Vis‑Viva Equation Calculator
Solve for orbital speed, distance, semi-major axis or gravitational parameter, with step-by-step math, an orbit diagram, a speed-vs-distance curve and a comparison of key orbital speeds.
The vis-viva equation
The vis-viva equation gives the speed of a body at any point on a Keplerian orbit using only its distance from the central body (r) and the orbit's semi-major axis (a). It follows from conservation of energy: kinetic plus potential energy per unit mass is constant and equal to −μ/(2a). It works for every conic section: ellipses (a > 0), parabolas (a → ∞) and hyperbolas (a < 0, entered as a negative number).
a = 1 / (2/r − v²/μ)
r = 2 / (v²/μ + 1/a)
μ = v² / (2/r − 1/a)
Circular orbit (r = a): v = √(μ/r) · Parabola (1/a = 0): v = √(2μ/r)
| Symbol | Meaning | Units used here |
|---|---|---|
| v | Orbital speed at distance r | km/s |
| μ | Gravitational parameter G·M of the central body | km³/s² |
| r | Distance from the centre of the central body | km |
| a | Semi-major axis (negative for hyperbolic paths) | km |
| R | Radius of the central body (for altitudes and the diagram) | km |
| γ | Flight-path angle at r (0° at perigee/apogee), used only for e, perigee and apogee | degrees |
Worked example
a = (6,678 + 42,164) / 2 = 24,421 km
vperigee = √(398,600 · (2/6,678 − 1/24,421)) = 10.15 km/s
vapogee = √(398,600 · (2/42,164 − 1/24,421)) ≈ 1.61 km/s
FAQ
Why does the equation not need the eccentricity?
Speed depends only on the orbital energy (which sets a) and the distance r. Eccentricity matters for where the orbit reaches its closest and farthest points, which is why the calculator uses the optional flight-path angle to find them.
What does a negative semi-major axis mean?
It describes a hyperbolic flyby: the body has more than escape energy and never returns. Enter a negative a (or solve for a with a speed above escape speed) and the same equation applies.
How is this related to circular orbital velocity?
Set r = a and the equation reduces to v = √(μ/r). Moving to a larger a at the same r gives a higher speed, and a = ∞ gives escape speed √(2μ/r).
What assumptions are made?
A two-body system with a point-mass central body, no drag, no thrust and no perturbations from other bodies.
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| Parameter | Value |
|---|---|
| r | 6.678×10⁶ m |
| a | 6.678×10⁶ m |
| Parameter | Value |
|---|---|
| r | 6.678×10⁶ |
| a | 2.4419×10⁷ |
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
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