Question

Pulsar astronomers parameterize pulsar spin-down in a model-independent way with the
relation O = On, where n is known as the braking index. Derive a functional form for n in terms of the three observables. What value is n for magnetic dipole radiation?

Answer 1

Answer on Question 55065, Physics / Astronomy | Astrophysics

$$\ddot{\Omega} = n \Omega^{n-1} \dot{\Omega}$$

But from the question:

$$\Omega^{n-1} = \frac{\dot{\Omega}}{\Omega}, \text{ and so:}$$ $$\ddot{\Omega} = n \frac{\dot{\Omega}}{\Omega} \dot{\Omega}, \text{ and therefore:}$$ $$n = \frac{\ddot{\Omega} \Omega}{\dot{\Omega}^2}$$

For magnetic dipole radiation:

$$\dot{E} = I \Omega \dot{\Omega} = \frac{2}{3c^2} (B R^2 \sin a)^2 \Omega^4$$

Therefore, $\dot{\Omega} \propto \Omega^3$, and therefore $n = 3$.

Answer: $n = 3$

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