Astronomy projects for calculus and differential equations

Question

Adapted from "Astronomy Projects for Calculus and Differential Equations" by Farshad Barman, portlnd community college, 2012.

You have been hired by NASA to determine the minimum distance between Mats and Earth at any time in the next 10 years. Assuming a Cartesian coordinate system with the sun at (0,0) and both planets orbiting in the (x, y)-plane, the location, (x,y) and velocity (v_x, v_y) of either planet is determined using the initial conditions and solving the following system of equations from Newton:

dx/dt = vx

dy/dt = vy

dvx/dt = - (G.M.x)/(√x² + y²)³

dvy/dt = - (G.M.y)/(√x² + y²)³

where G is the Universal Gravitational Constant, 6.67 x 10^-11 m³/s².Kg or 1.983979 x 10^-29 Au³/yr².Kg and M is the mass of the sun, 2.0 x 10³⁰ kg. Because of the large distance and time spans being simulated. it is recommended that the problem be solved using Astronomical Units (AU) as the units of distance and years (yr) as the units for time. There are 149.598 x 10⁹ m/AU and 3.15569 x 10⁷ s/yr.

The initial conditions for the Earth are: x(0) = 0.44503 AU, y(0) = 0.88106 AU, v,(0) = -5.71113 AU /yr, and vy(0) = 2.80924 AU /yr. The initial conditions for the Mars are: x(0) = 0.81449 AU, y(0) = 1.41483 AU, vx(0) = -4.23729 AU /yr, and vy (0) = -2.11473 AU/yr.

Your first project requirement is to solve the initial value problem for each planet and determine the location and velocity of each planet for the next 10 year period. NASA requests a plot from tu the orbits over that period. The second project requirement is to loop through each time 10-year solutions and determine the minimum distance between the two planets at any period. NASA recommends obtaining the solution for 10,000 time points during the 10-year period. Report the minimum distance in AU.