# Distance to pulsar with plasma dispersion relation

• Robsta
In summary, pulsars are rapidly rotating stars that emit a narrow beam of radiation. Distance estimates to pulsars were obtained by exploiting the dispersion of pulses in the interstellar medium, which has an electron density of about 105m−3. By using the dispersion relation and the delay between signals at different frequencies, the distance to a pulsar can be calculated. For the pulsar CP 0328, the distance was found to be 0.367s for signals at 610 and 408MHz, and 4.18s for signals at 408 and 151MHz.
Robsta

## Homework Statement

Pulsars are stars that have suffered gravitational collapse. They rotate rapidly and emit a narrow
beam of radiation. The pulse lengths, at the earth, are ∼1ms and the periods are ∼1s.
Within a few months of the discovery of pulsars distance estimates were obtained by exploiting
the dispersion of the pulses in the interstellar medium, which is ionised hydrogen with an electron
.
(a) Show that for ω2>>ωp2 where ωp is the plasma frequency of the interstellar medium, the time
delay ∆t as a function of f−2 − (f + ∆f)−2, where f is the pulse frequency, is a straight line
whose slope is a measure of the distance to the pulsar.
(b) For the pulsar CP 0328 the delay between signals at 610 and 408MHz was 0.367s; that
between signals at 408 and 151MHz was 4.18s. Find the distance to CP 0328.

## Homework Equations

I know the dispersion relation for the plasma. n2= 1 - (ω2p2)
Obviously speed * time = distance to pulsar
ω=2(pi)f
n*vp=c

## The Attempt at a Solution

So the pulsar emits a pulse at t=0 of two frequencies. Because of dispersion in the interstellar plasma, the waves travel at different speeds (because of their different frequencies).
So if one wave of frequency f arrives at time t, the one traveling with frequency (f + ∆f) arrives at time (t + ∆t)
I'm a bit confused because if ω2>>ωp2 then as per the formula for n above, n=1 and there would be no dispersion?

I'm really looking for some help formalising part 1

Check the fraction in your dispersion relation.
Robsta said:
I'm a bit confused because if ω2>>ωp2 then as per the formula for n above, n=1 and there would be no dispersion?
n is close to 1, but not exactly 1. The small difference leads to the small time differences: seconds, where the pulsar can be thousands of light years away so t is thousands of years.

## 1. What is the plasma dispersion relation?

The plasma dispersion relation is a mathematical equation that describes how electromagnetic waves propagate through a plasma. It takes into account the properties of the plasma, such as density and magnetic field strength, to determine the speed and direction of the wave.

## 2. How does the plasma dispersion relation affect the distance to a pulsar?

The plasma dispersion relation can be used to calculate the delay of radio signals from a pulsar due to their interaction with the surrounding plasma. This delay can then be used to estimate the distance to the pulsar.

## 3. What is the relationship between the frequency of the pulsar signal and the distance to the pulsar?

The frequency of the pulsar signal is inversely proportional to the distance to the pulsar. This means that as the distance increases, the frequency decreases. This relationship is based on the plasma dispersion relation and the properties of the plasma surrounding the pulsar.

## 4. Can the plasma dispersion relation be used to accurately measure the distance to a pulsar?

Yes, the plasma dispersion relation can be used to estimate the distance to a pulsar with a high degree of accuracy. However, it is important to take into account other factors that may affect the pulsar signal, such as interstellar medium and instrumental effects.

## 5. How does the distance to a pulsar affect our understanding of the universe?

The distance to a pulsar is a crucial piece of information for understanding the structure and evolution of the universe. Pulsars are used as cosmic clocks, providing precise measurements of time and distance. By accurately measuring the distance to a pulsar, we can gain insight into the distribution of matter and the expansion of the universe.

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