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How variable should we expect the orbital inclination of a typical black hole merger to be, relative to the galactic plane? The LIGO simulation seems to give the impression that the orbital inclination doesn’t change, but wouldn’t general relativity predict that there would be a very large precession of the perihelion? And wouldn’t the orbital inclination also precess due to the gravitational pull from the surrounding galaxy?

Asked a different way, what would the merger look like from an outside observer watching over billions of years? Would the orientation of the orbital inclination become extremely erratic, switching from horizontal to vertical and everywhere in between?

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  • $\begingroup$ An orbital plane depends on what orbiting object is considered. It is not a property of the black hole in itself. Do you mean the equatorial plane? It cannot change much due to angular momentum conservation. $\endgroup$ – Stéphane Rollandin Apr 4 '18 at 11:36
  • $\begingroup$ I mean the orbital plane of the two black holes, where they both orbit one central point. $\endgroup$ – Paul Apr 4 '18 at 11:46
  • $\begingroup$ For two isolated bodies, i.e. nothing outside the two body system is applying a force, the plane of the orbit cannot change since that would violate conservation of angular momentum. This applies to any binary system, relativistic or not. $\endgroup$ – John Rennie Apr 4 '18 at 11:55
  • $\begingroup$ @JohnRennie ok I guess I need to clarify that it’s the angle of the plane of the orbit relative to the galactic plane (or some external fixed point). $\endgroup$ – Paul Apr 4 '18 at 14:30
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    $\begingroup$ There is no evidence that binary orbits preferentially align themselves with the angular momentum vector of the galaxy. $\endgroup$ – Rob Jeffries Apr 4 '18 at 16:02
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If neither of the black holes is spinning (or if their spins are (anti-)aligned with the orbital angular momentum), then the orbital plane will stay fixed for the entire inspiral. There will be some effect due to the galactic potential, but that will be minute and will act on time scales much longer than those typical for the inspiral. You certainly would not expect anything dramatic.

However, if the spins are not aligned with the orbital angular momentum all sorts of the crazy effects can happen. In the weak field the dominant effect will be that the orbital plane will precess around the total angular momentum. The associated time scale is long compared to the orbital time scale, but short compared to the inspiral timescale. This affects the gravitational waveform by introducing a modulation of the signal. No clear signs of such modulations have been seen in the observed GW events, which tells us that the effective spin (a combination of the spins of the two objects weighted by their masses) of the observed systems is fairly low.

More extreme things can happen as well. For example, as the orbital angular momentum changes due to emission of gravitational waves, it is possible that we ended up in a situation where the total (vector) sum of the spins and the orbital angular momentum vanishes. As a result there is very little resistance to the direction of the orbital angular momentum being changed, and the system will transition from precessing around one direction to precessing around a different direction. This effect is known as "transitional precession".

Finally, in the strong field regime the precession time scale will become comparable to the orbital time scale and you can get some crazy looking orbits. For comparable mass binaries (such as observed by LIGO) this is not too noticeable, because at the same time the inspiral timescale also becomes similar to the orbital timescale and we can no longer really distinguish the effects from precession and the merger process. However, small mass-ratio sytems evolve much more slowly, and the crazy strong field orbits are much more apparent. Here is a picture of the an inclined orbit of a test mass around a spinning Kerr black hole: enter image description here You can even get into "resonant" situations where there is some rational ration between the orbital frequency, precession rate, and rate of pericenter advance. This leads to pretty pictures like this: enter image description here

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  • $\begingroup$ Thanks, that was what I was looking for. Is it fair to say that if the orbit starts becoming crazy that the orbital inclination would go through almost all possible angles in a "reasonable" timeframe - say 20,000 years (as seen by an outside observer)? $\endgroup$ – Paul Apr 5 '18 at 17:01
  • $\begingroup$ And if it does trace out almost all possible orbital inclinations, then how long would it take to do this? Are we talking seconds, or thousands of years, or even trillions of years - again, from the viewpoint of an outside observer. $\endgroup$ – Paul Apr 5 '18 at 17:49
  • $\begingroup$ In the test particle case shown in the picture the orbit will end up filling a toroidal-like region around the central black hole. How "thick" that torus is depends on the miss alignment of the spin and orbital angular momentum. The time scale for this is relatively short. The shown orbit would be about a days worth if the central object was $10^6$ solar mass black hole. $\endgroup$ – mmeent Apr 6 '18 at 6:27

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