A satellite of mass orbits under gravity

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics

Computational Physics, Ch Finding the best parameters: minimisation and fitting

Problem

A satellite of mass

m=500\,\mathrm{kg}

orbits under gravity, treated as a point mass with

GM=3.986\times10^{14}\,\mathrm{m^3\,s^{-2}}

for the Earth, and has a fixed angular momentum

L=2.60\times10^{13}\,\mathrm{kg\,m^2\,s^{-1}}

. Its effective potential energy at radius

r

U_{\rm eff}(r) = -\frac{GMm}{r} + \frac{L^2}{2mr^2}.

(a) Find the exact radius

r^*

at which

U_{\rm eff}

is minimised, by setting

\mathrm{d}U_{\rm eff}/\mathrm{d}r=0

. (b)

U_{\rm eff}(r)

is minimised numerically over the bracket

r\in[5.0\times10^6,\,9.0\times10^6]\,\mathrm{m}

(width

W_0=4.0\times10^6\,\mathrm{m}

) by golden-section search, which after its two starting evaluations shrinks the bracket by a factor of

0.618

at every further evaluation. How many of those further evaluations are needed to shrink the bracket to a width below

1.0\times10^3\,\mathrm{m}