How Does Calculus Explain the Changing Rotational Energy of Pulsars?

[AStaunton]

looking at my notes on pulsars I have:

$$K=\frac{2\pi^{2}I}{P^{2}}\implies dk=\frac{2\pi^{2}I}{P^{2}}dP\implies\frac{dk}{dt}=\frac{-4\pi^{2}I}{P^{3}}\frac{dP}{dt}$$

where K is kinetic rotational energy and P is momentum...

I don't quite follow the final expression (when the time derivative is taken), when this is done to the final equality, it feels like the chain rule should be used as both dP and 2pi^2I/P^2 are functions of t...clearly I'm mistaken but I don't see why..I'm saying that I would have expected:

$$\frac{dk}{dt}=\frac{-4\pi^{2}I}{P^{3}}dP+\frac{2\pi^{2}I}{p^{2}}\frac{dP}{dt}$$

 

[lanedance]

i think you should have applied the chain rule when you first move to differentials

 

[AStaunton]

one way I have reasoned the above result is that:

as $$\frac{dk}{dP}=\frac{-4\pi^{2}I}{P^{3}}$$ and as P is a function of t then can simply mulitiply both sides by dp/dt...

Although I reasoned thusly, it still feels pretty wishy-washy... and I don't know if that is a legit way of coming to that result..

it feels like the correct way of doing it should be to directly plug in the value of P into the equation.
ie as P=d(mx)dt plug that into the first equality I wrote up:

$$K=\frac{2\pi^{2}I}{(\frac{dmx}{dt})^{2}}$$

and then take the time derivitave of the above expression...the problem is I do not know how to take time derivative of (dx/dt)^-2...could you give advice here?



Also, the final piece of latex code I left at the bottom of my last post was completely incorrect!

 

[vela]

I think your notes are wrong. For one thing, if I is the moment of inertia, P has to have units of time, not momentum, for the units to work out. Your initial formula is

$$K = \frac{1}{2}I\omega^2 = \frac{1}{2}I\left(\frac{2\pi}{P}\right)^2 = \frac{2\pi^2I}{P^2}$$

P isn't the momentum but the period of rotation, and the angular velocity is ω=2π/P.

If you ignore the intermediate step you have written down, which is incorrect, you have

$$K = \frac{2\pi^2I}{P^2} \hspace{0.2in} \Rightarrow \hspace{0.2in} \frac{dK}{dt} = -\frac{4\pi^2I}{P^3}\frac{dP}{dt}$$

which you should recognize as a simple application of the chain rule.

The energy the pulsar radiates away ultimately comes from the rotational energy of the neutron star, which in turn affects the period of rotation. By measuring dP/dt, the rate at which a pulsar slows, you can determine the rate at which it radiates.