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.

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2 thoughts on “No sir.. we can’t get through the singularity (part 2)

    1. ember bu.. makanya daripada saya pake change of variables yg lain (walaupun lbh sederhana) mending saya pake yg ini.. soalnya “I just have to do copy paste from my own project and edit a little bit :lol:” 😉

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