plotting orbital trajectories in python

Question

How can I setup the three body problem in python? How to I define the function to solve the ODEs?

The three equations are
x'' = -mu / np.sqrt(x ** 2 + y ** 2 + z ** 2) * x,
y'' = -mu / np.sqrt(x ** 2 + y ** 2 + z ** 2) * y, and
z'' = -mu / np.sqrt(x ** 2 + y ** 2 + z ** 2) * z.

Written as 6 first order we have

x' = x2,

y' = y2,

z' = z2,

x2' = -mu / np.sqrt(x ** 2 + y ** 2 + z ** 2) * x,

y2' = -mu / np.sqrt(x ** 2 + y ** 2 + z ** 2) * y, and

z2' = -mu / np.sqrt(x ** 2 + y ** 2 + z ** 2) * z

I also want to add in the path Plot o Earth's orbit and Mars which we can assume to be circular. Earth is 149.6 * 10 ** 6 km from the sun and Mars 227.9 * 10 ** 6 km.

#!/usr/bin/env python                                                             
#  This program solves the 3 Body Problem numerically and plots the trajectories      

import pylab
import numpy as np
import scipy.integrate as integrate
import matplotlib.pyplot as plt
from numpy import linspace

mu = 132712000000  #gravitational parameter
r0 = [-149.6 * 10 ** 6, 0.0, 0.0]
v0 = [29.0, -5.0, 0.0]
dt = np.linspace(0.0, 86400 * 700, 5000)  # time is seconds

Answer 1

As you've shown, you can write this as a system of six first-order ode's:

x' = x2
y' = y2
z' = z2
x2' = -mu / np.sqrt(x ** 2 + y ** 2 + z ** 2) * x
y2' = -mu / np.sqrt(x ** 2 + y ** 2 + z ** 2) * y
z2' = -mu / np.sqrt(x ** 2 + y ** 2 + z ** 2) * z

You can save this as a vector:

u = (x, y, z, x2, y2, z2)

and thus create a function that returns its derivative:

def deriv(u, t):
    n = -mu / np.sqrt(u[0]**2 + u[1]**2 + u[2]**2)
    return [u[3],      # u[0]' = u[3]
            u[4],      # u[1]' = u[4]
            u[5],      # u[2]' = u[5]
            u[0] * n,  # u[3]' = u[0] * n
            u[1] * n,  # u[4]' = u[1] * n
            u[2] * n]  # u[5]' = u[2] * n

Given an initial state u0 = (x0, y0, z0, x20, y20, z20), and a variable for the times t, this can be fed into scipy.integrate.odeint as such:

u = odeint(deriv, u0, t)

where u will be the list as above. Or you can unpack u from the start, and ignore the values for x2, y2, and z2 (you must transpose the output first with .T)

x, y, z, _, _, _ = odeint(deriv, u0, t).T