Astronomy & Celestial Mechanics
the sky, computed
Concept

Tidal radius — where it appears

The distance from a massive body at which its tide across a smaller one exceeds that body's own self-gravity. For a star at a black hole it grows as the cube root of the hole's mass, and for a cluster in a galaxy it sets where escaping stars leave.

Named by 6 essays across 2 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.

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.

gravitation · Two-body relaxation Two lengths that cross at 1.1·10⁸ solar masses, above which nothing is seen. The radius at which a star of 1 solar radius and 1 solar mass is pulled apart by a black hole, and the hole's own horizon, both against the hole's mass and both on logarithmic axes. The tidal radius is the star's own radius times the cube root of the mass ratio, so it climbs with a slope of one third; the horizon is proportional to the mass, so it climbs with a slope of one. Two lines of different slope cross once, and this pair crosses at 1.14·10⁸ solar masses. Below that the star is torn apart outside the horizon, half of it is thrown out and half falls back, and the fallback is visible for months. Above it the star crosses the horizon while it is still a star, is swallowed whole, and produces no flare at all. The consequence is the reason these events are worth watching: a flare that is seen is an upper limit on the mass of the hole that made it, obtained without resolving anything, and it is the only such limit available for a hole that is not currently accreting.

A flare that puts a ceiling on a mass

A star torn apart by a black hole lights up for a year. The tidal radius grows as the cube root of the hole's mass and the horizon grows as the mass itself, so above about a hundred million suns the star is swallowed whole and nothing is seen — which makes the existence of a flare a measurement.

galaxies · Tidal disruption 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 A gap's depth is one number: 2GmT ÷ v b². The central density of the gap against impact parameter, 4 billion years after the encounter, for subhaloes of 10⁶, 10⁷, 10⁸ solar masses. Points are read off the drawn density profiles; the curves are 1/(1 + 2GmT/v b²), the stretch the map applies at the encounter point, and the two agree to 3.0 per cent wherever the histogram has enough stars left in the gap to measure a depth at all. Everything about the encounter enters the depth through that one combination. The consequence is the horizontal reading: a gap of a given depth is produced by every point along a locus on which the mass rises as the square of the impact parameter — half depth at 0.42, 1.33, 4.19 kiloparsecs for the three masses drawn, an exponent of 0.500 against the half the algebra requires. What breaks that particular degeneracy is the gap's width, which scales as the impact parameter itself while the depth does not: rescale s by b and the map is identical, so the profile is one shape stretched. Depth and width together give the impact parameter and the product mT. They do not give the mass.

A hole that says mass times time

A gap in a stellar stream is the strongest evidence available that dark subhaloes exist, and its depth depends on the perturber's mass, the impact parameter and the elapsed time only through one combination. Two of those three are unobservable, so a gap is a measurement of a product.

galaxies · Stellar streams The horizon and the orbit that cannot come back, against the hole's spin. Two radii round a rotating black hole, in gravitational radii GM/c², against its spin a from −1 (an orbit against the rotation) to +1 (with it), for orbits in the equatorial plane. The lower curve is the horizon, 1 + √(1 − a²), which is the same whichever way the orbit goes. The upper one is the marginally bound orbit, 2 − a + 2√(1 − a): the closest a body falling in from far away can pass and still escape back out. For a hole with no spin the horizon is at 2 and the marginally bound orbit at 4 — twice as far out. With maximal spin the marginally bound orbit comes in to 1.09 for a prograde orbit (a = 0.998) and moves out to 5.83 for a retrograde one. A star whose tidal radius lies inside this curve is swallowed whole, so the curve, not the horizon, is the line a disruption flare is measured against — and it moves by a factor of 5.3 with the spin.

The line a star is swallowed at is not the horizon

A star torn apart by a black hole makes a flare, and a star swallowed whole makes nothing, so the heaviest hole that can produce a flare is a measurement. That ceiling is set not at the horizon but at the closest orbit from which infalling matter can still come back out — twice as far out for a hole that does not spin, and moved by a factor of five by the hole's rotation.

galaxies · Tidal disruption Supply and demand in a cluster's core. The rate at which hard binaries heat a star cluster's core, against the core's central density, for binary fractions of 1%, 5%, 20%, both on logarithmic axes and in units that put the balance for five per cent at one. The heating rate per binary does not depend on how hard the binary is — the energy released per encounter is proportional to its binding, and the chance of an encounter to its size, so the two cancel — and summed over a core whose radius shrinks as its density rises, the total grows as the square root of the density. The horizontal line is the demand: the energy the rest of the cluster carries away by relaxation through its half-mass radius, which the core has no say in. Each supply curve meets it at one density: 25 for 1%, 1.0 for 5%, 0.063 for 20%. The core contracts or expands until it produces exactly what is demanded, and a core rich in binaries does so at a lower density.

A core that makes what the cluster asks for

After core collapse, a star cluster's binaries heat it — but how much they heat it is not theirs to decide. The rest of the cluster carries energy outward at a rate fixed by its own size and mass, and the core contracts or swells until its binaries supply exactly that. So the core's structure follows from the halo's demand, and the cluster expands at a rate that forgets how it was born.

gravitation · Binary heating

Named alongside it

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

Binary heatingCore collapseEvent horizonGlobular clusterGravothermal catastropheHalf-mass radiusNegative heat capacityOrbital energyRelaxation timeSupermassive black holeAccretionAccretion disc

All concepts