Documentation / @interstellar-tools/equations / specificAngularMomentum

Function: specificAngularMomentum()

function specificAngularMomentum(
  r: Vector3DTupleType,
  v: Vector3DTupleType
): number;

Defined in: orbits/specific-angular-momentum.ts:57

Specific angular momentum ($h$) (scalar magnitude).

Definitions

$$h=\Vert \mathbf{r}\times \mathbf{v}\Vert$$

Useful identities:

$$h=r{v}_{\perp },\text{(circular)}{v}_{\perp }=\sqrt{\mu /r}\Rightarrow h=\sqrt{\mu r}$$

(For orbital elements, one also has ($h=\sqrt{\mu a(1-e^2)}$), not used here.)

Units

• Inputs: ($\mathbf{r}$) in meters (m), ($\mathbf{v}$) in m/s.
• Output: ($h$) in m²/s.

INFO

• Returns the magnitude of ($\mathbf{h}=\mathbf{r}\times \mathbf{v}$) (not the vector).
• ($h=0$) iff ($\mathbf{r}\parallel \mathbf{v}$) (purely radial motion) or ($\mathbf{v}=\mathbf{0}$).
• In a two-body Keplerian model, ($h$) is constant along the orbit.

Invariants (floating-point tolerance aside)

• ($h\ge 0$)
• ($\mathbf{h}\perp \mathbf{r}$) and ($\mathbf{h}\perp \mathbf{v}$) (directional property; this function returns only ($\Vert \mathbf{h}\Vert$)).

Parameters

Parameter	Type	Description
r	Vector3DTupleType	Inertial position vector ($\mathbf{r}=[x,y,z]$) (m).
v	Vector3DTupleType	Inertial velocity vector ($\mathbf{v}=[v_x,v_y,v_z]$) (m/s).

Returns

number

Specific angular momentum ($h$) (m²/s).

Throws

Error If any component of r/v is non-finite; if r has zero length; or if v is invalid (NaN).

Example

// LEO-like state: r ⟂ v
const r: [number, number, number] = [6778e3, 0, 0]; // m
const v: [number, number, number] = [0, 7.67e3, 0]; // m/s
const h = specificAngularMomentum(r, v); // ≈ 5.20e10 m²/s

See

• https://en.wikipedia.org/wiki/Specific_angular_momentum
• specificAngularMomentumFromElements (if you compute from ($a,e,\mu$))