Two black holes colliding (classical mechanics)

Question

I had this question come up in my exam where two identical black holes are in orbit around each other. There is a loss of energy via gravitational waves : $$\frac{d E}{d t} = kr^4\omega^6$$ where $k$ is a constant, $r$ is the separation between the black holes and $\omega$ is the angular frequency of the orbit.

They asked to show $\frac{dE}{dt}$ is proportional to $\frac{1}{r^2}$.

But I found it to be $1/r^5$.

I know that this is circular motion where both bodies are on the same orbit revolving about the centre of mass (which would be the centre of the circle) and they both would have the same period so that the centre of mass is in the same place. I substituted $\omega$ = $2\pi/P$ with $P$ being the period of the orbit.

I used $$P^2 = \frac{4\pi^2 r^3}{G(m_1+m_2)}$$ to substitute $P$ for $r$. Where have I gone wrong? Is it possible that the dependence is $1/r^5$ after all?

Answer 1

Is it possible that the dependence is $1/r^5$ after all?

Yes. For confirmation, see equation (2.38) in [1], equation (3) in [2], and the combination of equations (41)-(42) in [3]. All agree with your result $dE/dt\propto 1/r^5$.

However, note that the same result can also be written $dE/dt\propto v^6/r^2$ where $v$ is the orbital speed, because $v^2\propto 1/r$. Equation (20) in [4] writes it both ways.

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References:

[1] Kokkotas (2009), "Gravitational wave physics," http://www.tat.physik.uni-tuebingen.de/~kokkotas/Teaching/NS.BH.GW_files/GW_Physics.pdf

[2] Miller (2008), "Binary Sources of Gravitational Radiation," https://www.astro.umd.edu/~miller/teaching/astr498/lecture25.pdf

[3] Hirata (2011), "Lecture XV: Gravitational energy and orbital decay by gravitational radiation," http://www.tapir.caltech.edu/~chirata/ph236/2011-12/lec15.pdf

[4] Kokkotas (2008), "Gravitational Wave Astronomy," https://arxiv.org/abs/0809.1602