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Orbital Energy Calculator

Enter the central body mass and orbit semi-major axis to find the specific orbital energy.

Result

-2.8471 × 10⁷ J/kg
3.9859 × 10¹⁴ m³/s²

How it is calculated

The tool first finds the standard gravitational parameter, μ = G · M, with G = 6.67430×10⁻¹¹ m³ kg⁻¹ s⁻² and M the mass of the central body in kilograms. The semi-major axis a is typed in kilometres and converted to metres (× 1000). The specific orbital energy is then ε = −μ / (2a), in joules per kilogram of the orbiting body. Both results are printed in scientific notation with five significant figures. Mass and semi-major axis must be greater than zero, otherwise no result is produced. Because the formula always gives a negative number for a positive a, the tool describes bound (elliptical or circular) orbits; the energy is the same wherever the body is on that orbit. Note that a is measured from the centre of the central body, not from its surface.

Worked example

Take Earth (M = 5.972e24 kg) and a satellite with a semi-major axis of 7000 km. Step 1: μ = 6.6743e-11 × 5.972e24, which the tool shows as 3.9859 × 10¹⁴ m³/s². Step 2: convert the axis, 7000 km = 7 000 000 m, so 2a = 1.4e7 m. Step 3: ε = −3.9859 × 10¹⁴ / 1.4e7. The specific orbital energy is -2.8471 × 10⁷ J/kg, the negative sign meaning the satellite is gravitationally bound to Earth.

Try this example →

Common mistakes

  • Typing the semi-major axis in metres. The field expects kilometres, so entering 7000000 for a low Earth orbit makes the energy 1000 times too small in size.
  • Entering the altitude above the surface instead of the semi-major axis. For Earth, a 600 km altitude orbit has a ≈ 6371 + 600 = 6971 km, because a is measured from the centre.
  • Putting the mass in grams, solar masses or Earth masses. The field takes kilograms, so Earth is 5.972e24, not 1.
  • Expecting the result to change with eccentricity or with the satellite's current position. Only a matters; a more elongated orbit with the same a has exactly the same energy.
  • Confusing specific energy (J/kg) with the total energy of the spacecraft. Multiply by the spacecraft's mass to get joules.

Sources

About this calculator

The specific orbital energy (ε = −μ/2a) of a two-body orbit is the sum of kinetic and potential energy per unit mass, where μ is the standard gravitational parameter (GM) of the central body and a is the orbit's semi-major axis. Because ε depends only on a — not on eccentricity or where the orbiting body currently sits along its path — it stays constant throughout the orbit under the vis-viva relation, which is exactly what makes it useful: it's a conserved quantity you can check against at any point.

This calculator takes the central body's mass (in kg) and the orbit's semi-major axis (in km), computes μ, and returns μ and the specific orbital energy. Both inputs must be positive, so the result is always a negative ε, which is the sign of a bound orbit; for reference, ε = 0 would be parabolic and positive ε a hyperbolic escape trajectory. Aerospace engineering students, mission planners sanity-checking transfer orbits, and astronomy enthusiasts modeling satellite or planetary motion use it to verify orbit energy and cross-check vis-viva calculations without deriving the energy equation from scratch each time.

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Comments (1)

  • Editorial team

    A circular orbit gives you a quick cross-check on orbital speed. Since ε = −v²/2 there, the speed is the square root of twice the size of ε. For the Earth example with a = 7000 km, ε is -2.8471 × 10⁷ J/kg, which gives about 7,546 m/s. The tool does not output speed itself, so do this last step by hand. Which central bodies would you like us to list the masses for?

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