Orbit eq. holds inside BH? - MTW sec. 25

[Jorrie]

MTW section 25, from eq. 25.16 onwards, derives an orbital equation (with G=c=1, u = M/r, E and L Schwarzschild constants for energy and angular momentum respectively):

$$\left(\frac{du}{d\phi}\right)^2 = \frac{M^2}{L^2}(E^2-1) + \frac{2M^2}{L^2}u - u^2 + 2u^3$$

This equation is readily differentiable to give

$$\frac{d^2u}{d\phi^2}= \frac{M^2}{L^2} - u + 3u^2$$

which is often used for numerical integration to obtain orbital plots of $r$ against $\phi$.

My question: since both equations seem to be well behaved for any $u < \infty$, can they be used to plot the 'infalling' orbit inside the horizon? Or are either E or L or both not valid there?

 

[Bill_K]

Yes, they're valid both inside and outside the horizon, but... why would you differentiate the equation if your purpose is to integrate it?? Calling the RHS of the first equation V(u), just solve for φ:

φ = ∫du/√V(u)

and evaluate this integral numerically.

 

[Jorrie]

Yes, they're valid both inside and outside the horizon, but... why would you differentiate the equation if your purpose is to integrate it??

Thanks Bill.

I think the second order equation is more 'integrator friendly' than the first one. I recall having had a few cases where the orbits 'locks' itself into either peri- or apo-apsis when trying the first one, while the second one seems to be immune from that.

 

[Mentz114]

The exact solution of the Schwarzschild orbital equation is here http://arxiv-web3.library.cornell.edu/abs/astro-ph/0305181v3

... the resulting geodesic is solved by the Weierstrass-Jacobi modular form.

 

[Bill_K]

Does anyone know the rate of perihelion advance for (equatorial, nearly circular) Kerr orbits?

 

[Jorrie]

Does anyone know the rate of perihelion advance for (equatorial, nearly circular) Kerr orbits?

Some information on Kerr orbit perihelion advance is given by Levin et. al in "A Periodic Table for Black Hole Orbits", Appendix A. It is essentially about finding the orbits that advance by rational multiples of $2\pi$.

 

[Bill_K]

Some information on Kerr orbit perihelion advance is given by Levin et. al in "A Periodic Table for Black Hole Orbits", Appendix A. It is essentially about finding the orbits that advance by rational multiples of $2\pi$.

Interesting paper, thanks.