On April 22nd, I attended the CU APS Colloquium at the JILA Auditorium. The speaker was Kelsey Johnson from the University of Virginia. The title of the talk was: "How were the most ancient objects in the universe formed? ". She was specifically talking about the formation of globular clusters. Although she has worked on a variety of research throughout her career, this particular thread of research actually began during her PhD at JILA over 25 years ago. During a lot of the talk, I became interested in developing an easy way to look at the likelihood of objects colliding given a few simple input parameters, so I thought I would share here, given that I have no other particular use for it.
I started down a bit of a rabbit hole when we were discussing the collisions of galaxies in the class I was auditing at CU. This theme came up multiple times in the class (originally when we were talking about the quantity, size and average spacing of stars vs. galaxies in the universe and then again when we were talking about the formation of elliptical galaxies). Galaxy collisions were one of the main explanations for yielding the correct rare conditions for the formation of globular galaxies, which was made clear during Kelsey’s talk.
An easy initial approximation for the trend of the likelihood of collisions can be made by taking the ratio of the overall volume of the objects that will be colliding and dividing that by the total volume of space that those objects are occupying. This gives a quick and dirty logic for why we had more galaxy collisions in the early universe, when the overall space was smaller (it hadn’t yet expanded so much) and also an explanation for the increased likelihood for the merging of galaxies compared to the merging of stars (if The Milky Way was grapefruit-sized and was located in Washington DC, our next nearest galaxy would be located ~3 meters from us but if our Sun was grapefruit-sized and located in Washington DC, then our next closest star would be located across the distance of the United States in California).
This very coarse approximation should yield a relative likelihood that looks roughly like the graph shown below in Figure 1.
I started down a bit of a rabbit hole when we were discussing the collisions of galaxies in the class I was auditing at CU. This theme came up multiple times in the class (originally when we were talking about the quantity, size and average spacing of stars vs. galaxies in the universe and then again when we were talking about the formation of elliptical galaxies). Galaxy collisions were one of the main explanations for yielding the correct rare conditions for the formation of globular galaxies, which was made clear during Kelsey’s talk.
An easy initial approximation for the trend of the likelihood of collisions can be made by taking the ratio of the overall volume of the objects that will be colliding and dividing that by the total volume of space that those objects are occupying. This gives a quick and dirty logic for why we had more galaxy collisions in the early universe, when the overall space was smaller (it hadn’t yet expanded so much) and also an explanation for the increased likelihood for the merging of galaxies compared to the merging of stars (if The Milky Way was grapefruit-sized and was located in Washington DC, our next nearest galaxy would be located ~3 meters from us but if our Sun was grapefruit-sized and located in Washington DC, then our next closest star would be located across the distance of the United States in California).
This very coarse approximation should yield a relative likelihood that looks roughly like the graph shown below in Figure 1.
This doesn’t really tell us much other than the trend that if objects have a smaller average spacing in comparison to their size, then they are less likely to collide. This does not consider the impact of speed, which was a question that one of the other students brought up in class.
I have some background in path planning and object avoidance algorithms for automated vehicles and silly robots, so I thought it might be easy and fun to re-purpose some of those pieces of code to consider the likelihood of objects in the universe colliding based on their speed of travel and their spacing relative to their size.
Problem Setup
I set up the following functions using MATLAB:
- Space / object initialization: place the set of objects in a 3D zone (when speed is 0, randomly set the initial positions of the objects; otherwise, set the objects in an organized grid with the correct spacing).
- Inputs: Spacing of the objects relative to their radius; Number of objects; Speed of objects (in radii traveled in each time step); Size of space within which the objects are moving.
- Object trajectory specification: randomly walk each object within the space (the speed is specified as an input by the user but the direction is randomized). Note: we talked about random walks of the photons out of the radiative zone of the Sun, so could re-purpose this function to look at the timing of those photons journey through the radiative zone or something else fun.
- Inputs: Objects’ positions at previous time step; Speed of Objects; Zone boundaries.
- Collision detection: detect and count the collisions between each object.
- Inputs: Objects’ current occupied zones.
- Monte Carlo simulation: perform a Monte Carlo analysis to assess the probability of collisions for each of the conditions.
- Inputs: Test case matrix.
Results:
The compiled results for the various simulations run are shown in Figure 3.
Conclusions and Next Steps:
This was a fun undertaking that showed me what I expected based on the definition of pressure. I consider the collision probability as being a kind of analog for gas pressure and I expect gas pressure to be impacted by temperature (which is the analog for speed in this simulation) and volume (which is the analog for object spacing in this simulation). The trend I was came from some manipulations of the ideal gas law and a few other relations:
If I plot this in a similar way to what I plotted above, I get a similar looking shape to the results of my simulation, which I have shown in Figure 4.
I think from this sanity check, we can see that the problem setup is working as expected and the conclusions that we can draw are:
- The likelihood of collisions of objects in a zone increases with speed (a squared relation)
- The likelihood of collisions of objects in a zone decreases as the average spacing between them relative to their size increases (an inverse cubic relation)
The next steps for this particular model would be to add would be some dynamics – adding gravity could show us some cool things, as well as adding more representative trajectories. Furthermore, running the simulation for more iterations would of course increase the fidelity of the results and having a dynamic expansion of the zone could be a nice representation of the Universe expansion, but that would be getting quite overly deep into the weeds of something that it seems we already understand.