Three circular orbits at

PHYSICS · Multi-part · Fermi University Physics · Computational Physics · Mechanics & Dynamical Systems · Simulating classical dynamical systems

Computational Physics, Ch An orbit that must not drift

Problem

a = 1.00

3.00

and

9.00\,\mathrm{AU}

are integrated by velocity Verlet at

\Delta t = 10^{-3}\,\mathrm{yr}

in units where

GM_\odot = 4\pi^2\,\mathrm{AU^3\,yr^{-2}}

, and their periods are measured from the trajectories as

T = 1.000013

5.196155

and

27.000000\,\mathrm{yr}

. ((a)) For each of the three pairs of orbits form the exponent

p = \ln(T_j/T_i)/\ln(a_j/a_i)

, and quote the exponent as a mean with an uncertainty taken from the spread of the three. ((b)) Recover

GM_\odot

from each orbit separately as

4\pi^2

times

a^3/T^2

, say which orbit gives the value furthest from

4\pi^2

, and explain why that one is the worst at a step that is the same for all three. ((c)) A fourth body moves on an orbit of

a = 2.00\,\mathrm{AU}

and eccentricity

e = 0.20

. Using Kepler's equation and two Newton steps from

E_0 = M

, find its position

(x,y)

a quarter of a period after perihelion.