Astronomy & Celestial Mechanics
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Stars

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.

Assumes Degeneracy, Pulsars and Gravitational waves.

15 min read 9 figures Measuring the unreachableScales that defeat intuitionTwo bodies, and no more

There is a state of matter that occurs in exactly one place and cannot be made anywhere else. At the centre of a neutron star the density is several times that of an atomic nucleus, the temperature is negligible compared with the Fermi energy, and the composition is unknown: neutrons certainly, protons and electrons in some proportion, and possibly hyperons or free quarks. What that material does when it is squeezed — how much pressure it produces for a given density, which is all that “an equation of state” means — is not calculable from first principles and is not measurable in any laboratory.

It is nevertheless measured, and the instrument is a star. The logic is not statistical and it is not a fit: every proposed equation of state, integrated through the equations of stellar structure, predicts a definite heaviest star it could hold up, and a single object above that mass falsifies it outright. A field with no laboratory has arranged for the sky to run the experiment.

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.
Fig. 1 Mass against radius for three candidate equations of state, each integrated through the relativistic equation of hydrostatic balance from the centre outwards until the pressure reaches zero. Every sequence rises, turns over and falls, and only the rising part is stable. The three maxima are 1.60, 2.10 and 2.57 solar masses, in order of increasing stiffness. The horizontal band is a pulsar whose mass has been measured to about three per cent, and it lies above the maximum of the softest of the three — which is not a disfavouring but a deletion.

Why a maximum exists at all

The existence of a heaviest possible white dwarf is a statement about relativity in the electrons: as they become relativistic the pressure they supply stops rising fast enough with density, the polytropic index falls to four-thirds, and the central density drops out of the mass entirely.

From 1.667 to 1.343, and the limit is in the second number. The local slope Γ = d ln P / d ln ρ of the exact degenerate electron equation of state, against density, differentiated numerically from the drawn pressure rather than quoted. It is 1.6666 at 10⁴ kg/m³ — the 5/3 of a non-relativistic Fermi gas — and 1.3430 at 10¹² — the 4/3 of a relativistic one — falling monotonically between, and passing 1.5 at 3.63·10⁹ kg/m³. Nothing gives way at either end; the electrons simply run out of room to go faster, because they are already near c. The mass limit is in the exponent and not in the strength. Hydrostatic balance for a polytrope of index Γ gives M ∝ ρ_c^(3Γ−4)/2, so at Γ = 4/3 the exponent is zero: the mass no longer depends on the central density at all, and there is exactly one mass a fully relativistic degenerate star can have — 1.459 M☉ at μ_e = 2. Above it, squeezing harder buys pressure that rises more slowly than gravity does, and there is nothing left to stop at.
Fig. 2 That softening, drawn: the local slope of pressure against density for a degenerate electron gas, falling from five-thirds to four-thirds as the electrons become relativistic. The limit is in the second number, and it is a property of quantum statistics with no free parameters — which is why a white dwarf’s maximum mass is predicted and a neutron star’s is not.

A neutron star’s maximum comes from somewhere else, and the difference is the point of this essay. Neutron degeneracy pressure alone would give a limit near seven-tenths of a solar mass, far below every neutron star ever weighed. What actually holds a neutron star up is the repulsive part of the nuclear force at short range, and that force is not known well enough to be written down.

The distinction is worth being blunt about. A white dwarf is held up by a gas whose properties are completely understood — free electrons obeying the exclusion principle, a star held up by a counting rule — so its structure is a solved problem and Chandrasekhar’s limit is a theorem. A neutron star is held up by something whose properties are the object of the inquiry, so its structure is not a prediction but a measurement, and the same equations that made the white dwarf case a theorem here make it an instrument.

There is also a second and more interesting reason for a maximum, which has nothing to do with the material. The consequence is severe. In Newtonian gravity a sufficiently stiff material can hold up any mass; in general relativity it cannot, because the pressure that would be needed to hold up more mass itself adds weight. Above some mass, adding pressure makes the situation worse. So a maximum exists for every equation of state, however stiff, and even for one so stiff that the speed of sound in it approaches the speed of light — the causality limit drawn on the figure at the top.

How much of the star is at unknown density

Not all of it, which is what makes the radius and the mass carry different information.

The outer kilometre or so of a neutron star is a crust of ordinary nuclei in a lattice, at densities laboratories reach; below that is a region of neutron-rich nuclei whose properties are extrapolated but not wildly so; and only the inner core, perhaps half the radius and most of the mass, sits at densities where nothing is settled. A star’s radius is set mostly by the pressure at around one to two times nuclear density, because that is where the outer layers that determine the size are. Its maximum mass is set by the pressure at four to seven times nuclear density, because that is what the centre of the heaviest star reaches.

So the two observables probe different parts of the same curve, and a description that fits one can fail the other. The most useful constraints have been the ones that pin both ends: a heavy star fixes the high-density end, and a small radius fixes the low-density end.

Weighing a star nobody can see

The masses come from timing, and the best of them come from a relativistic effect. In a binary pulsar the pulses from the neutron star pass close to its companion once per orbit, and the curved space near the companion delays them. The size of the delay depends on the companion’s mass, and its detailed shape through the orbit depends on the inclination — so the Shapiro delay measures both, and combining it with the Keplerian orbit gives the pulsar’s own mass.

The virtue of that route is what is absent from it. There is no model of the neutron star’s atmosphere, no assumed radius, no distance and no spectroscopy. It is a timing measurement of light propagating through a gravitational field, and its systematic error budget is essentially the accuracy of the solar-system ephemeris and the interstellar dispersion.

The heaviest star measured this way sits near 2.08 solar masses, with an uncertainty of a few per cent. That single number is the most informative constraint on the nuclear equation of state that exists, and it was obtained by counting radio pulses.

There is a second and older route to a mass, and its history is instructive. In a double neutron-star binary the orbit precesses relativistically, exactly as Mercury’s does around the Sun but by four degrees a year instead of forty-three arcseconds a century, and the precession rate depends on the total mass. Add the time dilation of the pulsar’s clock as it moves through the companion’s potential, which depends on a different combination, and two relativistic effects give two equations for two masses. That is how the first precise neutron-star masses were obtained, three decades before the Shapiro-delay systems, and it is another case of general relativity being used as a measuring instrument rather than tested as a theory.

Radius against mass for a degenerate star. The mass–radius relation for electron-degenerate matter. More mass gives a smaller star, and the radius reaches zero at 1.46 solar masses — the Chandrasekhar limit, solved from the same expression that draws the curve rather than quoted alongside it.
Fig. 3 The non-relativistic version of the same argument, one regime down. A white dwarf’s radius falls as its mass rises — the counting rule that holds it up stiffens more slowly than gravity — and the relation runs backwards all the way to the Chandrasekhar limit, where the radius goes to zero. A neutron star’s relation turns over instead of running to zero, because relativity softens the pressure before the counting rule runs out. Both maxima come from the same competition; only the equation of state differs.

What a radius adds

A mass constrains the equation of state at the highest densities the star reaches. A radius constrains it at densities a couple of times nuclear, where most of the star’s material actually sits, and the two constraints are largely independent.

A radius is far harder to get. The star subtends about ten nanoarcseconds, so nothing resolves it, and the two available routes are indirect.

The first is spectral: model the X-ray emission from the surface, fit the observed flux and spectrum, and extract the angular size, then combine with a distance. The difficulty is the atmosphere model — a neutron star’s atmosphere is a centimetre of hydrogen in a field of 101210^{12} gauss, and its opacity is not a solved problem — and different treatments have historically differed by kilometres.

The second is geometric and much cleaner. A rotating neutron star with a hot polar cap produces a pulsed X-ray light curve whose shape depends on how strongly the star’s own gravity bends the light from the far side into view. That bending depends on the compactness, the ratio of mass to radius, so a pulse profile is a compactness measurement — and with the mass known from timing, a radius falls out. Applied to the same two-solar-mass pulsar, this gives a radius near 12.4 kilometres. The compactness route has a feature worth admiring. It does not need a distance and it does not need an absolute flux — the shape of the pulse carries the information, and shapes survive calibration errors that amplitudes do not. It is the same structural advantage that makes a velocity read off a line’s shape more robust than one read off its position.

The measurement that arrived from a completely different direction

In 2017 two neutron stars merged, and the last few minutes of the inspiral were recorded as gravitational waves. Buried in the last cycles is a much smaller effect that is a measurement of the equation of state. Each star raises a tide on the other, and a deformed star has a different quadrupole moment, and a different quadrupole moment changes the rate at which the orbit shrinks. The size of that change depends on how easily the star deforms — the tidal deformability — which is a steep function of the radius at fixed mass: a large, distended star deforms readily and a compact one does not.

The measurement was an upper limit rather than a detection, and an upper limit on deformability is an upper limit on radius. It excluded the stiffest equations of state, which is precisely complementary to what the two-solar-mass pulsar does: the pulsar rules out the soft ones from below and the merger rules out the stiff ones from above. Two measurements, made with a radio telescope and an interferometer, converging on the same narrow band of nuclear physics from opposite sides.

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.
Fig. 4 Three stiffnesses, and the whole of what a radius measurement buys. A stiffer equation of state supports more mass at a larger radius; a softer one gives a smaller star and a lower maximum. The three curves are separated by a couple of kilometres at 1.4 solar masses — which is why measuring a neutron star’s radius to a kilometre is worth as much as measuring another maximum mass, and far harder.

A third constraint, from how fast the star cools

Masses and radii are the two mechanical observables. There is a thermal one, and it constrains a different property of the same matter — not how stiff it is, but what it is made of.

A neutron star is born at 101110^{11} kelvin and cools, almost entirely by neutrino emission from its interior for the first hundred thousand years and by photon emission from its surface thereafter. The neutrino emission rate depends on which reactions are available, and the available reactions depend on the proton fraction.

The efficient channel is a beta decay and its inverse operating in sequence — a neutron decaying to a proton, an electron and an antineutrino, and a proton capturing an electron to return to a neutron — with two neutrinos carried away per cycle. Both steps have to conserve energy and momentum among the participating particles, and in a degenerate medium only particles near the Fermi surface can take part. That imposes a triangle condition on the three Fermi momenta, and it is satisfied only when the proton fraction exceeds about eleven per cent.

The consequence is a switch. Below that threshold the fast channel is forbidden and cooling proceeds through a much slower process requiring a spectator nucleon, which is suppressed by a large factor. Above it the star cools by orders of magnitude faster, and its surface temperature at a given age is correspondingly lower.

So a measured surface temperature and an independent age is a statement about the proton fraction at the centre, which is a statement about the symmetry energy of nuclear matter — the same quantity that sets the radius. The observations are a scatter of young neutron stars with temperatures spanning a wide range at similar ages, which is read as some of them having crossed the threshold and some not, and therefore as the threshold falling within the range of central densities that real neutron stars reach.

The chain has a weak link and it is the atmosphere. What is observed is a spectrum, and converting it to a surface temperature requires knowing what the outermost centimetre is made of — hydrogen and helium give very different emergent spectra from an iron-rich surface, and the inferred temperature differs by tens of per cent between them. Since a neutron star’s atmosphere is whatever fell on it most recently, the composition is not predictable and is fitted alongside everything else.

That has produced the one case where a cooling curve has been watched changing. A young neutron star in a supernova remnant has been observed repeatedly over two decades, and its surface temperature appears to have fallen by a few per cent — which, if real, is a cooling rate rather than a temperature, and rates are far more diagnostic than values. Whether the decline is the star or the detector’s own calibration drift has been argued about for as long as the measurement has existed, which is the usual fate of a trend measured at the level of an instrument’s stability.

A cooling curve is a third instrument on the same material, and its systematics — the age, the distance, the atmospheric composition of a star whose surface is a centimetre of hydrogen — have nothing in common with the timing measurements or with the gravitational-wave one.

From 1.667 to 1.343, and the limit is in the second number. The local slope Γ = d ln P / d ln ρ of the exact degenerate electron equation of state, against density, differentiated numerically from the drawn pressure rather than quoted. It is 1.6666 at 10⁴ kg/m³ — the 5/3 of a non-relativistic Fermi gas — and 1.3430 at 10¹² — the 4/3 of a relativistic one — falling monotonically between, and passing 1.5 at 3.63·10⁹ kg/m³. Nothing gives way at either end; the electrons simply run out of room to go faster, because they are already near c. The mass limit is in the exponent and not in the strength. Hydrostatic balance for a polytrope of index Γ gives M ∝ ρ_c^(3Γ−4)/2, so at Γ = 4/3 the exponent is zero: the mass no longer depends on the central density at all, and there is exactly one mass a fully relativistic degenerate star can have — 1.459 M☉ at μ_e = 2. Above it, squeezing harder buys pressure that rises more slowly than gravity does, and there is nothing left to stop at.
Fig. 5 The same uncertainty as a pressure against a density, which is where it actually lives. Above nuclear density the published equations of state differ by factors of several in pressure at the same density, and every one of them is a different extrapolation of laboratory nuclear physics into a regime no experiment reaches. The mass–radius curves above are that spread, integrated; this is the spread itself.

Why the mass distribution is a separate puzzle

Weighing many neutron stars gives a distribution as well as a maximum, and the distribution has a shape nobody predicted.

Neutron-star masses cluster tightly near 1.35 solar masses, with a second and broader group near 1.8 in the systems that have accreted from a companion. The tight peak is not a selection effect and it is not the maximum: it is far below any candidate limit, so nothing about the equation of state is causing it. What causes it is the collapse itself — the iron core of a massive star reaches its own Chandrasekhar mass and collapses, and that mass depends only weakly on the mass of the star around it.

So the mass function of neutron stars is a statement about stellar evolution rather than about nuclear matter, and the two questions have to be kept apart. The equation of state fixes what is possible; the progenitors fix what is produced; and the heaviest star ever weighed is informative about the first precisely because it is an outlier of the second.

What the answer currently is

Assembled, the constraints put the radius of a 1.4-solar-mass neutron star at somewhere near 11 to 12.5 kilometres and the maximum mass at somewhere between 2.2 and 2.5 solar masses. That is a genuine measurement of matter at densities no experiment reaches, and it is worth pausing on how it was obtained: by timing radio pulses, by fitting X-ray pulse shapes, and by measuring the phase of a gravitational wave.

None of the three is a measurement of matter. Each is a measurement of gravity, in a regime where gravity is strong enough that the material’s pressure has become part of its own source. The nuclear physics is being read out through general relativity, because relativity is what converts an equation of state into a mass and a radius.

The same trick is what makes any of this collection’s remote measurements possible — nothing in the sky is weighed in kilograms, and everything in it is weighed by watching how something moves.

The equation of state enters through two curves, and it is worth reading each of them at a second setting because the two say quite different things about what is uncertain.

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 1.21, 1.92, 2.88 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 2 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.
Fig. 6 Three equations of state spanning a wider range of stiffness than the observations allow. The softest gives a maximum mass below the measured 2.08 solar masses and is therefore excluded outright, which is the sense in which a single well-measured mass deletes a family of nuclear models.
From 1.666 to 1.336, and the limit is in the second number. The local slope Γ = d ln P / d ln ρ of the exact degenerate electron equation of state, against density, differentiated numerically from the drawn pressure rather than quoted. It is 1.6664 at 10⁵ kg/m³ — the 5/3 of a non-relativistic Fermi gas — and 1.3356 at 10¹³ — the 4/3 of a relativistic one — falling monotonically between, and passing 1.5 at 3.98·10⁹ kg/m³. Nothing gives way at either end; the electrons simply run out of room to go faster, because they are already near c. The mass limit is in the exponent and not in the strength. Hydrostatic balance for a polytrope of index Γ gives M ∝ ρ_c^(3Γ−4)/2, so at Γ = 4/3 the exponent is zero: the mass no longer depends on the central density at all, and there is exactly one mass a fully relativistic degenerate star can have — 1.263 M☉ at μ_e = 2.15. Above it, squeezing harder buys pressure that rises more slowly than gravity does, and there is nothing left to stop at.
Fig. 7 The local slope of the degenerate equation of state for matter with 2.15 nucleons per electron. It slides from 5/3 to 4/3 exactly as before, because the slide is a property of the electrons rather than of the composition — the composition enters the limiting mass and not the shape of the curve.

Where the picture stops

The composition is not settled and may not be uniform. A star at the maximum mass has a central density where hyperons or deconfined quarks may appear, and either would soften the equation of state — which is the “hyperon puzzle”: the naive expectation that hyperons appear at a few times nuclear density is hard to reconcile with a two-solar-mass star existing. Something either suppresses them or stiffens what is left.

Rotation matters and has been ignored. Every curve here is for a non-rotating star. A rapidly rotating one can support up to about twenty per cent more mass, which means the maximum mass inferred from a merger remnant that survived briefly before collapsing is a maximum for a rotating configuration, and converting it to the non-rotating value is a model-dependent step.

Magnetic fields are absent from the whole account. A field of 101210^{12} gauss contributes negligible pressure compared with the material, so the structure is unaffected; a magnetar’s 101510^{15} gauss does not, and there the field is a term in the equilibrium. Since a pulsar’s field strength is itself inferred from its spin-down rather than measured, the magnitude of that correction is estimated from a quantity that is estimated.

And the radius measurements are the least secure part. The X-ray pulse-profile route depends on the geometry of the hot spots, which is fitted rather than known, and the published uncertainties are dominated by how much freedom the fit is given. Two groups analysing the same data have differed by more than their quoted errors.

And the white-dwarf relation at a composition no star has, drawn because it shows exactly where the limit’s dependence on composition lives.

Radius against mass for a degenerate star. The mass–radius relation for electron-degenerate matter. More mass gives a smaller star, and the radius reaches zero at 2.02 solar masses — the Chandrasekhar limit, solved from the same expression that draws the curve rather than quoted alongside it.
Fig. 8 The relation for 1.7 nucleons per electron rather than two. The limiting mass rises as the inverse square of that number, to nearly two solar masses, and the whole curve shifts with it — so the Chandrasekhar mass is a constant of nature divided by a fact about chemistry.

One more composition shows the direction the limit moves as the matter neutronises.

Radius against mass for a degenerate star. The mass–radius relation for electron-degenerate matter. More mass gives a smaller star, and the radius reaches zero at 0.93 solar masses — the Chandrasekhar limit, solved from the same expression that draws the curve rather than quoted alongside it.
Fig. 9 The relation for matter with 2.5 nucleons per electron. The limiting mass falls below one solar mass, which is the direction electron capture pushes a star approaching the limit — the star’s mass rises towards a limit that is itself coming down to meet it.

Where this ladder goes next

Later rungs on this anchor: the Shapiro delay in detail, and why an edge-on binary pulsar is worth more than any other; the pulse-profile modelling that turns a light curve into a compactness; tidal deformability and what a second, louder merger would add; the hyperon puzzle and the quark-matter alternative; and the rotational limit, where a star spinning near break-up is a different object from the one described here and the maximum mass has to be quoted with the spin attached.

Where this goes next

From 1.667 to 1.343, and the limit is in the second number. The local slope Γ = d ln P / d ln ρ of the exact degenerate electron equation of state, against density, differentiated numerically from the drawn pressure rather than quoted. It is 1.6666 at 10⁴ kg/m³ — the 5/3 of a non-relativistic Fermi gas — and 1.3430 at 10¹² — the 4/3 of a relativistic one — falling monotonically between, and passing 1.5 at 3.63·10⁹ kg/m³. Nothing gives way at either end; the electrons simply run out of room to go faster, because they are already near c. The mass limit is in the exponent and not in the strength. Hydrostatic balance for a polytrope of index Γ gives M ∝ ρ_c^(3Γ−4)/2, so at Γ = 4/3 the exponent is zero: the mass no longer depends on the central density at all, and there is exactly one mass a fully relativistic degenerate star can have — 1.459 M☉ at μ_e = 2. Above it, squeezing harder buys pressure that rises more slowly than gravity does, and there is nothing left to stop at. The mass a cold star cannot exceed A degenerate star's radius falls as its mass rises, and nothing in that relation suggests a limit. Making the electrons relativistic softens the pressure law until pressure and gravity scale the same way with radius — and the radius drops out of the balance, leaving one mass and no room to argue. Three ways for a timing model to be wrong, and three shapes that say which. Timing residuals over 6 years for the Crab pulsar, one curve per kind of error in the model, in microseconds. A position error of 1.2 mas leaves a sinusoid of period exactly one year — measured off the drawn curve as 1.000 — with amplitude (a/c)·δθ·cos β = 2.7 μs, because the error is being projected onto a baseline that is the Earth's own orbit and nothing about the pulsar. An unmodelled proper motion of 0.9 mas/yr leaves the same sinusoid with an envelope growing linearly: twice as large at 6 years as at 3. An error of one part in 10⁹ in Ṗ leaves a parabola, the second integral of a frequency drift, whose second derivative is constant to 4e-12 across the span — and that fractional error is deliberately minute, because anything larger produces a residual thousands of times the other two and draws them as flat lines. The shapes do not resemble each other, which is the whole reason a pulsar is an instrument rather than a clock: fitting them simultaneously delivers a position, a proper motion and — from the annual curvature term, not drawn here — a parallax, all from the arrival times of pulses and no image of anything. What is left when every known shape has been removed is the science: glitches, red noise, and the correlated residual between pairs of pulsars that a timing array exists to find. A clock read against a model of everything in between A pulsar delivers arrival times and nothing else. Everything else is a model, and what is measured is the difference between the model and the arrivals — a residual whose shape says which term is wrong, and whose annual sinusoid is a position measured from pulses rather than from an image. GW150914: 33 Hz to 250 Hz in 0.22 seconds. The strain of GW150914 — two black holes — through the last 0.22 seconds before merger, computed from the quadrupole sweep at the chirp mass its fit returned, 28.716 solar masses, and drawn at the luminosity distance it returned, 440 megaparsecs. Two things rise together and neither is free to rise on its own: the frequency goes from 33 Hz to 250 Hz, and the envelope — the outer curve — grows by a factor of 3.9, because the amplitude goes as f^2/3 and nothing else in it changes over so short a span. The vertical axis is in units of 10⁻²¹, so the peak here is a fractional length change of about 2.9·10⁻²¹: over the four kilometres of an interferometer arm that is 1.2·10⁻¹⁷ metres, a thousandth of the width of a proton. The chirp mass is not fitted to the amplitude at all — it comes from the spacing of these zero crossings, which is why it is the best-determined number in the whole event and why the distance, which does come from the amplitude, is the worst. A distance with no ladder under it The frequency sweep of an inspiral fixes the chirp mass with no distance in it, and the amplitude then gives the luminosity distance directly, because one expression fixes both. That is a distance measured with nothing calibrated beneath it — and its error budget is one angle.

About the same objects

Not linked from either essay — found by the objects both name.

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

Causality limitCompactnessDegeneracy pressureEquation of stateMaximum massMultimessengerNeutron starNuclear saturation densityThe Shapiro delayStiffnessTidal deformabilityTolman oppenheimer volkoff