REPORT OF THE IAU/IAG WORKING GROUP ON

CARTOGRAPHIC COORDINATES AND ROTATIONAL ELEMENTS OF

THE PLANETS AND SATELLITES: 2000

 

P.K. SEIDELMANN (CHAIR)
U.S. Naval Observatory, Washington, DC, U.S.A.

V.K. ABALAKIN
Institute for Theoretical Astronomy, St. Petersburg, Russia

M. BURSA
Astronomical Institute, Prague, Czech Republic

M.E. DAVIES
RAND, Santa Monica, CA, U.S.A
.

C. de BERGH
Observatoire de Paris, Paris, France

J.H. LIESKE
Jet Propulsion Laboratory, Pasadena, CA, U.S.A.

J. OBERST
DLR Berlin Adlershof, Berlin, Germany

J.L. SIMON
Institut de Mecanique Celeste, Paris, France

E.M. STANDISH
 
Jet Propulsion Laboratory, Pasadena, Ca, USA

P. STOOKE
Univ. of Western Ontario, London, Canada

P.C. THOMAS
Cornell Univ., Ithaca, NY, USA

 

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.


 


Sun    a0     286. 13
d0   = 63.87  
W    =  84.10 + 14.1844000d  
Mercury  a0      =  281.01 - 0.033T  
d0  =   61.45 - 0.005T
W    =   329.548 + 6.1385025d  (a)  
 
Venus   a0  =   272.76 
d0  =   67.16 
W   =   160.20 - 1.4813688d   (b)
 Earth a0       =     0.00 - 0.641T  
d0   =    90.00 - 0.557T
W    =   190.16 + 360.9856235d  (c)  
Mars a0       =   317.68143 - 0.1061T
d0   =     52.88650 - 0.0609T  
W    =   176.753 + 350.89198226d  (d)  
Jupiter      a0       =   268.05 - 0.009T
d0   =   64.49 + 0.003T  
W    =   284.95 + 870.5366420d  (e)
Saturn  a0       =   40.589 - 0.036T
d0   =   83.537 - 0.004T  
W    =   38.90 + 810.7939024d    (e)  
Uranus  a0       =   257.311  
d0   =    -15.175  
W    =    203.81 - 501.1600928d  (e)  
Neptune a0       =   299.36 + 0.70 sin N  
d0   =   43.46 - 0.51 cos N  
W    =   253.18 + 536.3128492d-0.48sin N (e)  
N    =   357.85 + 52.316T  
Pluto   a0       =    313.02  
d0   =   9.09  
W    =   236.77 - 56.3623195d (f)  


 


(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:                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 J          - 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 J        - 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             - 577.3628170      - 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

 


                        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 N          + 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


   

Planet   Satellite Mean
Radius
(km)
Subplanetary
equatorial
radius (km)
Along Orbit 
equatorial 
radius (km)
Polar 
radius 
 (km)  
RMS 
deviation
  from
 ellipsoid 
 
(km)
Maximum
  elevation 
(km) 
Maximum  
Depression
(km)


   

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.

               

 

 

 

References

 

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