How this calculation works
A bound two-body ellipse has different speeds at its closest and farthest points. Conservation of energy links each speed to radius and semi-major axis, while Kepler’s period depends on the semi-major axis alone. This page derives period, apsis speeds and eccentricity from the two extreme radii. It is a static orbital-geometry calculation, distinct from planning a transfer between two separate circular orbits.
a = (rp+ra)/2; v² = μ(2/r−1/a); period = 2π√(a³/μ)A worked example
For gravitational parameter 398600.4418 km³/s² and radii 7000 and 14000 km, the semi-major axis is 10500 km and eccentricity is one third. Periapsis speed is about 8.71343 km/s, while apoapsis speed is half that value because its radius is twice as large. Period is about 10707.7 seconds under the ideal two-body model.
Good to know
The central mass is treated as a point or spherical attractor. Atmospheric drag, oblateness, third bodies and non-gravitational forces are absent. A mathematical radius does not demonstrate clearance from the physical body. This page is educational and does not supply navigation commands, manoeuvre timing or an operational orbit determination.
Reference: Orbital Mechanics & Astrodynamics — elliptical orbits (opens in a new tab) ↗
What this calculation answers
A bound two-body ellipse has different speeds at its closest and farthest points. Conservation of energy links each speed to radius and semi-major axis, while Kepler’s period depends on the semi-major axis alone.
This page derives period, apsis speeds and eccentricity from the two extreme radii. It is a static orbital-geometry calculation, distinct from planning a transfer between two separate circular orbits.
The Hohmann coplanar circular orbit transfer delta v calculator connects two separate circular orbits through a half ellipse. A single bound ellipse instead has its own fixed apsides and period.
Use the right input conventions
Both radii are measured from the centre of the attracting body. Altitude above a surface must first be combined with that body’s radius. Apoapsis must be at least periapsis; equal values represent the circular limit.
The gravitational parameter uses km³/s² and distances use kilometres, producing speeds in km/s and periods in seconds. The default parameter is illustrative Earth data, but the calculation uses the value you enter rather than an automatic body lookup.
Interpret the result
Periapsis speed exceeds apoapsis speed for a noncircular ellipse. Multiplying each apsis radius by its speed gives the same specific angular momentum, a useful independent consistency check.
Eccentricity is the radius difference divided by their sum. It approaches zero for a circle and approaches one for a highly elongated bound ellipse; this page does not support parabolic or hyperbolic trajectories.
Work through the example
For gravitational parameter 398600.4418 km³/s² and radii 7000 and 14000 km, the semi-major axis is 10500 km and eccentricity is one third.
Periapsis speed is about 8.71343 km/s, while apoapsis speed is half that value because its radius is twice as large. Period is about 10707.7 seconds under the ideal two-body model.
Where the model stops
The central mass is treated as a point or spherical attractor. Atmospheric drag, oblateness, third bodies and non-gravitational forces are absent.
A mathematical radius does not demonstrate clearance from the physical body. This page is educational and does not supply navigation commands, manoeuvre timing or an operational orbit determination.
A few questions, answered
Can I enter altitude instead of radius?
No. Add the central body radius first so each distance is measured from its centre.
What changes in the comparison?
Set both apsides equal and inspect the circular speed and zero eccentricity.