M-sigma relation
The M-sigma relation is an empirical correlation between the stellar velocity dispersion
Velocity dispersion
In astronomy, the velocity dispersion σ, is the range of velocities about the mean velocity for a group of objects, such as a cluster of stars about a galaxy...

  of a galaxy
A galaxy is a massive, gravitationally bound system that consists of stars and stellar remnants, an interstellar medium of gas and dust, and an important but poorly understood component tentatively dubbed dark matter. The word galaxy is derived from the Greek galaxias , literally "milky", a...

Bulge (astronomy)
In astronomy, a bulge is a tightly packed group of stars within a larger formation. The term almost exclusively refers to the central group of stars found in most spiral galaxies...

 and the mass M of the supermassive black hole
Supermassive black hole
A supermassive black hole is the largest type of black hole in a galaxy, in the order of hundreds of thousands to billions of solar masses. Most, and possibly all galaxies, including the Milky Way, are believed to contain supermassive black holes at their centers.Supermassive black holes have...

the galaxy's center.

The relation can be expressed mathematically as

A recent study, based on a complete sample of published black hole masses in nearby galaxies,

The scatter about the mean relation is comparable to that of other empirical relations between black hole mass and galaxy properties. The small scatter is generally interpreted
to imply some source of mechanical feedback
Feedback describes the situation when output from an event or phenomenon in the past will influence an occurrence or occurrences of the same Feedback describes the situation when output from (or information about the result of) an event or phenomenon in the past will influence an occurrence or...

 between the growth of supermassive black holes and the growth of galaxy bulges, although the source of this feedback is still uncertain.

The M relation was discovered in 2000
2000 in science
The year 2000 in science and technology involved some significant events.-Astronomy and space exploration:* May 4 – A rare conjunction occurs on the New Moon including all seven of the traditional celestial bodies known from ancient times up until 1781 with the discovery of Uranus...

by two independent groups. Earlier work had demonstrated a possible relationship between galaxy luminosity and black hole mass, but that relationship had larger scatter and did not motivate the conclusion that all luminous galaxies are likely to have black holes. Discovery of the M relation produced a sea change in research on black holes in the nuclei of galaxies. Prior to that time the main concern had been the simple detection of black holes, while afterward the interest changed to understanding the role of supermassive black holes as a critical component of galaxies.
This led to the main uses of the relation to estimate black hole masses in galaxies that are too distant for direct mass measurements to be made, and to assay the overall black hole content of the Universe.


The tightness of the M relation suggests that some kind of feedback acts to maintain the connection between black hole mass and stellar velocity dispersion, in spite of processes like galaxy merger
Galaxy merger
Galaxy mergers can occur when two galaxies collide. They are the most violent type of galaxy interaction. Although galaxy mergers do not involve stars or star systems actually colliding, due to the vast distances between stars in most circumstances, the gravitational interactions between galaxies...

s and gas accretion that might be expected to increase the scatter over time.
One such mechanism was suggested by Joseph Silk and Martin Rees in 1998 . These authors proposed a model in which supermassive black holes first form via collapse of giant
gas clouds before most of the bulge mass has turned into stars. The black holes created in this way would then accrete and radiate, driving a wind which acts back on the accretion flow.
The flow would stall if the rate of deposition of mechanical energy into the infalling gas was large enough to unbind the protogalaxy in one crossing time. The Silk and Rees model predicts
a slope for the M relation of , which is somewhat larger than observed, but does predict approximately the correct normalization of the relation.


Before the M relation was discovered in 2000, a large discrepancy existed between black hole masses derived using three techniques.
Direct, or dynamical, measurements based on the motion of stars or gas near the black hole seemed to give masses that averaged ~1% of the bulge mass (the "Magorrian relation"). Two other techniques -- reverberation mapping
Reverberation mapping
Reverberation mapping is to-day a statistical method within astrophysics preferably used in connection with velocity analysis of gaseous discs around supermassive compact objects. The technique is used to measure the size of the broad emission-line region and mass of the alleged central black hole...

 in active galactic nuclei, and the Soltan argument
Soltan argument
The Sołtan argument is an astrophysical theory outlined in 1982 by Polish astronomer Andrzej Sołtan. It maintains that if quasars were powered by accretion onto a supermassive black hole, then such supermassive black holes must exist in our local universe as "dead" quasars.-History:As early as...

, which computes the cosmological density in black holes needed to explain the quasar
A quasi-stellar radio source is a very energetic and distant active galactic nucleus. Quasars are extremely luminous and were first identified as being high redshift sources of electromagnetic energy, including radio waves and visible light, that were point-like, similar to stars, rather than...

 light -- both gave a mean value of M/ that was a factor ~10 smaller than implied by the Magorrian relation. The M relation resolved this discrepancy by showing that most of the direct black hole masses published prior to 2000 were significantly in error, presumably because the data on which they were based were of insufficient quality to resolve the black hole's dynamical sphere of influence
Sphere of influence (astronomy)
The sphere of influence is a region around a supermassive black hole in which the gravitational potential of the black hole dominates the gravitational potential of the host galaxy....

The mean ratio of black hole mass to bulge mass is now believed to be approximately 0.1% , i.e., a bulge of one billion solar masses contains a black hole of approximately one million solar masses.

A common use of the M relation is to estimate black hole masses in distant galaxies using the easily-measured quantity Black hole masses
in thousands of galaxies have been estimated in this way. The M relation is also used to calibrate so-called secondary and tertiary mass estimators, which relate the black hole mass to the strength of emission lines from hot gas in the nucleus or to the velocity dispersion of gas in the bulge .

The tightness of the M relation has led to suggestions that every bulge must contain a supermassive black hole. However, the number of galaxies in which the effect of the black hole's gravity on the motion of stars or gas is unambiguously seen is still quite small. It is unclear whether the lack of black hole detections in many galaxies implies that these galaxies do not contain black holes; or that their masses are significantly below the value implied by the M relation; or that the data are simply too poor to reveal the presence of the black hole .

The smallest supermassive black hole with a well-determined mass has M≈106 solar masses (see figure). The existence of black holes in the mass range solar masses ("intermediate-mass black holes
Intermediate-mass black hole
An Intermediate-mass black hole is a black hole whose mass is significantly more than stellar black holes yet far less than supermassive black holes...

") is predicted by the M relation in low-mass galaxies, and the existence of intermediate mass black holes has been reasonably well established in a number of galaxies which contain active galactic nuclei, although the values of M in these galaxies are highly uncertain .
No clear evidence has been found for ultra-massive black holes with masses above 1010 solar masses, although this may be an expected consequence of the observed upper limit to .
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