Week 5: WS10 #1

1. In 1933, the astronomer Fritz Zwicky is observing a cluster of gravitationally bound galaxies called the Coma cluster. 
a) First, consider a spherical distribution of matter of uniform density \(\rho\), mass M, and radius R. Show that the potential energy U, is given by \[U=-\frac{3GM^2}{5R}.\]
We start with the general equation for gravitational potential energy: \(U=-\frac{GMm}{r}\).
We know the density of the matter is \(\rho\), so we can figure out the mass: \(M=\rho\frac{4}{3}\pi r^3\).
The mass of the infinitesimal piece of collapsing mass is \(dM=\rho4\pi r^2dr\), which allows us to calculate the infinitesimal amount of potential energy contributed by that piece of mass.
\(dU=-\frac{G(\rho\frac{4}{3}\pi r^3)(\rho4\pi r^2dr)}{r}=-\frac{16G\rho^2\pi^2r^4}{3}dr\)
Finally we just have to integrate.
\(\int^U_0dU=-\frac{16G\rho^2\pi^2}{3}\int^R_0r^4dr\)
\(U=-\frac{3G\rho^2\pi^2}{3}\frac{R^5}{5}\)
If we make the substitution \(\rho=\frac{M}{\frac{4}{3}\pi r^3}\), we can simplify to the final answer: \(U=-\frac{3GM^2}{5R}\).

b) The Coma cluster is approximately 1 million light years across (~10²⁴cm). The Coma cluster consists of some ~800 galaxies, each with mass ~10⁹ solar masses. What is the gravitational potential energy of the cluster? 

\(U=-\frac{3(6.67\times10^{-8}\frac{cm^3}{s^2g})(800\times10^9\times2\times10^{33}g)^2}{5\times0.5\times10^{24}cm}=-4.1\times10^{59}erg\)

c) By the Virial theorem, what should be the typical velocity of a galaxy in this system? 

Using the Virial theorem, we can say that \(\frac{1}{2}mv^2=-\frac{1}{2}U\).
\(\frac{1}{2}Mv^2=\frac{1}{2}\frac{3GM^2}{5R}\)
\(v=(\frac{3GM}{5R})^{1/2}\)
\(v=(\frac{3(6.67\times10^{-8}\frac{cm^3}{s^2g})(800\times10^9\times2\times10^{33}g)}{5\times0.5\times10^{24}cm})^{1/2}=1.1\times10^7cm/s=110km/s\)

d) Fritz Zwicky measured the Doppler shifts of the galaxies in the Coma cluster, and found that the typical velocity exceeds 1000km/s! Knowing what you know of the Virial theorem, and realizing that the mass M we calculated only took account of matter that we can see (galaxies), what is one probable cause of this discrepancy? 

It's likely that this discrepancy is caused by dark matter. If we use the Virial theorem to calculate the velocity but we use a mass that's too low, we'll end up with a velocity that's too low--which is what happened here. If there's a large amount of matter that we can't see, that could explain why our mass and our velocity are too low.