Vis-Viva Equation
Calculate orbital speed at any point in an elliptical orbit.
The vis-viva equation relates orbital speed to position and the semi-major axis.
The Formula
The vis-viva equation (Latin: "living force") gives the orbital speed of an object at any point in its orbit. It is derived from conservation of energy and works for any conic section orbit: circular, elliptical, parabolic, or hyperbolic. "Vis-viva" reflects its connection to kinetic energy, which 17th-century physicists called "living force."
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| v | Orbital speed at current position | m/s |
| G | Gravitational constant = 6.674 × 10⁻¹¹ N·m²/kg² | N·m²/kg² |
| M | Mass of the central body | kg |
| r | Current distance from the center of the central body | meters (m) |
| a | Semi-major axis of the orbit | meters (m) |
For a circular orbit: a = r, so v² = GM/r → v = √(GM/r)
For escape velocity from radius r: a → ∞, so v_esc = √(2GM/r)
Escape velocity is exactly √2 times the circular orbital velocity at the same radius.
Example 1 — Earth's Orbit Around the Sun
Earth orbits the Sun at an average distance of 1 AU = 1.496 × 10¹¹ m (nearly circular). Find Earth's orbital speed.
GM_Sun = 1.327 × 10²⁰ m³/s²
For circular orbit: r = a = 1.496 × 10¹¹ m
v² = GM/r = 1.327 × 10²⁰ / 1.496 × 10¹¹ = 8.87 × 10⁸ m²/s²
v ≈ 29,800 m/s ≈ 29.8 km/s — Earth orbits the Sun at about 107,000 km/h
Example 2 — Comet at Perihelion vs Aphelion
Halley's Comet has perihelion r_p = 0.586 AU and aphelion r_a = 35.08 AU. Semi-major axis a = (r_p + r_a)/2 = 17.83 AU. Find speeds at each extreme.
GM_Sun = 1.327 × 10²⁰ m³/s², 1 AU = 1.496 × 10¹¹ m
a = 17.83 × 1.496 × 10¹¹ = 2.667 × 10¹² m
At perihelion: v² = 1.327 × 10²⁰ × (2/(0.586×1.496×10¹¹) − 1/2.667×10¹²)
v_perihelion ≈ 54,500 m/s (54.5 km/s)
v_aphelion ≈ 910 m/s (0.91 km/s) — 60× slower at aphelion than perihelion, as required by conservation of angular momentum
When to Use It
Use the vis-viva equation when:
- Finding orbital speed at any point in an elliptical orbit, not just perihelion and aphelion
- Planning spacecraft maneuvers — how much delta-v is needed to change orbits
- Calculating hyperbolic excess velocity for interplanetary missions
- Comparing comet or asteroid speeds at different points in their orbits