Elliptical Orbit Calculator

An Earth orbit from 400 km to 4000 km altitude has a semi-major axis of 8571 km and a period of 131.615 min.

Central body:

EarthMoonMarsSun (radii from its centre)

Periapsis altitude (or radius for the Sun):

Apoapsis altitude (or radius for the Sun):

Orbital period T =

131.615480238621 min

Precision

Full answer, then cut to fewer decimals (not rounded):

131.615480238621 min
131.61548023862 min
131.6154802386 min
131.615480238 min
131.61548023 min
131.6154802 min
131.615480 min
131.61548 min
131.6154 min
131.615 min
131.61 min
131.6 min

Semi-major axis a: 8571 km
Eccentricity e: 0.2100105005
Speed at periapsis: 8.439895134 km/s
Speed at apoapsis: 5.510223696 km/s
Specific orbital energy −μ/2a: -23.25285508 MJ/kg
Specific angular momentum r_p v_p: 57146.52995 km²/s

Step-by-step solution

a = (r_p + r_a)/2 = 8571 km; e = (r_a − r_p)/(r_a + r_p) = 0.2100105005
T = 2π√(a³/μ) = 7896.928814 s (Kepler’s third law)
v = √(μ(2/r − 1/a)): v_p = 8439.895134 m/s, v_a = 5510.223696 m/s (vis-viva)

Elliptical Orbit Formula

T = 2π√(a³/μ), v = √(μ(2/r − 1/a))

a = (r_p + r_a)/2, e = (r_a − r_p)/(r_a + r_p), energy = −μ/(2a)

Worked Example

An Earth orbit with a 400 km perigee and a 4000 km apogee:

a = (r_p + r_a)/2 = 8571 km; e = (r_a − r_p)/(r_a + r_p) = 0.2100105005
T = 2π√(a³/μ) = 7896.928814 s (Kepler’s third law)
v = √(μ(2/r − 1/a)): v_p = 8439.895134 m/s, v_a = 5510.223696 m/s (vis-viva)

Eccentricity e: 0.2100105005 · Speed at periapsis: 8.439895134 km/s · Speed at apoapsis: 5.510223696 km/s · Specific orbital energy −μ/2a: -23.25285508 MJ/kg · Specific angular momentum r_p v_p: 57146.52995 km²/s

GATE tip

Orbital energy depends only on a. A parabolic escape trajectory has energy 0, so escape speed at radius r is √2 times the circular speed there.

More Space Dynamics Calculators

Hohmann Transfer Calculator

Δv₁ = √(μ/r₁)(√(2r₂/(r₁ + r₂)) − 1)

Δv₂ = √(μ/r₂)(1 − √(2r₁/(r₁ + r₂))); t = π√(a³/μ), a = (r₁ + r₂)/2

Example: From a 300 km circular orbit to geostationary altitude (35,786 km): Total Δv = 3.89523 km/s.

Open the hohmann transfer calculator →

Frequently asked questions

Why is the satellite fastest at perigee?

Angular momentum r × v is constant (Kepler’s second law), so speed is highest where r is smallest.

What does the vis-viva equation give?

The speed at any radius from the semi-major axis alone: v² = μ(2/r − 1/a). With r = a it gives the circular speed.

Sources

CODATA Recommended Values of the Fundamental Physical Constants: 2018, National Institute of Standards and Technology .
Planetary Fact Sheet, NASA Goddard Space Flight Center .

Last reviewed: 2026-10-15. Factors are stored exactly and computed with 50-digit decimal arithmetic. See how we verify conversion factors.

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Scientific Converters. (2026, October 15). Elliptical Orbit Calculator. Retrieved October 25, 2026, from https://scientificconverters.com/calculators/elliptical-orbit-calculator/

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