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A certain number of gravitational wave events have been detected. Is it possible to know how far away the mergers that caused those gravitational wave events are?

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Yes, it is possible to calculate (within an error range) the distance of observed gravitational wave events. It is known that a variety of parameters will affect how the amplitude and frequency of the observed gravitational waves will change over time as recorded in the "chirp" event from the interferometers: the parameters include distance of the event, the mass of each of the colliding objects, the angular momentum of each of the colliding objects, the orientation of the objects' angular momentum vectors with respect to each other and with their orbital plane. With general relativity, you can build a model that calculates the expected "chirp" given a value for all these parameters; when a chirp is observed, it is possible to determine the combination of these parameters that result in a chirp that best matches the observation.

The effect of a larger distance parameter is to decrease the amplitude of the expected waves from colliding objects of a given mass, as well as to "slow down" the entire event due to cosmological red shift.

From GWTC-1: A Gravitational-Wave Transient Catalog of Compact Binary Mergers Observed by LIGO and Virgo during the First and Second Observing Runs

Gravitational waves from compact binaries carry information about the properties of the source such as the masses and spins. These can be extracted via Bayesian inference by using theoretical models of the GW signal that describe the inspiral, merger, and ringdown of the final object for BBH [23–30] and the inspiral (and merger) for BNS [31–33]. Such models are built by combining post-Newtonian calculations [34–38], the effective-one-body formalism [39–44], and numerical relativity [45–50].

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  • $\begingroup$ Decreasing amplitude -- is this analogous to sounds getting softer due to distance, or is it some other process? $\endgroup$ – Barmar Jan 15 at 18:13
  • $\begingroup$ @Barmar yes, the normal thing — the further away you are, the more area the wavefront has to spread out over. $\endgroup$ – hobbs Jan 16 at 5:52
  • $\begingroup$ Thought so, but just making sure there's not some additional GR effect involved. $\endgroup$ – Barmar Jan 16 at 5:53
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Yes, it's possible, but less straightforward than for "normal" objects.

If the optical counterpart of the GW signal is located, as in the case of GW170817, the distance can be inferred by standard methods of observing the redshift of its host galaxy.

If not, the luminosity distance $d_L$ can still be inferred because the amplitude of the GW signal scales inversely with the $d_L$. This can then be converted to a redshift, assuming some cosmology. This was done for the first ever GW detection GW150914 (Abbott et al. 2016).

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  • $\begingroup$ So there are GW events we don't know the distance of? $\endgroup$ – usernumber Jan 14 at 16:23
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    $\begingroup$ @usernumber We always have some clue of the luminosity distance (as far as I know), but they typically have huge error bars, of the order of 50% or so. I think the problem is that the distance correlates with the orbital plane, but system typically precesses during the inspiral, so it's difficult to get an unambiguous value. $\endgroup$ – pela Jan 14 at 19:52
  • $\begingroup$ @Pela I believe that multiple detectors can significantly clear up the orientation ambiguity. $\endgroup$ – Rob Jeffries Jan 17 at 8:12
  • $\begingroup$ @RobJeffries Ah yes, I hadn't thought of that. It makes sense, although it doesn't really seem like the uncertainties went down when VIRGO joined. $\endgroup$ – pela Jan 17 at 13:44
  • $\begingroup$ If anything determining the distance of a GW event is a lot more straightforward then for EM events. $\endgroup$ – mmeent Jan 18 at 0:05
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To answer the question in your title (by following the links in the other answers):

GW170817 (two neutron stars): 40 Mpc

GW150914 (two black holes): 410 (+160 or -180) Mpc

antlersoft's link (GWTC-1: A Gravitational-Wave Transient Catalog of Compact Binary Mergers Observed by LIGO and Virgo during the First and Second Observing Runs): distances range from 320 (+120 -110) Mpc to 2840 (+1400 -1360) Mpc for binary black hole mergers.

One Mpc (megaparsec) is about 3.26 million light years.

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This is additional to the other answers. We now have three GW detectors (LIGO x2 + VIRGO). This allows the direction of the event to be deduced, by the relative timing of the arrival of the chirp, which is an effectively planar wave passing through the Earth at the speed on light. More accurately, deduce one of two possible directions: towards the event ot towards its celestial antipode (a fourth detector would eliminate this ambiguity).

I don't know the acccuracy to which this direction can be deduced. However, if one assumes that a black hole merger would not be taking place in intergalactic space, it might serve alongside the other information deduced from the chirp to identify the galaxy in which it took place, even if there were no visible light emission.


Direction, distance and GW170817

There are two ways by which the direction can improve estimates of distance. Both of these ways are demonstrated in the detection of GW170817, the signal from a binary neutron star merger.

  • Follow up searches for sources emitting light. In the case of GW170817 the searches for a light signal helped to pinpoint the origin of the source (NGC 4994) more precisely. This allows to improve the estimates of distance by including estimates of distance based on light sources.
  • Relation between source position and observed detector amplitude. The amplitude of the signal is depending both on the distance and the position. The amplitude of the waves will be larger when the source is closes, but also when the direction of the source is more perpendicular to the arms of the detector. This means that the amplitude of the signal is relating to two different unknown parameters. Therefore, being able to independently pinpoint the location, will be allow to better estimate the source distance.

Detailed article about pinpointing parameters: https://arxiv.org/abs/gr-qc/9402014

How using three detectors LIGO + VIRGO improved location for GW170817: https://www.ligo.caltech.edu/page/press-release-gw170817 (see the image for comparison with other sources that only used the two LIGO detectors and have an estimate of location in a ring shape)

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    $\begingroup$ Does this help to determine the distance at which the merger occurred? $\endgroup$ – usernumber Jan 16 at 10:51
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    $\begingroup$ It certainly does if there is only one galaxy at a plausible distance in the narrow cone in which the observed chirp might have emanated. Not if there are many. $\endgroup$ – nigel222 Jan 16 at 11:00
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    $\begingroup$ You're right that three detectors greatly reduces the positional accuracy, but we're still talking many degrees on the sky, covering thousands of galaxies. An optical counterpart is needed to really see the position. $\endgroup$ – pela Jan 16 at 13:39
  • $\begingroup$ @usernumber in a (not yet approved) edit I have added some information explaining how this (distance) helps to determine the distance. $\endgroup$ – Sextus Empiricus Jan 16 at 17:07
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    $\begingroup$ You haven't quite got it. Multiple detections can tell you the orientation of the binary by differing sensitivity to the two polarisation directions by detectors with differing orientations. $\endgroup$ – Rob Jeffries Jan 17 at 8:09

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