KPL/PCK
P_constants (PcK) SPICE kernel file
===========================================================================
By: Nat Bachman, Boris Semenov (NAIF) 2001 December 11
Organization
--------------------------------------------------------
The contents of this file are as follows.
Introductory Information:
-- Version description
-- Disclaimer
-- Sources
-- Explanation
-- Body numbers and names
Pck Data:
-- Orientation constants for Mars and its natural
satellites
-- Radii of Mars and its natural satellites
Version description
--------------------------------------------------------
This file was created on December 11, 2001. This version
incorporates new data from a preprint of the ``Report of
the IAU/IAG Working Group on Cartographic Coordinates
and Rotational Elements of the Planets and Satellites: 2000.''
Disclaimer
--------------------------------------------------------
This constants file may not contain the parameter values that
you prefer. Note that this file may be readily modified by
you or anyone else. NAIF suggests that you inspect this file
visually before proceeding with any critical or extended
data processing.
NAIF requests that you update the ``by line' and date if you
modify the file.
Sources
--------------------------------------------------------
The sources for the constants listed in this file are:
1. Preprint: ``Report of the IAU/IAG Working Group
on Cartographic Coordinates and Rotational Elements of
the Planets and Satellites: 2000.''
2. ``Planetary Geodetic Control Using Satellite
Imaging,'' Journal of Geophysical Research, Vol. 84,
No. B3, March 10, 1979, by Thomas C. Duxbury. This
paper is cataloged as NAIF document 190.0.
3. Letter from Thomas C. Duxbury to Dr. Ephraim
Lazeryevich Akim, Keldish Institute of Applied
Mathematics, USSR Academy of Sciences, Moscow, USSR.
This letter is catalogued as NAIF document number
195.0.
Most values are from [1]. All exceptions are commented
where they occur in this file. The exceptions are:
-- The second nutation precession angle (M2) for Mars is
represented by a quadratic polynomial in the 1994
IAU report. The SPICELIB subroutine BODEUL can not
handle this term (which is extremely small), so we
truncate the polynomial to a linear one.
-- Values for Mars' prime meridian offset are listed
as comments. These are taken from [2] and [3].
``Old values'' listed are from the SPICE PCK file dated April
24, 2000. Most of these values came from the 1997 IAU report.
Explanation
--------------------------------------------------------
The NAIF Toolkit software that uses this file is documented in
the NAIF ``Required Reading'' file PCK.REQ. See that document
for a detailed explanation of the NAIF text kernel file format.
PCK.REQ is available in both printed and electronic form.
This file, which is logically part of the SPICE P-kernel,
contains constants used to model the orientation and shape of
the Sun, planets, and satellites. The orientation models
express the direction of the pole and location of the prime
meridian of a body as a function of time. The shape models
represent all bodies as ellipsoids, using two equatorial
radii and a polar radius. Spheroids and spheres are obtained
when two or all three radii are equal.
Orientation models
The discussion below was taken from a PCK file containing
data for bodies for which data was provided in the 1997 IAU
report. While it refers to data that are not present in
this file, the explanation remains applicable to this file.
All of the orientation models use three Euler angles to
describe body orientation. To be precise, the Euler angles
describe the orientation of the coordinate axes of the
``Body Equator and Prime Meridian'' system with respect to
an inertial system. By default, the inertial system is
J2000 (also called ``EME2000''), but other frames can be
specified in the file. See the PCK Required Reading for
details.
The first two angles, in order, are the right ascension and
declination (henceforth RA and DEC) of the north pole of a
body as a function of time. The third angle is the prime
meridian location (represented by ``W''), which is expressed
as a rotation about the north pole, and is also a function of time.
For the Sun and planets, the expressions for the north pole's
right ascension and declination, as well as prime meridian location,
are always (as far as the models that appear in this file are
concerned) quadratic polynomials, where the independent variable is
time. Some coefficients may be zero.
In this file, the time arguments in expressions always refer
to Barycentric Dynamical Time (TDB), measured in centuries or
days past the a reference epoch. By default, the reference
epoch is the J2000 epoch, which is Julian ephemeris date
2451545.0, but other epochs can be specified in the file. See
the PCK Required Reading for details.
Example: 1991 IAU Model for orientation of the Earth. Note that
these values are used as an example only; see the data area below
for current values.
alpha = 0.00 - 0.641 T ( RA )
0
delta = 90.0 - 0.557 T ( DEC )
0
W = 190.16 + 360.9856235 d ( Prime meridian )
T represents centuries past J2000 ( TDB ),
d represents days past J2000 ( TDB ).
In this file, the polynomials' coefficients above are assigned to the
symbols
BODY399_POLE_RA
BODY399_POLE_DEC
BODY399_POLE_PM
as follows:
BODY399_POLE_RA = ( 0. -0.641 0. )
BODY399_POLE_DEC = ( +90. -0.557 0. )
BODY399_PM = ( 190.16 +360.9856235 0. )
Note the number ``399''; this is the NAIF ID code for the Earth.
You'll see an additional symbol grouped with the ones listed here; it
is
BODY399_LONG_AXIS
This term is zero for all bodies except Mars. It represents the offset
between the longest axis of the triaxial ellipsoid used to model a
body and the prime meridian of the body.
Expressions for satellites are a little more complicated; in addition
to polynomial terms, there are trigonometric terms. The arguments of
the trigonometric terms are linear polynomials. In this file, we call
the arguments of these trigonometric terms ``nutation precession
angles.''
In this file, the polynomial expressions for the nutation precession
angles are listed along with the planet's RA, DEC, and prime meridian
terms.
Example: 1991 IAU nutation precession angles for Earth. Note that these
values are used as an example only; see the data area below for current
values.
E1 = 125.045 - 0.052992 d
E2 = 250.090 - 0.105984 d
E3 = 260.008 + 13.012001 d
E4 = 176.625 + 13.340716 d
E5 = 357.529 + 0.985600 d
d represents days past J2000 ( TDB )
Because the NAIF Toolkit software expects the time units for the
angles to be CENTURIES (as in the IAU models for most bodies--the
Earth is an exception), the linear coefficients are scaled by 36525.0
in the assignments:
BODY3_NUT_PREC_ANGLES = ( 125.045 -1935.5328
250.090 -3871.0656
260.008 475263.336525
176.625 487269.6519
357.529 35999.04 )
As stated above, the satellite orientation models use polynomial and
trigonometric terms, where the arguments of the trigonometric terms
are the ``nutation precession angles.''
Example: 1988 IAU values for the Moon. Again, these values are used
as an example only; see the data area below for current values.
alpha = 270.000 + 0.003 T - 3.878 sin(E1) - 0.120 sin(E2)
0
+ 0.070 sin(E3) - 0.017 sin(E4) (RA)
delta = 66.541 + 0.013 T + 1.543 cos(E1) + 0.024 cos(E2)
0
- 0.028 cos(E3) + 0.007 cos(E4) (DEC)
W = 38.317 + 13.1763582 d + 3.558 sin(E1)
+ 0.121 sin(E2)
- 0.064 sin(E3)
+ 0.016 sin(E4)
+ 0.025 sin(E5) ( Prime
meridian )
d represents days past J2000.
E1--E5 are the nutation precession angles.
The polynomial terms are assigned to symbols by the statements
BODY301_POLE_RA = ( 270.000 0.003 0. )
BODY301_POLE_DEC = ( +66.541 0.013 0. )
BODY301_PM = ( 38.317 +13.1763582 0. )
The coefficients of the trigonometric terms are assigned to symbols by
the statements
BODY301_NUT_PREC_RA = ( -3.878 -0.120 +0.070 -0.017 0. )
BODY301_NUT_PREC_DEC = ( +1.543 +0.024 -0.028 +0.007 0. )
BODY301_NUT_PREC_PM = ( +3.558 +0.121 -0.064 +0.016 +0.025 )
Note that for the RA and PM (prime meridian) assignments, the ith term
is the coefficient of sin(Ei) in the expression used in the IAU model,
while for the DEC assignment, the ith term is the coefficient of
cos(Ei) in the expression used in the IAU model.
NAIF software expects the models for satellite orientation to
follow the form of the model shown here: the polynomial portions of the
RA, DEC, and W expressions are expected to be quadratic, the
trigonometric terms for RA and W (satellite prime meridian) are expected
to be linear combinations of sines of nutation precession angles, the
trigonometric terms for DEC are expected to be linear combinations of
cosines of nutation precession angles, and the polynomials for the
nutation precession angles themselves are expected to be linear.
Eventually, the software will handle more complex expressions, we
expect.
Shape models
There is only one kind of shape model supported by the NAIF Toolkit
software at present: the triaxial ellipsoid. The 1994 IAU report does
not use any other models.
For each body, three radii are listed: The first number is
the largest equatorial radius (the length of the semi-axis
containing the prime meridian), the second number is the smaller
equatorial radius, and the third is the polar radius.
Example: Radii of the Earth.
BODY399_RADII = ( 6378.14 6378.14 6356.75 )
Body numbers and names
--------------------------------------------------------
The NAIF ID codes used in this file are
4 Mars barycenter
499 Mars
401 Phobos
402 Deimos
See the NAIF_IDS Required Reading for the full list of ID
codes recognized by the SPICE system.
Orientation constants for Mars and its natural satellites
---------------------------------------------------------
Mars
Old values:
Values shown are from the 1997 IAU report.
body499_pole_ra = ( 317.681 -0.108 0. )
body499_pole_dec = ( +52.886 -0.061 0. )
body499_pm = ( 176.868 +350.8919830 0. )
Old nutation precession angles match those in the 2000 report.
Current values:
\begindata
BODY499_POLE_RA = ( 317.68143 -0.1061 0. )
BODY499_POLE_DEC = ( 52.88650 -0.0609 0. )
BODY499_PM = ( 176.630 350.89198226 )
\begintext
Source [2] specifies the following value for the lambda_a term
(BODY4_LONG_AXIS ) for Mars.
This term is the POSITIVE WEST LONGITUDE, measured from the prime
meridian, of the longest axis of the ellipsoid representing the ``mean
planet surface,'' as the article states.
body499_long_axis = ( 110. )
Source [3] specifies the lambda_a value
body499_long_axis = ( 104.9194 )
We list these lambda_a values for completeness. The IAU gives equal
values for both equatorial radii, so the lambda_a offset does not
apply to the IAU model.
The 2000 IAU report defines M2, the second nutation precession angle,
by:
2
192.93 + 1128.4096700 d + 8.864 T
We truncate the M2 series to a linear expression, because the PCK
software cannot handle the quadratic term.
Again, the linear terms are scaled by 36525.0:
-0.4357640000000000 --> -15916.28010000000
1128.409670000000 --> 41215163.19675000
-1.8151000000000000E-02 --> -662.9652750000000
We also introduce a fourth nutation precession angle, which
is the pi/2-complement of the third angle. This angle is used
in computing the prime meridian location for Deimos. See the
discussion of this angle below in the section containing orientation
constants for Deimos.
\begindata
BODY4_NUT_PREC_ANGLES = ( 169.51 -15916.2801
192.93 +41215163.19675
53.47 -662.965275
36.53 +662.965275 )
\begintext
Satellites of Mars
Phobos
Old values:
Values are unchanged in the 2000 report.
Current values:
The quadratic prime meridian term is scaled by 1/36525**2:
8.864000000000000 ---> 6.6443009930565219E-09
\begindata
BODY401_POLE_RA = ( 317.68 -0.108 0. )
BODY401_POLE_DEC = ( +52.90 -0.061 0. )
BODY401_PM = ( 35.06
+1128.8445850
6.6443009930565219D-09 )
BODY401_LONG_AXIS = ( 0. )
BODY401_NUT_PREC_RA = ( +1.79 0. 0. 0. )
BODY401_NUT_PREC_DEC = ( -1.08 0. 0. 0. )
BODY401_NUT_PREC_PM = ( -1.42 -0.78 0. 0. )
\begintext
Deimos
Old values:
Values are unchanged in the 2000 report.
New values:
The Deimos prime meridian expression is:
2
W = 79.41 + 285.1618970 d - 0.520 T - 2.58 sin M
3
+ 0.19 cos M .
3
At the present time, the constants kernel software (the routine
BODEUL in particular) cannot handle the cosine term directly,
but we can represent it as
0.19 sin M
4
where
M = 90.D0 - M
4 3
Therefore, the nutation precession angle assignments for Phobos
and Deimos contain four coefficients rather than three.
The quadratic prime meridian term is scaled by 1/36525**2:
-0.5200000000000000 ---> -3.8978300049519307E-10
\begindata
BODY402_POLE_RA = ( 316.65 -0.108 0. )
BODY402_POLE_DEC = ( +53.52 -0.061 0. )
BODY402_PM = ( 79.41 +285.1618970 -3.897830D-10 )
BODY402_LONG_AXIS = ( 0. )
BODY402_NUT_PREC_RA = ( 0. 0. +2.98 0. )
BODY402_NUT_PREC_DEC = ( 0. 0. -1.78 0. )
BODY402_NUT_PREC_PM = ( 0. 0. -2.58 0.19 )
\begintext
Radii for Mars and its natural satellites
--------------------------------------------------------
Mars
Old values:
Values are from the 1997 IAU report.
body499_radii = ( 3397. 3397. 3375. )
Current values:
Per the source [1], the polar radius is the average of the
north polar radius:
3373.19 km
and the south polar radius:
3379.21 km
\begindata
BODY499_RADII = ( 3396.19 3396.19 3376.20 )
\begintext
Satellites of Mars
Old values:
Values for Phobos and Deimos are unchanged in the 2000 report.
Current values:
\begindata
BODY401_RADII = ( 13.4 11.2 9.2 )
BODY402_RADII = ( 7.5 6.1 5.2 )
\begintext