Imprint of magnetic obliquity in apparent spin-down of radio pulsars

Numerical simulations predict that the spin-down rate of a single rotation-powered neutron star depends on the angle $\alpha$ between its spin and magnetic axes as $P\dot{P}\propto {\mu }^2(k_0+k_1{\sin }^2\alpha )$, where $P$ is the star spin period, $\mu$ is its magnetic moment, while $k_0\sim k_1\sim 1$. Here we describe a simple observational test for this prediction based on the comparison of spin-down rates of 50 nearly orthogonal (with $\alpha$ close to 90 deg) and 27 nearly aligned (with $\alpha$ close to 0 deg) pulsars. We found, that the apparent pulsar spin-down is consistent with the theory if assumed, that magnetic moments of orthogonal rotators are systematically larger than those of aligned ones for $\sim \mathrm{0.15..0.2}$ dex. Also, as a by-product of the analysis, we provide yet another constraint on the average braking index of radio pulsars as $1\le n\le 4$ with formal significance not worse than 99%.