Orbital Mechanics Problem Set 1- Two Body Problems

2-Body Problems in orbital mechanics -

Given two bodies with mass m1 and m2 at positions r1 and r2 with respect to an inertial frame (as pictured in fig 1) and assuming the only force acting on the two bodies is their mutual gravitational attraction:
(a) Derive the relative 2-body equations of motion show all steps and clearly state any assumptions made. (b) What is the definition of µ? Why can we often use the approximation µ = Gmplanet?

Write a simulation using ode45 that propagates a spacecraft trajectory around Earth (µ = 398600 km3/s2) with an initial condition of r and r dot (a) Plot the position trajectory on a single 3D plot (use plot3) for a single orbit period (b) For the combined system, plot the mass-specific orbit energy, scalar angular momentum, and eccentricity as a function of time for the following integration tolerances: abstol= 10−3 abstol= 10−6 abstol= 10−12 (c) Discuss the behavior of the energy, angular momentum, and eccentricity for the different absolute and relative tolerances. Are the the same? Different? Why?
(d) Include your code

Luke Skywalker entered the Dagobah system and maneuvered into a parking orbit to observe the swamp-covered planet, which has a gravitational parameter µDagobah = 4.5e4 km3/s2 The orbit of Luke’s X-wing can be described by the following scalar quantities relative to the planet Dagobah: E = −2.2093 [km2/s2]
h = 21, 252.877 [km2/s]