Documentation / @interstellar-tools/equations / computeAngle

Function: computeAngle()

function computeAngle(
  orbit: Pick<OrbitInterface, 'e' | 'angle' | 'period'>,
  timeStep: TimeStepInterface
): Radians;

Defined in: angle/compute-angle.ts:68

Advances a body along its orbit by one time step and returns its new true anomaly ($\nu$).

The orbit's angle is the starting true anomaly. There is no date input: call this once per frame or step, and feed the result back in as the next angle to move the body over time. To find a body at a given date instead, use trueAnomalyAt.

Step 1: Mean anomaly after the time step ($M$)

Done by advanceMeanAnomaly: $M=M_0+n\cdot \mathrm{\Delta }t,n=\frac{2\pi }{|P|}$ where:

• $M_0$ is the mean anomaly at the start, converted from the true anomaly angle (equal to it when $e=0$).
• $n$ is the mean motion, from the period $P$ converted to days. $P$ is a positive duration; a negative one, from data in the old 0.x convention, is read as its absolute value. A positive step always moves the body forward.
• $\mathrm{\Delta }t$ is timeStep.value in days. It can be any length, and negative to go back in time.

The result is wrapped to $[0,2\pi )$.

Step 2: Solve Kepler's equation for the eccentric anomaly ($E$)

$M=E-e\sin (E)$ is solved numerically by solveKepler.

Step 3: Convert the eccentric anomaly to the true anomaly ($\nu$)

Done by eccentricToTrueAnomaly: $\nu =2{\tan }^{-1}(\sqrt{\frac{1+e}{1-e}}\tan \left(\frac{E}{2}\right))$

Parameters

Parameter	Type	Description
orbit	Pick<OrbitInterface, "e"	"angle"
timeStep	TimeStepInterface	How far to advance, in days.

Returns

Radians

The true anomaly after the time step, in radians.

Throws

RangeError If the orbit's eccentricity is outside $0\le e<1$.

Example

import { PLANETS } from '@interstellar-tools/constants';
import { computeAngle } from '@interstellar-tools/equations';

const earth = PLANETS.find((planet) => planet.id === 'earth')!;
const timeStep: TimeStepInterface = { value: 1, unit: 'd' };

// Earth's true anomaly one day after its dataset position
const angle: Radians = computeAngle(earth.orbit!, timeStep);

See

• Kepler's Equation (Wikipedia)
• True Anomaly (Wikipedia)
• Mean Anomaly (Wikipedia)
• Orbital Mechanics (NASA)