Orbital Energy Calculator

The total orbital energy of a two-body system is negative for bound orbits and zero for escape trajectories. Energy depends on the semi-major axis, not on eccentricity.

🛸 Astrophysics📐 E = -GMm/2r🌍 Orbital Mechanics

Central Mass M (kg)

Orbiting Mass m (kg)

Orbital Radius r (m)

Please enter valid values.

Formula & Reference

Variable	Symbol	Formula	Units
Orbital Energy	E	E = −GMm/(2r)	J
Escape Condition	E = 0	r = ∞ (parabolic trajectory)	—
Virial Theorem	—	KE = −E, PE = 2E	—
Binding Energy	—	−E = GMm/(2a)	J

Step-by-Step Examples

Example 1

ISS Orbit

M=5.972e24 kg (Earth), m=420,000 kg (ISS), r=6.771e6 m.

E = -6.674e-11 * 5.972e24 * 420000 / (2*6.771e6)
E = -1.237e16 J = -12.37 PJ

✓ ISS total orbital energy = -12.37 petajoules

Example 2

Geostationary Orbit

r=42,164 km = 4.2164e7 m, m=5000 kg satellite.

E = -6.674e-11*5.972e24*5000/(2*4.2164e7)
E = -2.364e10 J = -23.64 GJ

✓ GEO satellite orbital energy = -23.64 GJ

Example 3

Energy to Circularize

Compare two orbital energies to find delta-E needed.

E(LEO,r=6.6e6m) - E(GEO,r=4.2e7m)
= -G*M*m/2 * (1/r_LEO - 1/r_GEO)

✓ Energy difference = work done by rocket engines

Real-World Applications

🛸

Satellite Design

Orbital energy determines fuel needed to reach and maintain an orbit.

🚀

Transfer Orbits

Hohmann transfer: two burns change orbital energy to reach target orbit.

🌌

Binary Stars

Two stars orbit their common center of mass; total orbital energy determines orbital period.

🔭

Exoplanet Detection

Orbital energy and period give stellar mass via Kepler third law.

Common Mistakes to Avoid

⚠️

Negative energy means bound orbit

E < 0: bound (circular or elliptical orbit). E = 0: escape parabolic. E > 0: hyperbolic flyby. Always negative for stable orbits.

⚠️

r is semi-major axis for ellipses

For elliptical orbits, replace r with semi-major axis a. Energy depends on a, not eccentricity.

⚠️

Virial theorem: KE = -E

Kinetic energy = magnitude of total energy. Potential energy = twice total energy. Important for stable bound systems.

Frequently Asked Questions

Why is orbital energy negative? ▾

Negative energy means the orbiting body is gravitationally bound. Zero energy is the minimum to escape to infinity. More negative = more tightly bound.

What is the vis-viva equation? ▾

v^2 = GM*(2/r - 1/a). Gives orbital speed at any point knowing position r and semi-major axis a.

How much energy to launch to GEO? ▾

From Earth surface (r=6.371e6 m) to GEO (r=42,164 km): dE = -GMm/2*(1/r_GEO - 1/r_surf) plus kinetic energy. About 58 MJ/kg payload.

What is the virial theorem? ▾

For stable gravitationally bound systems: time-averaged KE = -1/2 * PE. So total E = KE + PE = -KE = PE/2. Powerful for understanding clusters and galaxies.

What does orbital energy tell us about stability? ▾

More negative E = more energy needed to escape = more stable orbit. Low Earth orbit (E ~ -30 MJ/kg) is less stable than GEO (E ~ -4.7 MJ/kg) against perturbations.

Related Physics Calculators

Formula Explorer connections

Interpretation: This relationship connects motion, force, momentum, work or energy in a mechanical system. Assumption: Choose a consistent reference direction and unit system. The model may assume constant acceleration, rigid bodies, negligible losses or an isolated system.

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