Orbital Velocity

Derivation

For a circular orbit of radius r around a body of mass M, gravity provides the centripetal force:

GMm/r² = mv²/r

The satellite mass m cancels, giving:

v_orb = √(GM/r)

Higher orbits → larger r → slower orbital velocity. The ISS at 400 km altitude moves at ~7.66 km/s; geostationary satellites at 35,786 km altitude move at only ~3.07 km/s.

Formula and Variables

Kepler's Third Law

The orbital period T = 2πr/v_orb = 2πr/√(GM/r) = 2π√(r³/GM). Squaring:

T² = (4π²/GM) × r³

This is Kepler's third law: the square of the orbital period is proportional to the cube of the semi-major axis. For any two bodies orbiting the same central mass: T₁²/r₁³ = T₂²/r₂³.

Key Orbit Types

Worked Examples

Example 1 — ISS orbital speed

ISS altitude: h = 408 km. Earth's radius: R = 6,371 km. So r = 6,371 + 408 = 6,779 km = 6.779 × 10⁶ m.

v = √(GM/r) = √(6.674 × 10⁻¹¹ × 5.972 × 10²⁴ / 6.779 × 10⁶) = √(5.879 × 10⁷) ≈ 7,667 m/s ≈ 7.67 km/s ✓

Example 2 — Geostationary radius

For T = 24 h = 86,400 s: r³ = GM·T²/(4π²) = (6.674 × 10⁻¹¹ × 5.972 × 10²⁴ × 86400²)/(4π²). r = ∛(7.54 × 10²²) ≈ 42,241 km from Earth's centre, altitude ≈ 35,870 km ✓

Orbital vs Escape Velocity

At the same radius: v_orb = √(GM/r) and v_esc = √(2GM/r) = √2 × v_orb.

Escape velocity is always √2 ≈ 1.41 times the circular orbital velocity. Adding ~41% more speed at a given orbital radius converts a circular orbit into an escape trajectory.

Common Mistakes

• Using altitude instead of radius. r in the formula is measured from the centre of the planet, not from its surface. Always add the planet's radius.
• Forgetting period increases with altitude. Higher orbits are slower AND have longer circumferences — periods increase rapidly: T ∝ r^(3/2).
• Confusing orbital and escape velocity. Orbital velocity keeps you in orbit. Escape velocity gets you out. Escape = √2 × orbital at the same point.

Energy in Circular Orbits

A satellite in circular orbit has total mechanical energy E = KE + PE = ½mv² − GMm/r. Since v² = GM/r, KE = ½mv² = GMm/2r and PE = −GMm/r. Therefore:

E = GMm/2r − GMm/r = −GMm/2r

The total energy is negative (bound orbit) and equals −KE. To raise a satellite to a higher orbit, you must add energy (make E less negative). Paradoxically, the satellite ends up moving slower — it gains potential energy faster than it loses kinetic energy.

Hohmann Transfer Orbits

The most fuel-efficient way to move between two circular orbits is the Hohmann transfer — an elliptical orbit tangent to both circles. Two burns are needed: the first raises the apogee to the target orbit; the second circularises at the target. For a transfer from orbit r₁ to r₂:

• ΔV₁ = √(GM/r₁) × (√(2r₂/(r₁+r₂)) − 1)
• ΔV₂ = √(GM/r₂) × (1 − √(2r₁/(r₁+r₂)))

Hohmann transfers are used by virtually every interplanetary mission, from Mars rovers to the James Webb Space Telescope.

Orbital Mechanics and Special Relativity

At low altitudes, Newtonian orbital mechanics is accurate. However, general relativity predicts precession of orbital ellipses — the famous 43 arcseconds per century for Mercury that could not be explained classically. GPS satellites also require both special-relativistic (time dilation from velocity) and general-relativistic (gravitational time dilation) corrections to maintain nanosecond-level timing accuracy.

Related Topics

Perturbations and Real Orbits

In practice, orbits deviate from ideal Keplerian paths due to perturbations. For satellites in low Earth orbit, the dominant perturbations are: Earth's oblateness (J₂ term), which causes nodal precession; atmospheric drag, which slowly reduces orbital altitude; and lunar and solar gravity. Mission planners must account for all of these when designing orbital manoeuvres.

Atmospheric drag in low orbit causes orbits to decay gradually. Counterintuitively, a satellite experiencing drag speeds up as it spirals inward — losing potential energy faster than kinetic energy increases. The ISS requires reboost manoeuvres every few weeks to months to maintain altitude against drag.

At geostationary altitude, inclination drift from lunar and solar gravity is the main perturbation. Correcting this consumes station-keeping propellant, which is often the life-limiting factor for GEO satellites.

Vis-Viva Equation

For any conic-section orbit (circular, elliptical, parabolic, hyperbolic) the speed at any point is given by the vis-viva equation: v² = GM(2/r − 1/a), where r is the current distance from the central body and a is the semi-major axis. For a circular orbit (r = a): v² = GM/r — recovering the circular orbital velocity. For a parabolic (escape) trajectory (a → ∞): v² = 2GM/r — recovering escape velocity.

Frequently Asked Questions

References

1. Halliday, D., Resnick, R., & Krane, K. S. (2001). Physics (5th ed.). Wiley. Chapter 13.
2. Bate, R. R., Mueller, D. D., & White, J. E. (1971). Fundamentals of Astrodynamics. Dover Publications.
3. Vallado, D. A. (2013). Fundamentals of Astrodynamics and Applications (4th ed.). Microcosm Press.