Astronomy & Celestial Mechanics
the sky, computed
Concept

Isothermal sphere — where it appears

A mass distribution whose density falls as the inverse square of radius, giving a flat rotation curve. Its potential grows logarithmically and never converges, so an escape speed from it exists only once an outer edge is imposed.

Named by 4 essays across 4 fields — each of them below, with the objects they name alongside it.

Where the stars that are torn apart come from, round a 10⁶ solar-mass hole. The rate at which stars are delivered onto orbits reaching the tidal radius, per logarithmic interval of their orbital radius, for a hole of 10⁶ solar masses at the centre of a nucleus that is isothermal outside the hole's influence and a Bahcall–Wolf cusp within it, with a velocity dispersion of 54 km/s set by the black hole mass–dispersion relation. The radius is in units of the hole's influence radius, GM/σ² = 1.47 pc. Close in, a star's orbit is so short and its angular momentum changes so slowly that the loss cone empties every orbit, and the rate is limited by relaxation diffusing stars into it; far out, the angular momentum wanders across the whole cone in one orbit, the cone is full, and the rate is limited by how many stars there are and how long they take to come in. The flux peaks at 0.27 influence radii, near where the two regimes meet (0.19), and the total is 7.4·10⁻⁵ disruptions a year. The nucleus model is the simplest there is and real nuclei differ from it by factors of several; the shape — a narrow band of radii supplying most of the flares — is what survives.

The stars a black hole eats come from a narrow band

A star is torn apart only if its orbit happens to point almost exactly at the hole — within a millionth of the possible directions of its angular momentum. Stars on those orbits are gone within one orbit, and the only way new ones arrive is by the slow random walk of encounters with other stars. Close to the hole that walk is too slow; far from it the orbits take too long; and most of the stars a hole destroys come from a narrow band between.

galaxies · Tidal disruption The ratio that is √2 only for a point. Escape speed divided by circular speed, against distance in units of the Sun's, for a point mass and for flat-rotation-curve haloes truncated at four different radii. The point-mass value is √2 at every distance and it is a floor rather than a rule. Mass outside an orbit contributes nothing to the circular speed — a spherical shell exerts no force inside itself — and everything to the depth of the well, so any extended distribution has a ratio above √2, by an amount that measures how much lies beyond. For a flat rotation curve the escape speed is not even defined without an outer edge: the potential grows as the logarithm of radius and diverges, so the ratio is √(2 ln(R/r)) with R the truncation radius, and the answer depends on a boundary nobody can see. The measured value near the Sun is about 2.27, which on this curve puts the edge at 13 times the Sun's distance — and the strength of that inference is exactly the weakness of the method, because a ratio of 2.50 would put it at 23.

The ratio that is not √2

Escape speed is exactly √2 times circular speed at every distance from a point mass, and the Sun's own escape speed from the Galaxy is 2.3 times its orbital speed. The excess is not an error in either number. It is a measurement of the mass that lies outside the Sun's orbit, which pulls on nothing and holds everything.

spaceflight · Escape The most pressure a core can supply, and what its envelope asks for. The pressure at the boundary of a star's isothermal core, on a logarithmic scale, against the fraction of the star's mass the core holds. The coloured curves are the most pressure a core of that mass can supply at its edge — the maximum of the isothermal sphere, 1.40 c⁸/(G³M²) for a core of mass M — for cores of mean molecular weight 0.6, 1.0, 4/3; the flat grey line is the pressure the envelope, of molecular weight 0.6, exerts on the core at the same temperature. The core's ceiling falls as the square of its mass, and it can hold the envelope up only while it lies above the line. The curves cross it at core fractions of 0.370, 0.133, 0.075: a core of the same molecular weight as its envelope could hold 37 per cent of the star, and a fully ionised helium core, μ = 4/3, under this envelope only 7.5 per cent. The limit exists because the core is heavier per particle than the gas it has to hold up.

A core too heavy for the gas around it

When a star's centre runs out of hydrogen, the helium left behind stops making energy, settles to one temperature, and becomes the one member of the polytrope family with no surface of its own. It can hold up the envelope around it only while it is less than about a tenth of the star. Past that there is no balance to be had, and the star crosses to the giant branch in a few per cent of its lifetime — which is why that part of the diagram is empty.

stars · Polytropes A cold slab collapsing into a spiral. 800 self-gravitating sheets in phase space — position across, velocity up — at times 0, 3, 8, 120 in units where a sheet released from the edge reaches the centre in about 1.4. They start as a cold rectangle, uniform in position, with 10 per cent of the kinetic energy equilibrium needs. The slab falls in, overshoots, and the sheets, each orbiting at a period set by its own energy, wind into a spiral whose arms grow thinner and closer until they can no longer be told apart. By the last panel the system is in a steady state: a dense core and a halo of sheets that were flung out in the first collapse. Nothing in the dynamics is an encounter; every change of a sheet's energy comes from the potential changing in time.

A relaxation that stops before it is finished

A cold, collapsing system of stars reaches virial equilibrium in a few crossing times without any star meeting another, because the potential changing in time reshuffles their energies. Lynden-Bell's theory predicted the state it should reach — the most probable one. Computed exactly, in a slab of self-gravitating sheets where every force is known, the collapse reaches equilibrium and not that state: the fluctuations that drive the mixing are what the mixing destroys, and they die before the job is done, leaving a profile and an ordering of energies that remember how the collapse began.

gravitation · Virial theorem

Named alongside it

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

Dark matter haloBahcall wolf cuspCollisionless systemColour magnitude diagramCoulomb logarithmDegeneracy pressureDisruption rateEntropyEscape velocityExtrapolationGravitational potentialHigh-velocity star

All concepts