Deep Crustal Heating by Neutrinos from the Surface of Accreting Neutron Stars

Deep Crustal Heating by Neutrinos from the Surface of Accreting Neutron Stars


We present a new mechanism for deep crustal heating in accreting neutron stars. Charged pions () are produced in nuclear collisions on the neutron star surface during active accretion and upon decay they provide a flux of neutrinos into the neutron star crust. For massive and/or compact neutron stars, neutrinos deposit of heat per accreted nucleon into the inner crust. The strength of neutrino heating is comparable to the previously known sources of deep crustal heating, such as from pycnonuclear fusion reactions, and is relevant for studies of cooling neutron stars. We model the thermal evolution of a transient neutron star in a low-mass X-ray binary, and in the particular case of the neutron star MXB 1659-29 we show that additional deep crustal heating requires a higher thermal conductivity for the neutron star inner crust. A better knowledge of pion production cross sections near threshold would improve the accuracy of our predictions.

25.40.Qa, 25.40.Sc, 26.60.Gj, 26.60.Kp, 97.60.Jd

I Introduction

Neutron stars in X-ray binaries accrete matter from their companion stars. As matter accretes, the crust is continually compressed and undergoes a series of non-equilibrium nuclear reactions such as electron captures, neutron emissions and pycnonuclear fusion reactions that release per accreted nucleon Bisnovatyĭ-Kogan and Chechetkin (1979); Sato (1979); Haensel and Zdunik (1990, 2003, 2008). Energy release mainly occurs in the inner crust at mass densities between and is referred to as deep crustal heating. During an accretion outburst, deep crustal heating brings the entire crust out of thermal equilibrium with the core. When accretion ends and the neutron star enters quiescence, crust cooling powers an observable X-ray light curve Ushomirsky and Rutledge (2001); Rutledge et al. (2002); Shternin et al. (2007). Thermal evolution models of accreting neutron stars that include deep crustal heating successfully reproduce most observed quiescent X-ray light curves Brown and Cumming (2009). The cooling light curves of several sources, however, require an additional heat deposition in the outer crust during outburst to reach observed quiescent temperatures Cumming et al. (2006); Deibel et al. (2015). The source of extra heating remains unknown, but must be comparable in strength to the heat release from non-equilibrium nuclear reactions.

Here we discuss a new source of heating in neutron star crusts from the decay of charged pions on the neutron star surface. The neutron star’s strong gravity accelerates incoming particles to kinetic energies of several hundred MeV per nucleon before they reach the neutron star surface. The accreted matter, usually consisting of hydrogen or helium, undergoes nuclear collisions with the nuclei on the neutron star surface. Nuclear collisions produce pions, in particular , that upon decay emit a flux of neutrinos. Approximately half of these neutrinos carry their energy into the crust, where they experience multiple scatterings and are eventually absorbed in the inner crust. This neutrino heating provides an additional source for deep crustal heating. In this work we present the first calculations of deep crustal heating by neutrinos from the decay of stopped pions on the surface of neutron stars.

The paper is organized as follows. In Section II we develop the formalism required to discuss the energy deposition by neutrinos from the stopped pion decays. We will first briefly review the main mechanism of deep crustal heating by neutrinos in II.1. Following this discussion, we review the main steps involved in calculating the energy deposited from pion production on the surface of neutron stars in II.2. This section is closed by discussing at what depth in the crust neutrinos will deposit their energy in II.3. We then proceed to Section III to display the results of our calculations using various equations of state (EOS) as well as pion production in three possible nuclear collisions. This section ends with a discussion of the observational implications of deep crustal heating in understanding cooling light curves of neutron stars in X-ray transients. Finally, in Sec. IV we offer our summary and conclusions.

Ii Formalism

ii.1 The Main Mechanism

Neutron stars in low-mass X-ray binaries typically accumulate hydrogen-rich or helium-rich matter from the surface of their companions in an accretion disk that is later accreted onto the neutron star surface during an accretion outburst. During outburst, the incoming particles collide with the nuclei on the neutron star surface Bildsten et al. (1992) and can produce pions if the particle’s kinetic energy is above the pion production threshold of . Neutral pions decay almost instantaneously via releasing their energy at the surface. Neutrinos from the decay of negative pions may be strongly suppressed because are often absorbed, via strong interactions, before they can undergo a weak decay. Positively charged pions slow down and stop near the neutron star surface and decay into muons and muon neutrinos through . This produces monoenergetic muon neutrinos, , of energy MeV. The anti-muon subsequently decays through on a muon-decay time scale of , with a well-determined neutrino energy spectrum Scholberg (2006). Approximately half of the neutrinos produced escape the neutron star and the other half move into the crust carrying a total energy of


per accreted nucleon, where is the total number of ’s produced per accreted nucleon. In addition to gravitational acceleration, accreting particles may undergo electromagnetic acceleration in the strong electric and magnetic fields that are likely present. This could significantly increase pion and neutrino production, but we will explore this in later work.

ii.2 Pion Production per Accreted Nucleon

We now calculate the number of charged pions produced from infalling matter. Assuming that the infalling matter has zero velocity at infinity (free-falling), we estimate the kinetic energy of the accreted matter at the surface of the neutron star Bildsten et al. (1992) using


where is the Schwarzschild radius, is the mass of the infalling particle, and and are the neutron star mass and radius, respectively. If the kinetic energy of the incoming particles is sufficiently large, they will collide with the nuclei on the surface of the neutron star and can produce pions. The multiplicity of pion production, defined as the number of pions produced per collision event, strongly depends on the initial kinetic energy of the incoming particle as well as on the target nuclei that is composed of the mixture of light-to-medium nuclei. Since both incoming protons and -particles are charged particles, before they undergo a hard nuclear collision they partially lose energy due to interaction with atmospheric electrons. The energy loss of charged particles can be calculated using the Bethe-Bloch equation


where MeV mol cm, is the charge number of the incident particle (projectile), is the atomic number of the target, is the atomic mass of the target in g mol, , is the relativistic Lorentz factor, is the mean excitation energy in eV, and is the density effect correction to ionization energy loss, which is negligible for energies under consideration. Here


is the maximum kinetic energy which can be imparted to a free electron in a single collision. A complete description of the electronic energy loss by heavy particles can be found in Chapter 32 of Ref. Amsler et al. (2008). A similar study of the incident-beam particles deceleration through repeated Coulomb scatters from atmospheric electrons was also carried out in Ref. Bildsten et al. (1992). The energy of the particle that undergoes a hard nuclear collision is therefore


where is the strong interaction mean free path, is the number density of scattering centers, and is the strong collision cross section. Note that the energy loss depends on the initial beam energy through Lorentz parameters. Therefore, Eqn. (5) takes an exact form if one replaces with , where , and solves the equation


iteratively for all -values. The probability density function for the interaction of a particle after traveling a distance in the medium is given by Tavernier (2010)


If the incident beam energy per nucleon during hard collision is above the threshold energy of MeV pions are produced. The pion production multiplicity, , depends greatly on the kinetic energy of the incident particles as given by Eqn. (6). Here is the pion production cross section, whereas is total reaction cross section. We discuss in Sec. III. The total number of pions produced per infalling particle can be calculated as


where is the range, or the maximum possible distance the incoming charged particle can penetrate the matter before losing all its kinetic energy through electromagnetic energy loss.

ii.3 Optical Depth and Deep Crustal Heating

The neutrinos moving into the neutron star crust are first scattered and/or absorbed at mass densities which can be determined by the neutrino transport optical depth


where () is the neutrino-ion (neutrino-free neutron) transport cross section, is the ion number density in the crust, is the number of nucleons in the unit cell, is the number density of free neutrons, and is the number of free neutrons in a unit cell. The transport cross section is defined as


with the free-space differential cross section for neutrino-nucleon elastic scattering given by Horowitz (2002)


where is the scattering angle, is the incoming neutrino energy, is the vector coupling constant, and is the axial vector coupling constant. The neutrino-ion elastic scattering differential cross section is


where is the total weak charge of the ion with and , is the ground state elastic form factor of the ions, and is the four momentum transfer squared.

Notice that both and are functions of the neutrino energy. The neutrino energy spectrum from stopped pions is well known Scholberg (2006). To determine the neutrino-ion and neutrino-free neutron elastic scattering cross sections we use the root-mean-square neutrino energies calculated as


where is the neutrino flux with energy . In particular, we use the root-mean-squared values of and for electron and muon neutrinos, respectively (see Eqn. (1)) and .

By definition, represents the number of mean free paths for the neutrino traveling from the surface of the star at to some inner depth . Neutrinos are first scattered (absorbed) at a depth of corresponding to . We assume neutrinos are eventually absorbed near this optical depth. Electron neutrinos are most likely absorbed in the crust via inelastic neutrino charged current interactions, whereas muon neutrinos deliver most of their energies through muon neutrino-electron scatterings, and may leave the star once their energy is low corresponding to the large neutrino mean free path. The energy of as given by the Eqn. (1) is therefore delivered to the crust at depth of .

Iii Results

iii.1 Equations of State of Neutron-Star Matter

Model (km) (MeV) (km) (MeV)
FSU2 (soft)
FSU2 (stiff)
HLPS (soft)
HLPS (stiff)
Table 1: The radii of a 1.4 and neutron stars as well as the incoming kinetic energies of a nucleon (with mass of MeV) at the surface of neutron stars predicted by the four equations of state discussed in the text.

The equation of state adopted in this work is composed of several parts. Matter in the outer crust of the neutron star is organized into a Coulomb lattice of neutron-rich nuclei embedded in a degenerate electron gas. The composition in this region is solely determined by the masses of neutron-rich nuclei in the region of and the pressure support is provided primarily by the degenerate electrons. For this region we adopt the equation of state by Haensel, Zdunik and Dobaczewski (HZD) Haensel et al. (1989). The inner crust begins at the neutron-drip density of . The EOS for the inner crust at mass densities is, however, highly uncertain and must be inferred from theoretical calculations. In addition to a Coulomb lattice and an electron gas, the inner crust now includes a dilute vapor of quasi-free neutrons. Moreover, at the bottom layers of the inner crust, complex and exotic structures with almost equal energies referred to as “nuclear pasta” have been predicted to emerge Ravenhall et al. (1983); Hashimoto et al. (1984); Lorenz et al. (1993). For this region we use the EOS by Negele and Vautherin Negele and Vautherin (1973). The inner crust ends at a mass density near , beyond which the neutron star matter becomes uniform. For this uniform liquid core region we assume two equations of state that cover a wide range of uncertainties that currently exist in the determination of the equation of state of nuclear matter at normal and supra nuclear densities:

  • The relativistic mean-field model by Chen and Piekarewicz Chen and Piekarewicz (2014) (FSU2), whose parameters were calibrated to reproduce the ground-state properties of finite nuclei and their monopole response, as well as to account for the maximum neutron star mass observed to date Demorest et al. (2010); Antoniadis et al. (2013). Due to the lack of stringent isovector constraints, the original FSU2 predicts a relatively stiff symmetry energy of MeV with density slope of MeV. It is known that by tuning two purely isovector parameters of the RMF model one can generate a family of model interactions that have varying degrees of softness in the nuclear symmetry energy without compromising the success of the model in reproducing ground-state properties Horowitz and Piekarewicz (2001); Fattoyev et al. (2012). Following this scheme we tuned the purely isovector parameters of the FSU2 model to get MeV and MeV and refer to this model as the FSU2 (soft).

  • The soft and stiff equations of state that agree with the lower and upper limits of the EOS band derived from microscopic calculations of neutron matter are based on nuclear interactions from chiral effective field theory by Hebeler et al.Hebeler et al. (2010) (HLPS). Notice that the symmetry energy parameters in this model are MeV and MeV.

A recent survey on the mass spectrum of compact objects in X-ray binaries from 19 sources shows that their masses can be anywhere in the range of  Casares et al. (2017). Note that stars made with stiff equations of state can accelerate particles to near the pion-production threshold, whereas those with soft equations of state allow particles to gain kinetic energies significantly larger than the pion-production threshold even for low-mass neutron stars.

iii.2 Production of in -, - and - Collisions

Figure 1: (Color online) The multiplicity of pion production as a function of beam energy for - collisions (dash-dotted black line) estimated using the Monte-Carlo results of Ref. Burman et al. (1990) and in - collisions (dashed blue line) which is a polynomial fit to the experimental data from Daehnick et al. (1995); Hardie et al. (1997); Flammang et al. (1998); Schwaller et al. (1979) (blue diamond symbols). Also shown is multiplicities from the IBUU transport simulations in the - collision is also shown (solid red line).
Figure 2: (Color online) Same as Fig. 1 but now for ‘subthreshold’ energies.
Figure 3: (Color online) The multiplicity in -Fe collisions versus impact parameter for incident energies per nucleon of MeV calculated using IBUU transport simulations.

Charged pion production from the interaction of proton beams with some selected nuclei have been measured at incident energies of 585 MeV Crawford et al. (1980), 730 MeV  Cochran et al. (1972), as well as at 800 and 1600 MeV Denes et al. (1983); Lemaire et al. (1991). Inclusive pion production at lower incident energies of 330, 400, and 500 MeV from proton-nucleus collisions ( and ) nuclei have also been measured. However, measurements of pion production cross sections at medium beam energies for proton-nucleus collisions are still incomplete. On the other hand, the surface composition of neutron stars remains an outstanding problem Chang et al. (2010). For accreting neutron stars, the upper layer is likely composed of lighter elements such as hydrogen or helium, depending on the composition of accreted material from the companion star. For the sake of simplicity, instead of a range of target nuclei, we assume only two types of the target nuclei: protons and .

Based on the available experimental data Ref. Burman et al. (1990) performed a Monte-Carlo simulation to evaluate the total pion production cross sections at various proton beam energies on selected nuclei. Using these Monte-Carlo data and the - reaction cross sections we estimated pion multiplicities for - collision at incident beam energies above 325 MeV, see Fig. 1. Note that the pion production cross section in Monte-Carlo simulations is assumed to go to zero at energies below 325 MeV Burman et al. (1990). It is important to mention, however that pions can be produced at subthreshold energies via the excitation and decay of -resonances (See Ref. Tsang et al. (2017) and references therein). The Fermi motion of nucleons in nuclei can also greatly enhance the pion production cross sections in the vicinity of the threshold energy Bertsch (1977); Sandel et al. (1979). Despite efforts to measure subthreshold pion production in the past (See, Ref. Miller et al. (1993) and references therein), regrettably, experimental data on this front still remains incomplete.

In the case of accreting hydrogen the - collision becomes an interesting case Bildsten et al. (1992, 1993) to study pion production. Fortunately, there are sufficient experimental data available on the pion production in collisions. Using the experimental data from Daehnick et al. (1995); Hardie et al. (1997); Flammang et al. (1998); Schwaller et al. (1979) we plot the multiplicity of pion production as a function of the proton beam energy from the - collisions (see Fig. 1). The current experimental error-bars are in the order of 25% for most of these measurements, except for few cases when beam energies are in the range of MeV, the relative error-bars are as large as 80%.

On the other hand, experimental measurements of pion production in in - collisions are still missing. For this we use the isospin-dependent Boltzmann-Uehling-Uhlenbeck (IBUU) transport model Li (2002a, b) to calculate multiplicities at various incident beam energies per nucleon and impact parameters. In this model all subthreshold pions are produced from decays of low-mass (1232) resonances formed in nucleus-nucleus inelastic collisions. While the pion production cross sections drop sharply when the energy per nucleon is below threshold, there is an appreciable pion production cross section at incident beam energies as low as 150 MeV per nucleon mostly due to Fermi motion of nucleons in . Note that pion production in heavy-ion collisions depends on the EOS and the ratio of charged pions on the nuclear symmetry energy used. In this exploration study, we use a momentum-independent potential corresponding to a stiff EOS with MeV and a symmetry energy that is linear in density. In Fig. 2 we plot multiplicities as a function of the incident beam energies per nucleon for subthreshold energies. In our calculation for - collisions, we use multiplicities averaged over the impact parameter , whose dependence is plotted in Fig. 3. For head-on collisions with the pion-production cross section is obviously much larger. In generating these data we run events for most cases, except for low incident energies, where we used up to events to have a better statistics. The statistical error-bars for these simulations are . It is worth noting that the model dependence on the EOS and the density dependence of the symmetry energy is of the same order as the statistical errors quoted above. The detailed model dependencies near the pion production threshold are addressed in Ref. Li (2015).

iii.3 Neutrino Energy Deposition in the Inner Crust

Model (MeV) (MeV) (MeV)
FSU2 (stiff) [] [] []
FSU2 (soft) [] [] []
HLPS (stiff) [] [] []
HLPS (soft) [] [] []
Table 2: The total energy per accreted nucleon deposited by neutrinos in the inner crust for a 2.0 [1.4] neutron star using the four equations of state discussed in the text and for three possible reactions: , -Fe, and -Fe.
() () () (km) () () (km) () () (km)
FSU2 (soft)
HLPS (soft)
Table 3: The location in the neutron star inner crust, where neutrino transport optical depth equals . Here , , and are the mass density of the neutron-star matter in the crust where neutrinos with different flavors are first scattered (or absorbed), and are the corresponding column depth Brown and Cumming (2009) and radial depth, where is the local pressure and is the local gravitational acceleration Thorne (1977) at .
Figure 4: (Color online) The total energy deposited by neutrinos into the inner crust as a function of neutron start mass for the soft EOSs discussed in the text: HLPS (soft) and FSU2 (soft).
Figure 5: (Color online) The total energy deposited by neutrinos into the inner crust as a function of neutron star compactness, .

We now calculate the total energy carried by neutrinos into the inner crust. In Table 2 we present results for a and for a (in square brackets) neutron star and the four equations of state discussed in the text. In Fig. 4 we display the full results as a function of the neutron star mass for soft equations of state only. As is evident from Table 1, low-mass neutron stars can only accelerate the infalling matter to energies of about the pion-production threshold. Therefore the result is highly sensitive to the pion production cross section around threshold energies. This result calls for improved experimental measurements of pion production in proton-proton collisions, as well as for pion production in - and - collisions for beam energies per nucleon in the range of to MeV.

Moreover, there is a strong sensitivity of the pion production to the equation of state employed in determination of stellar structure. In particular, if the equation of state is very stiff—such as the HLPS (stiff)—then even for a neutron star the incoming particles are not accelerated enough to produce pions (See Tables 1 and  2). On the other hand, if the equation of state is soft, then for a neutron star the energy deposited by neutrinos can be as large as MeV per accreted nucleon. The result is more pronounced if helium is being accreted onto the surface of neutron star, mainly because the IBUU simulations suggest that a substantial amount of pions can be produced at subthreshold beam energies.

We also investigate the impact of the neutron star’s compactness, with compactness defined as . For example, while stars built with the HLPS (stiff) equation of state may not accelerate the infalling matter to high kinetic energies for low-mass stars, it can certainly do so for very massive neutron stars. In particular, for a , the total energy deposit is MeV, when -Fe (-Fe) collisions take place at the surface. To cover all possible equations of state in Fig. 5 we plot the result as a function of the compactness.

We find that the heat deposition from neutrinos is comparable with other previously known sources of deep crustal heating such as from pycnonuclear fusion reactions. In Table 3 we present our results for the more interesting cases of soft equations of state only, where is significant even for moderate-mass neutron stars. Depending on the mass of the star and the EOS used, the energy of is delivered to the regions of the inner crust where mass densities are of order to . For example, for a neutron star this would correspond to mass densities of in units of , or equivalently to baryon densities of , where g cm is the nuclear saturation density.

Figure 6: (Color online) The location in the neutron star inner crust, where neutrino transport optical depth equals for different angles of incidence .

Table 3 assumes that half of the neutrinos produced from the decay of stopped pions travel radially inward. In reality, the decay is isotropic and therefore it is worth to analyze the location of heat delivery as an angle of incidence of neutrinos. The fraction of the number of neutrinos within a cone with apex angle to the total number of neutrinos is equal to . Here corresponds to the angle of incidence in the radial direction, whereas corresponds to the direction horizontal to the surface. In Fig. 6 we display the location of heat deposition as a function of for a neutron star using HLPS (soft) EOS. The result shows that most of neutrinos are delivered to the deep region of the crust, and only a small fraction of them scatter at shallower regions.

Note that in our calculations above we did not take into account additional redshift effects as neutrinos go deeper into the crust. The effective neutrino energy should slightly increase due to the gravitational redshift by a factor of , where is the local gravitational potential Thorne (1977). However our calculations show this effect is because the crust is thin.

iii.4 Observational Implications

Cooling neutron stars

Non-equilibrium nuclear reactions during active accretion heat the neutron star crust out of thermal equilibrium with the core. When accretion stops, the crust cools toward thermal equilibrium with the core Brown et al. (1998); Ushomirsky and Rutledge (2001); Rutledge et al. (2002); Cackett et al. (2008). Crust cooling is observed as a quiescent X-ray light curve, with one of the most well studied examples being the cooling transient MXB 1659-29 Brown et al. (1998); Ushomirsky and Rutledge (2001); Cackett et al. (2008); Brown and Cumming (2009). Cooling observations at successively later times into quiescence probes successively deeper layers in the crust with increasingly longer thermal times Brown and Cumming (2009). In particular, it was shown that about a year into quiescence the shape of the cooling light curve is sensitive to the physics at mass densities greater than neutron drip corresponding to the inner crust Page and Reddy (2012). This suggests that cooling light curves of neutron stars in low-mass X-ray binaries one-to-three years after accretion outbursts should be sensitive to the additional deep crustal heating by neutrinos Brown and Cumming (2009).

Comparing our results with the heat released from pycnonuclear fusion reactions Haensel and Zdunik (2008) we notice that not only are they of the same order, but also the heat is deposited in the same density regions (crust layer). Subsequently we calculated the column depths where neutrinos are first scattered,  Brown and Cumming (2009). Here is the local pressure and is the local gravitational acceleration defined as Thorne (1977)


We find that the column depth values lie in the range of (See Table 3). Since the amount of heat deposited for massive stars is comparable to the heat released from pycnonuclear reactions, the observation of cooling light curves, in particular, could be used to help distinguish massive stars from the low-mass stars.

Figure 7: (Color online) The redshifted surface temperature in units of as a function of time into quiescence. For the details of each curve please refer to the text.

To analyze the sensitivity of crustal heating by neutrinos on the cooling curves, we simulate the thermal evolution of a neutron star crust using the thermal evolution code dStar Brown (2015), which solves the general relativistic heat diffusion equation. The detailed microphysics of the crust is discussed in Ref. Brown and Cumming (2009) and the parameters of the cooling model are described in Ref. Deibel et al. (2017). In particular, this model assumes an impurity parameter of throughout the crust, which is defined as


where is the number density of the nuclear species with number of protons, and is the average proton number of the crust composition.

In Fig. 7 we display the crust cooling curves for four possible cases. The solid black curve corresponds to the case without heat deposition from neutrinos in the inner crust with . The red dashed curve corresponds to the case when a MeV per accreted nucleon heat source is deposited at density regions of . The crust temperature is marginally increased by the neutrino heating because most of the additional heat is transported into the core.

We then examine two cases, with and without neutrino heating, but including a nuclear pasta layer in the inner crust. It is expected that nuclear pasta forms at densities above corresponding to the bottom layers of the inner crust. The thermal conductivity of nuclear pasta could be small, corresponding to a large impurity parameter Horowitz et al. (2015). The black short-dashed curve shows the case of no neutrino heating, but at densities of corresponding to nuclear pasta. Finally, in blue dash-dotted line we display a cooling curve that includes both nuclear pasta and the heat depositiion from neutrinos. The crust temperature is higher in these two cases, because the low thermal conductivity of the nuclear pasta layer prevents a large portion of heat from diffusing into the core.

As evident from Fig. 7, the additional heat source can make a noticeable change in the cooling light curves. The cooling rate depends on many other factors and in particular strongly depends on the crust thickness, which is usually small for massive stars. Moreover, as illustrated in Fig. 7 the low thermal conductivity corresponding to the nuclear pasta can strongly affect thermal diffusion time maintaining a temperature gradient between the neutron star’s inner crust and core for several hundred days into quiescence Deibel et al. (2017). Note that all of the above models use the same pairing gap model of neutron superfluid in the singlet state with the critical temperature profile given by Schwenk et al. Schwenk et al. (2003). We have also tested other superfluid pairing gap models, such as the one by Gandolfi et al. Gandolfi et al. (2008), and the results are qualitatively similar to Fig. 7.

Figure 8: (Color online) Observed cooling light curve of MXB 1659-29 (black data points) and thermal evolution models for three cases of neutrino heating. (a) Crust cooling models with the S03 pairing gap and no nuclear pasta. The models use an impurity parameter of for the entire crust and a core temperature of . (b) Crust cooling models with the G08 pairing gap and nuclear pasta. The models use an impurity parameter of for the crust and a layer representing nuclear pasta at mass densities . These models have a core temperature of .

Crust cooling in MXB 1659-29

As described above, neutrino deep crustal heating will noticeably increase the crust temperature and the shape of the cooling light curve. Here we investigate the impact of extra heating from neutrinos on the particular case of MXB 1659-29 that entered quiescence after an accretion outburst Wijnands et al. (2003, 2004) and cooled for before entering outburst once more Negoro et al. (2015). The late time cooling observations probe the thermal properties of the inner crust and make MXB 1659-29 an interesting test case for neutrino heating.

Our thermal evolution model of MXB 1659-29 uses a and neutron star at the observed outburst accretion rate of . The model includes a per accreted nucleon shallow heat source between and , consistent with the findings from Brown and Cumming (2009). Using a model without nuclear pasta, the cooling light curve is fit with an impurity parameter for the entire crust of and the S03 pairing gap Schwenk et al. (2003). We then test representative values of neutrino heating: and per accreted nucleon. As can be seen in Fig. 8, the model fit with becomes inconsistent with the observational data once neutrino heating is added to the inner crust. In order to reestablish a fit, the crust impurity parameter must be lowered to (corresponding to a higher crust thermal conductivity) as increases.

Alternatively, the cooling of MXB 1659-29 may be fit with a nuclear pasta layer in the crust if the G08 pairing gap model is used Gandolfi et al. (2008), as we demonstrate in panel (b) of Fig. 8. In this case, the low thermal conductivity of the nuclear pasta maintains a higher crust temperature during quiescence and a layer of normal neutrons forms at the base of the crust Deibel et al. (2017). Without neutrino heating, the cooling observations of MXB 1659-29 are fit with a crust impurity of and a pasta impurity parameter of . We find that, similar to the model without nuclear pasta, as neutrino heating is increased in the inner crust, the pasta impurity parameter must decrease to reestablish a fit to the observations.

Note that the cooling of MXB 1659-29 may be fit with other neutron star masses and radii Brown and Cumming (2009) and the results in Fig. 8 are for a fixed neutron star gravity (and crust thickness). Cooling light curve shapes are degenerate in several parameters, for example: the neutron star gravity, the crust impurity parameter, and the mass accretion rate. Because the effect of neutrino heating is difficult to delineate from the effects of other model parameters we therefore can not determine if neutrino heating is present during outburst. It is worth noting, however, that if deep crustal heating from neutrinos is present then existing constraints derived from cooling light curves will need to be revisited, likely requiring a higher crust thermal conductivity or a different neutron star gravity.

Iv Conclusion and Outlook

We presented a new mechanism of deep crustal heating of neutron stars in mass-transferring binaries by neutrinos that are decay remnants of charged pions produced at the surface of neutron stars. Our calculations showed that massive and compact stars can accelerate infalling matter to energies substantially larger than the pion-production threshold resulting in ample generation of neutrinos. Approximately half of these neutrinos travel into the inner crust and deposit per accreted nucleon for massive and compact stars.

The deep crustal heating from neutrinos is comparable in strength to pycnonuclear fusion reactions and other non-equilibrium nuclear reactions taking place during active accretion. Additional deep crustal heating will affect the cooling light curves of accreting neutron stars at late times into quiescence. The effect is most pronounced when the star is massive and might help distinguish high-mass stars from low-mass stars. In general, for a fixed neutron star gravity we find that additional deep crustal heating requires a higher thermal conductivity for the crust and the crust impurity parameter must be lowered. In the particular case of MXB 1659-29, for a model without nuclear pasta and the S03 pairing gap, is required if any neutrino heating is added. For a model with nuclear pasta and the G08 pairing gap, for nuclear pasta is needed if neutrino heating is present.

Our calculation of pion production assumes that the incoming protons are slowed by Coulomb collisions with atmospheric electrons Zel’dovich and Shakura (1969). Plasma instabilities or a collisionless shock may instead stop the proton beam (e.g. see Ref. Shapiro and Salpeter (1975)), reducing the rate of nuclear collisions. In addition, depending on the accretion geometry, the incoming particles may not have the full free-fall velocity, e.g. in disk accretion if the disk reaches all the way to the neutron star surface. Neutrino heating may operate only with a quasi-spherical accretion flow or if the neutron star lies within the last stable orbit (e.g. see discussion in Ref. Bildsten et al. (2003)).

There is also a strong sensitivity of our results to the pion production cross sections at near threshold energies. Pion production may play a significant role in stellar environments and in particular, a better knowledge of pion production cross sections in -, -, -, and - collisions for beam energies - MeV/nucleon may help to better understand the structure and transport properties of neutron star crusts from cooling observations.

We thank Professors Hans-Otto Meyer, Hendrik Schatz, and Rex Tayloe for many helpful discussions. FJF, CJH, and ZL are supported by the U.S. Department of Energy (DOE) grants DE-FG02-87ER40365 (Indiana University), DE-SC0008808 (NUCLEI SciDAC Collaboration) and by the National Science Foundation through XSEDE resources provided by the National Institute for Computational Sciences under grant TG-AST100014. EFB is supported by the US National Science Foundation under grant AST-1516969. AC is supported by an NSERC Discovery Grant and is a member of the Centre de Recherche en Astrophysique du Quebec (CRAQ). BAL is supported by the U.S. Department of Energy, Office of Science, under Award Number DE-SC0013702 (Texas A&M University-Commerce) and the National Natural Science Foundation of China under Grant No. 11320101004.


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