Gravitational Waves! A Comprehensive Explanation of the Significance of Detecting GW170814

Introduction

On September 27th 2017, a group of Astronomers from both of the LIGO announced they have observed a fourth instance of gravitational waves. This time they also received a third confirmation from the Virgo detector in Italy. This detection known as GW170814 (for the date it was detected) is quite possibly the most important of the four confirmed detection since the original GW150914. As we will see, by recording the event from a third location astronomers are now capable of determining approximately where the gravitational-waves originated!

The above graph is from LIGO’s publication depicting where the origin of the gravitational-waves could be in light-hours.

Trilateration

One method used to find the location of a specific point is called trilateration. In trilateration you start with a series of equations for three circles in which their radius is the distance from their centers to a point A. Next you solve the series of equations to find their point of intersection (point A). This concept can be applied in higher dimensions by using n+1
n spheres.

Here is a graphical representation for trilateration:

Bayesian Parameter Estimation

Unfortunately, in practice we are unable to use trilateration because of the experimental error and the randomness associated with the experiment. Instead, the study relied on Bayesian parameter estimation. Bayesian parameter estimation uses a similar approach to trilateration except for that it calculates a range the point is likely be be within.

I do not claim to be an expert on Bayesian parameter estimation and found the following explanation in the report Bayesian Methods of Parameter Estimation by Aciel Eshky at the University of Edinburgh.

With the addition of a third detector the above form of Bayes Rule can be expanded to consider a third variable C which in turn will help us look for the source of the gravitational waves in a smaller region. Finally, by applying Bayes Rule on the data from each of the three detectors, the possible region can be decreased even further by looking at where they overlap.

Additional Information

Here is some more of the data collected for anyone who is interested:

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Work Cited:
https://dcc.ligo.org/LIGO-P170814/public/main
http://homepages.inf.ed.ac.uk/rbf/CVonline/LOCAL_COPIES/AV0809/eshky.pdf