Editing Virial theorem (section)

=== Dark matter ===
In 1933, Fritz Zwicky applied the virial theorem to estimate the mass of [[Coma Cluster]], and discovered a discrepancy of mass of about 450, which he explained as due to "dark matter".<ref name=":1">{{Cite journal |last=Zwicky |first=Fritz |date=1933 |others=Translated by Heinz Andernach |title=The Redshift of Extragalactic Nebulae |url=https://resolver.caltech.edu/CaltechAUTHORS:20190108-092708978 |journal=Helvetica Physica Acta |language=en |volume=6 |pages=110–127 |issn=0018-0238}}</ref> He refined the analysis in 1937, finding a discrepancy of about 500.<ref>{{Cite journal |last=Zwicky |first=F. |date=October 1937 |title=On the Masses of Nebulae and of Clusters of Nebulae |journal=The Astrophysical Journal |volume=86 |pages=217 |doi=10.1086/143864 |bibcode=1937ApJ....86..217Z |issn=0004-637X|doi-access=free }}</ref><ref>{{Cite journal |last1=Bertone |first1=Gianfranco |last2=Hooper |first2=Dan |date=2018-10-15 |title=History of dark matter |url=https://link.aps.org/doi/10.1103/RevModPhys.90.045002 |journal=Reviews of Modern Physics |language=en |volume=90 |issue=4 |page=045002 |doi=10.1103/RevModPhys.90.045002 |arxiv=1605.04909 |bibcode=2018RvMP...90d5002B |s2cid=18596513 |issn=0034-6861}}</ref>

==== Theoretical analysis ====
He approximated the Coma cluster as a spherical "gas" of <math>N</math> stars of roughly equal mass <math>m</math>, which gives <math display="inline">\langle T \rangle= \frac 12 Nm \langle v^2 \rangle</math>. The total gravitational potential energy of the cluster is <math>U = -\sum_{i < j} \frac{Gm^2}{r_{i,j}}</math>, giving <math display="inline">\langle U\rangle = -Gm^2 \sum_{i < j} \langle {1}/{r_{i,j}}\rangle</math>. Assuming the motion of the stars are all the same over a long enough time ([[ergodicity]]), <math display="inline">\langle U\rangle = -\frac{1}2 N^2 Gm^2\langle {1}/{r}\rangle</math>.

Zwicky estimated <math>\langle U\rangle</math> as the gravitational potential of a uniform ball of constant density, giving <math display="inline">\langle U\rangle = -\frac 35 \frac{GN^2m^2}{R}</math>.

So by the virial theorem, the total mass of the cluster is<math display="block">Nm = \frac{5\langle v^2\rangle}{3G\langle \frac{1}{r}\rangle}</math>

==== Data ====
Zwicky<math>_{1933}</math><ref name=":1" /> estimated that there are <math>N = 800</math> galaxies in the cluster, each having observed stellar mass <math>m = 10^9 M_{\odot}</math> (suggested by Hubble), and the cluster has radius <math>R = 10^6 \text{ly}</math>. He also measured the radial velocities of the galaxies by doppler shifts in galactic spectra to be <math>\langle v_r^2\rangle = (1000 \text{km/s})^2</math>. Assuming [[Equipartition theorem|equipartition]] of kinetic energy, <math>\langle v^2\rangle = 3 \langle v_r^2\rangle</math>.

By the virial theorem, the total mass of the cluster should be <math>\frac{5R \langle v_r^2\rangle}{G} \approx 3.6\times 10^{14} M_\odot</math>. However, the observed mass is <math>Nm = 8 \times 10^{11} M_\odot</math>, meaning the total mass is 450 times that of observed mass.