= Solution
A <radio pulsar> is a rapidly rotating, magnetized <neutron star> whose emission beam periodically crosses the line of sight. Collapse of a massive stellar core can leave such a compact remnant. Angular-momentum conservation during contraction helps explain rapid <rotation>, while amplified <magnetic fields> and a plasma-filled magnetosphere enable particle acceleration and beamed emission. The pulses are not repeated stellar contractions. Not every newborn <neutron star> is seen as a <radio pulsar>, because emission, beaming and observational selection matter.
<rotation> supplies <energy> to <radiation> and the magnetized wind. With approximately constant <moment of inertia> $I$,
$$
E_{\rm rot}=\frac12I\Omega^2,\qquad
\boxed{L_{\rm sd}=4\pi^2I\frac{\dot P}{P^3}}.
$$
An ideal fixed magnetic-dipole torque has $\dot\Omega=-K\Omega^3$. Its integration gives
$$
t=\frac{P}{2\dot P}\left[1-\left(\frac{P_{\rm birth}}P\right)^2\right],
$$
so the <pulsar characteristic age> $\tau_c=P/(2\dot P)$ approximates the true age only for a short birth period, constant torque coefficient and <braking index> three. Real magnetospheres, field evolution and kinematic contributions to observed $\dot P$ complicate both age and inferred <magnetic field>.
As the <star> <spins> down, the available accelerating potential falls. If particle acceleration and <pair production> can no longer sustain the radio-emitting magnetosphere, the object crosses a model-dependent <pulsar death line> and ceases to be detectable as an ordinary <radio pulsar>. It still exists as a cooling <neutron star>; “death” refers to the emission state, not disappearance of the <star>. The death boundary is not a precisely universal spin period.
In a surviving binary, later <Roche-lobe overflow> from a companion can create an <accretion disc>. Transferred <angular momentum> <spins> the old <neutron star> up, and the <accretion> <luminosity> can produce an <X-ray binary>. Long <accretion> episodes can produce millisecond <rotation>, commonly accompanied by a smaller inferred external <magnetic field>. After <mass> transfer declines, the <radio pulsar> can switch on again as a <recycled pulsar>, often with a <white dwarf> companion. Other binary histories can leave a second <neutron star>, an ablated companion or an isolated recycled object; there is no single obligatory evolutionary sequence.
A direct observational bridge is the coherent approximately $2.49\ \mathrm{ms}$ X-ray pulsation discovered with RXTE in SAX J1808.4-3658, an accretion-powered binary. It demonstrated an accreting <neutron star> rotating at a rate appropriate to recycled millisecond <radio pulsars>.<Discovery of the first accretion-powered millisecond X-ray pulsar by Rudy Wijnands and Michiel van der Klis>[Abstract, coherent pulsations and interpretation]{ref} Timing analysis must correct orbital <Doppler shifts>, telescope and solar-system timing, and instrumental aliases. Radio dispersion, sky coverage and beaming selection affect demographic inferences. \b[The sequence is rotational <pulsar spin-down>, possible loss of radio activity, and possible accretion-driven recycling—not a necessary path for every <neutron star>.]
\Image[/past-exam-of-the-relativistic-astrophysics-and-cosmology-course-of-the-university-of-cambridge/2015/paper-1-pulsar-recycling.png]
{title=Original schematic of ordinary pulsar spin-down and the conditional binary-recycling route}
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