- #1
Dick Taid
- 6
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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:
<br /> d\tau^2 = \left(1 - \frac{2M}{r}\right) dt^2<br /> + \frac{4Ma}{r} dt d\phi<br /> - \frac{r^2}{\Delta} dr^2<br /> - R_a^2 d\phi^2,<br />
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): <br /> \frac{d\phi}{d\tau} = \frac{1}{\Delta}\left[\left(1 - \frac{2M}{r}\right)\ell + \frac{2Ma}{r}e\right]<br />
and (15.19): <br /> \frac{e^2 - 1}{2} = \frac{1}{2}\left(\frac{dr}{d\tau}\right)^2 + V_\text{eff}(r,e,\ell)<br />
where
<br /> 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}.<br />
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
(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 (http://www.eftaylor.com/exploringblackholes/SpinBH111107v2.pdf). 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
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:
<br /> d\tau^2 = \left(1 - \frac{2M}{r}\right) dt^2<br /> + \frac{4Ma}{r} dt d\phi<br /> - \frac{r^2}{\Delta} dr^2<br /> - R_a^2 d\phi^2,<br />
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): <br /> \frac{d\phi}{d\tau} = \frac{1}{\Delta}\left[\left(1 - \frac{2M}{r}\right)\ell + \frac{2Ma}{r}e\right]<br />
and (15.19): <br /> \frac{e^2 - 1}{2} = \frac{1}{2}\left(\frac{dr}{d\tau}\right)^2 + V_\text{eff}(r,e,\ell)<br />
where
<br /> 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}.<br />
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
(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 (http://www.eftaylor.com/exploringblackholes/SpinBH111107v2.pdf). 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
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