# Pulsar distance estimation

This is quite an interesting problem in astrophysics so I thought it would be a good idea to ask here so we can archive the solution for future reference.

Consider a pulsar that emits pulses of frequencies $$\omega_1$$ and $$\omega_2$$. Due to the interstellar space, these pulses will each take a different time to arrive to Earth, thus we will observe a delay between signals.

If we know the dispersion relation $$\epsilon(\omega)$$, how can one find the delay between these two signals? For example, let's consider the most simple dispersion relation:

$$\epsilon(\omega)=1-\frac{\omega_p^2}{\omega^2}$$

where $$\omega_p$$ is the plasma frequency (for simplicity, let's take it as constant).

My idea is to start from the wave equation for a wave packet in a dispersive media and take an arbitrary distance $$L$$ for the wave to travel, and calculate the times $$t_1$$ and $$t_2$$ for the wave to take using each frequency, and somehow introduce the dispersion relation somewhere on the wave equation. However, while I know the evolution for a wave packet $$\psi(x,t)$$, I don't know how to find the time it takes to travel a distance $$L$$.

Edit: I found the solution using a less complicated method. We know that the wave number $$k$$ follows the relation,

$$k^2=\epsilon \mu\frac{\omega^2}{c^2}$$

Substituting the dispersion relation and considering non-magnetic media ($$\mu=1$$),

$$k^2=\frac{\omega^2}{c^2}(1-\frac{\omega_p^2}{\omega^2})$$

Thus we get,

$$\omega^2=k^2c^2+\omega_p^2$$

Taking the derivative and remembering the definition of group velocity,

$$\frac{d\omega}{dk}=\frac{kc^2}{\omega}=v_g$$

Substituting $$k$$ in the previous expression, we have:

$$v_g=c\sqrt{1-\frac{\omega_p^2}{\omega}}$$

If we consider that the wave travel a distance $$L$$, each pulse takes a time:

$$t_1=\frac{L}{v_1}=\frac{L\omega_1}{c\sqrt{\omega_1^2-\omega_p^2}}$$

$$t_2=\frac{L}{v_2}=\frac{L\omega_2}{c\sqrt{\omega_2^2-\omega_p^2}}$$

And thus the time difference is a function of the distance and frequencies,

$$\Delta t=\frac{L}{c}\left ( \frac{\omega_1}{\sqrt{\omega_1^2-\omega_p^2}} - \frac{\omega_2}{\sqrt{\omega_2^2-\omega_p^2}} \right )$$

If anyone can tell me it's correct so I can post it as an answer to my question, and thus close this thread.