devNotes 7-08-2017 Orbital Mechanics

https://nssdc.gsfc.nasa.gov/planetary/factsheet/

 

Pluto Mean Orbital Elements (J2000)

Semimajor axis (AU)                 39.48168677  
Orbital eccentricity                 0.24880766  
Orbital inclination (deg)           17.14175  
Longitude of ascending node (deg)  110.30347  
Longitude of perihelion (deg)      224.06676  
Mean longitude (deg)               238.92881

 

#include "sofa.h"

int iauPlan94(double date1, double date2, int np, double pv[2][3])
/*
**  - - - - - - - - - -
**   i a u P l a n 9 4
**  - - - - - - - - - -
**
**  This function is part of the International Astronomical Union's
**  SOFA (Standards Of Fundamental Astronomy) software collection.
**
**  Status:  support function.
**
**  Approximate heliocentric position and velocity of a nominated major
**  planet:  Mercury, Venus, EMB, Mars, Jupiter, Saturn, Uranus or
**  Neptune (but not the Earth itself).
**
**  Given:
**     date1  double       TDB date part A (Note 1)
**     date2  double       TDB date part B (Note 1)
**     np     int          planet (1=Mercury, 2=Venus, 3=EMB, 4=Mars,
**                             5=Jupiter, 6=Saturn, 7=Uranus, 8=Neptune)
**
**  Returned (argument):
**     pv     double[2][3] planet p,v (heliocentric, J2000.0, AU,AU/d)
**
**  Returned (function value):
**            int          status: -1 = illegal NP (outside 1-8)
**                                  0 = OK
**                                 +1 = warning: year outside 1000-3000
**                                 +2 = warning: failed to converge
**
**  Notes:
**
**  1) The date date1+date2 is in the TDB time scale (in practice TT can
**     be used) and is a Julian Date, apportioned in any convenient way
**     between the two arguments.  For example, JD(TDB)=2450123.7 could
**     be expressed in any of these ways, among others:
**
**            date1          date2
**
**         2450123.7           0.0       (JD method)
**         2451545.0       -1421.3       (J2000 method)
**         2400000.5       50123.2       (MJD method)
**         2450123.5           0.2       (date & time method)
**
**     The JD method is the most natural and convenient to use in cases
**     where the loss of several decimal digits of resolution is
**     acceptable.  The J2000 method is best matched to the way the
**     argument is handled internally and will deliver the optimum
**     resolution.  The MJD method and the date & time methods are both
**     good compromises between resolution and convenience.  The limited
**     accuracy of the present algorithm is such that any of the methods
**     is satisfactory.
**
**  2) If an np value outside the range 1-8 is supplied, an error status
**     (function value -1) is returned and the pv vector set to zeroes.
**
**  3) For np=3 the result is for the Earth-Moon Barycenter.  To obtain
**     the heliocentric position and velocity of the Earth, use instead
**     the SOFA function iauEpv00.
**
**  4) On successful return, the array pv contains the following:
**
**        pv[0][0]   x      }
**        pv[0][1]   y      } heliocentric position, AU
**        pv[0][2]   z      }
**
**        pv[1][0]   xdot   }
**        pv[1][1]   ydot   } heliocentric velocity, AU/d
**        pv[1][2]   zdot   }
**
**     The reference frame is equatorial and is with respect to the
**     mean equator and equinox of epoch J2000.0.
**
**  5) The algorithm is due to J.L. Simon, P. Bretagnon, J. Chapront,
**     M. Chapront-Touze, G. Francou and J. Laskar (Bureau des
**     Longitudes, Paris, France).  From comparisons with JPL
**     ephemeris DE102, they quote the following maximum errors
**     over the interval 1800-2050:
**
**                     L (arcsec)    B (arcsec)      R (km)
**
**        Mercury          4             1             300
**        Venus            5             1             800
**        EMB              6             1            1000
**        Mars            17             1            7700
**        Jupiter         71             5           76000
**        Saturn          81            13          267000
**        Uranus          86             7          712000
**        Neptune         11             1          253000
**
**     Over the interval 1000-3000, they report that the accuracy is no
**     worse than 1.5 times that over 1800-2050.  Outside 1000-3000 the
**     accuracy declines.
**
**     Comparisons of the present function with the JPL DE200 ephemeris
**     give the following RMS errors over the interval 1960-2025:
**
**                      position (km)     velocity (m/s)
**
**        Mercury            334               0.437
**        Venus             1060               0.855
**        EMB               2010               0.815
**        Mars              7690               1.98
**        Jupiter          71700               7.70
**        Saturn          199000              19.4
**        Uranus          564000              16.4
**        Neptune         158000              14.4
**
**     Comparisons against DE200 over the interval 1800-2100 gave the
**     following maximum absolute differences.  (The results using
**     DE406 were essentially the same.)
**
**                   L (arcsec)   B (arcsec)     R (km)   Rdot (m/s)
**
**        Mercury        7            1            500       0.7
**        Venus          7            1           1100       0.9
**        EMB            9            1           1300       1.0
**        Mars          26            1           9000       2.5
**        Jupiter       78            6          82000       8.2
**        Saturn        87           14         263000      24.6
**        Uranus        86            7         661000      27.4
**        Neptune       11            2         248000      21.4
**
**  6) The present SOFA re-implementation of the original Simon et al.
**     Fortran code differs from the original in the following respects:
**
**       *  C instead of Fortran.
**
**       *  The date is supplied in two parts.
**
**       *  The result is returned only in equatorial Cartesian form;
**          the ecliptic longitude, latitude and radius vector are not
**          returned.
**
**       *  The result is in the J2000.0 equatorial frame, not ecliptic.
**
**       *  More is done in-line: there are fewer calls to subroutines.
**
**       *  Different error/warning status values are used.
**
**       *  A different Kepler's-equation-solver is used (avoiding
**          use of double precision complex).
**
**       *  Polynomials in t are nested to minimize rounding errors.
**
**       *  Explicit double constants are used to avoid mixed-mode
**          expressions.
**
**     None of the above changes affects the result significantly.
**
**  7) The returned status indicates the most serious condition
**     encountered during execution of the function.  Illegal np is
**     considered the most serious, overriding failure to converge,
**     which in turn takes precedence over the remote date warning.
**
**  Called:
**     iauAnp       normalize angle into range 0 to 2pi
**
**  Reference:  Simon, J.L, Bretagnon, P., Chapront, J.,
**              Chapront-Touze, M., Francou, G., and Laskar, J.,
**              Astron. Astrophys. 282, 663 (1994).
**
**  This revision:  2013 June 18
**
**  SOFA release 2016-05-03
**
**  Copyright (C) 2016 IAU SOFA Board.  See notes at end.
*/
{
/* Gaussian constant */
   static const double GK = 0.017202098950;

/* Sin and cos of J2000.0 mean obliquity (IAU 1976) */
   static const double SINEPS = 0.3977771559319137;
   static const double COSEPS = 0.9174820620691818;

/* Maximum number of iterations allowed to solve Kepler's equation */
   static const int KMAX = 10;

   int jstat, i, k;
   double t, da, dl, de, dp, di, dom, dmu, arga, argl, am,
          ae, dae, ae2, at, r, v, si2, xq, xp, tl, xsw,
          xcw, xm2, xf, ci2, xms, xmc, xpxq2, x, y, z;

/* Planetary inverse masses */
   static const double amas[] = { 6023600.0,       /* Mercury */
                                   408523.5,       /* Venus   */
                                   328900.5,       /* EMB     */
                                  3098710.0,       /* Mars    */
                                     1047.355,     /* Jupiter */
                                     3498.5,       /* Saturn  */
                                    22869.0,       /* Uranus  */
                                    19314.0 };     /* Neptune */

/*
** Tables giving the mean Keplerian elements, limited to t^2 terms:
**
**   a       semi-major axis (AU)
**   dlm     mean longitude (degree and arcsecond)
**   e       eccentricity
**   pi      longitude of the perihelion (degree and arcsecond)
**   dinc    inclination (degree and arcsecond)
**   omega   longitude of the ascending node (degree and arcsecond)
*/

   static const double a[][3] = {
       {  0.3870983098,           0.0,     0.0 },  /* Mercury */
       {  0.7233298200,           0.0,     0.0 },  /* Venus   */
       {  1.0000010178,           0.0,     0.0 },  /* EMB     */
       {  1.5236793419,         3e-10,     0.0 },  /* Mars    */
       {  5.2026032092,     19132e-10, -39e-10 },  /* Jupiter */
       {  9.5549091915, -0.0000213896, 444e-10 },  /* Saturn  */
       { 19.2184460618,     -3716e-10, 979e-10 },  /* Uranus  */
       { 30.1103868694,    -16635e-10, 686e-10 }   /* Neptune */
   };

   static const double dlm[][3] = {
       { 252.25090552, 5381016286.88982,  -1.92789 },
       { 181.97980085, 2106641364.33548,   0.59381 },
       { 100.46645683, 1295977422.83429,  -2.04411 },
       { 355.43299958,  689050774.93988,   0.94264 },
       {  34.35151874,  109256603.77991, -30.60378 },
       {  50.07744430,   43996098.55732,  75.61614 },
       { 314.05500511,   15424811.93933,  -1.75083 },
       { 304.34866548,    7865503.20744,   0.21103 }
   };

   static const double e[][3] = {
       { 0.2056317526,  0.0002040653,    -28349e-10 },
       { 0.0067719164, -0.0004776521,     98127e-10 },
       { 0.0167086342, -0.0004203654, -0.0000126734 },
       { 0.0934006477,  0.0009048438,    -80641e-10 },
       { 0.0484979255,  0.0016322542, -0.0000471366 },
       { 0.0555481426, -0.0034664062, -0.0000643639 },
       { 0.0463812221, -0.0002729293,  0.0000078913 },
       { 0.0094557470,  0.0000603263,           0.0 }
   };

   static const double pi[][3] = {
       {  77.45611904,  5719.11590,   -4.83016 },
       { 131.56370300,   175.48640, -498.48184 },
       { 102.93734808, 11612.35290,   53.27577 },
       { 336.06023395, 15980.45908,  -62.32800 },
       {  14.33120687,  7758.75163,  259.95938 },
       {  93.05723748, 20395.49439,  190.25952 },
       { 173.00529106,  3215.56238,  -34.09288 },
       {  48.12027554,  1050.71912,   27.39717 }
   };

   static const double dinc[][3] = {
       { 7.00498625, -214.25629,   0.28977 },
       { 3.39466189,  -30.84437, -11.67836 },
       {        0.0,  469.97289,  -3.35053 },
       { 1.84972648, -293.31722,  -8.11830 },
       { 1.30326698,  -71.55890,  11.95297 },
       { 2.48887878,   91.85195, -17.66225 },
       { 0.77319689,  -60.72723,   1.25759 },
       { 1.76995259,    8.12333,   0.08135 }
   };

   static const double omega[][3] = {
       {  48.33089304,  -4515.21727,  -31.79892 },
       {  76.67992019, -10008.48154,  -51.32614 },
       { 174.87317577,  -8679.27034,   15.34191 },
       {  49.55809321, -10620.90088, -230.57416 },
       { 100.46440702,   6362.03561,  326.52178 },
       { 113.66550252,  -9240.19942,  -66.23743 },
       {  74.00595701,   2669.15033,  145.93964 },
       { 131.78405702,   -221.94322,   -0.78728 }
   };

/* Tables for trigonometric terms to be added to the mean elements of */
/* the semi-major axes */

   static const double kp[][9] = {
    {   69613, 75645, 88306, 59899, 15746, 71087, 142173,  3086,    0 },
    {   21863, 32794, 26934, 10931, 26250, 43725,  53867, 28939,    0 },
    {   16002, 21863, 32004, 10931, 14529, 16368,  15318, 32794,    0 },
    {    6345,  7818, 15636,  7077,  8184, 14163,   1107,  4872,    0 },
    {    1760,  1454,  1167,   880,   287,  2640,     19,  2047, 1454 },
    {     574,     0,   880,   287,    19,  1760,   1167,   306,  574 },
    {     204,     0,   177,  1265,     4,   385,    200,   208,  204 },
    {       0,   102,   106,     4,    98,  1367,    487,   204,    0 }
   };

   static const double ca[][9] = {
    {       4,    -13,    11,   -9,    -9,   -3,     -1,     4,     0 },
    {    -156,     59,   -42,    6,    19,  -20,    -10,   -12,     0 },
    {      64,   -152,    62,   -8,    32,  -41,     19,   -11,     0 },
    {     124,    621,  -145,  208,    54,  -57,     30,    15,     0 },
    {  -23437,  -2634,  6601, 6259, -1507,-1821,   2620, -2115, -1489 },
    {   62911,-119919, 79336,17814,-24241,12068,   8306, -4893,  8902 },
    {  389061,-262125,-44088, 8387,-22976,-2093,   -615, -9720,  6633 },
    { -412235,-157046,-31430,37817, -9740,  -13,  -7449,  9644,     0 }
   };

   static const double sa[][9] = {
    {     -29,    -1,     9,     6,    -6,     5,     4,     0,     0 },
    {     -48,  -125,   -26,   -37,    18,   -13,   -20,    -2,     0 },
    {    -150,   -46,    68,    54,    14,    24,   -28,    22,     0 },
    {    -621,   532,  -694,   -20,   192,   -94,    71,   -73,     0 },
    {  -14614,-19828, -5869,  1881, -4372, -2255,   782,   930,   913 },
    {  139737,     0, 24667, 51123, -5102,  7429, -4095, -1976, -9566 },
    { -138081,     0, 37205,-49039,-41901,-33872,-27037,-12474, 18797 },
    {       0, 28492,133236, 69654, 52322,-49577,-26430, -3593,     0 }
   };

/* Tables giving the trigonometric terms to be added to the mean */
/* elements of the mean longitudes */

   static const double kq[][10] = {
    {   3086,15746,69613,59899,75645,88306, 12661,  2658,    0,     0 },
    {  21863,32794,10931,   73, 4387,26934,  1473,  2157,    0,     0 },
    {     10,16002,21863,10931, 1473,32004,  4387,    73,    0,     0 },
    {     10, 6345, 7818, 1107,15636, 7077,  8184,   532,   10,     0 },
    {     19, 1760, 1454,  287, 1167,  880,   574,  2640,   19,  1454 },
    {     19,  574,  287,  306, 1760,   12,    31,    38,   19,   574 },
    {      4,  204,  177,    8,   31,  200,  1265,   102,    4,   204 },
    {      4,  102,  106,    8,   98, 1367,   487,   204,    4,   102 }
   };

   static const double cl[][10] = {
    {      21,   -95, -157,   41,   -5,   42,  23,  30,      0,     0 },
    {    -160,  -313, -235,   60,  -74,  -76, -27,  34,      0,     0 },
    {    -325,  -322,  -79,  232,  -52,   97,  55, -41,      0,     0 },
    {    2268,  -979,  802,  602, -668,  -33, 345, 201,    -55,     0 },
    {    7610, -4997,-7689,-5841,-2617, 1115,-748,-607,   6074,   354 },
    {  -18549, 30125,20012, -730,  824,   23,1289,-352, -14767, -2062 },
    { -135245,-14594, 4197,-4030,-5630,-2898,2540,-306,   2939,  1986 },
    {   89948,  2103, 8963, 2695, 3682, 1648, 866,-154,  -1963,  -283 }
   };

   static const double sl[][10] = {
    {   -342,   136,  -23,   62,   66,  -52, -33,    17,     0,     0 },
    {    524,  -149,  -35,  117,  151,  122, -71,   -62,     0,     0 },
    {   -105,  -137,  258,   35, -116,  -88,-112,   -80,     0,     0 },
    {    854,  -205, -936, -240,  140, -341, -97,  -232,   536,     0 },
    { -56980,  8016, 1012, 1448,-3024,-3710, 318,   503,  3767,   577 },
    { 138606,-13478,-4964, 1441,-1319,-1482, 427,  1236, -9167, -1918 },
    {  71234,-41116, 5334,-4935,-1848,   66, 434, -1748,  3780,  -701 },
    { -47645, 11647, 2166, 3194,  679,    0,-244,  -419, -2531,    48 }
   };

/*--------------------------------------------------------------------*/

/* Validate the planet number. */
   if ((np < 1) || (np > 8)) {
      jstat = -1;

   /* Reset the result in case of failure. */
      for (k = 0; k < 2; k++) {
         for (i = 0; i < 3; i++) {
            pv[k][i] = 0.0;
         }
      }

   } else {

   /* Decrement the planet number to start at zero. */
      np--;

   /* Time: Julian millennia since J2000.0. */
      t = ((date1 - DJ00) + date2) / DJM;

   /* OK status unless remote date. */
      jstat = fabs(t) <= 1.0 ? 0 : 1;

   /* Compute the mean elements. */
      da = a[np][0] +
          (a[np][1] +
           a[np][2] * t) * t;
      dl = (3600.0 * dlm[np][0] +
                    (dlm[np][1] +
                     dlm[np][2] * t) * t) * DAS2R;
      de = e[np][0] +
         ( e[np][1] +
           e[np][2] * t) * t;
      dp = iauAnpm((3600.0 * pi[np][0] +
                            (pi[np][1] +
                             pi[np][2] * t) * t) * DAS2R);
      di = (3600.0 * dinc[np][0] +
                    (dinc[np][1] +
                     dinc[np][2] * t) * t) * DAS2R;
      dom = iauAnpm((3600.0 * omega[np][0] +
                             (omega[np][1] +
                              omega[np][2] * t) * t) * DAS2R);

   /* Apply the trigonometric terms. */
      dmu = 0.35953620 * t;
      for (k = 0; k < 8; k++) {
         arga = kp[np][k] * dmu;
         argl = kq[np][k] * dmu;
         da += (ca[np][k] * cos(arga) +
                sa[np][k] * sin(arga)) * 1e-7;
         dl += (cl[np][k] * cos(argl) +
                sl[np][k] * sin(argl)) * 1e-7;
      }
      arga = kp[np][8] * dmu;
      da += t * (ca[np][8] * cos(arga) +
                 sa[np][8] * sin(arga)) * 1e-7;
      for (k = 8; k < 10; k++) {
         argl = kq[np][k] * dmu;
         dl += t * (cl[np][k] * cos(argl) +
                    sl[np][k] * sin(argl)) * 1e-7;
      }
      dl = fmod(dl, D2PI);

   /* Iterative soln. of Kepler's equation to get eccentric anomaly. */
      am = dl - dp;
      ae = am + de * sin(am);
      k = 0;
      dae = 1.0;
      while (k < KMAX && fabs(dae) > 1e-12) {
         dae = (am - ae + de * sin(ae)) / (1.0 - de * cos(ae));
         ae += dae;
         k++;
         if (k == KMAX-1) jstat = 2;
      }

   /* True anomaly. */
      ae2 = ae / 2.0;
      at = 2.0 * atan2(sqrt((1.0 + de) / (1.0 - de)) * sin(ae2),
                                                       cos(ae2));

   /* Distance (AU) and speed (radians per day). */
      r = da * (1.0 - de * cos(ae));
      v = GK * sqrt((1.0 + 1.0 / amas[np]) / (da * da * da));

      si2 = sin(di / 2.0);
      xq = si2 * cos(dom);
      xp = si2 * sin(dom);
      tl = at + dp;
      xsw = sin(tl);
      xcw = cos(tl);
      xm2 = 2.0 * (xp * xcw - xq * xsw);
      xf = da / sqrt(1  -  de * de);
      ci2 = cos(di / 2.0);
      xms = (de * sin(dp) + xsw) * xf;
      xmc = (de * cos(dp) + xcw) * xf;
      xpxq2 = 2 * xp * xq;

   /* Position (J2000.0 ecliptic x,y,z in AU). */
      x = r * (xcw - xm2 * xp);
      y = r * (xsw + xm2 * xq);
      z = r * (-xm2 * ci2);

   /* Rotate to equatorial. */
      pv[0][0] = x;
      pv[0][1] = y * COSEPS - z * SINEPS;
      pv[0][2] = y * SINEPS + z * COSEPS;

   /* Velocity (J2000.0 ecliptic xdot,ydot,zdot in AU/d). */
      x = v * (( -1.0 + 2.0 * xp * xp) * xms + xpxq2 * xmc);
      y = v * ((  1.0 - 2.0 * xq * xq) * xmc - xpxq2 * xms);
      z = v * (2.0 * ci2 * (xp * xms + xq * xmc));

   /* Rotate to equatorial. */
      pv[1][0] = x;
      pv[1][1] = y * COSEPS - z * SINEPS;
      pv[1][2] = y * SINEPS + z * COSEPS;

   }

/* Return the status. */
   return jstat;

/*----------------------------------------------------------------------
**
**  Copyright (C) 2016
**  Standards Of Fundamental Astronomy Board
**  of the International Astronomical Union.
**
**  =====================
**  SOFA Software License
**  =====================
**
**  NOTICE TO USER:
**
**  BY USING THIS SOFTWARE YOU ACCEPT THE FOLLOWING SIX TERMS AND
**  CONDITIONS WHICH APPLY TO ITS USE.
**
**  1. The Software is owned by the IAU SOFA Board ("SOFA").
**
**  2. Permission is granted to anyone to use the SOFA software for any
**     purpose, including commercial applications, free of charge and
**     without payment of royalties, subject to the conditions and
**     restrictions listed below.
**
**  3. You (the user) may copy and distribute SOFA source code to others,
**     and use and adapt its code and algorithms in your own software,
**     on a world-wide, royalty-free basis.  That portion of your
**     distribution that does not consist of intact and unchanged copies
**     of SOFA source code files is a "derived work" that must comply
**     with the following requirements:
**
**     a) Your work shall be marked or carry a statement that it
**        (i) uses routines and computations derived by you from
**        software provided by SOFA under license to you; and
**        (ii) does not itself constitute software provided by and/or
**        endorsed by SOFA.
**
**     b) The source code of your derived work must contain descriptions
**        of how the derived work is based upon, contains and/or differs
**        from the original SOFA software.
**
**     c) The names of all routines in your derived work shall not
**        include the prefix "iau" or "sofa" or trivial modifications
**        thereof such as changes of case.
**
**     d) The origin of the SOFA components of your derived work must
**        not be misrepresented;  you must not claim that you wrote the
**        original software, nor file a patent application for SOFA
**        software or algorithms embedded in the SOFA software.
**
**     e) These requirements must be reproduced intact in any source
**        distribution and shall apply to anyone to whom you have
**        granted a further right to modify the source code of your
**        derived work.
**
**     Note that, as originally distributed, the SOFA software is
**     intended to be a definitive implementation of the IAU standards,
**     and consequently third-party modifications are discouraged.  All
**     variations, no matter how minor, must be explicitly marked as
**     such, as explained above.
**
**  4. You shall not cause the SOFA software to be brought into
**     disrepute, either by misuse, or use for inappropriate tasks, or
**     by inappropriate modification.
**
**  5. The SOFA software is provided "as is" and SOFA makes no warranty
**     as to its use or performance.   SOFA does not and cannot warrant
**     the performance or results which the user may obtain by using the
**     SOFA software.  SOFA makes no warranties, express or implied, as
**     to non-infringement of third party rights, merchantability, or
**     fitness for any particular purpose.  In no event will SOFA be
**     liable to the user for any consequential, incidental, or special
**     damages, including any lost profits or lost savings, even if a
**     SOFA representative has been advised of such damages, or for any
**     claim by any third party.
**
**  6. The provision of any version of the SOFA software under the terms
**     and conditions specified herein does not imply that future
**     versions will also be made available under the same terms and
**     conditions.
*
**  In any published work or commercial product which uses the SOFA
**  software directly, acknowledgement (see www.iausofa.org) is
**  appreciated.
**
**  Correspondence concerning SOFA software should be addressed as
**  follows:
**
**      By email:  sofa@ukho.gov.uk
**      By post:   IAU SOFA Center
**                 HM Nautical Almanac Office
**                 UK Hydrographic Office
**                 Admiralty Way, Taunton
**                 Somerset, TA1 2DN
**                 United Kingdom
**
**--------------------------------------------------------------------*/
}

 

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