Astronomy & Celestial Mechanics
the sky, computed
Concept

Loss cone — where it appears

The range of pitch angles for which a trapped particle's mirror point would lie inside a planet's atmosphere, so that it is absorbed rather than reflected. Its width falls with distance from the planet, and scattering particles into it is what empties a radiation belt.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

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 Three motions, three decades apart, and that is the point. The three periods of a trapped 1-MeV electron's motion against the shell it is trapped on, on a logarithmic time axis. It gyrates about a field line in 0.22 milliseconds, bounces between its mirror points in 0.33 seconds, and drifts right round the planet in 16 minutes — ratios of 1498 and 3015 at L = 4. Each motion carries a conserved quantity: the magnetic moment, the longitudinal invariant, and the magnetic flux the drift shell encloses. The separation is what makes them conserved. An invariant survives anything that changes slowly compared with its own period, so a disturbance lasting minutes destroys the third and leaves the first two untouched — and a particle that keeps its magnetic moment while being moved inward to a stronger field must gain energy. That is not a loophole; it is how the belts are filled.

Three clocks and nothing to fall onto

A charged particle in a dipole field gyrates, bounces and drifts, on timescales a millisecond, a second and a quarter of an hour. The three periods are three decades apart, and that separation is not a curiosity — it is the reason each motion has a conserved quantity, and the reason a magnetic storm can accelerate particles rather than merely stir them.

spaceflight · Radiation belts 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

Named alongside it

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

Two-body relaxationAdiabatic invariantAngular momentumBahcall wolf cuspBremsstrahlungCore scouringCoulomb logarithmDisruption rateDrift shellDynamical frictionFinal parsec problemGalaxy merger

All concepts