deep.cpp

#include <math.h>
#include "norad.h"
#include "norad_in.h"

      /* omega_E = number of (sidereal) rotations of the earth per UT day: */
const double omega_E = 1.00273790934;
#ifdef USE_ACCURATE_ANOMALISTICS
         /* The anomalistic month is the mean time it takes the moon to go
            from perigee to perigee.  The anomalistic year is the mean time
            it takes the earth to go from perihelion to perihelion.
            The following lines compute the "correct" mean motions of
            the earth and sun: zns_per_day is the rate of change of
            the earth's mean anomaly,  in radians per day,  and the 'znl'
            quantities give similar rates for the moon.
               Problem is,  the original SxPx sources give values that are
            close to,  but not exactly equal to,  these values.  The
            "new" values are probably improvements from further observations,
            but if you actually used them,  you'd break compatibility with
            older implementations,  and wouldn't match up with the way
            NORAD and others actually compute TLEs.  So the following few
            lines should be regarded as explanatory;  we're stuck with using
            the older,  less accurate SxPx values.  */
const double days_per_anomalistic_month =  27.554551;
const double days_per_anomalistic_year = 365.259635864;
const double zns_per_day = twopi / days_per_anomalistic_year;
const double zns_per_min = zns_per_day / minutes_per_day;
const double znl_per_day = twopi / days_per_anomalistic_month;
const double znl_per_min = znl_per_day / minutes_per_day;
         /* thdt = angular velocity of the earth,  in radians/minute. */
         /* Again,  we have to use a less accurate value from the original */
         /* SxPx,  to replicate everybody else's results.                  */
const double thdt = twopi * omega_E / minutes_per_day;
#else
const double zns_per_min = 1.19459E-5;
const double zns_per_day = 0.017201977;
const double znl_per_day = 0.228027132;
const double znl_per_min = 1.5835218E-4;
const double thdt =   4.37526908801129966e-3;
#endif
         /* zes = mean eccentricity of earth's orbit */
         /* zel = mean eccentricity of the moon's orbit */
#define zes      0.01675
#define zel      0.05490

/* thetag: computes Greenwich sidereal time,  as an angle in radians
from 0 to 2*pi,  for a given UT0 JD. */

static inline double ThetaG( const double jd)
{
                 /* Reference:  The 1992 Astronomical Almanac, page B6. */
                 /* Earth rotations per sidereal day (non-constant) */
  const double UT = fmod( jd + .5, 1.);
  const double seconds_per_day = 86400.;
  const double jd_2000 = 2451545.0;   /* 1.5 Jan 2000 = JD 2451545. */
  double t_cen, GMST, rval;

  t_cen = (jd - UT - jd_2000) / 36525.;
  GMST = 24110.54841 + t_cen * (8640184.812866 + t_cen *
                           (0.093104 - t_cen * 6.2E-6));
  GMST = fmod( GMST / seconds_per_day + omega_E * UT, 1.);
  if( GMST < 0.)
     GMST += 1.;
  rval = twopi * GMST;

  return( rval);
} /*Function thetag*/

      /* Previously,  the integration step was given as two variables:      */
      /* 'stepp' (positive step = +720) and 'stepn' (negative step = -720). */
      /* Exactly why this should be made a variable,  much less _different_ */
      /* variables for positive and negative,  is entirely unclear...       */
      /* (8 Apr 2003) INTEGRATION_STEP is now a maximum integration step.   */
      /* The code in 'dpsec' splits the integration range into equally-sized */
      /* pieces of 720 minutes (half a day) or smaller.                      */
      /* (25 Aug 2006) INTEGRATION_STEP is now the variable                  */
      /* 'dpsec_integration_step' so I can experiment with different         */
      /* integration techniques & evaluate their errors.                     */

static double dpsec_integration_step = 720.;
static int dpsec_integration_order = 2;
static int is_dundee_compliant = 0;

void DLL_FUNC sxpx_set_implementation_param( const int param_index,
                                              const int new_param)
{
   switch( param_index)
      {
      case SXPX_DPSEC_INTEGRATION_ORDER:
         dpsec_integration_order = new_param;
         break;
      case SXPX_DUNDEE_COMPLIANCE:
         is_dundee_compliant = new_param;
         break;
      }
}

void DLL_FUNC sxpx_set_dpsec_integration_step( const double new_step_size)
{
   dpsec_integration_step = new_step_size;
}

static inline double eval_cubic_poly( const double x, const double constant,
               const double linear, const double quadratic_term,
               const double cubic_term)
{
   return( constant + x * (linear + x * (quadratic_term + x * cubic_term)));
}

/* DEEP */
void Deep_dpinit( const tle_t *tle, deep_arg_t *deep_arg)
{
   const double sinq = sin(tle->xnodeo);
   const double cosq = cos(tle->xnodeo);
   const double aqnv = 1/deep_arg->aodp;
   const double c1ss   =  2.9864797E-6;
           /* 1900 Jan 0.5 = JD 2415020. */
   const double days_since_1900 = tle->epoch - 2415020.;
           /* zcosi, zsini start as cos & sin of obliquity of earth's  */
           /* orbit = 23.444100 degrees... matches obliquity in 1963; */
           /* probably just a slightly inaccurate value: */
   const double zcosi0 = 0.91744867;
   const double zsini0 = 0.39785416;
   double zcosi = zcosi0;
   double zsini = zsini0;
           /* zcosg, zsing start as cos & sin of -78.779197 degrees */
   double zsing = -0.98088458;
   double zcosg =  0.1945905;
   double bfact, cc = c1ss, se;
   double ze = zes, zn = zns_per_min;
   double sgh, sh, si;
   double zsinh = sinq, zcosh = cosq;
   double sl;
   int iteration;

   deep_arg->thgr = ThetaG( tle->epoch);
   deep_arg->xnq = deep_arg->xnodp;
   deep_arg->xqncl = tle->xincl;
   deep_arg->omegaq = tle->omegao;

   /* If the epoch has changed,  recompute (or initialize) the lunar and */
   /* solar terms... except that now that zcosil, etc. are within the    */
   /* deep_arg structure,  instead of static,  they must _always_ be     */
   /* recomputed.  So I've commented out the 'if' part.  (Revision made  */
   /* 14 May 2005)                                                       */

/* if( days_since_1900 != deep_arg->preep)   */
      {
      const double lunar_asc_node = 4.5236020 - 9.2422029E-4 * days_since_1900;
      const double sin_asc_node = sin(lunar_asc_node);
      const double cos_asc_node = cos(lunar_asc_node);
      const double c_minus_gam = znl_per_day * days_since_1900 - 1.1151842;
            /* gam = longitude of perigee for the moon,  in radians: */
      const double gam = 5.8351514 + 0.0019443680 * days_since_1900;
      double zx, zy;

      deep_arg->preep = days_since_1900;
      deep_arg->zcosil = 0.91375164 - 0.03568096 * cos_asc_node;
      deep_arg->zsinil = sqrt(1. - deep_arg->zcosil * deep_arg->zcosil);
      deep_arg->zsinhl = 0.089683511 * sin_asc_node / deep_arg->zsinil;
      deep_arg->zcoshl = sqrt(1. - deep_arg->zsinhl*deep_arg->zsinhl);
      deep_arg->zmol = FMod2p( c_minus_gam);
      zx = zsini0 * sin_asc_node / deep_arg->zsinil;
      zy = deep_arg->zcoshl * cos_asc_node + zcosi0 * deep_arg->zsinhl * sin_asc_node;
      zx = atan2( zx, zy) + gam - lunar_asc_node;
      deep_arg->zcosgl = cos( zx);
      deep_arg->zsingl = sin( zx);
      deep_arg->zmos = FMod2p( 6.2565837
                     + zns_per_day * days_since_1900);
      } /* End if( days_since_1900 != deep_arg->preep) */

   /* Do solar terms */
   deep_arg->savtsn = 1E20;

   /* There was previously some convoluted logic here,  but it boils    */
   /* down to this:  we compute the solar terms,  then the lunar terms. */
   /* On a second pass,  we recompute the solar terms,  taking advantage */
   /* of the improved data that resulted from computing lunar terms.     */
   for( iteration = 0; iteration < 2; iteration++)
      {
      const double c1l = 4.7968065E-7;
      const double a1 = zcosg * zcosh + zsing * zcosi * zsinh;
      const double a3 = -zsing * zcosh + zcosg * zcosi * zsinh;
      const double a7 = -zcosg * zsinh + zsing * zcosi * zcosh;
      const double a8 = zsing * zsini;
      const double a9 = zsing * zsinh + zcosg * zcosi * zcosh;
      const double a10 = zcosg * zsini;
      const double a2 = deep_arg->cosio * a7 + deep_arg->sinio * a8;
      const double a4 = deep_arg->cosio * a9 + deep_arg->sinio * a10;
      const double a5 = -deep_arg->sinio * a7 + deep_arg->cosio * a8;
      const double a6 = -deep_arg->sinio * a9 + deep_arg->cosio * a10;
      const double x1 = a1 * deep_arg->cosg + a2 * deep_arg->sing;
      const double x2 = a3 * deep_arg->cosg + a4 * deep_arg->sing;
      const double x3 = -a1 * deep_arg->sing + a2 * deep_arg->cosg;
      const double x4 = -a3 * deep_arg->sing + a4 * deep_arg->cosg;
      const double x5 = a5 * deep_arg->sing;
      const double x6 = a6 * deep_arg->sing;
      const double x7 = a5 * deep_arg->cosg;
      const double x8 = a6 * deep_arg->cosg;
      const double z31 = 12 * x1 * x1 - 3 * x3 * x3;
      const double z32 = 24 * x1 * x2 - 6 * x3 * x4;
      const double z33 = 12 * x2 * x2 - 3 * x4 * x4;
      const double z11 = -6 * a1 * a5 + deep_arg->eosq * (-24 * x1 * x7 - 6 * x3 * x5);
      const double z12 = -6 * (a1 * a6 + a3 * a5) +  deep_arg->eosq *
                (-24 * (x2 * x7 + x1 * x8) - 6 * (x3 * x6 + x4 * x5));
      const double z13 = -6 * a3 * a6 + deep_arg->eosq * (-24 * x2 * x8 - 6 * x4 * x6);
      const double z21 = 6 * a2 * a5 + deep_arg->eosq * (24 * x1 * x5 - 6 * x3 * x7);
      const double z22 = 6 * (a4 * a5 + a2 * a6) +  deep_arg->eosq *
                (24 * (x2 * x5 + x1 * x6) - 6 * (x4 * x7 + x3 * x8));
      const double z23 = 6 * a4 * a6 + deep_arg->eosq * (24 * x2 * x6 - 6 * x4 * x8);
      const double s3 = cc / deep_arg->xnq;
      const double s2 = -0.5 * s3 / deep_arg->betao;
      const double s4 = s3 * deep_arg->betao;
      const double s1 = -15 * tle->eo * s4;
      const double s5 = x1 * x3 + x2 * x4;
      const double s6 = x2 * x3 + x1 * x4;
      const double s7 = x2 * x4 - x1 * x3;
      double z1 = 3 * (a1 * a1 + a2 * a2) + z31 * deep_arg->eosq;
      double z2 = 6 * (a1 * a3 + a2 * a4) + z32 * deep_arg->eosq;
      double z3 = 3 * (a3 * a3 + a4 * a4) + z33 * deep_arg->eosq;

      z1 = z1 + z1 + deep_arg->betao2 * z31;
      z2 = z2 + z2 + deep_arg->betao2 * z32;
      z3 = z3 + z3 + deep_arg->betao2 * z33;
      se = s1*zn*s5;
      si = s2*zn*(z11+z13);
      sl = -zn*s3*(z1+z3-14-6*deep_arg->eosq);
      sgh = s4*zn*(z31+z33-6);
      if (deep_arg->xqncl < pi / 60.)      /* pi / 60 radians = 3 degrees */
         sh = 0;
      else
         sh = -zn*s2*(z21+z23);
      deep_arg->ee2 = 2*s1*s6;
      deep_arg->e3 = 2*s1*s7;
      deep_arg->xi2 = 2*s2*z12;
      deep_arg->xi3 = 2*s2*(z13-z11);
      deep_arg->xl2 = -2*s3*z2;
      deep_arg->xl3 = -2*s3*(z3-z1);
      deep_arg->xl4 = -2*s3*(-21-9*deep_arg->eosq)*ze;
      deep_arg->xgh2 = 2*s4*z32;
      deep_arg->xgh3 = 2*s4*(z33-z31);
      deep_arg->xgh4 = -18*s4*ze;
      deep_arg->xh2 = -2*s2*z22;
      deep_arg->xh3 = -2*s2*(z23-z21);

      if( !iteration)   /* we compute lunar terms only on the first pass: */
         {
         deep_arg->sse = se;
         deep_arg->ssi = si;
         deep_arg->ssl = sl;
         deep_arg->ssh = (deep_arg->sinio ? sh / deep_arg->sinio : 0.);
         deep_arg->ssg = sgh-deep_arg->cosio*deep_arg->ssh;
         deep_arg->se2 = deep_arg->ee2;
         deep_arg->si2 = deep_arg->xi2;
         deep_arg->sl2 = deep_arg->xl2;
         deep_arg->sgh2 = deep_arg->xgh2;
         deep_arg->sh2 = deep_arg->xh2;
         deep_arg->se3 = deep_arg->e3;
         deep_arg->si3 = deep_arg->xi3;
         deep_arg->sl3 = deep_arg->xl3;
         deep_arg->sgh3 = deep_arg->xgh3;
         deep_arg->sh3 = deep_arg->xh3;
         deep_arg->sl4 = deep_arg->xl4;
         deep_arg->sgh4 = deep_arg->xgh4;
         zcosg = deep_arg->zcosgl;
         zsing = deep_arg->zsingl;
         zcosi = deep_arg->zcosil;
         zsini = deep_arg->zsinil;
         zcosh = deep_arg->zcoshl*cosq+deep_arg->zsinhl*sinq;
         zsinh = sinq*deep_arg->zcoshl-cosq*deep_arg->zsinhl;
         zn = znl_per_min;
         cc = c1l;
         ze = zel;
         }
      }

   deep_arg->sse += se;
   deep_arg->ssi += si;
   deep_arg->ssl += sl;
   deep_arg->ssg += sgh;
   if( deep_arg->sinio)
      {
      deep_arg->ssg -= sh * deep_arg->cosio / deep_arg->sinio;
      deep_arg->ssh += sh / deep_arg->sinio;
      }

         /* "if mean motion is 1.893053 to 2.117652 revs/day, and ecc >= .5" */
   if( deep_arg->xnq >= 0.00826 && deep_arg->xnq <= 0.00924 && tle->eo >= .5)
      {           /* start of 12-hour orbit, e >.5 section */
                  /* 'root##' variables are somewhat inaccurate values for */
                  /* a few fully normalized sectorial/tesseral spherical   */
                  /* harmonics of the Earth's gravitational potential:     */
      const double root22 = 1.7891679E-6;
      const double root32 = 3.7393792E-7;
      const double root44 = 7.3636953E-9;
      const double root52 = 1.1428639E-7;
      const double root54 = 2.1765803E-9;
      const double g201 = -0.306 - (tle->eo - 0.64) * 0.440;
      const double sini2 = deep_arg->sinio*deep_arg->sinio;
      const double f220 = 0.75*(1+2*deep_arg->cosio+deep_arg->cosio2);
      const double f221 = 1.5 * sini2;
      const double f321 = 1.875 * deep_arg->sinio * (1 - 2 *\
               deep_arg->cosio - 3 * deep_arg->cosio2);
      const double f322 = -1.875*deep_arg->sinio*(1+2*
               deep_arg->cosio-3*deep_arg->cosio2);
      const double f441 = 35 * sini2 * f220;
      const double f442 = 39.3750 * sini2 * sini2;
      const double f522 = 9.84375*deep_arg->sinio*(sini2*(1-2*deep_arg->cosio-5*
                 deep_arg->cosio2)+0.33333333*(-2+4*deep_arg->cosio+
                 6*deep_arg->cosio2));
      const double f523 = deep_arg->sinio*(4.92187512*sini2*(-2-4*
                 deep_arg->cosio+10*deep_arg->cosio2)+6.56250012
                 *(1+2*deep_arg->cosio-3*deep_arg->cosio2));
      const double f542 = 29.53125*deep_arg->sinio*(2-8*
                 deep_arg->cosio+deep_arg->cosio2*
                 (-12+8*deep_arg->cosio+10*deep_arg->cosio2));
      const double f543 = 29.53125*deep_arg->sinio*(-2-8*deep_arg->cosio+
                 deep_arg->cosio2*(12+8*deep_arg->cosio-10*
                 deep_arg->cosio2));
      double g410, g422, g520, g521, g532, g533;
      double g211, g310, g322;
      double temp, temp1;

      deep_arg->resonance_flag = 1;       /* it _is_ resonant... */
      deep_arg->synchronous_flag = 0;     /* but it's not synchronous */
             /* Geopotential resonance initialization for 12 hour orbits: */
      if (tle->eo <= 0.65)
         {
         g211 = 3.616-13.247*tle->eo+16.290*deep_arg->eosq;
         g310 = eval_cubic_poly( tle->eo, -19.302, 117.390, -228.419, 156.591);
         g322 = eval_cubic_poly( tle->eo, -18.9068, 109.7927, -214.6334, 146.5816);
         g410 = eval_cubic_poly( tle->eo, -41.122, 242.694, -471.094, 313.953);
         g422 = eval_cubic_poly( tle->eo, -146.407, 841.880, -1629.014, 1083.435);
         g520 = eval_cubic_poly( tle->eo, -532.114, 3017.977, -5740.032, 3708.276);
                               /* NOTE: quadratic coeff was 5740 */
         }
      else
         {
         g211 = eval_cubic_poly( tle->eo, -72.099, 331.819, -508.738, 266.724);
         g310 = eval_cubic_poly( tle->eo, -346.844, 1582.851, -2415.925, 1246.113);
         g322 = eval_cubic_poly( tle->eo, -342.585, 1554.908, -2366.899, 1215.972);
         g410 = eval_cubic_poly( tle->eo, -1052.797, 4758.686, -7193.992, 3651.957);
         g422 = eval_cubic_poly( tle->eo, -3581.69, 16178.11, -24462.77, 12422.52);
         if (tle->eo <= 0.715)
            g520 = eval_cubic_poly( tle->eo, 1464.74, -4664.75, 3763.64, 0.);
         else
            g520 = eval_cubic_poly( tle->eo, -5149.66, 29936.92, -54087.36, 31324.56);
         } /* End if (tle->eo <= 0.65) */

      if (tle->eo < 0.7)
         {
         g533 = eval_cubic_poly( tle->eo, -919.2277, 4988.61, -9064.77, 5542.21);
         g521 = eval_cubic_poly( tle->eo, -822.71072, 4568.6173, -8491.4146, 5337.524);
         g532 = eval_cubic_poly( tle->eo, -853.666, 4690.25, -8624.77, 5341.4);
         }
      else
         {
         g533 = eval_cubic_poly( tle->eo, -37995.78, 161616.52, -229838.2, 109377.94);
         g521 = eval_cubic_poly( tle->eo, -51752.104, 218913.95, -309468.16, 146349.42);
         g532 = eval_cubic_poly( tle->eo, -40023.88, 170470.89, -242699.48, 115605.82);
         } /* End if (tle->eo <= 0.7) */

      temp1 = 3 * deep_arg->xnq * deep_arg->xnq * aqnv * aqnv;
      temp = temp1*root22;
      deep_arg->d2201 = temp * f220 * g201;
      deep_arg->d2211 = temp * f221 * g211;
      temp1 *= aqnv;
      temp = temp1*root32;
      deep_arg->d3210 = temp * f321 * g310;
      deep_arg->d3222 = temp * f322 * g322;
      temp1 *= aqnv;
      temp = 2*temp1*root44;
      deep_arg->d4410 = temp * f441 * g410;
      deep_arg->d4422 = temp * f442 * g422;
      temp1 *= aqnv;
      temp = temp1*root52;
      deep_arg->d5220 = temp * f522 * g520;
      deep_arg->d5232 = temp * f523 * g532;
      temp = 2*temp1*root54;
      deep_arg->d5421 = temp * f542 * g521;
      deep_arg->d5433 = temp * f543 * g533;
      deep_arg->xlamo = tle->xmo+tle->xnodeo+tle->xnodeo-deep_arg->thgr-deep_arg->thgr;
      bfact = deep_arg->xmdot + deep_arg->xnodot+
                   deep_arg->xnodot - thdt - thdt;
      bfact += deep_arg->ssl + deep_arg->ssh + deep_arg->ssh;
      }        /* end of 12-hour orbit, e >.5 section */
   else if( deep_arg->xnq < 1.2 * twopi / minutes_per_day &&
            deep_arg->xnq > 0.8 * twopi / minutes_per_day)
      {                        /* "if mean motion is .8 to 1.2 revs/day" */
      const double q22    =  1.7891679E-6;
      const double q31    =  2.1460748E-6;
      const double q33    =  2.2123015E-7;
      const double cosio_plus_1 = 1. + deep_arg->cosio;
      const double g200 = 1+deep_arg->eosq*(-2.5+0.8125*deep_arg->eosq);
      const double g300 = 1+deep_arg->eosq*(-6+6.60937*deep_arg->eosq);
      const double f311 = 0.9375*deep_arg->sinio*deep_arg->sinio*
             (1+3*deep_arg->cosio)-0.75*cosio_plus_1;
      const double g310 = 1+2*deep_arg->eosq;
      const double f220 = 0.75 * cosio_plus_1 * cosio_plus_1;
      const double f330 = 2.5 * f220 * cosio_plus_1;

      deep_arg->resonance_flag = deep_arg->synchronous_flag = 1;
      /* Synchronous resonance terms initialization */
      deep_arg->del1 = 3*deep_arg->xnq*deep_arg->xnq*aqnv*aqnv;
      deep_arg->del2 = 2*deep_arg->del1*f220*g200*q22;
      deep_arg->del3 = 3*deep_arg->del1*f330*g300*q33*aqnv;
      deep_arg->del1 *= f311*g310*q31*aqnv;
      deep_arg->xlamo = tle->xmo+tle->xnodeo+tle->omegao-deep_arg->thgr;
      bfact = deep_arg->xmdot + deep_arg->omgdot + deep_arg->xnodot - thdt;
      bfact = bfact+deep_arg->ssl+deep_arg->ssg+deep_arg->ssh;
      } /* End of geosych case */
   else              /* it's neither a high-e 12-hr orbit nor a geosynch: */
      deep_arg->resonance_flag = deep_arg->synchronous_flag = 0;

   if( deep_arg->resonance_flag)
      {
      deep_arg->xfact = bfact-deep_arg->xnq;

      /* Initialize integrator */
      deep_arg->xli = deep_arg->xlamo;
      deep_arg->xni = deep_arg->xnq;
      deep_arg->atime = 0;
      }
   /* End case dpinit: */
}

static inline void compute_dpsec_derivs( const deep_arg_t *deep_arg,
                                           double *derivs)
{
   const double sin_li = sin( deep_arg->xli);
   const double cos_li = cos( deep_arg->xli);
   const double sin_2li = 2. * sin_li * cos_li;
   const double cos_2li = 2. * cos_li * cos_li - 1.;
   int i;

                /* Dot terms calculated,  using a lot of trig add/subtract */
                /* identities to reduce the computational load... at the   */
                /* cost of making the code somewhat hard to follow:        */
   if( deep_arg->synchronous_flag )
      {
/*    const double fasx2 = 0.1313091 radians =   7.523456 degrees */
/*    const double fasx4 = 2.8843198 radians = 165.259351 degrees */
/*    const double fasx6 = 0.3744809 radians =  21.456173 degrees */
      const double c_fasx2 =  0.99139134268488593;
      const double s_fasx2 =  0.13093206501640101;
      const double c_2fasx4 =  0.87051638752972937;
      const double s_2fasx4 = -0.49213943048915526;
      const double c_3fasx6 =  0.43258117585763334;
      const double s_3fasx6 =  0.90159499016666422;
      const double sin_3li = sin_2li * cos_li + cos_2li * sin_li;
      const double cos_3li = cos_2li * cos_li - sin_2li * sin_li;
      double term1a = deep_arg->del1 * (sin_li  * c_fasx2  - cos_li  * s_fasx2);
      double term2a = deep_arg->del2 * (sin_2li * c_2fasx4 - cos_2li * s_2fasx4);
      double term3a = deep_arg->del3 * (sin_3li * c_3fasx6 - cos_3li * s_3fasx6);
      double term1b = deep_arg->del1 * (cos_li  * c_fasx2  + sin_li  * s_fasx2);
      double term2b = 2. * deep_arg->del2 * (cos_2li * c_2fasx4 + sin_2li * s_2fasx4);
      double term3b = 3. * deep_arg->del3 * (cos_3li * c_3fasx6 + sin_3li * s_3fasx6);

      for( i = 0; i < dpsec_integration_order; i += 2)
         {
         *derivs++ = term1a + term2a + term3a;
         *derivs++ = term1b + term2b + term3b;
         if( i + 2 < dpsec_integration_order)
            {
            term1a = -term1a;
            term2a *= -4.;
            term3a *= -9.;
            term1b = -term1b;
            term2b *= -4.;
            term3b *= -9.;
            }
         }
      }        /* end of geosynch case */
   else
      {        /* orbit is a 12-hour resonant one: */
/*    const double g22    =  5.7686396;      */
/*    const double g32    =  0.95240898;     */
/*    const double g44    =  1.8014998;      */
/*    const double g52    =  1.0508330;      */
/*    const double g54    =  4.4108898;      */
      const double c_g22 =  0.87051638752972937;
      const double s_g22 = -0.49213943048915526;
      const double c_g32 =  0.57972190187001149;
      const double s_g32 =  0.81481440616389245;
      const double c_g44 = -0.22866241528815548;
      const double s_g44 =  0.97350577801807991;
      const double c_g52 =  0.49684831179884198;
      const double s_g52 =  0.86783740128127729;
      const double c_g54 = -0.29695209575316894;
      const double s_g54 = -0.95489237761529999;
      const double xomi =
                deep_arg->omegaq + deep_arg->omgdot * deep_arg->atime;
      const double sin_omi = sin( xomi), cos_omi = cos( xomi);
      const double sin_li_m_omi = sin_li * cos_omi - sin_omi * cos_li;
      const double sin_li_p_omi = sin_li * cos_omi + sin_omi * cos_li;
      const double cos_li_m_omi = cos_li * cos_omi + sin_omi * sin_li;
      const double cos_li_p_omi = cos_li * cos_omi - sin_omi * sin_li;
      const double sin_2omi = 2. * sin_omi * cos_omi;
      const double cos_2omi = 2. * cos_omi * cos_omi - 1.;
      const double sin_2li_m_omi = sin_2li * cos_omi - sin_omi * cos_2li;
      const double sin_2li_p_omi = sin_2li * cos_omi + sin_omi * cos_2li;
      const double cos_2li_m_omi = cos_2li * cos_omi + sin_omi * sin_2li;
      const double cos_2li_p_omi = cos_2li * cos_omi - sin_omi * sin_2li;
      const double sin_2li_p_2omi = sin_2li * cos_2omi + sin_2omi * cos_2li;
      const double cos_2li_p_2omi = cos_2li * cos_2omi - sin_2omi * sin_2li;
      const double sin_2omi_p_li = sin_li * cos_2omi + sin_2omi * cos_li;
      const double cos_2omi_p_li = cos_li * cos_2omi - sin_2omi * sin_li;
      double term1a =
               deep_arg->d2201 * (sin_2omi_p_li*c_g22 - cos_2omi_p_li*s_g22)
             + deep_arg->d2211 * (sin_li * c_g22 - cos_li * s_g22)
             + deep_arg->d3210 * (sin_li_p_omi*c_g32 - cos_li_p_omi*s_g32)
             + deep_arg->d3222 * (sin_li_m_omi*c_g32 - cos_li_m_omi*s_g32)
             + deep_arg->d5220 * (sin_li_p_omi*c_g52 - cos_li_p_omi*s_g52)
             + deep_arg->d5232 * (sin_li_m_omi*c_g52 - cos_li_m_omi*s_g52);
      double term2a =
               deep_arg->d4410 * (sin_2li_p_2omi*c_g44 - cos_2li_p_2omi*s_g44)
             + deep_arg->d4422 * (sin_2li * c_g44 - cos_2li * s_g44)
             + deep_arg->d5421 * (sin_2li_p_omi*c_g54 - cos_2li_p_omi*s_g54)
             + deep_arg->d5433 * (sin_2li_m_omi*c_g54 - cos_2li_m_omi*s_g54);
      double term1b =
              (deep_arg->d2201 * (cos_2omi_p_li*c_g22 + sin_2omi_p_li*s_g22)
             + deep_arg->d2211 * (cos_li * c_g22 + sin_li * s_g22)
             + deep_arg->d3210 * (cos_li_p_omi*c_g32 + sin_li_p_omi*s_g32)
             + deep_arg->d3222 * (cos_li_m_omi*c_g32 + sin_li_m_omi*s_g32)
             + deep_arg->d5220 * (cos_li_p_omi*c_g52 + sin_li_p_omi*s_g52)
             + deep_arg->d5232 * (cos_li_m_omi*c_g52 + sin_li_m_omi*s_g52));
      double term2b = 2. *
              (deep_arg->d4410 * (cos_2li_p_2omi*c_g44 + sin_2li_p_2omi*s_g44)
             + deep_arg->d4422 * (cos_2li * c_g44 + sin_2li * s_g44)
             + deep_arg->d5421 * (cos_2li_p_omi*c_g54 + sin_2li_p_omi*s_g54)
             + deep_arg->d5433 * (cos_2li_m_omi*c_g54 + sin_2li_m_omi*s_g54));

      for( i = 0; i < dpsec_integration_order; i += 2)
         {
         *derivs++ = term1a + term2a;
         *derivs++ = term1b + term2b;
         if( i + 2 < dpsec_integration_order)
            {
            term1a = -term1a;
            term2a *= -4.;
            term1b = -term1b;
            term2b *= -4.;
            }
         }
      } /* End of 12-hr resonant case */
}

void Deep_dpsec( const tle_t *tle, deep_arg_t *deep_arg)
{
   double temp, xni, xli;
   int final_integration_step = 0;

   deep_arg->xll += deep_arg->ssl*deep_arg->t;
   deep_arg->omgadf += deep_arg->ssg*deep_arg->t;
   deep_arg->xnode += deep_arg->ssh*deep_arg->t;
   deep_arg->em = tle->eo+deep_arg->sse*deep_arg->t;
   deep_arg->xinc = tle->xincl+deep_arg->ssi*deep_arg->t;
   if( !deep_arg->resonance_flag ) return;

            /* If we're closer to t=0 than to the currently-stored data
               from the previous call to this function,  then we're
               better off "restarting",  going back to the initial data.
               The Dundee code rigs things up to _always_ take 720-minute
               steps from epoch to end time,  except for the final step.
               So if we'd have to integrate "backwards" (toward the epoch),
               we gotta do a restart if we're to be Dundee-compliant.  */
   if( fabs( deep_arg->t) < fabs( deep_arg->t - deep_arg->atime)
            || (is_dundee_compliant && fabs( deep_arg->t) < fabs( deep_arg->atime)))
      {                                    /* Epoch restart */
      deep_arg->atime = 0.;
      xni = deep_arg->xnq;
      xli = deep_arg->xlamo;
      }
   else                          /* use xni, xli from previous runs: */
      {
      xni = deep_arg->xni;
      xli = deep_arg->xli;
      }

   while( !final_integration_step)
      {
      double xldot, derivs[20], xlpow = 1., delt_factor;
      double delt = deep_arg->t - deep_arg->atime;
      int i;

      deep_arg->xni = xni;
      deep_arg->xli = xli;
      compute_dpsec_derivs( deep_arg, derivs);
      if( delt > dpsec_integration_step)
         delt = dpsec_integration_step;
      else if( delt < -dpsec_integration_step)
         delt = -dpsec_integration_step;
      else
         final_integration_step = 1;

      xldot = xni+deep_arg->xfact;

      xli += delt * xldot;
      xni += delt * derivs[0];
      delt_factor = delt;
      for( i = 2; i <= dpsec_integration_order; i++)
         {
         xlpow *= xldot;
         derivs[i - 1] *= xlpow;
         delt_factor *= delt / (double)i;
         xli += delt_factor * derivs[i - 2];
         xni += delt_factor * derivs[i - 1];
         }
      if( !is_dundee_compliant || !final_integration_step)
         {
         deep_arg->xni = xni;
         deep_arg->xli = xli;
         deep_arg->atime += delt;
         }
      }

   deep_arg->xn = xni;

   temp = -deep_arg->xnode + deep_arg->thgr + deep_arg->t * thdt;

   deep_arg->xll = xli + temp
         + (deep_arg->synchronous_flag ? -deep_arg->omgadf : temp);
   /*End case dpsec: */
}

void Deep_dpper( const tle_t *tle, deep_arg_t *deep_arg)
{
   double sinis, cosis;

            /* If the time didn't change by more than 30 minutes,      */
            /* there's no good reason to recompute the perturbations;  */
            /* they don't change enough over so short a time span.     */
            /* However,  the Dundee code _always_ recomputes,  so if   */
            /* we're attempting to replicate its results,  we've gotta */
            /* recompute everything,  too.                             */
   if( fabs(deep_arg->savtsn-deep_arg->t) >= 30. || is_dundee_compliant)
      {
      double zf, zm, sinzf, ses, sis, sil, sel, sll, sls;
      double f2, f3, sghl, sghs, shs, sh1;

      deep_arg->savtsn = deep_arg->t;

            /* Update solar perturbations for time T: */
      zm = deep_arg->zmos+zns_per_min*deep_arg->t;
      zf = zm+2*zes*sin(zm);
      sinzf = sin(zf);
      f2 = 0.5*sinzf*sinzf-0.25;
      f3 = -0.5*sinzf*cos(zf);
      ses = deep_arg->se2*f2+deep_arg->se3*f3;
      sis = deep_arg->si2*f2+deep_arg->si3*f3;
      sls = deep_arg->sl2*f2+deep_arg->sl3*f3+deep_arg->sl4*sinzf;
      sghs = deep_arg->sgh2*f2+deep_arg->sgh3*f3+deep_arg->sgh4*sinzf;
      shs = deep_arg->sh2*f2+deep_arg->sh3*f3;

            /* Update lunar perturbations for time T: */
      zm = deep_arg->zmol+znl_per_min*deep_arg->t;
      zf = zm+2*zel*sin(zm);
      sinzf = sin(zf);
      f2 = 0.5*sinzf*sinzf-0.25;
      f3 = -0.5*sinzf*cos(zf);
      sel = deep_arg->ee2*f2+deep_arg->e3*f3;
      sil = deep_arg->xi2*f2+deep_arg->xi3*f3;
      sll = deep_arg->xl2*f2+deep_arg->xl3*f3+deep_arg->xl4*sinzf;
      sghl = deep_arg->xgh2*f2+deep_arg->xgh3*f3+deep_arg->xgh4*sinzf;
      sh1 = deep_arg->xh2*f2+deep_arg->xh3*f3;

            /* Sum the solar and lunar contributions: */
      deep_arg->pe = ses+sel;
      deep_arg->pinc = sis+sil;
      deep_arg->pl = sls+sll;
      deep_arg->pgh = sghs+sghl;
      deep_arg->ph = shs+sh1;
#ifdef RETAIN_PERTURBATION_VALUES_AT_EPOCH
      if( deep_arg->solar_lunar_init_flag)
         {
         deep_arg->pe0 = deep_arg->pe;
         deep_arg->pinc0 = deep_arg->pinc;
         deep_arg->pl0 = deep_arg->pl;
         deep_arg->pgh0 = deep_arg->pgh;
         deep_arg->ph0 = deep_arg->ph;
         }
      deep_arg->pe  -= deep_arg->pe0;
      deep_arg->pinc -= deep_arg->pinc0;
      deep_arg->pl  -= deep_arg->pl0;
      deep_arg->pgh -= deep_arg->pgh0;
      deep_arg->ph  -= deep_arg->ph0;
      if( deep_arg->solar_lunar_init_flag)
         return;        /* done all we really need to do here... */
#endif
      }

               /* In Spacetrack 3, sinis & cosis were initialized       */
               /* _before_ perturbations were added to xinc.  In        */
               /* Spacetrack 6,  it's the other way around (see below). */
#ifndef SPACETRACK_3
   deep_arg->xinc += deep_arg->pinc;
#endif
   sinis = sin( deep_arg->xinc);
   cosis = cos( deep_arg->xinc);
#ifdef SPACETRACK_3
   deep_arg->xinc += deep_arg->pinc;
#endif

         /* Add solar/lunar perturbation correction to eccentricity: */
   deep_arg->em += deep_arg->pe;
   deep_arg->xll += deep_arg->pl;
   deep_arg->omgadf += deep_arg->pgh;
   if( tle->xincl >= 0.2)
      {
             /* Apply periodics directly */
      const double sinis = sin(deep_arg->xinc);
      const double cosis = cos(deep_arg->xinc);
      const double temp_val = deep_arg->ph / sinis;

      deep_arg->omgadf -= cosis * temp_val;
      deep_arg->xnode += temp_val;
      }
   else
      {
        /* Apply periodics with Lyddane modification */
      const double sinok = sin(deep_arg->xnode);
      const double cosok = cos(deep_arg->xnode);
      const double alfdp = deep_arg->ph * cosok
                    + (deep_arg->pinc * cosis + sinis) * sinok;
      const double betdp = - deep_arg->ph * sinok
                    + (deep_arg->pinc * cosis + sinis) * cosok;
      double dls, delta_xnode;

//    deep_arg->xnode = FMod2p(deep_arg->xnode);
      delta_xnode = atan2(alfdp,betdp) - deep_arg->xnode;

       /* This is a patch to Lyddane modification suggested */
       /* by Rob Matson, streamlined very slightly by BJG, to */
       /* keep 'delta_xnode' between +/- 180 degrees: */

      if( delta_xnode < - pi)
         delta_xnode += twopi;
      else if( delta_xnode > pi)
         delta_xnode -= twopi;

      dls = -deep_arg->xnode * sinis * deep_arg->pinc;
#ifdef SPACETRACK_3
      deep_arg->omgadf += dls
               + cosis * deep_arg->xnode -
               - cos( deep_arg->xinc) * (deep_arg->xnode + delta_xnode);
#else
      deep_arg->omgadf += dls - cosis * delta_xnode;
#endif
      deep_arg->xnode += delta_xnode;
      } /* End case dpper: */
}