Astronomy & Celestial Mechanics
the sky, computed
Concept

Gravitational collapse — where it appears

The inward fall of a body whose pressure can no longer support it against its own gravity, proceeding on roughly a free-fall time. It forms stars from clouds and, at the end of a massive star's life, neutron stars and black holes from its core.

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

A fall into the Sun from a standing start. The distance of a body released at rest 1 AU from the Sun, as a fraction of its starting distance, against time as a fraction of the fall's total duration, 64.6 days: the exact radial orbit (solid) and a fall at the constant acceleration felt at the start (dashed). The exact fall begins exactly like the constant one and finishes far faster, because the pull grows as the inverse square of the shrinking distance: it reaches the Sun at 79 per cent of the time the constant pull would take. The curve is a cycloid, and the total time is a fifth of the circular orbital period at the starting distance — exactly 1/(4√2) of it — because a straight fall is half of an ellipse whose semi-major axis is half the starting distance.

A fall is half an orbit with no width

Stop a planet dead and it falls into the Sun, in a time that is not a quarter of its year or a half but exactly one over four times the square root of two. The fall is an ellipse squeezed to a line, it spends four-fifths of its duration in the outer half of its path, and in a uniform cloud every shell of it arrives at the centre at the same moment — which is where the free-fall time of star formation comes from.

orbits · Conic sections The centre goes soft at 19 solar masses; the whole core only at 27. Two criteria for the pair instability, against the mass of the oxygen core, each the deepest value reached as the core contracts through the pair-producing range. The orange curve is local: the lowest Γ₁ − 4/3 at the core's centre. It goes negative at 19.1 solar masses — the centre of every such core passes through gas that, taken by itself, cannot hold itself up. The blue curve is the criterion that actually decides the matter, the pressure-weighted mean over the whole core, and it stays positive until 27.5 solar masses of oxygen. In the shaded band between the two the centre goes soft and the core does not go unstable, because the stiffer gas around the centre outvotes it. The threshold is computed for a core built as an n = 3 polytrope with the pairs ignored in its structure; detailed evolutionary models find the first pair-instability pulses in stars whose helium cores exceed roughly thirty solar masses, so the construction lands in the right neighbourhood rather than on a precise number.

A soft centre the rest of the core outvotes

Above a billion kelvin, light in a star's core starts turning into electron–positron pairs, and gas that spends its heat on making mass cannot push back hard enough to hold itself up. But stability belongs to the whole core, not to its centre. An oxygen core goes soft in the middle from about nineteen solar masses and holds; it gives way only near twenty-eight, when the soft gas carries two-thirds of the pressure-weighted volume.

stars · Polytropes

Named alongside it

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

Adiabatic indexAngular momentumCycloidDegeneracy pressureDynamical instabilityEccentricityFree-fall timeKepler equationKepler's third lawPair productionPolytropeRadial orbit

All concepts