Documentation / @interstellar-tools/equations / eccentricToTrueAnomaly

Function: eccentricToTrueAnomaly()

function eccentricToTrueAnomaly(E: Radians, e: number): Radians;

Defined in: anomalies/eccentric-to-true-anomaly.ts:43

Converts Eccentric Anomaly ($E$) to True Anomaly ($V$) for an orbit.

Mathematical Explanation:

The eccentric anomaly ($E$) and the true anomaly ($V$) are related through:

$$\tan \frac{V}{2}=\sqrt{\frac{1+e}{1-e}}\tan \frac{E}{2}$$

where:

• $E$ is the eccentric anomaly (in radians).
• $V$ is the true anomaly (in radians).
• $e$ is the orbital eccentricity ($0\le e<1$ for elliptical orbits).

This function handles:

• Circular orbits ($e=0$): True anomaly is equal to eccentric anomaly.
• Parabolic orbits ($e=1$): Uses the special case:$$V=2{\tan }^{-1}\left(\frac{E}{2}\right)$$
• Elliptical orbits ($0<e<1$): Uses the standard conversion equation.

Additionally, numerical stability is ensured when $E\approx \pi$.

Parameters

Parameter	Type	Description
E	Radians	Eccentric anomaly ($E$) in radians.
e	number	Orbital eccentricity ($0\le e<1$).

Returns

Radians

True anomaly ($V$) in radians.

Throws

RangeError If eccentricity is out of the range $0\le e\le 1$.

Example

console.log(eccentricToTrueAnomaly(1.0, 0.5)); // Output: True anomaly in radians
console.log(eccentricToTrueAnomaly(Math.PI, 0)); // Output: Math.PI (circular orbit)
console.log(eccentricToTrueAnomaly(0, 1)); // Output: 0 (parabolic orbit)

See

True Anomaly (Wikipedia)