Colloquium: Measuring the neutron star equation of state using x-ray timing

One of the primary science goals of the next generation of hard x-ray timing instruments is to determine the equation of state of matter at supranuclear densities inside neutron stars by measuring the radius of neutron stars with different masses to accuracies of a few percent. Three main techniques can be used to achieve this goal. The first involves waveform modeling. The flux observed from a hotspot on the neutron star surface offset from the rotational pole will be modulated by the star’s rotation, and this periodic modulation at the spin frequency is called a pulsation. As the photons propagate through the curved spacetime of the star, information about mass and radius is encoded into the shape of the waveform (pulse profile) via special and general-relativistic effects. Using pulsations from known sources (which have hotspots that develop either during thermonuclear bursts or due to channeled accretion) it is possible to obtain tight constraints on mass and radius. The second technique involves characterizing the spin distribution of accreting neutron stars. A large collecting area enables highly sensitive searches for weak or intermittent pulsations (which yield spin) from the many accreting neutron stars whose spin rates are not yet known. The most rapidly rotating stars provide a clean constraint, since the limiting spin rate where the equatorial surface velocity is comparable to the local orbital velocity, at which mass shedding occurs, is a function of mass and radius. However, the overall spin distribution also provides a guide to the torque mechanisms in operation and the moment of inertia, both of which can depend sensitively on dense matter physics. The third technique is to search for quasiperiodic oscillations in x-ray flux associated with global seismic vibrations of magnetars (the most highly magnetized neutron stars), triggered by magnetic explosions. The vibrational frequencies depend on stellar parameters including the dense matter equation of state, and large-area x-ray timing instruments would provide much improved detection capability. An illustration is given of how these complementary x-ray timing techniques can be used to constrain the dense matter equation of state and the results that might be expected from a $10{\mathrm{m}}^2$ instrument are discussed. Also discussed are how the results from such a facility would compare to other astronomical investigations of neutron star properties.

Figure 1

Schematic structure of a neutron star. The outer layer is a solid ionic crust supported by electron degeneracy pressure. Neutrons begin to leak out of ions (nuclei) at densities $\sim 4\times {10}^{11}\mathrm{g}/{\mathrm{cm}}^3$ (the neutron drip density, which separates the inner from the outer crust), where neutron degeneracy also starts to play a role. At densities $\sim 2\times {10}^{14}\mathrm{g}/{\mathrm{cm}}^3$, the nuclei dissolve completely. This marks the crust-core boundary. In the core, densities reach several times the nuclear saturation density ${\rho }_{\mathrm{sat}}=2.8\times {10}^{14}\mathrm{g}/{\mathrm{cm}}^3$ (see text).

Figure 2

Hypothetical states of matter accessed by neutron stars and current or planned laboratory experiments (Large Hadron Collider and other heavy-ion collision experiments, shown by black arrows), in the parameter space of temperature against baryon chemical potential (1–2 GeV corresponds to $\sim 1-6$ times the density of normal atomic nuclei). Quarkyonic matter: a hypothesized phase where cold dense quarks experience confining forces ([170]; [75]). The stabilizing effect of gravitational confinement in neutron stars permits long-time-scale weak interactions (such as electron captures) to reach equilibrium, generating matter that is neutron rich [see Fig. 2 of 261] and may involve matter with strange quarks. This means that neutron stars access unique states of matter that can be created only with extreme difficulty in the laboratory: nuclear superfluids, strange matter states with hyperons, deconfined quarks, and color superconducting phases.

Figure 3

The pressure density relation (EOS, left) and the corresponding $M\text{−}R$ relation (right) based on models with different microphysics. The three individual models that extend to the highest pressures and masses) (red): nucleonic EOS. From [148]. Solid black curves: hybrid models (strange quark core, [273]). Dashed black curves: hyperon core models ([29]). Dash-dotted magenta curves: a self-bound strange quark star model ([148]). Shaded gray bands: range of a parametrized family of nucleonic EOS based on chiral effective field theory at low densities, which provides a systematic expansion for nuclear forces that allows one to estimate the theoretical uncertainties involved, combined with using general extrapolations to high densities [see Fig. 12 of [113] for examples of specific representative EOS lying within this band].

Figure 4

Left: As the neutron star rotates, emission from a surface hotspot generates a pulsation. The observer inclination is $i$ and hotspot inclination is $\alpha$. The invisible surface is smaller than a hemisphere due to relativistic light bending. Right: A schematic from [210] to illustrate the effects on the waveform arising from relativistic effects: The waveform is modified from a pure sinusoid, and the temperature (indicated by the color, which is the ratio of the number of photons with energies above to those below the blackbody temperature) also varies even though the underlying hotspot has a uniform temperature. The shape of the waveform and its energy dependence can be used to recover $M$ and $R$. See also [260] for examples of waveforms that include the expected variation in x-ray polarization.

Figure 5

Constraints on $M$ and $R$ obtained by fitting a waveform model to synthetic observed waveform data. [177] The black squares indicate the mass and radius used to construct the synthetic data, the solid red lines show the $1\sigma$ constraints, and the dashed blue lines show the $2\sigma$ constraints. Here the spin rate $f_s=600\mathrm{Hz}$, hotspot inclination $\alpha$, and observer inclination $i$ are 90°, the fractional rms amplitude of the oscillations is 10.8%, and the total number of counts is ${10}^7$. $M$ and $R$ are tightly constrained: $\mathrm{\Delta }M/M=2.8\%$ and $\mathrm{\Delta }R/R=2.9\%$. The parameter values used to generate the waveform data used in (b) are the same as in (a), except the rotation rate, which is much lower (300 Hz), causing the constraints on $M$ and $R$ to be weaker: $\mathrm{\Delta }M/M=5.6\%$ and $\mathrm{\Delta }R/R=7.6\%$. (c) When the spot is at an intermediate colatitude (here 60°), the constraints on $M$ and $R$ are weaker ($\mathrm{\Delta }M/M=6.5\%$ and $\mathrm{\Delta }R/R=6.7\%$), even if the star has a large radius (here 15 km) and is rapidly rotating (here at 600 Hz). From [177].

Figure 6

$1\sigma$ confidence regions illustrating the constraints on the EOS that would be expected from a $\sim 10{\mathrm{m}}^2$ hard x-ray timing telescope. For illustration, the regions are assumed to trace the $M\text{−}R$ curve of a proposed EOS that produces a neutron star with a strange matter core. The larger regions assume observing time has been dedicated to reach $\mathcal{R}\approx 400$ for the 50% of known burst oscillation systems (here 10) that have optimal spot and observer inclinations ($i=60°–90°$) if their orientations are randomly distributed. The size and shape of these regions are from the Bayesian analyses of waveform data by [162] and [177] that assume the hotspots and observers are in the equatorial plane and that there is no independent knowledge of any of the model waveform parameters. The smaller $1\sigma$ confidence regions show the constraints that could be achieved by a deep follow-up to halve the fractional uncertainties in $M$ and $R$ for the 3–4 stars that are most constraining, in terms of their locations in the $M\text{−}R$ plane, the potential for reducing their uncertainties, and the availability of complementary constraints from other observations. Constraints like those shown will exclude many of the EOS models. In this illustration, the models shown as gray lines are excluded.

Figure 7

Spin limits on the EOS. Neutron stars of a given spin rate must lie to the left of the relevant limiting line in the $M\text{−}R$ plane (shown, in blue, for various spins as labeled). The current record holder which spins at 716 Hz ([119]) is not constraining. However, given a high enough spin, individual EOS can be ruled out. Between 1 and 1.25 kHz, for example, some individual EOS in the gray band [which represents a parametrized family of EOS models from 113] would be excluded. From [261].

Figure 8

Spin distributions of neutron stars with rotation rates above 100 Hz. Top panel: radio pulsars. Lower panel: accreting neutron stars (accretion-powered millisecond pulsars and burst oscillation sources).

Figure 9

$M\text{−}R$ diagram showing the seismological constraints for the soft gamma-ray repeater SGR 1806–20 using the relativistic torsional crust oscillation model of [223], in which the 29 Hz QPO is identified as the fundamental and the 625 Hz QPO as the first radial overtone. The neutron star lies in the box where the constraints from the two frequency bands overlap. Once QPOs are detected, frequency measurement errors are negligible for this purpose. This model is very simple (it does not include crust-core coupling), but it gives some idea of the type of constraints that might result from the detection of a harmonic sequence of seismic vibrations. More sophisticated models that take into account coupling and the other relevant physical effects are under development.