Energy of Orbiting Satellites: Kinetic and Potential

A satellite in orbit carries two kinds of energy at once: the kinetic energy of its high-speed motion and the gravitational potential energy of its height. Added together they give the satellite's total mechanical energy — and for a circular orbit this total is always negative, the mathematical stamp of being gravitationally bound.

This matters because the energy ledger controls every orbital manoeuvre. Raising a satellite to a higher orbit means adding energy (making the total less negative), and the neat 2-to-1 split between kinetic and potential energy lets engineers compute fuel needs with one formula instead of a simulation.

Key Concepts

• Kinetic energy: KE = (1/2)mv². With v² = GM/r for circular orbits, KE = GMm/(2r) — positive, as motion always is.
• Potential energy: PE = −GMm/r — negative, twice the magnitude of KE, measuring how deep the satellite sits in Earth's well.
• Total energy: E = KE + PE = −GMm/(2r) — negative, so the satellite is bound; zero would mean just barely escaping.
• The virial-style split: |PE| = 2·KE and |E| = KE. Memorise one, get all three.

Formulas

• KE = GMm/(2r), PE = −GMm/r, E = −GMm/(2r).
• r is measured from the planet's centre (r = R + h).
• Energy to move between circular orbits: ΔE = E₂ − E₁ = (GMm/2)(1/r₁ − 1/r₂) — positive when r₂ > r₁.
• Units: joules (J).

Worked Example

How much energy does it take to raise a 1000 kg satellite from a 400 km orbit to an 800 km orbit?

1. r₁ = 6.771 × 10⁶ m; r₂ = 6.371 × 10⁶ + 0.8 × 10⁶ = 7.171 × 10⁶ m.
2. E₁ = −GMm/(2r₁) = −3.986 × 10¹⁷/(2 × 6.771 × 10⁶) = −3.986 × 10¹⁷/1.3542 × 10⁷ ≈ −2.943 × 10¹⁰ J.
3. E₂ = −3.986 × 10¹⁷/(2 × 7.171 × 10⁶) = −3.986 × 10¹⁷/1.4342 × 10⁷ ≈ −2.780 × 10¹⁰ J.
4. ΔE = E₂ − E₁ = (−2.780 + 2.943) × 10¹⁰ ≈ 1.64 × 10⁹ J.

About 1.6 billion joules (≈ 455 kWh) of extra energy — supplied by rocket burns — lifts the tonne satellite 400 km higher. The total energy got less negative, which is what "gaining energy" means for a bound object.

Common Misconceptions

• "Higher satellites have less energy." The opposite: total energy increases (toward zero) with altitude. Higher orbits are more energetic — that is why reaching them costs fuel.
• "Faster means more total energy." Not here: the higher, slower orbit has the greater total energy because potential energy dominates the trade.
• "Negative energy means something is wrong." Negative total energy is perfectly physical — it is simply the definition of "bound". Only changes in energy need a physical source.

Real-World Applications

• Orbit raising: Hohmann transfers between orbits are budgeted entirely in ΔE; the formula above gives the minimum.
• Station-keeping: atmospheric drag steals total energy daily; the ISS burns fuel to top it back up.
• De-orbiting: lowering a dead satellite means removing energy — brakes, drag sails or retro burns do exactly that.

An orbiting satellite's books always balance as KE = GMm/(2r), PE = −GMm/r, E = −GMm/(2r): kinetic half the size of potential's magnitude, total negative and bound. Climb higher and the total rises toward zero — energy in, altitude out — the one rule behind every orbital manoeuvre.