Caption: A log-log plot illustrating the M-σ relation (AKA M-sigma relation) for a small sample of galaxies.

Features:

1. The M-σ relation is an empirical relation relating the mass of the supermassive black hole located at the center of galaxies and to the stellar dispersion velocity (which is described below).

   Actually, the number of galaxies which have been proven to have central supermassive black holes is rather small, but the tightness of the M-σ relation itself and fact there always or almost always is a central supermassive black hole when there adequate observations to know has has led to the widely-held hypothesis that nearly all large galaxies do have central supermassive black holes (see Wikipedia: Supermassive black holes Wikipedia: M-σ relation: Importance).

2. The vertical axis is black hole mass in solar masses (M_☉'s).

3. The horizontal axis is the dispersion velocity of the stars making up the galactic bulge in the case of spiral galaxies or lenticular galaxies and making up the overall galaxy in the case elliptical galaxies.

   The dispersion velocity is given the symbol σ (which is the Greek letter sigma).

   The dispersion velocity is a sort of average velocity and is taken over over the region interior to a radius specified in some reasonable way---which yours truly does NOT know for the plot.

4. Dispersion velocity of grouping of stars is related to the virial radius R and and the virial mass M (the mass of the grouping within virial radius of the grouping center) by the order-of-magnitude version of the virial theorem

         GM/R ∼ σ**2 ,

   where gravitational constant G = 6.67408(31)*10**(-11) (MKS units) (see Wikipedia: Virial theorem: Galaxies and cosmology (virial mass and radius)).

   The formula can be made exact if one specifies exactly how one is setting R for a fully specified grouping of stars.

   Exactly, how people assign R and M for determining the M-σ relation is probably a bit various---and yours truly does NOT know any of them at this moment.

5. A recent power-law function fit to the M-σ relation is

         M/(10**8 M_☉) = 1.9*[σ/(200 km/s)]**5.1

   or nearly

         M ∝ σ**5 ,

   as for the line shown in the plot.

6. The tightness of the M-σ relation over many orders of magnitude and its apparent universality shows that there is a strong connection between the galaxy evolution (and therefore large-scale structure evolution) and the evolution of central supermassive black holes.

   There is a theory of this connection (see Wikipedia: M-σ relation: Importance), but a complete understanding is NOT yet attained.

   The theory shows that there is a strong feedback of evolution of central supermassive black holes on galaxy evolution.

   The feedback is in the form of electromagnetic radiation (EMR) and ejected mass from the accretion disk around the black hole. Much of the ejected mass is in the form of bipolar jets which depend on magnetic fields.

7. The M-σ relation and the theory explaining it show that galaxies would NOT be the same without central supermassive black holes.

   So supermassive black holes are NOT only interesting exotic astronomical objects, but vital to the evolution of the observable universe---NOT like those puny stellar-mass black holes which are mostly just fascinating stellar cadavers.

8. The M-σ relation may extend down to intermediate-mass black holes (mass range 100 to 10**6 M_☉) in low-mass galaxies and dwarf galaxies (see Wikipedia: M-σ relation: Importance).

9. The galaxies in the sample for the plot are:
   Circinus, Milky Way, NGC 221 (M32), NGC 224 (Andromeda Galaxy, M31), NGC 4258 (M106), NGC 4374 (M84), NGC 4486 (M87), NGC 5128 (Centauras A),

   where NGC stands for New General Catalogue and M for Messier Catalog.