In this Year 1 Python coursework, I find the first Lagrangian point of the Earth and Moon using the Newton-Raphson method. I also create diffraction patterns for an Airy disk. I had to create a numerical integration function in order to solve the Bessel functions which was implemented into the diffraction pattern procedure.

A Lagrange point is the point between to celestial bodies where the gravitational attraction of the two bodies is equal. As such, there is no net gravitational force execrated by the two bodies and the orbiting body will remain in equilibrium.

The first Lagrangian point between the Earth and Moon is found by equating the gravitational forces and then using the Newton-Rhapson method to find the root of the rearranged equation.

L_1 = 326050 \textrm{ km}

Using a numerical integration method, Bessel functions are numerically evaluated.

J_m(x)=\frac{1}{\pi}\int_0^x \cos(m\vartheta)-x\sin(\vartheta)dx

From these solutions the Airy disk patterns are calculated for a small circular aperture.

I( r)=I_0 (\frac{2 J_1(x)}{x})^2

I plotted this diffraction pattern in both 1 dimension and two dimensions. I believe the 2D image is more ‘relevant’ as unlike the 1D pattern it is seen directly by the human eye.

Finding the Lagrangian point and producing the diffraction pattern were each an exercise in this Python coursework.