Geostationary Orbit Calculator Formula

Understand the math behind the geostationary orbit calculator. Each variable explained with a worked example.

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Formulas Used

Orbit Radius

orbit_radius_km = pow(6.674e-11 * planet_mass * pow(rotation_period, 2) / (4 * pow(pi, 2)), 1/3) / 1000

Altitude Above Surface

altitude_km = (pow(6.674e-11 * planet_mass * pow(rotation_period, 2) / (4 * pow(pi, 2)), 1/3) - planet_radius) / 1000

Orbital Speed

orbital_speed = 2 * pi * pow(6.674e-11 * planet_mass * pow(rotation_period, 2) / (4 * pow(pi, 2)), 1/3) / rotation_period

Variables

VariableDescriptionDefault
planet_massPlanet Mass(kg)5.972e+24
rotation_periodRotation Period(s)86164
planet_radiusPlanet Radius(m)6371000

How It Works

Geostationary Orbit

A geostationary orbit has a period equal to the planet's rotation, so the satellite appears fixed above one equatorial spot.

Formula

r = (G M T² / 4π²)^(1/3)

For Earth: T = 86 164 s (sidereal day), giving r ≈ 42 164 km, altitude ≈ 35 786 km.

Worked Example

Find Earth's geostationary orbit.

planet_mass = 5.972e+24rotation_period = 86164planet_radius = 6371000
  1. 01r = (G M T² / 4π²)^(1/3)
  2. 02G M T² = 3.986e14 × (86164)² = 2.960e23
  3. 03r = (2.960e23 / 39.478)^(1/3) ≈ 42 164 km
  4. 04Altitude = 42 164 - 6 371 = 35 793 km
  5. 05Speed = 2π × 42 164 000 / 86 164 ≈ 3 075 m/s

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