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restrepo
GitHub Repository: restrepo/ComputationalMethods
 Path: blob/master/activities/radial-poisson.ipynb
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Task 6

This homework is an activity intended to apply the differentiation formulas given in class. Here we want to solve a radial Poisson's equation for estimating the density field of a rocky planet.

Due to: Feb 1

Radial Poisson's Equation

The Poisson's equation relates the matter content of some physical distribution with the gravitational force or equivalently, a exerted gravitational potential. It is given by

∇2ϕ(r⃗)=−4πGρ(r⃗)\nabla^2 \phi(\vec r) = -4\pi G \rho(\vec r)

where ϕ(r⃗)\phi(\vec r) is the gravitational potential, GG the Cavendish constant and ρ(r⃗)\rho(\vec r) the density field.

In the case of a radially-symmetric distribution, the angular part vanishes, yielding a radial part given by:

1r2ddr(r2dϕ(r)dr)=−4πGρ(r)\frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{d\phi(r)}{dr}\right)= -4\pi G \rho(r)

1. Taking these data and using the three-point Midpoint formula, find the density field from the gravitational potential (seventh column in the file) and plot it against the radial coordinate (second column).

Questions:

  • How many data points do you get from this procedure as compared with the original number of data of the potential? Why do you get this?

  • How do you explain the discontinuity in the obtained density profile?

Remeber:

use GG with SI units, ϕ\phi is shown in SI units of potential, i.e. Joules/Kg, and radial coordinates in Earth radius (63716371 km).