# Characteristic age of pulsars

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## Main Question or Discussion Point

Hi everyone.
I'm trying to derive the formula for the characteristic age of a pulsar.

What i'm starting with is the following differential equation.
dP/dt=K*P2-n

What i think is odd, is several places they say solving this differential equation gives the following solution.
T=(P/((n-1)*dP/dt))*(1-(P0/P)n-1

Here is a picture of the equation too: But how do you get from equation 1. to this equation.
Please help me out, if you could explain it step by step, i would really appreciate it.
Because it doesn't make sense if you're trying to seperate both variables and integrate?

Last edited:

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Hope this helps:
$\dot{P}=\frac{dP}{dt}=kP^{2-n}$
$dt=\frac{dP}{kP^{2-n}}$
$\tau=t-t_{0}=\int_{P_{0}}^P \frac{dP}{kP^{2-n}}=\frac{1}{k}\frac{P^{n-1}-P_{0}^{n-1}}{n-1}=\frac{P^{n-1}}{k(n-1)}[1-(\frac{P_{0}}{P})^{n-1}]$
$k=\frac{\dot{P}}{P^{2-n}}$
$\tau=\frac{P}{(n-1)\dot{P}}[1-(\frac{P_{0}}{P})^{n-1}]$

Student100