Astronomy & Celestial Mechanics
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The m-sigma generator

The M–σ relation, over three decades of black-hole mass
The M–σ relation, over three decades of black-hole mass. 40 galaxies drawn from log M = 8.12 + 4.24 log(σ/200) with 0.31 dex of intrinsic scatter, the slope then fitted back off the drawn points at 4.19 with 0.35 dex of residual. What makes the relation remarkable is the scale mismatch: the black hole is about a thousandth of the bulge's mass, and the radius inside which its gravity dominates the bulge's own is a few parsecs against several kiloparsecs. Almost every star whose speed contributes to σ has never been anywhere near it, and cannot have been influenced by it. The correlation is therefore a statement about how the two grew, not about how they pull on each other.

40 galaxies drawn from log M = 8.12 + 4.24 log(σ/200) with 0.31 dex of intrinsic scatter, the slope then fitted back off the drawn points at 4.19 with 0.35 dex of residual. What makes the relation remarkable is the scale mismatch: the black hole is about a thousandth of the bulge's mass, and the radius inside which its gravity dominates the bulge's own is a few parsecs against several kiloparsecs. Almost every star whose speed contributes to σ has never been anywhere near it, and cannot have been influenced by it. The correlation is therefore a statement about how the two grew, not about how they pull on each other.

2 essays call m-sigma. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Every figure listed here is the same construction drawn at different numbers, so a correction to one is a correction to all of them.

7 decades of relaxation time, and a Hubble time between them. Crossing time and two-body relaxation time for 5 self-gravitating systems, against the number of bodies in each. The lower marks are the crossing time R/σ, which spans 3 decades; the upper ones are t_relax ≈ 0.1N/ln N times it, which spans 8. The horizontal rule is a Hubble time, and the same expression puts these systems on opposite sides of it. Two of them relax — an open cluster and a globular cluster — so their stars have exchanged energy, their heavy stars have sunk, and their present structure is not the one they were born with. The rest do not: a dwarf spheroidal, an elliptical galaxy, a cluster of galaxies, the elliptical missing it by a factor of 10⁶, which is why a galaxy is modelled as a smooth potential with no stars in it at all. The dependence is on N and hardly at all on anything else, because the Coulomb logarithm makes every decade of impact parameter contribute equally. A cluster that relaxes also evaporates: about a stellar mass in a hundred and forty leaves per relaxation time, so the globular cluster empties in 6.1·10¹¹ years. Gravitation

How long a system takes to forget

A star crossing a galaxy is deflected by every other star, and the deflections add as a random walk. The time for that walk to change a star's energy by its own amount is a hundred million billion years for a galaxy and a billion for a globular cluster, and everything about how the two are modelled follows from which side of a Hubble time that falls on.

 S2's orbit, and the mass it implies. The orbit of S2 about the centre of the Milky Way, drawn from its measured elements: a semi-major axis of 0.1251 arcseconds, which at 8.28 kpc is 1036 AU, an eccentricity of 0.8843, and a period of 16.05 years watched round twice. Kepler's third law in solar units — a³/P² — gives 4.31 million solar masses, by exactly the calculation that weighs a planetary system. At periapsis the star is 120 AU from the focus, 1407 Schwarzschild radii, and the mean density inside that radius is 5.3e+15 M☉ per cubic parsec — a million times the densest star cluster known. That density, not the mass, is the argument: there is no configuration of stars that would fit. Galaxies

The mass at the centre that is not stars

One star at the centre of the Milky Way has been watched round a complete orbit. Its semi-major axis and its period give four million solar masses by Kepler's third law — and the volume that mass occupies is small enough to rule out every alternative to a black hole.

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