Constants & Accuracy

Physical constants, astronomical constants, and accuracy bounds for every computational theory used in pg_orbit. All constants are compiled from their original sources and embedded at compile time — no runtime configuration files, no external data dependencies.

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Physical Constants

WGS-72 (SGP4/SDP4 Only)

The SGP4/SDP4 propagator uses WGS-72 constants internally. This matches the reference frame in which TLEs are generated. Using WGS-84 with SGP4 would introduce systematic errors.

Constant	Symbol	Value	Unit
Gravitational parameter	mu	398600.8	km^3/s^2
Equatorial radius	ae	6378.135	km
J2	J2	0.001082616	—
J3	J3	-0.00000253881	—
J4	J4	-0.00000165597	—

Caution

These WGS-72 values are only used inside the SGP4/SDP4 propagator. All coordinate output (geodetic, topocentric) uses WGS-84. Mixing WGS-72 and WGS-84 constants in the same computation is a common source of systematic error in satellite tracking software; pg_orbit handles this boundary correctly.

WGS-84 (Coordinate Output)

All geodetic and topocentric coordinate conversions use WGS-84 constants.

Constant	Symbol	Value	Unit
Semi-major axis (equatorial radius)	a	6378.137	km
Flattening	f	1/298.257223563	—
Semi-minor axis (polar radius)	b	6356.752314245	km
First eccentricity squared	e^2	0.00669437999014	—

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Astronomical Constants

Constant	Symbol	Value	Unit	Source
Astronomical Unit	AU	149597870.7	km	IAU 2012
Obliquity of the ecliptic at J2000	epsilon	23.4392911	degrees	IAU 1976
Gaussian gravitational constant	k	0.01720209895	AU^(3/2) / (day * Msun^(1/2))	IAU 1976
Julian century	—	36525.0	days	—
J2000 epoch	—	2451545.0	JD	2000-01-01T12:00:00 TT
Speed of light	c	299792.458	km/s	IAU 2012
Earth rotation rate	omega_e	7.292115e-5	rad/s	WGS-84

Jupiter-Specific Constants

Constant	Value	Unit	Source
System III rotation period	9h 55m 29.711s	—	IAU 1965
System III rotation rate	870.536	deg/day	Derived

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Theory Accuracy Bounds

Each computational theory in pg_orbit has well-characterized accuracy limits. The bounds below are drawn from the original theory publications and validated against JPL ephemerides where possible.

SGP4/SDP4 (Satellite Propagation)

Orbit Class	Typical Error at Epoch	Error Growth Rate	Source
LEO (< 2000 km)	< 1 km at epoch	1-3 km/day	Vallado et al., 2006
MEO (2000-35786 km)	< 5 km at epoch	5-10 km/day	Vallado et al., 2006
GEO (~35786 km)	< 10 km at epoch	10-50 km/day	Vallado et al., 2006
HEO (Molniya-type)	< 10 km at epoch	Highly variable	Vallado et al., 2006

Note

These errors assume the TLE is freshly generated. TLE age is the dominant error source. The tle_age accessor function reports how old a TLE is relative to the current time. For LEO conjunction screening, TLEs older than 3 days should be treated with caution.

Valid epoch range: TLEs are typically valid for +/- 7 days from epoch for LEO, +/- 14 days for GEO. Beyond this, errors grow rapidly and propagation may fail outright (returning a fatal error code).

Deep space selection: The SGP4/SDP4 algorithm switch is based on orbital period. Orbits with period >= 225 minutes (~3.75 hours, corresponding to an altitude of roughly 5,900 km for circular orbits) use the SDP4 deep-space model, which includes lunar and solar perturbation terms.

VSOP87 (Planetary Positions)

Position accuracy relative to JPL DE405 ephemeris:

Planet	Max Error (within +/- 2000 yr of J2000)	Max Error (within +/- 4000 yr of J2000)
Mercury	0.6”	1”
Venus	0.3”	2”
Earth-Moon barycenter	0.4”	2”
Mars	0.8”	4”
Jupiter	0.3”	7”
Saturn	0.4”	10”
Uranus	0.2”	20”
Neptune	0.3”	40”

Tip

For dates within a few centuries of J2000, VSOP87 is accurate to sub-arcsecond levels for all planets. For most observational purposes (telescope pointing, rise/set times, conjunction identification), this exceeds requirements.

Valid epoch range: The series coefficients are fitted over the interval J2000 +/- 4000 years. Outside this range, accuracy degrades unpredictably. For observational planning (present-day queries), accuracy is well within 1 arcsecond.

ELP2000-82B (Lunar Position)

Quantity	Accuracy
Geocentric longitude	< 2” for dates within +/- 500 years of J2000
Geocentric latitude	< 1”
Distance	< 0.5 km

Valid epoch range: Nominally +/- 4000 years from J2000, though accuracy degrades beyond +/- 500 years. For present-day lunar observations, the error is well under 1 arcsecond and 1 km in distance.

Lieske L1.2 (Galilean Moons)

Quantity	Accuracy
Position relative to Jupiter	< 500 km (typical), < 1000 km (worst case)
Differential positions (moon-to-moon)	< 200 km

Valid epoch range: Fitted to observations spanning 1891-2000. Accuracy degrades outside this range. For present-day observations and near-term predictions, the theory is reliable.

TASS17 (Saturn Moons)

Quantity	Accuracy
Position relative to Saturn	< 500 km (inner moons), < 2000 km (Hyperion)
Titan position	< 300 km

Notes: Hyperion has the largest uncertainty due to its chaotic rotation and irregular orbital perturbations. Titan, as the most massive moon, is the best-determined.

Valid epoch range: Fitted to observations spanning 1886-1985. The theory uses secular terms that limit extrapolation.

GUST86 (Uranus Moons)

Quantity	Accuracy
Position relative to Uranus	< 1000 km (Titania, Oberon), < 2000 km (Miranda)

Notes: The Uranian system was significantly improved by Voyager 2 encounter data (1986). Pre-Voyager observations constrain secular rates; Voyager data constrains short-period terms.

Valid epoch range: Most reliable within +/- 50 years of the Voyager encounter (1986). Present-day accuracy is good.

MarsSat (Mars Moons)

Quantity	Accuracy
Phobos position relative to Mars	< 10 km
Deimos position relative to Mars	< 20 km

Notes: The Mars moon theories benefit from spacecraft tracking data (Viking, Mars Express). Phobos is better determined than Deimos due to more frequent close encounters with Mars orbiters.

Kepler Propagation (Comets & Asteroids)

Orbit Type	Limitation
Elliptic (e < 1)	Two-body only. No planetary perturbations. Error grows with distance from perihelion epoch.
Parabolic (e = 1)	Barker’s equation. Exact for the two-body case.
Hyperbolic (e > 1)	Two-body only. Valid for interstellar objects near perihelion.

Caution

Keplerian propagation ignores gravitational perturbations from planets, non-gravitational forces (outgassing, radiation pressure), and relativistic effects. For long-period comets far from perihelion, the two-body approximation is reasonable. For short-period comets with close planetary encounters (e.g., Jupiter-family comets), errors accumulate over time. Use fresh osculating elements.

Lambert Solver (Transfer Orbits)

Quantity	Notes
Transfer orbit accuracy	Exact for the two-body (patched conic) approximation
Planet position accuracy	Limited by VSOP87 (sub-arcsecond for present-day)
C3 accuracy	Departure C3 values are typically within 0.1 km^2/s^2 of JPL trajectory tools for well-posed transfers

Limitations: The Lambert solver assumes patched conic trajectories (two-body between planets). It does not account for:

• Gravity assists
• Solar radiation pressure
• Finite thrust arcs
• N-body perturbations during the transfer

For preliminary mission design and pork chop plot generation, these limitations are standard and expected.

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Reference Publications

Theory	Publication
SGP4/SDP4	Vallado, D.A., Crawford, P., Hujsak, R., Kelso, T.S. “Revisiting Spacetrack Report #3.” AIAA 2006-6753, 2006.
VSOP87	Bretagnon, P., Francou, G. “Planetary theories in rectangular and spherical variables. VSOP87 solutions.” Astronomy and Astrophysics, 202, 309-315, 1988.
ELP2000-82B	Chapront-Touze, M., Chapront, J. “The lunar ephemeris ELP-2000.” Astronomy and Astrophysics, 124, 50-62, 1983.
Lieske L1.2	Lieske, J.H. “Galilean satellites of Jupiter.” Astronomy and Astrophysics Supplement Series, 129, 205-217, 1998.
TASS17	Vienne, A., Duriez, L. “TASS1.7: An ephemeris generator for the major satellites of Saturn.” Astronomy and Astrophysics, 297, 588-605, 1995.
GUST86	Laskar, J., Jacobson, R.A. “GUST86: An analytical ephemeris of the Uranian satellites.” Astronomy and Astrophysics, 188, 212-224, 1987.
MarsSat	Jacobson, R.A. “The orbits and masses of the Martian satellites and the libration of Phobos.” The Astronomical Journal, 139, 668-679, 2010.
Carr source regions	Carr, T.D., Desch, M.D., Alexander, J.K. “Phenomenology of magnetospheric radio emissions.” In Physics of the Jovian Magnetosphere, Cambridge Univ. Press, 1983.
Lambert solver	Battin, R.H. “An Introduction to the Mathematics and Methods of Astrodynamics.” AIAA Education Series, Revised Edition, 1999.