# MTW Gravitation: Exersise 25.16 Periastron Shift

1. Apr 22, 2012

### luinthoron

Hello, I am trying to work out this exercise for my personal research connected with my bachelor thesis. The task is to compare equations (25.42) and (25.47) and express $u_0$ in terms of \tilde{L}. I have so far put the two equations together getting

12u^2u_0\tilde{L}^2-18uu_0^2\tilde{L}^2-u_0^2\tilde{L}^2+2uu_0\tilde{L}^2-\tilde{E}_0^2=2u-1

After this I tried putting some terms together but I think I am missing another equation since there are in fact two unknowns: $u_0$ and $\tilde{E}_0$ or is there some trick I am missing?

For those without access to MTW, here are the equations: \\

(25.42)

\left(\frac{\mathrm{d}u}{\mathrm{d}\varphi}\right)^2=\frac{\tilde{E}^2}{\tilde{L}^2}-\frac{1}{\tilde{L}^2}\left(1-2u\right)\left(1+\tilde{L}^2u^2\right)

and (25.47)

\left(\frac{\mathrm{d}u}{\mathrm{d}\varphi}\right)^2+\left(1-6u_0\right)\left(u-u_0\right)^2-2\left(u-u_0\right)^3=\frac{\tilde{E}^2-\tilde{E}_0^2}{\tilde{L}^2}

Thank you.

2. Apr 22, 2012

### Bill_K

Each equation expresses (du/dφ)2 in terms of a cubic in u. Expand out the cubics and equate the coefficients of each power of u.

3. Apr 22, 2012

### luinthoron

Thank you for the advice. I ended up with:

\tilde{E}_0^2=4\tilde{L}^2u_0^3-\tilde{L}^2u_0^2+1

3u_0^2-u_0=-\frac{1}{\tilde{L}^2}

The second one is a quadratic equation, so I can write the solution

{u_0}_{1,2}=\frac{1\pm\sqrt{1-\frac{12}{\tilde{L}^2}}}{6} .

But this would give a condition for \tilde{L}, which I find suspicious. Also which of the two roots is correct? It stays possitive in both cases and I can't come with any other clue to help me choose. Any additional hints, please?

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