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We complete our study of pulsars' non-uniform surface temperature and of itseffects on their soft X-ray thermal emission. Our previous work had shown that,due to gravitational lensing, dipolar fields cannot reproduce the strongpulsations observed in Vela, Geminga, PSR 0656+14, and PSR 1055-52. Assuming astandard neutron star mass of 1.4 Msol, we show here that the inclusion of aquadrupolar component, if it is suitably oriented, is sufficient to increasesubstantially the pulsed fraction, Pf, up to, or above, the observed values ifthe stellar radius is 13 km or even 10 km. For models with a radius of 7 km themaximum pulsed fraction obtainable with (isotropic) blackbody emission is ofthe order of 15% for orthogonal rotators (Vela, Geminga and PSR 1055-52) andonly 5% for an inclined rotator as PSR 0656+14. Given the observed values thisindicates that the neutron stars in Geminga and PSR 0656+14 have radiisignificantly larger than 7 km and, given the very specific quadrupolecomponents required to increase Pf, even radii of the order of 10 km may beunlikely in all four cases.  We confirm our previous finding that the pulsed fraction always increaseswith photon energy, below about 1 keV, when blackbody emission is used and showthat it is due to the hardenning of the blackbody spectrum with increasingtemperature. The observed decrease of pulsed fraction may thus suggest that theemitted spectrum softens with increasing temperature.  Finally, we apply our model to reassess the magnetic field effect on theouter boundary condition used in neutron star cooling models and show that, incontradistinction to several previous claims, it is very small and mostprobably results in a slight reduction of the heat flow through the envelope.Comment: 17 pages with 8 figures. Uses AASTeX v4.0 macro. Submitted to Ap.

Surface Temperature of a Magnetized Neutron Star and Interpretation of theROSATData. II.