Does Dark Matter?

Jed Rembold

April 12, 2026

Announcements

HW11 due tonight
HW12 posted: You have everything you need after today
Final Project Groups sent out via email?
No class on Wednesday! Go support your presenting peers!
- Presenters from this class:
  - Yumi @ 2:20 in Ford 202
  - Florian @ 3:10 in Ford 202
  - Saul & Cody @ 11:30 in Ford 202

Emcee or MCMCEnsemble can help in running MCMC analysis
Both utilize algorithms that adapt the step-size on the fly
- I know you are tired of trying to optimize step-sizes by now
Both also allow for running multiple walkers at once to check for convergence
The corners library or ggmcmc allow for easy creation of corner and pairwise plots

Rotation Curves
Intro to dark matter?
- Why do we think it exists?
- How can we measure it?
- What ramifications does it have?

Objects traveling in a circle need a force pulling them inwards
The amount of force, size of the circle, and speed are related: \[ F_{in} = \frac{mv_{circ}^2}{R} \]
For our orbits, the force must be gravity

We can work out how the velocity should vary with distance from the center: \[\begin{aligned} F_{grav} &= \frac{mv_{circ}^2}{R}\\ \frac{GMm}{R^2} &= \frac{mv_{circ}^2}{R}\\ \frac{GM}{R} &= v_{circ}^2 \end{aligned} \]
For nicely symmetric mass distributions, \(M\) can be taken to be the total mass internal to the radius \(R\), or \(M_{in}(R)\) \[ v_{circ}(R) = \sqrt{\frac{GM_{in}(R)}{R}} \]

I am providing you with two density profiles below:
- Solar System-like (density in \(kg/au^3\)) (1 au = 1.496E11 m)
- Galaxy-like (density in \(kg/ly^3\)) (1 ly = 9.461E15 m)
For each, you can assume that the density beside a distance describes the density between that distance and the previous distance
You task is to generate velocity curve profiles for each. You can assume a spherical distribution of the material.
How do they compare (both to one another and the distributions we just looked at)?
If using dataframes, lag/shift, coalesce/fillna, and cumsum may be helpful!
- Or you can just loop things

The second velocity curve is what astronomers expected to see when looking at the Milky Way and other galaxies
- Would be consistent with the visible light we saw from the galaxy and known star masses
But instead…

Determining how to reconcile the mass that we see, with the mass that the rotation curves predict, has been an ongoing struggle
Current estimates predict between 5-10 times as much mass as we see
- Spread somewhat evenly throughout the visible galaxy and far beyond
Definitely does not seem to have the majority of mass concentrated at the center
Are we missing dark objects?

What other forms of mass do we know of that would be faint / invisible?
Could the halo of the galaxy be filled with faint, dead stars?
- Massive Compact Halo Objects?
- Brown dwarfs
- Neutron stars
- Black Holes
How does one find an invisible object?