Question

Short pulses of microwave radiation are emitted from the Crab nebula at a distance
L from the earth. When they reach the earth, the frequency components of the pulses
are dispersed after a travel time T, such that waves of different frequencies arrive at
different times. The observed dispersion is due to the intervening interstellar plasma,
which enables us to estimate the electron density of this plasma. Assuming one knows L
by other means, obtain the expression for the electron density. Assume v p ≈ vg which is
a reasonable approximation for interstellar medium.

Answer 1

If we assume that the pulses emmited by the Crab Nebula travel at the speed of light, they'd take some finite amount of time to reach to the Earth. However, there's an observed time delay, accounted by the fact that the space is not a perfect vacuum. The interstellar environment is actually comprised of a low-density plasma that can be modeled as free electrons. This plasma is known as the interstellar medium (ISM). So, a signal traveling through the ISM cannot travel as fast as the speed of light, because the photons are dispersed, depending on their frequency, as if they were going through a prism. The refractive index os the ISM is given by:

Where the frecuency of the photons is a function of the electron density:

Where n_e is the electron density, e is the electron charge magnitude and m_e the mass of the electron. So the frequency of the photons traveling through the ISM is:

The time delay along a path dl can be calculated as:

Where L is the distance between the Crab Nebula and earth. But if we take

And work the expression out we get:

The integral is known as dispersion measure, DM. Finally we can write the expression in terms of the difference between low and high frequencies (signals with lower frequencies arrive later):

And since the DM proportional to the distance L, if we know this last one by other means, we can estimate the electron density of the ISM between the Crab Nebula and Earth as