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 &nbs | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
