Rotochemical Heating Research Articles

Internal heating mechanisms in neutron stars

The cooling mechanisms of a neutron star (hereafter referred to as NS) have the potential to reveal important features of superdense matter. The values of the surface temperatures are known for a good sample of NSs aged [Formula: see text] years and, with some exceptions, can be explained through standard cooling mechanisms (neutrinos and photons emissions without internal heating), as confirmed in our study. However, for older objects the surface temperatures are in some cases higher than expected, and it is necessary to consider some kind of internal heating to explain these results. With this objective, we revisit in this paper the kinetic mechanisms of heating in NSs considering fermionic dark matter (DM) heating, rotochemical heating and magnetic field decay. Our results indicate that NSs older than [Formula: see text] years, such as some “Black Widows” (BW — a subset of binary systems in which only the upper bounds of surface temperatures are known) and old pulsars, in contrast to younger NSs, exhibit much higher surface temperatures than the values predicted by these three heating mechanisms. Furthermore, by restricting the DM heating parameters to the current values that were fitted and/or measured for the local DM density, masses and NS radii, the models studied here also do not reproduce the upper limits of the temperatures from the surface of BWs or the actual temperatures of other ancient pulsars. We conclude that if the upper limits for BWs are close to real temperatures, dark heating will not represent a convincing explanation of these results, indicating that rotochemical mechanisms may be favored.

Timing and evolution of PSR B0950+08

ABSTRACT We present timing solutions of PSR B0950+08, using 14 years of observations from the Nanshan 26-m Radio Telescope of Xinjiang Astronomical Observatory. The braking index of PSR B0950+08 varies from –367 392 to 168 883, which shows an oscillation with large amplitude (∼105) and uncertainty. Considering the variation of braking indices and the most probable kinematic age of PSR B0950+08, a model with long-term magnetic field decay modulated by short-term oscillations is proposed to explain the timing data. With this magnetic field decay model, we discuss the spin and thermal evolution of PSR B0950+08. The uncertainties of its age are also considered. The results show that three-component oscillations are the more reasonable for the spin-frequency derivative distributions of PSR B0950+08, and the initial spin period of PSR B0950+08 must be shorter than $97\rm \ ms$ when the age is equal to the lower bound of its kinematic age. The standard cooling model could explain the surface temperature of PSR B0950+08 with its most probable kinematic age. Vortex creep heating with a long-term magnetic field decay could maintain a relatively high temperature at the later stages of evolution and explain the thermal emission data of old and warm pulsars. Coupling with the long-term magnetic field decay, an explanation of the temperature of PSR B0950+08 with roto-chemical heating needs an implausibly short initial rotation period ($P_0 \lesssim 17\rm { ms}$). The spin and thermal evolution of pulsars should be studied simultaneously. Future timing, ultraviolet or X-ray observations are essential for studying the evolution and interior properties of pulsars.

Long-lasting accretion-powered chemical heating of millisecond pulsars

ABSTRACT We analyse the effect of magnetic field in superconducting neutron star (NS) cores on the chemical heating of millisecond pulsars (MSPs). We argue that the magnetic field destroys proton superconductivity in some volume fraction of the stellar core, thus allowing for unsuppressed non-equilibrium reactions of particle mutual transformations there. The reactions transform the chemical energy, accumulated by an NS core during the low-mass X-ray binary stage, into heat. This heating may keep an NS warm at the MSP stage (with the surface temperature $\sim 10^5\, \rm K$) for more than a billion of years after ceasing of accretion, without appealing to the rotochemical heating mechanism.

Roto-Chemical Heating with Fall-Back Disk Accretion in the Neutron Stars Containing Quark Matter

Probing quark matter is one of the important tasks in the studies of neutron stars (NS). Some works explicitly consider the existence of quark matter in the appearance of hybrid star (HS) or pure quark star (QS). In the present work, we study the roto-chemical heating with accretion in HS and QS, and compare their chemical evolution and cooling features with pure NS. Different from HS and NS, there are two jumps in the chemical evolution of QS, which results from the fast direct Urca (Durca) reaction causing the fast recovery to chemical balance. However, the sudden change in the chemical evolution doesn’t provide an obvious heating effect in the thermal evolution. Differently, the roto-chemical heating effect appears both in the accretion phase and spin-down phase of the HS, and the heating platform in the accretion phase relies on the accretion rate. Larger accretion rate results in larger chemical deviation, higher and longer heating platform, and earlier appearance of the heating effect. Interestingly, with the disappearance of the heating effect in the accretion phase, the surface temperature drops fast, which is another possibility of the rapid cooling trend of the NS in Cas A. Furthermore, the surface temperature of the QS is obviously lower than the HS and NS, which is a latent candidate for the explanation of the old classical pulsar J2144-3933 with the lowest known surface temperature.

Cooling theory faced with old warm neutron stars: role of non-equilibrium processes with proton and neutron gaps

ABSTRACT Recent observations have found several candidates for old warm neutron stars whose surface temperatures are above the prediction of the standard neutron star cooling scenario, and, thus, require some heating mechanism. Motivated by these observations, we study the non-equilibrium beta process in the minimal cooling scenario of neutron stars, which inevitably occurs in pulsars. This out-of-equilibrium process yields the late-time heating in the core of a neutron star, called the rotochemical heating, and significantly changes the time evolution of the neutron star surface temperature. To perform a realistic analysis of this heating effect, we include the proton-singlet- and neutron-triplet-pairing gaps simultaneously in the calculation of the rate and emissivity of this process, where the dependence of these pairing gaps on the nucleon density is also taken into account. We then compare the predicted surface temperature of neutron stars with the latest observational data. We show that the simultaneous inclusion of both proton and neutron gaps is advantageous for the explanation of the old warm neutron stars since it enhances the heating effect. It is then found that the observed surface temperatures of the old warm millisecond pulsars, J2124−3358 and J0437−4715, are explained for various choices of nucleon gap models. The same set-up is compatible with the observed temperatures of ordinary pulsars, including old warm ones, J0108−1431 and B0950+08, by choosing the initial rotational period of each neutron star accordingly. In particular, the upper limit on the surface temperature of J2144−3933 can be satisfied if its initial period is $\gtrsim 10\, \mathrm{ms}$.

Dark matter heating vs. rotochemical heating in old neutron stars

Dark matter (DM) particles in the Universe accumulate in neutron stars (NSs) through their interactions with ordinary matter. It has been known that their annihilation inside the NS core causes late-time heating, with which the surface temperature becomes a constant value of Ts≃(2−3)×103 K for the NS age t≳106−7 years. This conclusion is, however, drawn based on the assumption that the beta equilibrium is maintained in NSs throughout their life, which turns out to be invalid for rotating pulsars. The slowdown in the pulsar rotation drives the NS matter out of beta equilibrium, and the resultant imbalance in chemical potentials induces late-time heating, dubbed as rotochemical heating. This effect can heat a NS up to Ts≃106 K for t≃106−7 years. In fact, recent observations found several old NSs whose surface temperature is much higher than the prediction of the standard cooling scenario and is consistent with the rotochemical heating. Motivated by these observations, in this letter, we reevaluate the significance of the DM heating in NSs, including the effect of the rotochemical heating. We then show that the signature of DM heating can still be detected in old ordinary pulsars, while it is concealed by the rotochemical heating for old millisecond pulsars. To confirm the evidence for the DM heating, however, it is necessary to improve our knowledge on nucleon pairing gaps as well as to evaluate the initial period of the pulsars accurately. In any cases, a discovery of a very cold NS can give a robust constraint on the DM heating, and thus on DM models. To demonstrate this, as an example, we also discuss the case that the DM is the neutral component of an electroweak multiplet, and show that an observation of a NS with Ts≲103 K imposes a stringent constraint on such a DM candidate.

Hubble Space Telescope Nondetection of PSR J2144–3933: The Coldest Known Neutron Star∗

Abstract We report nondetections of the ∼3 × 108 yr old, slow, isolated, rotation-powered pulsar PSR J2144–3933 in observations with the Hubble Space Telescope in one optical band (F475X) and two far-ultraviolet bands (F125LP and F140LP), yielding upper bounds , , and , at the pivot wavelengths 4940 Å, 1438 Å and 1528 Å, respectively. Assuming a blackbody spectrum, we deduce a conservative upper bound on the surface (unredshifted) temperature of the pulsar of 42,000 K. This makes PSR J2144–3933 the coldest known neutron star, allowing us to study thermal evolution models of old neutron stars. This temperature is consistent with models with either direct or modified Urca reactions including rotochemical heating, and, considering frictional heating from the motion of neutron vortex lines, it puts an upper bound on the excess angular momentum in the neutron superfluid, J < 1044 erg s.

Roto-chemical heating in a neutron star with fall-back disc accretion

Roto-chemical heating in a neutron star with fall-back disc accretion

Thermal evolution of neutron stars with decaying magnetic fields

Rotochemical heating originates in the deviation from beta equilibrium due to spin-down compression, which is closely related to the dipole magnetic field. We numerically calculate the deviation from chemical equilibrium and thermal evolution of neutron stars with decaying magnetic fields. We find that the power-law long term decay of the magnetic field slightly affects the deviation from chemical equilibrium and surface temperature. However, the magnetic decay leads to older neutron stars that could have a different surface temperature with the same magnetic field strength. That is, older neutron stars with a low magnetic field (108 G) could have a lower temperature even with rotochemical heating in operation, which probably explains the lack of other observations on older millisecond pulsars with higher surface temperature, except millisecond pulsar J0437–4715.

Rotochemical heating of millisecond and classical pulsars with anisotropic and density-dependent superfluid gap models

When a rotating neutron star loses angular momentum, the progressive reduction of the centrifugal force makes it contract. This perturbs each fluid element, raising the local pressure and originating deviations from beta equilibrium, inducing reactions that release heat (rotochemical heating). This effect has previously been studied by Fern\'andez & Reisenegger (2005) for non-superfluid neutron stars and by Petrovich & Reisenegger (2010) for superfluid millisecond pulsars. Both studies found that pulsars reach a quasi-steady state in which the compression driving the matter out of beta equilibrium is balanced by the reactions trying to restore the equilibrium. We extend previous studies by considering the effect of density-dependence and anisotropy of the superfluid energy gaps, for the case in which the dominant reactions are the modified Urca processes, the protons are non-superconducting, and the neutron superfluidity is parametrized by models proposed in the literature. By comparing our predictions with the surface temperature of the millisecond pulsar PSR J0437-4715 and upper limits for twenty-one classical pulsars, we find the millisecond pulsar can be only explained by the models with the effectively largest energy gaps (type B models), the classical pulsars require with the gap models that vanish for some angle (type C) and two different envelope compositions. Thus, no single model for neutron superfluidity can simultaneously account for the thermal emission of all available observations of non-accreting neutron stars, possibly due to our neglect of proton superconductivity.

Rotochemical heating in hybrid stars with modified Urca reactions in operation

We have recently shown that, as a compact star containing mixed-phase matter slows down, the compression can cause deconfinement phase transition, and thus enhance the chemical deviations and raise the chemical heating efficiency. In a previous study, only the direct Urca processes in nucleon and quark matter were considered. In this work, we extend the previous analysis to the case where the much slower modified Urca processes operate in nucleon matter. We find a fast promotion in the surface effective temperature of hybrid stars, and that the cooling process is dominated by both the nucleon and quark channels.

Long-period thermal oscillations in superfluid millisecond pulsars

In previous papers, we have shown that, as the rotation of a neutron star slows down, it will be internally heated as a consequence of the progressively changing mix of particles (rotochemical heating). In previously studied cases (non-superfluid neutron stars or superfluid stars with only modified Urca reactions), this leads to a quasi-steady state in which the star radiates thermal photons for a long time, possibly accounting for the ultraviolet radiation observed from the millisecond pulsar J0437-4715. For the first time, we explore the phenomenology of rotochemical heating with direct Urca reactions and uniform and isotropic superfluid energy gaps of different sizes. We first do exploratory work by integrating the thermal and chemical evolution equations numerically for different energy gaps, which uncovers a rich phenomenology of stable and unstable solutions. To understand these, we perform a stability analysis around the quasi-steady state, identifying the characteristic times of growing, decaying, and oscillating solutions. For small gaps, the phenomenology is similar to the previously studied cases, in the sense that the solutions quickly converge to a quasi-steady state. For large gaps ($\gtrsim 0.05$ MeV), these solutions become unstable, leading to a limit-cycle behavior of periodicity $\sim 10^{6-7} yr, in which the star is hot ($T_s\gtrsim 10^5$ K) for a small fraction of the cycle ($\sim 5- 20 %$), and cold for a longer time.

Cooling of Akmal-Pandharipande-Ravenhall neutron stars with a rotochemical heating source

Employing phenomenological density-dependent critical temperatures of strong singlet-state proton pairing and of moderate triplet-state neutron pairing, we investigate the effects of rotochemical heating on the thermal evolution of superfluid neutron stars whose cores consist of npe matter with the Akmal-Pandharipande-Ravenhall equation of state. Since the star is not quite in the weak interaction equilibrium state during spin-down, the departure from the chemical equilibrium leads to the rotochemical heating in a rotating NS which will increase the stellar's temperature. Our calculations show that the rotochemical heating delays the cooling of superfluid neutron stars considerably and makes the previous classification of NS cooling ambiguous. What's more, our model is currently consistent with all the observational data, and in particular some middle-aged and cold NSs (PRS J0205+6449 in 3C 58, PRS J1357-6429, RX J007.0+7303 in CTA 1, Vela) can be better explained when taking into account rotochemical heating.

Internal heating of old neutron stars: contrasting different mechanisms

Context: The standard cooling models of neutron stars predict temperatures $T<10^{4}$ K for ages $t>10^{7}$ yr. However, the likely thermal emission detected from the millisecond pulsar J0437-4715, of spin-down age $t_s \sim 7\times10^9$ yr, implies a temperature $T\sim 10^5$ K. Thus, a heating mechanism needs to be added to the cooling models in order to obtain agreement between theory and observation. Aims: Several internal heating mechanisms could be operating in neutron stars, such as magnetic field decay, dark matter accretion, crust cracking, superfluid vortex creep, and non-equilibrium reactions ("rotochemical heating"). We study these mechanisms in order to establish which could be the dominant source of thermal emission from old pulsars. Methods: We show by simple estimates that magnetic field decay, dark matter accretion, and crust cracking mechanism are unlikely to have a significant effect on old neutron stars. The thermal evolution for the other mechanisms is computed using the code of Fern\'andez and Reisenegger. Given the dependence of the heating mechanisms on the spin-down parameters, we study the thermal evolution for two types of pulsars: young, slowly rotating "classical" pulsars and old, fast rotating millisecond pulsars. Results: We find that magnetic field decay, dark matter accretion, and crust cracking do not produce detectable heating of old pulsars. Rotochemical heating and vortex creep can be important both for classical pulsars and millisecond pulsars. More restrictive upper limits on the surface temperatures of classical pulsars could rule out vortex creep as the main source of thermal emission. Rotochemical heating in classical pulsars is driven by the chemical imbalance built up during their early spin-down, and therefore strongly sensitive to their initial rotation period.

Rotochemical heating in millisecond pulsars: modified Urca reactions with uniform Cooper pairing gaps

Context: When a rotating neutron star loses angular momentum, the reduction in the centrifugal force makes it contract. This perturbs each fluid element, raising the local pressure and originating deviations from beta equilibrium that enhance the neutrino emissivity and produce thermal energy. This mechanism is named rotochemical heating and has previously been studied for neutron stars of nonsuperfluid matter, finding that they reach a quasi-steady configuration in which the rate at which the spin-down modifies the equilibrium concentrations is the same at which neutrino reactions restore the equilibrium. Aims: We describe the thermal effects of Cooper pairing with spatially uniform energy gaps of neutrons \Delta_n and protons \Delta_p on the rotochemical heating in millisecond pulsars (MSPs) when only modified Urca reactions are allowed. By this, we may determine the amplitude of the superfluid energy gaps for the neutron and protons needed to produce different thermal evolution of MSPs. Results: We find that the chemical imbalances in the star grow up to the threshold value \Delta_{thr}= min(\Delta_n+ 3\Delta_p, 3\Delta_n+\Delta_p), which is higher than the quasi-steady state achieved in absence of superfluidity. Therefore, the superfluid MSPs will take longer to reach the quasi-steady state than their nonsuperfluid counterparts, and they will have a higher a luminosity in this state, given by L_\gamma ~ (1-4) 10^{32}\Delta_{thr}/MeV \dot{P}_{-20}/P_{ms}^3 erg s^-1. We can explain the UV emission of the PSR J0437-4715 for 0.05 MeV<\Delta_{thr}<0.45 MeV.

Internal heating and thermal emission from old neutron stars

The equilibrium composition of neutron star matter is achieved through weak interactions (direct and inverse beta decays), which proceed on relatively long time scales. If the density of a matter element is perturbed, it will relax to the new chemical equilibrium through non-equilibrium reactions, which produce entropy that is partly released through neutrino emission, while a similar fraction heats the matter and is eventually radiated as thermal photons. We examined two possible mechanisms causing such density perturbations: (1) the reduction in centrifugal force caused by spin-down (particularly in millisecond pulsars), leading to rotochemical heating, and (2) a hypothetical time-variation of the gravitational constant, as predicted by some theories of gravity and current cosmological models, leading to gravitochemical heating. If only slow weak interactions are allowed in the neutron star (modified Urca reactions, with or without Cooper pairing), rotochemical heating can account for the observed ultraviolet emission from the closest millisecond pulsar, PSR J0437-4715, which also provides a constraint on |dG/dt| of the same order as the best available in the literature.

Rotochemical Heating of Neutron Stars: Rigorous Formalism with Electrostatic Potential Perturbations

The electrostatic potential that keeps approximate charge neutrality in neutron star matter is self-consistently introduced into the formalism for "rotochemical heating" presented in a previous paper by Fernández and Reisenegger. Although the new formalism is more rigorous, we show that its observable consequences are indistinguishable from those of the previous one, leaving the conclusions of the previous paper unchanged.

Rotochemical Heating in Millisecond Pulsars: Formalism and Nonsuperfluid Case

Rotochemical heating originates in a departure from beta equilibrium due to spin-down compression in a rotating neutron star. The main consequence is that the star eventually arrives at a quasi-equilibrium state, in which the thermal photon luminosity depends only on the current value of the spin-down power, which is directly measurable. Only in millisecond pulsars the spin-down power remains high long enough for this state to be reached with a substantial luminosity. We report an extensive study of the effect of this heating mechanism on the thermal evolution of millisecond pulsars, developing a general formalism in the slow-rotation approximation of general relativity that takes the spatial structure of the star fully into account, and using a sample of realistic equations of state to solve the non-superfluid case numerically. We show that nearly all observed millisecond pulsars are very likely to be in the quasi-equilibrium state. Our predicted quasi-equilibrium temperatures for PSR J0437-4715 are only 20% lower than inferred from observations. Accounting for superfluidity should increase the predicted value.

Constraining Dense Matter Superfluidity through Thermal Emission from Millisecond Pulsars

As a neutron star spins down, the gradual decrease of the centrifugal force produces a progressive increase of the density of any given fluid element in its interior. Since the ``chemical'' (or ``beta'') equilibrium state is determined by the local density, this process leads to a chemical imbalance quantified by a chemical potential difference, e.g., \delta\mu=\mu_n-\mu_p-\mu_e, where n, p, and e denote neutrons, protons, and electrons. In the presence of superfluid energy gaps, in this case \Delta_n and \Delta_p, reactions are strongly inhibited as long as both \delta\mu and kT are much smaller than the gaps. Thus, no restoring mechanism is available, and the imbalance will grow unimpeded until \delta\mu=\delta\mu_{thr}=\Delta_n+\Delta_p. At this threshold, the reaction rate increases dramatically, preventing further growth of \delta\mu, and converting the excess chemical energy into heat. The thermal luminosity resulting from this ``rotochemical heating'' process is $L\sim 2\times 10^{-4}(\delta\mu_{thr}/0.1\MeV)\dot E_{rot}$, similar to the typical x-ray luminosity of pulsars with spin-down power \dot E_{rot}. The threshold imbalance, and therefore the luminous stage, are only reached by millisecond pulsars. A preliminary study of eleven millisecond pulsars with reported ROSAT observations shows that the latter can already be used to start constraining superfluid energy gaps in the theoretically interesting range, ~ 0.1 - 1 MeV.