Cooling of Accretion-Heated Neutron Stars

We present a brief, observational review about the study of the cooling behaviour of accretion-heated neutron stars and the inferences about the neutron-star crust and core that have been obtained from these studies. Accretion of matter during outbursts can heat the crust out of thermal equilibrium with the core and after the accretion episodes are over, the crust will cool down until crust-core equilibrium is restored. We discuss the observed properties of the crust cooling sources and what has been learned about the physics of neutron-star crusts. We also briefly discuss those systems that have been observed long after their outbursts were over, i.e, during times when the crust and core are expected to be in thermal equilibrium. The surface temperature is then a direct probe for the core temperature. By comparing the expected temperatures based on estimates of the accretion history of the targets with the observed ones, the physics of neutron-star cores can be investigated. Finally, we discuss similar studies performed for strongly magnetized neutron stars in which the magnetic field might play an important role in the heating and cooling of the neutron stars.     

Time-variable emission from transiently accreting neutron stars in quiescence due to deep crustal heating

Transiently accreting neutron stars in quiescence ( L _X ≲10³⁴ erg s⁻¹) have been observed to vary in intensity by factors of few, over time-scales of days to years. If the quiescent luminosity is powered by a hot neutron star core, the core cooling time-scale is much longer than the recurrence time, and cannot explain the observed, more rapid variability. However, the non-equilibrium reactions which occur in the crust during outbursts deposit energy in isodensity shells, from which the thermal diffusion time-scale to the photosphere is days to years. The predicted magnitude of variability is too low to explain the observed variability unless – as is widely believed – the neutrons beyond the neutron-drip density are superfluid. Even then, the variability due to this mechanism in models with standard core neutrino cooling processes is less than 50 per cent – still too low to explain the reported variability. However, models with rapid core neutrino cooling can produce a variability by a factor as great as 20, on time-scales of days to years following an outburst. Thus, the factors of ∼ few intensity variability observed from transiently accreting neutron stars can be accounted for by this mechanism only if rapid core cooling processes are active.     

Pycnonuclear burning of 34Ne in accreting neutron stars

We study the pycnonuclear burning of ³⁴Ne in the inner crust of an accreting neutron star. We show that the associated energy production rate can be calculated analytically for any arbitrary temporal variability of the mass accretion rate. We argue that the theoretical time-scale for ³⁴Ne burning is currently very uncertain and ranges from a fraction of a millisecond to a few years. The fastest allowable burning may change the composition of the accreted crust while the slowest burning leads to a time-independent nuclear energy generation rate for a variable accretion. The results are important for constructing self-consistent models of the accreted crust and deep crustal heating in neutron stars which enter soft X-ray transients.     

A young cooling neutron star in the remnant of Supernova 1987A

We describe the cooling theory for isolated neutron stars that are several tens of years old. Their cooling differs greatly from the cooling of older stars that has been well studied in the literature. It is sensitive to the physics of the inner stellar crust and even to the thermal conductivity of the stellar core, which is never important at later cooling stages. The absence of observational evidence for the formation of a neutron star during the explosion of Supernova 1987A is consistent with the fact that the star was actually born there. It may still be hidden in the dense center of the supernova remnant. If, however, the star is not hidden, then it should have a low thermal luminosity (below ～10³⁴ erg s^?1) and a short internal thermal relaxation time (shorter than 13 yr). This requires that the star undergo intense neutrino cooling (e.g., via the direct Urca process) and have a thin crust with strong superfluidity of free neutrons and/or an anomalously high thermal conductivity.     

Small-scale Gradients and Large-scale Diffusion of Charged Particles in the Heliospheric Magnetic Field

We present a simple spin-evolution model that predicts that rapidly rotating accreting neutron stars will be confined mainly to a narrow range of spin frequencies: P=1.5-5 ms. This is in agreement with current observations of neutron stars in both the low-mass X-ray binaries and the millisecond radio pulsars. The main ingredients in the model are (1) the instability of r-modes above a critical spin rate, (2) the thermal runaway that is due to the heat released as viscous damping mechanisms counteract the r-mode growth, and (3) a revised estimate of the strength of the dissipation that is due to the presence of a viscous boundary layer at the base of the crust in an old and relatively cold neutron star. We discuss the gravitational waves that are radiated during the brief r-mode-driven spin-down phase. We also briefly touch on how the new estimates affect the predicted initial spin periods of hot young neutron stars.     

Crust–core coupling and r-mode damping in neutron stars: a toy model

r-modes in neutron stars with crusts are damped by viscous friction at the crust–core boundary. The magnitude of this damping, evaluated by Bildsten & Ushomirsky (BU) under the assumption of a perfectly rigid crust, sets the maximum spin frequency for neutron stars spun up by accretion in low-mass X-ray binaries (LMXBs). In this paper we explore the mechanical coupling between the core r-modes and the elastic crust, using a toy model of a constant-density neutron star having a crust with a constant shear modulus. We find that, at spin frequencies in excess of ≈50 Hz, the r-modes strongly penetrate the crust. This reduces the relative motion (slippage) between the crust and the core compared with the rigid-crust limit. We therefore revise down, by as much as a factor of 10²–10³ , the damping rate computed by BU, significantly reducing the maximal possible spin frequency of neutron stars with solid crusts. The dependence of the crust–core slippage on the spin frequency is complicated, and is very sensitive to the physical thickness of the crust. If the crust is sufficiently thick, the curve of the critical spin frequency for the onset of the r-mode instability becomes multivalued for some temperatures; this is related to avoided crossings between the r-mode and higher-order torsional modes in the crust. The critical frequencies are comparable to the observed spins of neutron stars in LMXBs and millisecond pulsars.     

Nuclear deflagration on the surface of accreting neutron stars

We analyze the structure of a nuclear deflagration front in the crust of accreting neutron stars. Models of quasi-stationary deflagration fronts are calculated and subsequently evolved in time and space in order to check their stability. Unlike white dwarfs, where the velocity of aninwards propagating combustion front is governed mainly by energy losses to the gravitational field, the structure of a deflagration front in neutron stars is determined essentially by the heat fluxes into the stellar core.     

Thermal relaxation in young neutron stars

The internal properties of the neutron star crust can be probed by observing the epoch of thermal relaxation. After the supernova explosion, powerful neutrino emission quickly cools the stellar core, while the crust stays hot. The cooling wave then propagates through the crust, as a result of its finite thermal conductivity. When the cooling wave reaches the surface (age 10–100 yr) , the effective temperature drops sharply from 250 eV to 30 or 100 eV, depending on the cooling model. The crust relaxation time is sensitive to the (poorly known) microscopic properties of matter of subnuclear density, such as the heat capacity, thermal conductivity, and superfluidity of free neutrons. We calculate the cooling models with the new values of the electron thermal conductivity in the inner crust, based on a realistic treatment of the shapes of atomic nuclei. Superfluid effects may shorten the relaxation time by a factor of 4. The comparison of theoretical cooling curves with observations provides a potentially powerful method of studying the properties of the neutron superfluid and highly unusual atomic nuclei in the inner crust.     

A numerical survey of neutron star crustal density profiles

An accurate numerical survey of the density profiles corresponding to the crusts of neutron stars for representative equation of state models is presented. This will find application in calculations of thermal and magnetic evolution of neutron stars.     

Seismic signatures of strange stars with crust

We study acoustic oscillations (eigenfrequencies, velocity distributions, damping times) of normal crusts of strange stars. These oscillations are very specific because of huge density jump at the interface between the normal crust and the strange matter core. The oscillation problem is shown to be self-similar. For a low (but non-zero) multipolarity l , the fundamental mode (without radial nodes) has a frequency of ∼300 Hz and mostly horizontal oscillation velocity; other pressure modes have frequencies ≳20 kHz and almost radial oscillation velocities. The latter modes are similar to radial oscillations (having approximately the same frequencies and radial velocity profiles). The oscillation spectrum of strange stars with crust differs from the spectrum of neutron stars. If detected, acoustic oscillations would allow one to discriminate between strange stars with crust and neutron stars and constrain the mass and radius of the star.     

The Crust Properties and Cooling of Strange Stars

We investigate the influence of the following parameters on the crust properties of strange stars: the strange quark mass (m _s), the strong coupling constant (α_c) and the vacuum energy density (B). It is found that the mass density at the crust base of strange stars cannot reach the neutron drip density. For a conventional parameter set of m _s=200 MeV, B ^1/4 = 145 MeV and α_c = 0.3, the maximum density at the crust base of a typical strange star is only 5.5 × 10¹⁰ gcm^-3, and correspondingly the maximum crust mass is 1.4 ×10^-6 M_⊙. Subsequently, we present the thermal structure and the cooling behavior of strange stars with crusts of different thickness, and under different diquark pairing gaps. Our work might provide important clues for distinguishing strange stars from neutron stars.     

Strange Stars： Can Their Crust Reach the Neutron Drip Density？

The electrostatic potential of electrons near the surface of static strange stars at zero temperature is studied within the frame of the MIT bag model. We find that for QCD parameters within rather wide ranges, if the nuclear crust on the strange star is at a density leading to neutron drip, then the electrostatic potential will be insufficient to establish an outwardly directed electric field, which is crucial for the survival of such a crust. If a minimum gap width of 200 fm is brought in as a more stringent constraint, then our calculations will completely rule out the possibility of such crusts. Therefore, our results argue against the existence of neutron-drip crusts in nature.     

Signal of quark deconfinement in thermal evolution neutron stars with deconfinement heating

As neutron stars spin-down and contract, the deconfinement phase transition can continue to occur, resulting in energy release (so-called deconfinement heating) in case of the first-order phase transition. The thermal evolution of neutron stars is investigated to combine phase transition and the related energy release self-consistently. We find that the appearance of deconfinement heating during spin-down result in not only the cooling delay but also the increase of surface temperature of stars. For stars characterized by intermediate and weak magnetic field strength, a period of increasing surface temperature could exist. Especially, a sharp jump in surface temperature can be produced as soon as quark matter appears in the core of stars with a weak magnetic field. We think that this may serve as evidence for the existence of deconfinement quark matter. The results show that deconfinement heating facilitates the emergence of such characteristic signature during the thermal evolution process of neutron stars.     

Spatial Distribution of Accreting Isolated Neutron Stars in the Galaxy

In the last few years, the spatial distribution of old, isolated neutron stars has become of great interest (see, for example, Treves and Colpi (1991)). Several sources of this size have been observed by ROSAT. We present here a computer model of the distribution of the luminosity produced by old, isolated neutron stars accreting from the interstellar medium.We use direct calculations of trajectories in the Galaxy potential, taken in the form given by Paczynski (1990). The system of differential equations was solved numerically. We made calculations on a grid with cell size 100 pc in the R direction and 10 pc in the Z direction (centered at R=50 pc, Z=5 pc and so on). Stars were born in the Galactic plane with a specified velocity distribution corresponding to non-symmetrical supernova explosions.In our model, we assumed that the birth rate of neutron stars is proportional to the square of the local density. The local density was calculated using the data and formulas of Bochkarev (1993) and Zane et al. (1995). We then calculated the luminosity using the Bondi formula (in the inner kiloparsec our results are only a rough estimate).We show that for various mean velocities for the old isolated neutron stars, the distribution of the luminosity has a torus-like structure, with the maximum at 5kpc. Since we made very general assumptions, we argue that this type of distribution is not unique to our Galaxy, and all spiral galaxies should have such a distribution of the luminosity density, associated with accreting old, isolated neutron stars.     

The de-excited energy of electron capture in accreting neutron star crusts

When a daughter nucleus produced by electron capture takes part in a level transition from an excited state to its ground state in accreting neutron star crusts, thermal energy will be released and heat the crust, increasing crust temperature and changing subsequent carbon ignition conditions. Previous studies show that the theoretical carbon ignition depth is deeper than the value inferred from observations because the thermal energy is not sufficient. In this paper, we present the de-excited energy from electron capture of rp-process ash before carbon ignition, especially for the initial evolution stage of rp-process ash, by using a level-to-level transition method. We find the theoretical column density of carbon ignition in the resulting superbursts and compare it with observations. The calculation of the electron capture process is based on a more reliable level-to-level transition, adopting new data from experiments or theoretical models(e.g., large-scale shell model and proton-neutron quasi-particle random phase approximation). The new carbon ignition depth is estimated by fitting from previous results of a nuclear reaction network. Our results show the average de-excited energy from electron capture before carbon ignition is ~0.026 Me V/u, which is significantly larger than the previous results. This energy is beneficial for enhancing the crust's temperature and decreasing the carbon ignition depth of superbursts.     

The effect of the neutron-star crust on the evolution of a core magnetic field

We consider the expulsion of the magnetic field from the super-conducting core of a neutron star and its subsequent decay in the crust. Particular attention is paid to a strong feedback of the distortion of magnetic field lines in the crust on the expulsion of the flux from the core. This causes a considerable delay in the core flux expulsion if the initial field strength is larger than 10¹¹ G. It is shown that the hypothesis on the magnetic field expulsion induced by the neutron-star spin-down is adequate only for a relatively weak initial magnetic field B ≈10¹¹ G. The expulsion time-scale depends not only on the conductivity of the crust, but also on the initial magnetic field strength itself. Our model of the field evolution naturally explains the existence of the residual magnetic field of neutron stars. Its strength is correlated with the impurity concentration in neutron-star crusts and anticorrelated with the initial field strengths.     

Cyclotron Lines: From Magnetic Field Strength Estimators to Geometry Tracers in Neutron Stars

With forty years since the discovery of the first cyclotron line in Her X-1, there have been remarkable advancements in the field related to the study of the physics of accreting neutron stars – cyclotron lines have been a major torchbearer in this regard, from being the only direct estimator of the magnetic field strength, a tracer of accretion geometry and an indicator of the emission beam in these systems. The main flurry of activities have centred around studying the harmonic separations, luminosity dependence, pulse phase dependence and more recently, the shapes of the line and the trend for long-term evolution in the line energy. This article visits the important results related to cyclotron lines since its discovery and reviews their significance. An emphasis is laid on pulse phase resolved spectroscopy and the important clues a joint timing and spectral study in this context can provide, to build a complete picture for the physics of accretion and hence X-ray emission in accreting neutron stars.     

Relaxation of the Vela pulsar’s angular velocity following its jumps

The dynamics of the vortex lattice in the inner crust of a neutron star is considered. A general equation of motion is obtained and solved under the assumption that there are regions of pinned and of free vortices. By comparing these solutions with observational data for the Vela pulsar, the relative moments of inertia of regions of relaxation with the corresponding characteristic times are calculated for two model stars with different equations of state. It is shown that the theory can be reconciled with observations of the relaxation of pulsar angular velocity only for model stars with extremely stiff equations of state. Translated from Astrofizika, Vol. 40, No. 1, pp. 67–76, January–March, 1997.     

A striking confluence between theory and observations of high-mass X-ray binary pulsars

We analyze the most powerful X-ray outbursts from neutron stars in eleven Magellanic high-mass X-ray binaries and three pulsating ultraluminous X-ray sources. Most of the outbursts rise to L_(max) which is about the level of the Eddington luminosity, while the remaining more powerful outbursts also appear to recognize that limit when their emissions are assumed to be anisotropic and beamed toward our direction. We use the measurements of pulsar spin periods P_S and their derivatives P_S to calculate the X-ray luminosities L_p in their faintest accreting("propeller-line") states. In five cases with unknown P_S, we use the lowest observed X-ray luminosities, which only adds to the heterogeneity of the sample. Then we calculate the ratios L_p/L_(max) and we obtain an outstanding confluence of theory and observations from which we conclude that work done on both fronts is accurate and the results are trustworthy: sources known to reside on the lowest Magellanic propeller line are all located on/near that line, whereas other sources jump higher and reach higher-lying propeller lines. These jumps can be interpreted in only one way, higher-lying pulsars have stronger surface magnetic fields in agreement with previous empirical results in whichP_S and L_p values were not used.     

Heating and cooling of magnetars with accreted envelopes

We study the thermal structure and evolution of magnetars as cooling neutron stars with a phenomenological heat source in an internal layer. We focus on the effect of magnetized (   B ≳ 10¹⁴  G) non-accreted and accreted outermost envelopes composed of different elements, from iron to hydrogen or helium. We discuss a combined effect of thermal conduction and neutrino emission in the outer neutron star crust and calculate the cooling of magnetars with a dipole magnetic field for various locations of the heat layer, heat rates and magnetic field strengths. Combined effects of strong magnetic fields and light-element composition simplify the interpretation of magnetars in our model: these effects allow one to interpret observations assuming less extreme (therefore, more realistic) heating. Massive magnetars, with fast neutrino cooling in their cores, can have higher thermal surface luminosity.