(You can find the paper on arXiv here!)

In November 2017, Dr. Scott Chapman approached me about simulating the observed system for an additional 1 billion years after the observation in order to determine whether the system would indeed produce a large BCG.

The following visualization shows the initial condition of the highly idealized disk galaxies that I produced. I approximate each galaxy with a gaseous and stellar disk, in addition to a dark matter halo. I then randomly sample positions within a sphere of radius 65kpc for each system, as well as randomly sample orientations of the disk spin axes.

In order to study the dynamics of the system we need initial velocities. I sample the bulk velocity of each galaxy from the observed distribution and assign this to the members at random. The following simulation shows one possible realization of the evolution of the cluster, for an additional 1 billion years after the observation.

Young stars are shown in blue, and older stars shown in red. Occasionally dust obscures the light from the stars during the collision.

Based on the above simulation these 14 galaxies smashing together do produce a large spheroidal system! In fact, the final state of the galaxy looks very similar to NGC 474 (shown below). We understand the origins of the * streams* and

The most interesting result from these simulations is that an intracluster medium (ICM) builds from supernova explosions ejecting gas from the galaxies. Below is a slice through the gas within 200kpc of the centroid, with temperatures shown in color between 100 000 K and 10 000 000 K! You can see shocks moving outward from the center as the gas rapidly mixes in the center. Eventually over time, the gas extends 1 Mpc from the centroid and builds something very similar to the ICM.

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I'm looking to solve an integral of them form:

This is a special case where the limits are the roots of the radicand. If we make the transformation:

We can now use this to replace one of the factors in the denominator:

and are found by the transformation equation, which we can also use to get rid of the in the denominator. I'll find them below, but leave them as and for simplicity.

Now since:

Substituting back into our integral:

Now another transformation can be made, which most conveniently can be chosen as:

Using this transformation on our integral we obtain:

This integral is a standard integral for , which we can derive by making the transformation:

Back into our integral we have:

Since . This is now a trivial integral, which is just the difference of the limits and . The integral becomes:

The task now is to find the values and by transforming back to the original equation. The limits are given by:

The limits are therefore:

and finally:

Which gives us the result:

This holds for all such that the limits are roots corresponding to the precise shape of given at the start. It's kind of amazing that it always comes out to , a result which has been in the back of my mind since I first did this integral about a month ago.

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