satbase

Equatorial Orbit Designer

Orbit geometry, apsidal velocities, period, and worst-case eclipse for an equatorial orbit around any body in the planetary database, from a periapsis and apoapsis altitude.

Inputs

Central body

Orbit

The two apsides, measured above the surface. Equal altitudes are a circular orbit; the eclipse assumes the Sun lies in the orbit plane.

km
km

Results

Periapsis radius 6,778km
Apoapsis radius 7,378km
Semi-major axis 7,078km
Eccentricity 0.0424
Periapsis velocity 7.83km/s
Apoapsis velocity 7.193km/s
Orbital period 98.8 min
Eclipse duration 36.5 min
Eclipse share of orbit 37%

Method

An equatorial orbit from its two apsides: the shape, the speeds at each end, the period, and how long each pass spends in the body's shadow.

e = (ra − rp) / (ra + rp), v = √(μ(2/r − 1/a)), T = 2π·√(a³ / μ) A two-body Keplerian orbit from its two apsides; the eclipse is the worst case, with the Sun in the orbit plane and the shadow centered on apoapsis.

Derivation

1
Measure the orbit from the center Altitudes are quoted above the surface, but gravity pulls from the center, so both apsidal radii are the body radius plus the altitude. rp = R + hp, ra = R + ha
2
Fix the shape from the two radii The semi-major axis is their mean and the eccentricity is their normalized difference. Equal altitudes give e = 0, a circular orbit. a = (rp + ra) / 2, e = (ra − rp) / (ra + rp)
3
Take the speed at each apsis Vis-viva gives the speed anywhere on the orbit. At the apsides the velocity is purely transverse, so rₚ·vₚ = rₐ·vₐ. The conserved angular momentum is a free check on both numbers. vp = √(μ(2/rp − 1/a)), va = √(μ(2/ra − 1/a))
4
Close the orbit with Kepler’s third law The period depends only on the semi-major axis, so two orbits with wildly different eccentricities can share one. The mean motion n = 2π/T is what converts angles into times in the last step. T = 2π·√(a³ / μ)
5
Find where the orbit leaves the shadow The umbra is modeled as a cylinder of the body’s radius pointing away from the Sun. Measuring u from apoapsis, the satellite is in shadow while its distance from the cylinder axis, r(u)·sin u, is under R. That boundary has no closed form, so it is solved numerically. r(u) = p / (1 − e·cos u), r(u)·sin u = R
6
Turn the angle into a duration Kepler’s equation converts the boundary’s true anomaly to a mean anomaly, and the mean motion turns that into a time. Subtracting the sunlit arc either side of apoapsis from the period leaves the eclipse. Centering it on apoapsis is what makes this the worst case: that is where the satellite is slowest, so it lingers in shadow longest. M = E − e·sin E, tecl = T − 2M(π − u*) / n

Symbols

Input Description Unit
μ Gravitational parameter GM of the body km³/s²
R Equatorial radius of the body km
hp Periapsis altitude above the surface km
ha Apoapsis altitude above the surface km
Derived Description Unit
rp Periapsis radius: R + hₚ km
ra Apoapsis radius: R + hₐ km
a Semi-major axis: the mean of the two radii km
e Eccentricity: 0 is circular
p Semi-latus rectum: a(1 − e²) km
vp Speed at periapsis, the orbit’s fastest km/s
va Speed at apoapsis, the orbit’s slowest km/s
T Orbital period s
u* Angle from apoapsis where the orbit leaves the umbra deg
tecl Time spent in shadow each orbit s

Reference values

First-order figures for sanity-checking inputs.

Body μ km³/s² Radius km
Mercury 22,033 2,441
Venus 324,869 6,052
Earth 398,614 6,378
Moon 4,903 1,738
Mars 42,830 3,396
Jupiter 126,690,000 71,492
Saturn 37,932,000 60,268
Uranus 5,794,100 25,559
Neptune 6,835,300 24,764
Pluto 870 1,188
Orbit Altitudes km Eclipse
Low Earth orbit 400 × 400 36.1 min
GPS altitude 20,200 × 20,200 55.4 min
Geostationary transfer 185 × 35,786 2 h 15 min
Geostationary 35,786 × 35,786 69.4 min
Low Mars orbit 400 × 400 41.7 min
Low lunar orbit 100 × 100 46.5 min

This is a two-body idealization of an equatorial orbit: no J2 or third-body effects, no drag, no orbital decay, and inclination is not modeled. The eclipse is the worst case: a cylindrical umbra with no penumbra, the Sun in the orbit plane (exact only at equinox, since the body's axial tilt is not modeled), and the shadow centered on apoapsis, where the satellite is slowest. A real orbit's eclipse varies through the year and is usually shorter than this. Treat these results as first-pass sizing, not a mission analysis.