Astronomy & Celestial Mechanics
the sky, computed
Concept

Globular cluster — where it appears

A gravitationally bound sphere of a hundred thousand to a million old stars, orbiting a galaxy's halo. Its relaxation time is under a Hubble time, which makes it the standard laboratory for mass segregation, evaporation and core collapse.

Named by 9 essays across 3 fields — each of them below, with the objects they name alongside it.

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.

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.

gravitation · Two-body relaxation Three clusters, three ages, one diagram. Isochrones for populations of 100 Myr, 1 Gyr, 12 Gyr, each drawn as the main sequence up to its own turnoff and then the post-main-sequence track of the turnoff mass. The turnoffs are at 5.52 M☉, 2.20 M☉, 0.94 M☉, from t = 10¹⁰ M/L with this file's own mass–luminosity relation. Nothing here is a track along which a star moves. Every point is a different star of a different mass, all the same age, and the bend is simply where the population runs out of stars that have had time to leave. That is why a cluster has an age and a field star does not: the bend needs a population, and one star is not one. The oldest globular clusters sit near the 12 Gyr line, and in the 1990s the same construction gave them ages of 16 to 18 Gyr against a universe measured at 10 — a two-standard-deviation contradiction that was resolved from the distance side, by Hipparcos, and not from this one.

An age read off a bend

A cluster is a population of one age and many masses, so the place where its stars leave the main sequence is a clock. The bend needs a population — which is why a cluster has an age and a single star does not.

stars · The HR diagram A stream that is not the orbit it came from. 456 stars released in pairs from the two saddles of a 10⁵ solar-mass cluster over 4.0 billion years, integrated in a halo whose circular speed is 220 kilometres a second, drawn with the progenitor's own orbit. The cluster runs between 10 and 25 kiloparsecs and the orbit is the thin closed-looking curve; the stars are everything else. The point of the figure is the discrepancy. Stars leaving through the inner saddle are on slightly smaller orbits, turning round at a median of 24.77 kiloparsecs rather than the progenitor's 25.00, and therefore running ahead; stars leaving through the outer saddle reach 25.22 and fall behind. The whole spread is 1.8 per cent of the apocentre, which is the number worth carrying: an offset far too small to see in this drawing builds the entire stream, because it acts for four billion years. The two arms are therefore not merely displaced along the orbit, they are on different orbits, and the track a survey measures is a family of them rather than any single one. Fitting a Galactic potential by demanding that a stream lie along an orbit is wrong by exactly this much, and the size of the error grows with the mass of the progenitor, because the mass is what sets the distance between the two doors.

A stream is not the orbit it came from

A cluster torn apart by a galaxy leaves a thin trail of stars across the sky, and the obvious thing to do with it is fit an orbit. That is wrong by a knowable amount, because stars leave through two doors with a small energy offset and end up on a family of orbits rather than on one.

galaxies · Stellar streams 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.

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.

gravitation · Binary heating 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 Mestel cooling for white dwarfs of 0.4 to 1 solar masses. Luminosity against age for white dwarfs of 0.4, 0.6, 0.8, 1 solar masses, both axes logarithmic. Each line is the whole of the star's later life: there is no nuclear source, so what is radiated is the thermal energy the ions had when the star was made, leaking out through a thin envelope whose opacity is Kramers'. That one sentence gives L ∝ t^(−7/5), and the lines are straight here because a power law is straight on these axes — the slope is the exponent and can be measured off the picture. Two features are worth more than the numbers. The first is how flat the sequence becomes: a factor of ten in luminosity costs a factor of 5.2 in age, so a white dwarf spends most of its existence faint, and any population of them piles up at the bottom. The second is that the heavier tracks lie above the lighter ones. A heavier white dwarf is smaller, so it has more ions to cool and less surface to lose them through, and at a given age it is the brighter object. The picture does not show what happens at the faint end, where the ions crystallise and the heat capacity stops being constant; that is a separate figure, and it is where this law stops being enough.

A clock with no fuel in it

A white dwarf has nothing left to burn, so its brightness is a record of how long it has been cooling. The faintest ones in the Galaxy are not faint because they are small — they are as faint as anything has had time to become, and the place where they stop is a date.

stars · White dwarf cooling Every pair arrives circular. Eccentricity against gravitational-wave frequency for four binaries of 30 and 30 solar masses, each starting at 0.2 astronomical units with an eccentricity of 0.3, 0.7, 0.9, 0.99. Both axes are logarithmic and the tracks run left to right as the orbit shrinks. The curves are Peters' closed solution a(e), and they are checked against Peters' differential equation at three eccentricities on each track rather than against the integral they came from. The ordering is preserved — a pair that starts rounder stays rounder — but by the time the orbit is radiating at 10 hertz, where a ground-based detector begins to hear it, the four eccentricities are 8.8·10⁻⁸, 5.6·10⁻⁷, 3.6·10⁻⁶, 1.5·10⁻⁴. All four are far below anything a detector could measure. That is the figure's whole content and it is a strong statement: radiation reaction removes angular momentum faster, relative to energy, than a circular orbit would need, so eccentricity is destroyed on the way in. A binary observed to be eccentric in band therefore cannot have spent long shrinking quietly, and must have been put on that orbit recently — by a third body, or in the crowded centre of a cluster. The figure assumes the two bodies are points and nothing else acts on them, which is exactly the assumption an eccentric detection would refute.

Every pair arrives circular

Gravitational radiation drains a binary's energy and its angular momentum at rates that do not keep step, and the mismatch destroys eccentricity far faster than it shrinks the orbit. A pair that starts at 0.99 and spirals in from a fifth of an astronomical unit is round to better than a part in a hundred million by the time a detector can hear it.

gravitation · Gravitational waves 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 The kick falls as b⁻² and the heating as b⁻⁴. Two quantities delivered by the same distant encounter, against impact parameter in units of the target's half-mass radius, both logarithmic and both scaled to cross near one. The upper line is the impulse itself, 2Gm/vb, which every star in a bound system receives almost equally — so it moves the system and changes nothing inside it. The lower line is what is left after the common part is subtracted: the difference of the impulse across the system, which is its gradient multiplied by the system's own size, one power of b smaller and squared in the energy. The slopes are −2 and −4 exactly, and they are measured off the drawn curves rather than quoted. What follows is the point. Encounters at impact parameter b arrive at a rate proportional to b db, so summing the impulse over all of them gives ∫b⁻¹ db, which diverges logarithmically and is the origin of the Coulomb logarithm that appears in every treatment of relaxation. Summing the heating gives ∫b⁻³ db, which converges: extending the population from ten half-mass radii out to 300 multiplies the summed impulse by 2.48 and the summed heating by 1.010. Distant encounters diffuse velocities and heat nothing, and a tidal-heating calculation therefore needs no cutoff at large impact parameter, where a relaxation calculation cannot proceed without one.

The part of a kick that heats nothing

Most of the impulse a passing mass delivers to a star cluster is delivered equally to every star in it, so the cluster moves and nothing inside it changes. What heats it is the difference across it — one power of the impact parameter smaller, squared in the energy — and that one distinction decides which encounters matter and which cannot.

gravitation · Impulse approximation

Named alongside it

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

Coulomb logarithmImpulse approximationRelaxation timeTwo-body relaxationCore-collapseCrossing timeDynamical frictionGravitational focusingMass segregationAdiabatic invariantAngular momentumBinary black hole

All concepts