# Computer plot of particle orbits around Kerr black holes

1. Jul 6, 2012

### Dick Taid

Hi there,

First post, so be gentle. I'm currently doing a project on Kerr black holes, part of which is based on a project in Edwin F. Taylor's book "Exploring Black Holes" (the chapter is found here) as well as some of Hartle's book (chapter 15). Part of my project, and as a way of better understanding some principles, as well as getting up to speed with MATLAB again, is making a little program that can compute and plot the orbits of particles.

Basically I'm using the terms from Hartle (eqs. (15.18b) and (15.19) (I won't bother with my derivations of them here) derived from the Kerr metric in the equatorial plane:
$$d\tau^2 = \left(1 - \frac{2M}{r}\right) dt^2 + \frac{4Ma}{r} dt d\phi - \frac{r^2}{\Delta} dr^2 - R_a^2 d\phi^2,$$

where $a$ is the spin parameter, $R_a^2 = r^2 + a^2 + \frac{2Ma^2}{r}$ is the reduced circumference, and $\Delta \equiv r^2 - 2Mr + a^2$. Calculations give (15.18b): $$\frac{d\phi}{d\tau} = \frac{1}{\Delta}\left[\left(1 - \frac{2M}{r}\right)\ell + \frac{2Ma}{r}e\right]$$

and (15.19): $$\frac{e^2 - 1}{2} = \frac{1}{2}\left(\frac{dr}{d\tau}\right)^2 + V_\text{eff}(r,e,\ell)$$

where
$$V_\text{eff}(r,e,\ell) = -\frac{2M}{r} + \frac{\ell^2 - a^2(e^2 - 1)}{r^2} - \frac{2M(\ell - a\cdot e)^2}{r^3}.$$

Now, following Hartle's example (I think it's in Taylor as well), one finds the orbit of a particle, moving radially inwards, by $(dr/d\tau)/(d\phi/d\tau)$, which is somewhat simple enough. I haven't written an expression for it, as it isn't very pretty, but by solving that differential equation, I get a nice plot of what I wanted. That is
http://dicktaid.com/kerrorbit.jpg [Broken]

(here $a=M$, $e = 1$ and $\ell = 0$). The same can be achieved by using the program Taylor supplies on his webpage (the GRorbits.jar file) which he uses in his drafts of the second edition of his book (chapter 16). I want to be able to plot bound orbits as depicted in fig. 16.9-10. Inserting those values in my MATLAB code gives an error (the values he uses for angular momentum, energy and spin parameter implies $(dr/d\tau)^2$ is negative).

I'm trying to understand the coding done in the GRorbits.jar file, that is, I'm trying to understand how those orbits are achieved. I've looked around in the books, as well as the net, to find something that helps me make sense of it. Perhaps I'm searching for the wrong things, but I'm not able to locate what I'm looking for. Any pointers you guys can give me is appreciated.

EDIT: Fig. 16.9 from Taylor's drafts so you don't have to download it:
http://dicktaid.com/boundorbit.jpg [Broken]

Last edited by a moderator: May 6, 2017
2. Jul 6, 2012

### m4r35n357

Beginner here, but I think I have been there or thereabouts recently ;)
I might be wrong, but I think GROrbits no longer uses those equations. I looked at the code myself, and I believe it now uses code developed along with the second edition of that book, the text of which is open for review at the book's site.
That said, the existing code is still pretty difficult to reverse-engineer (well I couldn't fathom it, and I thought I knew what to look for!), so you might be most successful if you develop your code "alongside" the two methods already there using your working algorithm, making use of the vast quantities of GUI code provided.
As for your parameter issues, sometimes the a, E and L values really are unphysical. That's when I use GROrbits to check if there really is a solution.

3. Jul 6, 2012

### Dick Taid

Thank you for your suggestions. I've just looked at the source code of GROrbits, and I think I've suffered a massive brain aneurysm -- JAVA apparently makes less sense than I remembered.

Looking at Taylor's draft again, it doesn't appear they have used another way of computing the orbits, than they did in the first edition. As it says on page 21
which is pretty much what I'm doing.

4. Jul 6, 2012

### m4r35n357

Sure, just checked page & equation nos. It's the same method (effective potential), just with different equations (well, they look very different to me!).
I ended up doing my own equations based on generic metric coefficients because I didn't like the way they reduce the symbol m to the value "2". That's just confusing IMO.

5. Jul 6, 2012

### Dick Taid

The only difference between the two editions, equation-wise, is that they in first edition confine themselves to the maximal Kerr black hole ($a=M$), while in the second edtion they generalize for all values of $a$, as well as converting to unitless entries by dividing through with $M$. But other than that, it's the same. :)

6. Jul 7, 2012

Anyone else?