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The vis-viva equation gives the speed magnitude at a point on a bound Kepler ellipse from the gravitational parameter, current radius and semimajor axis.
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Formula
v in km/s or m/s · μ in km3/s2 or m3/s2 · r and a in the same length unit
| Symbol | Meaning | Unit |
|---|---|---|
| Magnitude of instantaneous orbital velocity, not its direction | km/s or m/s | |
| System gravitational parameter; for a test body near a central mass, μ=GM | km3/s2 or m3/s2 | |
| Instantaneous center-to-center separation | km or m | |
| Semimajor axis of the bound elliptical orbit | km or m |
v^2=\mu\left(\frac{2}{r}-\frac{1}{a}\right)The equation follows from conservation of specific mechanical energy: v2/2−μ/r=−μ/(2a). For a test body around a spherical central mass, μ=GM; for relative two-body motion, μ=G(M+m). It assumes an ideal Newtonian two-body system without perturbations, drag or thrust. An ellipse has a>0 and r ranges between periapsis and apoapsis. Circular orbital speed is the special case r=a; escape speed belongs to the parabolic energy boundary, not an elliptical orbit.
Applies to an unperturbed Newtonian two-body problem on a bound ellipse with a>0. For a test body, μ=GM of the central mass; for relative motion of two bodies, μ=G(M+m). Use the same length unit for r and a.
Specific orbital energy is conserved and, for an ellipse, depends only on semimajor axis a.
Speed
For an ellipse, the radicand must be nonnegative.
Semimajor axis
For a bound ellipse, a is positive.
An Earth orbit has μ=398600 km3/s2, a=10000 km and periapsis radius r=7000 km. What is its speed?
v2=398600(2/7000−1/10000)=74.0 km2/s2; v≈8.60 km/s.
Units reduce to km2/s2 under the square root.
How does speed change between periapsis and apoapsis on a bound ellipse?
It is greater at periapsis: r is smaller there, and the body moves faster near the central focus, consistent with Kepler’s area law.
Apply the circular-orbit expression √(μ/r) at an elliptical orbit point.
Use 2/r−1/a; only for a circle, r=a, does it reduce to 1/r.
Use different units for r and a or mistake a for the instantaneous separation.
Convert both lengths to one unit; a is constant for the entire orbit.
Formula cluster
The vis-viva equation generalizes circular speed to points on an ellipse and complements Kepler’s third law with a local speed.
For an Earth ellipse with μ=398600 km3/s2, a=10000 km and r=7000 km, v=√[398600(2/7000−1/10000)]≈8.60 km/s. At apoapsis r=13000 km, v≈4.63 km/s. Check: the body moves faster at the nearer periapsis.
Find speed at periapsis, apoapsis or another known orbital point and compare elliptical transfer orbits energetically.
Here is how to work through a typical Vis-Viva Equation for Elliptical Orbits (v² = μ(2/r − 1/a)) task step by step:
Task
An Earth orbit has μ=398600 km3/s2, a=10000 km and periapsis radius r=7000 km. What is its speed?
Solution path
v2=398600(2/7000−1/10000)=74.0 km2/s2; v≈8.60 km/s.
Units reduce to km2/s2 under the square root.
Task
How does speed change between periapsis and apoapsis on a bound ellipse?
Solution path
It is greater at periapsis: r is smaller there, and the body moves faster near the central focus, consistent with Kepler’s area law.
It calculates the speed magnitude at a particular point on an ideal Kepler orbit. The inputs are the system gravitational parameter μ, the instantaneous distance r from the force center and the semimajor axis a of the entire orbit: v2=μ(2/r−1/a). The equation follows from conservation of specific mechanical energy. It does not give the direction of motion; that requires orbital elements or velocity components as well. On an ellipse, r changes along the path while a stays constant. This explains why a body moves faster at periapsis and slower at apoapsis.
Circular speed v=√(μ/r) assumes a constant orbital radius. The vis-viva equation also uses semimajor axis a and works at different points on an ellipse. In the circular case r=a, so 2/r−1/a becomes 1/r and the circular formula follows exactly. On an ellipse, r and a are generally different. Do not substitute circular speed at periapsis or apoapsis: vis-viva gives a different speed there, depending on eccentricity and orbital position. At the point r=a, its instantaneous speed equals circular speed at that radius; this does not make the rest of the orbit circular.
A bound ellipse has negative total energy and positive a. The parabolic boundary has zero specific total energy, formally corresponding to 1/a=0. In that limit, v2=2μ/r, the familiar local escape speed. This is a limiting case, not an ellipse with finite semimajor axis. Escape speed is the minimum launch speed that reaches infinity in the ideal model; vis-viva instead gives the speed of a body whose orbit shape and energy are already specified. At a given radius, this is the boundary speed at which the path is no longer closed and the body does not return.
Use μ=398600 km3/s2, a=10000 km and r=7000 km. First, 2/7000−1/10000≈0.0001857 km−1. Multiplying by μ gives v2≈74.0 km2/s2, so v≈8.60 km/s. Check the units: km3/s2 times km−1 yields km2/s2. At apoapsis on the same ellipse, r=13000 km and v≈4.63 km/s. The lower value is consistent with a bound body moving more slowly at the farther point. As a cross-check, the opposite apsis at r=13000 km gives 4.63 km/s; both apsides lie symmetrically around a=10000 km. Negative specific energy confirms the orbit is bound.
The derivation assumes a Newtonian two-body problem: a central spherical mass, no other gravitational perturbations, no drag and no ongoing thrust. For a test body, μ=GM of the central mass; for relative motion of two comparable bodies, use μ=G(M+m). Orbital parameters must share a reference frame and compatible length units. Real satellites can deviate because of oblateness, moons, radiation pressure or atmosphere. The equation alone is therefore insufficient for precision navigation; it is a basic energy relation and a useful plausibility check.
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