Documentation / @interstellar-tools/equations / solveKeplerNewtonRaphson

Function: solveKeplerNewtonRaphson()

function solveKeplerNewtonRaphson(
  M: Radians,
  e: number,
  maxIter: number,
  tolerance: number
): number | Radians;

Defined in: kepler/solve-kepler-newton-raphson.ts:85

Solves Kepler's Equation for the Eccentric Anomaly ($E$) using the Newton-Raphson method with Householder acceleration for fast convergence.

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Mathematical Explanation:

Kepler's equation relates the mean anomaly ($M$), the eccentric anomaly ($E$), and the orbital eccentricity ($e$) as:

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

Since this equation cannot be solved algebraically, iterative numerical methods are required.

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Solving Strategy:

1. Handle Special Cases:

   • If eccentricity is out of range ($e<0$ or $e\ge 1$), a RangeError is thrown.
   • Kepler's equation is $2\pi$-periodic ($M+2\pi k\mapsto E+2\pi k$), so the equation is solved for $Mmod2\pi$ and the whole turns are added back. Any finite $M$ works, including negative values and many revolutions.

2. Initial Approximation (Danby 1987, good for all $0\le e<1$):

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

3. Newton-Raphson Iteration with Householder Acceleration:

   • With $f(E)=E-e\sin (E)-M$, $f^{\prime }(E)=1-e\cos (E)$, $f^{″}(E)=e\sin (E)$ and $f^{‴}(E)=e\cos (E)$, each step refines the Newton correction to third order:$${\delta }_1=-\frac{f}{f^{\prime }},{\delta }_2=-\frac{f}{f^{\prime }+\frac{1}{2}{\delta }_1{f}^{″}},\mathrm{\Delta }E=-\frac{f}{f^{\prime }+\frac{1}{2}{\delta }_2{f}^{″}+\frac{1}{6}{\delta }_2^2{f}^{‴}}$$

4. Convergence Check:

   • The iteration stops when:$$|E_{n+1}-E_n|<\text{tolerance}$$(default tolerance is 1e-9).

5. Failure Handling:

   • If the method does not converge, NaN is returned, signaling that a fallback method should be used.

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Performance Considerations:

• Typically converges in 3-4 iterations, including $e\to 1$.
• Time complexity: $O(1)$ for Newton-Raphson.

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Parameters

Parameter	Type	Description
M	Radians	Mean anomaly ($M$) in radians.
e	number	Orbital eccentricity ($0\le e<1$).
maxIter	number	Maximum number of iterations before failure.
tolerance	number	Convergence criterion for stopping the iteration.

Returns

number | Radians

The eccentric anomaly ($E$) in radians (or NaN if the method fails).

Throws

RangeError If the eccentricity ($e$) is invalid ($e<0$ or $e\ge 1$).

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Example

const M = Math.PI / 4; // 45 degrees in radians
const e = 0.1; // Orbital eccentricity
console.log(solveKeplerNewtonRaphson(M, e, 50, 1e-9)); // Output: Eccentric anomaly in radians

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See

• Kepler's Equation (Wikipedia)
• Newton-Raphson Method (Wikipedia)
• Eccentric Anomaly (Wikipedia)
• Danby, J. M. A. (1987). The solution of Kepler's equation, III. Celestial Mechanics, 40, 303–312.