# Interstellar Medium and Pulsars

• Norman
In summary, the conversation is about a problem in Jackson related to the partially ionized interstellar medium and its response to optical frequencies in a weak magnetic field. The individual is seeking help in solving the problem and has tried to use the relevant section in Jackson's work. Another individual has attempted to offer assistance but suggests posting in a different forum for better help. The original poster then presents their solution to the problem.
Norman
Originally posted in College Level Homework help but I got no responses there. Please help if you can.

I am studying for my qualifier and doing problems out of Jackson.
I am stuck on this one... any help would really be appreciated... I am unsure how to begin:
Jackson 7.15
The partially ionized interstellar medium (mostly hydrogen) responds to optical frequencies as an electronic plasma in a weak magnetic field. The broad-spectrum pulses from a pulsar allow determination of some average properties of the interstellar medium. The treatment of an electronic plasma in a magnetic field of Section 7.6 is pertinent.
a) Ignoring the weak magnetic field and assuming that $max(w_p) \ll w$, show that c times the transit time of a pulse of mean frequency w from a pulsar a distance R away is
$$ct(w) \approx R+\frac{e^2}{2 \epsilon_0 m_e w^2} \int n_e (z) dz$$
where $n_e (z)$ is the electron density along the path of light.

so this is what I have so far:
ignoring the weak B-field the position has a solution of:
$$x=\frac{e}{m_e w^2}E$$
and obviously ct(w) is a distance, but now I am lost...
Please help, I have been stumbling with this problem for a couple of days and it is turning into a monster that I need to solve.
(ps. I have read the pertinent section of Jackson over and over... I don't see any help in it.)

I had a go at trying to be of help, but it's been too long since I did this stuff. Try posting (again) in the Stellar Astrophysics forum

I think I actually solved it...

if $$t=\int^r_0 \frac{1}{v_g} dz$$

and I write
$$v_gv_p=c^2$$

Assumming that the the electron density is slowly varying over a wavelength of radiation, so that it is reasonable to think about a slowly varying index of refraction n(w,z) is can write:

$$v_p=\frac{c}{n(w,z)}$$

which implies that $$v_g=n(w,z) c$$

for an electronic plasma: $$n(w,z)=\sqrt{1 - \frac {w_p^2}{w^2} }$$

where $$w_p^2 =\frac{ n_e (z) e^2}{\epsilon_0 m_e}$$
where $n_e (z)$ is the electron density

so therefore $$n(w,z)=\sqrt{1-\frac{n_e (z ) e^2}{\epsilon_0 m_e w^2}}$$

and then $$v_g=c\sqrt{1-\frac{n_e (z ) e^2}{\epsilon_0 m_e w^2}}$$

which implies that:
$$t=\frac{1}{c} \int_0^R (1-\frac{n_e (z ) e^2}{\epsilon_0 m_e w^2})^{-\frac{1}{2}} dz$$

since $$w_p \ll w$$:

$$ct(w) \approx \int_0^R (1+\frac{n_e (z) e^2}{2 \epsilon_0 m_e w^2}) dz$$

finally:

$$ct(w) \approx R+\frac{e^2}{2 \epsilon_0 m_e w^2} \int^R_0 n_e(z ) dz$$

Last edited:
Chi Meson said:
I had a go at trying to be of help, but it's been too long since I did this stuff. Try posting (again) in the Stellar Astrophysics forum

Chi,

Thanks a lot for atleast trying... does the above look correct?
Thanks a lot,
Norm

## What is the interstellar medium?

The interstellar medium is the material that exists between stars in a galaxy. It is made up of gas, dust, and cosmic rays, and plays an important role in the formation and evolution of stars and planetary systems.

## What are pulsars?

Pulsars are highly magnetized, rapidly rotating neutron stars that emit beams of electromagnetic radiation from their magnetic poles. These beams can be detected on Earth as a series of regular pulses, giving pulsars their name.

## How are pulsars formed?

Pulsars are formed when a massive star runs out of nuclear fuel and undergoes a supernova explosion, leaving behind a dense core of neutrons. If the core has a strong magnetic field and is rotating rapidly, it can become a pulsar.

## What is the significance of pulsars?

Pulsars are important objects to study because they provide scientists with a unique window into the extreme conditions of the universe. They also serve as natural laboratories for testing theories of gravity and the behavior of matter under extreme pressures and temperatures.

## How do pulsars help us understand the interstellar medium?

Pulsars are used as probes of the interstellar medium because their signals can be affected by the material they pass through on their journey to Earth. By studying the properties of pulsar signals, scientists can learn about the composition, density, and magnetic fields of the interstellar medium.

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