A fit to an asteroid's orbit gives best-fit semi-major axis and eccentricity

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Probability, statistics and stochastic methods · Probability, statistics and stochastic methods

Error, Chance and the Machine, Ch Monte Carlo integration and variance reduction

Problem

A fit to an asteroid's orbit gives best-fit semi-major axis

a_0 = 2.50\ \mathrm{AU}

and eccentricity

e_0 = 0.100

, with covariance matrix (in units of

\mathrm{AU^2}

\mathrm{AU}

, and dimensionless respectively)

\mathsf{V} = \begin{pmatrix} 0.0009 & 0.00015 \\ 0.00015 & 0.0004 \end{pmatrix}.

((a)) Find the correlation coefficient between

a

and

e

. ((b)) Find the lower-triangular Cholesky factor

\mathsf{L}

\mathsf V

, with

\mathsf V=\mathsf L\mathsf L^{\rm T}

, that is

\mathsf L=\mathsf R^{\rm T}

for the upper-triangular

\mathsf R

with

\mathsf V=\mathsf R^{\rm T}\mathsf R

. ((c)) Using independent standard normal draws

z_1 = 1.00

z_2 = 0.50

, generate a correlated sample

(a,e)

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