The M–σ Relation

An impossible-looking correlation

Take a galaxy with a stellar bulge — an elliptical, or the central swelling of a big spiral like Andromeda. Measure two numbers. First, the mass of the supermassive black hole sitting at its very center, found by watching how stars or gas orbit within a few light-years of it. Second, the velocity dispersion σ of the bulge as a whole: the spread of random stellar speeds across a region thousands of light-years wide, measured from the broadening of absorption lines in the combined starlight. These two numbers describe utterly different scales, separated by a factor of a thousand or more in size. There is no reason they should know about each other.

And yet they do. Plot the logarithm of black-hole mass against the logarithm of σ for every galaxy with a well-measured black hole, and the points fall on a strikingly tight straight line. The slope is steep — black-hole mass scales as σ to roughly the fourth or fifth power, so a galaxy with twice the dispersion hosts a black hole roughly 20 to 30 times heavier. The scatter about that line is under 0.3 dex, less than a factor of two in mass. This is the M–σ relation, and it is the single most important scaling law in the study of how black holes and galaxies grow together.

The reason it matters so much is precisely that it should not exist as a coincidence. The black hole's gravity dominates the stars only out to its sphere of influence, a radius of a few parsecs to a few tens of parsecs. The bulge whose σ we measure is hundreds of times larger. The outer bulge stars feel essentially nothing of the central black hole. For the two to track each other so tightly, black-hole growth and bulge growth must have been wired together by some physical process operating during the galaxy's formation. The leading candidate for that wiring is feedback from the active galactic nucleus (AGN) — the energy a feeding black hole pours back into its host.

How the relation is built

The relation is purely empirical: a fit to measured data. Building it requires two hard measurements per galaxy.

The velocity dispersion σ. The bulge contains millions of stars, each on its own orbit with its own line-of-sight velocity. Their combined spectrum smears every stellar absorption line into a broadened profile. Fit that broadening against a stellar template and the width gives σ, usually quoted within an aperture of about one effective radius R_e and corrected to a common scale. Bulges run from σ ≈ 60 km/s in small systems to σ ≈ 350–400 km/s in the giant ellipticals at cluster centers.

The black-hole mass M_BH. This demands resolving — or at least dynamically constraining — the sphere of influence, the radius r_infl ≈ G M_BH / σ² inside which the black hole dominates the gravitational potential. The gold-standard methods are stellar-dynamical (Schwarzschild orbit) modeling, gas-disk rotation, and water-maser disks like the one in NGC 4258 that pins M_BH to a few percent. For the Milky Way, decades of tracking individual stars orbiting Sagittarius A* give M_BH = 4.3 × 10⁶ M_⊙. In distant active galaxies, reverberation mapping times the light-travel lag between the variable continuum and the broad emission lines to size the broad-line region and weigh the black hole.

Fit a power law to the resulting cloud of points and you get the canonical form. Two widely used 2013 calibrations agree closely:

M_BH / (10⁸ M_⊙) ≈ 0.31 × (σ / 200 km/s)^4.4   (Kormendy & Ho 2013)
M_BH / (10⁸ M_⊙) ≈ 0.31 × (σ / 200 km/s)^5.6   (McConnell & Ma 2013, ellipticals)

The intrinsic scatter is the headline number: about 0.28–0.30 dex in M_BH at fixed σ for classical bulges and ellipticals. No other black-hole scaling relation is that tight, which is why σ — not luminosity, not bulge mass — is the preferred predictor of black-hole mass when a direct dynamical measurement is impossible.

Worked example: predicting two black holes

Use the Kormendy & Ho calibration M_BH/(10⁸ M_⊙) = 0.31 (σ/200)^4.4 to predict a black-hole mass from σ alone, then compare to the measured value.

The Milky Way. The bulge velocity dispersion is about σ ≈ 105 km/s (the Galactic bulge is small). Then

M_BH = 0.31 × 10⁸ × (105 / 200)^4.4  M_⊙
     = 0.31 × 10⁸ × (0.525)^4.4
     = 0.31 × 10⁸ × 0.072
     ≈ 2.2 × 10⁶  M_⊙

The directly measured mass of Sagittarius A* is 4.3 × 10⁶ M_⊙ — within a factor of two of the prediction, comfortably inside the 0.3 dex scatter. A relation with no free parameters per galaxy nails our own black hole to within a factor of two using only the wobble speed of bulge stars.

M87. The giant elliptical at the heart of the Virgo Cluster has σ ≈ 320 km/s. Then

M_BH = 0.31 × 10⁸ × (320 / 200)^4.4  M_⊙
     = 0.31 × 10⁸ × (1.60)^4.4
     = 0.31 × 10⁸ × 7.6
     ≈ 2.4 × 10⁹  M_⊙

The Event Horizon Telescope and stellar-dynamical work put M87*'s mass at 6.5 × 10⁹ M_⊙. The σ-only prediction lands within a factor of three — again broadly consistent, and a reminder that the most massive ellipticals tend to lie a touch above the relation. Notice the dynamic range: a 3× change in σ between the Milky Way and M87 corresponds to a 1000× change in black-hole mass, exactly what a fourth-to-fifth-power law demands.

The same calibration also fixes the bulge-mass fraction. With M_BH ≈ 0.002 M_bulge, a bulge of 10¹¹ M_⊙ should host a black hole near 2 × 10⁸ M_⊙ — so the central black hole carries only about 0.1–0.2% of the stellar mass it grew alongside, a tiny tail wagging a very large dog.

Why the slope is 4 or 5: feedback self-regulation

The steep slope is not arbitrary; it falls out of energy bookkeeping between an accreting black hole and the gas of its host. Two limiting arguments bracket the data.

Energy-driven limit (Silk & Rees 1998) → M_BH ∝ σ⁵. Suppose the black hole grows until the energy it radiates over a dynamical time can unbind the surrounding bulge gas. The black hole's luminosity at the Eddington limit scales as L_Edd ∝ M_BH. The binding energy of the gas in the bulge scales with its mass and depth of potential, ∝ M_gas σ². Tying the gas mass to the potential (M_gas ∝ σ²/G × scale) and balancing the deposited energy against the binding energy yields M_BH ∝ σ⁵. Roughly:

feedback energy  ~  binding energy of bulge gas
   M_BH c² (efficiency)  ~  M_gas σ²
   with M_gas ∝ f_gas σ⁴ / (G²)   ⇒   M_BH ∝ σ⁵

Momentum-driven limit (King 2003) → M_BH ∝ σ⁴. If instead the outflow is driven by radiation pressure on dust and the relevant balance is of momentum rather than energy, the black hole grows until its radiation momentum L/c can drive the gas shell against gravity. Setting the Eddington momentum flux equal to the weight of the swept-up gas in an isothermal potential gives

M_BH ≈ (f_gas κ) / (π G²) × σ⁴
     ≈ 3 × 10⁸ M_⊙ × (σ / 200 km/s)⁴

where κ is the electron-scattering opacity and f_gas the gas fraction. Remarkably, plugging in standard values gives a normalization of a few × 10⁸ M_⊙ at σ = 200 km/s — within a factor of a few of the observed zero point, from first principles and with no fitting. The observed slope of 4–5 sits squarely between the momentum-driven (4) and energy-driven (5) predictions, which is the strongest theoretical reason to believe AGN feedback sets the relation.

Variants and related scaling relations

M–σ is the tightest member of a family of black-hole scaling relations, each with its own slope and scatter:

• M_BH–M_bulge. The black hole tracks the bulge stellar mass at M_BH ≈ 0.002–0.005 M_bulge (0.1–0.5%), but with larger scatter (~0.3–0.5 dex) because decomposing bulge from disk light is hard and merger remnants are messy.
• M_BH–L_bulge. The luminosity version, historically the first noticed (Kormendy & Richstone 1995), now superseded by the mass and σ versions which are less sensitive to the stellar population.
• M_BH–n (Sérsic index) and M_BH–(number of globular clusters), both of which correlate but more loosely; the globular-cluster count is surprisingly tight for ellipticals.
• The fundamental plane of black-hole activity, a separate 3D relation linking radio luminosity, X-ray luminosity, and mass for accreting black holes across nine orders of magnitude in mass.

Different galaxy types populate M–σ differently. Classical bulges and ellipticals — built by mergers — define the canonical relation. Pseudobulges, the disky central components assembled by slow secular processes, show weak or absent correlation, evidence that their black holes did not grow through the merger-driven channel that ties M–σ together.

Outliers and what they teach

Galaxy	Type	σ (km/s)	M_BH (M_⊙)	Position on M–σ
Milky Way	SBb (small bulge)	~ 105	4.3 × 10⁶	slightly high, on relation
NGC 4258	SABbc (maser disk)	~ 115	4.0 × 10⁷	slightly high
M31 (Andromeda)	Sb	~ 160	1.4 × 10⁸	on relation
M87	cD elliptical	~ 320	6.5 × 10⁹	slightly above
NGC 1277	compact lenticular	~ 333	~ 1.7 × 10¹⁰	far above (overmassive)
NGC 4486B	compact elliptical	~ 180	~ 6 × 10⁸	above
Holm 15A	BCG elliptical	~ 350	~ 4 × 10¹⁰	above (dry mergers)
NGC 4395	bulgeless dwarf Seyfert	~ 30	~ 3 × 10⁵	low-mass anchor

The overmassive black holes at the top — NGC 1277, Holm 15A, and the brightest cluster galaxies generally — carry far more than 0.2% of their bulge mass; NGC 1277's black hole was claimed at ~14% of the bulge. These almost certainly grew by repeated dry (gas-poor) mergers that pile up stars and black-hole mass without the gas needed for star formation to keep pace. At the bottom, bulgeless dwarfs like NGC 4395 extend the relation down to M_BH ~ 10⁵ M_⊙, probing whether the co-evolution machinery still operates when there is barely a bulge at all. Neither the high nor the low end falsifies M–σ; they map the boundaries of where merger-driven co-evolution turns on.

Common pitfalls and misconceptions

• "The black hole's gravity sets σ." It does not. The sphere of influence (r_infl ≈ G M_BH/σ², a few to tens of parsecs) is hundreds of times smaller than the bulge. The black hole's mass is negligible compared with the bulge mass it lives in, so it cannot directly govern the bulge-wide dispersion. The correlation is a fossil of shared growth, not a live gravitational effect.
• Confusing σ with rotation speed. σ is the random (pressure-supported) component of stellar motion, not ordered rotation. For disks you must remove the rotational contribution; using a raw line width inflates σ and biases the inferred mass.
• Quoting one slope as definitive. Published slopes range from ~3.7 to ~5.6 depending on the sample (all bulges vs. ellipticals only), the fitting method, and how upper limits are treated. Cite the calibration you used.
• Ignoring resolution bias. If the sphere of influence is unresolved, dynamical models tend to overestimate M_BH, artificially steepening the relation at low mass. Reliable points require r_infl to be resolved.
• Treating pseudobulges and classical bulges alike. They follow different (or no) relations. Lumping them together smears the intrinsic tightness and hides the physics.
• Assuming the relation is fixed in time. Whether M–σ holds, shifts, or scatters more at high redshift is unsettled; AGN selection makes distant black holes look overmassive even if the underlying relation is unchanged.

Observational status and applications

Since the 2000 discovery papers by Ferrarese & Merritt and by Gebhardt and collaborators, the local sample of dynamically measured black holes has grown from a couple of dozen to roughly 100, compiled in datasets like the Kormendy & Ho (2013) and McConnell & Ma (2013) reviews. The relation has held up across that fivefold growth, and its applications now reach well beyond demography:

• Estimating black-hole masses cheaply. Where dynamics are impossible, a single bulge spectrum yields σ and hence M_BH to within ~0.3 dex — the workhorse method for surveying black-hole demographics across thousands of galaxies.
• Calibrating reverberation mapping. The virial factor f that converts broad-line widths into masses is set by requiring reverberation-mapped AGN to lie on the local quiescent M–σ relation.
• The Soltan argument. Comparing the local black-hole mass density (built from M–σ demographics) with the integrated light of quasars confirms that most black-hole mass was assembled by luminous accretion — closing the books on how black holes grew.
• Tuning galaxy-formation simulations. Cosmological models (EAGLE, IllustrisTNG, SIMBA) implement AGN feedback recipes whose strength is tuned so that simulated galaxies reproduce the observed M–σ slope and normalization — making the relation a primary benchmark for subgrid physics.
• Probing co-evolution across cosmic time. JWST is now measuring M_BH and host properties for active galaxies at z > 4, testing whether black holes lead or lag their bulges and whether the relation evolves.

The bottom line is unchanged from the discovery papers: a number measured a few parsecs from a galaxy's center and a number measured across thousands of light-years of bulge are locked together to within a factor of two. Something coupled the growth of the black hole to the growth of the galaxy, and the steep slope points squarely at the feedback the black hole itself supplies.