Documentation / @interstellar-tools/equations / solveKeplerHighEccentricity

Function: solveKeplerHighEccentricity()

function solveKeplerHighEccentricity(
  M: Radians,
  e: number,
  maxIter?: number,
  tolerance?: number
): Radians;

Defined in: kepler/solve-kepler-high-eccentricity.ts:104

Solves Kepler's Equation for the Eccentric Anomaly ($E$) in highly eccentric orbits ($e>0.9$) using an iterative numerical approach.

Mathematical Explanation:

Kepler's equation for eccentric anomaly ($E$) is:

$$M=E-e\sin (E)\text{(for elliptical orbits, }0<e<1\text{)}$$

$$M=e\sinh (E)-E\text{(for hyperbolic orbits, }e>1\text{)}$$

For orbits with high eccentricity ($e\approx 1$), standard Newton-Raphson solvers struggle due to extreme changes in orbital velocity near perihelion. Instead, we apply a fixed-point iteration method that is more stable for these cases.

---

Solving Strategy:

1. Initial Guess:
   • Elliptical ($e<1$): the equation is $2\pi$-periodic, so $M$ is first reduced to one turn with its sign kept ($Mmod2\pi$), then Danby's starting value is used:

$$E_0=M+0.85e\mathrm{sign}(\sin M)$$

• Hyperbolic ($e>1$):

$$E_0=M\pm \frac{e\sin (M)}{1-e\cos (M)}$$

2. Iterative Refinement:
   • The method updates $E$ using:

$$\mathrm{\Delta }E=\frac{E-e\sin (E)-M}{1-e\cos (E)}$$

for elliptical orbits.

$$\mathrm{\Delta }E=\frac{e\sinh (E)-E-M}{e\cosh (E)-1}$$

for hyperbolic orbits.

• Iteration stops when:

$$|E_{n+1}-E_n|<\text{tolerance}$$

(default tolerance is 1e-9).

3. Angle Wrapping (Elliptical Only):
   • The result is wrapped using wrapAngle(), which keeps the sign: $E\in (-2\pi ,2\pi )$ with the sign of $M$. Use solveKepler() for a result normalized to $[0,2\pi )$.

Performance Considerations:

• Typically converges in fewer than 10 iterations for $e>0.9$.
• Time complexity: $O(1)$ for standard convergence.

Parameters

Parameter	Type	Default value	Description
M	Radians	undefined	Mean anomaly ($M$) in radians.
e	number	undefined	Orbital eccentricity ($e>0.9$ for high-eccentricity orbits).
maxIter?	number	...	Maximum number of iterations before failure.
tolerance?	number	1e-9	Convergence criterion for stopping the iteration.

Returns

Radians

The eccentric anomaly ($E$) in radians (for elliptical orbits in $(-2\pi ,2\pi )$, with the sign of $M$).

Examples

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

const M = Math.PI / 4; // 45 degrees in radians
const e = 0.95; // High orbital eccentricity
const result = solveKeplerHighEccentricity(M, e);
console.log(result); // Output: Eccentric anomaly in radians

// Hyperbolic orbit example (e > 1)
const M_hyperbolic = 2.0; // Mean anomaly for hyperbolic orbit
const e_hyperbolic = 1.2; // Hyperbolic eccentricity
console.log(solveKeplerHighEccentricity(M_hyperbolic, e_hyperbolic));

See

• Kepler's Equation (Wikipedia)
• Hyperbolic Anomaly (Wikipedia)