Documentation / @interstellar-tools/equations / solveKeplerBisection

Function: solveKeplerBisection()

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

Defined in: kepler/solve-kepler-bisection.ts:35

Solve Kepler's equation for the eccentric anomaly ($E$) via bisection (elliptic case: ($0\le e<1$)).

Equation

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

Bracketing & normalization

• Normalize ($M$) to ($[0,2\pi )$).
• Use the bracket ($[0,2\pi ]$). Then ($F(0)=-M\le 0$) and ($F(2\pi )=2\pi -M\ge 0$).
• For ($0\le e<1$), ($F^{\prime }(E)=1-e\cos E\ge 1-e>0$) ⇒ strictly increasing ⇒ unique root.

Stopping criteria (either)

• Residual: ($|F(E)|<\text{tolerance}$)
• Bracket width: ($\frac{1}{2}(E_{\text{high}}-E_{\text{low}})<\text{tolerance}$)

Parameters

Parameter	Type	Description
M	Radians	Mean anomaly (radians). Can be any real; will be normalized to ($[0,2\pi )$).
e	number	Eccentricity, must satisfy ($0\le e<1$) (elliptic).
maxIter	number	Maximum iterations (must be > 0).
tolerance	number	Convergence tolerance for both residual and half-interval (must be > 0, radians).

Returns

Radians

Eccentric anomaly ($E$) in radians, normalized to ($[0,2\pi )$).

Throws

If inputs are non-finite or out of domain.

Example

const E = solveKeplerBisection((Math.PI / 3) as Radians, 0.5, 100, 1e-12);