Astronomy & Celestial Mechanics
the sky, computed
Concept

Tidal deformability — where it appears

A dimensionless measure of how much a body's shape responds to a tidal field, equal to two-thirds of its Love number divided by the fifth power of its compactness. For neutron stars it is effectively a radius, read from a merger's waveform.

Named by 2 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 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.

CompactnessEquation of stateCausality limitChirp massDegeneracy pressureGravitational wavesLove numbersMaximum massMultimessengerNeutron starNeutron starsNuclear saturation density

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