How well can we localize gravitational wave sources?

Question

A recent question cited a story about the recent gravitational wave detection saying that we can use the gravitational wave sensing to find supernova earlier in their process of collapse:

[with the discovery of gravitational waves, we will be able to] Track Supernovas hours before they're visible to any telescope because the waves arrive Earth long before any light does, giving astronomers time to point telescopes like Hubble in that direction

From what I can tell we currently have two detectors operating, which gives us a rough idea for localization of the source:

By looking at the time of arrival of the signals—the detector in Livingston recorded the event 7 milliseconds before the detector in Hanford—scientists can say that the source was located in the Southern Hemisphere.

This article goes on to talk about plans for a third detector in India, presumably to perform trilateration on future measurements.

With all three detectors in place, Livingston, Hanford, and (somewhere in) India, how precisely will we be able to localize the source of future gravitational waves? Enough to be pointed at the supernova when continues its collapse and lights up? Will more detectors significantly improve the precision?

Answer 1

To expand on Ernie's answer, there are in fact qualitative changes in the posterior distribution of the sky location as you add more detectors to a network, beyond just shrinking the contours.

With a single detector, you can't localise the source at all; with two detectors, timing triangulation across detector sites allows you to localise the source to a ring-shaped locus on the sky; a third detector reduces this ring to two antipodean spots on the sky; and a fourth breaks the degeneracy entirely into a single spot. Of course, the degree to which these degeneracies are broken depends on the baseline distance between detectors; having four in the same place won't help very much!

There are some nice slides about GW sky localisation here, with more details on triangulation in this paper by the same author.

There are more sophisticated ways of localising the source than simple triangulation, though; see this paper for a comparison. For example, the phase difference between detector sites carries some useful information about sky location. There are also strong correlations between the sky location, distance, and the mass of the source. The "correct" way to infer sky location is therefore a coherent analysis that fits all of the source parameters simultaneously – but this is much more computationally costly.

---

As for localisation prospects, I'm not sure about supernovae, but I can tell you about compact binaries like GW150914! Assuming a network comprising the two LIGO detectors and the Virgo detector, all at design sensitivity, most events will be localised to within about $100\,\mathrm{deg}^2$ on the sky (to 95% confidence), with the very loudest perhaps being $1-10\,\mathrm{deg}^2$. With the current network configuration of two detectors, this increases to several hundred $\mathrm{deg}^2$ (GW150914 was about $600\,\mathrm{deg}^2$).

In any case, compact binaries are the most promising sources for detectors like LIGO; supernovae are far, far weaker and will likely only be detectable within the Milky Way (i.e., within tens of kiloparsecs, compared to the gigaparsecs to which LIGO is sensitive to compact binary mergers).

You quite rightly suggest that with sufficient information about the sky location of a source, we could point a telescope at the source and observe its electromagnetic counterpart (although this will probably require one of the component masses to be a neutron star; black holes are rather... black). This is in fact a major science goal for the gravitational wave community, but it's a tricky one and there are many variables.

It depends – among other things – on the wavelength band you're interested in, but the biggest challenge here is the large uncertainty on sky location. Typical telescopes used for following up these events have fields-of-view of the order $0.1\,\mathrm{deg}^2$, making it difficult to target the right place in the sky with a reasonable number of telescope pointings.

Unfortunately, most compact binary detections probably won't be followed up with an EM observation. The EM counterpart of a compact binary merger is fairly faint (especially at such large distances), and it can be difficult to distinguish it from background contaminants. And then there's the logistic difficulty of coordinating telescope observations to cover the uncertainty region on the sky promptly enough to catch the EM counterpart.

Answer 2

The time difference between the detections indicates that the gravitational wave came roughly from the direction of the earlier detection. With more detectors operating in widely dispersed locations on the Earth's surface, re-construction of the source direction will be more precise, as more points on the gravitational wave's vector, from the earlier detections to the later detections, can be plotted as the wave front passes through the Earth.

A map of directional probability has been constructed for the two-detector array currently operative. The probability contours will shrink with more detectors in operation at more locations.

Here's a link to a paper on the Science and Detection of Gravitational waves by Barry Barish, former director of LIGO. If you read through the paper, you'll find much food for thought.