IPN Progress Report 42-196 • February 15, 2014

The Planetary and Lunar Ephemerides
DE430 and DE431
William M. Folkner,* James G. Williams,† Dale H. Boggs,†
Ryan S. Park,* and Petr Kuchynka*

abstract. — The planetary and lunar ephemerides DE430 and DE431 are generated by fitting

numerically integrated orbits of the Moon and planets to observations. The present-day
lunar orbit is known to submeter accuracy through fitting lunar laser ranging data withan updated lunar gravity field from the Gravity Recovery and Interior Laboratory (GRAIL)
mission. The orbits of the inner planets are known to subkilometer accuracy through fitting
radio tracking measurements of spacecraft in orbit about them. Very long baseline interferometry measurements of spacecraft at Mars allow the orientation of the ephemeris to betied to the International Celestial Reference Frame with an accuracy of 0′′.0002. This orientation is the limiting error source for the orbits of the terrestrial planets, and corresponds toorbit uncertainties of a few hundred meters. The orbits of Jupiter and Saturn are determinedto accuracies of tens of kilometers as a result of fitting spacecraft tracking data. The orbits
of Uranus, Neptune, and Pluto are determined primarily from astrometric observations,for which measurement uncertainties due to the Earth’s atmosphere, combined with starcatalog uncertainties, limit position accuracies to several thousand kilometers. DE430 and
DE431 differ in their integrated time span and lunar dynamical modeling. The dynamical
model for DE430 included a damping term between the Moon’s liquid core and solid mantle that gives the best fit to lunar laser ranging data but that is not suitable for backward
integration of more than a few centuries. The ephemeris DE431 is similar to DE430 but was
fit without the core/mantle damping term, so the lunar orbit is less accurate than in DE430
for times near the current epoch, but is more suitable for times more than a few centuries
in the past. DE431 is a longer integration (covering years –13,200 to +17,191) than DE430(covering years 1550 to 2650).

I. Introduction

The planetary and lunar ephemeris DE430 succeeds the ephemeris DE421 [1] and its precursor DE405 [2] as a general purpose ephemeris. Several interim ephemerides have beenreleased since DE421 with specific improvements in estimates of the orbits of Mercury,Mars, and Saturn intended primarily to support the Mercury Surface, Space Environment,* Mission Design and Navigation Section.
† Tracking Systems and Applications Section.

1

The research described in this publication was carried out by the Jet Propulsion Laboratory, California Institute ofTechnology, under a contract with the National Aeronautics and Space Administration. © 2014 California Instituteof Technology. U.S. Government sponsorship acknowledged.

Geochemistry, and Ranging (MESSENGER), Mars Science Laboratory, and Cassini missions.
These interim ephemerides had small deficiencies in other areas, particularly the lunar
librations. DE430 has been updated for all bodies, including the Moon.
The positions and velocities of the Sun, Earth, Moon, and planets, along with the orientation of the Moon, result from a numerically integrated dynamical model. They are stored
as Chebyshev polynomial coefficients fit in 32-day-long segments [3]. DE430 also includes
Chebyshev polynomial coefficients fit to a numerically integrated difference between
ephemeris coordinate time and terrestrial time. Parameters of the dynamical model havebeen adjusted to fit measurements of the relative positions of the planets and Moon withrespect to Earth. The dynamical model is similar to that used for DE405 [4], but includes
a model for the fluid core of the Moon [5,6] and an updated treatment of asteroids. The
coordinate system and units are similar to those used for DE421 and DE405, but include
changes to the definition of coordinate time and the astronomical unit.
Perturbations from 343 asteroids have been included in the dynamical model. The asteroid
orbits were iteratively integrated with the positions of the planets, the Sun, and the Moon.
The set of 343 asteroids is identical to the set used in DE421. The set represents 90 percentof the total mass of the main belt and contains the asteroids with the most significant effects on the orbit of Mars in terms of perturbation amplitude and frequency [7]. For DE421,
a limited number of individual asteroid mass parameters were estimated, with the rest of
the asteroid mass parameters determined by dividing them into three taxonomic classes,
estimating a constant density for each class, and using volumes estimated from Infrared
Astronomical Satellite (IRAS) observations [8]. For DE430, we estimated the mass parameterfor each asteroid individually subject to a priori values and uncertainties for volume anddensity derived from the Wide-field Infrared Survey Explorer (WISE) and the Supplemental
IRAS Minor Planet Survey (SIMPS) [9,10].
The lunar dynamical model for DE430 includes interaction between the rotation of thefluid core and the mantle. The effect of this interaction is clearly seen in lunar laser ranging(LLR) data and characterizes a damping in the differential rotation between the core andmantle. Because the initial conditions of the lunar core cannot be determined perfectly, error grows in backward integrations when this interaction model is included. Consequently,
the DE430 time span has been limited to the years 1550 to 2650. Further information on
the lunar coordinate system and data reduction is available elsewhere.1
In order to provide a general ephemeris covering several thousand years, a lunar model
without the core–mantle interaction term has been used for the ephemeris DE431. DE430
and DE431 have been fit to the same data. The difference in the orbits of the planets between DE430 and DE431 is less than 1 m over the DE430 time span, which is well below
the estimated uncertainties from the fit. The difference in the orbit of the Moon between
DE430 and DE431 is less than 1 m during the time span of the LLR data, 1970 to 2012, but
grows over longer times primarily due to a difference in estimated tidal acceleration. Figure 1 shows a plot of the difference between the lunar orbit in DE430 and DE431.

1 J. G. Williams, D. H. Boggs, and W. M. Folkner, “DE430 Lunar Orbit, Physical Librations and Surface Coordinates,” JPLInteroffice Memorandum 335-JW,DB,WF-20130722-016 (internal document), Jet Propulsion Laboratory, Pasadena, California, July 22, 2013. 	
2

rM [DE431] – rM [DE430], m

40
∆R

20

∆T

∆N

0
–20
–40
1900.01950.02000.0Year

2050.02100.0Figure 1. Difference in lunar orbit of DE431 and DE430 in radial (R),along-track (T), and normal (N) components.

DE430 should be used to analyze modern data. DE431 is suitable for the analysis of earlier
historical observations of the Sun, Moon, and planets. The DE431 time span from the year
–13,200 to the year 17,191 extends far beyond historical times and caveats are offered.
For the planets, uncertainties in the initial conditions of the orbits will cause errors in
the along-track directions that increase at least linearly with time away from the present.
Resonances including, but not limited to, those between Jupiter and Saturn, and between
Uranus, Neptune, and Pluto, may complicate the propagation of errors. Typically, thealong-track component will degrade faster than the other two components. For the Moon,
the uncertainty given for the tidal acceleration causes a 28 m/century2 along-track uncertainty. But there are other concerns, e.g., the theory for the orientation of Earth includes
polynomial expressions that are adequate for thousands of years, but are not designed formuch longer times.
Compared with DE421, DE430 and DE431 have been fit to additional data for the Moonand planets. The lunar orbit has been improved through use of additional LLR data,
and an improved gravity field of the Moon from the Gravity Recovery and Interior Laboratory (GRAIL) mission. The orbit of Mercury has been improved through use of range
measurements to the MESSENGER spacecraft after it entered orbit about Mercury. The orbits
of Venus, Earth, and Mars have been improved though additional very long baseline interferometry (VLBI) observations and additional range measurements to Venus Express, Mars
Express, Mars Odyssey, and Mars Reconnaissance Orbiter. A more accurate orbit of Saturnhas been achieved primarily due to improved treatment of range measurements to the Cassini spacecraft. The orbit of Pluto has been improved through additional observations fromseveral observatories. The orbits of Jupiter, Uranus, and Neptune are not significantly different from those in DE421, although additional astrometric observations have been included.
Improvements for the orbits of Mercury and Pluto are expected in the next year. The
MESSENGER data used for DE430 and DE431 are from the first year in orbit about Mercury.
During that time, the spacecraft orbit pericenter was always over the northern hemisphere.
Lack of Doppler measurements at low altitude over the southern hemisphere limited theaccuracy of the estimated Mercury gravity field and spacecraft trajectory. Data from a second year with the pericenter over the southern hemisphere are available but had not beenprocessed at the time of the DE430 and DE431 fit. An improved orbit for Pluto is needed for3the New Horizons mission. Additional measurements and improved processing of existingdata are in progress.
The coordinate system for DE430 and DE431 is described in Section II. The dynamical
model for translation is described in Section III. The dynamical model for the orientationof the Moon is described in Section IV. Section V describes the data to which the dynamicalparameters have been adjusted. The initial conditions and dynamical constants are given in
Section VI.

II. Planetary Ephemeris Coordinates
A. Orientation of Axes

The coordinate system is defined by axes aligned with the International Celestial Reference System (ICRS), with the XY plane close to the mean equator of epoch J2000.0 and
the X‑axis close to the intersection of the mean equator of J2000.0 with the mean ecliptic
plane.2 The ICRS is currently realized by the positions of extragalactic radio sources given in
the Second Realization of the International Celestial Reference Frame (ICRF2) [11] adopted
by the International Astronomical Union (IAU) in 2009.3 The orbits of the inner planets
are aligned with ICRF2 with an accuracy of 0′′.0002 through use of VLBI observations ofspacecraft in orbit about Mars.4
Analysis of VLBI observations relative to ICRF2 indicates a secular drift in aberration consistent with the solar system orbiting about the center of the galaxy [12]. This effect is notmodeled in ICRF2 or in the ephemerides DE430 and DE431. It may need to be taken into
account in future ephemerides as measurement accuracies improve.

B. Solar System Barycenter

The origin of the ICRS is the solar system barycenter.5 For DE430 and DE431, the barycenter is approximated using the invariant quantities of the n-body metric used to model the
point mass interactions of the Sun, Moon, planets, and asteroids (see Section III.A). The
mass/energy of the system M is a conserved quantity where M is defined by6

M = | n *A

(1)

A

where the summation is over all bodies (Sun, Moon, planets, and asteroids) andn *A = GM A * 1 +

GM B
1 2
1
4
2 vA 2 | r
2c2c B ! A AB

(2)

2 21st International Astronomical Union General Assembly, Resolution B2, 1991.
3 27th International Astronomical Union General Assembly, Resolution B3, 2009.
4 W. M. Folkner and J. S. Border, “Linking the Planetary Ephemeris to the International Celestial Reference Frame,”

Highlights of Astronomy, vol. 16 (in press).
5 24th International Astronomical Union General Assembly, Resolution B1.3, 2000.
6 F. B. Estabrook, “Derivation of Relativistic Lagrangian for n-body Equations Containing Relativity Parameters β and g,”

JPL Interoffice Memorandum (internal document), Jet Propulsion Laboratory, Pasadena, California, 1971. 	
4

where GM A is the mass parameter of body A, rAB = | rA - rB | is the distance between body A
at position rA and body B at position rA , v A = | v A | is the magnitude of the velocity of body

A , and c is the speed of light. The momentum P is also conserved where P is given byd
P = dt ;| n *A rAE .

(3)

A

The position of the center of mass/energy R is given by

R = b | n *A rA l / b | n *A l .
A

(4)

A

The velocity of the center of mass/energy V = dR/dt = P/M is invariant for the n-bodymetric since P and M are invariant.
For DE430 and DE431, the initial positions and velocities of the Moon and planets withrespect to the Sun were estimated, and the positions and velocities of the asteroids withrespect to the Sun were estimates from the Horizons online solar system data service [13].
The initial position and velocity of the Sun were computed by setting R and V to zero. The
positions of the Sun, Moon, planets, and asteroids were then integrated using the equationsof motion given in Sections III and IV. Because the equations of motion include extended
body effects not included in the n-body metric, the center of mass/energy as defined by
Equation (4) is not an exact invariant. The position of the center of mass/energy moves
with respect to the origin of the coordinate system by less than 1 mm/century, as shownin Figure 2. This motion is near the numerical noise of the stored ephemeris and is small
compared with current measurement accuracy.

Position, mm

1.0
0.5

X
Y
Z

0.0
–0.5
–1.0
1950.0 1960.0

1970.01980.01990.02000.0Year

2010.02020.02030.02040.0 2050.0

Figure 2. Coordinates of the solar system barycenter of DE430 computed using
a translational invariant for the n-body metric.

Previous JPL ephemerides have implemented the barycenter in different manners. For
DE421, the positions of the Moon and planets were integrated using the equations of motion while the position and velocity of the Sun were computed at each integration time
step such that R and V remained zero. For DE405, the position of the Sun was integrated,with the initial position and velocity of the Sun computed by setting R and V to zero,
except that the rate of change of n *A was neglected in computing the initial velocity of theSun. As a consequence, the position of the barycenter as computed using Equation (4) for5

DE405 changed with time by approximately 0.5 m/century [14]. This drift has no measurable consequences on the relative motions of the Sun, Moon, and planets.
The location of the barycenter relative to the Sun, Moon, and planets depends on the set ofbodies modeled. In particular, inclusion of trans-Neptunian objects such as Sedna and Erisin the ephemerides from the Institute of Applied Astronomy [15] causes a difference of thelocation of the barycenter with respect to the Sun of about 100 km. The change in barycenter does not significantly affect the relative positions of the Sun, Moon, and planets.

C. Ephemeris Coordinate Time

The coordinate time scale used for DE430 and DE431 is Barycentric Dynamical Time (TDB)
as defined in terms of Barycentric Coordinate Time (TCB).7,8 For purposes of measurement
reduction, the difference between International Atomic Time (TAI) and TDB is needed at
the point the measurement is made. An intermediate time, Terrestrial Time (TT), is introduced, where TT = TAI+32.184s. The quantity TDB–TT as a function of TDB is given by

LG - L B
1 - LG
TDB - TT = 1 L (TDB - T0) + 1 - L TDB 0
- B
B
+

2
1 - LG TDB
1 e vE1#+ w 0E + w LE o dt + 2 v E $ _rS - rE i
1 - L B T0 + TDB0 c 2 2
c4
1 - L G TDB
1 e ve 3 2
1 2- 1-L #
- 2 v E w 0E + 4v E $ w AE + 2 w 0E + D E o dt
4
8
+
T
TDB
0 c
B 0

(5)

vE
1
+ 4 e 3w 0E + 2 o v E $ (rS - rE)c2

where, for this expression, TDB and TT are measured in Julian days, T0 is the Julian
day 2443144.5003725, TDB 0 = 65.5 × 10–6/86400 days, c is the speed of light, L G =6.969290134 × 10–10 defines the rate of TT with respect to Geocentric Coordinate Time
(TCG), L B = 1.550519768 × 10–8 defines the rate of TDB with respect to TCB, v E is thevelocity of the Earth, rS is the position of the measurement station, and rE is the positionof the Earth. Positions and velocities are all with respect to the solar system barycenter.

w 0E is the potential at the geocenter due to external point masses that is given by
GM A
w 0E = | r
AE
A!Ewith the summation over all modeled bodies A other than the Earth. w LE is the potential
at the Earth due to external oblate figures of external bodies, where for computation of
TDB–TT for DE430 and DE431, only the effect of the oblateness of the Sun is included, asgiven by

7 21st International Astronomical Union General Assembly, Resolution B2, 1991.
8 24th International Astronomical Union General Assembly, Resolution B1.3, 2000. 	

6(6)

w LE =-

2 1
2`j
3 R 9 2 3 sin { E, 9 - 1

GM9 J2 9| r9 - rE |(7)

where J 2 9 is the unnormalized second-degree zonal harmonic of the Sun’s gravitational
potential, R 9 is the radius of the Sun, and { E, 9 is the latitude of the Earth relative to theSun’s equator. w AE is the sum of the product of the potential of external point masses times
their velocities given by

GM A v A
rAE
A!E

w AE = |(8)

and D E is given by

GM A
GM B 1 v A $ (rE - rA)2
o + 1 a A $ _rE - rA iH
D E = | r >- 2v A + | r + 2 e
r
2
AE
AB
AE
A!E
B!A
2

(9)

where a A is the acceleration of body A. Equation (5) is based on [16]. It is essentially thesame as used in [17]. The relations between the various time scales are summarized in the
IERS 2010 Conventions [18]. Several of the terms included, specifically the contributionsof the asteroids and the solar oblateness, are small compared with current measurement
accuracies.
For DE430 and DE431, the negative of the quantity TDB–TT from Equation (5) evaluated at
the geocenter has been numerically integrated and saved as a set of Chebyshev polynomial
coefficients in a format similar to the positions and velocities of the bodies. For measurement reduction, TT–TDB as a function of TT is also needed; it can be computed by a simple
iterative technique.
Figure 3 shows the difference in the integrated value of TT–TDB and the numerically integrated time ephemeris based on DE405 (TE405) [19]. A constant difference of 65.5 ms wasintroduced in the current definition of TDB and has been subtracted from the difference.
The slope of ~ 5.7 ns/century is another consequence of the current definition of TDB. At
the time TE405 was integrated, the definition of TDB was different and not strictly suitablefor use as ephemeris coordinate time. The ephemeris coordinate time was labeled Teph andthe definition of Teph included an adjusted constant to ensure no average rate of Teph withrespect to TAI. The slope in the difference between TT–TDB for DE430 and TE405 seen in
Figure 3 is similar to the results for the INPOP08 ephemeris [17]. This slope causes differences in the reduction of planetary data that are small compared with current measurement
accuracies. The small “ripple” about the mean slope of the difference is due to inclusion of
the effects of asteroids in Equation (5) that are small and were not included in TE405.

D. Astronomical Unit

The JPL planetary ephemerides have been integrated with position coordinates in astronomical units. For ephemerides prior to DE430, the astronomical unit (au) was estimated7

TE430 – TE405 –65.5 mS, ns

5

0

–5
1950.02000.0Year

2050.0Figure 3. Difference between TT–TDB at the geocenter integrated from DE430 (TE430) and thecomparable quantity integrated from DE405 [19] after removing a constant 65.5 ms.

(in km) for each ephemeris, basically enabling conversion of the mass parameter of the Sunfrom units of au3/day2 to units of km3/s2. The mass parameter of the Sun was defined by

GM9 = k 2, where Gauss’s constant k = 0.01720209895 is a defined value. With the introduction of the TCB time scale, and with modern data nearly accurate enough to measure
the rate of change of GM 9 , the astronomical unit is now defined to be 149597870.700 km
in all time scales.9 GM 9 in units of au3/day2 is now an estimated quantity. For DE430 and
2

DE431, GM 9 has been set to k since our current estimate is consistent with this value given
the current value of the au [20].
While the JPL ephemerides have been integrated with coordinates given in au, the positions for each ephemeris have been scaled by the appropriate value of the au to units of kmand fit to Chebyshev polynomials for distribution to users. Many programs that read the
JPL ephemerides automatically convert the interpolated values of coordinates from units of
km to au. Users should now be aware of the value of the au they prefer to be used for thisconversion.

E. Orientation of the Moon

The orientation of the lunar exterior (mantle and crust, hereafter referred to by mantle)
is parameterized by Euler angles, z m , i m , and } m, that relate the Moon-centered, rotating
lunar mantle to the inertial frame. The Moon is distorted by tides and rotation. The mantle
coordinate system is defined by the principal axes of the undistorted mantle in which themoment of inertia matrix of the undistorted mantle is diagonal. The directions of the principal axes are estimated from analysis of LLR data. The Euler angles that define the rotationfrom the principal axis (PA) frame to the inertial ICRF2 frame are: z m , the angle from theX-axis of the inertial frame along the XY plane to the intersection of the mantle equator;

im , the inclination of the mantle equator from the inertial XY plane; and }m , the longitude
from the intersection of the inertial XY plane with the mantle equator along the mantle
equator to the prime meridian.
Position vectors expressed in coordinates along the principal axes, rPA , can be expressed ascoordinate vectors in inertial space rI using the relation

9 28th International Astronomical Union General Assembly, Resolution B2, 2012.

8

rI = R z _- zmi R x _- imi R z _- }mi rPA(10)where the rotation matrices R x, R y, and R z are right-handed rotations of frame orientations
defined by

JK1
0
0 NOO
KK
O
R x _ai = KKK0 cos a sin a OOOKO
L0 - sin a cos aP(11)

KKJcos a 0 - sin aNOOKO
R y _ai = KKK 0 1
0 OOO
K
O
L sin a 0 cos a P(12)

JK cos a sin a 0NO
KK
OO
R z _ai = KKK- sin a cos a 0OOO .
K
O
0 1P
L 0

(13)

The mantle Euler angles are numerically integrated along with the positions of the bodies.
Instead of integrating the Euler angles using second-order equations, they are integratedusing first-order equations with time derivatives computed in terms of the angular velocity of the mantle. The angular velocity of the mantle is integrated using its time derivativethat is computed from the torques acting on the Moon, as described in Section IV. The time
derivatives of the Euler angles are related to the angular velocity of the mantle expressed in
the mantle frame, ~ m , byzo m = (~m, x sin }m + ~ m, y cos }m) / sin im
io m = ~m, x cos }m - ~ m, y sin }m(14)}o m = ~m, z - zo m cos im .

The model for the Moon includes a fluid core. The orientation of the core with respect tothe inertial frame is described by the Euler angles z c, i c, and } c that are defined and numerically integrated in the same manner as the Euler angles for the mantle. As was done forthe mantle, the Euler angles for the core are integrated using coupled first-order equations
along with the core angular velocity. However, since the shape of the core is modeled as
fixed to the frame of the mantle, it is more convenient to express and integrate the core angular velocity expressed in the mantle frame. The time derivatives of the core Euler angles
are then computed byzo c = ~ @c, z - }o c cos ic
io c = ~ @c, x(15)}o c =- ~ c, y / sin ic@

where the coordinate vector of core angular velocity in the mantle frame ~ c is related tothe coordinate vector ~ @c in a frame defined by the intersection of the core equator with
the inertial XY plane by9

~ @c = R z (zc - zm) R x (- im) R z (- }m) ~ c .

(16)

Alternatively, the time derivatives of the core Euler angles can be computed using the angular velocity of the core expressed in the core frame ~ Ac byzo c = `~ Ac, x sin }c + ~ Ac, y cos }cj / sin ic
io c = ~ Ac, x cos }c - ~ cA , y sin }c(17)}o c = ~ cA , z - zo c cos ic
and the core angular velocity expressed in the core frame is given by(18)~ Ac = R z (}c) R x (ic) R z (zc) R z (- zm) R x (- im) R z (- }m) ~ c .
While the orientation for the mantle and core are both numerically integrated, only the
mantle Euler angles and mantle angular velocity are distributed with the planetary eph-emerides. The initial values for the core Euler angles and the core angular velocity, with thelatter expressed in the mantle frame, are given in Table 7 (see page 49).
The locations of features on the lunar crust are usually given by coordinates expressed in
the mean-Earth/mean-rotation (MER) frame, where the X axis is defined by the body-fixedaxis that points toward the mean Earth direction and the Z axis points toward the meanrotation axis direction. Coordinate vectors in the MER frame are related to coordinate vectors in the PA frame by a fixed rotation. However, the definition of the MER frame depends
on the approximations used to estimate the X and Z directions. For DE430, an updated
estimate of the MER frame has been made. Coordinate vectors in the DE430 PA frame areconverted to the DE430 MER frame by10

rMER, DE430 = R x _- 0.285”i R y _- 78.580”i R z _- 67.573”i rPA, DE430 .

(19)

Several recent lunar missions have been producing cartographic data products in the MER
frame as estimated from DE421. It may be convenient for continuity to retain that coordinate system. Coordinate vectors in the DE430 PA frame are converted to the DE421 MER
frame byrMER, DE421 = R x _- 0.295”i R y _- 78.627”i R z _- 67.737”i rPA, DE430 .

(20)

F. Orientation of the Earth

Only the long-term change of the Earth’s orientation is modeled in the ephemeris integration. The Earth orientation model used for the DE430 and DE431 integration is based on
the International Astronomical Union (IAU) 1976 precession model [21,22] with an estimated linear correction and on a modified IAU 1980 nutation model [23] including only
terms with a period of 18.6 years.

10 J. G. Williams, D. H. Boggs, and W. M. Folkner, “DE430 Lunar Orbit, Physical Librations, and Surface Coordinates,”

JPL Interoffice Memorandum 335-JW,DB,WF-20130722-016 (internal document), Jet Propulsion Laboratory, Pasadena,
California, July 22, 2013. 	
10The mean longitude of the ascending node of the lunar orbit measured on the ecliptic
plane from the mean equinox of date is calculated by
2

X = 125c02l 40ll .280 - 1934c08l 10ll .549T + 7ll .455T + 0ll .008T

3

(21)

where T is the TDB time in Julian centuries (36525 days) from J2000.0. The nutations inlongitude D} and obliquity De are given by

D} =- 17ll .206262 sin _Xi
De = 9ll .205348 cos _Xi .

(22)

The true pole of date unit vector, pd , is computed by rotating the Earth-fixed pole vector by
the effect of the 18.6-year nutation term to giveJKNO
sin (D}) sin (er + De)
KK
OO
pd = KKKcos (D}) sin (er + De) cos (er) - cos (er + De) sin (er)OOO
KK
O
cos (D}) sin (er + De) sin (er) + cos (er + De) cos (er)O
L
P

(23)

where the mean obliquity er is given by2

3

er = 84381.ll 448 - 46.ll 815T - 0.ll 00059T + 0.ll 001813T .

(24)

The pole unit vector in the inertial frame p E is computed by precessing the pole of datewith an estimated linear correction,

p E = R z _giR y _- HiR z _ z i R x _- U xiR y _- U yipd

(25)

where U x = U x0 + 100T # dU x /dt and U y = U y0 + 100T # dU y /dt are estimated linear cor-

rections with offsets and rates listed in Table 10 (see page 50) and the precession angles are
given byg = 2306ll .2181T + 0ll .30188T 2 + 0ll .017998T 3
H = 2004ll .3109T - 0 l .42665T 2 - 0ll .041833T 3
z = 2306ll .2181T + 1ll .09468T 2 + 0ll .018203T 3 .

(26)

III. Translational Equations of Motion

The translational equations of motion include contributions from: (a) the point mass interactions among the Sun, Moon, planets, and asteroids; (b) the effects of the figure of the Sunon the Moon and planets; (c) the effects of the figures of the Earth and Moon on each otherand on the Sun and planets from Mercury through Jupiter; (d) the effects upon the Moon’s
motion caused by tides raised upon the Earth by the Moon and Sun; and (e) the effects on
the Moon’s orbit of tides raised on the Moon by the Earth.
The point mass interactions are described in Section III.A. The effects of the static figuresof bodies are described in Section III.B. Also in Section III.B, the effects of the Moon’s static
11

figure plus its time-varying figure due to tides are included. The effects of tides raised on the
Earth acting on the Moon are described in Section III.C.

A. Point Mass Mutual Interaction

The gravitational acceleration of each body due to external point masses is derived from theisotropic, parametrized post-Newtonian (PPN) n-body metric [24–26]. For each body A , theacceleration due to interaction with other point masses, a A, pm - pm, is given by

a A, pm - pm = |B!AGM B (rB - rA)
r 3AB

*1 -

GM
GM
2 (b + c)| rACC - 2b -2 1 | rBCC
2
c
c
C!A
C!B

vA 2
v B 2 2 (1 + c)vA $ vB+ c b c l + (1 + c) b c l c2
2
3 (rA - rB) $ v B G
1- 2=+ 2 (rB - rA) $ a B 4
r
AB
2c
2c

(27)

GM B
1
87rA - rBA $ 7(2 + 2c) v A - (1 + 2c) v BAB (v A - v B)
2 |c B ! A r 3AB(3 + 4c)| GMrABBaB
+
2
2c
B!A

+

where b is the PPN parameter measuring the nonlinearity in superposition of gravity andc is the PPN parameter measuring space curvature produced by unit rest mass. The summation is over all bodies: Sun, Moon, planets and asteroids. The acceleration a B of body B appears in two terms on the right-hand side of Equation (27). Since these terms are multiplied-2-2

by c , using the Newtonian acceleration for these terms is accurate to O (c ).

B. Point Mass Interaction with Extended Bodies

The modeled accelerations of bodies due to interactions of point masses with the gravitational field of nonspherical bodies include: (a) the interaction of the zonal harmonics of the
Earth (through fourth degree) and the point mass Moon, Sun, Mercury, Venus, Mars, andJupiter; (b) the interaction between the zonal, sectoral, and tesseral harmonics of the Moon
(through sixth degree) and the point mass Earth, Sun, Mercury, Venus, Mars, and Jupiter;(c) the second-degree zonal harmonic of the Sun ( J 2 ) interacting with all other bodies.
The contribution to the inertial acceleration of an extended body arising from the interaction of its own figure with an external point mass is expressed in the phg coordinate
system, where the p-axis is directed outward from the extended body to the point mass, thepg -plane contains the figure (rotational) pole of the extended body, and the h-axis completes the right-handed system.
In that system, the acceleration due to the extended body is given by11

11 T. D. Moyer, Mathematical Formulation of the Double-Precision Orbit Determination Program, Technical Report 32-1527(internal document), Jet Propulsion Laboratory, Pasadena, California, 1971. 	

12

RS(n + 1) P (sin {)VW
SRSppWVW
n1
WW
SS WW
n
nS
WW
SShpWW =- GM * | J b R l SSS
0
SS
W
SS WW
n r
2
r n=2
SS- cos {P l (sin {)WWW
SSgpWW
n
T X
X
RS T
V_
S- (n + 1) P nm (sin {) [+ Cnm cos mm + S nm sin mm]WWWbbbb
n2
n n S
S
WWbb
R
S
+ | b r l | SSS m sec {P nm (sin {) [- C nm sin mm + S nm cos mm] WWW`bb
m=1S
n=2
SS cos {P l m (sin {) [+ C cos mm + S sin mm] WWWbbbb
n
nm
nm
T
Xa

(28)

where r is the center-of-mass separation between the two bodies; n 1 and n 2 are the maximum degrees of the zonal and tesseral expansions, respectively; Pn (sin {) is the Legendrempolynomial of degree n; P n (sin {) is the associated Legendre function of degree n and orderm; J n is the zonal harmonic coefficient for the extended body; C nm, S nm are the tesseral
harmonic coefficients for the extended body; R is the equatorial radius of the extended
body; { is the latitude of the point mass relative to the body-fixed coordinate system inwhich the harmonics are expressed; and m is the east longitude of the point mass in thesame body-fixed coordinate system. The primes denote differentiation with respect to theargument sin { . The accelerations are transformed into the inertial frame by application of
the appropriate rotation matrix.
The interaction between the figure of an extended body A and a point mass B also induces
an acceleration of the point mass. If a A, figA - pmB denotes the acceleration of extended body

A interacting with point mass external body B given by Equation (28) when expressed ininertial coordinates, then the corresponding acceleration of the point mass, a B, figA - pmB, isgiven bymA
a B, figA - pmB =- m a A, figA - pmB .
B(29)

For the Moon, the second-degree gravity field is time varying due to distortion by tides andspin and the spherical harmonic coefficients of the gravity field are computed from themoment of inertia tensor, which in turn is computed as a function of time, as described in
Section IV.B. The coefficients are given by

J 2, M (t) =C 22, M (t) =1
I 33, T (t) - 2 [I 11, T (t) + I22, T (t)]2mM RM
I22, T (t) - I11, T (t)24m M R M(30)

2
C 21, M (t) =- I13, T (t) /m M R M
2
S 21, M (t) =- I32, T (t) /m M R M
2
S 22, M (t) =- I 21, T (t) /2m M R M

where the I ij, T are the elements of the total lunar moment of inertia matrix (defined in Section IV); m M is the lunar mass, and R M is the lunar radius.

13

C. Acceleration of the Moon from Earth Tides

The tides raised upon the Earth by the Sun and Moon affect the motion of the Moon. The
distortion of the Earth by the Sun and Moon are characterized by the degree-2 Love numbers k 2j, E , where order j = 0, 1, and 2 correspond to tides with long-period, diurnal, andsemi-diurnal periods, respectively.
We apply a time-delay tidal model to account for dissipation. The distorted response of the
Earth is delayed with respect to the tide-raising forces from the Moon or Sun. The appropriate time delay depends on the period of each tidal component. Consequently, we employ
different time delays for each order j . To allow for time delays shifting across the diurnaland semidiurnal frequency bands, separate time delays are associated with the Earth’s rotation and the lunar orbit.
The acceleration of the Moon due to the Earth tides is evaluated separately for the tides
raised by the Sun and the tides raised by the Moon. The Earth tides depend on the positionof the tide-raising body with respect to Earth rT , where T can denote either the Sun or the
Moon. The position of the tide-raising body is evaluated at an earlier time t - xjl for longperiod, diurnal, and semi-diurnal responses. The distortion of the Earth is delayed bya response time xj , so that the distortion leads the direction to the tide-raising body byo j , where io is the rotation rate of the Earth. The long-period zonal tides ( j = 0)an angle ix
do not depend on the rotation of the Earth, so x0 = 0 . The acceleration of the Moon due tothe distorted Earth depends on the position of the Moon with respect to the Earth r and on
the modified position vector for the tide-raising body rj*, which is given for each order j byo j krT `t - xjl j
rj* = R z a - ix

(31)o j ) here means a right-handed rotation of the vector rT (t - xjl ) by the angle
where R z (- ix
o j about the Earth’s rotation axis with R z , as defined in Equation (13).
ix

The vectors r and rj* are expressed in cylindrical coordinates with the Z axis perpendicular
to the Earth’s equator, so that r = t + z and the time-delayed position of the tide-raisingbody is given by r *j = t *j + z *j . The acceleration of the Moon with respect to Earth, a M, tide,for each tide-raising body is then given by

Z]
2
2
1
5]
]k
5 9_zz 0) i + 2 _tt 0) i C r
m
m
Gm
R
+
20, E f
3d E
M
T E]
2
2
)
)2
)
[]
n
82z 0 z + t 0 tB a M, tide =+ r0 r p
2
5
)5
]
mE
2
]
r
r
] r0
\
k21, E
10zz 1) _t $ t 1) i r
) )
) )
p
+ ) 5 f 2 8_t $ t 1 i z 1 + zz 1 t 1B 2
rr1_b
9_t $ t 2) i2 - 21 _tt 2) i2C r bbb
5
)
)
)2
p`bb
+ ) 5 f82 _t $ t 2 i t 2 - t 2 tB 2
bb
r
r2
a
k22, E(32)

where m T is the mass of the tide-raising body.

14

The tidal acceleration due to tidal dissipation is implicit in the above acceleration. Tides
raised on the Earth by the Moon do not influence the motion of the Earth–Moon barycenter. The effect of Sun-raised tides on the barycentric motion is not considered.
The tidal bulge leads the Moon and its gravitational attraction accelerates the Moon forward and retards the Earth’s spin. Energy and angular momentum are transferred from the
Earth’s rotation to the lunar orbit. Consequently, the Moon moves away from the Earth,
the lunar orbit period lengthens, and the Earth’s day becomes longer. Some energy is dissipated in the Earth rather than being transferred to the orbit.
The estimated tidal acceleration in orbital mean longitude is –25.82 ± 0.03′′/century2
for DE430 and –25.80 ± 0.03′′/century2 for DE431. The semimajor axis recession rates are
38.08 ± 0.04 mm/yr and 38.05 ± 0.04 mm/yr for DE430 and DE431, respectively. The uncertainties reflect an uncertainty in extrapolating the lunar position beyond the span of the fit.
The uncertainty in converting the Love numbers and time delays to tidal acceleration andrecession rate might be as large as 0.5 percent.

IV. Evolution of the Lunar Orientation

The Moon is modeled as an anelastic mantle with a liquid core. The orientation of the coreand mantle are integrated from the differential equations for the core and mantle angularvelocities. The angular momentum vectors of the mantle and core are the product of theangular velocities and the moments of inertia. The angular momentum vectors change withtime due to torques and due to distortion of the mantle.

A. Rate of Change of Lunar Angular Velocities

In a rotating system, the change in angular velocity ~ is related to torques N byd
N = dt _I~ i + ~ # I~

(33)

where I is the moment of inertia tensor. The second term on the right side puts the time derivative into the rotating system. The total lunar moment of inertia I T , which is the sum of
the moment of inertia of the mantle I m and the moment of inertia of the core I c , is proportional to the mass m M times the square of the radius R M . Because the fractional uncertainty
in the constant of gravitation G is much larger than that for the lunar mass parameter Gm M ,Equation (33) is evaluated in the integration with both sides multiplied by G.
The components of vectors can be given in the inertial frame, mantle frame, or otherframes. Since the moment of inertia matrices are nearly diagonal in the mantle frame, there
is great convenience to inverting matrices and performing the matrix multiplications in
the mantle frame. The resulting vector components can then be rotated to other frames if
desired.

15

The moment of inertia of the mantle varies with time due to tidal distortions. The distortions are functions of the lunar position and rotational velocities computed at time t - xm ,where xm is a time lag determined from the fits to the LLR data. The time delay allows fordissipation when flexing the Moon [5]. The time derivative of the angular velocity of the
mantle is given by~o m = I m1 ' | N M, figM - pmA + N M, figM - figE - Io m ~ m - ~ m # Im ~ m + N cmb 1
A!M

(34)

where N M, figM - pmA is the torque on the lunar mantle from the point mass of body A,
N M, figM - figE is the torque on the mantle due to the extended figure of the Moon interactingwith the extended figure of the Earth, and N cmb is the torque due to interaction between
the mantle and core. The torques are given in Section IV.C.
The fluid core is assumed to be rotating like a solid and constrained by the shape of thecore–mantle boundary at the interior of the mantle, with moment of inertia constant in theframe of the mantle [6]. The time derivative of the angular velocity of the core expressed in
the mantle frame is given by~o c = I c-1 # - ~ m # Ic ~ c - N cmb - .

(35)

B. Lunar Moments of Inertia

In the mantle frame, the undistorted moment of inertia of the mantle and the moment ofinertia of the core are diagonal. The undistorted total moment of inertia uIT is given by

RSA 0 0 VW
SS TWWuIT = SSS 0 BT 0 WWW
SSWW
S 0 0 CTW
T
X

(36)

2 (1 - b L c L)
2
m R Ju(2b L - c L + b L c L) M M 2, M
2 ( 1 + c L)2BT =m R Ju(2b L - c L + b L c L) M M 2, M
2 (1 + b L)2
CT =m R Ju(2b L - c L + b L c L) M M 2, M

(37)

with A T , B T , and C T given by

AT =where Ju 2,M is the second-degree zonal harmonic of the undistorted Moon and b L and c L
are ratios of the undistorted moments of inertia given byb L = (C T - AT ) /B T
c L = (B T - A T ) /C T .

16

(38)

The undistorted total moment of inertia and the second-degree zonal harmonic of theundistorted Moon are not the same as the mean values since the tidal distortions have
non-zero averages.
The moment of inertia of the core I c is given by

RS1 - f
0 0WVW RSSAc 0 0 WVW
SS
c
W
W S
S
Ic = ac C T SS 0 1 - fc 0WWW = SSS 0 Bc 0 WWW
W
SS
W S
S 0
0 1WW SS 0 0 C cWW
T
X T
X

(39)

where a c = C c /C T is the ratio of the core polar moment of inertia to the undistorted total
polar moment of inertia and fc is the core oblateness. Distortion of the core moment ofinertia is not considered.
The undistorted moment of inertia of the mantle is the difference between the undistorted
total moment of inertia and the core moment of inertia,

uIm = uIT - Ic .

(40)

The moment of inertia of the mantle varies with time due to tidal distortion by the Earthand spin distortion,

RS
VW
SSx 2 - 1 r 2
WW
x
z
y
x
S
WW
3
S
WW
k2, M m E R 5M SSS
1
2
2
SS xy
y - 3ryz WWWIm (t) = uIm 5
SS
WW
r
SS
1 2WWW2
SS xz
yzz - 3r W
TX
RS
VW
SS~ 2 - 1 (~ 2 - n 2)
WW
~
~
~
~
,
,
,
,
m, x
m
m
x
m
y
m
x
m
z
S
WW
3
5 S
WW
k2, M R M SSS
1
2
2
2
WW
~ m, x ~ m, y
~ m, y - 3 (~ m - n )~m, y ~m, z+ 3G SS
WW
SS
W
SS
1
2
2
2 W
SS
~m, x ~m, z~m, y ~m, z~ m, z - 3 (~ m + 2n )WWW
T
X
(41)
where the position of the Moon relative to Earth r and the angular velocity of the mantle~ m, are evaluated at time t - xm; k2,M is the lunar potential Love number; m E is the mass of
the Earth; R M is the equatorial radius of the Moon; r is the Earth-Moon distance; x, y, z are
the components of the position of the Moon relative to Earth referred to the mantle frame;~m, x, ~m, y, ~ m, z are the components of ~ m in the mantle frame; and n is the lunar mean
motion.
The rate of change of the mantle’s moment of inertia is given by

17

SRS 2 1 2
WWVSSx - r
WW
xy
xz
3
S
WW
5
5k2, M m E R M r $ ro SSS
W
1 2
2
oIm =
S
y - 3ryz WWW
SS xy
7
WW
r
SS
1 2WWW2
SS xz
yzz - 3r W
S
X
RS T
VW
SS2 (xxo - 1 r $ ro )
WW
o
o
o
o
xy
+
xy
xz
+
xz
S
WW
3
5 S
S
WW
k2, M m E R M SS
1
WW
S
o
o
o
o
o
o
+
$
+
xy
xy
2
yy
r
r
yz
yz
(
)
WW
3
SSS
r5
WW
SS
1
SS xzo + xzoo
2 (zzo - 3 r $ ro )WWW
yzo + yzX
RS T
VW
SS2 (~ ~o - 1 ~ $ ~o )
W
o
o
o
o
~
~
+
~
~
~
~
+
~
~
m, x m, xmm
m, x m, y
m, x m, y
m, x m, z
m, x m, z W
S
WW
3
5 S
WW
k2, M R M SSS
1
2 (~m, y ~o m, y - 3 ~m $ ~o m)+ 3G SS ~ m, x ~o m, y + ~o m, x ~m, y~m, y ~o m, z + ~o m, y ~m, z WWW .
SS
WW
SS
W
1
SS ~m, x ~o m, z + ~o m, x ~m, zo2 (~m, z ~o m, z - 3 ~ m $ ~ m)WWW~m, y ~o m, z + ~o m, y ~m, z
T
X
(42)

C. Lunar Torques

The torque on the Moon due to an external point mass A is given by

N M, figM - pmA = M M rAM # a M, figM - pmA(43)

where rAM is the position of the point mass relative to the Moon and a M, figM - pmA is the acceleration of the Moon due to the interaction of the extended figure of the Moon with thepoint mass A, as described in Section III.B. Torques are computed for the figure of the Moon
interacting with the Earth, Sun, Mercury, Venus, Mars, and Jupiter.
Yoder [27] and Eckhardt [28] showed that torques due to the interaction of the figure of theMoon with the figure of the Earth are important for the orientation of the Moon. The threemost significant terms of the torque are

N M, figM - figE =15GM E R 2E J2, E
2r 5EM%`1 - 7 sin 2 { M, E j7rtEM # I M rtEMA+ 2 sin { M, E 7rtEM # I M pt E + pt E # I M rtEMA
2- 7pt E # I M pt EA1
5(44)

t E is the direction vector of the Earth’s pole and rtEM is the direction vector of thewhere pEarth from the Moon; I is the lunar moment of inertia tensor; R E is the equatorial radius oft E.
the Earth; and { M, E is defined by sin { M, E = rtEM $ p

The torque on the mantle due to the interaction between the core and mantle is evaluatedin the mantle frame and is given by

N cmb = k v _~ c - ~m i + _C c - Ac i_zt m $ ~ c i_zt m # ~ c i(45)where zt m is a unit vector in the mantle frame aligned with the polar axis. The torque on thecore is the negative of the torque on the mantle.
18

V. Observational Data Fit by the Planetary and Lunar Ephemerides

The observations that have been used to estimate the dynamical parameters for DE430 and
DE431 are summarized in Tables 1, 2, and 3. The observations are sorted by body, classification, and type.
LLR data are measurements of the round-trip light-time from an observatory to retroreflectors on the Moon at the Apollo 11, 14, or 15 landing sites or the Lunokhod 1 and 2 rovers.
These measurements began in 1970 following the first landings of astronauts and continue
to the present (e.g. [29,30]). Residuals between the measured round-trip light-time t meas andthe value computed from the model t comp are typically expressed as one-way range residuals Dt = (t meas - t comp) c/2 . The measurement accuracy has improved with time as technology for producing short-duration high-energy laser pulses and timing measurements has
advanced. The earliest measurements have root-mean-square (rms) residuals of ~ 25 cm
while the most recent measurements have rms residuals of ~ 2 cm. The most accurate measurement residuals are limited by dynamical modeling and by ambiguity in knowledge of
which corner cube in the retroreflector array returned the photon that was timed [31]. LLR
data are the only data used for estimation of the orbit and orientation of the Moon. The
majority of LLR measurements are from McDonald Observatory (e.g., [32,33]), Observatoirede la Côte d’Azur (e.g., [34]) and Apache Point Observatory (e.g., [35]). The LLR measurement residuals for DE430 are shown in Figure 4.
Spacecraft measurements are based on radio range, Doppler, and VLBI measurements. For
spacecraft in orbit about the planet, the Doppler measurements are used to estimate theposition of the spacecraft with respect to the planet and range and VLBI measurements arethen used to estimate the orbit of the planet. For spacecraft flying by a planet, the range,
Doppler, and VLBI data, as available, are used to estimate both the trajectory of the spacecraft and a 3-dimensional (3–D) position of the planet, given as range, right ascension, anddeclination.
Range measurements to spacecraft are usually made at regular intervals during a tracking
pass, typically every 10 min, while Doppler measurements are made more frequently, typically every minute. Both range and Doppler measurements are based on measurement of
the phase of a radio signal, with the carrier signal used for Doppler and a ranging modulation signal used for range. Since the carrier signal is at a much higher frequency and usually
has much higher signal strength, Doppler measures changes in range much more accurately than the range measurements. Because of the shorter wavelength associated with thehigher frequency, the integer number of carrier wavelengths cannot be resolved, so Doppler
measurements do not allow estimation of absolute range. Range measurements are morecorrectly measurements of round-trip light-time. For plotting residuals, the residual lighttime (measured minus computed) is multiplied by the speed of light, and divided by two
to give approximate residual distance in meters. The range measurement accuracy is often
limited by a calibration of the signal path delay in the tracking station prior to each tracking pass [20,36]. Since this calibration error is common to all range measurements in the
tracking pass, there is only one statistically independent range point per pass. We therefore
use only one range point per tracking pass in our data reduction, and the number of range
measurements per spacecraft in Tables 1–3 reflects this.

19

Table 1. Observational data for the Moon and inner planets. The columns contain the observatory/source,
the time coverage, and the number of observations.
Planet

Class

Type

LLR

Range

Observatory/Spacecraft

Span

NumberMoon
		McDonald 2.7 m

1969–1985

3451

				

MLRS/saddle

1984–1988

275

				

MLRS/Mt Fowlkes

1988–2007

2746

				

Haleakala

1984–1990

694

				

Observatoire de la Côte d’Azur

1984–2012

9635

				

Matera2004

				Apache Point

2006–2012111557

Mercury		Mariner 10

1974–1975				Spacecraft

Range

MESSENGER

2011

2

		

Spacecraft

3–D

MESSENGER

2008–2009

3

		

Radar

Range

Arecibo

1967–1982

242

				

Goldstone

1972–1997

283

				

Haystack

1966–1971

217

				
		

242

Eupatoria

1980–1995

75

Radar

ClosureGoldstone

1989–199739

		Spacecraft

Range

Venus Express

2006–2013

2158

		

Spacecraft

VLBI

Venus
Venus Express

2007–2013

54

				

Magellan

1990–1994

18

		

Arecibo

1967–1970

227

				

Radar

Range

Goldstone

1970–1990

512

				

Haystack

1966–1971

330

				

Eupatoria

1962–1995

1134

Viking Lander 1

1976–1982

1178

				

Viking Lander 2

1976–1977

80

				

Mars Pathfinder

1997

90

				

Mars Global Surveyor

1999–2006

3067

				

Mars Odyssey

2002–2012

6009

				

Mars Reconnaissance Orbiter

2006–2012

1475

				

Mars Express

2005–2013

5321

		

Mars
		

20

SpacecraftSpacecraft

Range

Mars Global Surveyor

2001–2003

15

				

VLBI

Mars Odyssey

2002–2013

77

				

Mars Reconnaissance Orbiter

2006–201359

Table 2. Observational data for Jupiter and Saturn. The columns contain the observatory/source,
the time coverage, and the number of observations.
Planet

Class

Type

Observatory/SpacecraftSpacecraft

3–D

Span

NumberJupiter		

Pioneer 10

1973

1

				

Pioneer 11

1974

1

				

Voyager 1

1979

1

				

Voyager 2

1979

1

				

Ulysses

1992

1

				

Cassini

2000

1

		

Spacecraft

VLBI

Galileo

1996–199724

		Astrometric

CCD

Flagstaff

1998–2012

342

				

Nikolaev

1962–1998

2586

		

La Palma

1986–1997

658

Washington

1914–1994

1705

AstrometricTransit				
Saturn21

		Spacecraft

Range

Cassini

2004–2013131

		Spacecraft

VLBI

Cassini

2004–201112		Spacecraft

3–D

Voyager 1

1980

1

				

Voyager 2

1981

1

		

Flagstaff

1998–2012

5635

				

Astrometric

CCD

Table Mountain

2002–2009

1374

				

Nikolaev

1972–1994

1176

		

Astrometric

Relative

Yerkes1910–192118

		

AstrometricTransitBordeaux

1987–1993

119

				

La Palma

1986–1997

730

				

Washington

1926–1993

1422

Table 3. Observational data for Uranus, Neptune, and Pluto. The columns contain the observatory/source,
the time coverage, and the number of observations.
Class

Type

		

Spacecraft

3–D

		

Astrometric

CCD

PlanetObservatory/Spacecraft

Span

Number

Voyager 2

1986

1

UranusFlagstaff

1995–2012

3892

				

Table Mountain

1998–2009

645

				

Nikolaev

1961–1998430

		Astrometric

Relative

Yerkes1908–192221		AstrometricTransitBordeaux

1985–1993

238

				

La Palma

1984–1997

1030

				

Washington

1926–1993

2043

Neptune		Spacecraft

3–D

Voyager 2

1989

		

Astrometric

CCD

Flagstaff

1995–2012

4259

1

				

Table Mountain

1999–2012

832

				

Nikolaev

1961–1998

436

		

Astrometric

Relative

Yerkes1904–192233

		AstrometricTransitBordeaux

1985–1993

183

				

La Palma

1984–1998

1106

				

Washington

1926–1993

1838

Flagstaff

1995–2012

995

				

Table Mountain

2001–2012

365

				

Pico dos Dias

1995–2012

5489

		

AstrometricPulkovo

1930–1992

53

		Occultation		

Various2005–2012

19

Pluto
		

Astrometric

CCD

Photographic

McDonald / 2.7 m
McDonald / Saddle
McDonald / Mt. Fowlkes

Cote d’Azur
HaleakalaMateraApache Point

100

Dr, cm

50

0

–50

–100
1970.01980.01990.0Year

2000.0Figure 4. Lunar laser ranging measurement residuals.

22

2010.0Spacecraft VLBI measurements are usually made using two widely separated tracking stations. The measurements are made using a modulation on the carrier signal (delta-differential one-way range) and give one component of the direction to the spacecraft [37]. The
angular component direction depends on the baseline used. The baseline from Goldstone,
California, to Madrid, Spain, is nearly parallel to the Earth’s equator, so measurements onthat baseline measure an angular component that is close to right ascension. The baselinefrom Goldstone to Canberra, Australia, has an angle of about 45 deg relative to the equatorand so measures an angular component that is approximately mid-way between the rightascension and declination directions. Residuals for single-baseline measurements are given
for each baseline. For Cassini, the Very Long Baseline Array (VLBA) was used, where the difference in time of arrival of the spacecraft carrier signal was used to determine both components of the direction to the spacecraft [38].
Radar measurements are measurements of the round-trip light-time from a radio telescopeto the surface of the inner planets. The topography of the planet was originally a significant
error source. Radar closure measurements are the difference in range measurements to thesame point on the surface made at different times in order to cancel the error in topography. The topography of the inner planets is now well known from radio and laser altimetry measurements from spacecraft. However, the radio range and VLBI measurements of
those spacecraft determine the orbits of the inner planets much more accurately than theplanetary radar measurements, so the planetary radar data are included here primarily forhistorical comparison.
Astrometric measurements record the direction to the planet, namely, right ascension anddeclination, based on imaging relative to a star field. The accuracy of the star catalog isoften the largest source of measurement error. We here use CCD to indicate more modern
observations using electronic detectors, generally referred to star catalogs based on the
Hipparcos mission launched in 1991 [39] that are referred to the ICRF2 through estimation of the positions of radio stars using VLBI [40]. Older measurements were taken using
photographic plates or transit methods, often referred to older star catalogs, though correctedto the Hipparcos catalog in some fashion. Barnard [41–57] measured the angular separation
between the outer planets and some of their satellites relative to angularly nearby stars at
Yerkes Observatory. The positions of those stars are taken from modern star catalogs, withaccuracies limited by knowledge of stellar proper motion.12 Transit observations cover a longer time span than the more modern spacecraft and astrometric measurements. Since themeasurement noise is relatively large for the transit measurements, they do not contributesignificantly to the ephemeris solution. The transit measurements are included mainly forhistorical comparison.
Occultation measurements of Pluto are included here, where the right ascension and declination are determined from the timed disappearance and reappearance of a star by Pluto [58].

12 R. A. Jacobson, “Visual Observations of the Outer Planets by E. E. Barnard,” American Astronomical Society, Division ofPlanetary Sciences meeting, Puerto Rico, 2009.

23

A. Mercury

For DE430 and DE431, the orbit of Mercury is primarily determined by range measurements to the MESSENGER spacecraft after entering orbit in early 2011 [59]. The residuals
for measurements processed are shown in Figure 5. The measurement residuals show somesignature at the Mercury orbit period that cannot be removed by the ephemeris dynamical model. The signature is due to limitations in the estimated spacecraft orbits relative to
Mercury. This will be improved with data from the second year of MESSENGER operations
that includes orbits with pericenter over the southern hemisphere.
Previously, the orbit of Mercury was best determined by range measurements to the Mariner 10 spacecraft during its first two flybys in 1974 and 1975 [60], radar range from 1967
to 1997 [61–64], and radio tracking of MESSENGER for three flybys in 2008 and 2009.
Residuals for these measurements are shown in Figures 6–8.

B. Venus

The orbit of Venus is primarily determined by range measurements to the Venus Express
spacecraft since 2006. Range residuals are shown in Figure 9. VLBI measurement of the
Magellan spacecraft from 1991 to 1995 and of the Venus Express spacecraft help determine
the orientation of the orbits of Earth and Venus with respect to the ICRF2 frame. These
VLBI data are less accurate than the Mars spacecraft VLBI measurements, primarily because
the modulation signal used on Magellan and Venus Express has a narrower bandwidththan the signal used on the Martian orbiters. The Magellan and Venus Express VLBI residuals are shown in Figures 10–11. Radar range measurements [61,65–67] were the primarydata for determining the orbit of Venus prior to Venus Express mission. The radar measurement residuals are shown in Figure 12.

C. Mars

VLBI measurements of Mars Global Surveyor, Mars Odyssey, and Mars Reconnaissance
Orbiter provide the best determination of the orientation of the solar system with respect
to the ICRF2.13 The current accuracy is approaching the accuracy of the knowledge of thepositions of the radio sources used to define the frame. VLBI measurement accuracy has
been improved from 2001 to 2013 through digitization of signals at the radio telescopes
at intermediate frequencies and the availability of wider digital data bandwidths that allow an improved signal-to-noise ratio. Residuals for the VLBI measurements are shown in
Figures 13–14.
Radio tracking of spacecraft landed on Mars or in orbit about Mars extends from 1976 to
the present. The large amount of data available and the proximity of the Mars orbit to theasteroid belt provide a continuously increasing sensitivity to asteroid mass parameters[7,17,20,68–70]. Although the increasing set of range measurements is improving the ability to predict the orbit of Mars, continued tracking is required to maintain the current orbit accuracy. Residuals for range measurements to Martian landers and orbiters are shownin Figures 15–20.

13

24

W. M. Folkner and J. S. Border, “Linking the Planetary Ephemeris to the International Celestial Reference Frame,”
Highlights of Astronomy, vol. 16 (in press). 	

100

Dr, m

50
0
–50
–100
2011.02011.2

2011.42011.6

Year

2011.82012.0Figure 5. Range measurement residuals for MESSENGER spacecraft in orbit about Mercury.
(b)

200

4

100

2
Dr, km

Dr, m

(a)

0

0

–100

–2

–200

–4

1974.01975.0Year

1976.01990.01995.0Year

2000.0Figure 6. (a) Range residuals for Mariner 10 spacecraft; (b) Mercury radar closure residuals.

10

AreciboDr, km5

Haystack

Goldstone

Eupatoria

0
–5

–10
1965.01975.01985.01995.0

6

100

2

3

50

0
–2
–4

Dr, m

4

Dd, masDa, masFigure 7. Residuals for radar range measurements for Mercury.

0–3

2008.02009.02010.0–6

0–502008.02009.02010.0–100

Figure 8. Residuals for MESSENGER flybys of Mercury.

25

2008.02009.02010.0

40

Dr, m

20
0
–20
–40
2006.02007.02008.02009.0Year

2010.02011.02012.02013.0Figure 9. Residuals for range measurement to Venus Express.

10

(a)

10
5
Dq, masDq, mas

5

(b)

0
–5

0
–5

–10

1991.0–10
1994.01994.0Year

1995.0Year

Figure 10. VLBI residuals for Magellan. (a) Goldstone–Canberra baseline; (b) Goldstone–Madrid baseline.

8

(a)

8
4
Dq, masDq, mas

4

(b)

0
–4

0
–4

–8
2007.02008.0–8
2007.02009.0Year

2011.02013.0Year

Figure 11. VLBI residuals for Venus Express. (a) Goldstone–Canberra baseline; (b) New Norcia–Cebreros baseline.

10

Dr, km5

Arecibo

Haystack

Goldstone

Eupatoria

0
–5

–10
1965.01975.01985.0Figure 12. Residuals for radar range measurements for Venus.

26

1995.02

Dq, mas10–1–22001.02003.02005.02007.0Year

2009.02011.02013.0Figure 13. Mars orbiter VLBI residuals on Goldstone-Madrid baseline.
2

Dq, mas10–1–22001.02003.02005.02007.0Year

2009.02011.02013.0Figure 14. Mars orbiter VLBI residuals on Goldstone-Canberra baseline.

40
Dr, m

20
0
–20
–40
1976.01977.01978.01979.01980.0Year

1981.01982.01983.01984.0Figure 15. Viking Lander 1 range residuals.
(a)

20

40

15
10
Dr, m

Dr, m

20
0
–20

5
0
–5
–10
–15

–40
1976.0(b)1977.0Year

1978.01979.0–20
1977.0Year

Figure 16. (a) Viking Lander 2 and (b) Mars Pathfinder range residuals.
27

1998.0

20
Dr, m

10
0

–10
–20
1999.02001.02003.02005.0Year

2007.02009.02011.02013.02011.02013.02011.02013.0Figure 17. Residuals for range measurement to Mars Global Surveyor.

20
Dr, m

10
0

–10
–20
1999.02001.02003.02005.0Year

2007.02009.0Figure 18. Residuals for range measurement to Mars Odyssey.

20
Dr, m

10
0

–10
–20
1999.02001.02003.02005.0Year

2007.02009.0Figure 19. Residuals for range measurement to Mars Reconnaissance Orbiter.

20
Dr, m

10
0

–10
–20
1999.02001.02003.02005.0Year

2007.02009.0Figure 20. Residuals for range measurement to Mars Express.

28

2011.02013.0

D. JupiterThe orbit of Jupiter is less well determined than the orbits of the inner planets and Saturn.
The only spacecraft to orbit Jupiter, Galileo, was not able to use the high-gain antenna
and so was not able to support range measurements. VLBI measurements of the Galileo
spacecraft were made, but with relatively low accuracy due to limitations on the modulations available on the low-gain antenna. The orbit of Jupiter is determined by radio tracking of spacecraft that flew past Jupiter, and by astrometric data, with each of these data
sets contributing approximately equally. Residuals for the spacecraft flyby data are shownin Figures 21–23 and for the Galileo VLBI data in Figure 24. Because Jupiter is resolved bytelescopes, its astrometric position is better determined by observations of its satellites. We
include measurements of the four Galilean satellites, which have orbits with respect to Jupiter that are most accurately determined. Residuals for CCD astrometric measurements areshown in Figures 25–32 from the U. S. Naval Observatory [71–77], Table Mountain Observatory, and Nikolaev Observatory [78]. Figures 33–34 show residuals for transit observations
from the La Palma Observatory [79] and the U. S. Naval Observatory in Washington, D.C.

E. SaturnThe orbit of Saturn is most accurately determined from radio range and VLBI measurementsof the Cassini spacecraft. The period of the Cassini orbit about Saturn varies from 14 days
to 28 days, which is much longer than the tracking passes and longer than the period ofspacecraft orbits about the inner planets. For DE421, the process used to determine ranges
to Saturn from Cassini data for the first three years after entering orbit about Saturn in 2004
showed systematic effects in range residuals. These were determined to be caused by useof the range measurements along with Doppler measurements to determine the spacecraft
orbits relative to Saturn. For DE430, we have used spacecraft orbits that were fit without therange data, allowing the range measurements to be used to determine the orbit of Saturn.
We have found that range measurements are strongly correlated for each orbit segment,
where the ends of the orbit segments are defined by a Saturn pericenter or a major maneuver used to change the spacecraft orbit. We therefore use only one range measurement per
orbit segment. Figure 35 shows the Cassini range measurement residuals. The range residuals are larger for the latter part of the mission, where the spacecraft orbit period is generally
shortest, so there is less Doppler data in each orbit segment, leading to larger uncertainties
in spacecraft orbit determination. The residuals for the VLBA observations of Cassini areshown in Figures 36–37.
Figure 38 shows residuals for the estimated position of Saturn from radio measurementsof the Voyager spacecraft during their flybys. Astrometric observations of Saturn do notcontribute much to the planetary ephemeris, but are included to assess them relative to theradio measurements. Figures 39–50 show residuals for CCD astrometric measurements ofSaturn. Figure 51 shows residuals from relative astrometric measurements from Yerkes Observatory. Figures 52–54 show residuals for transit observations from Bordeaux Observatory,
the La Palma Observatory, and the U. S. Naval Observatory in Washington, D.C.

29

Da, mas

40

Pioneer 10

20

Voyager 10CassiniUlysses
Voyager 2–20

Pioneer 11–401970.01980.01990.0Year

2000.0Figure 21. Right ascension residuals from spacecraft flybys of Jupiter.

Dd, mas40

Pioneer 10

20

Voyager 1

0

Voyager 2–20–40

UlyssesCassini

Pioneer 11

1970.01980.01990.0Year

2000.0Figure 22. Declination residuals from spacecraft flybys of Jupiter.

4

Dr, km2
0–2–4

Voyager 1

Ulysses
Voyager 2

Cassini

Pioneer 11

1970.01980.01990.0Year

2000.0Figure 23. Range residuals from spacecraft flybys of Jupiter.

30

(a)

30
15
Dq, masDq, mas

15
0

–15
–30
1996.0(b)

0–151997 .0
Year

1998.0–30
1996.01997 .0

1998.0Year

Figure 24. VLBI residuals for Galileo spacecraft in orbit about Jupiter. (a) Goldstone–Madrid baseline;(b) Goldstone–Canberra baseline.
30

0.5

Dd, ’’

Da, ’’

0.5

0.0

–0.5

0.0

–0.5

1995.02000.02005.02010.02015.0Year

1995.02000.02005.02010.02015.0Figure 25. Residuals from astrometric observations of Io by U. S. Naval Observatory at Flagstaff.

0.5

Dd, ’’

Da, ’’

0.5

0.0

–0.5

0.0

–0.5

1995.02000.02005.02010.02015.0Year

1995.02000.02005.02010.02015.0Figure 26. Residuals from astrometric observations of Europa by U. S. Naval Observatory at Flagstaff.

0.5

Dd, ’’

Da, ’’

0.5

0.0

–0.5

0.0

–0.5

1995.02000.02005.02010.02015.0Year

1995.02000.02005.02010.02015.0Figure 27. Residuals from astrometric observations of Ganymede by U. S. Naval Observatory at Flagstaff.

0.5

Dd, ’’

Da, ’’

0.5

0.0

–0.5

0.0

–0.5

1995.02000.02005.02010.02015.0Year

1995.02000.02005.02010.02015.0Figure 28. Residuals from astrometric observations of Callisto by U. S. Naval Observatory at Flagstaff.

31

0.5

Dd, ’’

Da, ’’

0.5

0.0

–0.5

0.0

–0.5

1960.01970.01980.01990.02000.0Year

1960.01970.01980.01990.02000.0Figure 29. Residuals from astrometric observations of Io by Nikolaev Observatory.

0.5

Dd, ’’

Da, ’’

0.5

0.0

–0.5

0.0

–0.5

1960.01970.01980.01990.02000.0Year

1960.01970.01980.01990.02000.0Figure 30. Residuals from astrometric observations of Europa by Nikolaev Observatory.

0.5

Dd, ’’

Da, ’’

0.5

0.0

–0.5

0.0

–0.5

1960.01970.01980.01990.02000.0Year

1960.01970.01980.01990.02000.0Figure 31. Residuals from astrometric observations of Ganymede by Nikolaev Observatory.

0.5

Dd, ’’

Da, ’’

0.5

0.0

–0.5
1960.00.0–0.5
1970.01980.01990.02000.0Year

1960.01970.01980.01990.0Figure 32. Residuals from astrometric observations of Callisto by Nikolaev Observatory.

32

2000.02

1

1
Dd, ’’

Da, ’’

2

0
–1

0
–1

–2
1985.01990.01995.0–2
1985.02000.01990.01995.02000.0

Year

2

2

1

1
Dd, ’’

Da, ’’

Figure 33. Residuals from astrometric observations of Jupiter by La Palma Observatory.

0
–1

0
–1

–2
1925.01960.0–2
1925.01995.01960.01995.0Year
Figure 34. Residuals from transit observations of Jupiter by U. S. Naval Observatory in Washington.

100

Dr, m

50
0
–50
–100
2004.02006.02008.0Year

2010.02012.02014.0Figure 35. Range residuals from Cassini in orbit about Saturn.

2

Da, mas10–1–22004.02006.02008.0Year

2010.0Figure 36. Right ascension residuals from VLBA observations of Cassini at Saturn.

33

2012.02

Da, mas10–1–22004.02006.02008.02010.0Year

2012.0Figure 37. Declination residuals from VLBA observations of Cassini at Saturn.

Voyager 2

0
–20
–40

500
Voyager 1

0
Voyager 2

Voyager 1

1980.01981.0Year

1982.0–200
1980.0

Voyager 1

250
Dr, m

Da, mas20

Dd, mas200

40

1981.0Year

1982.0

Voyager 2

0

–250
–500
1980.01981.0Year

1982.0Figure 38. Residuals from Voyager 1 and 2 flybys of Saturn.

0.5

Dd, ’’

Da, ’’

0.5

0.0

–0.5
1995.00.0–0.5
2000.02005.02010.02015.0Year

1995.02000.02005.02010.0Figure 39. Residuals from astrometric observations of Dione by U. S. Naval Observatory at Flagstaff.

0.5

Dd, ’’

Da, ’’

0.5

0.0

–0.5

0.0

–0.5

1995.02000.02005.02010.02015.0Year

1995.02000.02005.02010.0Figure 40. Residuals from astrometric observations of Rhea by U. S. Naval Observatory at Flagstaff.

34

2015.00.5

Dd, ’’

Da, ’’

0.5

0.0

–0.5

0.0

–0.5

1995.02000.02005.02010.02015.0Year

1995.02000.02005.02010.02015.0Figure 41. Residuals from astrometric observations of Titan by U. S. Naval Observatory at Flagstaff.

0.5

Dd, ’’

Da, ’’

0.5

0.0

–0.5
1995.00.0–0.5
2000.02005.02010.02015.0Year

1995.02000.02005.02010.02015.0Figure 42. Residuals from astrometric observations of Iapetus by U. S. Naval Observatory at Flagstaff.

0.5

Dd, ’’

Da, ’’

0.5

0.0

–0.5
1960.00.0–0.5
1970.01980.01990.02000.0Year

1960.01970.01980.01990.02000.0Figure 43. Residuals from astrometric observations of Tethys by Nikolaev Observatory.

0.5

Dd, ’’

Da, ’’

0.5

0.0

–0.5
1960.00.0–0.5
1970.01980.01990.02000.0Year

1960.01970.01980.01990.0Figure 44. Residuals from astrometric observations of Dione by Nikolaev Observatory.

35

2000.00.5

Dd, ’’

Da, ’’

0.5

0.0

–0.5

0.0

–0.5

1960.01970.01980.01990.02000.0Year

1960.01970.01980.01990.02000.0Figure 45. Residuals from astrometric observations of Rhea by Nikolaev Observatory.

0.5

Dd, ’’

Da, ’’

0.5

0.0

–0.5

0.0

–0.5

1960.01970.01980.01990.02000.0Year

1960.01970.01980.01990.02000.0Figure 46. Residuals from astrometric observations of Titan by Nikolaev Observatory.

0.5

Dd, ’’

Da, ’’

0.5

0.0

–0.5

0.0

–0.5

1998.02002.02005.02009.02013.01998.02002.02005.02009.02013.0Year
Figure 47. Residuals from astrometric observations of Rhea by Table Mountain Observatory.

0.5

Dd, ’’

Da, ’’

0.5

0.0

–0.5

0.0

–0.5

1998.02002.02005.02009.02013.01998.02002.02005.02009.0Year
Figure 48. Residuals from astrometric observations of Titan by Table Mountain Observatory.

36

2013.00.5

Dd, ’’

Da, ’’

0.5

0.0

–0.5

0.0

–0.5

1998.02002.02005.02009.02013.01998.02002.02005.02009.02013.0Year
Figure 49. Residuals from astrometric observations of Hyperion by Table Mountain Observatory.

0.5

Dd, ’’

Da, ’’

0.5

0.0

–0.5

0.0

–0.5

1998.02002.02005.02009.02013.01998.02002.02005.02009.02013.0Year
Figure 50. Residuals from astrometric observations of Iapetus by Table Mountain Observatory.

0.8

0.8
d Dp, ’’

1.5

Dd, ’’

1.5

0

–0.8
–1.5
1910.00–0.8
1915.01920.01925.0–1.5
1910.01915.01920.01925.0Year
Figure 51. Residuals of angular distance d and distance times position angle p of Saturn from Yerkes Observatory.

37

2

1

1
Dd, ’’

Da, ’’

2

0

0

–1

–1

–2
1985.01987.5

1990.01992.5–2
1985.01995.01987.5

1990.01992.5

1995.0

Year

2

2

1

1
Dd, ’’

Da, ’’

Figure 52. Residuals from transit observations of Saturn by Bordeaux Observatory.

0

0

–1

–1

–2
1985.01990.01995.0–2
1985.02000.01990.01995.02000.0

Year

2

2

1

1
Dd, ’’

Da, ’’

Figure 53. Residuals from astrometric observations of Saturn by La Palma Observatory.

0
–1

0
–1

–2
1925.01960.0–2
1925.01995.01960.0Year
Figure 54. Residuals from transit observations of Saturn by U. S. Naval Observatory in Washington.

38

1995.0F. Uranus and Neptune

All measurements of Uranus and Neptune are astrometric except for flybys by the Voyager 2
spacecraft in 1986 and 1989, respectively, that give a three-dimensional radio-tracking position for one time for each planet. The astrometric data include measurements of the planets
themselves and of their major satellites that have accurate enough orbits with respect tothe planet: Titania and Oberon for Uranus and Triton for Neptune. The measurements for
Uranus cover more than one orbit, giving a fairly well constrained orbit estimate. Measurements of Neptune cover less then one orbit, so the uncertainty in some orbital elements
is more than one order of magnitude larger than for Uranus. The residuals of the Uranus
measurements are shown in Figures 55–66 and for Neptune in Figures 67–76.

G. Pluto

Pluto’s orbit has been determined from ground-based astrometric data only. Measurementsfrom the Hubble Space Telescope (HST) are used to determine the relative orbits of Pluto
and its satellites. Most HST observations of the Pluto system do not give positions relative
to stars, and those that do are limited by the accuracy of star catalogs and are not muchbetter than ground-based measurements. Like Neptune, measurements of Pluto cover lessthan one orbit, so several orbit elements, such as the semi-major axis, are relatively poorly
known. The astrometric data used here are from the U. S. Naval Observatory [71–77], thePulkovo Observatory [80], the Observatório do Pico dos Dias,14 and Table Mountain Observatory. Other astrometric data for Pluto used for DE421 were not included here, since they
were of limited time span and used star catalogs not easily corrected to the ICRF2 frame. In
addition to the astrometric data, occultation timing measurements that determine the angular position of Pluto have been included [58]. Residuals for the astrometric data are given
in Figures 77–80 and for the occultation measurements in Figure 81.

VI. Initial Conditions and Constants

The values of many of the constants used in the equations of motion are estimated alongwith the initial conditions for the Sun, Moon, and planets where alternative estimates are
not sufficiently accurate. Values used for the integration of DE430 are given below. The values are given in double precision, which is sufficient to reproduce the integration. However,most values have uncertainties that are much larger than the number of digits indicate.
Table 4 lists the values of the speed of light and the astronomical unit that are defining
constants for DE430 and DE431. The values of the PPN parameters b and c were held fixedat their nominal values. The initial positions and velocities of the Moon and planets relative to the Sun were estimated as part of the ephemeris fit. The position and velocity of the
Sun were adjusted to have the barycenter at the integration origin at the initial time, whichis the TDB-compatible Julian day 2440400.5. The positions and velocities of the Earth andMoon with respect to the integration center were input using the Earth–Moon barycenterwith respect to the solar system barycenter and the Moon with respect to the Earth. The
initial conditions for the Sun and planets are given in Table 5, and for the Moon in Table 6.
14 G. Benedetti-Rossi et al., Astronomy and Astrophysics, submitted, 2014.

39

20

20

75

0
–20
–40
1985.0Dr, m

150

Dd, mas40

Da, mas40

0–20
1986.0Year

1987.00–75–401985.01986.0Year

1987.0–150
1985.01986.0Year

1987.0Figure 55. Residuals from Voyager 2 flyby of Uranus.

0.5

Dd, ’’

Da, ’’

0.5

0.0

–0.5

0.0

–0.5

1995.02000.02005.02010.02015.0Year

1995.02000.02005.02010.02015.0Figure 56. Residuals from astrometric observations of Uranus by U. S. Naval Observatory at Flagstaff.

0.5

Dd, ’’

Da, ’’

0.5

0.0

–0.5

0.0

–0.5

1995.02000.02005.02010.02015.01995.0Year

2000.02005.02010.02015.0Figure 57. Residuals from astrometric observations of Titania by U. S. Naval Observatory at Flagstaff.

0.5

Dd, ’’

Da, ’’

0.5

0.0

–0.5
1995.00.0–0.5
2000.02005.02010.02015.0Year

1995.02000.02005.02010.02015.0Figure 58. Residuals from astrometric observations of Oberon by U. S. Naval Observatory at Flagstaff.

40

0.5

Dd, ’’

Da, ’’

0.5

0.0

–0.5

0.0

–0.5

1960.01970.01980.01990.02000.01960.01970.01980.01990.02000.0Year
Figure 59. Residuals from astrometric observations of Uranus by Nikolaev Observatory.

0.5

Dd, ’’

Da, ’’

0.5

0.0

–0.5

0.0

–0.5

1998.02002.02005.02009.02013.01998.02002.02005.02009.02013.0Year
Figure 60. Residuals from astrometric observations of Uranus by Table Mountain Observatory.

0.5

Dd, ’’

Da, ’’

0.5

0.0

–0.5

0.0

–0.5

1998.02002.02005.02009.02013.01998.02002.02005.02009.02013.0Year
Figure 61. Residuals from astrometric observations of Titania by Table Mountain Observatory.

0.5

Dd, ’’

Da, ’’

0.5

0.0

–0.5

0.0

–0.5

1998.02002.02005.02009.02013.01998.02002.02005.02009.0Year
Figure 62. Residuals from astrometric observations of Oberon by Table Mountain Observatory.

41

2013.00.8

0.8
d Dp, ’’

1.5

Dd, ’’

1.5

0

–0.8

0

–0.8

–1.5
1900.01910.0–1.5
1900.0Year

1920.01910.01920.02

2

1

1
Dd, ’’

Da, ’’

Figure 63. Residuals of angular distance d and distance times position angle p of Uranus from Yerkes Observatory.

0
–1

0
–1

–2
1985.01987.5

1990.01992.5–2
1985.01995.01987.5

1990.01992.5

1995.0

Year

2

2

1

1
Dd, ’’

Da, ’’

Figure 64. Residuals from transit observations of Uranus by Bordeaux Observatory.

0
–1

0
–1

–2
1985.01990.01995.0–2
1985.02000.01990.01995.02000.0

Year

2

2

1

1
Dd, ’’

Da, ’’

Figure 65. Residuals from astrometric observations of Uranus by La Palma Observatory.

0–1–2
1925.00–11960.0–2
1925.01995.01960.0Year
Figure 66. Residuals from transit observations of Uranus by U. S. Naval Observatory in Washington.

42

1995.0

20

20

20

0
–20

Dr, m

40

Dd, mas40

Da, mas

40

0

–20

–40
1989.0Year

1990.00–20–401989.01990.0Year–40
1989.0Year

1990.0Figure 67. Residuals from Voyager 2 flyby of Neptune.

0.5

Dd, ’’

Da, ’’

0.5

0.0

–0.5

0.0

–0.5

1995.02000.02005.02010.02015.0Year

1995.02000.02005.02010.02015.0Figure 68. Residuals from astrometric observations of Neptune by U. S. Naval Observatory at Flagstaff.

0.5

Dd, ’’

Da, ’’

0.5

0.0

–0.5

0.0

–0.5

1995.02000.02005.02010.02015.0Year

1995.02000.02005.02010.02015.0Figure 69. Residuals from astrometric observations of Triton by U. S. Naval Observatory at Flagstaff.

0.5

Dd, ’’

Da, ’’

0.5

0.0

–0.5
1960.00.0–0.5
1970.01980.01990.02000.0Year

1960.01970.01980.01990.0Figure 70. Residuals from astrometric observations of Neptune by Nikolaev Observatory.

43

2000.00.5

Dd, ’’

Da, ’’

0.5

0.0

–0.5

0.0

–0.5

1998.02002.02005.02009.02013.01998.02002.02005.02009.02013.0Year
Figure 71. Residuals from astrometric observations of Neptune by Table Mountain Observatory.

0.5

Dd, ’’

Da, ’’

0.5

0.0

–0.5

0.0

–0.5

1998.02002.02005.02009.02013.01998.02002.02005.02009.02013.0Year
Figure 72. Residuals from astrometric observations of Triton by Table Mountain Observatory.

0.8

0.8
d Dp, ’’

1.5

Dd, ’’

1.5

0

–0.8

0

–0.8

–1.5
1900.01910.01920.0–1.5
1900.0Year

1910.01920.02

2

1

1
Dd, ’’

Da, ’’

Figure 73. Residuals of angular distance d and distance times position angle p of Neptune
from Yerkes Observatory.

0–1–2
1985.00–1

1987.5

1990.01992.5–2
1985.01995.01987.5

1990.01992.5Year
Figure 74. Residuals from transit observations of Neptune by Bordeaux Observatory.

44

1995.02

1

1
Dd, ’’

Da, ’’

2

0

0

–1

–1

–2
1985.01990.01995.0–2
1985.02000.01990.01995.02000.0

Year

2

2

1

1
Dd, ’’

Da, ’’

Figure 75. Residuals from astrometric observations of Neptune by La Palma Observatory.

0

0

–1

–1

–2
1925.01960.0–2
1925.01995.01960.01995.0Year
Figure 76. Residuals from transit observations of Neptune by U. S. Naval Observatory in Washington.

0.5

Dd, ’’

Da, ’’

0.5

0.0

–0.5

0.0

–0.5

1995.02000.02005.02010.02015.0Year

1995.02000.02005.02010.02015.0Figure 77. Residuals from astrometric observations of Pluto by U. S. Naval Observatory at Flagstaff.

0.5

Dd, ’’

Da, ’’

0.5

0.0

–0.5

0.0

–0.5

1998.02002.02005.02009.02013.0Year

1998.02002.02005.02009.0Figure 78. Residuals from astrometric observations of Pluto by Table Mountain Observatory.

45

2013.00.5

Dd, ’’

Da, ’’

0.5

0.0

–0.5

0.0

–0.5
1995.01999.02003.02007.02011.01995.0Year

1999.02003.02007.02011.0

1.0

1.0

0.5

0.5
Dd, ’’

Da, ’’

Figure 79. Residuals from astrometric observations of Pluto by Pico dos Dias Observatory.

0

–0.5
–1.0

0

–0.5
1940.01960.01990.0–1.0
Year

1940.01960.01990.0

0.6

0.6

0.4

0.4

0.2

0.2
Dd, ’’

Da, cosd, ’’Figure 80. Residuals from astrometric observations of Pluto by Pulkovo Observatory.

0

0

–0.2

–0.2

–0.4

–0.4

–0.6
1995.02000.02005.02010.0–0.6
1995.0Year

2000.0Figure 81. Residuals from occultation observations of Pluto.

46

2005.02010.0Table 4. Defining constants.
c299792.458 km/s

Speed of lightau149597870.700 km

Astronomical unitβ

1.0

PPN parameterγ

1.0

PPN parameter

Table 5. Initial positions (au) and velocities (au/day) of the Sun and planets at Julian day (TDB) 2440400.5(June 28, 1969), given with respect to the integration origin in the ICRF2 frame.
Sun		x, y, z0.00450250878464055477

v x , v y, v z

x, y, zv x , v y, v z

0.00336749397200575848

0.02489452055768343341

x, y, z0.61275194083507215477–0.34836536903362219295 –0.19527828667594382236

v x , v y, v z

0.01095206842352823448

0.01561768426786768341

x, y, z0.12051741410138465477–0.92583847476914859295 –0.40154022645315222236

Mercury
		
Venus
		

EM Bary

0.00076707642709100705

0.00026605791776697764–0.00000035174953607552

0.00000517762640983341

0.00000222910217891203

0.36176271656028195477–0.09078197215676599295 –0.08571497256275117236
0.01294630040970409203
0.00633110570297786403

		v x , v y, v z

0.01681126830978379448

0.00174830923073434441

Marsx, y, zv x , v y, v z–0.11018607714879824523–1.32759945030298299295 –0.60588914048429142236

0.01448165305704756448

0.00024246307683646861 –0.00028152072792433877

x, y, zv x , v y, v z–5.37970676855393644523–0.83048132656339789295 –0.22482887442656542236

0.00109201259423733748–0.00651811661280738459 –0.00282078276229867897

x, y, zv x , v y, v z

7.89439068290953155477

4.59647805517127300705

1.55869584283189997764

0.00433581034174662541

0.00192864631686015503

		
Jupiter		
Saturn		
Uranus		–0.00321755651650091552

x, y, z –18.26540225387235944523
v x , v y, v z
0.00022119039101561468

0.00075820289738312913–1.16195541867586999295 –0.25010605772133802236–0.00376247500810884459 –0.00165101502742994997

x, y, z –16.05503578023336944523 –23.94219155985470899295 –9.40015796880239402236
v x , v y, v z
0.00264276984798005548 –0.00149831255054097759 –0.00067904196080291327

Neptune
		
Pluto
		

x, y, z –30.48331376718383944523
v x , v y, v z
0.00032220737349778078–0.87240555684104999295

8.91157617249954997764–0.00314357639364532859 –0.00107794975959731297

Table 6. Initial position (au) and velocity (au/day) of the Moon at Julian day (TDB) 2440400.5(June 28, 1969), given with respect to Earth in the ICRF2 frame.
Moon
		

47

x, y, z–0.00080817735147818490–0.00199462998549701300 –0.00108726268307068900

v x , v y, v z

0.00060108481561422370–0.00016744546915764980 –0.00008556214140094871

The initial Euler angles and angular velocities for the lunar mantle and the initial angularvelocities for the core were estimated as part of the ephemeris fit and are given in Table 7.
The Euler angles rotate from the ICRF2 frame to the mantle or core frame, respectively. The
angular velocities of the mantle and core are given in the mantle frame.
The mass parameters of the planetary systems are most accurately estimated from radio
tracking data from spacecraft encounters, while the mass of the Pluto system is most accurately determined by observations of the Pluto system by the Hubble Space Telescope.
The planetary system mass parameters that were used are listed in Table 8, and are consistent with the values adopted by the IAU [81]. The mass parameter of the Sun was held fixed
as described in Section II.D. The mass parameters of the Earth and Moon were estimated.
LLR data are sensitive to the sum of the Earth and Moon mass parameters, while ranging
measurements to spacecraft are sensitive to the ratio of the mass of the Earth to the massof the Moon. The estimated mass of the Earth agrees within uncertainties with estimatesfrom satellite laser ranging [82–84]. The estimated mass parameter of the Moon agrees withindependent estimates from the GRAIL mission within uncertainties [85,86].15
The extended body parameters of the Sun are given in Table 9. The solar radius is a nominal
value [87] and the pole direction is from [88]. J 2 9 was estimated. However, since the effectof Lense–Thirring precession due to the angular momentum of the Sun was not includedin the dynamical model, the value estimated should be reduced by ~ 7 percent to represent
the actual solar gravity field [89].
The extended body parameters of the Earth and Moon are given in Tables 10 and 11. These
are described in more detail elsewhere.16 The radius and zonal gravity coefficients for the
Earth, including a rate of change in J 2, were taken from [18]. The Earth Love numbers k 20,E ,k 21,E , and k 22,E include a solid body contribution taken from [18] and a correction froman ocean model. Two time-lags for tidal deformation, the correction to the direction of
the rotation axis direction at epoch J2000.0 and the linear rates in rotation axis direction,
were estimated. Of the lunar extended body parameters, the quantities, b L, c L, x m, fc, andk vc have been estimated from the LLR data fit. Ju 2,M , k2M , and the higher degree and order

lunar gravity harmonic coefficients are estimates from the GRAIL mission [85]. The timedependent values of the degree-2 gravity harmonic coefficients, J 2,M , C 2, 2M , C 2, 1M , S 2, 1M ,and S 2, 2M were computed from the (time-varying) lunar moment of inertia tensor. The ratioof the core moment of inertia to the total moment of inertia of the undistorted Moon, a c ,was constrained to a previously determined value.
A set of 343 asteroids that have the largest perturbation on the orbit of Mars has been included in the integration. For DE430 and DE431, the positions of the asteroids were integrated iteratively with the positions of the Sun, Moon, and planets. The mass parametersof the 343 asteroids were estimated individually with a priori values and uncertainties. The
majority of the constraints were based on radiometric measurements of asteroid diameters
15 J. G. Williams, A. S. Konopliv, D. H. Boggs, R. S. Park, D.-N. Yuan, et al., “Lunar Interior Properties from the GRAIL Mis-sion,” Journal of Geophysical Research, submitted.
16 J. G. Williams, D. H. Boggs, and W. M. Folkner, “DE430 Lunar Orbit, Physical Librations and Surface Coordinates,”

JPL Interoffice Memorandum 335-JW,DB,WF-20130722-016 (internal document), Jet Propulsion Laboratory, Pasadena,
California, July 22, 2013.  	

48

Table 7. Lunar mantle and core initial Euler angles (radian) and angular velocities (radian/day) at Julian day(TDB) 2440400.5 (June 28, 1969). Note that the core angular velocity is expressed in the mantle frame.

z m, i m, } m~ m, x, ~ m, y, ~ m, z

0.38239278420173690000

1.29416700274878300000

0.00004573724185991433 –0.00000218986174567295

0.22994486018992250000

z c, i c , } c~ c, x , ~ c, y , ~ c, z

–0.00661836772247824400 –0.00107295445159005100

0.00512830031411853500–0.00241990927040684100

0.41101946488652730000 –0.46309468558363680000
0.22964879652299730000

Table 8. Mass parameters of major bodies.

49

GM, km3/s2

Body

GM, au3/day2Sun

0.295912208285591100E–03

1.000000

132712440041.939400

Mercury

0.491248045036476000E–106023682.155592

22031.780000

Venus

0.724345233264412000E–09

408523.718658

324858.592000

Earth0.888769244512563400E–09

332946.048834

398600.435436

Mars

0.954954869555077000E–103098703.590291

42828.375214

Jupiter

0.282534584083387000E–06

1047.348625

126712764.800000

Saturn0.845970607324503000E–07

3497.901768

37940585.200000

Uranus0.129202482578296000E–07

22902.981613

5794548.600000

Neptune0.152435734788511000E–07

19412.259776

6836527.100580

Pluto

0.217844105197418000E–11

135836683.768617

977.000000

Moon

0.109318945074237400E–10

27068703.241203

4902.800066

GM 9 /GM bodyTable 9. Extended body parameters for the Sun.

R9

696000.0

J 2, 9radius, km		2.1106088532726840E–07

Dynamical form–factor of the Sunap, 9268.13Right ascension of spin axis direction, deg

dp, 9

63.87

Declination of spin axis direction, deg

Table 10. Extended body parameters for the Earth.

RE

506378.1363

J2E

0.00108262545

J3E–0.00000253241

J4E–0.000001616

d (J2E ) /dt–2.60E–11

Radius, km
Zonal harmonics of the Earth

Rate of change of J2E , yr–1

k20,E

0.335

Potential Love number for long–period deformation

k21,E

0.32

Potential Love number for diurnal deformation

k22,E

0.32

Potential Love number for semi–diurnal deformation

x0l

0.0640

Orbital time–lag for long–period deformation, days

x1l

–0.0044

Orbital time–lag for diurnal deformation, daysx2l

–0.1000

Orbital time–lag for semi–diurnal deformation, days

x0
x1

0.0

Rotational time–lag for long–period deformation, days

7.3632190228041890E–03

Rotational time–lag for dirunal deformation, days

x2

2.5352978633388720E–03

Rotational time–lag for semi–diurnal deformation, days

U x0

5.6754203322893470E–03

X–axis rotation at J2000.0, ′′U y0–1.7022656914989530E–02

Y–axis rotation at J2000.0, ′′d Ux /dt

2.7689915574483550E–04

Negative obliquity rate correction, ′′/yrd Uy /dt–1.2118591216559240E–03

Precession rate correction times sine of obliquity, ′′/yrTable 11. Extended body parameters for the Moon.
RM

1738.0Radius, km

bL
cL

6.3102131934887270E–04

Lunar moment parameters

k 2,M0.024059

Potential love number

xM

9.5830547273306690E–02

Time–lag for the solid–body tide, days

ac

0.0007

Ratio of polar moment of inertia of core to mean total

2.2773171480091860E–04

				polar moment of inertia

51

fc

2.4623904789198150E–04

Oblateness of corek v /CT

1.6365616533709530E–08

Friction coefficient between core and mantle, radian/day

uJ2,M

2.0321568464952570E–04

Undistorted 2nd zonal harmonic coefficient

J3,M

8.4597026974594570E–06

Zonal harmonic coefficients

J4,M–9.7044138365700000E–06

J5,M

7.4221608384052890E–07

J6,M–1.3767531350969900E–05

S3, 1,M

5.8915551555318640E–06

S3, 2,M

1.6844743962783900E–06

S3, 3,M–2.4742714379805760E–07

S4, 1,M

1.5789202789245720E–06

S4, 2,M–1.5153915796731720E–06

S4, 3,M–8.0349266627431070E–07

S4, 4,M

8.2964257754075220E–08

S5, 1,M–3.5272289393243820E–06

S5, 2,M

1.7107886673430380E–07

S5, 3,M

2.8736257616334340E–07

S5, 4,M

5.2652110720146800E–10

S5, 5,M

–6.7824035473995330E–09

S6, 1,M–2.0453507141252220E–06

S6, 2,M–2.6966834353574270E–07

S6, 3,M

–7.1063745295915780E–08

S6, 4,M–1.5361616966632300E–08

S6, 5,M–8.3465073195142520E–09

S6, 6,M

1.6844213702632920E–09

C3, 1,M

2.8480741195592860E–05

C3, 2,M

4.8449420619770600E–06

C3, 3,M

1.6756178134114570E–06

C4, 1,M–5.7048697319733210E–06

C4, 2,M–1.5912271792977430E–06

C4, 3,M–8.0678881596778210E–08

C4, 4,M–1.2692158612216040E–07

C5, 1,M–8.6629769308983560E–07

C5, 2,M

7.1199537967353330E–07

C5, 3,M

1.5399750424904520E–08

C5, 4,M

2.1444704319218450E–08

C5, 5,M

7.6596153884006140E–09

C6, 1,M

1.2024363601545920E–06

C6, 2,M–5.4703897324156850E–07

C6, 3,M

–6.8785612757292010E–08

C6, 4,M

1.2915580402925160E–09

C6, 5,M

1.1737698784460500E–09

C6, 6,M–1.0913395178881540E–09

Tesseral harmonic coefficients

provided by the WISE and SIMPS surveys [9,10]. In a similar previous analysis [7], information on individual asteroid mass parameters was excluded, to allow independent estimates
based on the range data to Martian spacecraft. For DE430 and DE431, asteroid mass parameter estimates from other techniques were included as a priori constraints. Estimates were
included based on close encounters between asteroids [90,91], masses of binary asteroids[92–95], and masses determined from radio tracking of spacecraft directly affected by thegravity of individual asteroids [96–99]. The mass parameters used for DE430/DE431 aregiven in Table 12. The initial positions and velocities of the asteroids were taken from the
Horizons online solar system data service [13] and are given in Table 13.

Acknowledgments

The planetary ephemeris accuracies are limited by the accuracy of measurements to which
they are fit. These measurements are the results of the efforts of dozens of observers andhundreds of dedicated scientists and engineers operating the many spacecraft missions to
the Moon and planets. We would like to thank especially the following for directly delivering measurements for inclusion in the ephemeris development: Tom Murphy, Peter
Shelus, Randy Ricklefs, Jerry Wiant, Jean-Marie Torre, and colleagues for recent lunar laser
ranging data; Tony Taylor, Maria Zuber, and Dave Smith for detailed information on the
MESSENGER spacecraft dynamics; Alex Konopliv for producing range measurements adjusted to the central body for MESSENGER, Mars Global Surveyor, Mars Odyssey, and Mars Reconnaissance Orbiter; Jim Border for the Magellan and Mars spacecraft VLBI measurements;
Trevor Morely, Frank Budnik, and colleagues for Venus Express and Mars Express range andVenus Express VLBI measurements; Bob Jacobson for reduction of Voyager, Pioneer, andCassini spacecraft tracking data; Hugh Harris and Alice Monet for observations of the outerplanets at the U. S. Naval Observatory in Flagstaff; Bill Owen for observations of the outerplanets from Table Mountain Observatory; Julio Camargo, Gustavo Benedetti-Rossi, Felipe
Braga Ribas, Marcelo Assafin, Alex Dias-Oliveira, and Roberto Vieira Martins for observations of Pluto from Observatório do Pico dos Dias; Bruno Sicardy and Marcelo Assafin forrecent occultation measurements of the position of Pluto. This work is also greatly indebted
to the earlier work by Myles Standish and Skip Newhall, who created much of the software
for the development of the ephemerides.

52

Table 12. Mass parameters of asteroids (1 of 7).

53

GM, au3/day2

GM ast /GM 9

GM, km3/s2

0.140047655617234400E–12

4.73E–1062.809393

0.310444819893871300E–13

1.05E–10

13.923011

Juno

0.361753831714793700E–14

1.22E–11

1.622415

4

Vesta

0.385475018780881000E–13

1.30E–10

17.288009

5

Astraea

0.374873628455203200E–15

1.27E–12

0.168126

6

Hebe

0.831241921267337200E–15

2.81E–12

0.3728

7

Iris

0.213643444257140700E–14

7.22E–12

0.958161

8

Flora0.589425652970690800E–15

1.99E–12

0.264349

9Metis

0.107784100424073000E–14

3.64E–12

0.483396

10

Hygiea0.123580078729412500E–13

4.18E–11

5.542392

11

Parthenope

0.133153625545997500E–14

4.50E–12

0.597175

12

Victoria

0.193177578518292000E–15

6.53E–13

0.086637

13

Egeria0.179700489450744600E–14

6.07E–12

0.805931

14

Irene

0.110064567957506800E–14

3.72E–12

0.493624

15

Eunomia

0.467830741835090500E–14

1.58E–11

2.098155

16

Psyche0.341158682619381200E–14

1.15E–11

1.530048

17

Thetis

0.208150639646973800E–15

7.03E–13

0.093353

18

Melpomene

0.200892773665113200E–15

6.79E–13

0.090098

19

Fortuna0.103564484013119400E–14

3.50E–12

0.464472

20

Massalia

0.919980747763091100E–16

3.11E–13

0.04126

21

Lutetia0.252944287204099900E–15

8.55E–13

0.113442

22

Kalliope

0.120262444348346000E–14

4.06E–12

0.53936

23

Thalia0.189533176041978300E–15

6.41E–13

0.085003

24

Themis

0.189390166752538200E–14

6.40E–12

0.849388

25

Phocaea

0.723984152236621100E–16

2.45E–13

0.03247

26

Proserpina

0.163734395226108400E–15

5.53E–13

0.073433

27

Euterpe0.388800389854578800E–15

1.31E–12

0.174371

28

Bellona

0.292627274429452800E–15

9.89E–13

0.131239

29

Amphitrite

0.197584236512452000E–14

6.68E–12

0.886137

30

Urania0.148201901643752900E–15

5.01E–13

0.066466

31

Euphrosyne

0.634328047364860200E–14

2.14E–11

2.844872

32

Pomona0.119958501623344000E–15

4.05E–13

0.053834

Circe

0.294454129152128600E–15

9.95E–13

0.132059

35

Leukothea

0.235225617324184100E–15

7.95E–13

0.105495

36

Atalante

0.169706001840970900E–15

5.74E–13

0.076111

37

Fides

0.218562057711305600E–15

7.39E–13

0.09802238

Leda

0.132328596474676800E–15

4.47E–13

0.059348

39

Laetitia

0.149751968255670100E–14

5.06E–12

0.671616

40

Harmonia

0.295241408030842200E–15

9.98E–13

0.132412

41Daphne0.932422376219886900E–15

3.15E–12

0.418178

42

Isis

0.276531664347438100E–15

9.35E–13

0.124021

43

Ariadne0.727539385334071200E–16

2.46E–13

0.032629

44

Nysa

0.468864072012922000E–16

1.58E–13

0.021028

45

Eugenia

0.842567801856793400E–15

2.85E–12

0.37788

46

Hestia0.327280000000000000E–15

1.11E–12

0.14678

47

Aglaja0.554352356159888900E–15

1.87E–12

0.248619

48

Doris

0.253109172601506800E–14

8.55E–12

1.135159

49

Pales

0.754948162931440200E–16

2.55E–13

0.03385850

Virginia

0.163332639111751800E–15

5.52E–13

0.073252

Number

Name

1

Ceres

2

Pallas3

Table 12. Mass parameters of asteroids. (Continued: 2 of 7)

54

GM, au3/day2

GM ast /GM 9

GM, km3/s2

0.257054911335314500E–15

8.69E–13

0.115285

0.247678810125586700E–14

8.37E–12

1.110804

Kalypso0.623924331077516500E–16

2.11E–13

0.027982

54

Alexandra

0.562417365019245900E–15

1.90E–12

0.252236

56

Melete0.369928831270212600E–15

1.25E–12

0.165908

57

Mnemosyne

0.368060192063965100E–15

1.24E–12

0.16507

58

Concordia

0.848117391146600300E–16

2.87E–13

0.038037

59

Elpis

0.633944272758765100E–15

2.14E–12

0.28431560

Echo0.509113678301446400E–16

1.72E–13

0.022833

62

Erato

0.108904819196005700E–15

3.68E–13

0.048842

63

Ausonia

0.564040174397624300E–16

1.91E–13

0.025296

65

Cybele0.318065928265254100E–14

1.07E–11

1.426481

68

Leto0.343102659123796900E–15

1.16E–12

0.153877

69

Hesperia

0.514461002087673500E–15

1.74E–12

0.230728

70

Panopaea

0.276888884015784600E–15

9.36E–13

0.124181

71

Niobe

0.142449274635095600E–15

4.81E–13

0.063886

72

Feronia

0.799505104491654100E–16

2.70E–13

0.035857

74

Galatea

0.350737445129561400E–15

1.19E–12

0.157301

75

Eurydike

0.435737462507712700E–16

1.47E–13

0.019542

76

Freia

0.831220000000000100E–15

2.81E–12

0.37279

77

Frigga0.493129550950072900E–16

1.67E–13

0.022116

78

Diana

0.840190625346388700E–16

2.84E–13

0.037681

79

Eurynome

0.835182433140794000E–16

2.82E–13

0.037457

80

Sappho0.116144395411310800E–15

3.92E–13

0.052089

81

Terpsichore

0.102236755455613400E–15

3.45E–13

0.045852

82

Alkmene0.660126076693077000E–16

2.23E–13

0.029606

83

Beatrix

0.109683489062603500E–15

3.71E–13

0.049191

84

Klio

0.125731265563188600E–15

4.25E–13

0.05638985

Io0.925408545301853800E–15

3.13E–12

0.415033

86

Semele0.215239955702289100E–15

7.27E–13

0.096532

87

Sylvia0.219929517357407300E–14

7.43E–12

0.986353

88

Thisbe0.257711412731104700E–14

8.71E–12

1.155799

89

Julia

0.340203115743942900E–15

1.15E–12

0.152576

90

Antiope0.123519636282849100E–15

4.17E–13

0.055397

91

Aegina0.244046167770100600E–15

8.25E–13

0.109451

92

Undina0.403694351768607300E–15

1.36E–12

0.181051

93

Minerva

0.564773071797647600E–15

1.91E–12

0.253293

94

Aurora0.127923000000000000E–14

4.32E–12

0.573717

95

Arethusa

0.271661970839325900E–15

9.18E–13

0.121837

96

Aegle

0.154656769562432500E–14

5.23E–12

0.693614

97

Klotho0.103149563583763100E–15

3.49E–13

0.046261

98

Ianthe0.244283174173206900E–15

8.26E–13

0.109558

99

Dike0.735266201384159100E–16

2.48E–13

0.032976

100

Hekate0.136597719646869700E–15

4.62E–13

0.061262

102

Miriam0.128126622566059800E–15

4.33E–13

0.057463

103

Hera

0.100111801685864600E–15

3.38E–13

0.044899

104

Klymene0.481223966780187300E–15

1.63Ev12

0.215822

105

Artemis0.371546679753414500E–15

1.26E–12

0.166633

106

Dione

0.539739999999999800E–15

1.82E–12

0.242066

NumberName

51

Nemausa52

Europa53

Table 12. Mass parameters of asteroids. (Continued: 3 of 7)

55

GM, au3/day2

GM ast /GM 9

GM, km3/s2

0.167172099170064400E–14

5.65E–12

0.749743

0.108261858615819300E–15

3.66E–13

0.048554

Lydia

0.140769857221050400E–15

4.76E–13

0.063133

111

Ate

0.335191928112805600E–16

1.13E–13

0.015033

112

Iphigenia

0.579603970155323500E–16

1.96E–13

0.025994

113

Amalthea

0.255802139245781900E–16

8.64E–14

0.011472

114

Kassandra

0.170500000000000000E–15

5.76E–13

0.076467

115

Thyra

0.552582419038552700E–16

1.87E–13

0.024783

117

Lomia

0.447136801784178900E–15

1.51E–12

0.200535

118

Peitho0.270007596259813500E–16

9.12E–14

0.012109

120

Lachesis0.118898494995200800E–14

4.02E–12

0.533243

121

Hermione0.700790692204134300E–15

2.37E–12

0.314295

124

Alkeste0.887727082656233800E–16

3.00E–13

0.039813

127

Johanna

0.366116824306172300E–15

1.24E–12

0.164198

128

Nemesis0.965012951054875100E–15

3.26E–12

0.432795

129

Antigone

0.465424524739797500E–15

1.57E–12

0.208736

130

Elektra0.993662954590924800E–15

3.36E–12

0.445644

132

Aethra0.131961412217015600E–16

4.46E–14

0.005918

134

Sophrosyne

0.336204654208871600E–15

1.14E–12

0.150783

135

Hertha0.951560504184620700E–16

3.22E–13

0.042676

137

Meliboea

0.856126059955389200E–15

2.89E–12

0.38396

139

Juewa

0.422428821437744500E–15

1.43E–12

0.189453

140

Siwa

0.313167325322240600E–15

1.06E–120.140451

141

Lumen

0.376614060167083300E–15

1.27E–12

0.168906

143

Adria

0.112608611474888300E–15

3.81E–13

0.050503

144

Vibilia

0.699516335498308700E–15

2.36E–12

0.313723

145

Adeona0.755823292622889200E–15

2.55E–12

0.338976

146

Lucina0.394160000000000000E–15

1.33E–12

0.176775

147

Protogeneia

0.406817291771643100E–15

1.37E–12

0.182452

148

Gallia0.165952595163453000E–15

5.61E–13

0.074427

150

Nuwa

0.453364684879681200E–15

1.53E–12

0.203328

154

Bertha0.829838876716369400E–15

2.80E–12

0.372171

156

Xanthippe

0.263360000000000000E–15

8.90E–13

0.118113

159

Aemilia

0.288240661838050000E–15

9.74E–13

0.129272

160

Una

0.102484870587440500E–15

3.46E–13

0.045963

162

Laurentia

0.237290753793420100E–15

8.02E–13

0.106422

163

Erigone0.865800000000000100E–16

2.93E–13

0.03883164

Eva0.418326020317249000E–15

1.41E–12

0.187613

165

Loreley

0.106845990270469300E–14

3.61E–12

0.479189

168

Sibylla0.534709999999999800E–15

1.81E–12

0.23981

171

Ophelia

0.151098849092935500E–15

5.11E–13

0.067766

172

Baucis

0.431771576449205300E–16

1.46E–13

0.019364

173

Ino0.391242854708310100E–15

1.32E–12

0.175467

175

Andromache

0.289909719621158200E–15

9.80E–13

0.13002

176

Iduna

0.167134755200936100E–15

5.65E–13

0.074958

177

Irma

0.453728862942509900E–16

1.53E–13

0.020349

181

Eucharis

0.167967668335482500E–15

5.68E–13

0.075331

185

Eunike0.113558589448392200E–14

3.84E–12

0.509294

187

Lamberta

0.941936446359531100E–15

3.18E–12

0.422445

NumberName

107

Camilla109

Felicitas

110Table 12. Mass parameters of asteroids. (Continued: 4 of 7)

56

GM, au3/day2

GM ast /GM 9

GM, km3/s2

0.172525552363828900E–15

5.83E–13

0.077375

0.251194126565783300E–15

8.49E–13

0.112657

Prokne0.272305872890598200E–15

9.20E–13

0.122125

195

Eurykleia

0.835481985073780800E–16

2.82E–13

0.03747

196

Philomela0.450141365611792800E–15

1.52E–12

0.201882

198

Ampella0.205385768290833100E–16

6.94E–14

0.009211

200

Dynamene

0.175546167825261400E–15

5.93E–13

0.07873201

Penelope

0.101472968957165600E–15

3.43E–13

0.045509

203

Pompeja

0.184924720803005400E–15

6.25E–13

0.082936

205

Martha0.915149992259589400E–16

3.09E–13

0.041043

206

Hersilia

0.117896698493573100E–15

3.98E–13

0.052875

209

Dido

0.259310730988729300E–15

8.76E–13

0.116297

210

Isabella

0.131453781488021000E–15

4.44E–13

0.058955

211

Isolda0.304651155649041000E–15

1.03E–12

0.136632

212

Medea

0.386749965775167900E–15

1.31E–12

0.173452

213

Lilaea0.952238918721723900E–16

3.22E–13

0.042707

216

Kleopatra

0.690797124746742500E–15

2.33E–12

0.309813221

Eos0.181684062017330300E–15

6.14E–13

0.081483

223

Rosa

0.139060396851984600E–15

4.70E–13

0.062367

224

Oceana0.395205903434527200E–16

1.34E–13

0.017724

225

Henrietta

0.414565717253527800E–15

1.40E–12

0.185927

227

Philosophia

0.120484595454650200E–15

4.07E–13

0.054036

230

Athamantis

0.176089871551513500E–15

5.95E–13

0.078974

233

Asterope

0.197159196662545500E–15

6.66E–13

0.088423

236

Honoria

0.113632939011338100E–15

3.84E–13

0.050963

238

Hypatia

0.529666935807391100E–15

1.79E–12

0.237548

240

Vanadis0.110486225287326500E–15

3.73E–13

0.049552

241

Germania

0.300548362594600500E–15

1.02E–12

0.134792

247

Eukrate0.270972707241621300E–15

9.16E–13

0.121527

250

Bettina

0.196885501859924200E–15

6.65E–13

0.0883259

Aletheia

0.628085855393638300E–15

2.12E–12

0.281688

266

Aline

0.184133018752378200E–15

6.22E–13

0.082581

268

Adorea0.526690110925434800E–15

1.78E–12

0.236213

275

Sapientia

0.186540000000000000E–15

6.30E–13

0.083661

276

Adelheid

0.224815504884368300E–15

7.60E–13

0.100827

283

Emma

0.205281017798695800E–15

6.94E–13

0.092066

287

Nephthys

0.480337378560961000E–16

1.62E–13

0.021542

303

Josephina

0.288486840395101100E–15

9.75E–13

0.129382

304

Olga

0.688327109660540100E–16

2.33E–13

0.03087

308

Polyxo0.724054788525813900E–15

2.45E–12

0.324728

313

Chaldaea

0.115845913374079600E–15

3.91E–13

0.051955

322

Phaeo

0.827634952130435800E–16

2.80E–13

0.037118

324

Bamberga

0.138862658985619900E–14

4.69E–12

0.622779

326

Tamara0.189152474665620900E–15

6.39E–13

0.084832

328

Gudrun0.291270000000000000E–15

9.84E–13

0.13063

329

Svea

0.511430939429605700E–16

1.73E–13

0.022937

334

Chicago0.326249337943456200E–15

1.10E–12

0.146318

335

Roberta

0.147567692588891100E–15

4.99E–13

0.066182

336

Lacadiera

0.564300000000000000E–16

1.91E–13

0.025308

Number

Name

191

Kolga

192

Nausikaa

194

Table 12. Mass parameters of asteroids. (Continued: 5 of 7)

57

GM, au3/day2

GM ast /GM 9

GM, km3/s2

0.345150745118659600E–16

1.17E–13

0.015480.514076834286322000E–16

1.74E–13

0.023056

Desiderata

0.536890970425833500E–15

1.81E–120.240788

345

Tercidina0.123175211698841300E–15

4.16E–13

0.055242

346

Hermentaria

0.220991660771765600E–15

7.47E–13

0.099112

347

Pariana0.219491221778173700E–16

7.42E–14

0.009844

349

Dembowska

0.700787392713029200E–15

2.37E–12

0.314293

350

Ornamenta

0.141000784057630900E–15

4.76E–13

0.063237

354

Eleonora

0.158509865715968900E–14

5.36E–12

0.710894

356

Liguria

0.268161319613518200E–15

9.06E–13

0.120267

357

Ninina0.176195602610025700E–15

5.95E–13

0.079021

358

Apollonia

0.121780979958317900E–15

4.12E–13

0.054617

360

Carlova

0.336776057435300700E–15

1.14E–12

0.151039

362

Havnia0.808707928810328300E–16

2.73E–13

0.036269

363

Padua

0.698419453021186600E–16

2.36E–13

0.031323

365

Corduba

0.776955520962180000E–16

2.63E–13

0.034845

366

Vincentina

0.115639547136458300E–15

3.91E–13

0.051863

369

Aeria

0.573458156633685800E–16

1.94E–13

0.025719

372

Palma

0.174595572627050000E–14

5.90E–12

0.783036

373

Melusina

0.137206175961377900E–15

4.64E–13

0.061535

375

Ursula0.455859924880692500E–15

1.54E–12

0.204447

377

Campania

0.141442289208980800E–15

4.78E–13

0.063435

381

Myrrha0.359349348076176100E–15

1.21E–120.161163

385

Ilmatar0.110318518229682800E–15

3.73E–13

0.049476

386

Siegena

0.150793337119651900E–14

5.10E–12

0.676287

387

Aquitania

0.100446598396309400E–15

3.39E–13

0.045049

388

Charybdis

0.343334441285916900E–15

1.16E–12

0.153981

389

Industria

0.622510183873803300E–16

2.10E–13

0.027919

393

Lampetia

0.155860000000000000E–15

5.27E–13

0.069901

404

Arsinoe

0.145216789548421900E–15

4.91E–13

0.065128

405

Thia

0.557648047680853900E–15

1.88E–12

0.250097

407

Arachne0.156785041404748300E–15

5.30E–13

0.070316

409

Aspasia

0.821449999999999900E–15

2.78E–12

0.368409

410

Chloris0.361865469702973400E–15

1.22E–12

0.162292

412

Elisabetha

0.146557349756319100E–15

4.95E–13

0.065729

415

Palatia

0.540012140874342300E–16

1.82E–13

0.024219

416

Vaticana

0.148698562934496100E–15

5.03E–13

0.066689

419

Aurelia

0.367820000000000000E–15

1.24E–12

0.164962

420

Bertholda

0.483520000000000000E–15

1.63E–12

0.216852

423

Diotima

0.211243836059995200E–14

7.14E–12

0.947399

424

Gratia0.737998534437522900E–16

2.49E–13

0.033098

426

Hippo

0.229797581279714500E–15

7.77E–13

0.103061

431

Nephele0.634077834952169700E–16

2.14E–13

0.028437

432

Pythia0.187801229589892800E–16

6.35E–14

0.008423

433

Eros

0.990000118979590300E–18

3.35E–15

0.000444

442

Eichsfeldia

0.555346562874691200E–16

1.88E–13

0.024907

444

Gyptis

0.907080484411450500E–15

3.07E–12

0.406813

445

Edna

0.115342108493158500E–15

3.90E–13

0.051729

449

Hamburga

0.135581713067348800E–15

4.58E–13

0.060806

NumberName

337

Devosa338

Budrosa344

Table 12. Mass parameters of asteroids. (Continued: 6 of 7)

58

GM, au3/day2

GM ast /GM 9

GM, km3/s2

0.229555939063746200E–14

7.76E–12

1.029526

0.768088477469999900E–16

2.60E–13

0.034448

Bruchsalia

0.228615933395781000E–15

7.73E–13

0.102531

464

Megaira0.806351842900557800E–16

2.72E–13

0.036164

465

Alekto0.653992577244023600E–16

2.21E–13

0.029331

466

Tisiphone

0.270714167365278100E–15

9.15E–13

0.121411

469

Argentina

0.322840000000000000E–15

1.09E–12

0.144789

471

Papagena

0.845943072895968300E–15

2.86E–12

0.379394

476

Hedwig0.241931605646406600E–15

8.18E–13

0.108503

481

Emita

0.334070052970451900E–15

1.13E–12

0.149826

485

Genua

0.367379607992341800E–16

1.24E–13

0.016476

488

Kreusa0.191515627988507800E–15

6.47E–13

0.085892

489

Comacina

0.548467242911317100E–15

1.85E–12

0.24598

490

Veritas0.364149739778329100E–15

1.23E–12

0.163316

491

Carina0.148589708382528900E–15

5.02E–13

0.06664

498

Tokio

0.129879292391702200E–15

4.39E–13

0.058249

503

Evelyn0.111871833145007600E–15

3.78E–13

0.050173

505

Cava

0.341769850174723900E–15

1.15E–12

0.153279

506

Marion0.197353139110484100E–15

6.67E–13

0.08851508

Princetonia

0.341067697975348700E–15

1.15E–12

0.152964

511

Davida0.519812697945749800E–14

1.76E–11

2.331286

514

Armida0.293765207953148200E–15

9.93E–13

0.13175

516

Amherstia

0.696000000000000000E–16

2.35E–13

0.031215

517

Edith

0.108862665088002000E–15

3.68E–13

0.048823

521

Brixia0.181244939644586100E–15

6.12E–13

0.081286

532

Herculina

0.931594859406562000E–15

3.15E–12

0.417807

535

Montague

0.725386164346611400E–16

2.45E–13

0.032533

536

Merapi0.109756310328225100E–14

3.71E–12

0.492242

545

Messalina

0.194849861155712500E–15

6.58E–13

0.087387

547

Praxedis

0.281235774586576800E–16

9.50E–14

0.012613

554

Peraga0.274856688034015000E–15

9.29E–13

0.123269

566

Stereoskopia

0.626740000000000000E–15

2.12E–12

0.281084

568

Cheruskia

0.859780218818313100E–16

2.91E–13

0.03856569

Misa

0.611232197266831400E–16

2.07E–13

0.027413

584

Semiramis

0.217605956141594600E–16

7.35E–14

0.009759

585

Bilkis

0.117169915406794000E–16

3.96E–14

0.005255

591

Irmgard0.200383935095216300E–16

6.77E–14

0.008987

593

Titania

0.501889934520749200E–16

1.70E–13

0.022509

595

Polyxena

0.222111336158284500E–15

7.51E–13

0.099614

596

Scheila

0.383508917087800200E–15

1.30E–12

0.171998

598

Octavia

0.615306415407552400E–16

2.08E–13

0.027596

599

Luisa

0.691187747327401000E–16

2.34E–13

0.030999

602

Marianna0.215014674336122500E–15

7.27E–13

0.096431

604

Tekmessa

0.642603995331864300E–16

2.17E–13

0.02882618

Elfriede

0.463868801607939000E–15

1.57E–12

0.208039

623

Chimaera

0.144979767976932900E–16

4.90E–14

0.006502

626

Notburga

0.166437787258821500E–15

5.62E–13

0.074645

635

Vundtia

0.169956695263313900E–15

5.74E–13

0.076223

654

Zelinda0.359530000000000000E–15

1.21E–120.161244

449

Hamburga

0.135581713067348800E–15

4.58E–13

0.060806

Number

Name

451

Patientia

454

Mathesis455

Table 12. Mass parameters of asteroids. (Continued: 7 of 7)

59

GM, au3/day2

GM ast /GM 9

GM, km3/s2

0.120495494885781800E–15

4.07E–13

0.054041

0.909429441579076000E–16

3.07E–13

0.040787

Rachele0.143837480044678900E–15

4.86E–13

0.064509

675

Ludmilla0.159234708844922700E–15

5.38E–13

0.071415

680

Genoveva

0.995816201002551800E–16

3.37E–13

0.044661

683

Lanzia0.104361036125065600E–15

3.53E–13

0.046804

690

Wratislavia

0.476375353165424700E–15

1.61E–12

0.213648

691

Lehigh

0.945370522582010200E–16

3.19E–13

0.042399

694

Ekard

0.149386680301335600E–15

5.05E–13

0.066998

696

Leonora0.111777344239748500E–15

3.78E–13

0.050131

702

Alauda0.889506728492704500E–15

3.01E–12

0.398931

704

Interamnia

0.525616867849366200E–14

1.78E–11

2.357317

705

Erminia

0.291750035064825000E–15

9.86E–13

0.130846

709

Fringilla

0.154560085386599100E–15

5.22E–13

0.069318

712

Boliviana

0.502042422129256500E–15

1.70E–12

0.225159

713

Luscinia

0.142681578483330100E–15

4.82E–13

0.063991

735

Marghanna

0.859762026371732300E–16

2.91E–13

0.038559

739

Mandeville

0.713824339752934800E–16

2.41E–13

0.032014

740

Cantabia

0.118928144393176900E–15

4.02E–13

0.053338

747

Winchester

0.621574606623668800E–15

2.10E–12

0.278767

751

Faina

0.179893748114765000E–15

6.08E–13

0.08068

752

Sulamitis

0.499820481258924800E–16

1.69E–13

0.022416

760

Massinga

0.581679210057327800E–16

1.97E–13

0.026087

762

Pulcova

0.208180128309056600E–15

7.04E–13

0.093366

769

Tatjana0.197251539856937600E–15

6.67E–13

0.088465

772

Tanete0.252061449253389500E–15

8.52E–13

0.113046

773

Irmintraud

0.195761609064233300E–15

6.62E–13

0.087796

776

Berbericia

0.305671194466365300E–15

1.03E–12

0.137089

778

Theobalda

0.477435477387998300E–16

1.61E–13

0.021412

780

Armenia0.148574053476563100E–15

5.02E–13

0.066633

784

Pickeringia

0.163082309665980900E–15

5.51E–13

0.07314786

Bredichina

0.158301988046999200E–15

5.35E–13

0.070996

788

Hohensteina

0.215815568176360100E–15

7.29E–13

0.09679

790

Pretoria

0.175589918332470000E–14

5.93E–12

0.787496

791

Ani

0.121964875174107500E–15

4.12E–13

0.0547804

Hispania

0.226710334145990500E–15

7.66E–13

0.101676

814

Tauris

0.455595956637723700E–15

1.54E–12

0.204328

849

Ara0.103549725015275900E–15

3.50E–13

0.046441

895

Helio

0.378647503308948400E–15

1.28E–12

0.169818

909

Ulla

0.336297549276103900E–15

1.14E–12

0.150825

914

Palisana

0.411726826839954200E–16

1.39E–13

0.018465

980

Anacostia

0.153529788855662000E–15

5.19E–13

0.068856

1015

Christa

0.125837659264500500E–15

4.25E–13

0.056436

1021

Flammario

0.761714098728426500E–16

2.57E–13

0.034162

1036

Ganymed

0.646776721323746100E–16

2.19E–13

0.029007

1093

Freda

0.287164048267019600E–15

9.70E–13

0.128789

1107

Lictoria

0.954912821588764700E–16

3.23E–13

0.042826

1171

Rusthawelia

0.624856870846376100E–16

2.11E–13

0.028024

1467

Mashona0.111528013303481700E–15

3.77E–13

0.050019

NumberName

663

Gerlinde

667

Denise674

Table 13. Initial positions (au) and velocities (au/day) of the asteroids with respect to the Sun at
Julian day (TDB) 2440400.5 (June 28, 1969) in the ICRF2 frame (1 of 15).
1

Ceres

x,y,z
vx,vy,vz

2

Pallas

			
3

Juno

			
4
5

Vesta
Astraea

6

Hebe

7

Iris

8

Flora

9
10

Metis
Hygiea11

Parthenope

13

Egeria

14

Irene

15

Eunomia

16

Psyche

19
20
21
22
23
24
25
27

Melpomene
Fortuna
Massalia
Lutetia
Kalliope
ThaliaThemis
Phocaea
Euterpe0.000466157738595739

0.004684247977335608

0.203832272462290465 –3.209619436062307152

0.623843179079393351

vx,vy,vz

0.008534313855651248 –0.000860659210123161–0.000392901992572746

x,y,z

0.461207259670432135 –3.006098959780790114–0.580164049296942208

vx,vy,vz

0.008395458298285176

0.003111908045571209

0.000273059675893248

x,y,z

0.182371836377417107

2.386628211277654010

0.924596062836265498

vx,vy,vz–0.010174496747119257

0.000041478190529952

0.001344157634155624

x,y,z

2.489297359488491956

1.036395265106434982

0.210563198822894787

vx,vy,vz–0.005569115604615741

0.007959732929200320

0.003113959705731406

x,y,z

1.339049495814490065

1.442775542206668815

0.079273672077323748

vx,vy,vz–0.008775983793258694

0.009426820472364830

0.003535716141864189

1.892475267790300286 –0.848414748075139946–0.157159319044464590

vx,vy,vz

0.002786950314570632

0.011314057384917047

0.004975132577079665

x,y,z–2.119655892430383659

0.808466233981220750

0.533397871779012545

vx,vy,vz–0.005818098320155988 –0.008811943327809565–0.002835330893491553

x,y,z–2.424658333778681119 –0.125325594242630484

0.185966267773321953

vx,vy,vz–0.001166914352512719 –0.009845348309999084–0.004559667512580128

x,y,z

2.444257569754775261

2.180591649726027814

1.162855082530954043

vx,vy,vz–0.005924505919570356

0.005979686441736038

0.002286438610925529

x,y,z–1.231933494613708824 –1.941584219024864888–0.648652135887457293

vx,vy,vz

0.010146729938442466 –0.004310578383941682–0.002342316786027452

x,y,z

1.110470035635196862 –1.956883630909689531–1.669729176939996007

x,y,z

vx,vy,vz

0.008850484510873332

0.004168221460244788

0.000793489279402694

x,y,z

2.968959552750401354

0.179534605878148124–0.441493212476667563

vx,vy,vz

0.000114431097969882

0.008313751393872647

0.003664818027075008

x,y,z–1.438397661546355177

2.001287639171841271

0.767257269566094968

vx,vy,vz–0.009735666014582665 –0.002981534818366074–0.003694080871985814

x,y,z

1.459069269212805553 –2.194286961057163143–0.872045420720389042

0.008193866970882068

0.002167069206448382

0.006328897736618455

x,y,z–2.742133025533653790 –0.012766315168109336

0.239819478175997858

vx,vy,vz–0.000996361773223022 –0.008953551423450837–0.002237582813053089

x,y,z–2.421846633153337702 –1.337431649320276250–0.581397545560000673

vx,vy,vz

0.004769326689664416 –0.007520729608529603–0.002981882787056496

1.855382149831718674

0.778094121440758957

vx,vy,vz–0.012511421939296488 –0.002534666129817137–0.001141856361378328

x,y,z–0.446957693127368383

x,y,z–0.401500082512939060 –2.034737356634178429–0.878857903014080954

vx,vy,vz

0.011859546440634965 –0.000138986487715931–0.000750181020252093

x,y,z–1.360169904369056937 –2.672544709790108275–1.131055224812947957

vx,vy,vz

0.008015423798989894 –0.002573715371792212–0.003469704484884123

x,y,z–1.381180931801666123 –2.177231877010123551–0.869547456324116008

vx,vy,vz

0.007230267698506810 –0.005803206866356805–0.004345034311979445

1.713080040332319420

0.782177241944561841

vx,vy,vz–0.007376541658456220 –0.007515395858827745–0.003291592430864249

x,y,z

x,y,z

–1.986326887180129397

1.862187339641438788 –0.320827452239953126

0.370298833532256677

vx,vy,vz

0.004047653066899725

0.012783082342996164

0.002077625994146512

x,y,z–1.260198559466741086

1.418639340820153416

0.648923808375896249–0.010819473664968254 –0.006731978199858915–0.002575089185167887

vx,vy,vz

60–1.326397853361325874

0.008465406136316316

x,y,z

vx,vy,vz
18

1.438681809676469747 –2.204373633189407045

Table 13. Initial positions (au) and velocities (au/day) of the asteroids with respect to the Sun at
Julian day (TDB) 2440400.5 (June 28, 1969) in the ICRF2 frame. (Continued: 2 of 15.)
28
29
30
31
41
42
45
51

Bellona
Amphitrite
UraniaEuphrosyne
DaphneIsis
Eugenia
Nemausa52

Europa

60

Echo

63
65
69
78
94

Ausonia
CybeleHesperia
Diana
Aurora97

Klotho98

Ianthe1.739643135937173790

0.633933941012944979

vx,vy,vz–0.009328149750934787 –0.007427778551081034–0.001181230605742862

x,y,z
x,y,z

–0.146988273583742862 –2.362796893883333915–1.340108464209891492

vx,vy,vz

0.010080728492092296 –0.000192196827316014–0.000039573406685344

x,y,z–2.606330961010526526 –0.080313889098223457–0.119443301999580323

vx,vy,vz–0.000271687607297601 –0.009151555703778012–0.004227028420836736

x,y,z–2.344572196151709420 –2.202248246583482949–1.499371924448428484

vx,vy,vz

0.004929226633590931 –0.003894954486494073–0.005777736992660197

x,y,z

2.186522806828394838

2.716000834079776549

0.352083724354172178

vx,vy,vz–0.006362487833486645

0.004588916254801226

0.000658460597293756

x,y,z–2.311906318139493433

1.564709507254086418

1.081663757432244743

vx,vy,vz–0.005375422471534914 –0.006572477821943680–0.002070451318821641

x,y,z–1.286740660767531352 –2.064333827345071093–0.581533957673131763

vx,vy,vz

0.009624183869937625 –0.005447390675875109–0.002359148283535982

x,y,z

2.325725710714735328

0.919631243406134669

0.195346323188101051

vx,vy,vz–0.004226342069996713

0.009366245889668433

0.002292473770765156

1.630504392123616197 –2.807391275567590050–1.120111554599330095

vx,vy,vz

0.007545901962567781

0.004385628274804682

0.000681057876133087

x,y,z–2.053758095409557338

0.542618601186906857

0.167604428744862921

vx,vy,vz–0.005053755262744046 –0.010634893001978403–0.003929518991233821

x,y,z

x,y,z

–0.711507633861145239 –1.727049410575609123–0.980245752936226511

vx,vy,vz

0.011957874480153950 –0.003468676332191662–0.001404339015793962

x,y,z–2.818818186481110377 –1.452785968979383169–0.459547659952798004

vx,vy,vz

0.005491212616137729 –0.007709765186399763–0.002980907967655813

x,y,z–2.731373586077875171 –0.173990396273168008–0.091512829062557879

vx,vy,vz–0.000835829539403114 –0.010410309605217057–0.002792056931756292

x,y,z–2.230308915264555658 –0.612004762065796459–0.543275243058195634

vx,vy,vz

0.001875816436270626 –0.009912063988588916–0.005874203423004298

x,y,z

1.217204321473684070

2.230213797067274850

1.349204301622864088

vx,vy,vz–0.009480919363038220

0.003964160607269546

0.002537794862740618–1.909756234729403035 –2.678800550963929439–0.452067580136410296

0.006291707122836601 –0.005039334023207890–0.001538832193933371

x,y,z
vx,vy,vz
x,y,z
vx,vy,vz

105
111
135
139

ArtemisAte
Hertha
Juewa
Adeona187

Lamberta

192

Nausikaa

0.226114692352493402

0.112525782901568755–0.002121664740945849 –0.009612549934188979–0.007843089126949381–2.150697286825749899

0.541875356226059646–0.099139411741416952–0.000933995664699450 –0.011865009015818844–0.000510278917224607

x,y,z–2.065235972301910916 –1.395192740371126838–0.851171268109567225

vx,vy,vz

0.005618108019220444 –0.008131544741012933–0.003600368200049119

x,y,z–1.852065839438821637 –1.414537091163847737–0.701764695699997887

vx,vy,vz

0.008706738823410954 –0.006122907739411309–0.002836580152272733

x,y,z

0.129793402866451107

0.113110508166727894–0.001144921173599620 –0.010129177653436634–0.006915213503334226

x,y,z

1.101693094201525502

2.046278461562542006

0.733533233415469499

vx,vy,vz–0.010477714288306773

0.002702890747395121

0.003883525044571004

1.552362680005133111 –1.667017286215509042–1.224506107407864253

x,y,z

x,y,z

–2.286169743218249728

0.004083994537401527

0.001832035781453411

x,y,z–2.180257484088014674 –1.605283294502194646–1.009730329227873646

vx,vy,vz

0.006837112554701223 –0.005228654546530322–0.002756824225065406

vx,vy,vz

61–2.189380245469819020

vx,vy,vz

vx,vy,vz
145–1.473796460584195778

0.010026916904808853

Table 13. Initial positions (au) and velocities (au/day) of the asteroids with respect to the Sun at
Julian day (TDB) 2440400.5 (June 28, 1969) in the ICRF2 frame. (Continued: 3 of 15.)
194

Proknex,y,z			

vx,vy,vz

216

Kleopatra

x,y,z

230

Athamantis

324

Bamberga
Devosa344

Desiderata

354
372
405
409
419

Eleonora
Palma
Thia
Aspasia
Aurelia

0.000134167930615169–2.623737352933921407 –2.141208444909570296–0.848460319932531903

vx,vy,vz

0.005369324233200630 –0.005867957040039504–0.000585039109819107

x,y,z–2.280008375035697199 –0.880463965508790358–0.645752311103840460

vx,vy,vz

0.004331702501765179 –0.009223170407939286–0.002470255504286102

x,y,z

1.398759064223541682 –1.287476729008325105–0.669098428660833799

0.009219958777618218

0.006857861727407507

2.057440180433961707 –1.322162885949879740–0.780976069801601058

vx,vy,vz

0.005188028188451134

0.007591643038099468

0.004670779035477783

x,y,z–1.436281403678767443

2.243805682206950092

2.055782461490350155

vx,vy,vz–0.006473875571808206 –0.004308997045040672–0.001132197443002879

x,y,z

Patientia

488

Kreusa511

Davidax,y,z0.400598302733801259 –2.845496057705452220–0.363223167203308106

0.009411661670619676 –0.002169498446596223–0.002270974488656075

x,y,z–2.501245479525921134 –1.473356596063249313–2.186348442117559632

vx,vy,vz

0.004414109643858683 –0.005694498088384558–0.004127573934799787

x,y,z–1.680813989369112971 –0.720962772281693809–0.636265292808031457

vx,vy,vz

0.006053497060520836 –0.012004513735606900–0.003193078939729851

x,y,z

2.627942640819607600

0.032856722306475961

0.499378874814495721

vx,vy,vz–0.000121147420399894

0.009800913286716541

0.003214451046459611

x,y,z–0.832205073692585229

2.925043505666568677

1.070613454084370586–0.007845381994993020 –0.002434064039560803–0.001376245101202946

x,y,z

1.259008304010221346

2.454041267595273634

0.690482754870751503

vx,vy,vz–0.009063543500060710

0.003448045666261337

0.004183420491116245–2.067224538805315426 –1.883392328874813781–0.393159264399483011

0.005745445324864604 –0.007721753293565463–0.004781672638324677

x,y,z
vx,vy,vz

532
554

Herculina
Peraga1.486363108760649254

1.096959097247329984

vx,vy,vz–0.007390695920479255 –0.007779023504823501–0.000485540781565076

x,y,z

Zelindax,y,z–0.293159404090669762 –2.481691048455131110–0.725934583722743554

0.010153432932361773 –0.002073902764508174–0.003634391919195946

x,y,z

1.567515331928776900 –1.756839671087627286–0.730145653545691942

0.007168202681128191

0.007126474516266886

0.003646321528209388

2.454317123300634673 –1.156415934845627769

0.311816838296415377

0.001774157409192030

0.007979676971587255

0.004592146292516224

2.462835232264114715 –0.115048121589235539

0.784219288788080227

vx,vy,vz–0.001309568805636822

0.010348987192429257

0.004812522889055394

x,y,z–0.834902081339142454

2.059697173467183617

0.613292775285328884–0.012134738848543374 –0.002004140070559451

0.002575815432470426

x,y,z
vx,vy,vz

704
747

Interamnia
Winchester

x,y,z

vx,vy,vz
12
17
26
32
34

Victoria
ThetisProserpina
PomonaCirce

2.607456081000619541

0.819717059989116925

vx,vy,vz–0.009185378920496397 –0.000281429685500620–0.001275015521881669

x,y,z–0.224334228105830036 –1.994047200796976282–0.713424849893356972

vx,vy,vz

0.012457106084712540 –0.000945089974559068–0.001398683846484221

x,y,z–0.507844534173903961

1.125007532904078289

0.661062309969335460

vx,vy,vz–0.004431095916543229 –0.008819825820419791–0.004062402214390301

x,y,z–0.901728147437201644 –2.125115524966149305–0.803811944165582326

vx,vy,vz

0.010194219116656746 –0.004774494291409794–0.000989390504731257

x,y,z

x,y,z
vx,vy,vz

62–2.160191561573574504

vx,vy,vz
vx,vy,vz
654

0.007164363244556328

vx,vy,vz

vx,vy,vz
451

0.008680667325646010–0.299878060343942709

0.010437900139680028

vx,vy,vz
337

1.495439006518376468 –1.277354772575368669–2.278602546659383776–2.391157252691995083

0.282992268249411760

0.070390275547792716–0.001758506796666345 –0.010942908417041593–0.003566166537380931

Table 13. Initial positions (au) and velocities (au/day) of the asteroids with respect to the Sun at
Julian day (TDB) 2440400.5 (June 28, 1969) in the ICRF2 frame. (Continued: 4 of 15.)
35
36

Leukothea
Atalante

x,y,z–2.185818508920677861 –0.647312389706184987–0.427069919068639725

vx,vy,vz

0.004403906713249404 –0.010009270813997396–0.006061965347001481

x,y,z

2.077067513068779459 –0.589147154522667860–0.516882702948444517

vx,vy,vz
37

Fides

38

Leda

x,y,z
vx,vy,vz

Laetitia

40

Harmonia

43

Ariadne

44
46
47
48
49
50

Nysa
HestiaAglajaDoris
Pales
Virginia			53

Kalypso54

Alexandra

56
57
58
59
62
68
70

MeleteMnemosyne
Concordia
Elpis
Erato
Leto
Panopaea
Niobe
Feronia

0.003566522119529630 –0.007751964656304541–0.003895555791525518
0.766832361464905965

x,y,z

2.336193952026537790

0.759000796514275100

0.022176605809523606

vx,vy,vz–0.003257699513685890

0.010731650305157095

0.002872519477197957

x,y,z–0.690601749410825527 –2.038615544126432422–0.813975363544205632

vx,vy,vz

0.010841792489889901 –0.002359683762911624–0.001881496483239468

x,y,z

2.009602895794647015 –0.308045882132810533

0.000115161076006508

vx,vy,vz

0.003565080131794708

0.010950285369215265

0.004918573106272521

x,y,z–1.180941766350430466

1.554866402256739777

0.656611442582409666

vx,vy,vz–0.010685573544758461 –0.006831046309965641–0.002065889635700970–1.235289907955999134

x,y,z–0.474973466355436691 –2.296851721557127579–0.887903017375627823

vx,vy,vz

0.010895877228602094 –0.000193379625778825–0.000061860987657291

x,y,z

1.067741941627274427 –2.006425563476340379–1.093089277938812831

vx,vy,vz

0.010164990750471822

0.004729454222497436

0.002493051086595212

x,y,z–1.620193164505671524

2.388346628469387944

0.711322099748724956

vx,vy,vz–0.008761794768022678 –0.004974889431640200–0.001595006590297288

x,y,z–3.160577305628162748 –1.841530354628577504–1.017849520123809803

vx,vy,vz

0.004122755888684796 –0.006077379179740559–0.002511188490367351

x,y,z

1.333238507267154427 –1.602799470048592578–0.610763870330498060

0.009223622006417765

0.003430026595662231–0.624015677324980311 –2.887904714321112287–0.973890349097378372

vx,vy,vz

0.008425652319494320 –0.002165971738316451–0.001225638323642568

x,y,z–1.135653934893979367 –1.669000255199023952–1.229167471346461094

vx,vy,vz

0.011014262220981971 –0.004397975120144764–0.000793087156657538

x,y,z–2.509110608187361979 –0.599826488136790070–0.256184765903608447

vx,vy,vz

0.005001190631127405 –0.009156744414684281–0.002380258455330665

x,y,z–2.391692801502097954 –2.476663754241630322–0.613024391539033098

vx,vy,vz

0.006142650670634989 –0.006122207358406149–0.000426188587336327

x,y,z

1.970644363722096504

1.898617820025757519

0.583347727462360521

vx,vy,vz–0.007322186137912932

0.006519792084242099

0.002412465398938618

x,y,z–2.174043173151056330

1.754505167862759452

0.521206662638497353

vx,vy,vz–0.007194625470952607 –0.006696283399313014–0.001604669222795236

vx,vy,vz
x,y,z

0.007993780997143859

x,y,z–0.974153514232837803 –3.190444885566684707–1.264820437591952285

vx,vy,vz

0.008304485139178936 –0.001176603633891707–0.000760024384788282

x,y,z

2.150832185782553196 –0.472704842227096689–0.504675337037531091

vx,vy,vz

0.003447895159962464

0.010633207649093934

0.005531722529065961

x,y,z–0.512732908153528344

2.523947930349827118

1.619878869724124648–0.008876233401680499 –0.001390534286671486

0.000743421116570644

x,y,z
x,y,z
vx,vy,vz

63–0.637055179528812743–0.003684794791298678

vx,vy,vz
72

0.008367140895652419–2.571178583038596699 –1.322068718023980072–0.010386680301307009 –0.004838101200633434

vx,vy,vz
71

0.009307317089508693

1.834720116439384707

x,y,z
vx,vy,vz

390.001483717270391386–2.102963729152515615

1.401034628378233560

0.430525013472648932–0.004028656274637795 –0.007045329097739382–0.007586343532703573–0.698145332366829519

2.293578863469888596

0.742562228176233741–0.009984751110793636 –0.002036674761053411–0.001163542296488656

Table 13. Initial positions (au) and velocities (au/day) of the asteroids with respect to the Sun at
Julian day (TDB) 2440400.5 (June 28, 1969) in the ICRF2 frame. (Continued: 5 of 15.)74
75

Galatea
Eurydike

x,y,z

2.002901140763306831

0.640809773926249604

0.274754411294356571

vx,vy,vz–0.003966716076468380

0.011826725481734445

0.004119277914981450

x,y,z–3.200842305137657107

1.220975490656682982

0.662752781812484315–0.003148422634162045 –0.006154368263463862–0.003334345423877219

x,y,z

2.635360768423828937

1.437884630898934590

0.615131758192210332

vx,vy,vz–0.006649269126403664

0.007319514417658025

0.002778582333891908–2.690246721423859899 –1.025385948689222948–0.492898536579899738

vx,vy,vz

0.002901323944706134 –0.008214417661668621–0.003990796639863591

x,y,z–0.855375263619381876 –2.558706256410499424–0.931566610102818493

vx,vy,vz

0.009085151348455443 –0.001894651404025254–0.000307424335130962

x,y,z

1.608537185631021771 –0.946571596217173794–0.123702344488760718

vx,vy,vz
76

Freia

77

Frigga79

Eurynome

80

Sapphox,y,zvx,vy,vz
81

Terpsichore

82

Alkmene83

Beatrix

x,y,z

Klio

85

Io
Semele87

Sylvia

89
90
91
92
93
95
96
99

ThisbeJulia
Antiope
AeginaUndinaMinerva
Arethusa
Aegle
Dike

			
100

Hekate0.000185079802975043

vx,vy,vz

0.000372867010599650

0.007874899115079248

0.003828789513816485

x,y,z

2.537243040037000963

0.677406600928586178

0.239904468813102290

vx,vy,vz–0.002746008298400088

0.008573152131379970

0.004656090987115207

x,y,z–2.867387466643116500

0.123974227957666397–0.213371959277415285

Miriam0.000843918676535950 –0.007659425932435818–0.004589100376152419

2.196277362267318622

0.413880157287253858

0.281126843760316159

vx,vy,vz–0.001029040843236626

0.011980721214998944

0.002569598620931192

2.434775829626800281 –0.701653670472491275–0.530519782084361902

0.002147727796190763

0.004381939368423997

x,y,z

0.010457520401758034

x,y,z–3.660362899261072389 –0.885526882959144834

0.312168187323252744

vx,vy,vz

0.001817747464846094 –0.007261471114931983–0.003982770842554424

x,y,z

0.351034977545771065

2.726034310628392632

1.255821730738119202

vx,vy,vz–0.009085998946397741

0.002504365492940864

0.000217136735120159

x,y,z

0.668287897791156604

1.784401165953789503

1.391068013944257720

vx,vy,vz–0.010183706148394674

0.005465198051083607

0.001141423763308724

x,y,z–2.527596410164981489

2.393097637265075850

1.176158237649570948

vx,vy,vz–0.005785028160541267 –0.005353734747955876–0.002166792449198574

x,y,z–0.087297607412163220 –2.585707806328295000–1.234058824506656826

vx,vy,vz

0.009596957524554405 –0.000169138133959666–0.000157156795888277

x,y,z–1.042907269635494405 –2.875324060333224629–0.930751854561716052

vx,vy,vz

0.009168658604880101 –0.001743405265776949–0.002351877046242883

x,y,z

0.585579552533172509

2.601997108680597304

1.615157088233035720

vx,vy,vz–0.008803902414859047

0.001797983571727375

0.001256003779502425

x,y,z

2.542078873825008856

0.235650434494874467

0.625470306990500435

vx,vy,vz–0.002373003952808578

0.010704165151006678

0.002903697708982936

x,y,z–2.188255307668921734

1.381773051127626495

0.553799995799386546

vx,vy,vz–0.005576550150666639 –0.007277112779745350–0.006521139973971324

x,y,z

2.444770853104723862 –0.615836362049607455–0.892420122928066495

vx,vy,vz

0.005698847921220206

0.007805584032288921

0.004109108981978736

x,y,z–2.808466634462525313

1.998234760159137258

0.967652214689530998–0.004808839856292163 –0.006555049055444871–0.001867365486725437

x,y,z
vx,vy,vz

64

0.008642357666503482

x,y,z

vx,vy,vz
102–1.711616630966411812–0.330346049201625136

vx,vy,vz

880.004089724891808015–0.299527408394068329 –2.820136290562028858
0.000392975155467740

x,y,z

vx,vy,vz

86

0.011656883717961289

3.160151220673873329 –0.517688310655634809

vx,vy,vz

84

0.005907288892718746–0.802979209614599232

2.471520362371176116

0.813851431023047911–0.010278712148820868 –0.000277333514340038–0.000609446487533852

Table 13. Initial positions (au) and velocities (au/day) of the asteroids with respect to the Sun at
Julian day (TDB) 2440400.5 (June 28, 1969) in the ICRF2 frame. (Continued: 6 of 15.)
103
104

Hera
Klymene

106

Dione

107

Camilla109

Felicitas

x,y,z–1.624169502869726278 –2.061934384710811141–0.618220991356126959

vx,vy,vz

0.008815595236548105 –0.005135987476149960–0.002424171731408047

x,y,z

2.604625804466561778

0.829861198407666256

0.297592323238799750

vx,vy,vz–0.004595135750289872

0.008958088103164813

0.004449410220144689

1.186951890089662287 –2.641679471918139566–1.357909134882829605

x,y,z

0.004854178905816693

0.001679190825653463–2.814482356149371256 –2.125615900337749942–0.466315209535505482

vx,vy,vz

0.005016321991106533 –0.007283280755178675–0.001833852210438387

x,y,z–2.272223718831077210 –2.256013715954339638–1.348118693877923713

vx,vy,vz

0.005498298783944774 –0.004679670760771524–0.002914631877323399

vx,vy,vz

110

Lydia

x,y,z

x,y,z
vx,vy,vz

112
113

Iphigenia
Amalthea
Kassandra

115

Thyra

117

Lomia
Peitho120

Lachesis121

Hermione124

Alkeste127

Johanna
Nemesis
Antigone130

Elektra132

Aethra0.423210543438176212

0.263994487858637827

0.011003103680878389

0.005201699139327664

x,y,z–0.845649935641415329

2.041704300606039801

0.834445540016974907–0.010061995082559373 –0.004862835207405551–0.001044300650685426

x,y,z

1.003458535932775675

2.302229181420444970

0.755029808593063345

vx,vy,vz–0.010372485650250500

0.002522166795368615

0.001098397718920128

x,y,z–2.132402872023013263 –1.407030770831705802–1.230888073340032474

vx,vy,vz

0.005718496513543697 –0.006581478727881280–0.002888515609029340

x,y,z2.270382224122074355 –1.589433030843294015–1.113929084766832167

0.005926784306392254

0.005024462275567959

2.285956748706956976 –0.605001706866287292–0.594092775417243146

vx,vy,vz

0.001960243453552381

0.009608024580505914

0.005023973278452842

x,y,z

1.349895569447842503

2.571965525676553455

1.547065365110619428

vx,vy,vz–0.008469966093690717

0.003282200584511912

0.001536525822077163

1.334266785923234888 –2.574571237468854079–1.410921022512830580

x,y,z

x,y,z

137
140
141

65

Sophrosyne
Meliboea
Siwa
Lumen

0.006230326658274393

0.005070569717777809

0.001205938930940049

x,y,z–2.030985757945451553 –1.291015452637479966–0.500252551887808372

vx,vy,vz

0.008404347099665236

0.006742453819955392 –0.008540992687351152–0.003142188428784029

x,y,z

0.258596371131005198

2.254521103975096530

1.302780291910047072

vx,vy,vz–0.010847729029494223

0.000155121847092949

0.001051653942880798

x,y,z–1.950614158630362915

1.800692994261967828

1.063538206084451820–0.007974900712854933 –0.005725400853334638–0.001714798269305732

0.173250908378126567

0.822759553307659597

x,y,z

3.306268588514498230

vx,vy,vzz–0.008289119212281256 –0.000667809291620081

0.001070909483343232

x,y,z

0.808865897608793016 –2.931039369795729321–0.391590456276819932

vx,vy,vz

0.008508223312724572

0.004812568858903731–0.001548417459331940

x,y,z–0.480193840132689453

1.653722981508809253

0.319743825909667823–0.011837731432955296 –0.005329218981514342–0.007488534336320045

vx,vy,vz
134–0.004265106303010358

2.101268187945320776

vx,vy,vz
129

0.356482845934635972–0.000701525427796519 –0.008631983066179964–0.002085691485880794

vx,vy,vz

128

0.137720105704674162

x,y,z

vx,vy,vz
118–2.913671600037018194

vx,vy,vz
vx,vy,vz
114

0.008062335558122698

1.214628009571321776

0.731179432571293253

vx,vy,vz–0.007359702096515807 –0.006723867635909852–0.005074877395244866

x,y,z

x,y,z

–2.011298631478445564

2.185732502051722648 –1.145086389525992177–0.022766323557119300

vx,vy,vz

0.006166935083607356

0.010027540696906241

0.002522495066251339

x,y,z

2.349717450635818938

0.565740387115059251

0.098983118555951627

vx,vy,vz–0.000542241084460003

0.010767145886680794

0.004484878043585674

x,y,z

2.001263396461105071

0.319678076806489797

0.526316075837815478

vx,vy,vz–0.003483509074753528

0.010893891066401524

0.006389004515172658

Table 13. Initial positions (au) and velocities (au/day) of the asteroids with respect to the Sun at
Julian day (TDB) 2440400.5 (June 28, 1969) in the ICRF2 frame. (Continued: 7 of 15.)
143

Adria

144

Vibilia

146

Lucina147
148

Protogeneia
Galliax,y,z

2.882421698483517147

0.042452435493852865

vx,vy,vz–0.000265760440917905

0.008189549513733274

0.005447974152432384–3.202644502264647475 –0.708133625995427840–0.034778692392227049

vx,vy,vz

0.001746328337486271 –0.007334586431819497–0.003503786791915789

x,y,z–0.811168586625490406 –2.275206201114596816–0.838736686431402201

vx,vy,vz

0.010109426082854705 –0.002536990665401319–0.003737855276866437

x,y,z–0.135653717885019648 –2.925539832035890520–1.233404347151997538

vx,vy,vz

0.009577394154474565 –0.000281048647997959

0.000208962558114589–2.002954304100418081

1.803600909370170813

0.562676288618840337–0.007969397030290469 –0.006450526750365626

0.001741973397159357

x,y,z

x,y,z
vx,vy,vz

150
154
156
159

Nuwa
BerthaXanthippe
Aemilia

2.766574982586356590

1.080652555085234834

vx,vy,vz–0.009840605686461229 –0.000320540289433122–0.000310864131154414

x,y,z
x,y,z

–2.074683900631499434 –1.800589925977038375–0.921944736183594959

vx,vy,vz

0.006609578012061634 –0.005176399972958075–0.006376303490757747

x,y,z

2.923916135163274088 –0.729762768653841376

0.218492453225095001

vx,vy,vz

0.003649642497220172

0.007950919425059762

0.003317878320056731

x,y,z–0.410767564198755042

2.618954330535652186

0.940439073970279371–0.010553589480557164 –0.001942707083642232

0.000168561775632459

x,y,z

1.905452096025585851

1.518779824511213183

0.757828350346318791

vx,vy,vz–0.007533973461247138

0.007222883364505983

0.003809404100424659

x,y,z

0.262514031180226981

2.400281690284028269

1.275022530741664051–0.010860656605817488 –0.000837744879289118

0.000338095109459147

x,y,z

1.974914535093617252

0.857554972320634268

0.236073131153163474

vx,vy,vz–0.006815687100308090

0.009482254283590926

0.003447148757209308

vx,vy,vz
160
162

Una
Laurentia

vx,vy,vz
163
164
165
168

ErigoneEva
Loreley
Sibyllax,y,z–3.040157320355995285 –1.519642316497472834

0.691762796895534549

vx,vy,vz

0.002976277102337891 –0.005406627734211578–0.004499142864886430

x,y,z–1.922737034954617341 –1.819614502509028009–1.400331048398380185

vx,vy,vz

0.007959370240945991 –0.006004630422629679–0.001928143622055610

x,y,z3.116554420775199397 –0.994993474666817534–0.224327412952326943

vx,vy,vz
171

Ophelia

172

Baucis

173

Ino

x,y,z

Andromache

176

Iduna

177

Irma
Eucharis

185

EunikeKolga–0.255374648951711414
0.003396706876313383–0.620470933066101904

vx,vy,vz

0.000240618327164227

0.004378392174801965 –0.008170672938126655–0.004757814902613166

x,y,z0.171242886080394535

2.394184022514683097

0.451640900560661918

vx,vy,vz–0.011142161189368808

0.002538588490091165

0.001950356324201864–0.474252178163027949 –2.648333222299060541–1.305222351194791353

x,y,z
vx,vy,vz

0.010262122260410241

0.000548159272351288

0.000029475418796883

x,y,z–1.387628365703582611

2.833854776034395506–0.091557754074181458

vx,vy,vz–0.009225262571539295 –0.002671178335674209–0.001428340199372592

x,y,z

2.007796292614895339 –1.022926044579357097–0.462360572116189250

x,y,z

0.004104738603731208

0.010480904746312239

0.004870438746756638

2.120681612712470798 –2.907144330458012238–0.813722768281753606

0.005865675784688143–0.000342831237219019

0.005572377170526862

x,y,z–2.747368952622216920 –1.338746669764243169

0.422267924686840546

vx,vy,vz

0.003978039639761641 –0.008185576126933353–0.001011821450492685

x,y,z
vx,vy,vz

66

0.003163655969657688

3.452516715258505897 –0.216190515721613213
0.008055044745482545

vx,vy,vz

191

0.008691053371313727–2.385328686094084105 –0.620107743369059716

vx,vy,vz
181

0.002733580426163316

x,y,z

vx,vy,vz

175–0.513703300992534251

0.333727709970750852–2.879887570364938920

0.953404430384273138

0.413481905229524382–0.003754265666355869 –0.008618960919399883–0.001665556221000780

Table 13. Initial positions (au) and velocities (au/day) of the asteroids with respect to the Sun at
Julian day (TDB) 2440400.5 (June 28, 1969) in the ICRF2 frame. (Continued: 8 of 15.)
195

Eurykleia

196

Philomela

198

Ampella

200
201
203

Dynamene
Penelope
Pompeja

205

Martha206

Hersilia

209
210
211

Dido
Isabella
Isolda

212

Medea

213

Lilaea

221

Eos

223

Rosa

224

Oceanax,y,z

2.922594899011714986

0.199160094596518983

vx,vy,vz–0.001056479499979637

0.008538247223234249

0.005019069716262851

2.870866810351882403 –0.754636097994081934–0.748530242754146369

vx,vy,vz

0.003302144455379505

0.008603231629111158

0.003663627801932320

x,y,z

0.543891836565843279

1.941671690331441047

0.932327670839411482

vx,vy,vz–0.010661276142588431

0.005661564198512067

0.000542576864013208

x,y,z–0.109999438475556419 –2.621860198978502865–1.470414945946262097

vx,vy,vz

0.009376565385604106

0.000133656901059443

0.000810938484734692

x,y,z–2.800980089172922849

1.352246370925753505

0.554783862366244662

vx,vy,vz–0.004186345686485028 –0.007367963765518988–0.002243604557722191

x,y,z

1.500547182112841327 –2.086139267712977308–1.025020464889739635

x,y,z

vx,vy,vz

0.008286480532474490

0.005407135390513565

0.002806033412514854

x,y,z–2.770579982860186696

0.663732478579504792–0.116643403811919136

vx,vy,vz–0.002353576283912810 –0.009404587572302536–0.002658971716332037

x,y,z

1.238747961716257118 –2.376780403456169921–0.929349158559885868

vx,vy,vz

0.008918338308673593

0.004410268226602005

0.001280757530674212

x,y,z

2.268898815817652626

2.057453499016182352

1.209244033679382824

vx,vy,vz–0.006436569904961379

0.005714772045042667

0.003406952267267044

x,y,z

2.231740920050496513

0.842042304704040623

0.316984455862986181

vx,vy,vz–0.004831339830972945

0.009318806983407490

0.005200751457433266

x,y,z

2.711875240815070143

0.430216506380016683

0.382482385748132425

vx,vy,vz–0.003510234716571616

0.009467947330119904

0.003781089466930833

x,y,z–1.966750173220412679 –2.504391441888097258–1.362061306592617482

vx,vy,vz

0.007097449121202490 –0.004667784575521242–0.001897190061895615

x,y,z

2.419306968138564606

1.377064626174480333

0.236934520281399275

vx,vy,vz–0.003663586080577670

0.008842945171148394

0.003566597801239553

x,y,z

1.777939852260157272

2.264787669587976726

0.385115431224176685

vx,vy,vz–0.007376964281696770

0.006641442036154265

0.002628969249503958

x,y,z–2.914424443137129561 –0.575564316487753080–0.183166473580308381

vx,vy,vz

0.000831640269428134 –0.009184748787445272–0.004253682939508904

x,y,z

1.306840552514852316 –1.909320111535164788–1.052599637885983652

vx,vy,vz
225

Henrietta

227

Philosophia

x,y,z
vx,vy,vz
x,y,z
vx,vy,vz

233
236
238

Asterope
Honoria
Hypatia240

Vanadis241

Germania
Eukrate0.004737845515644764

0.002779620693421142–2.528098762364009211 –1.638702365555180007–0.383411150052558269

0.007735421152555965 –0.006833347198704122

0.000439444841229691–1.687868923923265596

2.423841903625734151

1.306304444440689938–0.006774671779471052 –0.005284654892018608–0.003871706711383315–2.905158595265389554

0.237101371710457365–0.201153100576835192–0.000352351008013982 –0.009087108811730400–0.002949940598791083

x,y,z–0.748198375227333456 –2.637125772058928508–0.758761586278685618

vx,vy,vz

0.010085435669044992 –0.000778370444183047–0.000057970932084369

x,y,z

2.410309982055119260

1.056479210705921545

0.250822081348433867

vx,vy,vz–0.004638599803102487

0.009855687835584624

0.001848375937922427

x,y,z–2.921723067002446772 –0.802623949197874942–0.228455192343604230

vx,vy,vz

0.001119414849423201 –0.008380246775381286–0.003517310270921982

x,y,z

2.157026156705858178 –1.650363596987127091–0.494242367750019174

x,y,z

x,y,z
vx,vy,vz

67

0.009528257077708791

vx,vy,vz

vx,vy,vz
247

0.060339478431695619

0.006296839602484864

0.007812769885222028

0.004076824994372825

2.169125903092219421 –0.866225171773255864–0.996634072113313341

0.003425982013346110

0.007955063682203814

0.007075076118810717

Table 13. Initial positions (au) and velocities (au/day) of the asteroids with respect to the Sun at
Julian day (TDB) 2440400.5 (June 28, 1969) in the ICRF2 frame. (Continued: 9 of 15.)250

Bettina

x,y,z

259

Aletheia

266

Aline

268
275

AdoreaSapientia

276

Adelheid

283

Emma

0.005535308622116215–1.627318570822305688 –2.150337559305796731–0.609408552011190818

vx,vy,vz

0.008749058066660119 –0.005149978486112262–0.004087199673172721

x,y,z

0.229960468787527295

2.407979990359146338

0.736809718189577856

vx,vy,vz–0.010856512774706088

0.002950023549757955–0.001382815037831516

0.639749669712347591 –2.794617398822625276–1.162862315045730011

vx,vy,vz

0.009746292661459431

0.000795802448212805–0.000055548373269038

x,y,z

2.660106410012033074

1.524565090452046245

0.389849845901316994

vx,vy,vz–0.005534317791085139

0.006792087695683437

0.002829793913036671

x,y,z–1.240280742637663192

2.649029697867973265–0.023688532806399230–0.009129755737320335 –0.004387217468802240–0.002182311342655559–3.467100975427897414

0.326845469549111034–0.273921918861556368–0.000408626057531563 –0.007480204767684386–0.004047747264448866

x,y,z

2.272270153229304146

0.682382283976578408–0.060915050989619822

vx,vy,vz–0.002758331417138625

0.010301650913710764

0.003144420571901631–2.714456487118279693 –1.569383554448778861–1.007607012785439293

0.004977413077900846 –0.006790226956868332–0.003754626824227870

x,y,z

x,y,z
vx,vy,vz

Nephthys303

Josephina

304

Olga

x,y,z
vx,vy,vz308
313

PolyxoChaldaea

322

Phaeo

326

Tamara

328
329
334

GudrunSvea
Chicago0.841743117969856125

0.394982651347284053

vx,vy,vz–0.002248382084316066 –0.008561565126572737–0.001074438237890270

x,y,z–1.042886921098909125 –2.332650336151194548–0.811036990846228489

vx,vy,vz

0.009903231347078257 –0.003677357201775901–0.001239523162032258

x,y,z

0.771884768362409268

1.939556783718940913

0.398378002792229235

vx,vy,vz–0.012098689536781022

0.002656162671110900

0.000678852180394095–0.002682348362129571 –2.507699408031491384–0.970531169598375820

vx,vy,vz

0.010244749836147331

0.001937251500163253

0.002228295777838911

x,y,z–2.071626956515489226

0.579660636027930942

1.193463031425092469

vx,vy,vz–0.003544442645081844 –0.007757868782935538–0.006335496904282256

x,y,z–2.573274034716255709 –1.465047934437442967–1.316844023039191125

vx,vy,vz

0.004832558218741573 –0.006280166157985758–0.004956463301670980

x,y,z

x,y,z

Roberta336

Lacadiera

338
345
346
347
349

68

Budrosa
Tercidina
Hermentaria
Pariana
Dembowska–2.768985967086891264

2.004856633119884712

0.272499885424718813

vx,vy,vz–0.008719664332914103 –0.006477652467348243–0.000805394135272350

x,y,z

3.443962847809292072 –1.324275053387580003–0.714472534127118086

x,y,z–1.442616581613682669

0.007676913891350206

0.002591553586693644–1.377939389741577925 –1.619477614777314223–0.493517119258204406

vx,vy,vz

0.010492320292650055 –0.005901393082463712–0.002553933297352890

x,y,z–0.658098578079305607 –1.795601132283736634–0.719216342800300446

vx,vy,vz

0.011950088864493176 –0.003907291040290498–0.000403494815576562–2.903591948167865588

0.470290381099696342–0.099200302996376888

vx,vy,vz–0.001425800173495607 –0.008860750970051479–0.004359767689390367

x,y,z–0.007889583388479199 –2.379738144165251423–0.650011796314688195

vx,vy,vz

0.010566766359888404 –0.000201418715010861

0.000963910315785457

x,y,z–2.982837107945091493 –0.700289625004069438

0.200506563052633757

vx,vy,vz

0.001487506028262646 –0.008384396621244858–0.003812447082362887

vx,vy,vz
335

0.000132313329422579–0.517813104094458687

x,y,z

vx,vy,vz

287

3.111197416227221346 –0.213218630746829368
0.007905211075541855

vx,vy,vz

x,y,z

x,y,z

0.003929882068712210

1.912864525007107019

1.057420519699953543

vx,vy,vz–0.010024705149477772 –0.006143851860785796–0.000494836214563871

x,y,z–3.156748018124499122 –0.130959673144165373

0.211799005012422992

vx,vy,vz–0.000235136952592180 –0.008001539901820919–0.004667985924570656

x,y,z–0.865269383843656370

Table 13. Initial positions (au) and velocities (au/day) of the asteroids with respect to the Sun at
Julian day (TDB) 2440400.5 (June 28, 1969) in the ICRF2 frame. (Continued: 10 of 15.)350

Ornamenta

356

Liguria

357

Ninina358

Apollonia

360

Carlova

362

Havnia

363

Padua

365

Corduba366

Vincentina

369
373
375
377
381
385

Aeria
Melusina
UrsulaCampania
MyrrhaIlmatar386

Siegena

387

Aquitania

388

Charybdis

389

Industria

x,y,z

1.897492886578636018

1.853115584753373568

vx,vy,vz–0.006802174405994125

0.006550715887629770

0.006201849595058421

2.403925905531135054 –1.080394157528443255–0.637740553547901712

x,y,z
vx,vy,vz

0.002474022842116686

0.008698179268340399

0.005392308524943403

x,y,z2.550147267653644079

1.285776955010184519–0.179678885435631430

vx,vy,vz–0.004469547804037060

0.009229935506104934

0.002753264074857219

2.637397004060416439 –0.033680969059450810–0.033009209749240563

x,y,z
vx,vy,vz–0.001238942541646334

0.010294117142742979

0.003742764415514056

x,y,z

2.365965905770136768

0.946364028961979908–0.104711173321544837

vx,vy,vz–0.005043923473529931

0.009797484839798252

0.003463941512246718

2.006016399583006038 –1.335233497908418832–0.941832432474522863

vx,vy,vz

0.006316106482288478

0.007613525568562483

0.004015883178030646

x,y,z

0.447395563619500725

2.418709764219945324

1.128484276493210148

vx,vy,vz–0.010190467851460631

0.001811104511745787

0.001955672995458056

x,y,z–0.946605401455762441 –2.952167141736141698–0.579774951531645599

vx,vy,vz

x,y,z

0.008854160018258806 –0.001893748322970214–0.000150172926506452

x,y,z

0.181803553220394987

2.727350474248680712

1.834488045213377383

vx,vy,vz–0.009184575428991438

0.001029954077974214

0.000256959389362181

x,y,z–2.461889264933398547

1.041415912802037491

1.026022247881923644

vx,vy,vz–0.004941491515371925 –0.008124405678199062–0.002138953770784814

x,y,z–0.690418739044735830 –2.379811252254516063–1.898265743321774224

vx,vy,vz

0.009694322965289898 –0.000476657325627097–0.000644192905393674

x,y,z–1.436446171450740827 –2.028484833217823802–1.788540836866821859

vx,vy,vz

0.009100906971536040 –0.003641283355002899–0.001642880584566145

x,y,z

0.269537079909927590

2.385864264311093219

0.778604188267153829

vx,vy,vz–0.011005666135096655

0.001823320353539882–0.000108001402849965

x,y,z–2.995154522909969863 –0.463362473245098105

0.425103778189245829

vx,vy,vz

0.002132868519734355 –0.009335695766624439–0.003072148504579520

x,y,z

2.204123556353274616

1.619262542204616961

1.359899523537142940

vx,vy,vz–0.007197099139091408

0.005191410685294344

0.003343985294701643

x,y,z–1.541442636926875664 –2.888691935130070387–0.066078605278559938

vx,vy,vz

0.008201852800212111 –0.003247965605600923–0.000826380013421358

x,y,z

1.673856039456203204

2.530440646483403278

0.105238509303274774

vx,vy,vz–0.006165312103874345

0.006354675190666645

0.002907264021179044

2.320935330770025118 –1.396656836528137102–0.777937804577914949

vx,vy,vz

0.005967878297330536

0.007526833190292071

0.004390084907098427

x,y,z–0.690870585159074957

2.285976832122341484

0.972909194639388297–0.010061709744287462 –0.002641850906012974–0.002795471716741464

x,y,z

vx,vy,vz
393
404

Lampetia
Arsinoex,y,z0.425228759538625878

3.223104999989440334

0.691457827571683836

vx,vy,vz–0.007669952102573410

0.003507362513697180–0.000448442801262118

x,y,z–1.528713317148434614

1.538431701390180706

1.056814160805905178–0.006833201218737411 –0.008984287880902874–0.001895810188518365

vx,vy,vz
407
410
412

Arachne
ChlorisElisabetha

1.294823426614003070

0.330302008894303445

vx,vy,vz–0.005206693346373313 –0.007456531256292877–0.004441540417027833

x,y,z–1.920013803213842740 –1.236776042431182887–0.102806516019325112

vx,vy,vz

0.007955416951132228 –0.007973149544052796–0.004854681062882307–0.660465882532509707

2.488779742331803213

1.036836462101317080–0.009611438891560443 –0.003517675356754544

0.001146633004327251

x,y,z

x,y,z
vx,vy,vz

69–0.168626022505561857–2.404168202561030032

Table 13. Initial positions (au) and velocities (au/day) of the asteroids with respect to the Sun at
Julian day (TDB) 2440400.5 (June 28, 1969) in the ICRF2 frame. (Continued: 11 of 15.)415
416

Palatia
Vaticana

2.248246074358777769 –0.101981217325653323–0.299618390748721652

vx,vy,vz–0.002060353514605755

0.011553860188848762

0.004093646216743197

x,y,z–0.968717353915249380

2.586243865176183654

1.726458358232957346–0.007646026045771258 –0.004063877953498320–0.000635412624451350–3.434992155705478112

x,y,z

vx,vy,vz
420
423
424

Bertholda
DiotimaGratia0.152957066714952405–0.330353631186631247

vx,vy,vz–0.000402294197824229 –0.008601469926580190–0.003283044092787496

x,y,z–1.836130012842030368 –2.232281627979681993–0.765829327150786887

vx,vy,vz

0.007840938442101581 –0.004797111494305463–0.004122385161061739

x,y,z

2.554859642210790316 –0.721209544624070054–0.683662530131363688

x,y,z

0.009552423334085201

0.003497050935833218

x,y,z–2.022547942292763690 –1.005817143532471203–1.399553685672040970

vx,vy,vz

vx,vy,vz
426

Hippo

431

Nephele432

Pythia

433

Eros

442

Eichsfeldia

444

Gyptis445

Edna0.005977729090057320 –0.008172230606490322–0.004212443868015491

x,y,z

2.126339087775298786

1.733215636283283123

0.656190187417207027

vx,vy,vz–0.005642781181932892

0.008360067976273988

0.003651458241242074

1.351686874096238222 –1.350586724406483530–0.907107472146849059

vx,vy,vz

0.009952768047903163

0.007416635165718826

0.000909632185035547

x,y,z

1.052635242977920882

0.652108246501082123

0.560644019635677004

vx,vy,vz–0.011649113662952394

0.009302582981661403

0.003194192782337934

x,y,z

2.438069471682043687

0.500402393682477853–0.018372380659463668

vx,vy,vz–0.001598988743082291

x,y,z

0.009827743844217883

0.003532527264151718–0.122162596591544409 –2.551357499130834761–0.622759399088688381

vx,vy,vz

0.010779467918873202

0.001166454438831065

0.000838720795358274

x,y,z–2.638867450404819515

2.337072213589934222

0.386737481707201902–0.005987522043549823 –0.003764390073667395–0.004911559441873959

x,y,z

1.415715061146446185

1.814956624851498335

0.711298139664957252

vx,vy,vz–0.010126512689018213

0.004544417521285093

0.002585888565144427

1.286986769213538828 –1.948980832970361732–1.156799971769590929

x,y,z

vx,vy,vz
449

Hamburga

454

Mathesis455

Bruchsalia

464

Megairax,y,z

0.003999746415627610

0.001524892144668299

x,y,z–2.056825944515058957 –2.004397979321892453–0.509262841989271742

vx,vy,vz

0.008122040874298211 –0.003536097158237312–0.003611496524712439

x,y,z

0.667444028746538631 –2.252197234941088144–0.993641094169664063

vx,vy,vz

0.009943729528332137

0.005243255029627681

0.000165611503527442

x,y,z

3.485008134627705534

0.426960893387227702

0.475918006579878372

vx,vy,vz–0.000224571561215830

0.007624192788766146

0.003685169590851310

x,y,z–2.667676861175595882

1.739553822505467640

0.048087007997351904

vx,vy,vz–0.004238745317538407 –0.006912875729415020–0.005683670800648313

vx,vy,vz

465
466
469
471
476
481
485
489

AlektoTisiphone
Argentina
Papagena
HedwigEmita
Genua
Comacina

0.009804719465114024

x,y,z–2.277066624323161648 –0.948290331570841527–0.878659731572872693

vx,vy,vz

0.004890309880280551 –0.008845224571058706–0.005480291439275048

x,y,z–2.365805692852444686 –2.593457491934213266–0.508479129862948076

vx,vy,vz

0.005830421752087237 –0.004151641950916174–0.003631216999264367

x,y,z–1.681308000287230264 –1.540816635314435912–1.120990466424186094

vx,vy,vz

0.008429343749817427 –0.006873524681428616–0.001716598086040729

x,y,z

0.974450886978483366

1.931290124495535832

0.819035276347056418

vx,vy,vz–0.010731270482033605

0.004061820639447179

0.004026303026251832

x,y,z–0.684382743244631553

2.097786304663209034

0.329093928144473968

vx,vy,vz–0.011962823721801429 –0.003569573771336420–0.001358782018689436

x,y,z

2.506508082475853438 –2.066222307541637271–0.511544638179387667

0.006047076129268948

0.001033759739079085

vx,vy,vz

700.002391618765227667

0.006964815651859941

Table 13. Initial positions (au) and velocities (au/day) of the asteroids with respect to the Sun at
Julian day (TDB) 2440400.5 (June 28, 1969) in the ICRF2 frame. (Continued: 12 of 15.)
490

Veritas

x,y,z

491

Carina

498

Tokio

503

Evelyn

505

Cava

506

Marion508

Princetonia
Armida0.000202399971558074–0.225387122172941129

vx,vy,vz

0.008172836509484513 –0.003408744893633426–0.000454450885764238

x,y,z–3.141633394765953380 –0.147256452035157803

0.502336236774096112

vx,vy,vz

0.000707169938473929 –0.007915881021043106–0.003339475619994648
0.196502828408131963

x,y,z

2.469457681649797731

0.897054685380899164

vx,vy,vz–0.005320146486079764

0.008280168851000714

0.004381003781921793

x,y,z

2.051381744115417849

0.715819257149662769–0.079692452727947352

vx,vy,vz–0.005500358097765526

0.010138687983931836

0.005377869001808236

2.338637506038451885 –2.236463709683592427–0.970896012702498035

vx,vy,vz

0.005454153770407211

0.004892127964045382

0.004916880315758551

x,y,z–1.834398047688895250

2.033726007866053731

1.680737578756668515–0.007530022629651348 –0.005443148627532285–0.002038851357460140

x,y,z

2.342984019721451094

0.883644880995315685–0.008098171175524310 –0.004726881362335257–0.002646867999807014

x,y,z

3.099051382763043705

0.296225047046435153

0.633503755281144221

vx,vy,vz

0.000113246728749009

0.007155674865686408

0.004973978514489579

0.558320524452791234 –3.192588255468263636–1.370798827031080469

x,y,z
vx,vy,vz

516

Amherstia

517

Edith

521

Brixia535

Montague

x,y,z
vx,vy,vz
x,y,z
vx,vy,vz

536
545

Merapi
Messalina
Praxedis

566

Stereoskopia

568

Cheruskia

584

Semiramis

585

Bilkis591

Irmgard593

Titania

595

Polyxena

71

Scheila0.001503486262573605–0.504053344408389448

0.005060051620143621 –0.004986735600834600–0.003179245956564224
0.617355942561913063–0.003755534201261866

x,y,z–2.750766666063489296 –2.439053500410161668–0.616255994545442776

vx,vy,vz

0.005689626184352026 –0.004264313174264865–2.316646842841627585–0.004857657183703378

1.764074066249742057

0.869124506509039163–0.003538173057074548 –0.006065837986010540–0.004381735842294599

x,y,z

1.676299422983591914

1.266883169004290322

0.274076519007798947

vx,vy,vz–0.007644163901651690

0.010654276917228497

0.000771636508649015

3.108217299055801242 –0.038924752084866489–0.306684557850228134

0.000163401251640635

0.009232636979490162

0.004135544106424715

2.499292317070373137 –0.704732562382201388

0.596817158342859422

x,y,z

x,y,z
x,y,z

–3.088249528284030543

0.010401955137544539

0.003335461022407042–0.314868951447094259 –2.848132605531735528–1.283448671165868582

vx,vy,vz

0.008734805915040946 –0.000862116582237407–0.000205188117724297

x,y,z–2.503710281861984743 –0.998160106756301402–0.996565071101295330

vx,vy,vz

0.004900032034025778 –0.007159666424074737–0.002463187315737590

x,y,z

1.912791286329316831

1.539478930510618904

0.440912149264970965

vx,vy,vz–0.007907529478716448

0.006999464858132820

0.001976597919510323

3.001474010271674153 –0.872392399734546298–0.271674947450289361

x,y,z

x,y,z

0.000384461732460371

0.006515819416529907

0.004916384720291039–0.426142954823842124 –2.875072915506421189–1.351850274140791841

vx,vy,vz

0.008043320855626214 –0.000800619582345174–0.003093655045037652

x,y,z–2.275279297558304492 –1.787230083745405596–1.091452381026450702

vx,vy,vz

0.006946976359411234 –0.004847860891067480–0.005228807026657581

x,y,z

2.068249321573481403

2.559630685610615686

0.793577660766494164

vx,vy,vz–0.006190047319295022

0.004343781071425732

0.004070640552158299

vx,vy,vz

596

0.008142074640245637–0.003949779425587768 –0.009578278138314685

vx,vy,vz
Misa

0.002287166228260473–2.549496878032234903 –2.363059656332096470

vx,vy,vz

vx,vy,vz

569–1.795059498387787666

0.712595568810880153

x,y,z

vx,vy,vz
547

0.009313455279943360–0.801007823894895110–1.361895485427903063 –3.165505741951435414

x,y,z

vx,vy,vz
514

0.036248932914226248 –3.171478757627256950
0.000902292183295022

vx,vy,vz

x,y,z

0.003412279465211854

Table 13. Initial positions (au) and velocities (au/day) of the asteroids with respect to the Sun at
Julian day (TDB) 2440400.5 (June 28, 1969) in the ICRF2 frame. (Continued: 13 of 15.)598

Octavia

x,y,z
vx,vy,vz

599

Luisa
Marianna604

Tekmessa

618

Elfriede

623

Chimaera

626

Notburga

663

Vundtia
Gerlinde

667

Denise674

Rachele1.640818403618105270

x,y,z

2.236421484533403614

0.256457028560368117

0.531502491866831517

vx,vy,vz–0.002987755039934792

0.010015170339145833

0.007115827480249712

x,y,z–2.443492364975899278 –2.553672554973274167–1.297289046740899066

vx,vy,vz

680
683

Genoveva
Lanzia690

Wratislavia

691

Lehigh

694

Ekard

696

Leonora702

Alauda705
709

Erminia
Fringilla

712

Boliviana713

Luscinia

0.006146762368787255 –0.004416770321035214–0.002442816546685167

2.516097127945998935

1.510176140872574457–0.283377085143805829

vx,vy,vz–0.004157210144504098

0.008684469448787927

0.003885209090645422–2.409466239800548415 –0.465525463994251487–0.851563289724215688

0.002342861813000729 –0.008762166195659019–0.005029967766286956–3.035450076806275543

0.421864008988716566–0.127583860826996742

vx,vy,vz–0.001866697145881276 –0.005612944658928348–0.006561241265170950

x,y,z

2.006725589468375315 –2.376415931764492395–0.496002046851244771

vx,vy,vz

0.006841157951777630

0.006652586659241708

0.001565035661244263

x,y,z

2.983897565601330903

1.397628190931970993

1.091574612026383173

vx,vy,vzz–0.004331655563956410

0.007414701609598470

0.000530966741235238–2.421777915332062658 –2.295475988992165650

0.452580813773379109

0.004815408543036114 –0.007668419938770380–0.000993922465387279–0.235827957791626630

2.008846377619600609

1.239571728159498454–0.011860879067687655 –0.002375626722544034

0.001425555621865033

x,y,z
x,y,z

x,y,z
x,y,z
x,y,z

0.001979483768247084

2.107337311005498925

0.870011327043579530

vx,vy,vz–0.012117274959326541

0.001965172863672976–0.001430686736150516

x,y,z–2.483498644334699357

1.961237812146513715

2.122017516184655506

vx,vy,vz–0.004535242841718457 –0.005563771866247991–0.003051609814162152–1.100369444712229416

x,y,z–0.038757248577513628 –2.929038122705282188

vx,vy,vz

0.009189739037992875 –0.001698156106930829

0.002602487296413445

x,y,z–1.572098580880649132 –2.754041913568673561–1.335506562200244884

vx,vy,vz

0.008204336231111772 –0.003237602143731363

0.000474631125854728

x,y,z–2.600679234384931426 –2.131333554500724148–0.280098141039464121

vx,vy,vz

0.005334054469881479 –0.005798095800458341–0.003891386736724764

x,y,z–2.073828833200439092 –1.723446996059183167–0.871272150977313808

vx,vy,vz

0.008389896606921710 –0.005221711697173815

0.000631908650981562

x,y,z–2.528679514939249806 –2.344092579990681102–1.934926268906861813

vx,vy,vz

0.005658575767356208 –0.004750757197783061–0.001445326359372374

x,y,z

0.731086409189228292

2.576019002727282015

1.840680315316418403

vx,vy,vz–0.008610612355498393

0.003701586218457099–0.001273396941220973

1.527810127063314871 –1.738330529507490452–2.020057855746948849

vx,vy,vz

0.008204195294184936

0.003409657750114686

0.003528767177190933

x,y,z

1.932009598635754166

1.322621695446154488

1.386519279952832973

vx,vy,vz–0.006619431762339829

0.007378506450047627

0.004203577713780057

x,y,z–2.181214001606280206

1.349361062281813073–0.026299365363868094

vx,vy,vz–0.006970304527136179 –0.007493618475270048–0.003330760378138190

x,y,z

0.602542358281345058 –2.935328109946095054–0.752927362734930772

x,y,z

0.002849533330894416

0.001959410111945836

x,y,z–2.970874542187170952 –1.503348928425136677–0.336696621923833705

vx,vy,vz

0.005083914736085787 –0.004623872868731655–0.004610217116933856

vx,vy,vz
735

72Marghanna–2.896166748961349846

x,y,z

vx,vy,vz
Ludmilla

0.000525696328445862–0.002708296082629357

vx,vy,vz

675

1.174114109004084749–0.004659866001601793 –0.005499407621023489

vx,vy,vz

635

1.946258357215855606–0.009625203890116693 –0.004100840847587676
1.273195801188978793

x,y,z
vx,vy,vz

602–1.508523042247438273

0.009618230119085674

Table 13. Initial positions (au) and velocities (au/day) of the asteroids with respect to the Sun at
Julian day (TDB) 2440400.5 (June 28, 1969) in the ICRF2 frame. (Continued: 14 of 15.)739

Mandeville

2.506148925696726870 –1.504824362532233950–0.834400633180084061

0.005428727641429763

0.007553927772582291–0.000302902354564315

3.314248241247279569 –0.064714374102698397–0.617044240710776459

vx,vy,vz

0.000448233529388049

0.008426445083442220

0.002733213589088293

x,y,z–2.301643137177721954

0.965649183773619191

1.182110401173152026–0.005854981025964248 –0.007733688166168566–0.002054122576185711

x,y,z

2.048812749653873055

1.241773705314024756

0.317962157290045455

vx,vy,vz–0.006453759688972923

0.008038950160103676

0.004297655236858543

3.670126642304055498 –1.151050350627846441–0.367902633175755500

x,y,z
vx,vy,vz

740
751

Cantabia
Faina

x,y,z

vx,vy,vz752

Sulamitis

760

Massinga762

Pulcova769

Tatjana772

Tanete773

Irmintraud

776

Berbericia

x,y,z
vx,vy,vz
x,y,z
vx,vy,vz
x,y,z

784
786

Armenia
Pickeringia
Bredichina

788

Hohensteina

790

Pretoria

804

Ani
Hispania
Tauris

849

Ara

895

Helio

909

Ulla

914

Palisana

980

Anacostia

0.000346838301451201

0.647310471297706802–1.267907728397291933

0.001010921533345003

x,y,z
x,y,z

0.007595842568837846

0.005064697781475233–1.324784143882017373 –1.754236649382118429–1.608060869413702854

0.009439627055445330 –0.004735182301384725–0.001496436978411633–2.430375861485484013

1.244119991480070464

1.499556777478691094–0.006193479474740193 –0.007012659876802435–0.001255056722427736

x,y,z

2.578770540715112691

0.484679459667908874

0.746658068287141652

vx,vy,vz–0.005606680738267385

0.008333234968279164

0.004809669391709696

x,y,z–2.947260333629879270

1.443734931655717313

0.745070083337502465

vx,vy,vz–0.004292145745643914 –0.007903853241575533–0.000226820581777773

x,y,z–1.216179274985955239 –1.725991913871303796–1.130983048638923849

vx,vy,vz

0.010795481346361395 –0.004416936010313539–0.003972330176354399–0.640973967234059860

1.482664510763615384

x,y,z

3.011946086754773155

vx,vy,vz–0.008114458019644099 –0.003635735760200800

0.000726237470920756

x,y,z–2.065161763450813925 –1.731735607044430303–0.269841031252473607

0.007166218568798618 –0.008400928402379429–0.001362268879167860–3.312660708096480278

0.159833898455113310–1.180393653213523963

vx,vy,vz

0.001800283505532356 –0.008606461983801833–0.001971983915071157

x,y,z–0.693006042192073846 –2.790965988787078800–0.471075353887803139

vx,vy,vz

0.010149322918663708 –0.000178409023167429–0.002316736158141167

x,y,z

1.180236347016857978 –1.755994298920930508–1.316225480612562171

x,y,z

0.004495021926921983

0.004245447086389762–2.643337763228612580 –3.043511779699754172–0.187636072682489108

vx,vy,vz

0.005375474685596784 –0.003186610433694088–0.003752983540055900

x,y,z–2.400436417716211412 –1.377376257048356312–0.903371236042467940

vx,vy,vz

0.006630709229090170 –0.008101548494299364

0.000290406058693144

x,y,z–1.153864786573528756 –2.970641315024131757–1.742880474756495213

vx,vy,vz

0.006900378729138242 –0.004383767421712856

0.001915953866110696

x,y,z

2.643585779564476823 –1.912997619870260957–0.715076842818698211

x,y,z

vx,vy,vz
x,y,z

0.009748138526187236

0.005129680510801807

0.008209415548612537

0.000169376652986247

0.598245936709027260 –1.827975513352569470–0.273954790847752927

vx,vy,vz

0.011446713245858879

0.003314424689681761

0.006488303174133064

x,y,z–3.175758333328849048

0.485972795063543628–0.719059100789527061–0.000142109533888624 –0.007460640782612650–0.004030038096555623

vx,vy,vz

73

0.004880661946113063–1.333719406992519119

0.002495895480057548

vx,vy,vz814

0.007234201884231036

0.509160148400431334 –2.303342379809931284
2.906025830739782823

vx,vy,vz

791

0.002109743357951093
0.010933982442492730

vx,vy,vz

780

0.004396658729641872–0.047622057373297624

x,y,z

vx,vy,vz

Theobalda

0.005888743311020824

3.175342326316428654 –1.199205892203630830

vx,vy,vz
vx,vy,vz

778

0.002369603147829039

Table 13. Initial positions (au) and velocities (au/day) of the asteroids with respect to the Sun at
Julian day (TDB) 2440400.5 (June 28, 1969) in the ICRF2 frame. (Continued: 15 of 15.)1015 Christa

x,y,z
vx,vy,vz

1021

Flammario

x,y,z
vx,vy,vz

1036
1093

Ganymed
Freda

1107

Lictoria

1171

Rusthawelia
Mashona

1.085692881000842158–0.000110280407398696–2.936407649398657060

0.239975615662415959

0.837347923291067331–0.003624193796013200 –0.008360760098837189–0.001485430871488255–2.506653200603659926

2.486748461463569093–0.616157838784046952–0.006637194622358507 –0.002518711399848999–0.001898585862499638

x,y,z–2.277804058924099539 –1.843336426423987318–0.338410936166434473

vx,vy,vz

0.007229516437050462 –0.003691334503305957–0.006389991099609126

x,y,z–2.569057933925188664 –1.937453059103700248–0.421229684865389586

vx,vy,vz

0.004720513971195381 –0.007449150358726622–0.003422150642625780

x,y,z

2.560542567362555477 –1.071816228353758005–0.550580177371690072

x,y,z

x,y,z
vx,vy,vz

74

2.568332206936112527–0.008675523560380710 –0.004175198720726467

vx,vy,vz

vx,vy,vz
1467–1.566656583058677832

0.002594814272586507

0.009717147787535695

0.003755086238400567

0.966768996929490920 –2.152669618944333507–1.688772902845859569

0.009795683288059361

0.004184325487804071

0.001634297002254343

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