Vis-Viva Orbital Velocity Equation
Computes the exact instantaneous orbital velocity of an orbiting satellite or spacecraft at any distance r along its elliptical trajectory of semi-major axis a.
\[ v^2 = \mu \left( \frac{2}{r} - \frac{1}{a} \right) \quad \text{where } \mu = G M \]
Plain Text Formula:
v = sqrt(mu * (2/r - 1/a)) where mu = G * M
📐 Variable Definitions & Parameters
| Symbol | Variable Name | Standard Unit | Physical Definition |
|---|---|---|---|
| \[ v \] | Orbital Speed | m/s or km/s | Instantaneous scalar velocity vector magnitude |
| \[ \mu \] | Standard Gravitational Parameter | m³/s² | G * M of primary body (Earth mu = 3.986004418×10¹⁴ m³/s²) |
| \[ r \] | Current Radial Distance | meters (m) | Distance from primary focus to orbiting body |
| \[ a \] | Semi-Major Axis | meters (m) | Half of major orbital axis (a > 0 ellipse, a = infinity parabola) |
💡 Worked Engineering Example
Problem Statement:
Find the speed of a satellite in circular Low Earth Orbit (r = a = 6,771 km = 6.771e6 m).
Step-by-Step Solution:
- In circular orbit r = a, so (2/r - 1/a) = 1/r
- v = sqrt(mu / r) = sqrt(3.986e14 / 6.771e6) = sqrt(58,868,704)
- v = 7,672 m/s = 7.67 km/s
🎯 Result: Orbital speed in circular 400 km LEO is 7.67 km/s (27,620 km/h).