Vis-Viva Equation

Gives the instantaneous orbital speed of a body at any point in an elliptical (or other conic) orbit.

Written by the Vorzep Team

Orbital MechanicsAstrodynamicsFundamental

Vis‑Viva Equation Calculator v = √(μ · (2/r − 1/a))

v = √( μ · (2 / r − 1 / a) )

v = orbital speed (m/s) · μ = standard gravitational parameter (m³/s²) · r = distance from central body (m) · a = semi‑major axis (m)

⟹ Solve ⟷ v, μ, r, a

v (speed)

m/s

μ (grav. parameter)

m³/s²

r (radius)

a (semi‑major axis)

Please fix the errors above.

Solve for: v (m/s) μ (m³/s²) r (m) a (m)

Presets:

Orbital Speed (v) —

—

v: — μ: — r: — a: —

✓ Copied!

Orbital Speed —

Low (< 3 km/s) Medium (3–10 km/s) High (> 10 km/s)

v = √(μ·(2/r − 1/a)) · The Vis‑Viva equation gives the speed of an orbiting body at any point in its orbit.

Variables

Symbol	Quantity	Unit
v	Orbital speed	m/s
μ	Standard gravitational parameter	m3/s2
r	Instantaneous radial distance	m
a	Semi-major axis	m

What it means

The vis‑viva equation is the energy equation for orbital motion, relating the speed v at a given radial distance r to the semi‑major axis a. It is derived from conservation of specific orbital energy (ε = v²/2 − μ/r = −μ/(2a)). This equation is one of the most important in astrodynamics, as it allows calculation of velocity at any point in an orbit (elliptical, parabolic, or hyperbolic). It is used for orbit determination, manoeuvre planning, and interplanetary trajectory design. For circular orbits (r=a), it reduces to v = √(μ/r). Understanding the vis‑viva equation is essential for orbital mechanics and space mission design.

Worked example

Vis‑Viva Equation – Two Examples

Real‑World

Scenario: At r = 6.678×10⁶ m, a = 6.678×10⁶ m (circular). Find orbital speed.

Parameter	Value
r	6.678×10⁶ m
a	6.678×10⁶ m

1v = √(μ(2/r - 1/a)) = √(3.986e14(2/6.678e6 - 1/6.678e6)) = √(3.986e14/6.678e6) = 7726 m/s

Result 7,726 m/s ✓ Circular

Scenario: At perigee of transfer orbit r = 6.678×10⁶, a = 2.4419×10⁷. Find speed.

Parameter	Value
r	6.678×10⁶
a	2.4419×10⁷

1v = √(3.986e14(2/6.678e6 - 1/2.4419e7)) = √(3.986e14(2.995e-7 - 4.095e-8)) = √(3.986e14 × 2.585e-7) = √(103,030,000) = 10,150 m/s

Result 10,150 m/s ✓ Transfer orbit

Key insight: Vis‑viva gives orbital speed for any position in any conic orbit.

Common mistakes

Vis‑viva equation: v = √(μ·(2/r − 1/a)).
a: Semi‑major axis of the orbit.
r: Radial distance at the point of interest.
Gives speed at any point in an elliptical orbit.
Units: consistent SI.

Applications

The Vis‑Viva equation, v = √(μ·(2/r − 1/a)), relates the speed of an object in an orbit to its radial distance and the semi‑major axis. It is the central equation of orbital mechanics, used to compute velocities at any point in an elliptical orbit. Engineers use it for trajectory design, orbit determination, and rendezvous planning. It allows calculation of the required Δv for orbit changes and is essential for interplanetary missions. By applying the Vis‑Viva equation, aerospace engineers can determine the energy state of a spacecraft and plan efficient manoeuvres, making it a cornerstone of astrodynamics.

Orbit trajectory design and analysis
Computation of spacecraft speeds at perigee, apogee
Mission planning for orbital insertion and transfers
Rendezvous and docking simulation
Interplanetary trajectory optimisation