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Interstellar Medium and Pulsars

  1. Apr 11, 2004 #1
    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 [itex] max(w_p) \ll w [/itex], show that c times the transit time of a pulse of mean frequency w from a pulsar a distance R away is
    [tex] ct(w) \approx R+\frac{e^2}{2 \epsilon_0 m_e w^2} \int n_e (z) dz [/tex]
    where [itex] n_e (z) [/itex] 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:
    [tex] x=\frac{e}{m_e w^2}E [/tex]
    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.)
     
  2. jcsd
  3. Apr 12, 2004 #2

    Chi Meson

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    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
     
  4. Apr 12, 2004 #3
    I think I actually solved it...

    if [tex] t=\int^r_0 \frac{1}{v_g} dz [/tex]

    and I write
    [tex] v_gv_p=c^2 [/tex]

    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:

    [tex] v_p=\frac{c}{n(w,z)} [/tex]

    which implies that [tex] v_g=n(w,z) c [/tex]

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

    where [tex] w_p^2 =\frac{ n_e (z) e^2}{\epsilon_0 m_e} [/tex]
    where [itex] n_e (z) [/itex] is the electron density

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

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

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

    since [tex] w_p \ll w [/tex]:


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

    finally:

    [tex] ct(w) \approx R+\frac{e^2}{2 \epsilon_0 m_e w^2} \int^R_0 n_e(z ) dz [/tex]
     
    Last edited: Apr 12, 2004
  5. Apr 12, 2004 #4
    Chi,

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