Documentation / @interstellar-tools/equations / trueToMeanAnomaly

Function: trueToMeanAnomaly()

function trueToMeanAnomaly(V: Radians, e: number): Radians;

Defined in: anomalies/true-to-mean-anomaly.ts:77

Converts true anomaly ($\nu$) to mean anomaly ($M$) using Kepler's equation.

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Mathematical Explanation:

In orbital mechanics, the true anomaly ($\nu$), the eccentric anomaly ($E$), and the mean anomaly ($M$) are related through Kepler's equation.

Step 1: Convert True Anomaly ($\nu$) to Eccentric Anomaly ($E$)

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

This transformation ensures that $E$ is computed correctly across all quadrants, using atan2(y, x) instead of atan(x) to avoid ambiguity in angle computation.

Step 2: Convert Eccentric Anomaly ($E$) to Mean Anomaly ($M$)

Kepler’s equation states:

$$M=E-e\sin (E)$$

Since anomalies are periodic over one full orbit ($0\le M<2\pi$), we apply wrapAngle(M) to ensure that the computed mean anomaly remains within this range.

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Why Use wrapAngle?

• Ensures that the mean anomaly is always wrapped within $[0,2\pi ]$.
• Corrects floating-point precision issues that may cause values slightly greater than $2\pi$.
• Prevents negative anomalies by shifting them into the valid range.

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Parameters

Parameter	Type	Description
V	Radians	True anomaly ($\nu$) in radians.
e	number	Eccentricity of the orbit ($0\le e<1$).

Returns

Radians

The mean anomaly ($M$) in radians, wrapped to the range $[0,2\pi ]$.

Throws

RangeError If the eccentricity ($e$) is outside the valid range $0\le e<1$.

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Examples

import { trueToMeanAnomaly } from '@interstellar-tools/equations';

// Example 1: Standard elliptical orbit
const V = Math.PI / 3; // 60 degrees in radians
const e = 0.1; // Eccentricity
console.log(trueToMeanAnomaly(V, e)); // Output: Mean anomaly in radians

// Example 2: Retrograde motion handling
const V_retrograde = -Math.PI / 2; // -90 degrees
const e_retrograde = 0.2;
console.log(trueToMeanAnomaly(V_retrograde, e_retrograde));

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See

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