# An extension of the M-Sigma relation to globular clusters

Title: MBH − $\sigma$ relation between SMBHs and the velocity dispersion of the globular cluster systems
Authors: Raphael Sadoun and Jacques Colin
First Author’s Institution: UPMC, CNRS, Institut d’Astrophysique de Paris

If you look at books written as recently as 1974, you can read lines like this: ‘At the time of writing there is no conclusive observational evidence to support the speculation that black holes are a common type of object in the universe.’  Thirty years later, we have pretty conclusive evidence that black holes are at the centers of most galaxies, and we know even more than that.  In particular, we know that there is a tight correlation between the mass of the central black hole and the velocity dispersion of  the “hot” (energetic) part of the host galaxy (see this paper for details).  What is the velocity dispersion, you ask?  It is just the standard deviation of the velocities about their mean.  Observational studies have shown that the black hole mass scales approximately as the 4th power of the velocity dispersion.  The paper I’m discussing today extends this relationship to parts of the host galaxy more distant from the central black hole: globular clusters.  This is an interesting result because it may shed light on how the $M-\sigma$ relation comes about, something that is not well understood.  I’ll get into that, but first, a bit of background.  What is a black hole?  What is a globular cluster? And what can the extended $M-\sigma$ relation tell us?

• A quick black hole refresher

When a star ceases nuclear burning, it no longer has a source of heat.  Most stars are supported against their own gravity by radiation pressure (pressure from photons), which is proportional to temperature to the fourth power. As the temperature falls because there is no more fusion, the star collapses.  Skipping some important intermediate steps, what happens then is this: the star’s outer layers plummet inwards until the density of the star is such as to be able to once more support it against its own gravity.  But for some stars, this is just not possible: no matter how dense the star becomes, it cannot produce enough pressure to support itself.  This is the case for stars with masses greater than $1.4$ solar masses because they can no longer support themselves by electron degeneracy pressure. Basically, two electrons can’t occupy the same quantum state, so you can only squeeze so many electrons into a given volume before they push back: that is what is meant by electron degeneracy pressure. Instead of stabilizing, these stars continue collapsing into black holes.
So this is the basic black hole formation mechanism for the black holes we do understand.  But these are stellar mass black holes. In contrast, the black holes dealt at the center of galaxies are supermassive black holes (SMBHs), which are much larger than the stellar mass black holes I describe the formation process for above (recall, black hole radius is proportional to black hole mass).  There are still unanswered questions about how these huge black holes form.

Three proposed mechanisms are:

1) They form by collapse of primordial, metal-free (so-called Population III) stars in the early Universe. In this case the details might parallel the formation of stellar mass BH’s as described above.

2) They form by “direct collapse of gas in isolated protogalaxies.”

3) They form by mergers of such protogalaxies.

Further, since an SMBH’s mass is correlated with its host galaxy’s velocity dispersion (which in turn correlates with the mass and other properties of that galaxy), it must grow in a way that is linked with its host galaxy’s growth.  How that happens is the central “mystery” behind the standard $M-\sigma$ relation. Sadoun and Colin’s work is interesting because the fact that a correlation also holds for globular clusters may shed light mechanisms for the standard $M-\sigma$ relation as well.

• And globular clusters?

A globular cluster is just a sphere of tightly gravitationally bound stars that orbits the center of a galaxy.  Globular clusters reside near the edges of galaxies and contain old stars as compared to stars found elsewhere in these galaxies.  There are around 150 globular clusters in the Milky Way, but larger galaxies can have even more: some large galaxies have on the order of 10,000 globular clusters!

• What can the extended M-Sigma relation tell us?

Sadoun and Colin’s paper takes the mass of the central black hole in 12 galaxies and correlates it with the velocity dispersion of the globular clusters.  They find a power law that is, like the standard $M-\sigma$ relation, around 4, but slightly on the lower side of that, with $M\propto \sigma ^{3.78\pm .53}$.  For the galaxies in their sample, they find that the correlation with the globular cluster velocity dispersion is just as good as the “standard” one with the velocity dispersion in the bulge (the inner, “hotter” part of the galaxy).

This plot from the paper shows the “M-sigma relation” for the black hole mass (y-axis) and globular cluster velocity dispersion (x-axis).

They also break up each data set to compute separate correlations for two different types of globular clusters: metal-poor (by metal, we mean anything heavier than helium!), old globular clusters far from the galaxy centers, and younger, metal-rich ones closer in.  They find a stronger correlation of the black hole mass with the velocity dispersion of the younger, closer in clusters than with the velocity dispersion of the older, farther-out ones.  Using this stronger correlation, they actually predict the (currently unknown) masses of the central black holes in five different galaxies.

Using the correlation between black hole mass and young, close globular clusters' velocity dispersions to predict the black hole mass for five galaxies where that is not yet known. From paper.

Even more interestingly, they use the correlation to constrain models of how the $M-\sigma$ relation arises.  The relation is thought to come about via feedback: globular clusters can undergo both tidal disruption (different gravitational forces on different parts of the cluster which can distort or disrupt it entirely) and gravitational shocks if they pass on eccentric (very non-circular) orbits near the central black hole.  These two mechanisms are possible causes of the $M-\sigma$ relation.  Sadoun and Colin’s work, by showing that this relation goes beyond the bulge, can constrain how effective these feedback processes are.

So, this paper is exciting for two reasons:

• it gives us a method to predict black hole masses that we don’t know, and
• it may shed light on how the mysterious M-sigma relation comes about.