A pulsar's spin-down can be written as a power law

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Astrophysics & Cosmology · Stellar death and compact objects

Stars from the Inside, Ch Neutron stars and pulsars: the densest matter, the maximum mass and the spinning magnet

Problem

A pulsar's spin-down can be written as a power law Ω˙=−KΩn\dot\Omega = -K\Omega^n, where K>0K>0 is a constant fixed by the star's moment of inertia II, magnetic moment and inclination angle, and nn is the braking index. Observationally nn is found from n≡ΩΩ¨Ω˙2.n \equiv \frac{\Omega\ddot\Omega}{\dot\Omega^{2}}. ((a)) Show that evaluating this ratio on the power law above returns nn itself, independent of KK and of the instantaneous value of Ω\Omega, so nn can be measured from Ω\Omega, Ω˙\dot\Omega and Ω¨\ddot\Omega alone without ever knowing KK. ((b)) Magnetic-dipole spin-down gives Ω˙∝−Ω3\dot\Omega \propto -\Omega^3. State the braking index a pure magnetic dipole predicts. ((c)) The Crab pulsar's rotation frequency and its first two time derivatives are measured to be ν=29.6 Hz\nu = 29.6\,\mathrm{Hz}, ν˙=−3.71×10−10 Hz s−1\dot\nu = -3.71\times10^{-10}\,\mathrm{Hz\,s^{-1}} and ν¨=1.17×10−20 Hz s−2\ddot\nu = 1.17\times10^{-20}\,\mathrm{Hz\,s^{-2}}. Because nn is a ratio of logarithmic derivatives it takes the same value whether computed from Ω=2πν\Omega = 2\pi\nu or from ν\nu directly. Compute the observed braking index and compare it with the pure-dipole prediction of part (b). ((d)) Integrate Ω˙=−KΩn\dot\Omega = -K\Omega^n from birth, when the period was P0P_0, to today, and show that the age is t=P(n−1)P˙[1−(P0P)n−1],t = \frac{P}{(n-1)\dot P}\left[1-\left(\frac{P_0}{P}\right)^{n-1}\right], which reduces to τc=P/2P˙\tau_c = P/2\dot P for n=3n = 3 and P0≪PP_0 \ll P. The Crab has P=33.4 msP = 33.4\,\mathrm{ms}, P˙=4.2×10−13\dot P = 4.2\times10^{-13}, τc=1260 yr\tau_c = 1260\,\mathrm{yr} and a true age of about 970 yr970\,\mathrm{yr}. Taking n=2.52n = 2.52, find its birth period P0P_0, and say whether n<3n < 3 or a non-negligible P0P_0 is what makes τc\tau_c exceed the true age.

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