visviva

Vis-viva, or the vis-viva equation, is a fundamental relation in orbital mechanics that gives the speed of a body in orbit at a distance r from the focus of its orbit around a much more massive primary. The equation is v^2 = μ (2/r − 1/a), where v is the instantaneous orbital speed, r is the distance to the central mass, a is the orbit’s semi-major axis, and μ is the standard gravitational parameter, equal to G M (G is the gravitational constant and M is the central body's mass). The form can be derived from the conservation of specific orbital energy ε = v^2/2 − μ/r, which for a Keplerian orbit equals − μ/(2a).

The vis-viva equation applies to conic-section orbits: ellipses (a > 0), hyperbolas (a < 0), and the parabolic limit (a → ∞). For a circular orbit, r = a and the equation reduces to v^2 = μ / r, the familiar circular orbital speed.

Applications of vis-viva are widespread in celestial and space mission analysis. It is used to compute orbital speeds at given positions, to determine velocity requirements for orbit transfers, and to diagnose orbital energy changes in trajectory design. While highly accurate for two-body dynamics, the equation assumes a point-mass central body and neglects perturbations from other bodies, non-spherical gravity, atmospheric drag, and relativistic effects. In practical use, μ is expressed in units compatible with the chosen distance and time units, such as km^3/s^2 for distance in kilometers and time in seconds.