The eccentric anomaly of an orbit satisfies Kepler's equation

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics

Error, Chance and the Machine, Ch Iteration: nonlinear systems and eigenvalues

Problem

The eccentric anomaly

E

of an orbit satisfies Kepler's equation

E = M + e\sin E

, where

M

is the mean anomaly (radians) and

e

is the orbital eccentricity. For an orbit with

e = 0.6

and

M = 1.2\ \mathrm{rad}

, the fixed-point iteration

E_{n+1} = M + e\sin E_n

is proposed to solve for

E

. ((a)) Find a bound, valid for all real

E

, on

|g'(E)|

where

g(E) = M + e\sin E

. ((b)) State whether the iteration is guaranteed to converge to the root from any starting value, and give the contraction factor implied by your bound.

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