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Astron. Astrophys. 342, 192-200 (1999)

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4. Cooling of magnetized neutron stars

In this section we illustrate the above results by cooling simulations. Let us consider a set of neutron star models with fixed equation of state but varying central density and magnetic field strength. Specifically, we take the EOS by Prakash et al. (1988) with the compression modulus [FORMULA] MeV and the symmetry energy in the form suggested by Page & Applegate (1992). It is assumed that matter in the stellar core consist of neutrons, protons and electrons (no muons and hyperons). The EOS predicts the monotonous increase of the proton fraction with increasing mass density. Therefore, if the central density [FORMULA] is higher than the certain threshold density [FORMULA] (corresponding to proton fraction [FORMULA]), or, equivalently, the stellar mass M is higher than the certain threshold mass [FORMULA] the direct Urca process becomes allowed in a central kernel of the inner stellar core. For the chosen EOS, the threshold parameters are [FORMULA] g cm-3 and [FORMULA].

As pointed out by Page and Applegate (1992) the cooling history of a neutron star in the field-free regime is extremely sensitive to the stellar mass: if the mass exceeds [FORMULA] the star cools rapidly via the direct Urca process, while the cooling of the low-mass star is mainly due to the modified Urca process, and, therefore, is strongly delayed. The effect of the magnetic field would be to speed up the cooling of the star with a mass below [FORMULA], because the strong field opens the direct Urca process even if the [FORMULA] condition [FORMULA] (or [FORMULA]) is not reached.

The magnetic field in the neutron star core may evolve on cooling time-scales. This may happen if the electric currents supporting the field are located in those regions of the core, where protons as well as neutrons are nonsuperfluid. If so, the electric currents transverse to the field may suffer enhanced ohmic decay due to magnetization of charged particles (e.g., Haensel et al., 1990). The consequences of the decay would be twofold. Firstly, if the strong field occupies a sufficiently large volume of the core, the decay produces an additional heating, which would delay the stellar cooling. Secondly, the field decay would reduce the direct Urca losses in the forbidden domain decreasing the factor [FORMULA]. If, however, the neutrons are strongly superfluid, the enhanced decay is absent (Haensel et al., 1990; Ostgaard & Yakovlev, 1992), and the ohmic decay time of the internal magnetic field is typically larger than the Universe age (Baym et al., 1969).

On the other hand, microscopic calculations of superfluid neutron gaps in the neutron star core suggest that the critical temperatures are rather high at not too large densities (say, below [FORMULA] g cm-3), but decrease at higher densities (see, e.g., Takatsuka & Tamagaki, 1997, and references therein). Thus the electric currents could persist in the outer stellar core, where the neutron superfluid is available, and the enhanced field-decay mechanism does not work, while the direct Urca reactions operate in the inner core and are not subject to superfluid reduction. This latter scenario we adopt. We assume the presence of the magnetic field B in the stellar kernel, where the direct Urca process can be allowed, and assume nonsuperfluid protons in the neutron star core. Thus, the entire core is nonsuperconducting, and the magnetic field does not vary over the cooling time (age [FORMULA] yr) being frozen into the outer core.

Let us analyse the cooling stage at which the neutron star interior is isothermal ([FORMULA] - [FORMULA] yr). The surface temperature [FORMULA] (seen by a distant observer, i.e., the gravitational redshift included) is then determined by the heat transport through the neutron star crust and is related uniquely to the internal temperature. We will use the cooling code described, for instance, by Levenfish & Yakovlev (1996), and Yakovlev et al. (1998). The effects of General Relativity are included explicitly. The neutrino luminosity is produced by the standard neutrino reactions in the entire stellar core complemented by the direct Urca process in the inner core. The effects of the neutron superfluidity on the neutrino reactions and neutron heat capacity in the outer core are taken into account as prescribed by Levenfish & Yakovlev (1996). In addition, we include the neutrino emission due to triplet-state Cooper pairing of neutrons (Yakovlev et al., 1998). The dependence of the surface temperature on the internal stellar temperature is taken from Potekhin et al. (1997) assuming no envelope of light elements at the stellar surface. To emphasize the effect of the internal magnetic fields on the neutron star cooling we neglect the presence of the surface magnetic fields and assume that the inner field does not affect the relationship between the surface and internal temperatures.

The density dependence of the neutron critical temperature (triplet-pairing) is given by the step-function: [FORMULA] K at [FORMULA] g cm-3, and [FORMULA] at [FORMULA] g cm-3 The resulting cooling curves, [FORMULA], are insensitive to the initial inner temperature provided the latter is sufficiently high ([FORMULA]K).

The cooling curves are depicted in Figs. 3 and 4. Dash line illustrates fast cooling due to direct Urca process allowed in a large portion of the core at [FORMULA]. While calculating this curve we have used exact factors [FORMULA] in all the neutron-star layers where the field-free direct Urca is either permitted or forbidden. We have verified that the results are insensitive to specific values of [FORMULA] in the permitted domain. In that domain, one can safely use the quasiclassical fit (23) or even set [FORMULA]. On the other hand, even a huge internal field is unimportant in the forbidden domain (the curves for [FORMULA] and [FORMULA] G coincide): new regions of the core, where the direct Urca is open by the field, amount for a negligible fraction of the total neutrino luminosity.

[FIGURE] Fig. 3. Logarithm of the surface temperature as seen by a distant observer as a function of neutron star age. The dash curve is for a star of mass [FORMULA], well above the threshold mass [FORMULA], with magnetic field [FORMULA] G. The dotted and solid curves are for the [FORMULA] star. Bar shows observations of the Geminga pulsar (Meyer et al., 1994).

[FIGURE] Fig. 4. Same as in Fig. 3. The dotted and solid curves are for the [FORMULA] star. The dash line is the same as in Fig. 3.

The upper dotted curves are calculated for the stars with masses 1.439 [FORMULA] (Fig. 3) and 1.320 [FORMULA] (Fig. 4) at [FORMULA]. They represent the slow cooling via the standard neutrino reactions (the direct Urca is forbidden). The solid curves illustrate the effect of the magnetic field for the stars of the same masses. If the stellar mass is slightly (by 0.2%) below [FORMULA] (Fig. 3) the cooling curve starts to deviate from the standard one for not too high fields, [FORMULA] G. Stronger fields, [FORMULA]-[FORMULA] G produce the cooling intermediate between the standard and rapid ones, while still higher fields [FORMULA] G open the direct Urca in a large fraction of the inner stellar core and initiate a nearly fully enhanced cooling. If, however, the mass is by about 8% below the threshold one (Fig. 4), only a very strong field [FORMULA] G could keep the direct Urca process slightly open to speed up the cooling.

The results indicate that the magnetic field in the very central stellar core can indeed enhance the cooling provided the stellar mass is close to the threshold mass [FORMULA]. If [FORMULA] G, the effect is significant in a mass range [FORMULA]. For lower fields the range becomes smaller. If, for instance, [FORMULA] G, the mass range becomes as narrow as [FORMULA].

Our results can be used for interpretation of observational data. By way of illustration, consider observations of the thermal radiation from the Geminga pulsar. Meyer et al. (1994) fitted the observed spectrum by the set of hydrogen atmosphere models. These fits yield rather low non-redshifted effective surface temperature [FORMULA] (2 - 3)[FORMULA] K. Introducing the appropriate redshift factor [FORMULA] (R is the stellar radius and [FORMULA] is the gravitational radius) one gets redshifted surface temperature [FORMULA] [K] [FORMULA] ([FORMULA]). Adopting the dynamical Geminga's age [FORMULA] yr we can place the Geminga's error bar in Figs. 3 and 4. Let us use our cooling model (with possible strong magnetic field B near the stellar center unrelated to the much weaker Geminga's surface magnetic field). If [FORMULA] Geminga is found between the lines of standard ([FORMULA]) and fast ([FORMULA]) cooling. It is clear that tuning the mass slightly above [FORMULA] we can force the cooling curve to cross the error bar. However, the mass range corresponding to the bar width (Fig. 5) would be tiny (about [FORMULA]), as the cooling rate is extremely sensitive to M in the domain just above [FORMULA]. The narrowness of the confidence mass range makes it fairly improbable that the Geminga's mass lies in this range. Accordingly the suggested interpretation of the Geminga's cooling is unlikely. The situation becomes strikingly different in the presence of the strong magnetic field. The field shifts the confidence range of M (faded area in Fig. 5) below [FORMULA] (cf. Fig. 3), where variation of [FORMULA] with M at a given t is much smoother. This widens considerably the confidence mass range. At [FORMULA] G it is about 0.02 [FORMULA], while at [FORMULA] G it is about 0.04 [FORMULA] so that the chances that the Geminga's mass falls into this range become much higher. This makes the proposed interpretation more plausible.

[FIGURE] Fig. 5. The allowed mass range for the Geminga pulsar as a function of the internal magnetic field

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© European Southern Observatory (ESO) 1999

Online publication: December 22, 1998