A Love number measured at nuclear density

A body pulled on from one side deforms, and how much it deforms, relative to how much it is pulled, is one dimensionless number — the second Love number, $k_2$. It is three halves for a uniform fluid, a few hundredths for a centrally condensed star, and for a moon with an ocean under its ice it is large enough to say that the ocean exists though not how deep it is. Its value is a statement about how the body’s mass is distributed inside it: the more concentrated toward the centre, the less the outer layers respond.

Nothing in that definition is specific to moons. A neutron star in a binary is pulled on by its companion, and in the last orbits before two neutron stars merge the pull is strong enough to raise a tide of several kilometres on each. The tide has a quadrupole moment, the quadrupole moment radiates and attracts, and both change how fast the orbit shrinks. The change is recorded in the gravitational wave the binary emits, and its size is set by the same $k_2$ — now evaluated for matter four times as dense as an atomic nucleus.

From a Love number to a deformability #

For a moon the Love number is all that matters, because its size and mass are known separately. For a neutron star neither is, and the quantity that enters the waveform is a combination. A star of mass $M$ and radius $R$ in a tidal field develops a quadrupole moment proportional to the field, and the constant of proportionality is

$$\lambda = \frac{2}{3}\,k_2\,\frac{R^5}{G},$$

the fifth power of the radius coming from the fact that the tidal field varies over the star’s size and the quadrupole moment is measured at that size. Divided by the appropriate power of the mass, to make it dimensionless, it becomes the tidal deformability

$$\Lambda = \frac{2}{3}\,k_2\,C^{-5}, \qquad C = \frac{GM}{Rc^2}.$$

The compactness $C$ is the ratio of the star’s gravitational radius to its actual radius. For a neutron star it is about 0.17 — the star’s surface sits about three times its Schwarzschild radius from the centre — and for the Earth it is a part in a billion.

The first figure separates the two factors. Both depend on the radius at fixed mass. A larger star is less compact, so $C^{-5}$ grows steeply; a larger star is also, for the equations of state that produce it, less centrally concentrated, so $k_2$ grows too, but slowly. Across the range of radii the candidate equations of state allow, the compactness term changes by a factor of about five and the Love number by about one and a half. The deformability is mostly a radius.

Why it is nearly the sixth power #

The steepness is the measurement’s chief virtue. A radius changes the deformability by its sixth or seventh power: five from the compactness, one or a little more from the Love number. So a deformability measured to a factor of two is a radius measured to about eleven per cent, and a deformability known to fifty per cent is a radius known to eight. Few measurements in astronomy convert a crude number into a precise one so efficiently.

The relation drawn here is not derived from any one equation of state. It is one of a family of nearly universal relations found by computing the structure of neutron stars for dozens of candidate equations of state and noticing that certain combinations of their properties — the deformability and the compactness, the moment of inertia and the quadrupole moment — fall on nearly the same curve for all of them, to a few per cent. The universality is partly an empirical fact and partly understood: the relations involve quantities dominated by the star’s outer layers, where the candidate equations of state differ least. Whatever the reason, it lets a measured deformability be converted to a radius without choosing an equation of state.

Why stiffness means size #

A neutron star’s radius is set by a contest between its gravity and the pressure of its matter, and the pressure at a given density is what an equation of state specifies. Matter that resists compression strongly — a stiff equation of state — holds the star up at lower density, so the star is larger and less compact for its mass. Matter that yields — a soft equation of state — lets the star contract until its density is high enough for the pressure to win, so the star is smaller.

The equation of state at the densities that matter is not known from any laboratory. Up to about the density of an atomic nucleus it is constrained by nuclear physics experiments: the binding energies of heavy nuclei, the thickness of their neutron-rich skins, the collisions of heavy ions. Above that, the nucleons are squeezed closer than they ever are on Earth, and whether they stay nucleons, or give way to hyperons, or dissolve into quarks, is a theoretical question with many competing answers. A star held up by a counting rule — the exclusion principle that supports white dwarfs and, in a much stronger form, neutron stars — sets the scale; the interactions between nucleons set the details, and the details are the radius.

So every equation of state predicts a curve of radius against mass, and each curve ends at a heaviest star the matter can hold up. The stiff curves sit at large radii and end at high masses; the soft ones sit at small radii and end at lower masses. A measured radius at one mass picks out a band of curves; a measured maximum mass cuts off the softest. The tidal deformability, being a radius to the sixth power, is one of the sharpest tools for the first.

The same number, a different job #

For a moon, $k_2$ measures the interior directly: a large Love number means a liquid layer, a small one means a rigid body, and the moon’s radius and mass are already known. For a neutron star the radius is what is unknown, and the Love number is a secondary effect riding on it.

General relativity reduces the Love number of a compact star below its Newtonian value. The reduction is roughly by the square of one minus twice the compactness, which for a neutron star is a factor of two or three, and it depends on the compactness itself — which is why the Love number in the first figure rises with radius: a larger, less compact star has less of its response suppressed. In the limit of a black hole, the compactness reaches a half and the Love number goes to zero. Black holes do not deform, and a measured non-zero deformability is evidence that the objects merging had surfaces.

That is the surprising connection. The same number that a spacecraft extracts from the wobble in a moon’s gravity field, to decide whether there is water under its crust, is extracted from a ripple in spacetime to decide how large an object of 1.4 solar masses can be, which is to decide how stiff matter is at four times nuclear density. The definitions are identical; only the factors that dominate them differ.

A tide that waits until the end #

The deformability enters the gravitational wave’s phase, and how it enters explains both why it can be measured at all and why the measurement is so difficult.

The orbital dynamics of a binary are expanded in powers of the orbital speed over the speed of light. The leading term is Newtonian gravity and the leading radiation is the quadrupole formula; corrections come at successive orders of $(v/c)^2$, the post-Newtonian orders. The tidal effect enters at the fifth order — formally ten powers of $v/c$ beyond the leading term — because it depends on the stars’ finite size, which matters only when they are close. That would make it unmeasurable, were it not multiplied by the deformability, which for a neutron star is several hundred.

The product of a very small factor and a large one is a phase shift that is tiny for most of the inspiral and grows steeply at the end. The detectors record a binary neutron star for about a minute and a few thousand cycles, from tens of hertz upward. Through nearly all of that time the tidal phase is a small fraction of a radian, and the waveform is set by the chirp mass alone. In the last few dozen cycles, above a few hundred hertz, the tidal phase reaches a radian, and only there can it be told from the other parameters.

That is also where the detectors are least sensitive. Their noise rises at high frequency, set by the quantum fluctuations of the laser light, so the part of the signal carrying the tidal information is the part most buried in noise. The first merger was loud — it was forty megaparsecs away, the closest gravitational-wave source yet — and even so it gave only an upper limit at first.

One number for two stars #

A binary contains two stars, each with its own deformability. The waveform does not measure them separately; it measures a mass-weighted combination,

$$\tilde\Lambda = \frac{16}{13}\,\frac{(m_1 + 12m_2)\,m_1^4\,\Lambda_1 + (m_2 + 12m_1)\,m_2^4\,\Lambda_2}{(m_1+m_2)^5},$$

which enters the leading tidal term of the phase. The individual stars’ deformabilities, and the mass ratio that would separate them, are poorly constrained.

That near-independence is fortunate and not accidental. At fixed chirp mass, making the binary more unequal lightens one star and raises its deformability steeply, while making the other heavier and lowering its deformability; the weighting in $\tilde\Lambda$ nearly balances the two. If both stars share a radius — as most equations of state predict for neutron stars between about 1.1 and 1.6 solar masses, whose radii vary by less than half a kilometre — the combined deformability is effectively a function of that radius alone.

What was actually measured #

The first binary neutron-star merger, detected in August 2017, had a chirp mass of 1.188 solar masses, measured to a part in a thousand from the thousands of cycles of its inspiral. Its combined deformability was first reported as an upper limit, $\tilde\Lambda < 800$ at ninety per cent confidence for the most plausible spins. A reanalysis with improved waveform models and the assumption that both stars obey the same equation of state gave a deformability for a 1.4 solar-mass star of about 190, with a range from about 70 to 580.

The upper limit alone excluded the stiffest equations of state — those predicting radii above about 13.5 kilometres for a 1.4 solar-mass star. That was the first constraint on neutron-star radii from a method that does not depend on modelling the emission from a star’s surface, and it is complementary to the pulsar masses, which exclude the softest equations of state from below by requiring that some stars weigh more than two solar masses.

The radius constraint was later joined by measurements of the X-ray pulses from hot spots on rotating neutron stars, whose shape depends on how strongly the star’s gravity bends light from its far side into view. Those give radii of about twelve to thirteen kilometres for stars of 1.4 and 2.0 solar masses. Within the errors, the tidal and pulse-profile radii agree.

The light that said the remnant did not collapse at once #

The first merger was also seen in light — a short gamma-ray burst two seconds after the merger, and then a kilonova, the glow of radioactive heavy elements thrown out in the collision, fading over a few weeks. The light carried its own constraint on the radius, from the opposite direction.

When two neutron stars merge, the product is a single object of about 2.7 solar masses — above the maximum mass of any non-rotating neutron star, but supported for a while by its rapid rotation and its heat. If the stars were very compact, the merged object would collapse to a black hole immediately, swallowing nearly all the matter and leaving little to be thrown out. If they were less compact, a massive, rapidly spinning remnant would survive for tens of milliseconds or longer, shedding a substantial mass of neutron-rich material before collapsing. The kilonova’s brightness required a few hundredths of a solar mass of ejecta, more than an immediate collapse could provide.

Simulations translate “no immediate collapse” into a lower bound on the radius: stars of 1.6 solar masses at least about ten and a half kilometres across. That bounds the radius from below, as the deformability bounds it from above. The two constraints come from the same event — one from the gravitational wave before contact, one from the light after it — and their agreement leaves a window of a few kilometres that later measurements have narrowed.

What loudness buys #

The precision of the tidal measurement depends on the signal-to-noise ratio in the last few hundred cycles, and that depends on the distance. A gravitational wave carries its own distance, and the first merger was close: forty megaparsecs, in a galaxy identified from the kilonova. A merger at twice the distance gives half the signal-to-noise, and the uncertainty on the deformability, which is measured from the high-frequency end where noise dominates, grows by more than that.

So the measurement is limited by rarity. Binary neutron-star mergers as close as the first happen perhaps once in several years within the detectors’ reach, and most detections are several times farther. Combining many quieter events helps — the deformability is a property of the equation of state, so every merger measures the same curve at different masses — but the gain is slow, as the square root of the number of events, while a single loud event can halve the uncertainty at once.

What the deformability cannot see #

The universal relation is good to a few per cent, not perfectly. Equations of state with a phase transition — a sudden softening at some density, as quark matter appearing in the core might produce — can depart from it, and a star whose core changes phase would have a different Love number at the same radius. The relation’s universality is tested against candidate equations of state, and a sufficiently exotic one could break it.

The leading tidal phase is not the only tidal effect. At higher orders the tide’s own dynamics enter: the star’s fundamental oscillation mode, whose frequency for a neutron star is about a kilohertz and below which the tide follows the orbit, is approached in the last orbits, and the tidal response grows as the orbital frequency nears it. Waveform models now include that dynamical tide, and its omission in the first analyses was one reason the deformability estimate moved.

Spin is degenerate with deformability. A spinning star is flattened by its own rotation and has a quadrupole moment that also changes the phase; the two effects have different frequency dependences, but not different enough to be separated in a single signal. The neutron stars in known binaries spin slowly, and assuming low spins tightens the deformability bound considerably.

The figures use the universal relation throughout. No equation of state is integrated here; the deformabilities, Love numbers and radii are those the relation implies, and the phase uses only the leading tidal term. They show the size and shape of each effect rather than a fit to any measurement.

A Love number at three scales #

The Love number now has measured values across a range no other dimensionless quantity in physics spans with the same definition. For the Earth, about 0.3, from the tides of the solid planet measured by gravimeters and satellite orbits. For Titan, about 0.6, enough to require an ocean. For the Sun, a few hundredths, implied by its central concentration. For a neutron star, about a tenth, entering a deformability of a few hundred because the star is so compact.

The first three are measured by watching a body respond to a slow, steady tide over many orbits. The fourth is measured once, in a tenth of a second, by a tide that rises from nothing to several kilometres and ends with the stars touching. The definitions are the same. What a measurement of each says about the body is different: for a moon, whether it has an ocean; for a planet, how its mantle flows; for a neutron star, what matter does at densities no laboratory reaches — through the size it lets a star of a given mass be.

Still open: a merger loud enough to see the Love number itself #

The first merger measured a radius through the deformability, and the radius dominated. A louder signal — closer, or detected by more sensitive instruments — would measure the deformability with enough precision to separate the small contribution of $k_2$ from the compactness, especially if combined with an independent radius from X-ray pulse profiles of the same kind of star. That separation is where the equation of state appears beyond the radius: two equations of state with the same radius at 1.4 solar masses can have different Love numbers, because one concentrates its mass more toward the centre.

The next generation of detectors, with sensitivity at the kilohertz frequencies where the tidal phase lives, will see hundreds of mergers a year, and the brightest of them will put the deformability at a few per cent. At that precision the measurement will distinguish how matter at nuclear density is arranged inside a star, not only how large the star is — the same distinction that a Love number draws for a moon between a body that is ice to the centre and one with an ocean beneath its crust.