The Maths Behind Dark Matter

Just Recently, I had the amazing opportunity to meet Jocelyn Burnell, and was blessed with a chance to present to her an area of astronomy that interested me. So I chose to talk about dark matter’s existence, and today I wanted to share my presentation with you.

This is a basic overview of dark matter, with added maths. It’s not complex, and ideal for those who want to take their understanding of this weird universe up a level.

The Two Equations That Will Help Us Today

A planet orbiting a star (or anything orbiting something) will have a centripetal force and a gravitational force. To find the centripetal force, we can use an adapted version of F=ma:

F = mv²/r

where:

  • m is the planet’s mass,
  • v is the planet’s velocity,
  • and r is the distance to the star.

We also need Newton’s law of gravitation:

F = GMm/r²

  • G is the universal gravitational constant
  • M (big one) is the star’s mass (not the planet’s)

These two equations will tell us enough about orbits for the case of dark matter, and can be applied to galaxy motion as well.

How are we going to use these equations?

In the case of galaxies, m will be the mass of a single star, and M will be the mass of the entire galaxy within that star’s orbit.

To find the velocity one of these stars should be orbiting with, we can equate both equations to make a new one:

mv²/r = GMm/r²

v²/r = GM/r²

v² = GM/r

What can we learn from this new equation? Firstly, M is the mass of the galaxy, so a larger M means a larger velocity. r is the star’s distance from the centre of the galaxy, so when we increase r we decrease the velocity.

Let’s graph it!

Graphs and pictures make life so much more simpler, so let’s make some pretty pictures!

Below I’ve graphed the solar system (Mercury-Pluto). All I care about for now is v² (the planet’s velocity) and r (distance to the centre of the sun), so all I’ve done is plot v² against r.

graph.png

If v² = GM/r and G and M are unchanging, then v² ∝ 1/r. This basically means the graph should look like a reciprocal curve, which it does.

But this isn’t the complete story. We want to know what happens when the semi-major axis is near 0. Of course, this isn’t possible with the solar system; planets can’t exist inside the Sun, but let’s assume it can.

If a planet were orbiting inside the Sun, the only mass of the sun (M) we need to worry about is the mass contained within the planet’s orbit. So if r is decreasing, M is decreasing by a larger magnitude, resulting in the velocity decreasing.

If the planet is in the very centre (r = 0), the velocity is 0, so our curve looks more like this:

jeb.png

Although this doesn’t exactly work for planets orbiting stars, it does work for stars in a galaxy.

The man who started it all

A Swiss astronomer, Fritz Zwicky, observed galaxies within Coma Berenices: a constellation rich in spiral, elliptical, and low-surface-brightness galaxies and most importantly, galaxy superclusters!

He noticed something odd about the speed at which outer stars were moving around the galaxy: they were moving incredibly quickly!

galrot_anim.gif
What we expected vs what we saw. taken from ircamera.as.arizona.edu/Astr2016/movies/garot_anim.gif

Wait, how do we know they’re orbiting too fast?

Remember the curve we found before? If we were to observe some stars in a galaxy, each increasingly further away from the centre, we expect that their velocities will also follow the curve i.e stars really close are moving slowly, shortly followed by much faster stars and then a drop in velocity as we get to the furthermost stars.

This isn’t what Zwicky observed.

He saw that the stars’ velocities followed a similar sharp increase as we move away from the centre, but beyond that their velocities were still increasing.

This is what both the observed and expected velocity curves look like:

Rotation_curve_of_spiral_galaxy_Messier_33_(Triangulum).png
Created by Mario de Leo

It turns out these outer stars are zooming around the galaxy! How weird!

Zwicky wasn’t the only astronomer to notice this; just about all galaxies also seem to have incredibly speedy stars, which baffled astronomers for ages.

How can this be?

With such high speeds, these stars should be flying out the galaxy, but they’re not.

Now, the problem may just be that out understanding of gravity is inaccurate. For centuries we believed that the further an object is, the less gravitational force it experiences. This idea could be wrong, which isn’t a stretch since we still don’t know much about the fundamental forces that govern physics, especially gravity.

Or, maybe our understanding of gravity is correct, and we’re just missing a piece of information.

When we looked at the model of a planet inside the sun, we said that the only mass (M) we need to worry about is the sun’s mass contained within the planet’s orbit. We can apply the same concept to a star in a galaxy; M will be the mass of the entire galaxy within the star’s orbit.

We also noted that the star’s velocity is governed by v² = GM/r.

G is a constant, so we don’t need to worry about it; it can’t change. v is what we’re measuring, so to try and find v we need to play around with M and r.

If we increase r, we’ll just be moving the star further away, which decreases v. That’s no good!

However, what if we increased M as we increased r? An increase in M will definitely increase v, so by deduction what’s probably happening is that the mass of the galaxy is increasing a lot as we get further out.

Isn’t that obvious? There are more stars as we get further away.

Yes, that is true. But look at where the bulk of most galaxies lie. In the centre, and don’t forget that many have black holes as the centre too (black holes are extremely massive!). Take the Milky way and Andromeda for example (pictures below). Most stars seem to be near the centre.

So to obtain such high results for v the further out we go, the mass needs to be increasing much more dramatically than what we’re observing. But that’s not what we’re seeing.

Perhaps there’s some missing galaxy mass that we can’t see!

An thus, the idea of dark matter was created!

 

 

 

Milky way (Image credit: NASA) & Andromeda (Image credit: Ivan Bok)

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