Heating of millisecond pulsars by magnetic field decay

Millisecond pulsars (MSPs) are believed to be very old neutron stars (NSs) whose age may exceed significantly ${10}^8$ years. Although cooling scenarios of isolated NSs predict for that age a surface temperature $T_s~{10}^4$ K, observations of the nearest MSP J0437-4715 indicate $T_s$ well above that value. Besides the heating of the polar cap surface by backflowing charged particles, Joule heating in the crust can contribute to the overall heat budget of MSPs. Because the dipolar field component, derived from $P$ and $\dot{P}$ measurements, is much too weak for remarkable heating, smaller-scale structures should be analyzed to understand whether they can supply the demanded heat. For this purpose, we study the small-scale field structure of radio pulsars. Magnetic field components, significantly stronger than the dipolar one, may exist especially at the surface of MSPs. We assign upper limits to the strength of single-field components up to a multipolarity of $l=10$ and the corresponding deviations from axial symmetry $m\le l$. Arguments are provided that the decay of the small-scale components with $l=3$ or $l=4$ of the crustal magnetic field may cause the relatively high surface temperature of isolated MSPs.