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    Orbital mechanics

    Orbital Energy Calculator

    Specific orbital energy and vis-viva speed at any point of an elliptical orbit.

    ε = −μ / (2a) · v = √(μ (2/r − 1/a))

    Inputs

    Results

    Specific orbital energy
    -8.16102 MJ/kg
    Speed at this distance (vis-viva)
    10.1516 km/s
    Total orbital energy
    -8.161 × 10^9 J
    Kinetic / potential per kg
    51,528,000 J/kg / -59,689,000 J/kg
    Worked solution
    1. ε = −μ/(2a) = −3.986 × 10^14 / (2 × 24,421,000) = -8,161,000 J/kg
    2. v = √(μ(2/r − 1/a)) = 10,152 m/s
    3. Check: ½v² − μ/r = -8,161,000 J/kg

    How it works

    The total energy per kilogram of an orbit depends only on its semi-major axis. Negative energy means the spacecraft is bound; zero energy is exactly escape speed.

    The vis-viva equation follows from conservation of energy and gives the speed anywhere along an ellipse — fastest at periapsis, slowest at apoapsis.

    Variables

    • ε — specific orbital energy (per kg)
    • a — semi-major axis
    • r — current distance from the centre

    Assumptions and limits

    • Bound elliptical orbit (a > 0)
    • Two-body dynamics; potential energy zero at infinity

    For education only. These models are simplified and are not suitable for spacecraft navigation or safety-critical engineering. See our scientific accuracy policy.