Flatland

LIGO recently released a new public page showing a summary of all the alerts from the 3rd Observing run so far. The page includes maps showing the estimated sky location of each event. This one comes from just this past Friday night:

S190720a

Seeing it reminded me of a question my father Steve had way back in 2015 after our first detection: Why do the regions have that funny banana shape?

At the time, I couldn't give him a good answer, since I had just started in the collaboration and didn't grasp all the details of the complicated matter of localization. At LAPP though, it's been a big part of my work, and I even gave a presentation on it a couple weeks ago at SFP. The simple answer to Steve's question is, they're not a funny banana shape, they're rings!

It's easier to understand this if you look at the image for GW170817 I showed last time:

LIGO/Virgo/NASA/Leo Singer

Each detector contributes a ring on the sky, and the overlap of those rings gives the estimated location of the wave source. The problem becomes showing data that lies on a sphere, on a flat surface like a computer screen instead.

Cartographers have been dealing with this issue for centuries in trying to show the surface of the Earth on a flat map. This is difficult because the geometry of a sphere is fundamentally different from a plane. The surface needs to be unravelled, or projected, in a way that introduces distortions. A common representation for the Earth is the Mercator projection:

Wikipedia

The problem with this is that areas near the poles are exaggerated, which results in Antarctica appearing to be almost as big as all the other landmass. This projection was designed for navigation, because a straight line on this map will always have the same angle with longitudinal lines – "lines of constant bearing", or loxodromes ("Two bagels enter, one bagel leaves!").

There are many, many different map projections, but the main one we use is called a Mollweide projection. This is designed to preserve the area of regions on the sky, while allowing distortions in the shape. That's important for multi-messenger astronomy, since it allows our EM partners to assess how much sky they need to search to find a companion signal.

Python's matplotlib includes the Mollweide projection, as well as several others, so I put together some code to show what it looks like when two rings with slightly different centers (like Livingston and Hanford) intersect on the sky. The first set of plots show the full rings, while the second set show their intersection, similar to how the LVC estimates source locations.

Thanks for a great question, Steve! I hope I can get around to others in a more timely fashion.