# No sir.. we can’t get through the singularity (part 2)

Hahaha.. I finally get back to this topic after leaving it for more than a month. This posting is the sequel of this one which discussed the gravitational singularity of the sun. Anyway, as I wrote in the first post, that I write because I remembered a scene in a movie titled Sunshine. Coincidentally, the movie was aired on Global TV few days after the post published 😀 Also, about that first post, a friend asked me about the simplification into two dimensional model. My defense was that the particle will always move on a fixed plane around the sun. To see this you can compute the angular momentum $m(X \times V)$, where $X$ is the position vector and $V$ is the velocity vector. Then, you can compute the derivative, which is equal to 0. Thus, the angular momentum is constant and then the position vector lies on a fixed plane orthogonal to angular momentum 😉

So, in the first post I have already stated that the sun is not a singularity since we can impose a change of coordinates such as the following:

The original ODE system is

$r'= v_r$

$\theta'= v_{\theta}/r$

$v_r'=-\frac{1}{r^{2}}+\frac{v_{\theta}^2}{r}$

$v_{\theta}'=-\frac{v_r v_{\theta}}{r}$

The proposed change of coordinates is to first introduce these two variables

$u_r=r^{-1/2}v_r$

$u_{\theta}=r^{-1/2}v_{\theta}$

so that the system becomes

$r'= r^{-1/2}u_r$

$\theta'= r^{-3/2}u_{\theta}$

$u_r'= r^{-3/2}(\frac{1}{2}u_r^2+u_{\theta}^2-1)$

$u_{\theta}'=r^{-3/2}(-\frac{1}{2}u_ru_{\theta})$

finally, multiplying the vector field by $r^{3/2}$ we get

$r'= ru_r$

$\theta'= u_{\theta}$

$u_r'= (\frac{1}{2}u_r^2+u_{\theta}^2-1)$

$u_{\theta}'=(-\frac{1}{2}u_ru_{\theta})$

and voila, the singularity disappear 😀 Of course some of you may argue that you can do simpler change of variables here to remove the singularity. I can only say that, there are reasons why McGehee introduced this change of variables but I suppose it is primarily for analysis on the so called collision surface. However, my point is, with this change of variables.. the plot stays the same, computable, and I just have to do copy paste from my own project and edit a little bit 😆

These are the plots by using Mathematica from my friend Thomas de Jong after the change of coordinates. Compare them with those from this one.