Astron. Astrophys. 335, 969-972 (1998)

2. Modified Urca process in strong magnetic fields

Throughout this paper, we consider the magnetic fields beyond , where G is the critical magnetic field strength by the relation . Hence, we must use the relativistic Dirac equation for the electrons, which yields energy eigenstates

where quantum number n is given by

for the Landau level and spin (For reference, see Mészáros 1993). The usual sum over electron states (per unit volume) in a zero field is

which should be replaced as follows when the magnetic field is nonzero,

where is the electron Compton wavelength. Notice that the cyclotron frequency of protons is about 1836 () times smaller than the electron cyclotron frequency. However, as shown by Lai and Shapiro (1991), the values of for electrons and protons are essentially the same for n-p-e system because of charge neutrality, here is the upper limit of the summation over n in Eq. (6). Therefore, we cannot ignore magnetic field effects on the proton of state when is small.

In the following calculations, we ignore the contribution of the proton branch of the modified Urca process in a magnetic field because the proton number density is about ten percent of neutron one in neutron star cores. According to Friman and Maxwell (1979), Yakovlev and Levenfish (1995) (), the neutrino energy production rate of the neutron branch in zero magnetic field is given by

where (j=1-4) are the nucleon momenta and is the electron momentum, are the corresponding Fermi momenta of particles, is the squared matrix element of the first process (2), s=2 is a symmetry factor for that process, T is the temperature of the equilibrium system, is the dimensionless momenta of neutrino and are those of other particles, here is the Fermi velocity and is an effective mass and is the Fermi-Dirac function. The integrations in A and I are standard and yield (e.g. Shapiro & Teukolsky 1983)

Taking the nucleon-nucleon interactions into account, Friman and Maxwell (1979) obtained

The previous studies demonstrated that the matrix elements for the neutron decay and electron capture reactions when magnetic field is nonzero are the same as those when magnetic field is zero (Fassio-Canuto 1969, Canuto & Ventura 1977, Lai & Shapiro 1991). We therefore need only consider the effect of a magnetic field on phase-space integration of the charged particles. Hence the energy loss rate of the modified Urca process in a magnetic field is written as

where is the rate in zero magnetic field, while R is the factor which describes the effect of the magnetic field on the modified Urca process. This factor can be written as follows according to Eqs.  (8) and (9):

After integrating over with j=1-3, we have

where

and in Eq. (16) are the magnetic field strengths in units of the critical magnetic field for electrons and protons, respectively.

Fig. 1 shows the ratio R as a function of density when T= K and T= K, respectively. The saw-teeth shape of the curves is due to the fact for B, we have more available phase space for electrons at given density, which causes the decay rate to increase. For the same reason, a nonzero magnetic field decreases the inverse -decay rate (Lai & Shapiro 1991), the competition of these two reaction rates therefore causes the energy loss rate to oscillate with the increase of density. The stronger the magnetic field is, the larger the amplitude is. But the tendency is to increase the reaction rate of the modified Urca process in the range of the core density. It is also evident from Fig. 1 that the temperature only influences the reaction rate when the electrons are in resonant states at which the electrons begin to fill the higher Landau levels.

Fig. 1. Ratio R versus the density of neutron star cores under the influence of different interior magnetic field strengths

© European Southern Observatory (ESO) 1998

Online publication: June 26, 1998
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