Astronomy & Celestial Mechanics
the sky, computed
Concept

Compactness — where it appears

The ratio 2GM/Rc² of a body's gravitational radius to its actual radius. It measures how strongly general relativity matters: millionths for the Sun, about a third for a neutron star and exactly one at a black hole's horizon.

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

A measured mass of 2.08 deletes an equation of state. Mass against radius for three neutron-star equations of state, each a polytrope P = Kρ² integrated through the Tolman–Oppenheimer–Volkoff equation from the centre outwards until the pressure reaches zero. Every sequence rises, turns over and falls; only the rising part is stable, because past the maximum adding mass makes the star smaller and the smaller star cannot hold itself up. The maxima here are 1.60, 2.10, 2.57 solar masses, in the order of increasing stiffness — the same nuclear matter with a slightly harder response to compression supports a heavier star, and nothing else in the calculation changes. The horizontal band is PSR J0740+6620, whose mass of 2.08 ± 0.07 solar masses comes from the Shapiro delay of its own pulses passing its companion, which is a timing measurement and involves no model of the star at all. It sits above the maximum of one of the three, and those are not disfavoured but excluded: an equation of state that cannot hold up two solar masses is wrong, whatever else recommends it. The two lines at the left are exact and no star may cross them — the Schwarzschild radius, and the bound above it inside which the speed of sound in the matter would exceed the speed of light. A caution about the curves themselves: a Γ = 2 polytrope is a stand-in for nuclear matter and runs a kilometre or two large in radius at fixed mass, so read the ordering and the maxima rather than the radii.

A radius that decides what matter can be

Nobody can make matter at four times the density of an atomic nucleus, and no calculation settles what it does there. What can be done is to weigh a neutron star — every candidate description of that matter predicts a heaviest star it could hold up, and a single measured mass above that value deletes the description permanently.

stars · Degeneracy One hot spot, seen through more and more gravity. The X-ray flux from a small hot spot at colatitude 60° on a rotating neutron star seen from inclination 60°, over one rotation, each curve divided by its own mean, for no bending, 2GM/Rc² = 0.20, 2GM/Rc² = 0.34, 2GM/Rc² = 0.48 — the last two are a 1.4 M☉ star of 12 km and a 2.0 M☉ star of 12.4 km. Light bending is computed exactly; Doppler effects are left out. With no bending the spot is behind the star for 39 per cent of the turn and the pulsed fraction is 1.00. As the compactness rises the star's gravity bends light from the far side into view: the dark interval shrinks to 27 per cent, then vanishes, and the pulsed fraction falls to 1.00, 0.99, 0.64. The more compact the star, the more of its own back it shows and the smoother its pulse.

The pulse that shows the back of the star

A hot spot on a spinning neutron star flashes once a turn, and the flash is weaker than it would be on an ordinary star because the star's gravity bends light from its far side into view. How much weaker measures the ratio of mass to radius. But on its own the flash cannot tell a compact star from a tilted one — the two trade exactly — and what separates them is the lean the spin puts into the pulse, which measures the radius directly.

stars · Degeneracy What a deformability is made of. The two factors of the tidal deformability Λ = ⅔ k₂ C⁻⁵ for a 1.4 solar-mass neutron star, each relative to its value at 12 km, on a logarithmic scale, against the star's radius. The Love number k₂ — the same number that measures a moon's response to a planet's tide — rises by a factor of 1.6 between 10 and 14 km; the inverse fifth power of the compactness rises by 5.4. Of the change in Λ across that range, 78 per cent, on a logarithmic scale, comes from the compactness: most of what a deformability measures is how large the star is for its mass. The Love number carries the rest: how centrally concentrated the matter is, which is where the equation of state enters beyond the radius.

A Love number measured at nuclear density

The number that told Cassini there was water under Titan's ice is defined the same way for a neutron star, and in the last tenth of a second before two of them merge it changes the shape of the gravitational wave they emit. What the wave measures is mostly not the Love number itself but the star's size, raised to the sixth power — which makes a faint tidal signal into one of the few direct measurements of how large matter at nuclear density lets a star be.

gravitation · Love numbers

Named alongside it

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

Equation of stateNeutron starThe Shapiro delayTidal deformabilityCausality limitChirp massDegeneracy pressureGeneral relativityGravitational lensingGravitational wavesLove numbersMass radius relation

All concepts