The following ODEs model the motion of a particle in the gravitational field of a star or a planet, y (22+ y2)3/2 (x2 +

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The Following Odes Model The Motion Of A Particle In The Gravitational Field Of A Star Or A Planet Y 22 Y2 3 2 X2 1 (31.19 KiB) Viewed 37 times

The following ODEs model the motion of a particle in the gravitational field of a star or a planet, y (22+ y2)3/2 (x2 + y2)3/2 The energy h and the angular momentum c are given by 1 ="y = " - - n = 5(+2 +32) and they are constant on each orbit of the system (prove it!). Consider the initial data r(0) = 1, y(0) = 0, 0(0) = 0, y(0) = 0.125, which corresponds to an elliptic orbit of eccentricity 0.984375. The period of the orbit is T = 2.247730547781942. Let us define ho and as the values of h and c at the initial data. We want to check the behaviour of some numerical integrators on a long simulation, i.e., up to T = 10%. (a) Use the leap-frog method, with constant step size. Try first for a single period of the orbit, with different time steps.