On September 14th, 2015, the LIGO gravitational wave observatory network detected the gravitational waves from the merger of two black holes. In moments, the LIGO team estimated (very broadly) where the black holes were located in the sky; these regions are highlighted in figure 1. Today I tell you how they figured this out. And why it’s important.

Electromagnetic Counterparts

First, let’s talk about why the direction of the waves is important. When LIGO detects gravitational waves, those waves can tell us an awful lot about their source. Just from the waveform, LIGO learned that the waves from December 14th came from the merger of two black holes, each about thirty times the mass of the sun, about 1.3 billion lightyears away. And from that information, the LIGO team was able to extrapolate a surprising amount about the astrophysics of the stars that became those black holes.

But we’d like to learn more. Are the black holes immersed in a disk of hot gas? Or are they alone, as we expect? What about other systems? Most astrophysicists expect that the merger of two neutron stars (or a black hole and a neutron star) will produce both a short gamma ray burst and a burst of gravitational waves. A detection of both at the same time would confirm this hypothesis. We know that when a star runs out of fuel, it may undergo a core-collapse supernova. Unfortunately, the precise mechanisms of the supernova explosion are unknown.

To extract this information, we can’t rely just on gravitational waves. We need electromagnetic waves, too. And this means looking at the source with optical telescopes, infrared telescopes, radio telescopes, gamma ray and X-ray telescopes, basically any telescope we can get our hands on. We even try to look with neutrino detectors. But looking means we need to know where to look. And that’s where LIGO’s skymap in figure 1 comes in.

So how did LIGO generate that? In the sections below, I’ll tell you about several pieces of information that LIGO can use to estimate the direction from which the waves came.

Time of Arrival

The most important piece of information LIGO has is the time the wave arrives at the detector. Or, more precisely, the times the waves arrive at the different detectors. LIGO currently has two gravitational wave detectors, one in Livingston, Louisiana, and one in Hanford, Washington. They’re about 3000 kilometers (1875 miles) apart, as shown in figure 2.

At normal human walking speed, it takes 758 hours to get between the two observatories. If you could travel at the speed of light and through the Earth (like gravitational waves), then you could make it from LIGO Livingston to LIGO Hanford in about 10 milliseconds. But there’s another reason why you can’t travel like a gravitational wave: You’re finite. You can only bet on Livingston OR Hanford, not both at the same time. Waves, on the other hand, are spread out. Just as, when you stand on the beach, an incoming wave can both tickle your toes and destroy your sister’s sand castle a few meters down the shore, the same gravitational wave can be present at both LIGOs Livingston and Hanford. This means that, if the waves came from “above” North America (or indeed any direction perpendicular to a straight line drawn between the two detectors on a map), as shown in figure 3, LIGOs Livingston and Hanford would detect the waves at exactly the same time, with no delay.

But this depends very much on the direction. If you draw a line between LIGOs Livingston and Hanford, and if a gravitational wave comes from a direction parallel to that line, as in figure 4, one of the detectors will measure the waves a full 14.4 milliseconds before the other!

The waves that LIGO detected on September 14th arrived at Livingston about 7 milliseconds before Hanford. So their direction of origin lies somewhere between the two extremes I’ve outlined. And this piece of information helped LIGO narrow down where they came from. Indeed, for the September 14th detection, this was the most significant directional hint by far.

Detector Sensitivity

We can’t point the LIGO detectors the same way you would point a telescope–they’re what we call “all-sky” detectors. However, they are most sensitive to waves coming from overhead. This is baked into how the detectors are built and how gravitational waves work. As I’ve described in the past, a gravitational wave is a distortion in distance that travels. But–and this is very important–the distortion is perpendicular to the direction of motion, as shown in figure 5. Just as an ocean wave travelling inland makes the water rise and fall, a gravitational wave travelling to your left might make you alternatively taller and shorter.

The LIGO detectors are designed to measure this distortion. They’re essentially two huge, perpendicular rulers. But this means that, for them to see anything, the distortion must be aligned with the arms of the detector. Figure 6 compares the two most extreme possibilities. If the gravitational wave (purple) comes from directly above LIGO, it is most sensitive. If the wave comes from a direction in the same plane as the two arms of the detector, LIGO will be significantly less sensitive.

Since LIGO Livingston and LIGO Hanford are at different places on the surface of the Earth (which is round), they have slightly different orientations with respect to an incoming gravitational wave. The LIGO team can use this information and the relative signal strengths at each detector to help infer the direction of the wave. With only two detectors, this piece of information is less significant than the delayed time of arrival, but every little bit helps. And in the future, when there are more gravitational wave detectors around the world (Virgo turns on next year!), orientation will help much more.

Polarization

There is a complication to the orientation story I told you above. The complication is, of course, that there is more than one way to orient a gravitational wave. The wave has a direction of motion, but it’s also possible to rotate the distortion that it produces around that direction. Gravitational waves have two fundamental “polarizations,” called “+” and “x” respectively. And you can get one by rotating the other. Figure 7 shows the action of a gravitational wave on a ring of test particles for the + and x polarizations of a gravitational wave. The motion is the same, but the direction differs by 45 degrees.

The wave can also be a combination of the two polarizations, in which case the rotation rotates around the axis, either clockwise or counter-clockwise, as shown in figure 8. These are called the “left” and “right” polarizations respectively.

If a wave is left- or right-polarized, LIGO will be able to see it. But if a LIGO detector is oriented for + polarization and the incoming wave is x-polarized, sensitivity would be reduced in a way that has little bearing on the direction the wave came from. It’s hard to distinguish loss of sensitivity from polarization to loss of sensitivity from orientation… especially with only two detectors. In the future, when we have more detectors, it will be easier to distinguish polarization from orientation.

Statistics

The LIGO team takes all this information (and more, such as the masses of the originating black holes) and plugs it into sophisticated computer codes that, given times of arrival and detector sensitivity, assign a probability to each piece of the sky indicating how likely it is that the gravitational waves came from that direction. Some of the codes (essentially) randomly generate a huge number of possible waveforms and directions and see which combination yields times of arrival and sensitivities that match the physical measurement. This is called a Monte Carlo algorithm and it has applications everywhere in physics. For example, I’ve used it in quantum gravity simulations. Other codes invert this question and ask what the probability of a direction is given the measurements we have. This is called a Bayesian algorithm.

To ensure accuracy (and to avoid putting their eggs all in one basket), the LIGO team uses several algorithms. The Bayesian algorithm is extremely fast, so the LIGO team can use it to inform telescopes where to look immediately. One of the Monte Carlo algorithms is almost as fast. Finally, there is a much slower algorithm which uses astrophysics information extracted from the waveform (such as the masses of the black holes) to make an inference. This algorithm is run later.

It is in these algorithms that the magic lies–magic which took many years of hard work by the LIGO team to develop. But that hard work was more than worth it.

The Results

Figure 1 shows the locations in the sky from which the gravitational waves most likely came from, as determined by LIGO’s algorithms. What it doesn’t show you is the more than sixty teams that used this information to search the sky for counterparts to the gravitational waves, optical or otherwise. Their efforts are summarized in figure 9. The green panes represent measurements by optical or infrared telescopes. The red, radio. The blue, X-ray. Two gamma ray detectors and the world’s most sensitive neutrino detectors, which are all-sky like LIGO, searched their data for correlations with LIGO. They didn’t find anything definitive. But with LIGO detecting gravitational waves and the search infrastructure in place, it’s only a matter of time until we see something amazing.

The LIGO collaboration consists of more than nine hundred scientists. But the combined search for counterparts, optical and otherwise, consists of at least as many. The team that looks for neutrinos is more than 300 people. (The neutrino team doesn’t need this sky map, since their detector is all-sky.) Figure 9 therefore captures the beginning of one of the biggest-ever collaborations in science. During a LIGO science run (when LIGO is actively taking data), scientists all over the world are poised, ready to aim their telescopes at the sky and search the sources of gravitational waves. It’s a stunning tribute to the collaborative spirit of science and to the things we can accomplish when we work together.

A Possible Gamma Ray Burst Counterpart?

All this hard work might already be bearing fruit. As I’ve discussed before, we believe that the merger of a neutron star and a black hole (or two neutron stars) will produce a gamma ray burst. Now, it’s hard to imagine that the merger of two black holes, which is what LIGO measured on September 14, could produce such a thing. But people were looking anyway. (They didn’t yet know that LIGO had seen a black hole merger… just that LIGO had seen something. And anyway, you never know.) And one of the gamma ray detectors, Fermi’s GBM, thought they saw something.

When LIGO announced their detection, the Fermi team went back through the data their detector had collected and looked for excess power from the detector. They found what looked like an event and, after much analysis, concluded that the probability it was a false alarm was approximately 0.22%. This is certainly exciting, but physicists are a cautious lot and the false alarm probability isn’t low enough to claim a definitive detection… especially when the other gamma ray detector looking for signal didn’t see anything.

To explain this event, there have been some crazy ideas, like the possibility that both of LIGO’s black holes emerged from the collapse of a single enormous star. I’m excited that something exotic might be happening… but I also urge caution. We need more gravitational wave detections to understand what’s going on.

We’ve entered the age of gravitational wave astronomy. It’s only a matter of time.

Corrections

In figure 6 I previously stated that when the waves come at LIGO from the in the same plane as the detector there is no signal. This isn’t quite true. The signal is just much weaker. Thanks to Google+ user Greg Roelofs for catching this.

I also said that it takes 14.4 milliseconds for gravitational waves to travel from LIGO Livingston to Hanford. This is wrong. It only takes about 10ms if they’re taking the straightest path. The straightest path distance is 3000km. Thanks to Prof. David Shoemaker(!) for the correction.

Related Reading

If you enjoyed this post, you may enjoy some of these, too:

• In this post, I describe some of the astrophysics we extracted from LIGO’s detection of gravitational waves.
• In this post, I attempt to capture some of the science–and poetry–of LIGO’s detection of gravitational waves.
• In this post, I describe how gravitational waves work.
• In this post, I describe research efforts to simulate a short gamma ray burst.
• Astronomy using gravitational waves, light, and subatomic particles is called multi-messenger astronomy. I wrote about that, and why it’s exciting, here.

Sources

Here are the papers I summarized in this post:

• This is the paper summarizing the generation of the skymap and the efforts to look for electromagnetic counterparts.
• This is the paper summarizing the search for neutrino counterparts of LIGO’s gravitational waves.
• This is the paper by the FERMI team describing their possible detection of a gamma ray burst counterpart.
• This is the original LIGO detection paper.
• There is a large body of literature on localizing the source of gravitational waves. This paper is on LALInference, the tool that uses astrophysics inferred from the gravitational waves to help pinpoint direction.
• This paper is on the fast Bayesian inference library that also searches for direction.
• This paper discusses two the performance of two algorithms: coherent WaveBurst and LALInference.
• This paper discusses extracting the polarization and including detector orientation in the inference.
• This paper contains the crazy idea of two black holes emerging from a single star.
• Astrophysicist Ethan Siegel evaluates the single star proposal in this popular article. Both Siegel and I think this proposal is pretty unlikely to be true.