The M–sigma (or ''M''–''σ'') relation is an empirical correlation between the stellar
velocity dispersion
In astronomy, the velocity dispersion (''σ'') is the statistical dispersion of velocity, velocities about the Arithmetic mean, mean velocity for a group of astronomical objects, such as an open cluster, globular cluster, galaxy, galaxy cluster, or ...
''σ'' of a
galaxy
A galaxy is a system of stars, stellar remnants, interstellar gas, dust, dark matter, bound together by gravity. The word is derived from the Greek ' (), literally 'milky', a reference to the Milky Way galaxy that contains the Solar System. ...
bulge
__NOTOC__
Bulge may refer to:
Astronomy and geography
*Bulge (astronomy), a tightly packed group of stars at the center of a spiral galaxy
*Equatorial bulge, a bulge around the equator of a planet due to rotation
* Tharsis bulge, vast volcanic pl ...
and the mass M of the
supermassive black hole
A supermassive black hole (SMBH or sometimes SBH) is the largest type of black hole, with its mass being on the order of hundreds of thousands, or millions to billions of times the mass of the Sun (). Black holes are a class of astronomical obj ...
at its center.
The ''M''–''σ'' relation was first presented in 1999 during a conference at the
Institut d'astrophysique de Paris The Institut d'Astrophysique de Paris (translated: Paris Institute of Astrophysics) is a research institute in Paris, France. The Institute is part of the Sorbonne University and is associated with the CNRS Centre national de la recherche scientifiq ...
in
France
France (), officially the French Republic ( ), is a country primarily located in Western Europe. It also comprises of Overseas France, overseas regions and territories in the Americas and the Atlantic Ocean, Atlantic, Pacific Ocean, Pac ...
. The proposed form of the relation, which was called the "Faber–Jackson law for black holes", was
:
where
is the
solar mass
The solar mass () is a standard unit of mass in astronomy, equal to approximately . It is often used to indicate the masses of other stars, as well as stellar clusters, nebulae, galaxies and black holes. It is approximately equal to the mass ...
. Publication of the relation in a refereed journal, by two groups, took place the following
year
A year or annus is the orbital period of a planetary body, for example, the Earth, moving in its orbit around the Sun. Due to the Earth's axial tilt, the course of a year sees the passing of the seasons, marked by change in weather, the hou ...
.
One of many recent studies, based on the growing sample of published black hole masses in nearby galaxies, gives
[McConnell, N. J. et al. (2011)]
Two ten-billion-solar-mass black holes at the centres of giant elliptical galaxies
''Nature'', 480, 215–218
:
Earlier work demonstrated a relationship between galaxy luminosity and black hole mass,
[
] which nowadays has a comparable level of scatter. The ''M''–''σ'' relation is generally interpreted as implying some source of mechanical
feedback
Feedback occurs when outputs of a system are routed back as inputs as part of a chain of cause-and-effect that forms a circuit or loop. The system can then be said to ''feed back'' into itself. The notion of cause-and-effect has to be handled ...
between the growth of supermassive black holes and the growth of galaxy bulges, although the source of this feedback is still uncertain.
Discovery of the ''M''–''σ'' relation was taken by many astronomers to imply that supermassive black holes are fundamental components of galaxies. Prior to about 2000, 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.
Origin
The tightness of the ''M''–''σ'' 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 mergers can occur when two (or more) galaxies collide. They are the most violent type of galaxy interaction. The gravitational interactions between galaxies and the friction between the gas and dust have major effects on the galaxies i ...
s and
gas accretion that might be expected to increase the scatter over time.
One such mechanism was suggested by
Joseph Silk
Joseph Ivor Silk FRS (born 3 December 1942) is a British-American astrophysicist. He was the Savilian Chair of Astronomy at the University of Oxford from 1999 to September 2011.
He is an Emeritus Fellow of New College, Oxford and a Fellow ...
and
Martin Rees
Martin John Rees, Baron Rees of Ludlow One or more of the preceding sentences incorporates text from the royalsociety.org website where: (born 23 June 1942) is a British cosmologist and astrophysicist. He is the fifteenth Astronomer Royal, ...
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''–''σ'' relation of , which is approximately correct. However, the predicted normalization of the relation is too small by about a factor of one thousand. The reason is that there is far more energy released in the formation of a supermassive black hole than is needed to completely unbind the stellar bulge.
A more successful feedback model was first presented by
Andrew King at the
University of Leicester
, mottoeng = So that they may have life
, established =
, type = public research university
, endowment = £20.0 million
, budget = £326 million
, chancellor = David Willetts
, vice_chancellor = Nishan Canagarajah
, head_labe ...
in 2003.
In King's model, feedback occurs through momentum transfer, rather than energy transfer as in the case of Silk & Rees's model. A "momentum-driven flow" is one in which the gas cooling time is so short that essentially all the energy in the flow is in the form of bulk motion. In such a flow, most of the energy released by the black hole is lost to radiation, and only a few percent is left to affect the gas mechanically. King's model predicts a slope of for the ''M''–''σ'' relation, and the normalization is exactly correct; it is roughly a factor times larger than in Silk & Rees's relation.
Importance
Before the ''M''–''σ'' relation was discovered in 2000, a large discrepancy existed between black hole masses derived using three techniques.
[Merritt, D. and Ferrarese, L. (2001), Relationship of Black Holes to Bulge]
/ref>
''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 (or Echo mapping) is an astrophysical technique for measuring the structure of the broad-line region (BLR) around a supermassive black hole at the center of an active galaxy, and thus estimating the hole's mass. It is consid ...
in active galactic nuclei
An active galactic nucleus (AGN) is a compact region at the center of a galaxy that has a much-higher-than-normal luminosity over at least some portion of the electromagnetic spectrum with characteristics indicating that the luminosity is not prod ...
, and the Sołtan argument The Sołtan argument is an astrophysical theory outlined in 1982 by Polish astronomer . 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 "d ...
, which computes the cosmological density in black holes needed to explain the quasar
A quasar is an extremely Luminosity, luminous active galactic nucleus (AGN). It is pronounced , and sometimes known as a quasi-stellar object, abbreviated QSO. This emission from a galaxy nucleus is powered by a supermassive black hole with a m ...
light—both gave a mean value of ''M''/''M''bulge that was a factor ≈10 smaller than implied by the Magorrian relation. The ''M''–''σ'' 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
In the field of international relations, a sphere of influence (SOI) is a spatial region or concept division over which a state or organization has a level of cultural, economic, military or political exclusivity.
While there may be a formal al ...
. The mean ratio of black hole mass to bulge mass in big early-type galaxies is now believed to be approximately , and increasingly smaller as one moves to less massive galaxies.
A common use of the ''M''–''σ'' relation is to estimate black hole masses in distant galaxies using the easily measured quantity σ. Black hole masses in thousands of galaxies have been estimated in this way. The ''M''–''σ'' 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''–''σ'' 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''–''σ'' 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 .[ The existence of black holes in the mass range ("]intermediate-mass black hole
An intermediate-mass black hole (IMBH) is a class of black hole with mass in the range 102–105 solar masses: significantly more than stellar black holes but less than the 105–109 solar mass supermassive black holes. Several IMBH candidate obje ...
s") is predicted by the ''M''–''σ'' relation in low-mass galaxies, and the existence of intermediate-mass black holes has been reasonably well established in a number of galaxies that contain active galactic nuclei
An active galactic nucleus (AGN) is a compact region at the center of a galaxy that has a much-higher-than-normal luminosity over at least some portion of the electromagnetic spectrum with characteristics indicating that the luminosity is not prod ...
, although the values of ''M''bh in these galaxies are very uncertain.
No clear evidence has been found for ultra-massive black holes with masses above , although this may be an expected consequence of the observed upper limit to ''σ''.[ Batcheldor, D. et al. (2007)]
How Special Are Brightest Cluster Galaxies?
''The Astrophysical Journal'', 663, L85–L88
See also
*Faber–Jackson relation
The Faber–Jackson relation provided the first empirical power-law relation between the luminosity L and the central stellar velocity dispersion \sigma of elliptical galaxy, and was presented by the astronomers Sandra M. Faber and Robert Ear ...
*Galaxy formation and evolution
The study of galaxy formation and evolution is concerned with the processes that formed a heterogeneous universe from a homogeneous beginning, the formation of the first galaxies, the way galaxies change over time, and the processes that have ge ...
*Sigma-D relation The Sigma-D relation, or Σ-D Relation, is the radio surface brightness to diameter relation of a supernova remnant.Urošević, D. et al. (2009Sigma-D relation for supernova remnants and its dependent on the density of the interstellar medium ''Ast ...
References
{{DEFAULTSORT:M-sigma relation
Astrophysics
Galaxies
Supermassive black holes
Equations of astronomy
Unsolved problems in astronomy