Astronomy & Celestial Mechanics
the sky, computed
Concept

Dynamical friction — where it appears

The drag a massive body feels from the overdensity of light bodies it gathers behind itself, which depends on the background's density and not on what it is made of. It is what sinks a globular cluster or a satellite galaxy to the centre of its host, and it acts even where the stars themselves have never exchanged energy with each other.

Named by 14 essays across 3 fields — each of them below, with the objects they name alongside 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.

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.

gravitation · Dynamical friction A bridge and a tail, integrated. A disc of 180 massless particles on circular orbits, and a companion of 1 times the primary's mass on a parabolic orbit with a pericentre of 1.6 disc radii, integrated from before the encounter to well after it. Times are in units of the disc's own outer orbital period, measured from pericentre. The outermost ring, drawn separately, is the one that produces both the bridge and the tail: at 3 its furthest particle is 31.4 disc radii from the centre, having started at 0.9; the panels are scaled to hold ninety per cent of the particles, so the very end of the tail is outside them. Nothing has been ejected and no material is new — every particle is on the orbit its own initial conditions and the two masses give it.

A bridge and a tail drawn by one force

The long thin streamers coming out of interacting galaxies look like debris thrown off by a collision. They are nothing of the kind — a simulation with no collisions in it, no gas and no self-gravity produces them from the tidal field alone, and only when the encounter runs the same way the disc turns.

galaxies · Galaxy interactions The whole kick, delivered in about two encounter times. The transverse force a body feels while a mass sweeps past it on a straight line, and the velocity that force has delivered so far, both against time in units of the impact parameter divided by the relative speed. The force is the component of the inverse-square attraction perpendicular to the path, which is the impact parameter over the cube of the distance, and it is drawn at its peak value of one at closest approach. The rising curve is its running integral, scaled by twice the gravitational constant times the mass over the impact parameter and the speed. Two things are visible and both are the point. The integral of the force over all time is exactly two in these units, so the kick is exactly 2GM/bv with no free constant anywhere — an answer to a three-body-shaped question obtained without solving anything. And it arrives quickly: 71 per cent of it within a single encounter time of closest approach and 98.6 per cent within the 6 drawn, which is what licenses calling the whole thing an impulse. On the scale of anything slower, the velocity changes discontinuously.

An answer obtained along a path that was not taken

Integrate the force of a passing mass along the straight line the body would have followed if the encounter had not happened, and out comes an exact deflection with no free constant in it. The approximation is circular, it is wrong in a known direction by a known amount, and it is the reason stellar dynamics has closed forms at all.

gravitation · Impulse approximation 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.

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.

gravitation · Dynamical friction 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.

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.

stars · Accretion Prograde and retrograde, the same encounter. The same encounter twice: a companion of 1 times the primary's mass passing at 1.5 disc radii, with the disc spinning the same way the companion orbits and then the other way. Nothing else differs. The prograde disc grows a bridge towards the companion and a tail away from it; the retrograde one is barely disturbed. The reason is a resonance rather than a force: in the prograde case the outer particles keep pace with the companion for a substantial part of the encounter and are pulled the whole time, and in the retrograde case they sweep past it and the impulses cancel.

Whether it merges is a ratio of two times

Two galaxies passing each other either merge or do not, and what decides it is not how close they come. It is whether the encounter lasts long enough for the internal motions to respond — a slow, prograde passage transfers orbital energy into stellar orbits and the pair is bound; a fast one leaves both galaxies heated and still moving.

galaxies · Galaxy interactions A halo is born with a spin of a few hundredths. The distribution of the dimensionless spin parameter λ = J|E|^½ ÷ (G M⁵ᐟ²) across dark matter haloes, drawn as a lognormal of median 0.035 and logarithmic width 0.5, with the disc scale length each λ implies printed along the lower axis. The distribution is required to integrate to one and to peak at 0.0272, which is the median times e raised to minus sigma squared, and is the signature of a lognormal rather than of a bell curve drawn to look like one. λ is small because a halo is supported by random motion rather than by rotation: a value of 0.035 means the halo turns at about three and a half per cent of the rate it would need to hold itself up centrifugally. It is also nearly independent of halo mass, which is what points at a common origin. Mapping it to a disc through R_d = λ R₂₀₀/√2 with all of the specific angular momentum retained, a 10¹²-solar-mass halo of radius 206 kiloparsecs gives 5.1 kiloparsecs at the median, against the 2.6 kiloparsecs the Milky Way's disc actually has. The gap is not a failure of the estimate; it is the measurement that the baryons arrived with less spin per unit mass than the halo they arrived in.

A disc the size its halo was born with

A galaxy's mass says how much light it makes. It does not say how big it is. What sets a disc's size is a single dimensionless number describing how fast the dark halo around it happens to be turning — a number the disc had no part in choosing, distributed the same way for every halo mass in the universe.

galaxies · Galaxy spin The last parsec, priced in stars. The two timescales that shrink a pair of 10⁸-solar-mass black holes at the centre of a merged galaxy, against their separation in parsecs, both axes logarithmic, for a stellar velocity dispersion of 200 km/s and a central density of 500 solar masses per cubic parsec. Ejecting stars hardens the binary at a rate proportional to the separation, so that timescale grows as the orbit shrinks; gravitational radiation goes as the fourth power of the separation, so its timescale collapses. Neither alone finishes the job and the crossing of the two is where the answer is set. With the loss cone kept full, the crossing is at 2.9e-2 parsecs and the total is 4.2·10⁸ years, inside a Hubble time. With the supply of low-angular-momentum orbits emptied by a factor of 100 — which is what happens in a smooth spherical nucleus, because the stars that could interact have already been thrown out and two-body relaxation refills the orbits far too slowly — the crossing moves outward to 7.3e-2 parsecs and the total becomes 1.7·10¹⁰ years. That is the final-parsec problem, and it is not a problem about gravity: it is a problem about supply. Real nuclei are not spherical, and the figure cannot show what a triaxial potential does, which is to keep feeding the binary orbits it has not already used.

The last parsec, and the stars that are not there

Two galaxies merge and their central black holes sink towards each other, and then stop. From about a parsec apart, friction no longer works and gravitational radiation is not yet strong enough — and the only mechanism in between throws away the stars it depends on faster than they can be replaced.

galaxies · Galactic nuclei 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.

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.

gravitation · Dynamical friction A disc galaxy walked down the plane by three dry mergers. Specific angular momentum against stellar mass, both logarithmic. The solid line is the relation disc galaxies follow, j proportional to M to the two-thirds, normalised to 1000 kpc km/s at 10¹¹ solar masses; the dashed line is the spheroids', a fifth as high. The points follow the median galaxy built from discs that start on the line at 2.51·10¹⁰ solar masses by successive equal-mass mergers, each between two galaxies with the same history as each other, adding spin vectors in random directions and keeping none of the orbital angular momentum. Each merger doubles the mass and moves the point right by 0.3 of a decade; the vector sum keeps on average only two thirds of the specific angular momentum while the line rises by the two-thirds power of two, so the point falls below it: to 0.45 of the line after one merger, 0.18 after two and 0.08 after three. Two dry major mergers carry a disc to where spheroids are measured.

Two spins that cancel, and an orbit that leaves

Disc galaxies and ellipticals of the same stellar mass differ by a factor of five in how much angular momentum each gram of their stars carries. Adding spin vectors in random directions explains most of it — two mergers of equals, with the orbit's angular momentum handed to the dark halo, carry a disc down to where ellipticals sit — and whether a merger makes a spheroid or rebuilds a disc turns on one number the vectors do not supply.

galaxies · Galaxy spin The separation of the Milky Way and Andromeda, run forward. The distance between the Milky Way and Andromeda against time from now, in billions of years, integrated from today's separation of 770 kpc and radial approach of 110 km/s, for four values of Andromeda's sideways (tangential) velocity: 0, 17, 57, 100 km/s. The haloes are extended Hernquist spheres of 1.15 and 1.5 trillion solar masses, and dynamical friction drains the orbit at each passage. At 0 km/s the first approach is so nearly head on that the centres pass within 10 kpc and the pair merges at 4.0 Gyr. At 17 km/s the first approach is so nearly head on that the centres pass within 10 kpc and the pair merges at 4.8 Gyr. At 57 km/s the first pericentre is 117 kpc at 4.5 Gyr, and the pair merges at 9.2 Gyr. At 100 km/s the first pericentre is 321 kpc at 4.8 Gyr, and the pair has not merged by 14 Gyr. The measured sideways speed runs from about 17 km/s (Hubble) to about 57 km/s (Gaia), and the outcomes across that range differ by billions of years.

A collision decided by a speed across the sky

Andromeda is coming toward the Milky Way at a hundred and ten kilometres a second, and that part is measured to a few per cent. Whether the two galaxies merge in the next ten billion years depends on how fast Andromeda is also moving sideways — a drift of fifteen millionths of an arcsecond a year, measured twice, with answers of seventeen and fifty-seven kilometres a second. One is a near head-on collision in under five billion years, the other a coin toss.

galaxies · Galaxy interactions How often a merged black hole is kicked faster than a given speed. The fraction of black-hole mergers whose remnant recoils faster than the speed on the horizontal axis, for mass ratios spread evenly in logarithm between 0.1 and 1 and spins of 0.9 of the maximum, spins within 10° of the orbit; spins within 30°; spins in random directions, from a fitting formula to numerical relativity over 4000 random mergers each. The median kicks are 113, 234, 534 km/s. Faster than 100 km/s: 62%, 85%, 96%. Faster than 300 km/s: 5%, 40%, 74%. Faster than 1000 km/s: 0%, 2%, 25%. Gas accreted before the merger is expected to drag both spins into line with the orbit, which is why a gas-rich merger keeps its black hole far more often than a gas-poor one.

A black hole thrown out by its own merger

When two black holes merge, the gravitational waves they radiate carry momentum as well as energy, and unless the two are identical and spinning in line the waves leave more in one direction than the other. The remnant recoils, at anything from a few tens of kilometres a second to several thousand — and whether a galaxy keeps its central black hole comes down to how its spins were pointing in the last orbit before the merger.

galaxies · Galactic nuclei 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.

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.

gravitation · Dynamical friction Velocities that stop pointing along the stream. The angle between the stream's direction on the sky, as seen from the Galactic centre, and the mean motion of its stars, along the stream, for Large Magellanic Cloud masses of 0.0, 0.5, 1.5, 2.5 × 10¹¹ solar masses. In a static halo the misalignment stays at 1° or below — the small offset any stream has from its progenitor's orbit. With the Cloud it grows, peaking at 4°, 13°, 32° for the three masses, on the stretch of stream that passed nearest the Cloud. The misalignment is a measurement: with the halo's own potential known, its size along the stream gives the Cloud's mass.

A stream that points the wrong way

In a halo that holds still, the stars of a tidal stream move along it. The Large Magellanic Cloud, falling into the Milky Way for the first time, pulls on a passing stream directly and drags the Galaxy's own centre towards itself, and the stream is bent sideways faster than its stars' motions can follow. Their velocities stop pointing along the track — and how far they stop is a measurement of the Cloud's mass.

galaxies · Stellar streams

Named alongside it

The objects these essays reach for when they reach for this one.

Galaxy mergerCoulomb logarithmImpulse approximationVelocity dispersionAngular momentumDark matter haloGlobular clusterGravitational focusingTidal tailAdiabatic invariantCross-sectionMass segregation

All concepts