Orbital Energy Formula
Calculate the total mechanical energy of an orbiting satellite or planet.
Negative orbital energy means the object is gravitationally bound.
The Formula
The total mechanical energy of an orbiting body is the sum of its kinetic energy and gravitational potential energy. The result is always negative for a bound orbit — negative energy means the object is gravitationally captured. A positive total energy means the object has enough speed to escape to infinity (hyperbolic trajectory).
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| E | Total mechanical energy of the orbit | Joules (J) |
| G | Gravitational constant = 6.674 × 10⁻¹¹ N·m²/kg² | N·m²/kg² |
| M | Mass of the central body (planet, star, etc.) | kg |
| m | Mass of the orbiting satellite or object | kg |
| a | Semi-major axis of the orbit (radius for circular orbits) | meters (m) |
For a circular orbit, this simplifies because a = r and the orbital speed is v = √(GM/r):
E = −½mv²
The orbital energy depends only on the semi-major axis — not the eccentricity of the orbit. Two orbits with the same semi-major axis have the same energy, even if one is circular and the other is highly elliptical.
Example 1 — International Space Station
The ISS orbits at approximately 408 km altitude. Calculate its orbital energy per kilogram of spacecraft mass.
a = 6371 + 408 = 6779 km = 6.779 × 10⁶ m
G × M_Earth = 3.986 × 10¹⁴ m³/s² (standard gravitational parameter)
E/m = −GM/(2a) = −3.986 × 10¹⁴ / (2 × 6.779 × 10⁶)
E/m ≈ −29.4 MJ/kg. The ISS has 29.4 megajoules of negative orbital energy per kilogram of its mass.
Example 2 — Changing Orbits
A 1000 kg satellite moves from a circular orbit at 300 km altitude to 600 km altitude. How much energy must be added?
a₁ = 6371 + 300 = 6671 km = 6.671 × 10⁶ m
a₂ = 6371 + 600 = 6971 km = 6.971 × 10⁶ m
E₁ = −GM × m/(2a₁) = −3.986 × 10¹⁴ × 1000 / (2 × 6.671 × 10⁶) = −29.85 GJ
E₂ = −3.986 × 10¹⁴ × 1000 / (2 × 6.971 × 10⁶) = −28.58 GJ
ΔE = E₂ − E₁ = 1.27 GJ must be added by rocket thrust to raise the orbit
When to Use It
Use the orbital energy formula when:
- Calculating the energy cost of launching satellites to different altitudes
- Determining whether an orbit is bound (E < 0), parabolic (E = 0), or hyperbolic (E > 0)
- Analyzing Hohmann transfers between circular orbits
- Computing orbital periods using Kepler's third law (T² ∝ a³)