Surface temperature of a magnetized neutron star and interpretation of   the ROSAT data. II
  We complete our study of pulsars' non-uniform surface temperature and of its effects on their soft X-ray thermal emission. Our previous work had shown that, due to gravitational lensing, dipolar fields cannot reproduce the strong pulsations observed in Vela, Geminga, PSR 0656+14, and PSR 1055-52. Assuming a standard neutron star mass of 1.4 Msol, we show here that the inclusion of a quadrupolar component, if it is suitably oriented, is sufficient to increase substantially the pulsed fraction, Pf, up to, or above, the observed values if the stellar radius is 13 km or even 10 km. For models with a radius of 7 km the maximum pulsed fraction obtainable with (isotropic) blackbody emission is of the order of 15% for orthogonal rotators (Vela, Geminga and PSR 1055-52) and only 5% for an inclined rotator as PSR 0656+14. Given the observed values this indicates that the neutron stars in Geminga and PSR 0656+14 have radii significantly larger than 7 km and, given the very specific quadrupole components required to increase Pf, even radii of the order of 10 km may be unlikely in all four cases.   We confirm our previous finding that the pulsed fraction always increases with photon energy, below about 1 keV, when blackbody emission is used and show that it is due to the hardenning of the blackbody spectrum with increasing temperature. The observed decrease of pulsed fraction may thus suggest that the emitted spectrum softens with increasing temperature.   Finally, we apply our model to reassess the magnetic field effect on the outer boundary condition used in neutron star cooling models and show that, in contradistinction to several previous claims, it is very small and most probably results in a slight reduction of the heat flow through the envelope. 
Dany Page and Antonio Sarmiento (Instituto de astronomia, UNAM,   Mexico)