How did LIGO estimate the distance of the black holes?

[JK423]

As we all know, the LIGO collaboration published a paper recently on the first direct observation of a binary merging black hole system. From the observed signal, they were able to infer the black holes' masses and their distance from Earth.

However, the fact that they can estimate masses and distance is completely non-intuitive to me. To my mind, the same signal could have been produced by two black holes that are much closer to us but have smaller masses.
What is it that singles out the particular black hole configuration they reported? Does anyone have any intuition on that?

 

[Orodruin]

The masses are inferred from the frequency of the wave and how it changes, the distance is inferred from the amplitude.

 

[JK423]

That makes sense! Thanks a lot

 

[Jorrie]

This 2014 paper by C. Messenger et. al describes the technique for binary neutron stars in detail. I'm not sure if it is directly applicable to BHs though.

 

[Inwicta]

The masses are inferred from the frequency of the wave and how it changes, the distance is inferred from the amplitude.

Do you know the specific formula they were using to count the distance. I am in high school and the physics course that i am having has only given me the basic equation of Waves, which is velocity = Hz x wavelength.

 

[pervect]

See https://dcc.ligo.org/public/0122/P150914/014/LIGO-P150914_Detection_of_GW150914.pdf

I'm not sure if they used a numerical simulation to get the "Keplerian effective black hole separation" or whether they just deduced that from the frequency of the chirp using Newton's laws. It sounds like it might be the later, but it wasn't too clear to me from reading the paper. There is a formula in the paper for the mass calculation of the pair, though. What's calculated is called the "chirp mass".

 

[pervect]

Let me expand my previous response in a bit more detail. Kepler's law is:

$$T^2 = \frac{4 \pi^2}{GM} a^3$$

where T is the orbital period, G is the gravitational constant, M is the mass (for a two body Newtonian system, the total mass) and a is the separation. So basically I'm assuming that when they calculate the ""Keplerian effective black hole separation", they are using Kepler's law to do it, just from the name they used. To calculate a, they would need T and M. The former they can get from 1/f, f being the frequency of the chirp, and they can estimate M from $\mathscr{M}$, the chirp mass. I should add that $\mathscr{M}$ is new to me, but they do give an explanation and a literature reference in the paper describing it's calculation.

 

[Inwicta]

Let me expand my previous response in a bit more detail. Kepler's law is:

$$T^2 = \frac{4 \pi^2}{GM} a^3$$

where T is the orbital period, G is the gravitational constant, M is the mass (for a two body Newtonian system, the total mass) and a is the separation. So basically I'm assuming that when they calculate the ""Keplerian effective black hole separation", they are using Kepler's law to do it, just from the name they used. To calculate a, they would need T and M. The former they can get from 1/f, f being the frequency of the chirp, and they can estimate M from $\mathscr{M}$, the chirp mass. I should add that $\mathscr{M}$ is new to me, but they do give an explanation and a literature reference in the paper describing it's calculation.

Thank you!