Documentation / @interstellar-tools/equations / visVivaSpeed

Function: visVivaSpeed()

function visVivaSpeed(r: number, a: number, mu: number): number;

Defined in: orbits/vis-viva-speed.ts:60

Vis-viva (orbital speed) - compute speed magnitude ($v$) from ($r$), ($a$), and ($\mu$).

Definition

$$v=\sqrt{\mu (\frac{2}{r}-\frac{1}{a})}$$

• ($r$): current distance from the focus (m)
• ($a$): semi-major axis (m). Elliptic ($a>0$), parabolic ($a\to \mathrm{\infty }$), hyperbolic ($a<0$)
• ($\mu$): gravitational parameter (m³/s²)

Units

• Inputs: ($r$) in m, ($a$) in m, ($\mu$) in m³/s².
• Output: ($v$) in m/s.

TIP

• For circular orbits (($a=r$)): ($v=\sqrt{\mu /r}$).
• For parabolic trajectories, pass a = Infinity (so ($1/a=0$)): ($v=\sqrt{2\mu /r}$).

In practiceDeparting for Mars

32.73 km/s

Speed at the start of a Hohmann transfer to Mars

Earth moves around the Sun at 29.78 km/s, so a spacecraft leaving on the transfer orbit needs 2.95 km/s on top of Earth's own speed.

const au = 1.495978707e11; // m
const muSun = 1.32712440018e20; // m³/s²
const a = ((1 + 1.524) / 2) * au; // transfer orbit

visVivaSpeed(au, a, muSun); // ≈ 32 731 m/s

Parameters

Parameter	Type	Description
r	number	Radius ($r$) (m), distance to the attracting body.
a	number	Semi-major axis ($a$) (m). Use Infinity for parabolic, negative for hyperbolic.
mu	number	Gravitational parameter ($\mu$) (m³/s²).

Returns

number

Speed magnitude ($v$) (m/s).

Throws

Error If inputs are non-finite / out of domain, or the radicand becomes negative.

Example

// LEO (~400 km): a ≈ r for near-circular
const muEarth = 3.986004418e14; // m³/s²
const r = 6378e3 + 400e3; // m
const a = r; // circular
const v = visVivaSpeed(r, a, muEarth); // ≈ 7670 m/s