Illustrated Physics
physics, drawn
Concept

Neutron star — where it appears

The collapsed core of a massive star, about twenty kilometres across and denser than an atomic nucleus, held up by degenerate neutrons. Spinning ones seen as pulsars keep time to parts in a million million and set bounds on gravitational waves.

Named by 9 essays across 3 fields — each of them below, with the objects they name alongside it.

The far end that has not been told yet. A rod 3 metres long, pushed at one end at time zero. On the left, position across and time upwards, for the two fastest disturbance speeds drawn here, with the light cone beside them: nothing may lean further to the right than that line, which crosses the rod in 10.0 nanoseconds. A rigid rod would be the vertical dashed line — the far end moving at the same instant as the near one — and it is not a limit that a hard material approaches. It is a signal at infinite speed. On the right, how long the far end actually waits, against how fast the disturbance travels, both logarithmic, with every material on it: 8.8 ms at 3.4e+2 m/s, 600.0 μs at 5.0e+3 m/s, 250.0 μs at 1.2e+4 m/s, 100.0 ns at 3.0e+7 m/s, 20.0 ns at 1.5e+8 m/s. The line has slope −1 and the light cone is a hard floor beneath it. Ordinary materials sit four to five decades above that floor, which is why rigidity is such a good approximation and why it is still not a limit: steel's delay is not small compared with light's, it is 6e+4 times larger. Everything usually derived from rigid bodies survives, because the delay is beneath notice in ordinary circumstances. What does not survive is the use of rigidity in an argument about simultaneity, which is where it does real damage: a rod pushed at one end is compressed for as long as the wave takes to cross it, and there is a frame in which its far end is still at rest while its near end is moving.

Nothing is allowed to be rigid

A rigid body would move its far end at the instant its near end was pushed, which is a signal at infinite speed. Relativity forbids it — not approximately, and not as a limit that a hard enough material approaches. What follows is a ceiling on how stiff matter may be, and that ceiling caps the mass of every neutron star.

relativity · Relativistic dynamics The proton fraction a cold star settles at. The fraction of nucleons that are protons in cold matter in beta equilibrium, where the neutron's chemical potential equals the proton's plus the electron's, against density on logarithmic axes, for free neutrons, protons and electrons with no nuclear forces. Below 1.22·10⁷ g/cm³ the electrons cannot pay the 1.29 MeV difference between a neutron and a proton and the matter is all protons. Above it the fraction collapses, to 0.014 per cent at its lowest near 7.9·10¹¹ g/cm³, and at nuclear density, 2.66·10¹⁴ g/cm³, it is 0.49 per cent. At higher densities it climbs slowly towards one ninth, 11 per cent, which it never reaches. The dashed part lies where real matter is made of nuclei rather than free nucleons.

The protons a star cannot afford

A free neutron decays in fifteen minutes into a proton, an electron and an antineutrino. Inside a neutron star it cannot, and the star is made of neutrons because of it. Beta decay is a reaction like any other, and at equilibrium the neutron's chemical potential must equal the proton's plus the electron's. In a crowded star an electron can only be added at the top of a sea tens of megaelectronvolts deep. So the matter settles where protons are rare: half a per cent of the nucleons at nuclear density, if nucleons were free, and never more than one in nine at any density.

thermodynamics · Chemical potential The strain a lumpy spinning star sends. The gravitational-wave strain at Earth from a neutron star one kiloparsec away, against the wave's frequency — twice the spin frequency — on logarithmic axes, for ellipticities of 10⁻⁹, 10⁻⁷, 10⁻⁵, 10⁻³, with the most each of three known pulsars could emit if all its observed spin-down were gravitational radiation. Crab: at most 1.4·10⁻²⁴ at 59.2 Hz, needing an ellipticity of 7.7·10⁻⁴; Vela: at most 3.3·10⁻²⁴ at 22.4 Hz, needing an ellipticity of 0.0018; J0437−4715: at most 1.6·10⁻²⁶ at 347.4 Hz, needing an ellipticity of 2·10⁻⁸. Searches have pushed the Crab's strain below about a hundredth of its limit, 1.4·10⁻²⁶. The strain grows as the square of the frequency, so a fast millisecond pulsar with a tiny mountain can send as much as a young pulsar with a large one.

The mountain a spinning star is allowed

A neutron star spinning thirty times a second with a bump on its surface a few centimetres high is a rotating mass quadrupole. It sends out gravitational waves at twice its spin frequency, as a steady tone lasting millions of years. None has been heard. The silence is a measurement. The Crab pulsar is slowing down, and if all the energy it loses went into gravitational waves they would have been detected many times over. Searches now say the waves carry less than a ten-thousandth of it. The mountain on the Crab is less than about ten centimetres high, on a star twenty-four kilometres across.

astrophysics · Gravitational waves The heaviest star free neutrons can hold up. Mass against radius for cold stars of free neutrons, each point a different central density, held up by the pressure of the neutrons' exclusion alone: in general relativity and, with the same gas, in Newton's gravity. Relativity's curve reaches a highest mass of 0.71 solar masses at a radius of 9.5 km and turns back; Newton's keeps rising towards 5.8 solar masses as the stars shrink. The measured masses of neutron stars reach about 2.08 (dashed), three times what free neutrons allow, so the matter inside has to be stiffer than a gas of free neutrons: the repulsion between nucleons at short range is holding them up.

The pressure that weighs down what it holds up

In Newton's gravity a star is held up by pressure, and stiffer matter can hold up more: a star of free neutrons could reach almost six solar masses. In general relativity the pressure itself has weight. Every layer squeezed harder to hold up the layers above adds to the pull it is resisting, and past a point pressing harder makes things worse. The same gas of neutrons then tops out at 0.71 solar masses. Neutron stars twice as heavy exist, which says what their insides must be like — and no matter of any kind can hold a static body smaller than nine-eighths of its Schwarzschild radius.

astrophysics · Self-gravity The band in which radiation feeds a mode instead of draining it. The speed of a mode's pattern as seen from far away, in units of the star's own spin rate, against the spin rate as a fraction of the fastest a star can turn. The shaded band, between zero and one, holds patterns that run forward across the sky and backward through the star; gravitational radiation drives any mode inside it and damps any mode outside. The r-modes with ℓ = m = 2, 3 and 4 sit at 0.667, 0.833 and 0.900 at every spin, inside the band however slowly the star turns. The dashed curves are schematic modes whose crests run backward on the star at a fixed 0.2, 0.5 and 0.9 of the Kepler rate: each is dragged into the band only once the star turns faster than its crest runs, which is why an ordinary oscillation needs a star near break-up and an r-mode does not.

The wobble that radiating makes larger

Radiation normally drains a vibration: a struck bell goes quiet because it sends its energy away as sound, and a ringing black hole settles in milliseconds because it sends its energy away as gravitational waves. A spinning star can have the opposite. A pattern that runs backward through the star's material, while the star carries it forward across the sky, has less angular momentum than the star would have without it, and every wave it radiates takes away angular momentum it did not have to give. The pattern grows. One family of such patterns, the r-modes, runs backward on every rotating star, so every spinning neutron star is unstable unless friction wins — and that may be why none has been found spinning faster than 716 turns a second.

astrophysics · Gravitational waves The stiffness a two-solar-mass star demands. The heaviest neutron star the model matter can support, in solar masses, against the constant square of the sound speed assumed above 300 MeV/fm³, as a fraction of c². At the conformal value, one-third (dashed), it is 1.92 solar masses; the heaviest star the model can make rises to 2.97 at the causal limit. To hold up the 2.08 solar masses of J0740+6620 (shaded) the square of the sound speed must exceed 0.394 of c² somewhere above that density — above the conformal one-third, by this model's reckoning. The conclusion depends on the model's low-density part, which the next figure varies.

The sound speed a neutron star needs

Heat any gas without limit and the speed of sound in it climbs towards a ceiling: not the speed of light, but light's speed divided by the square root of three, the sound speed of a gas of massless particles. For decades it was a natural guess that no matter could beat that ceiling, since the densest matter ought to behave like a gas of free quarks. Neutron stars say otherwise. A star of twice the Sun's mass, packed into a ball twenty-five kilometres across, can only be held up if its core is stiffer than the ceiling allows — unless nuclear physics fails at densities where it is believed to hold. Somewhere inside the heaviest neutron stars, sound almost certainly travels faster than it can in any free gas.

relativity · Relativistic thermodynamics The redshift of a white dwarf weighs it. The gravitational redshift at the surface of a cold white dwarf, expressed as the velocity that would give the same Doppler shift, against the star's mass; computed from the radius each mass is forced to have. The Sun's surface redshift is 636 m/s. A white dwarf of 0.6 solar masses shifts its light by 30.1 km/s, one of 1.0 by 77.6, of 1.2 by 128 and of 1.35 by 223. Because the radius is not free, the curve is single-valued and steepening: a measured shift returns one mass. The cold model puts Sirius B, at 1.018 solar masses, at 80.8 km/s; its measured mass and radius together give 80.0.

The redshift that weighs a dead star

Light leaving the surface of an ordinary star is reddened by gravity, but the shift alone says little, because a star's radius is free and the shift depends on mass over radius. A white dwarf has no free radius. Degenerate electrons fix its size from its mass, so one measured line shift returns one mass — and the gauge grows sharper the closer the star is to collapse. The first measurement of it, in 1925, was hailed as a confirmation of general relativity and was wrong by a factor of four, in agreement with a forecast that was wrong by the same factor.

astrophysics · Gravitational redshift A line carved into the X-rays of a magnetised star. A model X-ray spectrum of the kind fitted to accreting pulsars, drawn as energy per logarithmic interval on a logarithmic scale: a hard continuum that bends over above about 25 keV, with the photons near the cyclotron energy removed by resonant scattering. Dashed, the same continuum without the field. The fundamental is placed at 38 keV, where lines of this kind are seen; thermal motion of the scattering electrons along the field at kT = 8 keV broadens it to a width of about 4.8 keV. The second harmonic falls at 73.0 keV, a little below twice the fundamental. Read through ħeB/m with the relativistic level spacing and the redshift of a 1.4 solar-mass, 12 km star, z = 0.24, the fundamental says the field where the line forms is 4.24 × 10⁸ T. The depths and widths here are illustrative, not fitted to a star.

The circling that comes in quanta

An electron circling a magnetic field radiates at the frequency of its circling, and classically that is all: a continuous glow at whatever energy the circling has. On a neutron star the field is a hundred million tesla, the quantum of circling is a hard X-ray, and an electron lifted one step up falls back in a quarter of a femtosecond. So every electron there sits on the bottom step, and the X-rays that match the step are scattered out of the beam. The gap they leave is a line, and its energy is the field.

astrophysics · Radiating charge Rays that leave the far side of a compact star. Light rays leaving the surface of a star of radius 2.9 Schwarzschild radii — a neutron star of 1.4 solar masses and about 12 km — and travelling to an observer far off to the right, traced along the exact paths of Schwarzschild's geometry, shown above and below the line of sight. The last ray to escape leaves the surface tangentially from a point 121° round from the point facing the observer, so the bright arc — everything the observer sees — covers 76 per cent of the surface rather than the half that straight rays would show. The dashed circle is the horizon a star of the same mass would have if it were compressed inside it.

The star whose far side can be seen

Look at a ball and half of it faces you; the other half is hidden behind it. That half is a fact about straight lines, and near a neutron star the lines are not straight. Light leaving the surface is bent round the star on its way out, so a distant observer sees three-quarters of a typical neutron star at once — and a star squeezed to 1.76 times its own Schwarzschild radius would show every point of its surface, the point directly behind it included.

astrophysics · Light deflection

Named alongside it

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

Equation of stateCausalityChandrasekhar limitCompactnessDegenerate matterGravitational redshiftGravitational wavesQuadrupole radiationSpeed of soundTov equationAngular momentumBeta equilibrium

All concepts