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Integrating the solar wind equation

  1. May 11, 2016 #1
    Hi, eveyone have been struggling to do this problem for a long time now, figured it is something very simple I am missing so thought I should ask here.

    1. The problem statement, all variables and given/known data

    Parker's solar wind equation is given after some manipulation as:

    (v - (Cs2 / v) dv/dr = 2 (Cs2 / r2) (r - rc )


    where rc = GMm/4kT (the critical radius)
    and
    Cs2 = 2kT/m (coronal sound speed)
    3. The attempt at a solution
    So using the usual method I separate the variables and integrate this equation to get:

    v2 / 2 - Cs2 log(v) = 2Cs2 log(r) + (4Cs2 rc / r) + C

    where C is the constant of integration. However every solution

    However every solution I see around the internet and indeed in my notes quotes the solution:

    (v2 / Cs2) - log(v2/ Cs2) = 4log(r/rc) + 4(r/rc)

    and I cannot seem to figure out why. Am I missing something blindingly obvious here?

    Thanks for your time, John.
     
  2. jcsd
  3. May 11, 2016 #2

    haruspex

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    In your integration, the 4 should be 2.
    You can get your version a lot closer by mutiplying through by 2/Cs2.
    The v2 inside the log comes from bringing in the factor of 2 from outside, and the /Cs2 may come from your constant of integration. (Likewise the /rc in the other log.)
    Theonly discrepancy that leaves is r/rc versus rc/r. Are you sure you have quoted that correctly?
     
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