a grouping of available data points.

Yearly Illumination Patterns Over a Single Site

A straightforward application of the previous analyses is to establish a

satellite orbit that gives repetitive coverage over a selected site or group of

sites, determine one or more parameters of interest, and study their behavior

over the course of a year. This type of parametric study has practical applications in Earth resource management; pollution monitoring; land-use planning; and

many other research, survey, or inventory tasks. The example in this section is

based on a 1973 study of orbit requirements for observing the eastern coast of

the continental United States (ref. 7). In this study, an orbit was selected

that gives good coverage of the entire east coast and its performance was

assessed by reference to a single site at Norfolk, Virginia, which is roughly

midway along the Atlantic Coast.

The orbit inclination of 63 ° was selected by minimizing the off-nadir pointing required to cover 25 east coast sites of interest. This orbit has been

noted previously, in the section "Establishing Patterns of Surface Coverage From

a Satellite," as covering in latitude most of the inhabited areas of the Earth.

A ground track is plotted in figure 12; however, it is a "random" orbit in the

sense that its other elements were not selected to give a particular repetition

factor for regular coverage of a selected site. For a circular orbit, this

means just changing the semimajor axis to the appropriate value. This is easily

done with the algorithms described in appendix A. Table 9 gives a list of orbit

parameters for a 63 ° circular orbit which gives daily coverage over a particular

site (Q = 15). The list includes the starting conditions which will cause the

satellite to pass over Norfolk at the desired time on its initial orbit (local

clock noon). Note that the anomalistic and nodal periods are nearly identical,

as the inclination of 63 ° is very close to the value at which perigee precession

is zero. (See fig. 7.) The ground track of this orbit is plotted for the first

15 revolutions in figure 19, as seen from a vantage point in space directly over

Norfolk.

The parameter of interest in reference 7, and of general interest for any

similar mission, is the variation of Sun elevation angle during the repeated

passages over the site. Clearly, some of the passes will take place in darkness

because of the nodal precession of the orbit. The precession rate relative to

the Sun is about 4! deg/day; therefore, it would be reasonable to expect a com2

plete illumination cycle to take about 80 days. "On top" of this cycle should

34
la" = 28_7.371 kra

Q= 15

L o =-760.289

k o = 36°.853

."

/ .........i.....

Figure 19.- Ground track history of a 63° orbit started at Norfolk, Virginia,

with a l-day repeat cycle. Q = 15.

be an elevation-angle envelope resulting from the seasonal variation of the

Sun's declination. The actual variation in Sun elevation angle over Norfolk at

the time of satellite passage is illustrated in figure 20. The points show the

elevation angle for an orbit initially passing over Norfolk on Jan. I, 1981, at

17h UT (12:00 noon EST). Bear in mind that the curve is really not continuous

but is a set of 365 discrete points, one for each pass over Norfolk during the

year. The solid lines show the maximum and minimum available Sun elevation

35
8O

6O

Maximum available

elevation an

Minimum available

elevation angle

a = 6887.371 km

i = 63 °

Q= 15

to = 12:00 noon EST,

Jan. 1, 1981

Norfolk, Virginia

-8o I l I

0 30 60 90 120 150 210 240 270 300 330 360

Time, days from t o

Figure 20.- Sun elevation angle over Norfolk, Virginia,

for a satellite in a 63 ° circular orbit.

Lo = -760.289

;t o * 360.853

angles at this site as calculated from equation (23). It is interesting to

consider how little control can be exercised over the resulting illumination

pattern. By starting the orbit at a different time of day, the patterns can

be shifted within the envelope of available elevation angles; except for this,

there are no available parameters for altering the pattern in any significant

way. A different inclination will speed up or slow down the illumination cycle

without altering its basic behavior. A different value of Q will alter the

coverage of other places, all of which will have their own latitude-dependent

illumination patterns similar to those in figure 20. The price for repetitive

coverage over more sites is, as always, less frequent coverage of each site.

The resulting trade-off is a standard study for any Earth-oriented mission with

multiple objectives, each of which may have to be compromised for the total

optimization of the mission.

There is also the possibility of using a Sun-synchronous orbit. Then, most

of the rest of the east coast coverage will be lost because orbits at synchronous inclination no longer follow the coastline. In this case, the initial timing of the orbit is important, as it can be used to regulate the Sun elevation

pattern for the rest of the year, within the allowable envelope. If it is

desired to view the nadir at the maximum Sun elevation angle (not necessarily

a desirable situation), this condition can be approximately met by starting the

orbit at local clock noon. Indeed, the elevation angle for such an orbit started

at Norfolk would follow the top solid line in figure 20 and would be indistinguishable from it on the scale of this figure. Curves for other local times

would have the same shape as the maximum-minimum envelopes in figure 20 but

36
would be displaced within these envelopes by an amount proportional to the local

time, between noon and midnight.

Yearly Variations in Illumination Angle as a Function of Latitude

The specific example of a 63° orbit discussed in the previous section can

easily be generalized to predict the illumination conditions encountered by any

orbit. The envelope of available Sun elevation angles apparent in figure 20 is

but one of a family of such curves which are a function of latitude and time.

Figure 21 gives 10 such yearly curves for latitudes from 0o to 90° in 10° incre-

-"---C""-'---

I I I I I I i i I I f i 1 I _"_ "'=_

120 150 180 210 240 270 300 330 360

Time, days

Figure 21 .- Available Sun elevation angle as a function

of time at various latitudes,

ments. The data may be calculated with the aid of equation (23); they are an

extension of the data listed in table 8. It is within these envelopes, and of

course their corresponding southern hemisphere equivalents, that the Sun elevation angles vary.

As usual, Sun-synchronous orbits present a special case, as their orbital

planes do not precess on the average relative to the Sun. Assume that a satellite in a Sun-synchronous orbit has been started on the equator at the point

and time of the vernal equinox. The Sun elevation angles observed as the satellite passes several different latitudes are recorded in figure 22. These curves,

for a "high noon" configuration, closely resemble the elevation-angle envelopes

of figure 21, but small differences are detectable, especially at lower latitudes. These differences arise from the peculiarities of the Sun's apparent

motion around the Earth, as previously discussed in conjunction with figure 10.

It is often of interest to adjust the initial geometry of Sun-synchronous orbits

away from high noon to accommodate sensor_ - photographic, for example - for

which longer shadows might be desired to provide contrast in the observed surface features.

37
Nadir latitude,deg

:-..

80

•

J---4-_

f

\

\

\ //i \ / /

//

60 v /

50

'_ 40

,_ 30.

20

/

/

f

0 40 80 120 160 200 240 280 320 360

(T) Time from launch, days

Figure 22.- History of observed Sun elevation angle for a Sun-synchronous

orbit started at the location and time of the vernal equinox.

Solar Illumination Effects on Global Surface Coverage

One possible line of background justification for the examples in this section is the desire to monitor, on a global scale, the radiation balance of the

Earth. It is a problem which has widespread appeal because of the importance of

the radiation balance in predicting long-term trends in the Earth's weather and

climate. The present state of knowledge of the Earth's radiation processes is

such that significant advances require accurate measurements on a global scale

of the emitted and reflected radiation, in particular. At least one of these,

the reflected radiation, presents an exceedingly complex problem in data interpretation and requires not just global coverage on a longitude-latitude grid of

5° or so but repetitive coverage of the same locations over a wide range of Sun

geometries. This particular problem has been explored in an analysis of orbit

requirements for Earth radiation monitoring (ref. 8). It can be viewed more

generally as a problem that imposes the most severe possible requirements on the

ability to extract adequate data from Earth-orbiting satellites.

For the moment, consider the performance of a single nadir-viewing satellite in covering longitude, latitude, and time (or equivalently, illumination

angle). At the outset it can be stated that coverage of the longitude-latitude

grid, within the latitude limits of the orbit, is easy to obtain. In fact, by

appropriately specifying the repetition factor Q, it is possible to get extensive coverage in these two dimensions over as fine a grid as required. The

problem lies in the time coverage, for reasons which are made qualitatively

clear by considering the relative slowness with which a satellite orbit moves

with respect to the Sun or, put another way, the slowness with which the satel38
lite orbit plane changes its position relative to the plane of the solar terminator. It has already been demonstrated in previous sections that it is possible

to cover a particular site at most on a once per day basis, thereby giving up a

fine longitude-latitude grid in favor of a coarse one (with longitude points

roughly 25 ° apart according to table 4). The illumination on subsequent passes

will change daily (recall the coverage of Norfolk, Virginia, detailed in

fig. 20), but the coverage does not even approach the requirements for establishing diurnal changes in radiation patterns. The best that can be hoped for is

that over a sufficiently long period of time, say a month, adequate coverage will

be achieved to allow extrapolation in both space and time to the desired quantities. This process requires extensive modeling of (for example) the radiation

environment and an appreciation for the orbit constraints; it poses a problem

sufficiently complex to have defied satisfactory solution up to the present time.

Since it is really the Sun elevation angle which is of interest for the

radiation problem, rather than clock time per se, the single satellite coverage

is first expressed in terms of an hour-angle system relative to the satellite,

as a function of latitude. (It is the same system previously defined for use

in eq. (23), where 8 is the hour angle.) For each part of table 10, the globe

has been divided into thirty-six 5° latitude strips and the horizontal scale is

divided into twenty-four "l-hour" segments. Thus, each step on the horizontal

scale represents 15 ° of separation between the subsatellite and subsolar meridians. The contents of a latitude-time "box" are increased by one each time a

measurement at the satellite nadir point is assumed to have been made within the

space and time boundaries of that box. It is assumed that measurements are made

from orbit at a constant rate, which amounts to one measurement every 5° of mean

anomaly. The results of 30 000 such measurements (about 30 days) are given in

tables 10(a), (b), and (c), for 50 ° , 80 °, and Sun-synchronous orbits, respectively, at an altitude of 800 km.

To make the orbit geometry as easy as possible to visualize, the initial

conditions are very special ones: in each case the satellite starts on the equator at the location and time of a vernal equinox so that the first measurement

records an hour angle of zero. The boxing algorithm truncates hour-angle values

rather than rounding them off, so that the hour-angle equivalent time of "zero"

might go into either the first or last box, depending on its precise value. It

can be seen that with the given initial condition, the hour angle will then be

6 (a zenith angle of 90 °) at a mean anomaly of 90o; 12, at 180°; 18, at 270o;

and 0(24) again, at 360 ° . The imprecision of ignoring the distinction between

anomalistic and nodal revolutions is of no noticeable consequence for making

this point. The change in the coverage pattern over 30 days is due to orbital

plane precession relative to the Sun and, to a smaller extent over this short

period of time, the seasonal motion of the Sun into the northern hemisphere.

For the 50 ° orbit this relative precession rate is about -5.2 deg/day; for the

80 ° orbit it is about -2.1 deg/day; and for the Sun-synchronous orbit it is,

by design, zero.

Eventually, the nonsynchronous satellites will make measurements at all

hour angles for each available latitude. The question is whether or not the

necessary extrapolations can be made to determine diurnal variations and average out or otherwise account for their effects. The difficulty in doing this

over the required time scale (say, within 30 days) provides the argument in

39
favor of multiple satellites, which will cause the latitude-time grid to be

filled in much more quickly than will a single satellite. The trade-off between

the cost of additional satellites and increased mission capability is at the

heart of the justification process for this type of spaceborne monitoring program. One attempt at presenting the options for multiple satellite missions

may be found in reference 8, where the ability of various satellite systems to

fill in the latitude-time grid in 30 days is examined. A two-satellite configuration of one 50 ° and one 80 ° orbit was recommended in this study for "zonal"

coverage over latitude bands, but such a system still may not be able to provide total latitude-time coverage as quickly as desired for accurate interpretation of radiation data. Another solution to this problem is to use multiple

Sun-synchronous satellites to give continuous coverage of a particular local

illumination condition. Four such satellites might be sufficient to allow confident modeling of the diurnal radiation cycle. However, the expense of such

an extensive system has so far been an insurmountable obstacle to its

implementation.

There is another way of looking at the latitude-illumination grid. It can

be argued that not only is clock time an insufficient measure of illumination

coverage, but relative solar time (hour angle) by itself is not much better, as

it still does not unambiguously specify the illumination conditions at the nadir

point. 3 Thus, an additional parameter of real interest is the solar zenith angle.

These data are shown in table 11 for the same three orbits previously used - 50 °,

80 °, and Sun-synchronous - and may be compared with the corresponding parts of

table 10. The pattern of zenith-angle coverage can best be visualized by first

considering the Sun-synchronous orbit from which is viewed a dawn or dusk condition (zenith angle of 90 ° ) near the South or North Pole, respectively. The other

orbits initially have the same type of pattern, but they precess away from this

geometry. In 30 days the 50 ° orbit has precessed about -156 ° relative to the

Sun, and thus has viewed dawn and dusk conditions over every available latitude.

The 80 ° orbit will fill in its "hole in the middle" in a similar fashion if

given enough time - it has precessed only about -63 ° in 30 days.

The foregoing discussion of the problem of radiation-balance coverage may

appear quite discouraging from the point of view of the mission analyst, who

apparently cannot exercise sufficient control over orbital properties to achieve

the desired results with reasonable satellite systems. However, the definition

of global coverage is to some extent a flexible one which can be expected to

relax considerably under different circumstances. (Such a situation will be

dealt with in the following section.) In the present case, it is clear that

sensors with sufficiently wide fields of view will allow global coverage in

some sense. Then, the problem becomes not one of obtaining sufficient data

but one of interpreting the available information in an unambiguous way. The

implied trade-off between flight hardware for measurements and computer software for analysis is an important consideration in mission planning but lies

outside the scope of this analysis.

3Interpretation of the reflections of solar energy by the Earth's surface

requires such an unambiguous two-parameter specification of solar position.

4O
Tangent-Point Distributions For Solar Occultation Experiments

The three previous sections have dealt with someof the coverage patterns

resulting from a nadir-looking satellite system. Another basic type of sensing

involves looking not at the Earth but at the Sun under specialized conditions.

It has long been recognized that in studies of atmospheric constituents, it is

profitable to use the Sun as a radiation source. It is easy to visualize a

process whereby solar radiation received by a satellite when the Sun is well

above the horizon is comparedwith the radiation attenuated by the atmosphere

as the Sun rises or sets through the Earth's atmosphere. In keeping with previous general discussions, it is important to examine not just one or a few of

these measurementsin isolation, but to view a set of such measurementsin the

context of a total mission.

The data to be presented here are applicable to any solar occultation

experiment; they were generated originally in support of a proposed mission for

measuring halogens in the stratosphere (ref. 9). The question in this case is

a typical one: Is the coverage resulting from a flight project (assumedto last

I year) sufficient in space and time to permit useful interpretation of the

underlying distributions of the variable of interest? The particular example of

stratospheric halogens allows an immediate but qualified "yes" to this question,

as do manyother proposed experiments, since information of any kind about the

distribution of many stratospheric constituents is virtually nonexistent. However, the problem of justifying the return of the mission relative to its cost

is another obstacle which must be overcome.

Becauseof the paucity of worldwide stratospheric data, the definition of

global coverage can indeed be relaxed, as suggested in the previous section. It

may at first be sufficient merely to establish average levels over various parts

of the globe, taking note of only the coarsest features of the underlying factors which drive the distributions. But it is at this point that the qualification of the "yes" becomesapparent: What is "coarse"? Howwell can time averages be made? Can spatial and temporal distributions be separated, and to what

extent? To help answer these questions, several pertinent sets of solar occultation data are presented in figure 23. In figure 23(a) the distribution of tangent latitudes during I year is presented for a 70° orbit, chosen because it

is approximately the lowest orbit inclination at this moderate altitude

(a = 6978 km) which will allow somepolar coverage. The curves appear to be

continuous, but in reality they are madeup of the discrete contributions of

the sunrises and sunsets viewed from the satellite throughout the year: 9600

separate occurrences at the rate of about 14.9 per day. The orbit is deliberately started during a period of total solar exposure, and its subsequent precession through cycles of occultation and exposure is dependent in a complex

way on the time of year of the launch and the precession rate of the orbit relative to the Sun. The small dots represent sunrises; the crosses, sunsets, a

convention which also applies to the rest of figure 23. Note that in contrast

to nadir-looking coverage, the latitude limits of the tangent point are not

restricted to ±70° , but extend northward and southward depending on the solar

position. The length of the cycles can be altered, similarly to surface illumination cycles previously discussed, by choosing orbits with faster or slower

precession rates. However, since the precession rate is mostly a function of

inclination, this choice cannot be madeindependently of the coverage in

41
• Sunrise

+ Sunset

9O

6O

il I}

-30

0 30 60 90 120 150 180 210 240 270 300 330 360

Time from launch, days

(a) Tangent latitude on the Earth's surface at sunrise or sunset.

nset

120

_ 0

-120

0 30 60 90 120 150 180 210 240

Time from launch, days

?

270 300 330 360

(b) Pointing (yaw) angle to the Sun relative to the plane

of satellite motion at sunrise or sunset.

Figure 23.- Solar occultation data during a l-year flight

for a 70° orbit at h = 600 km.

42
3.5

2.5

1.5

o

.5

-.5

oF-4 -1.5

>

-2.5,

Sunrise

4.

-_-

1

Sunset

k

i {

30 60 90 120 150 180 210 240 270 300

Time from launch, days

(c) Apparent vertical rate of the solar image through

the atmosphere at sunrise and sunset.

330 360

Figure 23.- Concluded.

latitude. Also, the exact shape of the cycles and the location of the periods

of total sunlight are determined by the timing and initial geometry of the

launch. Therefore, figure 23 must be considered to be only representative in

details, while retaining generality in overall cyclic patterns.

Figure 23(b) illustrates another important facet of solar occultation experiments which must be considered in mission design: tracking the Sun. The quantity plotted is a "yaw" angle, previously defined as B, which locates the solar

image on the Earth's horizon relative to the direction of satellite motion. The

constantly changing satellite-Sun geometry causes an expected variation of B

over a range of ±180 °, a characteristic feature of Sun-tracking measurements.

A question clearly framed by the figure is: What level of sophistication is

warranted in designing a Sun tracker? The problem of finding the Sun when it

is "up" and following it as it sets is easier than predicting where it will be

just as it rises - a differentiation which carries with it the prospect of losing half of all the occultation opportunities. The sunrise-sunset symmetry

evident in the pointing requirements relative to 900 and -90 o is encouraging,

however, with the possibility of utilizing a sunset to predict the value of B

for the following sunrise.

A third cyclic pattern of interest during the mission is the variation in

the vertical rate R at which the Sun rises or sets. A useful approximation

is to consider the value of R at the horizon as characteristic of a particu43
lar measurement opportunity, even though the rate is not actually constant during the entire residence time of the solar image within the vertical limits of

the atmosphere. This quantity is plotted in figure 23(c). It is clear from

previous discussions that the maximum values of IRI occur when the Sun is

ahead of or behind the satellite (B = 0° or ±180 ° ) and that the minimum values

occur at B = 90 ° or -90 ° . Whereas the significance of this number depends on

the nature of the experiment being performed, it is easy to see that vertical

profiles of the atmosphere over a well-defined spot on the Earth can best be

achieved when IRI is as large as possible. At the other extreme the longitudelatitude coordinates can vary considerably during the passage of the solar image

through the stratosphere. Without going into details of data reduction, it is

reasonable to conclude that the distribution of R over the life of a mission

will affect the overall accuracy of the resulting global distributions. The

distribution of measurements of IRI in 0.5 km/sec intervals is given in figure 24 for the 9600 measurements of this l-year mission.

3000

Total: 9600 measurements

2500

2OOO

Number of

measurements 1500

1000

500

0 .5 1.0 1.5 2.0 2.5 3.0 3.5

[R[, km/sec

Figure 24.- Distribution of apparent rate of the solar image vertically through

the atmosphere as viewed from a satellite during sunrise or sunset.

Up to now, no mention has been made of longitude coverage; in the previous

section it has been assumed to be less critical than latitude coverage. Tangent

latitude is plotted against tangent longitude in figure 25, where it is clear

that in some sense the coverage in longitude is uniform at least over a time

span of I year; that is, there are no obvious gaps in this plot. It is useful

44
90 Sunset

60

30

-_ 0

_ -30

-60

0 30 60 90 120 150 180 210 240 270 300 330 3G0

Tangent longitude, deg

Sunrise

90 _-_. ,....-, ... ¢ ,.,.. _... _, ..._.,., .,. _-..._• , ",,,,., ",. :, ..::, ,.,.9., ,:"t:...,:,_, .r. ,;,...-., ,-..

• •.' •.....-" -..-." -.': _: " "'-.' "./ .-'.•. ." ... .• m;. .. •,..:....'-.-..-':..... ..;-• - •%".". '.. .'. I._.';.._ .',,; • .,,,. , .-,..,, ,--...'., .....-,, ".,..-'.-.....'C ...... *-.... .-. .... ,.... ".,.. ".-.... -.-,,.

{:--""" .... " • i • "" _...... * "'* * ° • . • L. . - t... ., • t*...... %% 'klk._. . .P .. • %. ., . ".-. .. . ,. ... .

•. • . -. • ... -.. ,. . . • , * • - • -. . ..o... ..'... •.'•.. .°;. . .t.. .,,. -.. , -',, • -*. ".- . ",-• , "., . • •,'. • "-;.

• .< :... • :., . _: ".': ,.:,_:':,. ,.',_:., _:_.:_, ..;,..'..,..._,.:.., _,',.,,_: :.,.,-,¢ ,_,,,'--._,.'-,--.._:.,,,-., ,*..,,-., 60 _._:_,.;._;..;.._,•.,...;_.,.; .... '_',..'_.;,-._._.•'%.,, "&._" "4.-'..'T':4:."..%". N.-. _2.-.._."'_i."""

._m." .. ._r. % .-w.. "..._.- ...._. .. "'iLn ._ %. ;" .• •" .*_a "1 .JA I' o.%_ .m.[l" ;.*_e b .. - ._..

• ;'- ".'' "''' ", • ".*. • _.". ",';'" " :L"" _• •o'." ".'_" "," ".". ._4 ,,_.. •

_ 30-,-"_. - ,". ." • . " . • . ." • • . • . " .,. .,. ., :. '

0

.,_. • . ." . : • •. • ., .• • ,• , ,. ,• , . . :" ,• ;" . ,., , ... • ..

.4-) __ •. .. .. . .*. _ • • ; ... .. * . - • • • * . • . • . . . • .•. . 0 ". " .... • ........ ,'" ., •" .." "" ....

-30

c_

-60

-9O

...... • ;. -;.. ...._. . , •.. _,. :,. %. ,.. -. :,,- "_.' :. •

:. • ".,. " ..,,. -,,. "". "k .. . ,,." . .,... .. ,,... ." ,,.. ." -.' • .." ''.. " ",,,. .

. • , ... ... .• ... _• •.. .. -.o . ..% :. ° :.• • .. _ ; , • • ,.•. _. • . ..., .. .* ...... ,... . ... I.'. ...... ." . ."_ : .,: '_

_,,," . ." . .,." . ;" , .*. ., ,-. .; ,,. ,",., ,., • , •. , ,,,, , .-,, - :_ •

• .'.m _-_- _'"" t.''"' ." ;" m _'• .e _ .: ._ ..-"-%. ..:',I, . .'.._...,'...,, . ....%. °..._...... ...

...._'. .;,...._ . ,.,.* . ... .... _,a._ • .:,,..- .'¢:..: .._..,. ._:..',,: "._,* . _," :a,r • ,_ • "_,._ , "-'.¢.. *_;,;'. • :.,<. _. ,,,,. , ..,, ._.., .:. _.: .r.: _ ._,_ .. ..,, ..,. .. • _,_ "e,,_ "-_,¢, " ,¢. " "_ .,* • • • _ , ,_ " • -" -" _-" ., ,, ,*

lr""_l" .."7'". ".'7"_: .." ",. ..... -". "" - • .'' ,,'--'" -. • "" .." "" ." "" -'.*" ..-.'" . .,.o'" • .',,

•..:..,,-•:, :._;,".../_;.../_.o_..s. _.,_..__.:_-.._, ,.:_.4,:.:_ .:,_._ ":,_.:._.",,_.:_--_._-':5"__",::,-'_","_. [,;'"I'" ""'I'":" ": I'" "I I r 'I I ,,1 I I

0 30 60 90 120 150 180 210 240 270 300 330 360

Tangent longitude, deg

Figure 25.- Distribution of tangent longitude and latitude on the Earth's

surface at sunrise or sunset after a _-year, solar occultation mission

for a 70 °, 600-km orbit•

45
to look at smaller time intervals, however, to see how the longitude-latitude

coverage is built up. In figure 26, about 400 orbits from the beginning of the

A- Sunset

O Sunrise

90 I

_3 ,_ 4- " * 4- 4"4- 4. 44. -4-..,. 4- 4. 4.4. 4.4. .4.4. ..i. 4. F-. 4..4."

@,_ .4- 4- .4- -4- ..4- 4- 4- 4- -4- 4- .4- 4- .4- 4- .4-

30_- * * * * * * * +15 "14 +13 "12 *11 *10 *9

"_ *8 *7 *6 *5 +4

• 3 *2

01 q)l

__ 07 06 05 04 03 02 O9 O8

1 l 1 I l 1 l l I

0 30 60 90 120 150 180 210 240 270 300 330 360

Tangent longitude, deg

Figure 26.- Distribution of tangent longitude and latitude at sunrise and

sunset for about 400 orbits during the first period of occultation.

first period of occultation are considered. The first 15 occulted orbits are

numbered, starting from the point where the Sun just reaches the horizon so

that sunrise and sunset occur at essentially the same time and place. It can

be seen that the longitude points are fairly regularly spaced throughout a

day - a time interval during which the latitude changes relatively little.

This is encouraging from the point of view of taking averages over bands of

latitude, a procedure which is commonly used for modeling the stratosphere.

The ability of th_s mission to achieve yearly averages of a single stratospheric quantity as a function of latitude is shown in figure 27. The solid

line gives the actual yearly averages of a dimensionless parameter F, produced

by a hypothetical stratospheric model which varies in longitude, latitude, and

time and has random components to account for measurement and analysis errors

and natural random fluctuations. (It can be thought of as representing a total

vertical burden of some stratospheric constituent.) A l-year mission which

extracts 9600 measurements from this model is flown, and the resulting 36 latitude band averages are calculated with standard statistical procedures for

grouping and weighting the available data. Although a detailed statistical

analysis of the measurements is beyond the scope of this report, the lack of

agreement in figure 27 between "theory" (perfect sampling) and "experiment"

(available sampling) near both poles can readily be attributed to an insufficient number of temporally spaced measurements near the poles, as is evident

in figure 23(a). In any event, the solar occultation technique has obvious

potential for space-based stratospheric measurements and may allow global modeling in certain restricted circumstances.

46
4O

F 30

-- O A: atUsa_r (ePe_fs_:u lat_Sampling) .O

20 I ] ] I I I I I I I ] ] ] I ] ] ]

-90 -60 -30 0 30 60 90

Latitude, deg

Figure 27.- Actual and measured (simulated) yearly averages of a hypothetical stratospheric constituent as a function of latitude.

a

A,B

C

h

^

H

V. SYMBOLS

semimajor axis, km

pointing angles to Sun relative to a satellite, deg (see fig. 18(a))

location of a fictitious sun along the celestial ecliptic, deg (see

fig. 9)

Julian days since Jan. 1.0, 1900

= d/10000

eccentricity

eccentric anomaly, rad

true anomaly, deg or rad

dimensionless parameter used for evaluating ability of a solar occultation mission to extract information on global distribution of

stratospheric constituents

altitude above surface of a fictitious spherical earth of radius r_,

km

unit heading vector (see fig. 17)

47
i

I,O

J2

J.D.

L

m,n

M

P

P

Q

r,r

re

R

-9

S

t

T

.9

V,V

x,y,z

8

inclination of an orbit, deg

pointing angles to Sun relative to a satellite, deg (see fig. 18(b))

term in expansion of Earth's gravitational potential (see eqs. (5))

Julian date or day

longitude, deg

integers (see eq. (20))

mean anomaly, deg or tad

unit vector normal to an orbit (see fig. 17)

semilatus rectum, km

point on Earth's surface, along a satellite orbit, or on the celestial

sphere

orbit repetition factor

satellite distance from center of Earth or a position vector, km

Earth's mean equatorial radius, 6378.145 km

rate at which solar image appears to rise or set on the horizon as

viewed from a satellite; km/sec or deg/sec

vector in direction of Sun as viewed from a satellite, with length as

defined in sketch (a), km

time; days, min, or sec, as defined in context

Julian centuries since Jan. 1.0, 1900

speed, km/sec, or a velocity vector

unit vector pointing toward Sun, defined in right-ascension--

declination coordinate system

Cartesian coordinates

right ascension, deg

hour angle of subsolar meridian relative to a satellite meridian, deg

declination, deg

obliquity of Earth's equator to the ecliptic, deg

48
n

6

T

T

solar zenith angle, deg (see fig. 16)

direction from which Sun is shining relative to Earth's surface, deg

(see fig. 16)

rotation rate of Earth, 0.25068447 deg/min or 360.9856473 deg/day

latitude, deg

Earth's gravitational constant, 398601.2 kmS/sec 2

Sun elevation angle, deg (see fig. 16)

period, sec or orbits/day

referring to vernal equinox (as when indicating direction of the first

point of Aries)

argument of perihelion, deg

right ascension of ascending node, deg

Subscripts:

A anomalistic periods

g Greenwich meridian

m meridian

AM evaluated for a mean-anomaly step size of AM

N nodal periods

o initial or unperturbed (Keplerian) quantities

s Sun-synchronous quantities

Earth

® Sun

obs observer

x,y,z components of a position vector

Superscripts:

h,m,s hours, minutes, seconds

d/dt

degrees

49
A

dimensioned vector

unit vector

5O
APPENDIX A

"BASIC" LANGUAGE ALGORITHMS FOR CALCULATING ORBIT PARAMETERS

Many of the equations of previous sections on orbital dynamics may easily

be incorporated into small computer programs for calculating relevant orbit

properties. Table AI gives a BASIC language algorithm written for the Wang 2200

computer system. With an input of semimaJor axis, eccentricity, and inclination,

plus the desired mean-anomaly step size, the precession rates and other data are

calculated. The Sun-synchronous inclination is also calculated; all the orbit

data can be obtained for synchronous orbits by setting I = S and rerunning the

program. The last two lines of printout will show that the inclination for synchronous orbits as calculated from just one iteration does, in fact, give very

nearly the desired result. An example of the output is given in the table.

Table A2 lists another useful BASIC algorithm. With inputs of Q (in terms

of an integer plus a fraction), eccentricity, inclination, and a guess at the

semimaJor axis, a simple iterative technique causes the semimajor axis to converge to the correct value for the desired Q. For Sun-synchronous orbits the

inclination is calculated internally and adjusted to maintain the synchronous

precession rate as semimajor axis is varied. This algorithm will generate data

such as those presented in tables 5 and 6. It can be modified to vary eccentricity instead of semimajor axis, but simultaneous freedom of these two variables

would lead to an infinite number of solutions for each value of Q.

51
APPENDIXA

TABLEAI.- "BASIC" ALGORITHMFORGENERATINGORBITPARAMETERS

ONTHEWANG2200 SYSTEM

10SELECT ])

20XI=66058. 33128

30X2=6. 283185308

40](3=398601.2

50X4=57.29577951

60X5=.9856473

701NPUT "lEAN ANOIALY

80 INPUT "UNPERTURBED

90INPUT "ECCENTRICITY", E

1001NPUT "INCLINATION" , I

IIOPRINT "D ",D

120PRINT "A0" ,A0

130PRINT "E ",E

140PRINT "I ", I

150PO--(A0*(I-E*E)) !2

160T0=X2*AO*SQR(AO/X3)

170110=X2/T0

180'II=M0" (I+XI/P

190DI=XI/PO*MI* (

200D2=-XI/P0*MI *

210TI=X2/MI*IS0/

220T2=X2/(MI+OI)

230S0=-A0 !i. 5*P0

240S=S0/ (I+XI/PO

250SO--ARCCOS(O)

260S=ARCCOS(S)

270D3=XI/

28004=-XI

290D5=II0*

300D6=-XI

340PRINT

35OPRINT

36OPRINT

370PRINT

330PRINT

390PRINT

400PRINT

410PRINT

420PRINT

430PRINT

440PRINT

450PRINT

STEP",D

SEHI:IAJOR AXIS", A0

O*SQR(I-E*E)*(I-I.5*SIN(1)!2))*X4*86400

2-2.5"SIN(I)!2)

cos(z)

#PI.86400

.180/#PI.36400

*X5/_1/SQR(X3)/X4/86400

*SQR(I-E*E)*(I-I.5*SIN(ARCCOS(S0))!2))

P0"(2-2.5*SIN (I) !2)*D

/PO*COS (I)*D

I I+XI/P0*SQR(I-E*E)*(I-I.5*SIN(S) !2))

P0*D5*COS (S) *X4"36400

310DT=D6*365. 2422

320Q:(:II+DI)/(.7292115061E-4*ISO/#PI*86400-D2)

"UNPERTURBED PERIOD,SEC ",T0

"UNPERTURBED MEAN :IOTION,RAD/SEC ",MO

":lEAN IOTION,DEG/DAY ",MI

"PERIGEE RATE,DEG/DAY ",DI

"NODE RATE,DEG/DAY ",D2

"PERIGEE STEP,DEG/MEAN ANOMALY STEP ",D3

"NODE STEP,DEG/>IEAN ANO:'IALY STEP ",D4

"ANOMALISTIC PERIOD ",TI

"NODAL PERIOD ",T2

"REPETITION FACTOR Q ",Q

"UNPERTURBED SYNCHRONOUS INCLINATION "_SO

"SYNCHRONOUS INCLINATION ",S

455MI=MO*(I+XI/P0*SQR(I-E*E)*(I-I.5*SIN(S) !2))*X4"86400

460DI=XI/PO*MI*(2-2.5*SIN(S) !2)

470QI= (MI+DI)/360

480PRINT "SYNCIIRONOUS REPETITION FACTOR ",QI

490PRINT "DALLY SYNCHRONOUS ORBIT PRECESSION ",D6

500PRINT "YEARLY SYNCHRONOUS ORBIT PRECESSION ",D7

52
APPENDIXA

TABLEAI.- Concluded

D 360

AO 7000

E 0

I 60

UNPERTURBEDPERIOD,SEC 5828.5110951

UNPERTURBEDMEAN !_OTION,RAD/SEC 1.07800863E-03

MEAN_IOTION,DEG/DAY 5335.626538

PERIGEE RATE,DEG/DAY .8991392485551

NODERATE,I)EG/DAY -3.596556994221

PERIGEE STEP,DEG/MEANANOI_ALYSTEP 6.06658144E-02

NODE STEP,DEG/MEANANOIIALY STEP -.2426632577633

ANO:_ALISTIC PERIOD 5829.493459208

NODALPERIOD 5828.511260928

REPETITION FACTORQ 14.63737447681

UNPERTURBEDSYNCHRONOUSINCLINATION 97.87448384351

SYNCIIRONOUS INCLINATION 97.87952788402

SYNCHRONOUSREPETITION FACTOR 14.80520669859

DALLY SYNCHRONOUSORBIT PRECESSION .9856473476633

YEARLY SYNCHRONOUSORBIT PRECESSION 360.0000056847

(Set I=S and rerun - RUNIO - to get synchronous orbit data)

D 36O

A0 7000

E 0

I 97.87952788402

UNPERTURBEDPERIOD,SEC 5828.5110951

UNPERTURBEDIEAN :[OTION,RAD/SEC I®07800863E-03

MEAN_OTION,DEG/DAY 5333.131479134

PERIGEE RATE,DEG/DAY -3.25706764226

NODERATE,DEG/DAY .9856473476636

PERIGEE STEP,DEG/MEANANO_ALY STEP -.2198603870542

NODESTEP,DEG/_IEANANOMALYSTEP 6.65337141E-02

ANO_IALISTIC PERIOD 5832.22073668

NODALPERIOD 5835.78478641

REPETITION FACTORQ 14.80520831492

UNPERTURBEDSYNCIIRONOUSINCLINATION 97.87448384351

SYNCHRONOUSINCLINATION 97.87952788402

SYNCHRONOUSREPETITION FACTOR 14.80520669859

DALLY SYNCHRONOUSORBIT PRECESSION .9856473476633

YEARLY SYNCIIRONOUSORBIT PRECESSION 360.0000056847

53
APPENDIX A

TABLE A2.- "BASIC" ALGORITHM FOR GENERATING ORBITS WITH SPECIFIED

REPETITION FACTORS ON THE WANG 2200 SYSTEM

i0 PRIHT

20 SELECT D

30 XI=66058.33128:X2=6.283185308:X3=398601.2:X4=57.29577951

40 INPUT "REPETITION FACTOR: INTEGER,NUM,DENO_I",Q,N,U

50 FO--Q+N/U

60 II_PUT "GUESS AT SE_|IMAJOR AXIS",AI

70 A2=AI+IO

80 INPUT "ECCENTRICITY", E

90 INPUT "INCLINATION-USE 999 FOR SUN SYNC",I

i00 IF I=999THEN 130

Ii0 11=1:12=1:13=1

120 PRINT

130 PRII{TUSING 390,F0,N,U;

140 PI=(AI*(I-E*E)) !2:P2=(A2*(I-E*E)) !2

150 TI=X2*AI*SQR(AI/X3) :T2=X2*A2*SQR(A2/X3)

160 MI=X2/TI:M2=X2/T2:IF I[]999THEN 190

170 GOSUB 'I(AI,PI):II=S

180 GOSUB 'I(A2,P2):I2=S

190 SI=_II*(I+XI/PI*SQR(I-E*E)*(I-I.5*SI_4(II) E2))*X4*86400

200 S2--M2*(I+XI/P2*SQR(I-E*E)*(I-I.5*SIN(12) !2))*X4"86400

210 DI=XI/PI*SI*(2-2.5*SIN(II) !2) :D2=XI/P2*S2*(2-2.5*SIN(12) !2)

220 CI---XI/PI*SI*COS(II) :C2=-XI/P2*SI*COS(12)

230 F1 =(SI+DI)/(.729211506E-4*IS0/#PI*86400-Ci)-FO

240 F2=(S2+D2)/(.72921iSO6E-4*I80/#PI*86400-C2)-FO

250 IF ABS(FI)[IE-6 TIIEN 360:IF ABS(F2)[IE-6 THEN 370

260 A3=(A2-AI)*F2/ (FI-F2)+A2

270 P3 =(A3*(I-E*E)) !2:T3=X2*A3*SQR(A3/X3):A3=X2/T3:IF I[]999THEN

280 GOSUB 'I(A3,P3):I3=S

290 S3=II3*(I+XI/P3*SQR(I-E*E)*(I-I 5"SIl{(13)!2))*X4"84600

300 D3=XI/P3*S3*(2-2.5*SIN(13) !2) :C3=-XI/P3*S3*COS(13)

310 F3= ($3+D3) / (. 7 292115061E-4*IS0/#PI*86400-C3)-F0

320 IF ABS(F3)[IE-6 TIIEN 340

330 AI=A2 :A2=A3 :GOTO 140

340 A=A3:F=F3+F0

350 GOTO 380

360 A=AI:F=FI+F0:GOTO 380

370 A=A2 :F=F2+F0

380 I=I3:PRI_[TUSI:_G 400,A+.0000005;I+.0000005

390,'## ##/##

400Z ####.###### ###. #####_'

410 GOTO 40

420 END

430DEFFN'I(A,P)

440 SO=-A!I.5*Pe.9856473/XI/SQR(X3)/X4/86400

450 S=S0/(I+XI/P*SQR(I-E*E)*(I-I.5*SIN(ARCCOS(S0))!2))

460 S=ARCCOS(S)

470 RETU Rq

290

(Sample case for a = 7000, e = O, i -- 60, Q = 13 1/2)

13 II 2 7396.373144 60.000000

(i = 999)

13 i/ 2 7445.166714 q9.793197

54
APPENDIX B

ORBIT ELEMENTS FOR THE EARTH'S MOTION AROUND THE SUN

The equations for the Earth's orbit elements can be obtained by inverting

the equations for the apparent motion of the Sun obtained from astronomical

observation. The time-varying orbit elements of the Sun can be found in reference I, and inverted orbit elements for the Earth, or more specifically, mean

elements for the gravitational center of the Earth-Moon system (barycenter) are

given explicitly in reference 6:

d = J.D. - 241 5020.0

D = d/10000

T = d/36 525

a_ = 1.00000023 AU

e@ = 0.01675104 - 0.0000418T - 0.000000126T 2

= 101o.220833 + 0o.0000470684d + 0o.0000339D 2 + 0o.00000007D3

M_ = 358o.475845 + 0°.985600267d - 0°.0000112D 2 - 0°.O0000007D 3

e_ = 23°.452294 - 0°.0035626D - 0°.000000123D 2 + 0°.0000000103D 3

Reference 6 gives the value of I AU (astronomical unit) as 149 597 893 ± 5 km.

The true anomaly of the Earth can be obtained from implementation of any of the

many available iterative solutions to Kepler's equation, or by expansion

(ref. 3, eq. (6-15)):

f : M + 2e sin M + 5e--_ 2 sin 2M + e3(13 sin 3M - 3 sin M) + e4(I03 sin 4M

4 12 96

- 44 sin 2M) + e5 (I097 sin 5M - 645 sin 3M + 50 sin M) + e6 (1223 sin 6M

960 960

-902 sin 4M + 85 sin 2M) + .

In this expression, both f and M must be in radians.

55
APPENDIX C

DETERMINING THE CONDITIONS UNDER WHICH THE SUN IS OCCULTED BY THE EARTH

RELATIVE TO A SATELLITE HAVING FIXED ORBIT ELEMENTS

Solution of the occultation problem involves determining the points on its

orbit at which a satellite enters or leaves the Earth's shadow. The geometry

is shown in sketch (CI). The symbols used for this appendix form a selfSketch (CI)

contained set, and they are not all defined in or consistent with the symbol

list in the body of the report. All the vector quantities are expressed in a

system whose x-y plane lies in the plane of the satellite orbit, with x and

y oriented as indicated in sketch (CI). Such a coordinate system is called the

PQW system. Vectors can be rotated into the PQW system by standard procedures.

(See, for example, ref. 2.)

A0 to A4 constants as defined below

vector perpendicular to the terminator, extending from the terminator

to the satellite, km

e eccentricity

56
f

P

F

^

%

%

APPENDIX C

true anomaly, deg

semilatus rectum, km

radius vector to satellite, km

unit vector to the Sun

Earth radius vector to the terminator, km

unit vector in direction of the perigee of the satellite orbit

unit vector at right angles to _ along the semilatus rectum

^

angle between r® and 3, deg

The geometric constraint is the requirement that at a point where the satellite

enters or leaves the Earth's shadow,

r® d = -d = -(r 2 r@2) I/2 • - : r cos

That is, __ and d are antiparallel. This restriction can be written in

another way_

rQ^ • _d : _® (_ - F_) : _ • _ : rr®^ (_ cos f + 9 sin f) : r cos ¢

The x- and y-components of the unit vector to the Sun are

r(Dx = _® " _

r(Dy : r(D" 9

Then,

rGx cos f + r(Dy sin f : -

I/2

Since r : p/(1 + e cos f),

P ( cos f + F p2

(I + e cos f) rGx roy sin f) + k(1 + e cos f)2

This can be rewritten in terms of cos f:

A0 cos 4 f + AI cos3 f + A2 cos 2 f + A 3 cos f + A4 = 0

The constants are

(_---_)4e4 (_--_)2(r_y + (r_x+ r_y) 2

A0 = - 2 - r_x)e2

=0

57
A, r2

APPENDIX C

A2 = 6 - 2 - - r_y)

A3 = 4(____)4e- 4(____)2(I- r_y)e

4 2

The fourth-order equation may be solved by standard procedures. The two spurious roots may be rejected by noting that Ir@ × _I = r sin @ = I_@I and A

r@ • _/r < 0 for occultation to occur.

58
REFERENCES

I. Explanatory Supplement to the Astronomical Ephemeris and the American

Ephemeris and Nautical Almanac. H.M. Naut. Alm. Off., 1961.

2. Escobal, Pedro Ramon: Methods of Orbit Determination. John Wiley & Sons,

Inc., c.1965.

3. Orbital Flight Handbook. Part I - Basic Techniques and Data. NASA SP-33,

Pt. I, 1963.

4. Kozai, Yoshihide: The Motion of a Close Earth Satellite. Astron. J.,

vol. 64, no. 9, Nov. 1959, pp. 367-377.

5. McCuskey, S. W.: Introduction to Celestial Mechanics. Addison-Wesley Pub.

Co., Inc., c.1963.

6. Melbourne, William G.; Mulholland, J. Derral; Sjogren, William L.; and

Sturms, Francis M., Jr.: Constants and Related Information for Astrodynamic Calculations, 1968. Tech. Rep. No. 32-1306 (Contract NAS 7-100),

Jet Propulsion Lab., California Inst. Technol., July 15, 1968.

7. Harrison, Edwin F.; and Green, Richard N.: Orbit Analysis for Coastal Zone

Oceanography Observations. AIAA Paper No. 73-207, Jan. 1973.

8. Harrison, Edwin F.; Brooks, David R.; and Gibson, Gary G.: Mission Analysis

To Define Satellite Orbits for Earth Radiation Budget Measurements. AIAA

Paper 76-811, Aug. 1976.

9. Russell, James M., III; Park, Jae H.; and Drayson, S. Roland: Global Monitoring of Stratospheric Halogen Compounds From a Satellite Using Gas Filter

Spectroscopy in the Solar Occultation Mode. Appl. Opt., vol. 16, no. 3,

Mar. 1977, pp. 607-612.

59
TABLE I.- JULIAN DAY NUMBER, 1950-1999, OF DAY COF_4ENCING

AT GREENWICH NOON ON:

Year Jan. 0.5 a Feb, 0,5 Mar. 0.5 Apr. 0,5 May0.5 June0.5 JulyO.5 Aug. 0.5 Sept. 0.50ct, O.5 Nov, 0,5 Dec. 0._

1950 243 3282 3313 3341 3372 3402 3433 3463 3494 3525 3555 3586 3616

1951 3647 3678 3706 3737 3767 3798 3828 3859 3890 3920 3951 3961

1952 4012 4043 4072 4103 4133 4164 4194 4225 4256 4286 4317 4347

1953 4378 4409 4437 4468 4498 4529 4559 4590 4621 4651 4682 4712

1954 4743 4774 4802 4833 4863 4894 4924 4955 4986 5016 5047 5077

1955 243 5108 5139 5167 5198 5228 5259 5289 5320 5351 5381 5412 5442

1956 5473 5504 5533 5564 5594 5625 5655 5686 5717 5747 5778 5808

1957 5839 5870 5898 5929 5959 5990 6020 6051 6082 6112 6143 6173

1958 6204 6235 6263 6294 6324 6355 6385 6416 6447 6477 6508 6538

1959 6569 6600 6628 6659 6689 6720 6750 6781 6812 6842 6873 6903

1960 243 6934 6965 6994 7025 7055 7086 7116 7147 7178 7208 7239 7269

1961 7300 7331 7359 7390 7420 7451 7481 7512 7543 7573 7604 7634

1962 7665 7696 7724 7750 7785 7816 7846 7877 7908 7938 7969 7999

1963 8030 8061 8089 8120 8150 8181 8211 8242 8273 8303 8334 8364

1964 8395 8426 8455 8486 8516 8547 8577 8608 8639 8669 8700 8730

1965 243 8761 8792 8820 8851 8881 8912 8942 8973 9004 9034 9065 9095

1966 9126 9157 9185 9216 9246 9277 9307 9338 9369 9399 9430 9460

1967 9491 9522 9550 9581 9611 9642 9672 9703 9734 9764 9795 9825

1968 9856 9887 9916 9947 9977 _0008 _0038 _0069 50100 _0130 _0161 *0191

1969 244 0222 0253 0281 0312 0342 0373 0403 0434 0465 0495 0526 0556

1970 244 0587 0618 0646 0677 0707 0738 0768 0799 0830 0860 0891 0921

1971 0952 0983 IOll i042 1072 1103 i133 I164 I195 1225 1256 1286

1972 1317 1348 1377 1408 1438 1469 1499 1530 1561 1591 1622 1652

1973 1683 1714 1742 1773 1803 1834 1864 1895 1926 1956 1987 2017

1974 2048 2079 2107 2138 2168 2199 2229 2260 2291 2321 2352 2382

1975 244 2413 2444 2472 2503 2533 2564 2594 2625 2656 2686 2717 2747

1976 2778 2809 2838 2869 2899 2930 2960 2991 3022 3052 3083 3113

1977 3144 3175 3203 3234 3264 3295 3325 3356 3387 34_7 3448 3478

1978 3509 3540 3568 3599 3629 3660 3690 3721 3752 3782 3813 3843

1979 3874 3905 3933 3964 3994 4025 4055 4086 4117 4147 4178 4208

1980 244 4239 4270 4299 4330 4360 4391 4421 4452 4483 4513 4544 4574

1981 4605 4636 4664 4695 4725 4756 4786 4817 4848 4878 4909 4939

1982 4970 5001 5029 5060 5090 5121 5151 5182 5213 5243 5274 5304

1983 5335 5366 5394 5425 5455 5486 5516 5547 5578 5608 5639 5669

1984 5700 5731 5760 5791 5821 5852 5882 5913 5944 5974 6005 6035

1985 244 6066 6097 6125 6156 6186 6217 6247 6278 6309 6339 6370 6400

1986 6431 6462 6490 6521 6551 6582 6612 6643 6674 6704 6735 6765

1987 6796 6827 6855 6886 6916 6947 6977 7008 7039 7069 7100 7130

1988 7161 7192 7221 7252 7282 7313 7343 7374 7405 7435 7466 7496

1989 7527 7558 7586 7617 7647 7678 7708 7739 7770 7800 7831 7861

1990 244 7892 7923 7951 7982 8012 8043 8073 8104 8135 8165 8196 8226

1991 8257 8288 8316 8347 8377 8408 8438 8469 8500 8530 8561 8591

1992 8622 8653 8682 8713 8743 8774 8804 8835 8866 8896 8927 8957

1993 8988 9019 9047 9078 9108 9139 9169 9200 9231 9261 9292 9322

1994 9353 9384 9412 9443 9473 9504 9534 9565 9596 9626 9657 9687

1995 244 9718 9749 9777 9808 9838 9869 9899 9930 9961 9991 *0022 _0052

1996 245 0083 0114 0143 0174 0204 0235 0265 0296 0327 0357 0388 0418

1997 0449 0480 0508 0539 0569 0600 0630 0661 0692 0722 0753 0783

1998 0814 0845 0873 0904 0934 0965 0995 I026 i057 1087 Ill8 i148

1999 245 I179 1210 1238 1269 1299 1330 1360 1391 1422 1452 1483 1513

2000 245 1544 1575 1604 1635 1665 1696 1726 1757 1788 1818 1849 1879

ajan. 0.5 = Greenwioh noon (12 h) UT, Dec. 31.

60
TABLE 2.- KEPLERIAN AND PERTURBED ORBITAL PERIODS AS A FUNCTION OF ALTITUDE AND INCLINATION

h, To, TA' TN' TA " TO, and TN - To,a see, fop i, deE, of -

km sec

0 I0 20 30 40 50 60 70 80 90

200 5309.6 5301.6 5301.9 5303.0 5304.6 5306.6 5308.7 5310.7 5312.3 5313.3 5313.7

-3.I -7.7 -6.7 -5.1 -3.1 -i.0 1.0 2.6 3.7 4.1

5285.4 5286.4 5289.2 5293.5 5290.7 5304.4 5309.6 5314.0 5316.8 5317.8

-24.2 -23.3 -20.5 -16.2 -10.9 -5.3 0.O 4.3 7.1 3.1

300 5431.2 5423.1 5423.5 5424.6 5426.2 5428.1 5430.2 5432.2 5433.8 5434.8 5435.2

-8.0 -7.7 -6.6 -5.0 -3.1 -I.0 1.0 2.6 3.7 4.0

5407.1 5408.1 5410.9 5415.1 5420.4 5425.9 5431.2 5435.5 5438.3 5439.2

-24.1 -23.1 -20.3 -16 .i -i0.8 -5.2 0.0 4.3 7.1 8.1

400 5553.6 5545.7 5546.0 5547.1 5548.6 5550.6 5552.7 5554.6 5556.2 5557.3 5557.6

-8.0 -7.6 -6.6 -5.0 -3.0 -1.0 1.0 2.6 3.6 4.0

5529.8 5530.7 5533.5 5537.7 5542.9 5548.4 5553.6 5557.9 5560.7 5561.6

-23.9 -22.9 -20.2 -15.9 -10.7 -5.2 0.0 4.3 7.0 8.0

500 5677.0 5669.1 5669.4 5670.5 5672.0 5674.0 5676.0 5678.0 5679.6 5680.6 5680.9

-7.9 -7.6 -6.5 -4.9 -3.0 -0.9 1.0 2.6 3.6 4.0

5653.3 5654.2 5657.0 5661.2 5666.3 5671.8 5677.0 5681.2 5684.0 5684.9

-23.7 -22.8 -20.0 -15.8 -10.7 -5.2 0.0 4.2 7.0 7.9

600 5801.2 5793.4 5793.7 5794.8 5796.3 5798.2 5800.3 5302.2 5803.8 5304.8 5805.2

-7.9 -7.5 -6.5 -4.9 -3.0 -0.9 1.0 2.6 3.6 3.9

5777.7 5778.6 5781.4 5785.5 5790.6 5796.1 5801,2 5305.4 5808.2 5809.1

-23.5 -22.6 -19.9 -15.7 -I0.6 -5.1 0.0 4.2 6.9 7.9

700 5926.4 5918.6 5918.9 5919.9 5921.5 5923.4 5925.4 5927.4 5928.9 5929.9 5930.3

-7.8 -7.5 -6.4 -4.9 -3.0 -0.9 1.0 2.5 3.6 3.9

5903.0 5904.0 5906.6 5910.3 5915.9 5921.3 5926.4 5930.5 5933.3 5934.2

-23.4 -22.4 -19.7 -15.6 -10.5 -5.1 0.0 4.2 6.9 7.3

800 6052.4 6044.7 6045.0 6046.0 6047.6 6049.5 6051.5 6053.4 6054.9 6055.9 6056.3

-7.7 -7.4 -6.4 -4.8 -2.9 -0.9 1.0 2.5 3.5 3.9

6029.2 6030.1 6032.8 6036.9 6:)42.0 6047.4 6052.4 6056.5 6059.2 6060.2

-23.2 -22.3 -19.6 -15.5 -I0.4 -5.1 0.0 4.1 6.8 7.3

9006179.3 6171.6 6172.0 6173.0 6174.5 6"176.4 6173.4 6180.3 6181.8 6132.8 6183.2

-7.7 -7.3 -6.3 -4.8 -2.9 -0.9 1.0 2.5 3.5 3.9

6156.3 6157.2 6159.9 6164.0 6169.0 6174.3 6179.3 6183.4 6186.1 6187.0

-23.1 -22.1 -19.5 -15.4 -10.4 -5.0 0.0 4.1 6.8 7.7

1000!6307.1 6299.5 6299.8 6300.8 6302.3 6304.2 6306.2 6308.1 6309.6 6310.6 6311.0

-7.6 -7.3 -6.3 -4.8 -2.9 -0.9 1.0 2.5 3.5 3.8

6284.2 6285.1 6287.8 6291.8 6296.8 6302.1 6307.1 6311.2 6313.9 6314.8

-22.9 -22.0 -19.3 -15.3 -10.3 -5.0 0.0 4.1 6.7 7.7

1100!6435.3 6428.2 6428.5 6429.5 6431.0 6432.9 6434.9 6436.7 6438.3 6439.2 6439.6

-7.6 -7.3 -6.3 -4.7 -2.9 -0.9 1.0 2.5 3.5 3.8

6413.0 6414.0 6416.6 6420.6 6425.6 6430.8 6435.8 6439.8 6442.5 6443.4

-22.7 -21.8 -19.2 -15.2 -10.2 -5.0 0.0 4.0 6.7 7.6

1200 6565.3 6557.8 6558.1 6559.1 6560.6 6562.4 6564.4 6566.2 6567.8 6568.7 6569.1

-7.5 -7.2 -6.2 -4.7 -2.9 -0.9 0.9 2.5 3.4 3.8

6542.7 6543.6 6546.2 6550.2 6555.1 6560.4 6565.3 6569.3 6572.0 6572.9

-22.6 -21.7 -19.1 -15.1 -i0.2 -4.9 0.0 4.0 6.6 7.6

1300 6695.7 6688.2 6688.5 6639.5 6691.0 6692.8 6694.8 6696.6 6698.1 6699.1 6699.4

-7.5 -7.2 -6.2 -4.7 -2.9 -0.9 0.9 2.4 3.4 3.8

6673.2 6674.1 6676.7 6630.7 6685.6 6690.3 6695.7 6699.7 6702.3 6703.2

-22.4 -21.5 -19.0 -15.0 -10.1 -4.9 0.0 4.0 6.6 7.5

1400 6826.9 6819.5 6819.8 6820.8 6822.3 6824.1 6826.0 6827.8 6329.3 6330.3 6830.6

-7.4 -7.1 -6.1 -4.7 -2,8 -0.9 0.9 2.4 3.4 3.7

6804.6 6805.5 6808.1 6812.0 6816.9 6822.1 6826.9 6830.9 6833.5 6834.4

-22.3 -21.4 -18.8 -14.9 -i0.0 -4.9 0.0 4.0 6.6 7.5

1500 6959.0 6951.6 6951.9 6952.9 6954.4 6956.2 6953.1 6959.9 6961.4 6962.4 6962.7

-7.4 -7.1 -6.1 -4.6 -2.8 -0.9 0.9 2.4 3.4 3.7

6936.8 6937.7 6940.3 6944.2 6949.0 6954.2 6959.0 6962.9 6965.5 6966.4

-22.2 -21.3 -18.7 -14.8 -10.0 -4.8 0.0 3.9 6.5 7.4

akt each value of altitude h, the first llne of data is TA; the second line, _A - To; the third line,

TN; and the fourth line, z N - TO, all in seconds.

61
Z

O

h.-I

E-q

Z

H

,J

O

Z

H

r_

t.'l

:3

E-*

I--I

E-*

,-1

B_

0

Z

0

H

Z

r..

H

m:

0

.]

o

r_

Cz:

E_

gg

0

r_

z

Q

i-4

e_

!

r._

,-1

O _ _ _ _ _ _O _ _ O_ _O O_

el ee em el em ee ee ee ee el me O

I I I I I I I II II I I II

• • • • el ee le • ee ee ee

I I I I I I I II II I I

i. i. • .i • el le •

_ I _ I I I I I I I I I I I I

o _ _ _ _ _ _ _ _ _ _o _ o

• e e_ el ee oe i_ el e_ ee el ee

O_ O_ _ h_ _ _ O_ NN _O NO NO II

_1 _1 I I I I I II II I I

I "3

_O NO 0

NI NI I I I I I II II I I Z

O ee el ie el et el o_ le el el el

_ _e • e_ _e el el ee el _l eo el

NI NI HI I I I I II II I I ._

_ .e • • • !! _i _i el .e el el

NI NI NI NI I I I II II I I

o _ _ _,_e. o_• _e* • ie e* • e* *

NI NI HI HI I I I II II I I

O

• el ee _e ee e* el el ee el •

_1 NI NI NI I I I II II I I

o

o

(N

_.-I u'l ¢'_] _0PI O u"l IrlU_ _.-0--..'r.--I_ u'_,Q O_ u"1 ',DO 0_, u'l

I

,,-I I ,-I I vii I I I I I I I

-,-I "El

62
H

Ix:;

0

H

a:

,-I

r.r.1

r..)

H

rJ

r_

O

Z

O

I.-i

[-4

O

r_

0

m

,-.1

),-t

H

C3

Z

O

,-1

I

,-3

E-4

J ., .

0 0 ,,,T ',O O0 C _ ,_ _ O0 CD 0,1 -o" ._ CO C

O0 CO _0 0", 0'_ 0'_ 0'_ O_ 0 C) C C C

63
TABLE 5.- CIRCULAR ORBIT PARAMETERS FOR SPECIFIED REPETITION FACTORS

Orbit altitude, km, for inclination, i, deE, of -

Q 0 10 20 30 40 50 60 70" 80 90

12 1616.2 1616.5 1617.6 1619.6 1622.7 1627.1 1633.0 1640.6 1649.9 1660.7

12 1/ 2 1395.9 1396.3 1397.5 1399.8 1403.3 1408.2 1414.7 1423.O 1433.O 1444.6

12 I/ 3 1467.7 1468.1 1469.3 1471.5 1474.8 1479.5 1485.9 1493.9 1503.7 1515.0

12 2/ 3 1325.6 1326.0 1327.3 1329.7 1333.3 1338.3 1345.1 1353.6 1363.8 1375.7

12 i/ 4 1504.2i 1504.6 1505.8 1507.9 1511.2 1515.8 1522.0 1530.O 1539.6 1550.8

12 3/ 4'1291.01291.4 1292.8 1295.2 1298.8 1304.O 1310.8 1319.4 1329.8 1341.8

12 i/ 5 1526.3 1526.7 1527.8 1529.9 1533.2 1537.8 1543.9 1551.8 1561.4 1572.5

12 4/ 5 1270.4 1270.8 1272.2 1274.6 1278.3 1283.6 1290.4 1299.1 1309.6 1321.7

12 I/ 6 1541.1 1541.5 1542.6 1544.7 1547.9 1552.5 155_.6 1566.4 1575.9 1587.0

12 5/ 6 1256.7 1257.2 1258.6 1261.0 1264.8 1270.0 1276.9 1285.7 1296.2 1308.3

12 i/ 7 1551.7 1552.1 1553.2 1555.3 1558.5 1563.0 1569.1 1576.9 1586.4 1597.4

12 6/ 7 1247.0 1247.5 1248.9 1251.3 1255.1 1260.4 1267.3 1276.111286.6 1298.8

12 i/ 3 1559.7 1560.1 1561.2 1563.3 1566.5 1571.O 1577.1 1584.8 1594.3 1605.3

12 7/ 3 1239.8 1240.2 1241.6 1244.1 1247.9 1253.1 1260.1 1268.9:1279.5 1291.7

12 I/ 9 1565.9 1566.3 156;.4 1569.5 1572.7 1577.2 1583.2 1591.O 1600.4 1611.4

12 8/ 9 1234.1 1234.6 1236.0 1233.5 1242.2 1247.5 1254.5 1263.3i1274.O 1286.2

12 I/I0 1570.9 1571.3 1572.4 1574.5!1577.6 1532.1 15SJ.2 1595.911605.3 161b.3

12 O/lO 1229.6 1230.1 1231.5 1234.O 1237.8 1243.1 1250.1 1258.911269.5 1281.8

12 i/II 1575.0 1575.4 1576.5 157_.5 1531.7 1536.2 1592.2 1599.9 1609._ 1620.3

12 I0/ii 1225.9 1226.4 1227.8 1230.3 1234.1 1239.411246.4 1255.3 1265._ 127S.2

12 1/12 1578.4 1578.8 1579.9 1581.9 1585.1 1589.611595.6 1603.3 1612.7 1623.6

12 11/1211222.9 1223.3 1224.7 1227.2 1231.O 1236.4 1243.4 1252.2 1262.9 1275.2

12 1/13 1581.3 1581.7 1582.8 1584.8 1588.O 1592.5 1598.5 1606.2 1615.6 1626.5

12 12/13 1220.3 1220.7 1222.1 1224.7 1228.5 1233.8 1240.811249.7 1260.4 1272.6

12 1/14 1583.8 1584.1 1585.3 1587.3 1590.4 1594.9 1600.9 1608.6 1618.O 1628.9

12 13/14 1218.1 1218.5 1219.9 1222.4 1226.3 12_1.6 1238.6 1247.5 1258.2 1270.5

12 1/15 1535.9 1586.3 1587.4 1589.4 1592.6 1597.1 1603.1 1610.7 1620.1 1631.0

12 14/15 1216.1 1216.61218.O 1220.5 1224.3 1229.7 1236.7 1245.6 1256.3 1268.6

12 1/16 1587.8 1588.211589.3 1591.3 1594.5 1598.9 1604.9 1612.6!1622.0 1632.9

12 15/16 1214.4 1214.9 1216.3 1218.8 1222.7 1228.O 1235.1 1243.9 1254.6 1267.0

12 1/17 1589.5 15G9.8 1591.0 1593.0 1596.1 1600.6 1606.6 1614.2 1623.6 1634.5

12 16/17 1213.0 1213.4 1214.8 1217.4 1221.2 1226.5 1233.6 1242.5 1253.2 1265.5

12 I/I_ 1591.O 1591.3 1592.4 1594.4 1597.6 1602.0 1608.0 1615.7 1625.1 1636.0

12 17/18 1211.6 1212.1 1213.5 L216.1 1219.9 1225.2!1232.3 1241.2 1251.9 1264.2

12 1/19 1592.3 1592.6 1593.7 1595.8 1598.9 1603.411609.4 1617.0 1626.4'1637.2

12 18/19,1210.5 1210.9 1212.4.1214.9 1218.7 1224.1 1231.1 1240.0 1250.7 1263.1

12 1/20 1593.5 1593.S 1594.9 1597.O 16OO.1 1604.5 1610.5 1618.2 1627.5 163S.4

12 19/20 1209.4 1209.9 1211.3 1213.8 1217.7 1223.0 1230.1 1239.0 1249.7 1262.0

12 1/21 1594.5 1594.9 1596.0 1598.0 1601.2 1605.6 1611.6 1619.3 1628.6 1639.5

12 20/21 1208.5 1208.9 1210.3 1212.9 1216.7 1222.1 1229.1 1238.0 1248.7 1261.1

64
TABLE 5.- Continued

Orbit altitude, _, for inclination, ir de_1 of -

Q 0 10 20 30 40 50 60 70 80 90

13 1189.3 1189.8 1191.2 1193.8 1197.7 1203.1 1210.2 1219.2 1230.0 1242.4

13 I/ 2 995.2 995.7 997.4 1000.2 1004.5 1010.5 1018.2 1027.9 1039.5 1052.8

13 i/ 3 1058.6 1059.1 1060.7 1063.4 1067.6 1073.4 1080.9 1090.4 1101.7 1114.7

13 2/ 3 933.0 933.6 935.3 938.2 942.7 948.8 956.8 966.7 978.6 992.1

13 i/ 4 1090.8 1091.3 1092.8 1095.5 1099.6 1105.3 1112.7 1122.1 1133.2 1146.1

13 3/ 4 902.4 903.0 904.7 907.7 912.2 918.4 926.5 936.6 945.6 962.3

13 I/ 5 1110.3 1110.8 1112.3 1115.0 ii19.0 1124.6 1132.0 i141.3ii152.3 1165.1

13 4/ 5 884.2 884.7 386.5 889.5 894.1 900.4 908.5 918.7 930.7 944.5

13 I/ 6 1123.3 1123.8 1125.3 1128.0 1132.0 1137.6 1144.9 1154.1 1165.1 1177.8

13 5/ 6 872.1 872.7 874.4 877.5 882.1 888.4 896.6 906.8 918.9 932.7

13 I/ 7 i132.7 1133.1 1134.6 1137.3 1141.3 1146.8 1154.1 1163.3 1174.3 1187.0

13 6/ 7 863.5 864.0 865.8 868.9 873.5 879.8 888.1 898.3 910.4 924.3

13 i/ 8 1139.7 1140.2 1141.7 1144.3 1148.3 I153.8 1161.1 1170.2 1181.2 1193.9

13 7/ 3 857.0 857.6 859.4 862.5 867.1 873.41 881.7 891.9 904.1 918.0

13 i/ 9 1145.2 1145.7 1147.1 1149.8 1153.7 1159.2 1166.5 1175.6 1186.6 1199.2

13 8/ 9 852.0 352.6 854.4 857.5 862.1 868.5 876.8 857.0 899.2 913.2

13 I/i0 1149.6 1150.0 1151.5 1154.1 1158.1 1163.6 1170.9 i180.0 1190.9 1203.5

13 9/10 848.0 848.6 850.4 853.5 858.1 864.5 872.8 883.1 895.3 909.3

13 I/II 1153.2 1153.6 1155.1 1157.7 1161.7 1167.2 1174.4 I183.51194.4 1207.0

13 I0/ii 844.8 845.3 347.1 850.2 854.9 861.3 869.6 879.9 892.1 906.1

13 1/12 1156.2 1156.6 1158.1 1160.7 1164.7 1170.2 1177.4 1186.5 1197.4 1209.9

13 11/12 842.0 _42.6 844.4 847.5 852.2 858.6 866.9 877.2 889.5 903.4

13 1/13 1158.7 1159.2 1160.6 1163.2 1167.2 1172.7 1179.9 1189.0 1199.9 1212.4

13 12/13 _39.7 G40.3 842.1 845.2 849.9 856.3 864.6 874.9 887.2 901.2

13 1/14 1160.9 1161.3 1162.8 1165.4 1169.3 I174._ 1182.1 1191.1 1202.O 1214.6

13 13/14 837.8 838.4 840.Iu 843.3 847.9 854.3 862.7 873.0 885.3 899.3

13 1/15 1162.8 1163.2 1164.7 1167.3 1171.211176.7 1183.9 1193.0 1203.9 1216.4

13 14/15 836.1 836.6 838.4 841.6 846.2 852.7 861.0 871.3 883.6 897.6!

13 1/16 1164.4 1164.9 1166.4 1169.0 1172.9 1178.4 1185.6 1194.6 1205.5 1218.0

13 15/16 834.6 835.2 836.9 840.1 844.8 851.2 859.5 369.9 882.2i 896.2

13 1/17 1165.9 1166.4 1167.8 I170.4 1174.3 1179.8 I187.0 i196.1 1206.911219.5

13 16/17 333.3 333.8 835.6 838.8 843.4 849.9i 358.2 863.6 880.9 894_9

13 1/18 1167.2 I167.7 I169.1 1171.7 1175.6 1181.1 1188.3 1197.3 1208.2 1220.7

13 17/13 832.1 832.7 834.5 837.6 842.3 848.7 857.1 867.4 879.7 893.8

13 1/19 1168.3 1168.8 1170.3 1172.9 I176.8 1182.2 i189.4 1198.5 1209.4!1221.9

13 18/19 831.0 831.6 833.4 836.6 841.2 847.7 856.0 866.4 878.7 892.7

13 1/20 L169.4 1169.9 1171.3 I173.9 1177.8 I183.3 I_90.5 1199.5 1210,4 1222.9

13 19/20 830.1 830.7 832.5 835.6 840.3 046.7 855.1 865.5 877.8 891.8

13 1/21 1170.3!1170.8 1172.3 1174.9 1178.8 1184.2 1191.4 1200.5 1211.311223.8

13 20/21 829.3 829.8 831.6 834.8 839.5 845.9 354.3 864.6 877.0 891.0

65
TABLE 5.- Continued

Orbit altitude, km, for inclination, i, deg, of -

Q 0 10 20 30 40 50 60 70 80 90

14 812.3 812.9 814.7 817.9 322.6 829.1 837.5 348.0 860.3 _74.5

14 I/ 2 639.5 640.1 642.2 645.7 650.9 657.9 667.0 678.2 691.5 706.5

14 1/ 3 696.0 696.7 698.6 702.0 707.0 713.9 722.3 733.7 746.7 761.4

14 2/ 3 534.0 584.7 586.7 590.4 595.7 603.0 612.3 623.8 637.3 652.6

14 1/ 4 724.7 725.3 727.2 730.6 735.5 742.3 751.1 761.9 774.7 789.3

14 3/ 4 556.6 557.3 559.4 563.1 568.5 575.9 535.3 596.9 610.6 626.1

14 1/ 5 742.0 742.6 744.5 747.8 752.7 759.4 763.1 778.9 791.6 806.1

14 4/ 5 540.3 541.0 543.1 546.8 552.3 559.7 569.3 580.9 594.7 610.3

14 1/ 6 753.6 754.2 756.1 759.4 764.3 770.9 779.6 790.3 803.0 817.4

14 5/ 6 529.5 330.2 532.3 536.1 541.6 549.0 553.6 570.3 584.1 599.3

14 1/ 7 761.9 762.5 764.4 767.7 772.5 779.2 7°7.8 798.5 311.1 325.5

14 6/ 7 521.8 522.5 524.6 528.4 533.9 541.4 351.0 562.8 576.6 592.3

14 1/ 8 768.2 763.8 770.6 773.9 773.7 735.4 794.0 804.6 817.2 831.5

14 7/ 3 516.0 516.7 518.9 522.6 528.2 535.7 545.3 557.1 571.0 586.7

14 1/ 9 773.0 773.6 775.5 778.8 733.6 790.2 793.3 309.4 022.0 836.3

14 8/ 9 511.5 512.2 514.4 518.2 523.7 531.3 540.9 552.7 566.6 532.4

14 1/10 776.9 777.5 779.4 732.6 787.5 794.1 302.6 313.2 825.8 340.1

14 9/I0 507.9 508.6 510.8 514.6 520.2 527.7 537.4 549.2 563.2 573.9

14 1/11 780.1 780.7 782.6 785.3 790.6 797.2 805.8 316.4 828.9 843.2

14 I0/11 505.0 505.7 507.9 511.7 517.3 524.8 534.5 546.4 560.3 576.1

14 1/12 782.8 783.4 785.3 788.5 793.3 799.9 803.4 819.0 831.5 845.3

14 11/12 502.6 503.3 505.5 509.3 514.8 522.4 532.1 544.0 557.9 573.7

14 1/13 785.1 785.7 787.5 790.7 795.5 802.1 810.6 321.2 833.7 848.0

14 12/13 500.5 501.2 503.4 507.2 512.8 520.4 530.1 542.0 555.9 571.7

14 1/14 737.0 787.6 789.4 792.7 797.5 804.0 812.6 323.1 835.6 349.9

14 13/14 498.8 499.5 501.7 505.5 511.0 518.6 528.3 540.2 554.2 570.0

14 1/15 788.7 789.3 791.1 794.3 799.1 805.7 814.2 824.8 837.3 851.5

14 14/15 497.2 497.9 500.1 503.9 509.5 517.1 526.3 538.7 552.7 568.5

14 1/16 790.1 790.7 792.6 795.8 800.6 307 .2 315.7 826.2 838.7 352.9

14 15/16 495.9 496.6 498.8 502.6 508.2 515._ 525.5 537.4 551.4 567.2

14 1/17 791.4 792.0 793.9 797.1 801.9 808.4 816.9 827.5 840.0 354.2

i4 16/17 494.7 495.4 497.6 501.4 507.0 514.6 524.4 536.3 550.3 566.1

14 1/18 792.6 793.2 795.0 798.2 803_0 809.6 818.1 828.6 841.1 355.3

14 17/18 493.7 494.4 496.6 500.4 506.0 513.6 523.3 535.2 549.3 565.1

14 1/19 793.6 794.2 796.1 799.3 304.1 810.6 819.1 829.6 042.1 856.3

14 18/19 492.7 493.4 495.6 499.5 505.1 512.7 522.4 534.3 548.3 564.2

14 1/20 794.6 795.2 797.0 800.2 805.0 811.5 820.0 830.5 843.0 357.2

14 19/20 491.9 492.6 494.8 498.6 504.2 511.3 521.6 533.5 547.5 563.3

14 1/21 795.4 796.0 797.8 801.0 805.8 812.4 820.9 831.4 343.8 358.0

14 20/21 491.1 491.8 494.0 497.9 503.5 511.1 520.3 532.8 546.8 562.6

66
TABLE 5.- Concluded

Orbit altitude, km, for inclination, i, deg, of -

Q

0 10 20 30 40 50 60 70 80 90

13 475.9 476.0 47S.9 492.7 4,;g.4 496.0 505.9 517.9 532.0 547.9

15 i/ 2 320.g 321.6 324.0 328.2 334.3 342.6 353.2 366.0 381.0 397.S

15 I/ 3 371.6 372.4 374.7 378.8 3G4.g 392.9 403.2 415.7 430.4 447.0

15 2/ 3 270._ 271.6 274.1 27g.4 234.8 293.2 304.0 317.2 332.4 349.6

15 I/ 4 397.3 39S.i 400.4 404.4 410.4 418.3 428.5 440.9 455.5 471.8

15 3/ 4 246.1 246.9 249.5 253.9 260.3 268.9 279.3 293.0 303.5 325.8

15 I/ 5 412.9 413.6 415.9 419.9 425.8 433.7 443.8 456.1 470.6 486.9

15 4/ 5 231.4 232.2 234.8 239.2 245.7 254.3 265.3 278.7 294.2 311.6

15 I/ 6 423.3 424.1 426.3 430.3 436.1 444.0 454.1 466.3 480.7 497.0

15 5/ 6 221.6 222.5 225.1 229.5 236.0 244.7 255.7 269.1 284.7 302.2

15 I/ 7 430.G 431.5 433.8 437.7 443.5 451.4 461.4 473.6 438.0 504.2

15 6/ 7 214.7 215.5 218.1 222.6 229.1 237.8 248.9 262.3 27_.0 295.5

15 i/ 8 436.4 437.1 439.4 443.3 449.1 456.9 466.9 479.1 493.4 509.6

15 7/ 3 209.5 210.3 212.9 217.4 223.9 232.7 243.8 257.3 272.9 290.5

15 i/ 9 440.7 441.5 443.7 447.7 453.4 461.2 471.2 483.4 497.7 513.8

15 3/ 9 205.4 206.3 203.9 213.4 219.9 228.7 239.8 253.3 269.0 286.6

15 1/10 444.2 445.0 447.2 451.2 456.9 464.7 474.7 486.8 501.1 517.2

15 9/10 202.2 203.0 205.7 210.2 216.7 225.5 236.7 250.2 265.9 283.5

15 1/11 447.1 447.8 450.i 454.0 459.8 467.5 477.5 489.6 503.9 520.0

15 10/II 199.6 200.4 203.0 207.5 214.1 222.9 234.1 247.6 263.3 281.0

15 1/12 449.5 450.2 452.5 456.4 462.1 469.9 479.8 492.0 506.2 522.3

15 11/12 197.4 19G.2 200.8 205.3 211.9 220.7 231.9 245.4 261.2 278.8

15 1/13 451.5 452.3 454.5 458.4 464.1 471.9 481.8 494.0 508.2 524.3

15 12/13 195.5 196.4 199.0 203.5 210.1 218.9 230.1 243.6 259.4 277.0

15 1/14 453.3 454.0 456.2 460.i 465.9 473.6 483.5 495.7 509.9 525.9

15 13/14 193.9 194.8 197.4 201.9 208.5 217.3 228.5 242.1 257.8 275.5

15 1/15 454.8 455.5 457.7 461.6 467.4 475.1 435.0 497.1 511.4 527.4

15 14/15 192.5 193.4 196.0 200.5 207.1 216.0 227.2 240.7 256.5 274.2

15 1/16 456.1 456.8 459.1 463.0 468.7 476.4 486.3 498.4 512.6 523.7

15 15/16 191.3 192.2 194.8 199.3 205.9 214.8 226.0 239.6 255.3 273.0

15 1/17 '457.2 458.0 460.2 464.1 469.8 477.6 487.5 499.6 513.3 529.8

15 16/17 190.3 191.1 193.8 198..3 204.9 213.7 224.9 233.5 254.3 272.0

15 1/18 458.3 459.0 461.3 465.1 470.9 478.6 488.5 500.6 514.8 530.3

15 17/18 189.3 190.2 192.8 197.3 203.9 212.8 224.0 237.6 253.4 271.1

15 1/19 459.2 459.9 462.2 466.1 471.8 479.5 489.4 501.5 515.7 531.7

15 18/19 188.5 189.3 192.0 196.5 203.1 212.0 223.2 236.8 252.6 270.3

15 1/20 460.0 460.8 463.0 466.9 472.6 480.3 490.2 502.3 516.5 532.5

15 19/20 187.7 188.6 191.2 195.7 202.4 211.2 222.4 236.0 251.8 269.5

15 1/21 460.8 461.5 463.8 467.6 473.4 431.1 491.0 503.0 517.2 533.2

15 20/21 187.0 187.9 190.5 195.1 201.7 210.5 221.8 235.3 251.2 268.9

67
TABLE 6.- CIRCULAR SUN-SYNCHRONOUS ORBIT PARAMETERS FOR

SPECIFIED REPETITION FACTORS

Q

12

12 1/ 2

12 1/ 3

12 2/ 3

12 II 4

12 3/ 4

12 I/ 5

12 4/ 5

12 i/ 6

12 5/ 6

12 i/ 7

12 61 7

12 11 8

12 7/ 8

12 11 9

12 3/ 9

12 1/lO

12 9/10

12 1/14

12 13/14

12 1121

12 20/21

13

13 11 2

13 11 3

13 21 3

13 ii 4

13 31 4

13 i/ 5

13 4/ 5

13 11 6

13 5/ 6

13 1/ 7

13 61 7

13 1/ 8

13 7/ 8

13 1/ 9

13 81 9

13 1110

13 9/10

13 1114

13 13/14

13 1121

13 20121

as, kln is, deg

8054.630

7837.948

7908.546

7768.903

7944.445

1734.9A6

7966.181

7714.749

7980.754

7701. 357

7991.204

7691.827

7999.064

7684.698

8005. 190

7679.165

8010.100

7674.746

8022.761

7663.412

8033. 349

7653.999

7635. 259

7445.167

7507.209

7384. 389

7538.717

7354.462

7557.781

7336.650

7570.558

7324.835

7579.717

102.944

I01. 748

102.128

i01. 385

102.325

101.210

102.445

101.106

102.526

101 .O38

102.585

100.990

102.629

100.954

102.663

i00.926

102.691

lOO .904

102. 762

i00.84 7

102.822

100.800

100. 706

99.793

100.085

99.514

100.235

99,378

100. 327

99,298

100. 389

99.246

IO0.433

7316

7586

7310

7591

7305

7596

7301

7607

7291

7616.632

7283,023

.425 99.208

.604 100.467

.133 99.180

.972 100.493

.248 99.159

.273 100.514

.347 99.141

.361 I00.569

.338 99.097

100.614

99.061

68
TABLE6.- Concluded

14

14

14

14

14

14

14

14

14

14

14

14

14

14

14

14

14

14

14

14

14

14

15

15

15

15

15

15

15

15

15

15

15

15

15

15

15

15

15

15

15

15

15

15

16

Qas, km i s, deg

i/2

1/3

2/3

1/4

3/4

1/5

4/5

1/6

5/6

1/7

6/7

1/8

718

I/9

8/9

1/10

9/10

1/14

13114

i121

20/21

1/2

1/3

7266.465

7098.096

7153.130

7044.107

7181.049

7017.493

7197.932

7001.646

7209.242

6991.130

7217.348

6983.643

7223.442

6978.0t_0

7228.191

6973.691

7231.996

6970.216

7241.802

6961.301

7250 .O00

6953.893

6939.136

6788.770

6837.983

98.988

98.275

98.504

98.056

98.621

97.949

98.693

97.886

98.741

97.844

98.776

91.815

98.802

97.793

98.822

97.776

98.839

97.762

98.881

97.727

98.917

97.698

97.641

97.074

97.256

213 6740

114 6862

314 6716

1/5 6878

415 6702

1/6 6888

5/ 6!6692

1/ 76895

6/7 6686

1/8 6900

7/8 6681

1/9 6905

8/9 6677

i/i0 6908

9/10 6674

1/14 6917

13/14 6666

1/21 6924

20/21 6659

6646.

.430 96.899

.926 97.350

.58O 96.813

.001 97.407

.371 96.763

.097 97.445

.940 96.729

.331 97.473

.224 96.706

.769 97.493

.197 96.688

.005 97.5i0

.294 96.674

.400 97.523

.176 96.663

.147 97.556

.175 96.635

,457 97.584

.525 96.612

276 96.566

69
TABLE 7.- SOLAR POSITION DATA AS A FUNCTION OF TIME

J.D. Calendar Days from Days from x-, y-, and z-components of _;a

_44 xxxx.5 day 1981T Jan. 1.0, 1981 right ascension and declination

4605.5

4610.5

4615.5

4620.5

4625.5

4630.5

4635.5

4640.5

4645.5

4650.5

4655.5

4660.5

4665.5

4670.5

4675.5

4680.5

4685.5

4690.5

4695.5

4700.5

4705.5

4710.5

4715.5

4720.5

4725.5

4730.5

4735.5

4740.5

4745.5

4750.5

81/ 1/ 1/ 0

81/ 11 6/ 0

gl/ l/ill 0

_1/ 1/16/ 0

31/ 11211 0

_11 11261 n

_11 11311 o

_11 21 51 o

'_11 21101 o

811 21151 0

_1/ 2/20/ 0

81I 21251 0

_11 31 21 o

8II 3I 71 o

81/ 3/12/ 0

811 31171 0

811 31221 0

811 11271 0

all 41 II 0

811 41 61 0

_II 41111 0

_ll 41161 o

811 41211 0

81/ 4/26/ 0

81/ 51 1/ n

_1/ 51 _/ o

_11 51111 o

_11_51161 0

811 51211 0

811 5/261 0

-78.6992631

-73.6992631

-68.6992631

-67.6902631

-58.6Q92631

-Sq.6gg2631

-4R.6992631

-43.6gg2631

-3R.69g2631

-33.6992631

-2n.6992631

-23.6992631

-I_.6902631

-13.6992631

-8.6992631

-3.6992631

1.3007369

6.3007369

11.3007369

16.3007369

21.3007369

26.3007369

31.30_7368

36.300736g

41.3007369

46.3007369

51.3007369

56.3007369

61.3007369

66.3007369

0

5

i0

15

20

25

30

35

40

45

50

55

60

65

70

75

80

35

gO

95

I00

105

II0

115

120

125

130

135

140

145

0.1821703 -0.9021130 -0.3911599

-78.583370 -23.026691

0.2687963 -0.8836995 -0.3831757

-73.081722 -22.530532

0.3532844 -0.8583031 -0.3721637

-67.627542 -21.849120

0.4349470 -0.8261368 -0.3582162

-62.233954 -2o.g90690

0.5131237 -0.7874736 -0.3414516

-56.911432 -19.965343

0.5871874 -0.7426430 -0.3220129

-51.667547 -18.7R4701

0.6565507 -0.6920272 -0.3000657

-46.506911 -17.461549

0_72_6715 -0.6360570 -0.2757967

-41.43127" -16.009503

0.7780581 -0.5757062 -0.2494116

-36.439765 -14.442697

0.8_12727 -0.5099864 -0.2211320

-31.579148 -12.77553_

0.376Q352 -0.44OQ415 -0.1911938

-26.694212 -11.022466

0.9157256 -0.36R6411 -0.15g8441

-21.928099 -9.197850

0.9473857 -0.2936752 -0.1273386

-17.222644 -7.315830

0.9717197 -0.2166476 -0.0939392

-12.568691 -5.390267

0.9885947 -0.1381703 -0.0599111

-7.956371 -3.434713

0.9979401 -0.0588577 -0.0255209

-3.37534g -1.462399

0.9997459 0.0206790 0.0089665

1.184954 0.513750

0.9940616 0.0998369 0.0432896

5.735175 2.481089

0.980q934 0.1780259 0.0771926

10.285819 4.427217

0.9607014 0.2546732 0.1104271

14.847094 6.339941

0.9333973 0.3292274 0.1427541

19.428741 8.207247

0.8993399 0.4011617 0.1739450

24.039855 10.017274

0.85R8_27 0.4699771 0.2037836

2R.6R8705 11.75_304

n.812219_ 0.5352049 0.2320666

33.382532 13.418772

0.75q8_13 0.5g64090 0.2586048

38.127349 14.987297

0.7022320 0.6531877 0.2832242

42.927723 16.452734

0.6397148 0.7051751 0.3057661

47.786575 17.804262

0.5727992 0.7520423 0.3260878

52.704987 19.031497

0.5019768 0.7934983 0.3440633

57.682053 20.124629

0.4277578 0.8292908 0.3595830

62.7147R8 21.074589

aRespeetively (top line).

bin degrees (bottom line).

7O
TABLE 7.- Continued

J.D.

244 xxxx .5

4755.5

4760.5

4765.5

4770.5

4775.5

4780.5

4785.5

4790.5

4795.5

4800.5

4805.5

4810.5

4815.5

482O .5

4825.5

4_30.5

4a35.5

4840.5

4845.5

4850.5

4855.5

4860.5

4865,5

4870.5

4875.5

4880.5

4885.5

4890.5

4895.5

4900.5

Calendar

day

811 51311 0

811 61 51 o

811 61101 0

311 61151 0

811 6120/ 0

81/ 6125/ 0

81/ 613o/ o

811 7/ 5/ o

81/ 71101 0

811 7/15/ 0

811 7/20/ o

81/ 7/25/ 0

81/ 7/30/ 0

811 81 41 0

81/ 3/ 0/ 0

_I/ 8/14/ 0

all 8/19/ 0

:81/ 81241 o

811 81201 0

81/ 91 31 o

81/ 9/ 8/ o

Sll 91131 0

811 91181 0

811 91231 o

81/ 9/28/ 0

81/lO/ 3/ o

81/10/ 8/ 0

811101131 0

811101181 0

81110123/ 0

Days from

1981T

71.3007360

76.3007369

81.3007369

86.3007369

91.3007369

96.3007369

101.3007369

I06.3007360

111.3007369

116.3007369

121.3007369

126.3007369

131.3007369

136.3007369

141.3007360

146.1007369

151.3007360

156.3007369

161.3007360

166.3007360

171,3007369

176.3007369

181.3007369

186.3007369

191.3007369

196.3007369

201.3007369

20_.3007369

211.3007369

216.3007369

Days from

Jan. 1.0, 1981

150

155

160

165

170

175

180

185

190

195

200

205

210

215

220

225

230

235

240

245

250

255

260

265

270

275

280

285

2"0

295

X-_ y-, and z-components of _a

right ascension and declinatio_ u

0.3506685 0.8592n50 0.3725542

67.7_8107 71.373232

0.2712473 0.8830692 0,3829014

72.924913 22.513521

0.1900419 0.9007454 0.3905658

7_.086287 22.080713

0.1076068 0.91213_I _.3955057

81.271794 21.207518

0.0244908 0.9171809 0.3976061

88.469887 23,414235

-0.05_7201 0.9158823 0.39712QI

93.66_3q3 23.398829

-0.1414958 0.9082146 0.3938130

98.855048 23.1_1968

-0.2232742 0,8943046 0._877729

104.018036 22.815Q95

-0.3035090 0.8741872 0.3790499

109.146491 22.274847

-0.3816632 0.8480146 0.3677014

114.230930 21.573926

-0.4572113 0.8159553 0.3538003

119.263594 20.719940

-0.5296418 0.7782138 0.3374355

12_.238676 19.720709

-0.5984601 0.7350298 0.3187108

129.152429 18.584977

-0.6631906 0.6866775 0.2977450

134.003190 17.322218

-0.7233793 0.6334647 0.2746718

139.791302 15.942462

-0.7785067 0.5757321 0.2&963_8

143.518081 14.456144

-0.828_401 0.5138519 0.2228074

148.190143 12.87_081

-0.8725366 0.4482265 0.1943521

152.810199 II.20687_

-0.9105459 0.3792875 0.1644599

157.385851 9.465863

-0.9421629 0.3074934 0.1333298

161.924887 7.662054

-0.9671206 0.2033281 0.1011716

166.435989 5.806641

-0.9851929 0.15729_7 0.0682051

170.928554 3.910901

-0.9961975 0.0799329 0.0346590

175.412528 1.986217

-0.9999981 0.0017759 0.0007700

179.898247 0.044119

-0.9965074 -0.0766122 -0.0332192

-175.603707 -1.903671

-0.9856891 -0.1546602 -0.0670610

-171.082670 -3.845198

-0.9675600 -0.2317897 -0.1005044

-166.528047 -5.768222

-0.9421917 -0.3074189 -0.1332974

-161.929496 -7.660183

-0.9097123 -0.3809676 -0.1651883

-157.277130 -9.508174

-0.8703065 -0.4518614 -0.1959279

-152.561749 -11.298939

aRespectively (top line).

bin degrees (bottom line).

71
TABLE7.- Concluded

J.D. Calendar Daysfrom Daysfrom 244xxxx.5 day 19_IT Jan. 1.0, 1981

4905.5

4910.5

4915.5

4920.5

4925.5

4930.5

4935.5

4940.5

4945.5

495O.5

4055.5

4960.5

4965.5

4970.5

4975.5

4980.5

4985.5

4990.5

4995.5

5000.5

5005.5

5010.5

5015.5

5020.5

5O25.5

5030.5

5035.5

5040.5

5045.5

5050.5

81/10/28/ 0

81/ii/ 2/ 0

81/iii 7/ 0

81/11/12/ 0

81111/17/ 0

81/Ii/22/ 0

81/11/27/ 0

81/12/ 2/ o

811121 7/ o

81/12/12/ n

81112117/ 0

81/12/22/ 0

81112127/ 0

221.3007369

226.3007369

231. 3007369

236.3007369

241. 3007369

246.3007369

251.3O07369

256.3007369

261.3q07369

266.3007369

271.3007369

276.3007_69

281.3007369

82/ 1/ 1/ 0

82/ 11 61 0

821 1/11/ 0

821 11161 0

821 1/21/ 0

82/ 1/26/ 0

821 11311 o

!82/ 2/ 5/ o

s2/ 2/10/ 0

82/ 21151 0

82/ 2/20/ O

82/ 2/25/ O

s_l 31 21 0

82/ 31 7/ o

821 31121 o

821 3/17/ q

82/ 31221 0

286.3007369

291.30O7369

296.3007160

301. 3007369

306.3007369

311. 3007369

316.3007369

321. 3007369

326. 3007369

331.3O07369

336. 3007369

341. 3007369

346. 3007369

351.3o07369

356.3007360

361. 3007369

366. 3007369

30O

305

310

315

320

325

330

335

340

345

350

355

360

365

370

375

38O

385

390

395

400

405

410

415

420

425

430

435

440

445

aRespectively (top line).

bin degrees (bottom line).

^ a

x-, y-, and z-components of xc_

right ascension and declinatiofi u

-0.8242168 -0.5195372 -0.2252723

-147.775111 -13.018892

-0.7717430 -0.5834481 -0.2529841

-142.910226 -14.654172

-0.7132410 -0.6430690 -0.2788359

-137.961694 -16.190740

-0.6491219 -0.6979018 -0.3026114

-132.926046 -17.614522

-0.5798494 -0.7474810 -0.3241090

-127.802092 -18.911607

-0.5050370 -0.7913788 -0.3431431

-122.591224 -20.068490

-0.4279443 -0.8292102 -0.35q5469

-117.297654 -21.072375

-0.3464723 -0.8606381 -0.3731740

-111.92_537 -21.911505

-0.2621588 -0.8853772 -0.3839000

;106.493941 -22.575527

-0.1756723 -0.90319R0 -0.301_2_0

-101.006626 -23.055840

-0.007705 _ -0.q139303 -0.3962815

-95.4P1630 -23.345q26

O.OOl0301 -0.0174653 -0.397_143

-89.935665 -23.441613

o.0808142 -0.9137579 -0.3962067

-84.306367 -23.341258

0.1770221 -0.902fl273 -0.3914671

-78.851454 -23.0_5823

0.2646340 -0.8847572 -0.3R36319

-73.347fl86 -22.558835

0.3492_19 -0.8596054 -0.3727650

-67.891082 -21.886248

0.4310573 -0.8278510 -0.3599576

-62.494286 -21.036198

0.5094182 -0.7894969 -0.3423268

-57.168107 -20.018702

0.5836956 -0.7449571 -0.3230143

-51.920262 -18.845316

0.6533003 -0.6946124 -0.3011847

-46.755505 -17.528777

0.7176877 -0.6388912 -0.2770239

-41.67572_ -16.082665

0.7763635 -0.5782654 -0.2507365

-36.680157 -14.521100

0.828_872 -0.5132449 -0.2225435

-31.765689 -12.858472

0.8748758 -0.4443720 -0.1926802

-26.927193 -11.109239

0.9140064 -0.3722156 -0.1613930

-22.157883 -9.207763

0.9_60176 -0.2973645 -0.1289375

-17.449650 -7.408200

0.9707107 -0.2204221 -0.0055752

-12.793379 -5.484429

0.9879498 -0.1420002 -0.0615714

-8.179231 -3.530015

0.9976610 -0.0627131 -0.0271Q24

-3.596808 -1.558205

0.99O8317 0.0168276 0.0072964

0.964222 0.418060

72
TABLE8.- AVAILABLESUNELEVATION ANGLE AS A FUNCTION OF LATITUDE

FOR THE WINTER AND SUMMER SOLSTICES AND THE EQUINOXES

Sea son deg

Winter

solstice

0

i0

20

30

4O

50

60

7O

8O

90

Vernal or

autumnal

equinox

0

i0

20

30

4O

50

6O

7O

8O

90

Summer

solstice

o

i0

20

30

40

50

60

70

80

oO

Hour angle,

B, deg .....

Sun elevation angle, _, deg, at local time

relative to high noon, sidereal hours, of -

66.6

56.6

46.6

36.6

26.6

16.6

6.6

-3.5

-13.5

-23.4

52.6

45.5

37.6

29.3

20.7

11.9

3.t7

-5.9

-14.7

-23.4

27

22

17

11.4

5.5

-0.6

-6.6

-12.5

-18.2

-23.4

6

.3 0.0

.5 -4.0

.2 -7.8

-11.5

-14.8

-17.7

-20.2

-22.0

-23.1

-23.4

-27.3

-31.4

-3t,.6

-36.6

-37.4

-36.8

-35.0

-32.1

-2_q.1

-23.4

I0

-52.6

-58.4

-62.0

-62.5

-59.8

-54.6

-47.9

-40.2

-32.0

-23.4

90.0

80.0

70.0

60.0

50.0

40.0

30.0

20.0

i0.0

0.0

60.0

58.5

54.5

48.6

41.6

33.8

25.7

17.2

8.6

0.0

30.0

29.5

28.0

25.7

22.5

18.7

14.5

9.8

5.0

0.0

66.6

76,6

86.6

83.5

73.5

63.5

53.5

43.4

33.5

23.4

52.6

58.4

62.0

62.5

59.8

54.6

47.9

40.2

32.0

23.4

30

27

31

34

36

37

36

35

32

28

23

6O

.3

.4

.6

.6

.4

.8

.0

.I

.I

.4

0.0

4.0

7.8

ii .5

14.8

17.7

20.2

22.0

23.1

23.4

9O

0.0 -30.0

q. 0 -2q. 5

0.0 -28.0

0.0 -25.7

0.0 -22.5

0.0 -18.7

0.0 -14.5

0.0 -9.8i

0.0 -5.0

0.0 0.0

-27.3

-22.5

-17.2

-11.4

-5.5

0.6

6.6

12.5

18.2

23.4

120

-60.0

-5S.5

-54.5

-43.6

-41.6

-33.8

-25.7

-17.2

-8.6

0.0

-52.6

-45.5

-37.6

-20.3

-20.7

-11.9

-3.0

5.9

14.7

23.4

150

12

-66.6

-76.6

-86.6

-83.5

-73.5

-63.5

-53.5

-43.4

-33.5

-23.4

-90.0

-80.0

-70.0

-60.0

-50.0

-40.0

-30.0

-20.0

-10.0

0.0

-66.6

-56.6

-46.6

-36.6

-26.6

-16.6

-6.6

3.5

13.5

23.4

180

73
TABLE 9.- ORBIT PARAMETERS FOR A CIRCULAR ORBIT WHICH PASSES

OVER THE SAME SITES ON THE EAST COAST ONCE PER DAY,

WITH NORFOLK, VIRGINIA, AS INITIAL CONDITION

Inclination, deg ............................. 63.0

Eccentricity ............................... 0.0

Semimajor axis, km .......................... 6887.371

Anomalistic period, sec ....................... 5689.92

Nodal period, see .......................... 5689.80

Mean motion:

5446.5085 deg/day .............................

Anomalistic revolutions per day ................. 15..184746

Perigee precession rate:

deg/orbit ............................

deg/day .............................

0.OO7654

0.116232

Nodal precession rate:

deg/orbit ............................

deg/day .............................

-0.227599

-3.456029

Repetition factor, Q .......................... 15.0

Initial conditions to pass over Norfolk, Virginia (L = -76.289 ° ,

at 12:00 noon EST on Jan I, 1981:

Longitude of ascending node: -98°.741

J.D. = 244 4605.2083333

= 42o.309

k = 36.853 °)

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I, Relx_t No. 2. Government Accession No.

NASA RP-IO09

41 Title and Subtitle

AN INTRODUCTION TO ORBIT DYNAMICS AND ITS APPLICATION

TO SATELLITE-BASED EARTH MONITORING MISSIONS

7. Author(s}

David R. Brooks

9. Performing Organization Name and Addre_

NASA Langley Research Center

Hampton, VA 23665

12. S_nsoring Agency Name and Addr_s

_ationaIAeronautics and Space Administration

Washington, DC 20546

3. Recipient's Catalog No.

5. Report Date

November 1977

6. Performing Organization Code

8. Performing Orgamzation Report No.

L-11710

10. Work Unit NO.

683-75-33-02

'11. Contract or Grant No.

13. Type of Report and Period Covered

Reference Publication

14. Sponsoring Agency Code

15. Supplementary Notes

16. Abstract

A self-contained tutorial treatment is given to the long-term behavior of satellites at a level of complexity suitable for the initial planning phases of Earth

monitoring missions. First-order perturbation theory is used to describe in detail

the basic orbit dynamics of satellite motion around the Earth and relative to the

Sun. Surface coverage capabilities of satellite orbits are examined. Several

examples of simulated observation and monitoring missions are given to illustrate

representative applications of the theory. The examples stress the need for devising ways of maximizing total mission output in order to make the best possible use

of the resultant data base as input to those large-scale, long-term Earth monitoring activities which can best justify the use of satellite systems.

17. Key Words (Suggested by Authoris))

Orbit dynamics

Remote sensing

Satellite monitoring

18. Dislribution Statement

Unclassified - Unlimited

Subject Category 13

19. Security Cla_if. (of this report] 20. Security Classif. (of this page} 21. No, of Pages 22. Price"

Unclassified Unclassified 84 $5. O0

* For sale by the National Technical Information Serwce, Springfield, Virgima 22161

NASA-Langley, 1977