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Electron density

  1. Feb 26, 2016 #1
    1. The problem statement, all variables and given/known data
    Hi!

    I need to solve a continuity equation for electron density as a function of time in the E-region of Earth's atmosphere. I shall neglect vertical transport and that the ion production rate completely shuts off dusk.

    2. Relevant equations
    See below

    3. The attempt at a solution
    Here is my attempt:

    (1+λ)dNe/dt=p(t)-(1+λ)(αd-aλi)Ne2-Ne dλ/dt
    Ne is the electron density, dy, λ=ratio of negative ions to electrons , p(t) is the electron production rate, αd is the recombination coeff. for ions and ai is the recombination coeff. negative ions with positive ions.

    The effective recombination coefficient is: αeff=(1+λ)(αd+aλi).
    If we assume that the is no negative ions, we get:
    dNe/dt=p(t)-adNe2 (1)
    and aeff simply becomes ad

    Here comes the issue, I need to have a solution for equation (1) with respect to the density and time. . How should I integrate this? The density from the beginning is ≈ 1*109 m3 , which I calculated
     
  2. jcsd
  3. Feb 26, 2016 #2

    Zondrina

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    Homework Helper

    There is a typo in equation ##(1)##. If I understand correctly you meant to write ##\alpha_d## not ##a_d##.

    Equation ##(1)## has the form of a Riccati equation:

    $$\frac{dN_e(t)}{dt} = p(t) - \alpha_d N_e^2(t)$$
    $$y'(t) = p(t) + g(t)y(t) + f(t)y^2(t)$$

    Where ##N_e(t) = y(t)##, ##p(t) = p(t)##, ##g(t) = 0##, and ##f(t) = \alpha_d##.

    To learn how to deal with equations of this form, see: https://en.wikipedia.org/wiki/Riccati_equation#Reduction_to_a_second_order_linear_equation
     
  4. Feb 26, 2016 #3
    THANK YOU THANK YOU THANK YOU!
     
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