A&A 374, 213-226 (2001)
DOI: 10.1051/0004-6361:20010698
A. Y. Potekhin - D. G. Yakovlev
Ioffe Physico-Technical Institute, Politekhnicheskaya 26, 194021 St. Petersburg, Russia
Received 19 February 2001 / Accepted 17 April 2001
Abstract
The thermal structure of neutron stars with magnetized envelopes
is studied using modern physics input.
The relation between the internal (
)
and local surface temperatures is calculated
and fitted by analytic expressions
for magnetic field strengths B from 0 to 10^{16} G
and arbitrary inclination of the field lines to the surface.
The luminosity of a neutron star with dipole magnetic field
is calculated and fitted as a function
of B,
,
stellar mass and radius.
In addition, we simulate cooling
of neutron stars with magnetized envelopes.
In particular, we analyse ultramagnetized envelopes
of magnetars and also the effects of the magnetic field
of the Vela pulsar on the determination of
critical temperatures of neutron and proton superfluids
in its core.
Key words: stars: neutron - dense matter - conduction - magnetic fields
Most of the observed NSs possess magnetic fields -10^{13} G (e.g., Taylor et al. 1993). Some NSs are possibly magnetars, with B>10^{14} G (e.g., Thompson & Duncan 1995; Kouveliotou et al. 1998,1999; Mereghetti 2001). The internal NS magnetic field can be even higher. The strong magnetic field affects physical properties of all NS layers in many ways. For instance, the field G may affect the neutrino emissivity in the NS core (e.g., Baiko & Yakovlev 1999). The field G may change the electrical resistivity of NS cores, accelerating evolution of the internal field. In principle the thermal evolution may be coupled to the magnetic one (e.g., Urpin & Shalybkov 1995).
Thus modelling of the thermal structure and evolution of the magnetized NSs is a complicated task. In this paper we focus on two problems. First we consider the thermal structure of the outer magnetized NS envelope of density g cm^{-3}with the magnetic field G. These envelopes produce thermal insulation (blanketing) of NS interiors. We solve this problem using updated physics input described in Sect. 2. The solution (Sect. 3) relates the internal NS temperature with the local effective surface temperature and, therefore, with the integrated NS luminosity (or the mean effective surface temperature, ). Second, in order to illustrate the obtained relation, we simulate (Sect. 4) cooling of NSs which possess a given dipole or radial magnetic field in the outer envelopes.
The effects of a strong magnetic field
on thermodynamic and kinetic properties
of the outer NS layers
have been reviewed, for instance, by
Yakovlev & Kaminker (1994) and
Ventura & Potekhin (2001).
The field affects the properties
of all plasma components, the electron
component usually being affected most strongly.
Motion of free electrons perpendicular to the field lines
is quantized in Landau orbitals
with a characteristic transverse scale equal to
the magnetic length
,
where
is the electron Compton wavelength,
is the magnetic field strength expressed in relativistic units,
is the electron cyclotron frequency,
and
G).
Except for the outermost parts of the NS envelopes,
the electrons constitute degenerate, almost ideal gas.
The electron entropy,
magnetization, thermal and electrical conductivities,
and other quantities
exhibit quantum oscillations of the de Haas-van Alphen type.
The oscillations occur under variations of density
or Bwhenever the electron Fermi momentum reaches the characteristic values
associated with occupation of Landau levels
n=1,2,...These oscillations appreciably change the properties
of degenerate electrons in the
limit of a strongly quantizing field
(e.g., Yakovlev & Kaminker 1994)
in which almost all electrons populate the ground Landau level.
The latter case takes place
at temperature
and density
,
where
The thermal structure of magnetized NS envelopes has been studied by a number of authors mainly using a plane-parallel approximation. More attention has been paid to the case of the radial magnetic field (normal to the surface). It has been thoroughly considered by Hernquist (1985), Van Riper (1988), Schaaf (1990a), Heyl & Hernquist (1998b) for G. In another paper Heyl & Hernquist (1998a) analysed the case of higher fields, -10^{16} G. One can consult the cited papers for references to earlier works. The principal conclusion of these studies is that the magnetic field reduces thermal insulation of the blanketing envelope by increasing the longitudinal (along the field lines) thermal conductivity of degenerate electrons due to Landau quantization of electron motion.
The thermal structure of the envelope with the magnetic field tangential to the surface has been analysed by Hernquist (1985), Schaaf (1990a), and Heyl & Hernquist (1998a) for G. In this case the field increases thermal insulation of the blanketing envelope due to the classical effect of reduction of the electron thermal conductivity transverse to the field because of the Larmor rotation.
The case of arbitrary inclination of the field to the surface was studied by Greenstein & Hartke (1983) in the approximation of constant (density and temperature independent) longitudinal and transverse thermal conductivities. The authors proposed a very simple formula (Sect. 3.3) which relates the local surface and internal stellar temperatures, and . It is constructed from two relations obtained for the radial and tangential magnetic fields. Page (1995) presented arguments that the formula of Greenstein & Hartke is valid also for realistic, variable thermal conductivities. If so, one immediately gets the required relation for any magnetic field inclination using the realistic relations for the radial and tangential fields. This method has been used by several authors (e.g., Shibanov & Yakovlev 1996). Recently, the case of arbitrary field inclination has been studied also by Heyl & Hernquist (1998b). In Sect. 3 we reconsider the thermal structure of the blanketing envelopes for any magnetic field inclination and compare our results with those of earlier studies.
Early simulations of cooling of magnetized NSs were performed assuming the radial magnetic field everywhere over the stellar surface (e.g., Nomoto & Tsuruta 1987; Van Riper 1991). Since the radial magnetic field reduces the thermal insulation, these theories predicted acceleration of cooling of the magnetized NS accompanied by enhanced NS luminosity at the early (neutrino-dominated) cooling stage (at stellar ages yr). Page (1995) and Shibanov & Yakovlev (1996) simulated cooling of NSs with dipole magnetic fields. They showed that the decrease of thermal insulation of the stellar envelope near the magnetic equator partly compensated by its increase near the pole, and the magnetic field did not necessarily accelerate the cooling, in agreement with an earlier conjecture of Hernquist (1985). In a series of papers Heyl & Hernquist (1997a,1997b,1998a,1998b) proposed simplified models of cooling of magnetized NSs including the cases of ultrahigh surface magnetic fields, G.
We illustrate our new models of magnetized NS envelopes with simulation of cooling of NSs (Sect. 4). We briefly discuss cooling of ultramagnetized NSs as well as cooling models of the Vela pulsar with the dipole magnetic field and superfluid core.
Integration of Eq. (6) gives a temperature profile where is the optical depth, and the integration constant corresponds to the Eddington approximation ( at the radiative surface, where ). A more accurate boundary condition requires solution of the radiative transfer equation in the NS atmosphere (e.g., Shibanov et al. 1998).
At the general relativistic equation of hydrostatic equilibrium
can be reduced to the Newtonian form:
,
where
is the surface gravity.
Together with Eq. (6), it leads to the thermal structure equation
The thermal conductivity tensor of magnetized plasma is anisotropic.
It is characterized by the conductivities parallel ()
and transverse (
)
to the field, and by the
off-diagonal (Hall) component.
In the plane-parallel approximation Eq. (6)
contains the effective thermal conductivity
(8) |
Assuming the dipole field, we
can use the general-relativistic solution
(Ginzburg & Ozernoy 1964)
(11) |
The quantities , and Lrefer to a local reference frame at the NS surface. The redshifted ("apparent'') quantities as detected by a distant observer are (Thorne 1977): , , .
Crucial for the thermal evolution is the relation between and . For non-magnetic blanketing envelopes composed of iron, this relation was studied by Gudmundsson et al. (1983), while the non-magnetic envelopes of various chemical compositions were analysed by PCY. Ventura & Potekhin (2001) presented an analytic analysis of the relation for the envelopes without magnetic fields and with strong magnetic fields; Heyl & Hernquist (1998a) performed a semianalytic investigation of the case of strong magnetic field. The results of Ventura & Potekhin (2001) and Heyl & Hernquist (1998a) are rather approximate because of a number of simplifying assumptions discussed by Ventura & Potekhin (2001).
The validity of the one-dimensional approximation can be checked by direct two-dimensional simulation of the heat transport in the blanketing envelope. Such simulation has been attempted by Schaaf (1990b) for a homogeneously magnetized NS under many simplified assumptions, so that a more realistic study is required. The heat conduction from hotter to cooler zones along the surface or possible meridional and convective motions can smooth the temperature variations over the NS surface. Nevertheless, the one-dimensional approximation seems to be sufficient for our cooling calculations presented below.
The outer envelope is assumed to be composed of iron, which can be partially ionized at . Following PCY, we employ the mean-ion approximation and adjust an effective ion charge to a more elaborate EOS: the Opacity Library (OPAL) EOS (Rogers et al. 1996) at B=0 and the Thomas-Fermi EOS of Thorolfsson et al. (1998) at B=(10^{10}-10^{13}) G. The tabular entries of these EOSs in and T are interpolated in the same way as in PCY. When necessary, we interpolate the effective charge at intermediate B. Since no reliable EOS of iron has been published for B>10^{13} G, we use the effective charge obtained at B=10^{13} G for higher B.
Thus, in our model of the outer envelope, the pressure
is produced by magnetized Fermi gas of electrons and
by the gas of classical ions
with an effective charge.
We neglect the anomalous magnetic moments
of the nuclei, which is a good approximation at
G
(cf. Broderick et al. 2000; Suh & Mathews 2001).
The electron pressure can be expressed through
the chemical potential
and T
(e.g., Blandford & Hernquist 1982):
Typically, the radiative conduction dominates ( ) in the outermost non-degenerate layers of a NS, whereas the electron conduction dominates ( ) in deeper, moderately and strongly degenerate layers. The relation depends mainly on the conductivities in the sensitivity strip on the plane (Gudmundsson et al. 1983) placed near the turning point, where .
At B=0, the second process is described by
the well-known Thomson cross section
in the non-relativistic limit
(e.g., Berestetskii et al. 1992).
The corresponding opacity is
The free-free absorption coefficient
at B=0 was calculated
by Karzas & Latter (1961)
as a function of the electron
velocity and photon frequency .
Hummer (1988) produced an accurate fit to
its thermal average in the range
and
,
where
(18) |
Figure 1: relations at B=0 and cm s^{-2}, obtained using OPAL radiative opacities (solid line) and the free-free+Thomson opacities described in the text (dashed line). Lower window: fractional differences between the accurate numerical values of and those obtained using (i) the free-free+Thomson opacity (dashed line) and (ii) the older analytic formula of PCY (dot-dashed line). | |
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Interaction of photons with magnetized plasma
depends on their polarization and propagation direction.
Accordingly, radiative thermal conductivities
and associated opacities along and across the field
become different.
Rosseland mean opacities
and
were calculated by Silant'ev & Yakovlev (1980)
for various values
of T and B.
We have fitted their numerical results by
n | 1 | 2 | 3 | 4 | 5 | 6 |
a_{n} | 0.0949 | 0.1619 | 0.2587 | 0.3418 | 0.4760 | 0.2533 |
b_{n} | 0.0610 | 0.1400 | 0.1941 | 0.0415 | 0.3115 | 0.1547 |
c_{n} | 0.090 | 0.0993 | 0.0533 | 2.15 | 0.2377 | 0.231 |
Asymptotic behaviour of Eqs. (21) and (22) at agrees with theoretical results of Silant'ev & Yakovlev (1980): at f=0 (Thomson scattering); at f=1 (free-free transitions); and at any f.
At finite but large uthe radiative opacities of fully ionized matter are strongly reduced. The reduction is 10 times stronger for scattering than for free-free transitions. Inserting this factor 10 into Eq. (16) and taking into account that, at G, the NS radiative surface is pushed to (Ventura & Potekhin 2001), one can see that in deep, strongly magnetized photospheric layers Thomson scattering dominates the opacity only at .
We note that the Rosseland
mean opacity due to scattering by free ions,
,
can be obtained from that for electrons, ,
by the simple scaling:
From the asymptote of at large u we see that the Thomson opacity of ions is times larger than that of electrons, if the ions are strongly quantized into Landau levels, which occurs at . Since the main contribution into the radiative opacity at sufficiently low temperature comes from free-free processes (see above), we can conclude that the Thomson ion scattering cannot contribute appreciably to the Rosseland mean opacity of the photosphere unless .
Recently, Baiko et al. (1998) considerably improved the treatment of by taking into account multiphonon absorption and emission processes in Coulomb crystals and incipient quasiordering in a Coulomb liquid of ions. Using these results, Potekhin et al. (1999) constructed an effective scattering potential which allowed them to calculate the non-magnetic conductivity in the relaxation time approximation in an analytic form; these analytic expressions described accurately numerical results obtained beyond the framework of the relaxation time approximation. The effective potential has been used then at arbitrary magnetic fields in order to derive practical expressions for evaluation of electrical and thermal conductivities of degenerate electrons in magnetized outer envelopes of NSs (Potekhin 1999). Unlike previous treatments of the electrical conductivities perpendicular to the quantizing magnetic fields (Kaminker & Yakovlev 1981; Hernquist 1984; Schaaf 1988), Potekhin (1999) went beyond the assumption that by introducing an interpolation from the case to . All tensor components of the kinetic coefficients in magnetic fields have been calculated and fitted by analytic formulae; the corresponding Fortran code is available electronically at http://www.ioffe.rssi.ru/astro/conduct/. This code is used here to calculate the temperature profile in the outer envelope of a NS.
In the inner envelope, we need to know the thermal conductivity only in young NSs, yr, as long as the internal region ( ) remains non-isothermal (GYP). For this purpose, we will use the non-magnetic conductivities presented by GYP. They generalize the expressions by Potekhin et al. (1999) in two respects. First, the atomic nuclei in the inner envelope cannot be considered as pointlike: the size and shape of nuclear charge distribution significantly affect the conductivity. Second, the electron-phonon scattering in the inner crust at temperatures much below 10^{8} K changes its character: the so called umklapp processes cease to dominate and the normal processes (with electron momentum transfer within one Brillouin zone) become more important.
Figure 2: Electron magnetization parameter at the top and bottom of the inner crust (solid lines) and in the outer envelope (dashed line). The curves are marked with the values of [g cm^{-3}]. | |
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The internal temperature in young NSs is relatively high, K. Our neglect of the effects of the magnetic fields in the internal region is strictly justified for the case of small magnetization parameter, . This parameter is shown in Fig. 2 as a function of Tfor three values of density: near the top and the bottom of the inner envelope (solid lines) and in the middle of the outer envelope (dashed line). We see that the use of non-magnetic is justified for the NSs with the magnetic fields G in the inner envelope. However it cannot be justified for higher B in the inner envelope or for G in the outer envelope. In such cases the conductivity across the field is suppressed by the large factor and the Hall thermal conductivity may become important. We neglect these effects in the inner envelope. Therefore, our simulations (Sect. 4) of cooling of young NSs ( yr) are justified for G in the inner envelopes. The results for older NSs, for which the inner envelope is isothermal, are valid at larger B as well.
Figures 3 and 4 present the calculated profiles for a NS of mass and radius R=10 km ( and ). The curves in Fig. 3 are calculated at fixed K. The solid curves show the results of accurate calculations for the magnetic field normal and tangential to the surface (cases of parallel and transverse conduction, Sect. 2.1.2). The dashed lines are obtained for the parallel conduction using the classical thermal conductivity (neglecting the Landau quantization). This approximation becomes inaccurate with increasing B. The dot-and-dash lines are calculated for the tangential magnetic field using the transverse conductivity in the limit of , employed in previous papers (e.g., Hernquist 1984,1985; Schaaf 1988,1990a,1990b; Heyl & Hernquist 1998a,1998b). This approximation is inaccurate at lower B; a more accurate approximation (solid line) is given by the interpolation (Potekhin 1999) mentioned in Sect. 2.3.2.
Figure 3: Temperature profiles through an iron envelope of a NS with and R=10 km for the magnetic field B=10^{11} G (left panel) and 10^{13} G (right panel) at the fixed effective surface temperature K. Lower solid curves correspond to and upper ones to . Dashed lines are calculated with the classical thermal conductivity for , while dot-dashed ones are calculated for assuming which is traditional for the transverse conduction. | |
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Figure 4: Temperature profiles through an iron envelope of a NS with and R=10 km at (from left to right) B=10^{11}, 10^{12}, 10^{14}, and 10^{15} G. The internal temperature is fixed to K (solid lines) or 10^{9} K (dot-dashed lines). The lines of each group correspond to (the lowest line), 0.7, 0.4, 0.1, and 0 (the highest line). | |
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Figure 4 shows temperature profiles at four field strengths B and five inclinations for and 10^{9} K (we do not present the curves for at B=10^{15} G because the temperature is too small near the surface in the transverse conduction case). The curves show pronounced -dependence at high densities starting from the turning point (Sect. 2.3).
The endpoints of the curves in Figs. 3 and 4 lie at the radiative surface , where , and near the neutron drip point. Their behaviour qualitatively agrees with the results of the approximate analytic study of Ventura & Potekhin (2001). For instance, we confirm the approximate linear dependence and the shift of the turning point to higher densities with increasing B. The density marked on the graphs is defined by Eq. (1). It separates the strongly- and weakly-quantizing field regimes. We see that the increase of T at (neglected by Heyl & Hernquist 1998a) can be significant.
The higher the temperature, the wider is the density region, where and are of similar magnitude. Therefore the dependence of the profiles on the inclination and magnetic field strength Bis less pronounced at higher showing convergence to the B=0 case.
For the parallel and transverse conduction
cases (at the magnetic pole and equator)
we have, respectively,
Figure 5: Dependence of the surface temperature on the internal temperature at the magnetic fields B=0, 10^{13}, and 10^{14} G normal (left panel) or tangential (right panel) to the surface. Numerical data (heavy dots) are compared with the present fit (26)-(29) (solid lines) and with the earlier fits of Schaaf (1990a) (dot-dashed line, for B=10^{13} G) and Heyl & Hernquist (1998b) (dashed lines). | |
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Figure 6 shows the distribution of the effective temperature from the magnetic pole to the equator. Heavy dots show the results of numerical integration of profiles. Solid lines are obtained using our fit, Eqs. (26)-(30), while short-dashed lines correspond to the approximation of Greenstein-Hartke (1983). For comparison, dot-dashed lines on the left panel represent the fitting formulae of Schaaf (1990a,1990b).
Horizontal long-dashed lines in Fig. 6 show which would have been observed at the same if the magnetic field were absent. In qualitative agreement with the results of earlier works (Sect. 1), the thermal flux emergent from the NS interior is suppressed by the magnetic fields in the equatorial region and enhanced near the pole. For a rotating NS, such distribution leads to a pulsating light curve (e.g., Page 1995), which may be observed, for instance, with Chandra in soft X-rays (where the spectral flux of the thermal NS radiation has maximum) or with the HST in UV.
Figure 6: Dependence of the surface temperature on the inclination angle at constant B and . Left panel: B=10^{12} G, , 8.5, 8.8; right panel: B=10^{14} G, . Dots: numerical results; solid lines: fit; dashed lines: interpolation of Greenstein-Hartke between the accurate values at and ; long dashes: the case of B=0; dot-dashed lines on the left panel: combined fitting formulae of Schaaf (1990a, 1990b). | |
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= |
a_{1}= |
a_{2}= |
a_{3}= |
Note one important feature: the effect of the magnetic field on the F(B)/F(0)ratio becomes weaker with growing . It is explained by the arguments presented in Sect. 3.1. Accordingly, the luminosity of a hot NS cannot be strongly affected even by very high magnetic fields.
Let us discuss briefly the effects of magnetized envelopes on NS cooling. For illustration, we take the models of NSs with two masses, M=1.3 and . For the chosen EOS in the stellar core (Sect. 2), the NS has the radius R=11.86 km. Its central density g cm^{-3}is lower than the density g cm^{-3} at which the powerful direct Urca process of neutrino emission is switched on. Thus the main neutrino emission mechanisms in the NS core are a modified Urca process and neutrino bremsstrahlung in nucleon-nucleon collisions. Accordingly, the model gives us the example of slow cooling. For we have R=11.38 km and g cm . The direct Urca process is open in the central kernel of mass and radius 2.84 km. This is an example of fast cooling. Further details of these NS models may be found in GYP.
We have calculated the cooling curves of NSs with magnetized envelopes, assuming the dipole magnetic field, Eq. (10). For comparison with earlier papers (Sect. 1), we also consider a familiar model of the field which is radial and constant in magnitude throughout the blanketing envelope. In addition, we have switched on and off superfluidity of neutrons and protons in stellar interiors to demonstrate the combined effects of magnetized envelopes and superfluid interiors on NS cooling.
Figure 7: Cooling curves of non-superfluid and NSs with dipole magnetic fields of different strengths. | |
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The results for G are in good qualitative agreement with those obtained by Page (1995) and Shibanov & Yakovlev (1996). The thermal state of the stellar interior is almost independent of the magnetic field in the NS envelope at the neutrino cooling stage ( yr), but is affected by the magnetic field later, at the photon cooling stage. On the contrary, the surface temperature is always affected by the magnetic field. The dipole field G makes the blanketing envelope overall less heat-transparent. This lowers at the neutrino cooling stage and slows down cooling at the photon cooling stage. The dipole field G makes the blanketing envelope overall more transparent, increasing at the neutrino cooling stage and accelerating the cooling at the photon stage.
Contrary to the case of the dipole magnetic field, any radial field (e.g., Van Riper 1991) would always lower the thermal insulation of the blanketing envelope, increasing at the neutrino cooling stage and accelerating cooling at the photon cooling stage. The radial field would affect the cooling noticeably more than the dipole field.
Even very strong magnetic fields do not change significantly thermal insulation of a hot blanketing envelope (Sect. 3.3). Accordingly, the effects of magnetic fields in a young and hot NSs are not too strong. The strongest effects take place in cold NSs, at the photon cooling stage. Unfortunately, our knowledge of insulating properties of the blanketing envelopes is the poorest for these NSs (Sect. 2). In addition, the magnetic field evolution and reheating processes may become important for these stars, which we ignore for simplicity. Notice that the fields G appreciably accelerate the cooling and lower the duration of the neutrino cooling stage.
For the dipole field, however, the detected flux depends on observation direction. The dipole field G reduces the flux averaged over the entire NS surface (solid line) by a factor of 1.3, while the fields and 10^{16} G amplify it by factors of about 2 and 4, respectively. The dashed and dotted lines in Fig. 8 refer to the fluxes detected in the direction to the magnetic pole and equator, respectively (assuming the magnetic axis coincides with the rotational one, for simplicity). These lines show the largest difference of the fluxes detected under different angles. They are obtained taking proper account of the general relativity effect of bending of light rays propagating from the NS surface to a distant observer, in the same manner as was done by Page (1995). For comparison, we present also the fluxes calculated neglecting the gravitational ray-bending effect (as if space-time outside the NS were flat). In the absence of light bending effects, the difference of fluxes observed under different angles is quite noticeable. For instance, at G the largest difference would be about 65%. The light bending reduces the largest difference to 14% for G, making it practically negligible. This is explained (e.g., Page 1995) by the fact that gravitational bending of light rays increases the fraction of the NS surface visible by the observer. The observer detects the flux from the larger portion of the surface, that is closer to the flux averaged over observation directions. The light bending effect is strong for our and NS models. Thus we will neglect weak dependence of fluxes on the detection direction and use the average fluxes and associated effective temperatures in our analysis. Notice that here we mean total (spectral integrated) fluxes but not the fluxes observed in selected spectral bands taking into account interstellar absorption.
Figure 8: Effect of dipole or radial surface magnetic fields on the flux of electromagnetic radiation F(B) detected from a non-superfluid NS of age yr. Solid line shows the flux averaged over the entire NS surface for the dipole field. Dot-and-dashed lines present the flux observed in the direction towards the magnetic pole (assumed to be aligned with the rotational axis) or towards the magnetic equator. The dotted lines show the same fluxes calculated neglecting gravitational bending of light rays outside the NS. | |
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Figure 9 shows the effective temperatures of young (1000 yr) non-superfluid and NSs with dipole and radial magnetic fields of different strengths. The effects of magnetic fields are similar to those in Fig. 8. Since the NS undergoes fast cooling, its internal temperature ( K) is much lower than in the star ( K). Accordingly, the magnetic fields affect the surface temperature of the NS more significantly. Previously, it was suggested (e.g., Heyl & Hernquist 1997a) that the models of young and hot, slowly cooling NSs with ultramagnetized envelopes could serve as models of anomalous X-ray pulsars and soft gamma repeaters. Although those previous cooling models were rather simplified, they are in qualitative agreement with our more elaborate models. The main outcome of these studies is that even ultrahigh magnetic fields cannot change appreciably the average surface temperatures of young NSs with iron envelopes (as seen clearly from Figs. 8 and 9). We stress that the local surface temperatures in narrow strips near the magnetic equators appear to be much lower than near the magnetic poles (Sect. 3), but they do not contribute noticeably to the NS luminosity integrated over the surface.
Figure 9: Effect of dipole or radial surface magnetic fields on the averaged effective surface temperature of non-superfluid and NSs of age t=1000 yr. | |
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Figure 10: Cooling of NS with B=0, with the dipole magnetic field G or with the radial field of the same strength. The upper curves are calculated assuming no superfluidity in the NS interior while the lower ones are calculated for certain critical temperatures of the neutron and proton superfluids in the NS cores (the values of and are indicated near the curves in units of 10^{8} K). Error bars show the possible interval of the surface temperatures of the Vela pulsar (see text). | |
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It is well known that neutron or proton superfluidity in the NS core affects the cooling. For illustration, we adopt the same simplified model of nucleon superfluidity which has been used in a number of previous works (Yakovlev et al. 1999 and references therein). In this model, the neutron pairing is assumed to be in the triplet state, while the proton pairing occurs in the singlet state of a nucleon-nucleon pair. The critical temperatures and of the neutron and proton superfluids are assumed to be constant and treated as free parameters to be adjusted to observational data. Figure 10 demonstrates that taking the neutron and proton superfluidities with certain values of and we can lower of our Vela model to the observed values. This can be done for all three models of the envelope, with B=0 as well as with the dipole and radial fields. In the given examples, the lowering of for the dipole or field-free cases is produced by neutrino emission due to Cooper pairing of neutrons, while for the radial field the lowering is produced by neutrino emission due to Cooper pairing of neutrons and protons. In these three cases we need different critical temperatures and . In each case the choice of and is not unique.
Generally, if we fix the blanketing envelope model, we can determine the domains of and values in the - plane which force the NS to have within the given errorbar by the specified age t. These domains (hatched regions) are shown in Fig. 11 for the Vela pulsar with a non-magnetic envelope, and also for the envelope with dipole and radial magnetic fields.
Figure 11: Domains (hatched) of and for which the NS with the superfluid core has the surface temperature K by the age of the Vela pulsar. Solid lines enclose the domains for the NS with the dipole magnetic field ( G); dashed lines are for the non-magnetized NS and dots are for the radial field ( ). | |
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The domain for the non-magnetic envelope has a complicated X-shape. The dipole magnetic field facilitates a lowering of to the required errorbar. The domain of acceptable values of and becomes narrower overall and splits into two parts (Fig. 11). In principle, in these two cases we do not need proton superfluidity to explain the observations (Fig. 10): the neutrino emission due to Cooper pairing of neutrons is sufficiently strong by itself to lower . On the other hand, were the magnetic field radial, it would raise and complicate the adjusting of to observations. Neutrino emission due to Cooper pairing of neutrons is insufficient to lower , and the emission due to Cooper pairing of protons is needed. The domain acquires a qualitatively different shape: the proton superfluidity cannot be too weak ( ).
The domains of and obtained for the Vela pulsar can be combined further with analogous domains determined for other cooling NSs in attempt to impose restrictions on the values of and in the stellar cores. A preliminary analysis of such a kind has been done by Yakovlev et al. (1999). However, further work remains to be done to constrain strongly the critical temperatures: one should take into account new observational data on thermal radiation from isolated NSs and new theoretical models of cooling NSs with superfluid cores; particularly, one has to incorporate density dependence of and over the stellar core. Such work is beyond the scope of the present paper, but our illustrative examples show that proper analysis of the NS superfluidity may require consideration of the effects of the magnetic field on the NS blanketing envelope.
We have analysed the thermal structure of neutron star envelopes with typical pulsar magnetic fields 10^{11}-10^{13} G and with ultrahigh magnetic fields up to 10^{16} G. We have used (Sect. 2) modern data on equation of state and thermal conductivities of magnetized neutron star envelopes. In particular, we have proposed an analytic model of the radiative thermal conductivity limited by the Thomson scattering and free-free absorption of photons in a magnetized plasma. We have used the values of thermal conductivity of degenerate electrons, updated recently by proper treatment of dynamical ion-ion correlations which affect electron-ion scattering. We have calculated the temperature profiles (Sect. 3) in the neutron star envelope for any inclination of the magnetic field to the surface and obtained a fit expression which relates the internal temperature of the neutron star to the local effective surface temperature. We have also calculated and fitted the relation between the internal temperature and total surface luminosity (or effective temperature) for the dipole magnetic fields G.
Furthermore, we have performed (Sect. 4) simulations of cooling of neutron stars with dipole or radial magnetic fields in the envelopes. In agreement with the previous studies of Page (1995) and Shibanov & Yakovlev (1996), we have found that the effects of the two magnetic field configurations on neutron star cooling are qualitatively different. We have briefly discussed cooling of young and middle-aged neutron stars with ultramagnetized envelopes (magnetars as the models of soft gamma repeaters and anomalous X-ray pulsars) and the effect of surface magnetic fields on constraining critical temperatures of the neutron and proton superfluids in the cores of ordinary pulsars, considering the Vela pulsar as an example.
We stress that our results are less reliable for cold neutron stars ( K) with very strong magnetic fields G because of lack of knowledge of ionization state and thermal conductivity of the outermost parts of the cold blanketing envelopes. Further work is required to fill these gaps. In this paper, we have considered the blanketing envelope composed of iron and have not analysed the envelopes containing light elements. The presence of light elements generally reduces thermal insulation of the envelope. This effect has been studied in detail for non-magnetic envelopes (PCY) and is expected to be important for magnetic envelopes as well (Heyl & Hernquist 1997a). We are planning to address this matter in a future work.
Acknowledgements
We are grateful to G. Pavlov for encouragement and for providing us with the results of observations (Pavlov et al. 2001) before publication, to U. Geppert for stimulating suggestions, to Yu. A. Shibanov and K. P. Levenfish for helpful discussions, and to the referee, Dany Page, for useful remarks. This work was supported by RFBR (grant No. 99-02-18099).