REPORT OF THE IAU/IAG WORKING GROUP ON
CARTOGRAPHIC COORDINATES AND ROTATIONAL ELEMENTS OF
THE PLANETS AND SATELLITES: 2000
P.K. SEIDELMANN (CHAIR) V.K. ABALAKIN M. BURSA M.E. DAVIES C. de BERGH J.H. LIESKE J. OBERST J.L. SIMON E.M. STANDISH P. STOOKE P.C. THOMAS Abstract.
Every
three years the IAU/IAG Working Group on Cartographic Coordinates
and Rotational Elements of the Planets and Satellites revises tables
giving the directions of the north poles of rotation and the prime
meridians of the planets, satellites, and asteroids. Also presented
are revised tables giving their sizes and shapes. Changes since the
previous report are summarized in the Appendix. Key words: Cartographic
coordinates, rotation axes, rotation periods, sizes, shapes 1.
Introduction The
IAU Working Group on Cartographic Coordinates and Rotational
Elements of the Planets and Satellites was established as a
consequence of resolutions adopted by Commissions 4 and 16 at the
IAU General Assembly at Grenoble in 1976. The first report of the
Working Group was presented to the General Assembly at Montreal in
1979 and published in the Trans.
IAU 17B, 72-79, 1980.
The report with appendices was published in Celestial
Mechanics 22, 205-230,
1980. The guiding principles and conventions that were adopted by
the Group and the rationale for their acceptance were presented in
that report and its appendices will not be reviewed here. The second
report of the Working Group was presented to the General Assembly at
Patras in 1982 and published in the Trans.
IAU 18B, 15 1 162, 1983, and
also in Celestial Mechanics 29,
309-321, 1983. The third report on the Working Group was
presented to the General Assembly at New Delhi in 1985 and published
in Celestial Mechanics 39,
103-113, 1986. The fourth report of the Working Group was
presented to the General Assembly at Baltimore in 1988 and was
published in Celestial
Mechanics and Dynamical Astronomy 46,187-204, 1989. The fifth
report of the Working Group was presented to the General Assembly at
Buenos Aires in 1991 and was published in Celestial
Mechanics and Dynamical Astronomy
53, 377-397, 1992.
The sixth report of the Working Group was presented to the General
Assembly at the Hague in 1994 and was published in Celestial
Mechanics and Dynamical Astronomy 63,
127-148, 1996. The seventh report of the Working Group was presented
to the General Assembly at Kyoto, but the changes were sufficiently
minor that the report was not published. In
1984 the International Association of Geodesy (IAG) and the
Committee on Space Research (COSPAR) expressed interest in the
activities of the Working Group, and after reviewing alternatives,
the Executive Committees of all three organizations decided to
jointly sponsor the Working Group. In 1998 COSPAR informed the
Working Group that, while the reports and expertise of the Working
Group are appreciated, the Working Group does not follow the
scientific structure of COSPAR and they wish to terminate the formal
affiliation. This
report incorporates revisions to the tables giving the directions of
the north poles of rotation and the prime meridians of the planets
and satellites since the last report. Also, tables giving the sizes
and shapes of the planets, satellites, and asteroids are presented. 2. Definition of
Rotational Elements Planetary
coordinate systems are defined relative to their mean axis of
rotation and various definitions of longitude depending on the body.
The longitude systems of most of those bodies with observable rigid
surfaces have been defined by references to a surface feature such
as a crater. Approximate expressions for these rotational elements
with respect to the J2000 inertial coordinate system have been
derived. The J2000 coordinate system is defined by the FK5 star
catalog and has the standard epoch of 2000 January 1.5 (JD
2451545.0), TCB. The variable quantities are expressed in units of
days (86400 SI seconds) or Julian centuries of 36525 days. The
north pole is that pole of rotation that lies on the north side of
the invariable plane of the solar system. The direction of the north
pole is specified by the value of its right ascension a0
and declination d0,
whereas the location of the prime meridian is specified by the angle
that is measured along the planet's equator in an easterly direction with respect to the
planet's north pole from the node Q
(located at right ascension 90°
+ a0)
of the planet's equator on the standard equator to the point B
where the prime meridian crosses the planet's equator. The right
ascension of the point Q
is 90°
+ a0
and the inclination of the planet's equator to the standard equator
is 90° - d0.
Because the prime meridian is assumed to rotate uniformly with the
planet, W accordingly
varies linearly with time. In addition, a0,
d0,
and W may vary with time
due to a precession of the axis of rotation of the planet (or
satellite). If W increases
with time, the planet has a direct
(or prograde) rotation and if W
decreases with time, the rotation is said to be retrograde. In
the absence of other information, the axis of rotation is assumed to
be normal to the mean orbital plane; Mercury and most of the
satellites are in this category. For many of the satellites, it is
assumed that the rotation rate is equal to the mean orbital period. The
angle W specifies the
ephemeris position of the prime meridian, and for planets or
satellites without any accurately observable fixed surface features,
the adopted expression for W defines
the prime meridian and is not subject to correction. Where possible,
however, the cartographic position of the prime meridian is defined
by a suitable observable feature, and so the constants in the
expression W = W0 +
Wd, where d is the interval in days from the standard epoch, are chosen so
that the ephemeris position follows the motion of the cartographic
position as closely as possible; in these cases the expression for W may require emendation in the future. Recommended
values of the constants in the expressions for a0,d0, and
W, in standard equatorial
coordinates with equinox J2000 at epoch J2000, are given for the
planets, satellites, and asteroids in Tables I, II, and III. In
general, these expressions should be accurate to one-tenth of a
degree; however, two decimal places are given to assure consistency
when changing coordinates systems. Zeros are added to rate values (W) for computational consistency and are not an indication of
significant accuracy. Additional decimal places are given in the
expressions for the Moon, Mars, Saturn, and Uranus, reflecting the
greater confidence in their accuracy. Expressions for the Sun and
Earth are given to a similar precision as those of the other bodies
of the solar system and are for comparative purposes only. The
recommended coordinate system for the Moon is the mean Earth/polar
axis system (in contrast to the principal axis system). 3. Definition of
Cartographic Coordinate Systems In
mathematical and geodetic terminology, the terms 'latitude' and
'longitude' refer to a right-hand spherical coordinate system in
which latitude is defined as the angle between a vector and the
equator, and longitude is the angle between the vector and the plane
of the prime meridian measured in an eastern direction. This
coordinate system, together with Cartesian coordinates, is used in
most planetary computations, and is sometimes called the
planetocentric coordinate system. The origin is the center of mass. Because of astronomical tradition, planetographic coordinates (those used on maps) may or may not be identical with traditional spherical coordinates. Planetographic coordinates are defined by guiding principles contained in a resolution passed at the fourteenth General Assembly of the IAU in 1970. These guiding principles state that: (1)
The rotational pole of a planet or satellite which lies on
the north side of the invariable plane will be called north, and
northern latitudes will be designated as positive. (2)
The planetographic longitude of the central meridian, as
observed from a direction fixed with respect to an inertial system,
will increase with time. The range of longitudes shall extend from 0°
to 360°. Thus,
west longitudes (i.e., longitudes measured positively to the west)
will be used when the rotation is prograde and east longitudes
(i.e., longitudes measured positively to the east) when the rotation
is retrograde. The origin is the center of mass. Also because of
tradition, the Earth, Sun, and Moon do not conform with this
definition. Their rotations are prograde and longitudes run both
east and west 180°
instead of the usual 360°. Latitude
is measured north and south of the equator; north latitudes are
designated as positive. The planetographic latitude of a point on
the reference surface is the angle between the equatorial plane and
the normal to the reference surface at the point. In the
planetographic system, the position of a point (P) not on the reference surface is specified by the planetographic
latitude of the point (P')
on the reference surface at which the normal passes through P and by the height (h) of
P above P'. The
reference surfaces for some planets (such as Earth and Mars) are
ellipsoids of revolution for which the radius at the equator (A)
is larger than the polar semiaxis (C).
Calculations
of the hydrostatic shapes of some of the satellites (Io, Mimas,
Enceladus, and Miranda) indicate that their reference surfaces
should be triaxial ellipsoids. Triaxial ellipsoids would render many
computations more complicated, especially those related to map
projections. Many projections would loose their elegant and popular
properties. For this reason spherical reference surfaces are
frequently used in mapping programs. Many
small bodies of the solar system (satellites, asteroids, and comet
nuclei) have very irregular shapes. Sometimes spherical reference
surfaces are used for computational convenience, but this approach
does not preserve the area or shape characteristics of common map
projections. Orthographic projections often are adopted for
cartographic portrayal as these preserve the irregular appearance of
the body without artificial distortion. With
the introduction of large mass storage to computer systems, digital
cartography has become increasingly popular. These databases are
important to irregularly shaped bodies and other bodies where the
surface can be described by a file containing planetographic
longitude, latitude, and radius for each pixel. In this case the
reference sphere has shrunk to a point. Other parameters such as
brightness, gravity, etc., if known, can be associated with each
pixel. With proper programming, pictorial and projected views of the
body can then be displayed by introducing a suitable reference
surface. Table
IV contains data on the size and shapes of the planets. The first
column gives the mean radius of the body (i.e., the radius of a
sphere of approximately the same volume as the spheroid). The
standard errors of the mean radii are indications of the accuracy of
determination of these parameters due to inaccuracies of the
observational data. Because the shape of a rotating body in
hydrostatic equilibrium is approximately a spheroid, this is
frequently a good approximation to the shape of planets, and so the
second and third columns give equatorial and polar radii for
'best-fit' spheroids. The origin of these coordinates is the
center-of-mass with the polar axis coincident with the spin axis.
The fourth column is the root-mean-square (RMS) of the radii
residuals from the spheroid and is an indication of the variations
of the surface from the spheroid due to topography. The last two
columns give the maximum positive and negative residuals to bracket
the spread. Table
V contains data on the size and shape of the satellites. The first
column gives the mean radius of the body. The standard errors of the
mean radii are indications of the accuracy of determination of these
parameters due to inaccuracies of the observational data. Because
the hydrostatic shape of a body in synchronous rotation about a
larger body is approximately an ellipsoid, that shape has been
selected to describe the shape of the satellites. The next three
columns (2-4) give the axes of the best-fit ellipsoids in the order
equatorial subplanetary, equatorial along orbit, and polar. The
origin of these coordinates is the center-of-mass with the polar
axis coincident with the spin axis. The fifth column is the RMS of
the radii residuals from the ellipsoid and is an indication of the
variations of the surface from the ellipsoid due to topography. The
last two columns give the maximum positive and negative residuals to
bracket the spread. Table
I. Recommended values for the direction of the north pole of
rotation and the prime meridian of the Sun and planets (2000) a0,d0
are standard equatorial coordinates with equinox J2000 at epoch J2000.
Approximate
coordinates of the north pole of the invariable plane are a0
= 273 °.85,
d0=
66°.99. T =
interval in Julian centuries (of 36525 days) from the
standard epoch d =
interval in days from the standard epoch. The
standard epoch is 2000 January 1.5, i.e., JD 2451545.0 TCB.
(a) The 20°
meridian is defined by the crater Hun Kal. (b) The 0°
meridian is defined by the central peak in the crater Ariadne. (c)
The expression for W might be in error by as much as 0°.2
because of uncertainty in the length of the UT day and the TT UT on
1 January 2000. (d)
The 0°
meridian is defined by the crater Airy-0. (e)
The equations for W for Jupiter, Saturn, Uranus and Neptune refer to the rotation of
their magnetic fields (System III). On Jupiter, System I (WI = 67 °.1
+ 877°.900d)
refers to the mean atmospheric equatorial rotation; System II (WII
= 43°.3
+ 870°.270d) refers to the mean atmospheric rotation north of the south
component of the north equatorial belt, and south of the north
component of the south equatorial belt. (f) The 0°
meridian is defined as the mean sub-Charon meridian. Table
II. Recommended values for the direction of the north pole of
rotation and the prime meridian of the satellites (2000)
a0, d0, T,
and d have the same
meanings as in Table I (epoch 2000 January 1.5, i.e., JD 2451545.0
TCB).
Earth:
Moon
a0 = 269°.9949
+ 0°.0031T -
3°.8787sin El
- 0°.1204 sin E2
+ 0.0700 sin E3 - 0.0172 sin E4
+ 0.0072 sin E6
- 0.0052sin El0 + 0.0043sin E13
d0 = 66.5392
+ 0.0130T
+ 1.5419 cos E1 +
0.0239 cos E2
- 0.0278 cos E3 + 0.0068 cos E4
- 0.0029 cos E6
+ 0.0009 cos E7 + 0.0008 cos E10
- 0.0009cos E13
W = 38.3213
+ 13.17635815 d
- 1.4 x 10-12 d2
+ 3.5610 sin E1
+ 0. 1208 sin E2 - 0.0642 sin E3
+ 0.0158 sin E4
+ 0.0252 sin E5 - 0.0066 sin E6
- 0.0047 sin E7
- 0.0046 sin E8 + 0.0028 sin E9
+ 0.0052 sin E 10
+ 0.0040sin E11 + 0.0019 sin E12
- 0.0044 sin E l3
where
El = 125°.045
- 0°.0529921d,
E2 = 250°.089
- 0°.1059842d,
E3 = 260°.008
+ 13°.0120009d,
E4 = 176.625 +
13.3407154d,
E5 = 357.529 +
0.9856003d,
E6 = 311.589 + 26.4057084d,
E7 = 134.963 +
13.0649930d,
E8 = 276.617 +
0.3287146d,
E9 = 34.226+ 1.7484877d,
E10
= 15.134 - 0.1589763d,
E11 = 119.743 +
0.0036096d,
E12 = 239.961 +
0.1643573d,
E13
= 25.053 + 12.9590088d Mars: I Phobos
a0 = 317.68
- 0.108T
+ 1.79 sin Ml
d0 = 52.90
- 0.061T
- 1.08 cos M1
W = 35.06
+ 1128.8445850d
+ 8.864T 2
-
1.42 sin M1
- 0.78 sin M2
II Deimos
a0 = 316.65
- 0.108T
+ 2.98 sin M3
d0 = 53.52
- 0.061T
- 1.78 cos M3
W = 79.41
+ 285.1618970d
- 0.520T2
- 2.58 sin M3 + 0.19
cosM3
where
M1
= 169°.51 - 0°.4357640d,
M2 = 192°.93
+ 1128 °.4096700d
+ 8°.864T2,
M3
= 53°.47 - 0°.0181510d Jupiter: XVI
Metis
a0 = 268.05
- 0.009T
d0 =
64.49
+ 0.003T
W = 346.09 +
1221.2547301d
XV
Adrastea
a0 = 268.05
- 0.009T
d0 = 64.49
+ 0.003T
W = 33.29
+ 1206.9986602d
V
Amalthea a0 = 268.05
- 0.009T
- 0.84 sin J1
+ 0.01 sin 2J1
d0 = 64.49
+ 0.003T
- 0.36 cos J1
W =
231.67
+ 722.6314560d
+ 0.76 sin J1 - 0.01
sin 2J1
XIV Thebe
a0 =
268.05
- 0.009T
- 2.11 sin J2
+ 0.04 sin2J2
d0 = 64.49
+ 0.003T
- 0.91 cos J2
+ 0.01 cos 2J2
W =
8.56
+ 533.7004100d
+ 1.91 sin J2 - 0.04
sin 2J2
I
Io
a0
= 268.05
- 0.009T
+ 0.094 sin J3
+ 0.024 sin J4
d0 = 64.50
+ 0.003T
+ 0.040 cos J3
+ 0.011 cos J4
W = 200.39
+ 203.4889538d - 0.085 sin J3
- 0. 022 sin J4
II
Europa
a0
= 268.08
- 0.009T
+ 1.086 sin J4
+ 0.060 sin J5
+ 0.015 sin J6
+ 0. 009 sin J7
d0 = 64.51
+ 0.003T
+ 0.468 cos J4
+ 0.026 cos J5
+ 0.007 cos J6
+ 0.002 cos J7
W =
35.67
+ 101.3747235d
- 0.980 sin J4 - 0.054 sin J5
- 0.014 sin J6 -
0.008 sin J7
(a)
III
Ganymede a0
= 268.20
- 0.009T
- 0.037 sin J4
+ 0.431 sin J5
+ 0.091 sin J6
d0 =
64.57
+ 0.003T
- 0.016 cos J4
+ 0.186 cos J5
+ 0.039 cos J6
W = 44.04
+ 50.3176081d + 0.033 sin J4
- 0.389 sin J5
(b)
- 0.082 sin J6
IV
Callisto
a0
= 268.72
- 0.009T
- 0.068 sin J5
+ 0.590 sin J6
+ 0.010 sin J8
d0 =
64.83
+ 0.003T
- 0.029 cos J5 + 0.254 cos J6
- 0.004 cos J8
W =
259.73
+ 21.5710715d
+ 0.061 sin J5
- 0.533 sin J6
(c)
-
0.009 sin J8
where
J l = 73°.32
+ 91472°.9T
J 2 = 24°.62 + 45137°.2T,
J 3 = 283°.90 + 4850°.7T,
J 4
= 355.80 + 1191.3T,
J 5 = 119.90 + 262.1T,
J6 = 229.80 + 64.3T,
J 7 = 352.25 +
2382.6T,
J 8 =113.35 +
6070.0T Saturn: XVIII
Pan
a 0 = 40.6
- 0.036T
d0 = 83.5
- 0.004T
W = 48.8
+ 626.0440000d
XV
Atlas
a0 = 40.58
- 0.036T
d0 = 83.53
- 0.004T
W = 137.88
+ 598.3060000d
XVI Prometheus
a0 = 40.58
- 0.036T
d0 = 83.53
- 0.004T
W = 296.14 +
587.289000d
XVII Pandora
a0 = 40.58
- 0.036T
d0 = 83.53
- 0.004T
W = 162.92
+ 572.7891000d
XI
Epimetheus a0 = 40.58
- 0.036T
- 3.153 sin S1
+ 0.086 sin 2Sl
d0 = 83.52
- 0.004T
- 0.356 cos S1 + 0.005 cos 2SI
W = 293.87 +
518.4907239d
+ 3.133 sin S1 - 0.086
sin 2S1
(j)
X
Janus
a0 = 40.58
- 0.036T
- 1.623 sin S2 + 0.023 sin 2S2
d0 = 83.52
- 0.004T
- 0. 183 cos S2 + 0.001 cos 2S2
W =
58.83
+ 518.2359876d
+ 1.613 sin S2 - 0.023 sin 2S2
(j)
I Mimas
a0 = 40.66
- 0.036T
+ 13.56 sin S3
d0 = 83.52
- 0.004T
- 1.53 cos S3
W = 337.46 +
381.9945550d
- 13.48 sin S3
- 44.85 sin S5
(d)
II
Enceladus a0 =
40.66
- 0.036T
d0 = 83.52
- 0.004T
W = 2.82
+ 262.7318996d
(e)
III
Tethys
a0 = 40.66
- 0.036T
+ 9.66 sin S4
d0 = 83.52
- 0.004T
- 1.09 cos S4
W = 10.45
+ 190.6979085d
- 9.60 sin S4
+ 2.23 sin S5
(f)
XIII Telesto
a0 = 50.51
- 0.036T
d0 = 84.06
- 0.004T
W = 56.88
+ 190.6979332d
(j)
XIV Calypso a0 = 36.41
- 0.036T
d0 = 85.04
- 0.004T
W = 153.51 +
190.6742373d
(j)
IV
Dione
a0 = 40.66
- 0.036T
d0 = 83.52
- 0.004T
W = 357.00
+ 131.5349316d
(g)
XII Helene
a0 = 40.85
- 0.036T
d0 = 83.34
- 0.004T
W = 245.12 +
131.6174056d
V
Rhea
a0 = 40.38
- 0.036T
+ 3. 10 sin S6
d0 = 83.55
- 0.004T
- 0.35 cos S6
W = 235.16 +
79.6900478d
- 3.08 sin S6
(h)
VI Titan
a0 =
36.41
- 0.036T
+ 2.66 sin S7
d0 = 83.94
- 0.004T
- 0. 30 cos S7
W = 189.64 +
22.5769768d
- 2.64 sin S7
VIII Iapetus
a0 = 318.16
- 3.949T
d0 = 75.03
- 1.143T
W
= 350.20
+ 4.5379572d
IX
Phoebe
a0 = 355.00
d0
= 68.70
W = 304.70
+ 930.8338720d
where Sl
= 353°.32 + 75706°.7T, S2
= 28°.72 + 75706°.7T, S3
= 177°.40 - 36505°.5T
S4 = 300.00 -
7225.9T,
S5 =
316.45 + 506.2T,
S6
= 345.20 - 1016.3T,
S7 = 29.80 - 52.1T Uranus: VI Cordelia
a0 = 257.31
- 0. 15 sin U1
d0 = -15.18
+ 0.14 cos U 1
W = 127.69
- 1074.5205730d - 0.04 sinUl
VII
Ophelia
a0 =
257.31
- 0.09 sin U2
d0 = -
15.18
+ 0.09 cos U2
W = 130.35
- 956.4068150d
- 0.03 sin U2
VIII Bianca
a0
= 257.31
- 0.16 sin U3
d0 = -15.18
+ 0. 16 cos U3
W = 105.46
- 828.3914760d - 0.04 sin U3
IX
Cressida
a0
= 257.31
- 0.04 sin U4
d0 = - 15.18
+ 0. 04 cos U4
W = 59.16
- 776.5816320d - 0.01 sin U4
X
Desdemona a0
=
257.31
- 0. 17 sin U5
d0 = -15.18
+ 0. 16 cos U5
W = 95.08
- 760.0531690d - 0.04 sin U5
XI
Juliet
a0
= 257.31
- 0.06 sin U6
d0 = - 15.18
+ 0.06 cos U6
W = 302.56
- 730.1253660d - 0.02 sin U6
XII
Portia
a0
= 257.31
- 0.09 sin U7
d0 = -
15.18 + 0.09 cos U7
W = 25.03
- 701.4865870d - 0.02 sin U7
XIII Rosalind
a0
= 257.31
- 0.29 sin U8
d0 =
-15.18
+ 0.28 cos U8
W =
314.90
- 644.6311260d -
0.08 sin U8
XIV Belinda
a0
=
257.31
- 0.03 sin U9
d0 = -15.18
+ 0.03 cos U9
W = 297.46
- 577.3628170d -
0.01 sin U9
XV
Puck
a0
= 257.31
- 0.33 sin U10
d0 =
-15.18
+ 0.31 cos U10
W = 91.24
- 472.5450690d -
0.09 sin Ul0
V
Miranda
a0
= 257.43
+ 4.41 sin U11
- 0.04 sin 2U11
d0 =
-15.08
+ 4.25 cos U11
- 0.02 cos 2U11
W = 30.70
- 254.6906892d - 1.27 sin U12
+ 0.15 sin 2Ul2
+
1.15 sin U 11
- 0.09 sin 2U11
I
Ariel
a0
=
257.43
+ 0.29 sin U13
d0 = -15.10
+ 0.28 cos U13
W = 156.22
- 142.8356681d + 0.05 sin U12
+ 0.08 sin U13
II
Umbriel
a0
= 257.43
+ 0.21 sin U14
d0 =
-15.10 + 0.20 cos U14 W = 108.05
- 86.8688923d
- 0.09 sin U12
+ 0.06 sin U14
III
Titania
a0
=
257.43
+ 0.29 sin U15
d0 = -15.10
+
0.28 cos U15
W = 77.74
- 41.3514316d + 0.08 sin U15
IV
Oberon
a0
= 257.43
+ 0.16 sin U16
d0 = -15.10
+ 0.16 cos U16
W = 6.77
- 26.7394932d + 0.04 sin U16
where Ul
= 115°.75 + 54991°.87T, U2
= 141°.69 + 41887°.66T, U3
= 135°.03 + 29927°.35T,
U4 = 61.77 +
25733.59T,
U5
= 249.32 + 24471.46T,
U6 = 43.86 + 22278.41T,
U7 = 77.66 +
20289.42T
U8 = 157.36 +
16652.76T,
U9 = 101.81 +
12872.63T,
U10 = 138.64 +
8061.81T,
U11 = 102.23 - 2024.22T,
U12 = 316.41 + 2863.96T,
U13
= 304.01 - 51.94T,
U14 = 308.71 - 93.17T,
U15 = 340.82 - 75.32T,
U16 =
259.14 - 504.81T Neptune
III
Naiad
a0
= 299°.36
+ 0°.70
sin N - 6.49 sin N1
+ 0°.25
sin 2Nl
d0 = 43.36 -
0.51 cos N
- 4.75 cos Nl. + 0.09 cos 2Nl
W =
254.06
+ 1222.8441209d - 0.48 sin N
+ 4.40 sin N1 - 0.27 sin 2N1
IV
Thalassa
a0
=
299.36
+ 0.70 sin N
- 0.28 sin N2
d0
=
43.45
- 0.51 cos N
- 0.21 cos N2
W =
102.06
+ 1155.7555612d - 0.48 sin N
+ 0. 19 sin N2
V
Despina
a0
= 299.36
+ 0.70 sin N
- 0.09 sin N3
d0 = 43.45
- 0.51 cos N
- 0.07 cos N3
W = 306.51
+ 1075.7341562d - 0.49 sin N
+ 0.06 sin N3
VI
Galatea
a0
=
299.36
+ 0.70 sin N
- 0.07 sin N4
d0 = 43.43
- 0.51 cos N
- 0.05 cos N4
W =
258.09
+ 839.6597686d
- 0.48 sin N +
0.05 sin N4
VII
Larissa
a0
= 299.36
+ 0.70 sin N
- 0.27 sin N5
d0 = 43.41
- 0.51 cos N
- 0.20 cos N5
W =
179.41
+ 649.0534470d
- 0.48 sin N +
0. 19 sin N5
VIII Proteus
a0 =
299.27
+ 0.70 sin N
- 0.05 sin N6
d0 = 42.91
- 0.51 cos N
- 0.04 cos N6
W =
93.38
+ 320.7654228d
- 0.48 sin N +
0.04 sin N6
I
Triton
a0
= 299.36
- 32.35 sin N7
- 6.28 sin 2N7
- 2.08 sin 3N7
- 0.74 sin 4N7 - 0.28 sin 5N7
- 0.11 sin 6N7
- 0.07 sin 7N7 - 0.02 sin 8N7
- 0.01 sin 9N7
d0 = 41.17
+ 22.55 cos N7
+ 2.10 cos 2N7 + 0.55 cos 3N7
+ 0.16 cos 4N7 + 0.05 cos 5N7
+ 0.02 cos 6N7
+
0.01 cos 7N7
W = 296.53
- 61.2572637d + 22.25 sin N7
+ 6.73 sin 2N7
+ 2.05 sin 3N7 + 0.74 sin 4N7
+ 0.28 sin 5N7
+ 0.11 sin 6N7 + 0.05 sin 7N7
+ 0.02 sin 8N7
+ 0.01 sin 9N7
where
N =
357°.85 + 52.316T,
N l = 323°.92+62606°.6T, N2 = 220°.51+55064°.2T,
N3 = 354.27 + 46564.5T,
N4
= 75.31 + 26109.4T,
N5 = 35.36 + 14325.4T,
N6 = 142.61 + 2824.6T,
N7 = 177.85 +
52.316T Pluto I
Charon
a0 = 313.02
d0 = 9.09
W = 56.77
- 56.3623195d
(a)
The 182°
meridian is defined by the crater Cilix. (b)
The 128°
meridian is defined by the crater Anat. (c)
The 326°
meridian is defined by the crater Saga. (d)
The 162°
meridian is defined by the crater Palomides. (e)
The 5°
meridian is defined by the crater Salih. (f)
The 299°
meridian is defined by the crater Arete. (g)
The 63°
meridian is defined by the crater Palinurus. (h)
The 340°
meridian is defined by the crater Tore. (i)
The 276°
meridian is defined by the crater Almeric. (j)
These equations are correct for the period of the Voyager
encounters. Because of precession they may not be accurate at other
time periods. Satellites
for which no suitable data are yet available have been omitted from
this table. Nereid is not included in this table because it is not
in synchronous rotation. Table
III. Recommended rotation values for the direction of the north pole
of rotation and the prime meridian of selected asteroids (2000)
a0, d0, W,
and d have the
same meanings as in the Table I (epoch 2000 January 1.5, i.e., JD
2451545.0 TCB).
243 Ida
a0 = 348°.76
d0 = 87°.12
W =
265°.95 - 1864°.6280070d
(a) 951 Gaspra
a0 = 9°.47
d0
=
26°.70
W = 83°.67 + 1226°.9114850d
(b) 4 Vesta
a0
= 301°
d0
=
41 °
W = 292°
+ 1617°.332776d 433
Eros
a0 = 11°.9
d0
=
20°.8
W
= 324°.08
+ 1639°.3922d
(a)
The 0°
meridian is defined by the crater Afon. (b)
The 0°
meridian is defined by the crater Charax. Table
VI contains data on the size and shape of selected asteroids. The
first column gives the mean radius of the body and an estimate of
the accuracy of this measurement. The next three columns give
estimates of the radii measured along the three principal axes. The
fifth column gives the radii of a best-fit ellipsoid. These are
given because an ellipsoid is a common reference shape for
photometric analyses. The last column gives an estimate of the
maximum deviation of the body from the ellipsoid and is an estimate
of the goodness of fit.
The values of the radii and axes in Tables IV, V, and VI are
derived by various methods and do not always refer to common
definitions. Some use star or spacecraft occultation measurements,
some use limb fitting, others use altimetry measurements from
orbiting spacecraft, and some use control network computations. For
the Earth, the spheroid refers to mean sea level, clearly a very
different definition from other bodies in the Solar System.
The uncertainties in the values for the radii and axes in
Tables IV, V, and VI are generally those of the authors, and, as
such, frequently have different meanings. Sometimes they are
standard errors of a particular data set, sometimes simply an
estimate or expression of confidence.
The radii and axes of the large gaseous planets, Jupiter,
Saturn, Uranus, and Neptune in Table IV refer to a one-bar-pressure
surface.
The radii given in the tables are not necessarily the
appropriate values to be used in dynamical studies; the radius
actually used to derive a value of J2 (for example) should always be
used in conjunction with it. Table
IV Size and shape parameters of the planets
Planet Mean
radius
Equatorial
Polar radius
RMS Maximum
Maximum
(km)
radius
(km)
deviation
elevation
depression
(km)
from
(km)
(km)
spheroid
(km)
Mercury 2439.7 ± 1.0
same
same
1
4.6
2.5 Venus 6051.8
± 1.0
same
same
1
11
2
Earth
6371.00 ± 0.01
6378.14 ± 0.01
6356.75 ± 0.01 3.57
8.85
11.52
Mars
3389.508 ± 0.003
3396.200 ± 0.16
N 3376.189 ± 0.05 3.3
21.183 ± 0.005
7.825 ± 0.005
S 3382.582 ± 0.05
Jupiter* 69911 ± 6
71492 ± 4
66854 ± 10
62.1 31
102 Saturn* 58232 ± 6
60268 ± 4
54364 ± 10
102.9 8
205 Uranus* 25362 ± 7
25559 ± 4
24973 ± 20
16.8 28
0
Neptune*
24622 ± 19
24764 ± 15
24341 ± 30
8
14
0
Pluto
1195 + 5
same
same
*The
radii correspond to a one-bar surface. Table
V. Size and shape parameters of the satellites
Earth
Moon
1737.4 ± 1 same
same
same 2.5
7.5
5.6 Mars I
Phobos
11.1 ± 0.15 13.4
11.2
9.2
0.5
II
Deimos
6.2 ±
0.18 7.5
6.1
5.2 0.2 Jupiter
XVI Metis
21.5 ±
4 30
20
17
XV Adrastea
8.2 ±
4 10
8
7
V
Amalthea
83.5 ± 3
125
73
64
3.2
XIV Thebe
49.3 ± 4
58
49
42
I
Io
1818.1 ±
0.1 1826.5
1815.7
1812.2 1.4
5-10
3
II
Europa
1561
same
same
same
0.5
III
Ganymede
2634
same
same
same
0.6
IV
Callisto
2408
same
same
same
0.6
XIII Leda
5
VI
Himalia
85 ± 10
X
Lysithea
12
VII Elara
40 ± 10
XII Ananke
10
XI
Carme
15
VIII Pasiphae
18 IX
Sinope
14 Saturn
XVIII Pan
10 ± 3
XV Atlas
16 ± 4
18.5
17.2
13.5
XVI Prometheus
50.1 ± 3
74.0
50.0
34.0
4.1
XVII Pandora
41.9 ± 2
55.0
44.0
31.0
1.3
XI
Epimetheus 59.5
± 3
69.0
55.0
55.0
3.1
X Janus
88.8 ± 4
97.0
95.0
77.0
4.2
I
Mimas
198.6 ± 0.6 209.1 ± 0.5
196.2 ± 0.5
191.4 ± 0.5
0.6
II Enceladus
249.4 ± 0.3
256.3 ± 0.3 247.3 ± 0.3
244.6 ± 0.5
0.4
III
Tethys
529.8 ± 1.5
535.6 ± 1.2
528.2 ± 1.2 525.8 ± 1.2
1.7
XIII Telesto
11± 4
15 ± 2.5
12.5 ± 5 7.5 ±
2.5
XIV Calypso
9.5 ± 4
15.0
8.0
8.0
0.6
IV
Dione
560 ± 5
same
same
same
0.5
XII Helene
16
17.5 ± 2.5
0.7
V
Rhea
764 ±4
same
same
same
VI
Titan
2575 ± 2
same
same
same
VII Hyperion
141.5 ± 20
180 ± 20 140
± 20
112.5 ± 20
7.4
VIII Iapetus
718 ± 8
same
same
same
6.1
12
IX Phoebe
110 ± 10
115 ± 10 110
± 10
105 ±10
2.7 Uranus VI
Cordelia
13 ± 2
VII Ophelia
15 ± 2
VIII Bianca
21 ± 3
IX Cressida
31 ± 4
X
Desdemona 27
± 3
XI
Juliet
42 ± 5
XII Portia
54 ± 6
XIII Rosalind
27 ± 4
XIV Belinda 33 ± 4
XV Puck
77 ± 5
1.9
V
Miranda
235.8 ± 0.7
240.4 ± 0.6
234.2 ± 0.9 232.9 ± 1.2
1.6
5
8
I
Ariel
578.9 ± 0.6 581.1 ± 0.9
577.9 ± 0.6
577.7 ± 1.0
0.9 4
4
II
Umbriel
584.7 ± 2.8
same
same
same
2.6
6
III
Titania
788.9 ± 1.8
same
same
same
1.3
4
IV
Oberon
761.4 ± 2.6
same
same
same
1.5
12
2 Neptune
III
Naiad
29 ± 6
IV
Thalassa
40 ± 8
V
Despina
74 ± 10
VI
Galatea
79 ± 12
VII Larissa
96 ± 7
104
89
2.9
6
5
VIII Proteus
208 ± 8
218
208
201
7.9
18
13
I
Triton
1352.6 ± 2.4
II
Nereid
170 ± 25 Pluto
I
Charon
593 ± 13
Table VI. Size and shape parameters of selected asteroids
Asteroid Mean radius
Radii measured along
Radii of best-fit
Maximum
(km)
principal axes
Ellipsoid
deviation
(km) (km)
(km) (km)
from ellipsoid
(km)
243 Ida
15.65 ± 0.6
26.8 12.0
7.6
30.0, 12.6, 9.3 8.4 951 Gaspra
6.1 ± 0.4
9.1
5.2
4.4
9.1, 5.2, 4.7
2.1 216 Kleopatra
108.5
47
40.5
4. Appendix This
appendix summarizes the changes that have been made to the tables
since the 1994 report (Celestial Mechanics and Dynamical Astronomy 63, 127-148, 1996).
In Table I, the new value for the W0
of Mercury was the result of a new control network
computation by Robinson et al. (1999). The new values of a0 and
d0
for
Mars are due to Folkner et al (1997). The new value for W
of Mars was the result of a control network computation by Davies
et al (1999). The value for the d term in W
for Jupiter is from Higgins et al. (1996).
The new value for the d
term in W for Jupiter is a
new radio rotation period by Higgins et al. (1996).
In Table II the value of W
for Metis is from Lieske (1997).
In Table III the value for Vesta are from Thomas et al. (1997)
and the values of Eros are from Thomas et al. (2000).
In Table IV the Mars model is that determined by the Mars
Orbiter Laser Altimeter (MOLA) group from Smith et al. 1999. In
Table V the sizes of the inner satellites of Jupiter are from Thomas
et al. (1998). The sizes of th Galilean satellites are from Davies et
al (1998). In
Table VI the parameters for 216 Kleopatia are from Ostro et al 2000.
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