AMPLIFICATION OF NEUTRON-STAR MAGNETIC-FIELDS BY THERMOELECTRIC EFFECTS .6. ANALYTICAL APPROACH

From numerical calculations in previous papers it is known that in you ng, hot neutron stars small-scale toroidal magnetic field modes can be amplified. This amplification was discussed as a first step of a scen ario, where the typical 10(12) - 10(13) G dipole magnetic field of pul sars is generated after the birth of neutron stars. Two physical effec ts - the thermo-Hall effect and the thermoelectric effect - are respon sible for this phenomenon. The numerical calculations show that the gr owing magnetic field is concentrated in a thin layer (less than 100m t hick) just below the neutron star surface (less than 100m deep). The g rowth rate is in the order of years and reaches its maximum for multip olarities n similar or equal to 1000, what is equivalent to sinus-like field variations with meridional wavelengths of similar or equal to 1 00m. By applying an analytical approximation (the magnetic field and t emperature functions are assumed to be Gauss functions of the depth z) we derive simple formulas for the maximum position and width of the m agnetic field and temperature functions, which show the dependence of these parameters on the material quantities like heat and electric con ductivities and on input parameters like the neutron star's mass and r adius, the surface temperature and the multipolarity. Moreover, an ana lytical formula for the growth rate as a function of n is found, demon strating the competition of different physical effects and confirming qualitatively the results of previous numerical calculations. Finally, a general criterion is found: The thermoelectric field amplification acts as long as T-s6(4)/g(s14) greater than or similar to 100 (T-s6 - surface temperature in 10(6)K, g(s14) - surface gravity in 10(14)cm s( -2)).