Binary Star System Numerical Modelling and Simulation

This jupyter notebook is presented as report on the modelling of a binary star system using numerical computing. We solve the binary star system equation of motion using ODEINT differential solver from the scipy module

Description

The binary star system will consist of a G and K class stars. It can be shown, without loss of generality, that a two-body problem is always evolving in a two- dimensional orbital plane. The force that drives the motion of the two stars in this model is the gravitational force. Therefore, the force of an object of mass M at location (x_M , y_M ) acting on another object of mass m at location (xm , ym ) is given by the following two formulae (one for each2 coordinate, x and y):

$$ F_x = G \left ( \frac{M \cdot m}{R^2} \right ) \frac{r_x}{R} $$ $$ F_y = G \left ( \frac{M \cdot m}{R^2} \right ) \frac{r_y}{R} $$

where G = 6.67408 × 10−11 m3 kg−1 s−2 is the gravitational constant, R is the distance between each objects given as:

$$ R = \sqrt{r^2_x + r^2_y} $$

and r_x = x_M − x_m and r_y= y_M − y_m , the signed relative positions between the two objects. The acceleration resulting from this force acting on object m at location (x_m , y_m ) is then given by

$$ a_x = \frac{dv_x}{dt} = G \left ( \frac{M}{R^2} \right ) \frac{r_x}{R} $$

$$ a_y = \frac{dv_y}{dt} = G \left ( \frac{M}{R^2} \right ) \frac{r_y}{R} $$

Integrating equations a_x and a_y gives us information on the velocity of the star with mass m which we can use to the change in position given by:

$$ x(t) ≈ x_0 + v_x (t) · \Delta t $$ $$ y(t) ≈ y_0 + v_y (t) · \Delta t $$

derived from $v = \frac{dx}{dt}$

For the purpose of this report we will use the odeint() differential equation solver from the scipy module

Aim

To compute the forces on the planets as their orbits propogate and establish whether the two stars orbit each other in a stable fashion. we then use data from the orbits to plot a x position vs time graph in order to deduce the period of the orbits.

Author info

Bruce Mvubele
LinkedIn
e-mail: cmvubele@gmail.com