inertia $\simeq 10^{45}$ g cm$^2$), where $\theta_B$ is the angle between the magnetic axis and the spin axis. Of course, pulsar spindown power is not exactly given by the vacuum dipole radiation formula (Contopoulos & Spitkovsky 2006). For concreteness, we adopt $B_p\simeq 8.8\times 10^{10}$ G in our calculations (corresponding to $\theta_B=45^\circ$), although using a somewhat different value (within a factor of two) would only have a small effect on our result.

Given the magnetic field configuration, the effective temperature distribution on the NS surface is determined by heat conduction in the crust (e.g., Schaaf 1990; Heyl & Hernquist 1998; Potekhin & Yakovlev 2001) Since the magnetic field is symmetric about the magnetic axis, the effective temperature depends only on the polar angle, $T_{\rm eff} 
= T_\mathrm{eff}(\theta)$. We use the zero-accretion fitting formula of Potekhin et al. (2003) to calculate $T_{\rm eff}(\theta)$. This formula takes as inputs a core temperature and a crustal magnetic field of arbitrary strength and direction (assumed constant through the radial extent of the crust), and returns the effective surface temperature. We fix the core temperature at $T_c=10^9$K, corresponding to $T_{\mathrm{eff}}=3.8\times10^6$ K at the magnetic poles, which matches sufficiently the hot spot temperature expected from blackbody fits.

