6.1. Dark matter in the Coma cluster¶
We derived the virial theorem in Chapter 5.2 and demonstrated that it is a powerful way to estimate the mass of a gravitating system. Perhaps the most famous and powerful application of the virial theorem ever was its use by Zwicky (1933) to show that the Coma cluster’s high observed velocity dispersion implied an enormous amount of unseen matter. This is now often considered to be the first clear evidence for the existence of dark matter, although as we will see, the original measurement has been significantly modified over time.
Zwicky (1933) found that the line-of-sight velocity dispersion of galaxies in the Coma cluster was \(\sigma_\mathrm{los} \approx 1000\,\mathrm{km\,s}^{-1}\) and used the virial theorem to interpret this measurement. Approximating the galaxy distribution as a homogeneous density sphere, a straightforward application of Equation (5.22) gives \begin{equation}\label{eq-mass-spherical-virialtheorem-selfgravity-continuous-homog} \langle v^2\rangle = \frac{3}{5}\,{GM\over R}\,, \end{equation} where \(M\) and \(R\) at the mass and size of the system and \(\langle v^2\rangle\) is the spatially-averaged velocity dispersion (like in the globular-cluster application in Chapter 5.2, \(\langle v^2\rangle = 3\sigma^2_\mathrm{los}\) assuming an isotropic, non-rotating system with a Gaussian velocity distribution, a reasonable model for the galaxies in the Coma cluster). Zwicky (1933) then calculated the predicted \(\sqrt{\langle v^2\rangle}\) assuming that all of the mass is contained in the \(\approx 800\) Coma galaxies known at the time and assuming an average mass of \(10^9\,M_\odot\) for these galaxies by converting their luminosities to stellar masses using the usual stellar mass-to-light ratio \(M/L \approx 3\) (see Chapter 1.2.1). From the observed size of the cluster, Zwicky (1933) determined \(R \approx 10^{24}\,\mathrm{cm} \approx 3\times 10^5\,\mathrm{pc}\). Putting this all together, we then predict \(\sigma^*_\mathrm{los}= 46\,\mathrm{km\,s}^{-1}\). This is a whopping factor of 20 smaller than the observed \(\sigma_\mathrm{los}\)!
Turning the argument from the previous paragraph around, Zwicky (1933) determined the mass that would be necessary for the virial theorem to be satisfied for \(\sigma_\mathrm{los} = 1000\,\mathrm{km\,s}^{-1}\) and again assuming a homogeneous-density-sphere model for the matter distribution, this is \begin{equation}\label{eq-mass-spherical-coma-predictedmass-zwicky} M \approx 4\times 10^{14}\,M_\odot\,, \end{equation} which is about 500 times larger than the observed stellar mass and, thus, corresponds to a mass-to-light ratio of \(M/L \approx 1500\). Thus, a large amount of unseen matter must be present in the Coma cluster to explain the high observed velocity dispersion.
While Zwicky (1933)’s argument has largely stood the test of time, further research has significantly changed the numbers in these calculations. While the velocity dispersion has proven to be robust, the measurement of the distance \(D\) to the Coma cluster has increased by almost a factor of ten. This is entirely due to Zwicky using the Hubble law—\(\langle v_\mathrm{los}\rangle = H_0\,D\), relating the distance to the the average line-of-sight velocity of galaxies in the cluster—with the then-best estimate of the Hubble constant \(H_0 = 558\,\mathrm{km\,s}^{-1}\,\mathrm{Mpc}^{-1}\). Currently, we know that \(H_0 \approx 70\,\mathrm{km\,s}^{-1}\,\mathrm{Mpc}^{-1}\) with an uncertainty of a few percent (e.g., Riess et al. 2016), so the distance estimated from the average velocity increases by a factor of 8 as does the size \(R\), because it is determined from the angular extent of the cluster. With this increased size, Equation (6.2) becomes \begin{equation}\label{eq-mass-spherical-coma-predictedmass-current} M \approx 3\times 10^{15}\,M_\odot\,. \end{equation} The increased distance also raises the galaxy luminosities and stellar masses, because they scale as \(\propto D^2\), so the mass-to-light ratio actually drops to \(M/L \approx 175\) and the ratio of total-to-stellar mass drops to 60. The second important difference in interpreting the high inferred mass is that X-ray observations demonstrate that the mass of the intracluster gas far exceeds the total cluster stellar mass, with Coma containing about \(2.5\times 10^{14}\,M_\odot\) of hot gas (Hughes 1989) compared to \(\approx 0.5\times 10^{14}\,M_\odot\) of stars (both of these masses derived assuming \(H_0 = 70\,\mathrm{km\,s}^{-1}\,\mathrm{Mpc}^{-1}\)). Thus, while the ratio of total-to-stellar mass is 60, the ratio of total-to-baryonic mass is \(\approx 10\). While this number is far smaller than Zwicky (1933)’s original ratio of 500 between total and stellar mass, it is clear that dark matter dominates the mass of the Coma cluster.
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