Lifetime of pulsars

Question

According to many textbooks, pulsars have lifetime (the time it would take the pulses to cease if $d(P)/dt$ were constant).

Lifetime is $P/[d(P)/dt]$, $P=$period

Why do P/[d(P)/dt] means the lifetime?

Answer 1

You should be able to see from a dimensional basis that the time taken to reach a certain value of something is that value divided by the rate at which it changes (assuming that you started from zero). In this case, where period is measured in seconds, in dimensional terms: $$ \tau \equiv \frac{{\rm seconds}}{\rm seconds\ per\ second}$$

However, there are a couple of wrinkles to this. Firstly, $\dot{P}$ is positive and $P$ is getting bigger from some initially small value, but not zero. Secondly, $\tau = P/\dot{P}$ also assumes that the rate of change is itself constant with time. This is not thought to be true for pulsars because the power emitted in radiation is strongly dependent on rotation rate, and there must be a small numerical factor in front of this expression. For example, if the pulsar spins down by emitting magnetic dipole radiation then $$ \tau = \frac{1}{2}\left( \frac{P}{\dot{P}}\right)$$ e.g. see here, where $\tau$ is the time to reach the present period $P$ and spindown rate $\dot{P}$, so long as $P\gg P_0$, where $P_0$ was the initial (small) period.