Interstellar Medium and Pulsars

[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.
Thanks for any help you can give.
(ps. I have read the pertinent section of Jackson over and over... I don't see any help in it.)

 

[Chi Meson]

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

 

[Norman]

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:

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

 

[Norman]

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