  The effect of the magnetic field on the energy loss rate in the direct Urca
reactions is studied. The general expression for the neutrino emissivity at
arbitrary magnetic field B is derived. The main emphasis is laid on a case, in
which the field is not superstrong, and charged reacting particles (e and p)
populate many Landau levels. The magnetic field keeps the process operative if
Delta k / k_{Fn} < N_{Fp}^{-2/3} (N_{Fp} is the number of the Landau levels
populated by protons and Delta k = k_{Fn}-k_{Fp}-k_{Fe}), that is beyond the
well-known switch-on limit in the absence of the field, Delta k < 0. Cooling of
magnetized neutron stars with strong neutron superfluid in the outer cores and
nonsuperfluid inner cores is simulated. The magnetic field near the stellar
center speeds up the cooling if the stellar mass M is slightly less than the
minimum mass M_c at which the direct Urca reaction becomes allowed for B=0. If
B=3x10^{16} G the affected mass range is M_c-M < 0.1M_c, while for B=3x10^{15}
G the range is M_c-M < 0.015M_c. This may influence a theoretical
interpretation of the observed thermal radiation as illustrated for the Geminga
pulsar. The case of superstrong magnetic fields (B>10^{18} G), such that e and
p populate only the lowest Landau levels is briefly outlined.
