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