Theory-to-Code Mapping

Every equation in pg_orbit traces to a published, peer-reviewed source. This page provides the complete mapping between the celestial mechanics literature and the source files that implement each theory.

If a constant, algorithm, or formula appears in the code without a citation, that is a defect to be corrected.

SGP4/SDP4 propagation

The core satellite propagation theory, implemented by Bill Gray’s sat_code library.

Theory	Source Paper	What it computes	Code location
Mean element recovery	Brouwer (1959)	Original mean motion $n_0'$ and semi-major axis $a_0'$ from input TLE, removing secular $J_2$ perturbations	sat_code/common.cpp:sxpall_common_init()
Secular perturbations	Lane & Cranford (1969); Hoots & Roehrich STR#3	Secular rates of $M$, $\omega$, and $\Omega$ due to $J_2$, $J_4$	sat_code/common.cpp:sxpx_common_init()
Atmospheric drag	Hoots & Roehrich STR#3	$B^*$ formulation of drag; $C_1$, $C_2$, $C_4$ coefficients; perigee-dependent $s$ parameter	sat_code/common.cpp:sxpx_common_init(); sat_code/sgp4.cpp:SGP4_init()
Short-period perturbations	Lane & Cranford (1969); Brouwer (1959)	Oscillatory corrections to radius, argument of latitude, node, and inclination	sat_code/common.cpp:sxpx_posn_vel()
Kepler equation	Classical	Newton-Raphson with second-order correction, bounded first step	sat_code/common.cpp:sxpx_posn_vel()
Deep-space resonance	Hujsak (1979)	Lunar/solar gravitational perturbations; geopotential resonance for 12-hour and 24-hour orbits	sat_code/deep.cpp:Deep_dpinit(), Deep_dpsec(), Deep_dpper()
Near-earth propagation	Hoots & Roehrich STR#3	SGP4 main loop: secular + short-period + drag terms	sat_code/sgp4.cpp:SGP4()
Deep-space propagation	Hoots & Roehrich STR#3	SDP4: SGP4 core + deep-space secular/periodic corrections	sat_code/sdp4.cpp:SDP4()
Near/deep selection	Hoots & Roehrich STR#3	Period threshold: 225 minutes ($n < 2\pi/225$ rad/min)	sat_code/norad.h:select_ephemeris()

Primary reference

Hoots, F. R. & Roehrich, R. L. (1980). “Models for Propagation of NORAD Element Sets.” Spacetrack Report No. 3, Aerospace Defense Command, Peterson AFB.

This is the canonical SGP4/SDP4 reference. All subsequent implementations, including Vallado’s 2006 revision, trace back to this report.

Coordinate transforms

Theory	Source	What it computes	Code location
GMST	Vallado (2013) Eq. 3-47; IAU 1982	Greenwich Mean Sidereal Time from Julian date	src/sidereal_time.c:gmst_from_jd()
TEME to ECEF	Vallado (2013)	Z-axis rotation by $-\text{GMST}$; velocity cross-product correction	src/coord_funcs.c:teme_to_ecef()
Geodetic from ECEF	Bowring (1976)	Iterative latitude from ECEF Cartesian on WGS-84	src/coord_funcs.c:ecef_to_geodetic()
Topocentric transform	Standard SEZ	ECEF range vector rotated to South-East-Zenith; azimuth from north	src/coord_funcs.c:ecef_to_topocentric()
Observer to ECEF	Geodesy standard	WGS-84 ellipsoid surface point to Cartesian	src/coord_funcs.c:observer_to_ecef()
Range rate	Dot product	Projection of relative velocity onto line-of-sight unit vector	src/coord_funcs.c:eci_to_topocentric()
Semi-major axis from $n$	Kepler’s third law	$a = (k_e / n)^{2/3}$ in earth radii	src/tle_type.c:tle_perigee()
IAU 1976 precession	Lieske et al. (1977)	Three Euler angles $\zeta_A$, $z_A$, $\theta_A$ for precession from J2000 to date	src/precession.c:precess_j2000_to_date()
Ecliptic to equatorial	IAU	X-axis rotation by obliquity $\varepsilon_0 = 23.4392911\degree$	src/planet_funcs.c:ecliptic_to_equatorial()

Primary references

• Vallado, D. A. (2013). Fundamentals of Astrodynamics and Applications, 4th ed. Microcosm Press.
• Lieske, J. H. et al. (1977). “Expressions for the Precession Quantities Based upon the IAU (1976) System of Astronomical Constants.” Astronomy & Astrophysics, 58, 1-16.
• Bowring, B. R. (1976). “Transformation from Spatial to Geographical Coordinates.” Survey Review, 23, 323-327.

Planetary ephemerides

Theory	Source	Bodies	Accuracy	Code location
VSOP87	Bretagnon & Francou (1988)	Mercury through Neptune	~1 arcsecond	src/vsop87.c
ELP2000-82B	Chapront-Touze & Chapront (1988)	Moon	~10 arcseconds	src/elp82b.c

VSOP87

Bretagnon, P. & Francou, G. (1988). “Planetary Theories in Rectangular and Spherical Variables. VSOP87 Solutions.” Astronomy & Astrophysics, 202, 309-315.

pg_orbit uses the VSOP87 rectangular ecliptic J2000 variant. The truncated coefficient tables provide full accuracy within the validity range of the theory (roughly 4000 BCE to 8000 CE for the inner planets, with degradation for the outer planets beyond $\pm$2000 years from J2000).

ELP2000-82B

Chapront-Touze, M. & Chapront, J. (1988). “ELP 2000-85: A Semi-Analytical Lunar Ephemeris Adequate for Historical Times.” Astronomy & Astrophysics, 190, 342-352.

The 82B revision is the version implemented. It provides geocentric ecliptic coordinates for the Moon, accounting for the principal perturbations from the Sun but not the complete set of planetary perturbations available in modern lunar ephemerides like DE421.

Planetary moon theories

Theory	Source	Moons	Accuracy	Code location
L1.2	Lieske (1998)	Io, Europa, Ganymede, Callisto	~1 arcsecond	src/l12.c
TASS17	Vienne & Duriez (1995)	Mimas through Iapetus	~1-5 arcseconds	src/tass17.c
GUST86	Laskar & Jacobson (1987)	Miranda through Oberon	~5-10 arcseconds	src/gust86.c
MarsSat	Jacobson (2010)	Phobos, Deimos	~1 arcsecond	src/marssat.c

References

• Lieske, J. H. (1998). “Galilean Satellites of Jupiter.” Astronomy & Astrophysics Supplement Series, 129, 205-217.
• Vienne, A. & Duriez, L. (1995). “TASS1.7: An Analytical Theory of the Motion of the Main Satellites of Saturn.” Astronomy & Astrophysics, 297, 588-605.
• Laskar, J. & Jacobson, R. A. (1987). “GUST86: An Analytical Ephemeris of the Uranian Satellites.” Astronomy & Astrophysics, 188, 212-224.
• Jacobson, R. A. (2010). “The Orbits and Masses of the Martian Satellites and the Libration of Phobos.” Astronomical Journal, 139, 668-679.

Transfer orbits

Theory	Source	What it computes	Code location
Lambert solver	Izzo (2015)	Transfer velocity vectors given two positions and time of flight	src/lambert.c
Keplerian propagation	Classical	Two-body elliptic/hyperbolic orbit from elements	src/elliptic_to_rectangular.c

References

• Izzo, D. (2015). “Revisiting Lambert’s Problem.” Celestial Mechanics and Dynamical Astronomy, 121, 1-15.

The Izzo solver uses Householder iterations for fast convergence and handles both short-way and long-way transfers. pg_orbit uses the prograde (short-way) solution by default.

Radio emission

Theory	Source	What it computes	Code location
Carr source regions	Carr et al. (1983)	Jupiter-Io decametric burst probability from CML and Io phase	src/radio_funcs.c
CML computation	Standard	Jupiter System III Central Meridian Longitude	src/radio_funcs.c

Reference

Carr, T. D. et al. (1983). “Phenomenology of Magnetospheric Radio Emissions.” Physics of the Jovian Magnetosphere, Cambridge University Press, 226-284.

Vallado reference vectors

The Vallado 518 test vectors are the definitive verification dataset for SGP4 implementations. Each row specifies a NORAD ID, minutes since epoch, and expected position/velocity in TEME.

Verification standard

All 518 vectors must match to machine epsilon ($\sim 10^{-8}$ km position, $\sim 10^{-11}$ km/s velocity) before any other development proceeds. The test file lives at test/data/vallado_518.csv.

Sample from the verification suite:

0.039550

# NORAD 00005 (Vanguard 1) - LEO, low eccentricity
# Minutes:   360.00  Expected X: -7154.031380 Y:  -3783.176825 Z: -2073.655980

# NORAD 29238 (GPS BIIR-11) - MEO, near-circular
# Minutes:     0.00  Expected X: -22503.132440 Y: 14513.963880 Z: 180.989390

# NORAD 28350 (Galaxy 15) - GEO, deep-space SDP4
# Minutes:     0.00  Expected X: -33110.816260 Y: 26044.993650 Z: -20.725400

These vectors cover the full range of orbit types that pg_orbit handles: LEO (SGP4), MEO (SGP4), GEO (SDP4), high-eccentricity Molniya (SDP4), and deep-space GPS (SDP4). Any implementation that matches all 518 vectors is functionally equivalent to the Vallado reference.

Source file index

A quick reference for finding the implementation of a specific theory.

Source file	Theory/Function	Lines (approx)
src/vsop87.c	VSOP87 planet positions	~3000 (coefficient tables)
src/elp82b.c	ELP2000-82B Moon position	~2000 (coefficient tables)
src/l12.c	L1.2 Galilean moons	~800
src/tass17.c	TASS17 Saturn moons	~1200
src/gust86.c	GUST86 Uranus moons	~600
src/marssat.c	MarsSat Mars moons	~400
src/precession.c	IAU 1976 precession	~60
src/sidereal_time.c	GMST computation	~40
src/lambert.c	Izzo Lambert solver	~300
src/coord_funcs.c	Coordinate transforms	~650
src/pass_funcs.c	Pass prediction algorithm	~550
src/gist_tle.c	GiST altitude-band index	~400
src/planet_funcs.c	Observation pipeline	~250
src/radio_funcs.c	Jupiter radio emission	~200
sat_code/sgp4.cpp	SGP4 near-earth propagator	~300
sat_code/sdp4.cpp	SDP4 deep-space propagator	~200
sat_code/deep.cpp	Deep-space perturbations	~800
sat_code/common.cpp	Shared SGP4/SDP4 initialization	~250