Elliptical Orbit Calculator (Period, Eccentricity and Speed at Any Point)
Orbital period, eccentricity and the speed at the closest, farthest and any chosen point of an elliptical orbit around the Sun, Earth, the Moon, Mars or Jupiter.
Results
The dashed line marks the value you entered.
What this tool does
A satellite does not fly at one speed: it is fastest at its closest point and slowest at its farthest, and the vis-viva equation gives the speed anywhere on the ellipse from just the semi-major axis. The period depends only on that axis — not on the eccentricity — which is Kepler's third law. Enter altitudes above the surface and the tool adds the body's radius for you; enter distances from the centre if you already have them. A 500 km by 39 700 km orbit around Earth, for example, takes about 12 hours, the kind of orbit that lingers over one region for most of its pass.
Formula
a = (r_p + r_a) ÷ 2 · e = (r_a − r_p) ÷ (r_a + r_p) · v = √[μ (2 ÷ r − 1 ÷ a)] · T = 2π √(a³ ÷ μ)
Variables
| Symbol | Meaning | Unit |
|---|---|---|
b | Body being orbited | — |
ms | Distances below are measured from | — |
dp | Closest point of the orbit (periapsis) | km |
da | Farthest point of the orbit (apoapsis) | km |
dx | Distance at which you want the speed | km |
TH | Orbital period in hours | h |
TD | Orbital period in days | d |
A | Semi-major axis | km |
E | Eccentricity | — |
VP | Speed at the closest point | km/s |
VA | Speed at the farthest point | km/s |
VX | Speed at the distance you chose | km/s |
Worked example
- Body being orbited2
- Distances below are measured from2
- Closest point of the orbit (periapsis)500 km
- Farthest point of the orbit (apoapsis)39,700 km
- Distance at which you want the speed20,000 km
- Orbital period in hours11.906 h
- Orbital period in days0.4961 d
- Semi-major axis26,471 km
- Eccentricity0.7404
- Speed at the closest point10.048 km/s
- Speed at the farthest point1.499 km/s
- Speed at the distance you chose3.895 km/s
Limitations
- The formula assumes ideal conditions: no friction losses, no air resistance and no efficiency losses unless you enter them.
- The default values are typical reference figures, not measurements of your situation. Replace them with your own data whenever you have it.