Before the M–σ 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 in active galactic nuclei, and the Soltan argument, which computes the cosmological density in black holes needed to explain the quasar light—both gave a mean value of M/Mbulge 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. The mean ratio of black hole mass to bulge mass is now believed to be approximately 1:1000.
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 M≈106 solar masses. The existence of black holes in the mass range 104 - 106 solar masses ("intermediate-mass black holes") 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 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 σ.
Read more about this topic: M-sigma Relation
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