It can be hard to have missed the news last week of the detection of gravitational waves – an event known as GW150914 [GW = Gravitational Wave, followed by the date of the event in the YYMMDD format]. There was, understandably, an awful lot of excitement – it hit pretty much every major news network, and was even mentioned at the BAFTA award ceremony (which is how we know we’ve made it…)!

In the weeks in the run-up to the detection there was a lot of talk about analogies and comparisons of the event, and the black holes involved. But how is that all calculated, and how does that compare to other things in the Universe?

### Weighing black holes

General Relativity has no inherent length scale when describing black holes – in fact relativists often discuss everything in terms of mass – and convert between the two with the factor G/c^{2} (about 10^{-27} m/kg). [That’s why the Schwarzschild radius (the size of the event horizon around a black hole) is often quoted as 2M.]

One of the really nice things about binary black holes collisions is that they are “standard candles” in terms of the gravitational wave signal produced. If you take the right combination of the the gravitational wave frequency (*f*) and amplitude (*h*) at a given time before the collision (*t*), you get a number that is equal to various fundamental constants (π, G, c etc.) and the distance from the source. That’s independent of mass – so instantly gives you a distance to the event (be that measured in metres, astronomical units or, in this case, millions of light years). There’s a complication (when isn’t there?!), in that the inclination of the system to the line of sight (i.e. whether we’re seeing the orbits of the black holes face-on or edge-on) creates a factor of two uncertainty in all that, so the distance estimate for GW150914 is actually fairly uncertain (750-1850 million light years). It’s all related to the polarisation of the gravitational waves, and this is one area where future detectors, with different orientations and therefore different sensitivities to different polarisation, will really help.

Once you’ve got the distance, you can convert from the waveform measured here on Earth to that measured at the source – that includes both the amplitude of the gravitational waves (which drop off linearly with distance) and the frequency (which can be redshifted over such large distances). The resulting waveform also tells you lots of other things as well, such as the masses of the black holes. The way the frequency of the waves changes tells you the “chirp mass”, which is just a combination of the masses of the two initial black holes. An analysis of that, based on incredibly complex computer simulations (one of the things worked on in Cardiff), reveals that the two black holes were about 29 and 36 times the mass of the Sun. Studying the gravitational waves after the merger, during something called the “ringdown phase”, indicates that the final black hole was 62 times the mass of the Sun, and that it was spinning on its axis at 100 Hz.

*[ Update: as Chris Berry pointed out, the waveform is compared to full numerical simulations of general relativity and used to fit all the parameters of the system simultaneously, rather than in separate steps. ]*

*[ Update: added a link to Mark Hannam’s blog, where he explains this process in much more detail] *

Let’s just pause for a minute to appreciate the scale of this. Two black holes, each in the region of 30 times the mass of the Sun (that’s 10 million times the mass of the Earth), collided together to form an even bigger black hole, 60 times the mass of the Sun (20 million Earths), spinning on its axis 100 times per second.

There was a lot of talk about calculating the size of the black hole, where by size we mean the event horizon – the point beyond which nothing can escape, not even light. Unfortunately,for a spinning black hole it’s not strictly speaking possible to give a diameter – all that the equations provide is a surface area – in this case 370,000 km^{2} (about the size of Germany). If it weren’t spinning, it’s diameter would be about 366km (from a calculation of the Schwarzschild radius), but since it’s spinning it’s what’s known as a Kerr black hole. The event horizon is then actually smaller, but since it’s spinning there’s an equatorial bulge (not unlike that we see on planets and moons in the Solar System). A PhD student here in Cardiff, Gernot Heissel, went through the calculations (you can see them here if you like) and concluded that the black hole is around 366km across at the equator, and about 300km at the poles. That makes it pretty much the size of Iceland!

### A big bang

Those keeping track of numbers will notice that the two initial black hole masses (29 and 36 solar masses) add up to more than the final black mass (62 solar masses). So where did the extra 3 solar masses go? Well, that’s what was detected at LIGO in the moment they collided!

Bear in mind that all happened very fast as well – LIGO only saw the last 200 ms of the black holes’ lives before they merged, and in that time the orbited each other 5 times. They spiralled round each other, closing from a few thousand kilometers away, and collided at over half the speed of light. Doing that with two objects tens of times the mass of Sun is going to make a bang.

In this case, the bang was purely gravitational, with no light of any sort emitted. And that’s probably a good thing – in the moment they merged the black holes emitted more power than all the stars in all the galaxies in the observable Universe! (There are debates as to whether it’s 10 or 50 times more, but that mainly depends on how you count the stars). In day-to-day units it was 3.6×10^{49} W, equivalent to the power 100 billion trillion (10^{23}) suns! It’s also 200 billion times more powerful than the brightest supernova ever recorded (called ASASSN15lh), which itself was much brighter than most supernovae. It was even 100 times brighter than the brightest gamma ray burst (termed by Carole Mundell, and no doubt others, as the “biggest bangs since the big one”) – and that’s making all sorts of (unreasonable and unphysical) assumptions about gamma ray bursts which increase the predicted power by a factor of 10s or 100s.

But while it was very” bright”, it didn’t last very long. The energy emitted was a “mere” 5.6×10^{47} J. That’s still quite a lot, though. It’s 500 times the energy output by all the stars in the Milky Way galaxy in a year. It’s also 3 quadrillion (3×10^{15}) times the energy required to blow up the Earth, and a million billion trillion (10^{27}) times the world’s annual energy consumption. Just think of the possibilities if we could harness that power!

But sadly, that’s not to be, as estimates of the energy dissipated as it passed through the Earth are around 10^{-17} J (about the energy of an x-ray photon).

You can read more about the way the black hole masses are calculated in this LIGO Science Summary, or in the Parameter Estimation paper itself.

[Credit to a number of other members of the LIGO Education & Public Outreach team for some of those numbers.]

### What the team have to say

The scientists involved in the design of the instrument and the analysis of the data have written their personal accounts of the detection and its announcements – you can read a few of their thoughts on their various blog pages:

- Mark Hannam – How to decode gravitational waves from black holes
- Andrew Williamson – The Era of Gravitational Waves has begun
- Shane Larson – The Harmonies of Spacetime
- Christopher Berry – Advanced LIGO detects gravitational waves!
- Roy Williams – LIGO is Awesome and This is Why
- Matt Pitkin – The wait is over
- Daniel Williams – Riding the Wave
- Sean Leavey – First Detection of Gravitational Waves!
- Amber Stuver – LIGO Makes the First Direct Detection of Gravitational Waves
- Brynley Pearlstone – Einstein Was Right!
- Becky Douglas – Gravitational Waves: The Big Discovery

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