# B Characteristic age of pulsars

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1. Mar 14, 2016

### Nicklas

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: Mar 14, 2016
2. Mar 15, 2016

### mbond

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

3. Mar 15, 2016

### Student100

In some texts this is considered the "true" age of pulsar, although the true age would also have k and n as functions of time.

The characteristic age is typically when k is kept as constant, n = 3, and $\frac{P_0}{P} = 0$, this is an overestimation, but a decent approximation.