SGP & Deep-Space Models

Source

Spacetrack Report No. 3, Hoots & Roehrich (1980) — Pages 53–66 of the source listing

SGP — Simplified General Perturbations (109 lines)

The simplest model in the suite. SGP is a first-order secular theory with atmospheric drag, suitable for short-term predictions of near-earth objects. It was superseded by SGP4 for most applications but remains in the codebase for completeness and test validation.

SGP lacks the higher-order drag terms ($D_2$, $D_3$, $D_4$) and the modified atmospheric parameters that SGP4 uses for low-perigee orbits. It also uses a simpler form for the long-period periodics.

*         SGP                                         31 OCT 80
      SUBROUTINE SGP(IFLAG,TSINCE)
      COMMON/E1/XMO,XNODEO,OMEGAO,EO,XINCL,XNO,XNDT2O,XNDD6O,BSTAR,
     1          X,Y,Z,XDOT,YDOT,ZDOT,EPOCH,DS50
      COMMON/C1/CK2,CK4,E6A,QOMS2T,S,TOTHRD,
     1          XJ3,XKE,XKMPER,XMNPDA,AE
      DOUBLE PRECISION EPOCH, DS50
      IF(IFLAG.EQ.0) GO TO 19
*         INITIALIZATION
      C1= CK2*1.5
      C2= CK2/4.0
      C3= CK2/2.0
      C4= XJ3*AE**3/(4.0*CK2)
      COSIO=COS(XINCL)
      SINIO=SIN(XINCL)
      A1=(XKE/XNO)**TOTHRD
      D1=     C1/A1/A1*(3.*COSIO*COSIO-1.)/(1.-EO*EO)**1.5
      AO=A1*(1.-1./3.*D1-D1*D1-134./81.*D1*D1*D1)
      PO=AO*(1.-EO*EO)
      QO=AO*(1.-EO)
      XLO=XMO+OMEGAO+XNODEO
      D1O= C3 *SINIO*SINIO
      D2O= C2 *(7.*COSIO*COSIO-1.)
      D3O=C1*COSIO
      D4O=D3O*SINIO
      PO2NO=XNO/(PO*PO)
      OMGDT=C1*PO2NO*(5.*COSIO*COSIO-1.)
      XNODOT=-2.*D3O*PO2NO
      C5=.5*C4*SINIO*(3.+5.*COSIO)/(1.+COSIO)
      C6=C4*SINIO
      IFLAG=0
*         UPDATE FOR SECULAR GRAVITY AND ATMOSPHERIC DRAG
   19 A=XNO+(2.*XNDT2O+3.*XNDD6O*TSINCE)*TSINCE
      A=AO*(XNO/A)**TOTHRD
      E=E6A
      IF(A.GT.QO) E=1.-QO/A
      P=A*(1.-E*E)
      XNODES= XNODEO+XNODOT*TSINCE
      OMGAS= OMEGAO+OMGDT*TSINCE
      XLS=FMOD2P(XLO+(XNO+OMGDT+XNODOT+(XNDT2O+XNDD6O*TSINCE)*
     1 TSINCE)*TSINCE)

SDP4 and SDP8 — Deep-Space Extensions

SDP4 and SDP8 wrap the near-earth SGP4/SGP8 models with calls to the DEEP subroutine for lunar-solar perturbations. The pattern is identical for both:

1. Initialize the near-earth model (same as SGP4/SGP8)
2. Call DPINIT to set up deep-space coefficients
3. Each timestep: call DPSEC for secular perturbations, then DPPER for periodic
4. Apply the near-earth short-period corrections

SDP8 column 72 fix

The sdp8.f source required a column 72 correction during extraction. A continuation line was indented too deeply, pushing *COSI past column 72 to become *COS — a function call with no arguments instead of a variable reference. This is the exact class of transcription error that produced decades of divergent SGP4 implementations.

Model Selection Summary

Model	Period	Theory	Drag
SGP	< 225 min	1st order secular	$\dot{n}$, $\ddot{n}$ terms
SGP4	< 225 min	Brouwer mean elements	Lane atmospheric model
SDP4	$\geq$ 225 min	Brouwer + lunar-solar	Lane + DEEP
SGP8	< 225 min	Hoots drag theory	Improved drag model
SDP8	$\geq$ 225 min	Hoots + lunar-solar	Hoots drag + DEEP

In practice, NORAD uses SGP4 for near-earth and SDP4 for deep-space. SGP, SGP8, and SDP8 are historical alternatives provided for completeness. The Rev-1 paper focuses exclusively on SGP4/SDP4 as the operational standard.