Astronomy & Celestial Mechanics
the sky, computed
Generator

The wake-drag generator

The wake, computed from the streamlines that make it
The wake, computed from the streamlines that make it. Left: 26 streamlines past a point mass, in the mass's own frame, each integrated from far upstream with the same speed and a different impact parameter, and mirrored about the axis. Nothing is drawn to converge — every track is the hyperbola its own impact parameter gives it, and they cross downstream because an attraction focuses. Right: the density that focusing produces at 3.2 focusing radii behind the mass, as (b/y)(db/dy) — the Jacobian of the map from starting radius to arrival radius — which peaks at 15.53 times the background at 0.03 radii off the axis. The overdensity is behind the mass, and that is the entire mechanism: the wake pulls backwards on the body that made it. What the figure cannot show is the steady state, because it has no time in it: a real wake is continuously replenished, and the drag is the sum over an infinite train of these encounters, which is where Chandrasekhar's logarithm comes from.

Left: 26 streamlines past a point mass, in the mass's own frame, each integrated from far upstream with the same speed and a different impact parameter, and mirrored about the axis. Nothing is drawn to converge — every track is the hyperbola its own impact parameter gives it, and they cross downstream because an attraction focuses. Right: the density that focusing produces at 3.2 focusing radii behind the mass, as (b/y)(db/dy) — the Jacobian of the map from starting radius to arrival radius — which peaks at 15.53 times the background at 0.03 radii off the axis. *The overdensity is behind the mass, and that is the entire mechanism*: the wake pulls backwards on the body that made it. What the figure cannot show is the steady state, because it has no time in it: a real wake is continuously replenished, and the drag is the sum over an infinite train of these encounters, which is where Chandrasekhar's logarithm comes from.

10 essays call wake-drag. 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.

The wake, computed from the streamlines that make it. Left: 26 streamlines past a point mass, in the mass's own frame, each integrated from far upstream with the same speed and a different impact parameter, and mirrored about the axis. Nothing is drawn to converge — every track is the hyperbola its own impact parameter gives it, and they cross downstream because an attraction focuses. Right: the density that focusing produces at 3.2 focusing radii behind the mass, as (b/y)(db/dy) — the Jacobian of the map from starting radius to arrival radius — which peaks at 15.53 times the background at 0.03 radii off the axis. The overdensity is behind the mass, and that is the entire mechanism: the wake pulls backwards on the body that made it. What the figure cannot show is the steady state, because it has no time in it: a real wake is continuously replenished, and the drag is the sum over an infinite train of these encounters, which is where Chandrasekhar's logarithm comes from. Gravitation

A drag with nothing to drag against

A massive body moving through a sea of light ones raises an overdensity behind itself and is pulled back by it. The force does not depend on the masses of the background bodies at all — and it is strongest at one particular speed.

 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.

 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

A cluster that boils itself away

A star cluster has no thermostat. Encounters between its members push a few of them above escape speed, the cluster loses them, and losing them makes it contract — which makes it hotter, which makes more of them escape. A self-gravitating system heats up as it loses energy, and the process ends by destroying the system.

 Hard below 8.3 astronomical units, soft above, and nothing settles at the line. The binding energy of a binary of two 0.7 solar-mass stars against its separation, both axes logarithmic, with the mean kinetic energy of a single cluster star at a velocity dispersion of 5 kilometres a second drawn as a level. Where the curve is above the level the binary is bound more tightly than a passing star's motion, and encounters on average take energy out of the field and put it into the pair; where it is below, they do the reverse. The crossing at 8.3 astronomical units is the hard–soft boundary, and it is the entire content of Heggie's law: hard binaries harden and soft binaries soften. The arrows are the direction each side moves, and they point away from the crossing in both directions rather than toward it. That is not a coincidence but a negative heat capacity, the same property that makes a star contract when it radiates: taking energy out of a bound pair moves it closer together and speeds it up, so a hard binary that gives energy to the cluster becomes harder still and gives more. A cluster with binaries in it therefore has a heat source that turns itself up, and the boundary drawn here is a watershed rather than an equilibrium. Gravitation

A pair that heats what is trying to cool

A star cluster has a negative heat capacity, so it cannot reach equilibrium — its core contracts and gets hotter without limit. What stops it is a binary, and which way a binary exchanges energy with the stars around it is settled by one comparison of two energies.

 Dynamical friction is strongest at 0.97 times the dispersion, and weaker either side. The dynamical friction on a body moving through a Maxwellian sea of stars, against its own speed in units of the square root of two times the velocity dispersion. The dashed curve is the fraction of the field moving slower than the body, which is the only part of the field that contributes: a star overtaking from behind pulls the body forward exactly as often as one being overtaken pulls it back, so the fast tail cancels out entirely. That fraction rises from nothing to one. The drag is that fraction divided by the square of the speed, and the quotient of a saturating numerator and a growing denominator has a single maximum, here at 0.97. The consequences run in both directions. A body moving much faster than the stars around it is barely slowed at all, which is why a galaxy passing through a cluster at a thousand kilometres a second does not sink; and a body already at rest with respect to the field feels nothing either, because there is no wake to be behind it. Sinking is therefore fastest in the middle of the process rather than at the start of it. Gravitation

A drag that is strongest in the middle

Dynamical friction is not like air resistance. Only the stars moving slower than the body contribute at all, so the drag rises from nothing, peaks when the body has slowed to about the speed of everything around it, and falls away again — which is why a fast satellite is never captured and a slow one sinks in a hurry.

 Accretion from a medium a body is moving through, against how fast it moves. The rate at which a gravitating body captures gas out of a medium it is ploughing through, divided by the rate it would capture at rest, against its Mach number, both axes logarithmic. The accretion radius is set by where the body's escape speed matches the speed of the gas relative to it, and that relative speed combines the sound speed with the motion in quadrature; the rate carries the square of that radius times the speed, which leaves one plus the square of the Mach number to the power minus three halves. Subsonic motion therefore costs almost nothing — the curve is flat below Mach one third — while supersonic motion costs the inverse cube of the speed, drawn here with a measured logarithmic slope of -2.99. The same focusing produces the wake and the drag: the body pulls a denser column behind it, that column pulls back, and the material closest to the axis is captured. Accretion and dynamical friction are not two processes but two accounts of one, which is why a body that grows by this mechanism is also being slowed by it. Stars

The wake and the meal are one calculation

A body moving through gas gathers what passes inside the radius at which its escape speed matches the flow. That single radius sets both the drag it feels and the rate at which it grows, so accretion and friction are not two processes but two readings of one — and the rate falls as the inverse cube of the speed.

 Four defensible choices, and a factor of 2.9 between them. The Coulomb logarithm against the ratio of the two impact parameters it is cut off at. The curve is a logarithm, so it is flat — a factor of ten in the ratio buys 2.3 — and that is usually offered as the reason not to worry. The four marked conventions are all in current use and all defensible, and they give ln Λ from 3.4 to 9.9. Since the drag force is proportional to ln Λ and not to its logarithm, that is a factor of 2.9 in every sinking time computed from it. The flatness protects the answer from a wrong guess about the ratio; it does not protect it from there being no correct guess, which is the actual situation. Gravitation

A drag computed with a logarithm nobody can pin down

Chandrasekhar's drag formula is exact, derived from first principles, and contains a logarithm of a ratio of two lengths that the derivation does not supply. Every sinking time in astronomy is proportional to that logarithm, and the four conventions in current use differ by a factor of three.

 How fast a disc forgets, by the mass of the lumps it is made of. The two-body relaxation time of a star moving at 30 km/s among perturbers of mass m, against m, for a fixed mass density of 0.015 solar masses per cubic parsec — about the density of molecular gas in the plane near the Sun — with a Coulomb logarithm of 5. The time falls as one over the perturber's mass: the same mass in heavier lumps relaxes a population proportionally faster. The orange point is the stars themselves, 0.04 solar masses per cubic parsec in half-solar-mass pieces: 4.9·10⁹ million years, 3.5·10⁵ times the age of the universe (dashed). Molecular clouds of 5·10⁵ solar masses give 13 billion years as point masses — comparable to the age of the disc. Fewer than a hundred-thousandth of the bodies, carrying less than half the mass, relax the disc 3.8·10⁵ times faster than all the stars together. Gravitation

A few thousand clouds outrun a hundred billion stars

Encounters between stars relax the Galactic disc in a time hundreds of thousands of times the age of the universe, so in practice they do nothing. Put a fraction of the same mass into a few thousand molecular clouds and the time falls to the age of the disc — because relaxation counts the mass of each perturber, not their number. The disc's present thickness is the record of that, and of a bound on anything heavier hiding in the halo.

 Five clocks around the black hole at the centre of the Galaxy. Timescales against distance from a black hole of 4.3·10⁶ solar masses at the centre of a cluster whose stellar mass inside r is 10⁶ (r/1 pc)^1.5 solar masses, made of stars of 1 solar mass. Ordinary two-body relaxation (orange) is slow everywhere, 8.4·10¹⁰ years at 0.01 pc. Orbits that close are nearly Keplerian and keep their orientation until they precess — through the stars' own mass farther out, through relativity close in — and over that coherence time the torques between them add rather than random-walk. That makes scalar resonant relaxation (blue) faster than two-body relaxation inside about 0.57 pc, and vector resonant relaxation (green), which turns orbital planes and is not stopped by precession within a plane, faster still: 3.5·10⁷ years at 0.1 pc, against the 6·10⁶-year age of the young stars there. Coefficients of order one are set to one; the slopes are the content. Gravitation

Torques that add because the orbits hold still

Near a massive black hole every star's orbit is almost a fixed ellipse, repeated thousands of times before it precesses. The torques the other stars exert on it therefore point the same way orbit after orbit, and they add in proportion to time instead of cancelling in a random walk — so angular momentum near the hole relaxes hundreds of times faster than ordinary encounters allow, and orbital planes faster still.

 A cluster sinking by Chandrasekhar's formula alone. The orbital radius of a 2.0 × 10⁵ solar-mass globular cluster released on a circular orbit at 1 kpc in the two Fornax halo models, decaying under Chandrasekhar's dynamical friction with a velocity dispersion of 11.7 km/s and a Coulomb logarithm of 3. In the cusp it reaches the centre in 6.7 billion years. In the core the formula gives no arrival within 12 billion years, crossing into the core at 2.8. The formula already slows the cluster inside the core, because the circular speed there drops below the stars' velocity dispersion and few stars move slower than the cluster; but it still lets the cluster creep inward without limit. N-body experiments of clusters sinking into cored haloes find something stronger: the decay stops altogether near the core, which the local formula cannot produce at any setting of its logarithm. Gravitation

A drag that stops where every orbit keeps time

Chandrasekhar's formula reads the density and the velocities where a sinking body is, and nothing about the orbits those stars are on. Inside a constant-density core every orbit has the same frequency, the differential motion that drags a body inward disappears, and the sinking stops — which is why the five globular clusters of the Fornax dwarf galaxy are still where they are.

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