Pulsar spin-down ($L_{\mathrm{sd}}=4{\pi }^{2}I\dot{P}/P^3$)

Loss of a rotating neutron star's angular momentum and rotational energy through electromagnetic torque and a magnetized outflow. For constant moment of inertia $I$, differentiating $E_{\mathrm{rot}}=I(2\pi /P{)}^2/2$ gives the displayed positive loss rate when $\dot{P}>0$. It is not the same as the measured radiation luminosity in any one observing band.

Extrapolate an anchored magnetic dipole field to the light cylinder $R_c=c/\Omega$ and estimate outward power as magnetic energy density times $c{R}_c^2$. The result fixes the displayed scaling but not the geometric coefficient. An aligned vacuum dipole does not radiate, while a current-carrying plasma magnetosphere can have nonzero aligned spin-down. Corotation at the light cylinder is a notional limiting construction, not a material surface moving rigidly at light speed.

For a power-law torque $\dot{\Omega }=-K{\Omega }^n$ with constant $K$, the displayed combination equals the exponent. A measured value need not be three: magnetospheric torques, evolving fields and timing variability complicate inference.

For $\dot{\Omega }=-K(t){\Omega }^q$, differentiation gives the displayed braking index. Inferring the torque power $q$ from a measured index therefore requires an effectively constant coefficient. Field evolution, changing inclination and moment-of-inertia changes can all contribute to the difference.

For a constant-coefficient braking law $\dot{\Omega }=-K{\Omega }^3$, integration gives $t={\tau }_c[1-(P_{\mathrm{birth}}/P{)}^2]$. The characteristic age approximates the actual elapsed time only when the birth period was short and torque evolution and kinematic timing contributions are negligible.