Orbital Energy Calculator (Specific and Total)
Calculate specific and total orbital energy of a satellite from semi-major axis, or from current speed and distance.
Classifies bound, parabolic, hyperbolic.
One number that tells you everything about an orbit’s fate
The specific orbital energy is the single quantity that decides whether something stays in orbit or escapes forever. It is the sum of kinetic energy and gravitational potential energy, per unit mass:
epsilon = v² / 2 − mu / r
where v is the current speed, r is the current distance from the center of the body, and mu is the standard gravitational parameter (mu = G × M, the gravitational constant times the central body’s mass). The result is energy per kilogram, in joules per kilogram (J/kg), or equivalently m²/s².
The remarkable thing is that for a given orbit, epsilon stays constant everywhere. As a satellite falls toward periapsis it speeds up (kinetic energy rises) but drops deeper into the gravity well (potential energy falls by exactly the same amount). The two always trade off so the total never changes. This is conservation of energy in its purest orbital form.
The semi-major axis form
Because epsilon is constant, it can be written purely from the orbit’s shape:
epsilon = − mu / (2a)
where a is the semi-major axis. This is often the cleaner form because it does not require knowing the speed at any particular point. It also reveals the deep fact that orbital energy depends only on a, not on eccentricity. A nearly circular orbit and a wildly elliptical one with the same semi-major axis have identical energy. They take the same time to complete (same period) and require the same energy to establish.
Total versus specific energy
Multiply by the orbiting mass m to get total mechanical energy:
E = − G × M × m / (2a) = m × epsilon
For most astrodynamics work, specific energy (per kg) is preferred because it is independent of the spacecraft’s mass: a marble and a space station in the same orbit have the same specific energy. This calculator reports both when you supply the orbiting mass.
The sign of epsilon classifies the trajectory
This is the whole point of the quantity:
| epsilon | Trajectory | Meaning |
|---|---|---|
| epsilon < 0 | Elliptical (or circular) | Bound. The object is gravitationally captured and orbits forever (absent drag or perturbations). |
| epsilon = 0 | Parabolic | The exact escape boundary. The object reaches infinity with zero leftover speed. |
| epsilon > 0 | Hyperbolic | Unbound. The object escapes and still has speed left over at infinity. |
Escape velocity is just the speed that makes epsilon = 0 at your current distance: v_escape = sqrt(2 mu / r). Plug that in and the kinetic term exactly cancels the potential term.
Standard gravitational parameters
You can enter the central body’s mass directly, or use these common mu values (mu = GM, in m³/s²), which are known far more precisely than G or M individually:
- Earth: 3.986 × 10¹⁴
- Sun: 1.327 × 10²⁰
- Moon: 4.903 × 10¹²
- Mars: 4.283 × 10¹³
- Jupiter: 1.267 × 10¹⁷
Worked example, the ISS
The International Space Station orbits at roughly 408 km altitude. Earth’s radius is about 6,371 km, so a = 6,779 km = 6.779 × 10⁶ m. With Earth’s mu = 3.986 × 10¹⁴:
epsilon = − 3.986 × 10¹⁴ / (2 × 6.779 × 10⁶) = − 29.4 × 10⁶ J/kg = − 29.4 MJ/kg
The negative sign confirms the ISS is bound. The magnitude, 29.4 megajoules per kilogram, is the energy you would need to add to each kilogram to bring it exactly to escape (epsilon = 0). For the station’s roughly 420,000 kg mass, that is about 12 terajoules to fully unbind, which is why nobody pushes the ISS to escape velocity and instead deorbits it downward.
Why this matters for mission design
Every orbit transfer is really an energy change. Raising an orbit (going to a higher a) means making epsilon less negative, which costs propellant. A Hohmann transfer is the minimum-energy way to move between two circular orbits: two small burns, each nudging epsilon up by just the right amount. Interplanetary missions push epsilon all the way to positive (hyperbolic escape from Earth) and then let the Sun’s gravity recapture the craft into a new heliocentric orbit. Gravity assists are clever exchanges of energy with a planet, changing the spacecraft’s epsilon relative to the Sun without burning fuel. Understanding the sign and magnitude of orbital energy is the foundation underneath all of it.
A note on units
Distances must be from the center of the body, not the surface. For an Earth satellite at 400 km altitude, r is 6,371 + 400 = 6,771 km, not 400 km. This is the single most common mistake. This calculator expects r and a in kilometers and converts internally.