Braking index ($n=\Omega \ddot{\Omega }/{\dot{\Omega }}^2$)

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.