1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Astronomy - Greenhouse Effect (integral troubles mainly)

  1. Nov 14, 2005 #1


    User Avatar

    Given equations:
    [tex]\frac{d T^4}{dr} = - \frac{3 \kappa \rho}{a c} F_{rad}[/tex]
    where [tex]a = 4 \sigma_{B} / c[/tex] with [tex]\sigma_{B}[/tex] being the Boltzmann constant.
    Also, define the optical depth [tex]\tau = \int \kappa \rho dr[/tex]
    Optical depth measured at ground level is [tex]\tau_{g}[/tex]
    Where [tex]\tau_{g} = \int^\infty_{ground} \kappa \rho dr[/tex] and the optical depth at the photosphere equals 2/3
    Also [tex]F_{rad} = \frac {L_{r}}{4 \pi r^2} = \frac {A \sigma_{B} T_{g}^{4}}{4 \pi r^2} = \sigma_{B} T_{g}^{4}[/tex]
    Now, I have to use the first equation given to find:
    [tex]T^4_{g} = T^4_{p} [1 + \frac{3}{4} (\tau_{g} - \frac {2}{3})][/tex]
    So what I did was take the integral of the first equation and try to work from there but I am having difficulty understanding what to do with the left hand side of the integral but here's what I have so far:
    1) [tex]\frac{d T^4}{dr} = - \frac{3 \kappa \rho}{a c} F_{rad}[/tex]
    2) [tex]\int^{photosphere}_{ground} d T^4 = - \frac{3}{a c} F_{rad} \int^{photosphere}_{ground} \kappa \rho dr[/tex] (because F_rad is constant)
    3) [tex](T^4_{g} - T^4_{p}) = - \frac{3}{a c} F_{rad} ( \int^\infty_{ground} \kappa \rho dr - \int^\infty_{photosphere} \kappa \rho dr)[/tex]
    4) [tex](T^4_{g} - T^4_{p}) = - \frac{3}{a c} F_{rad} (\tau_g - \frac{2}{3})[/tex]
    5) [tex](T^4_{g} - T^4_{p}) = - \frac{3}{4} T_{g}^{4} (\tau_g - \frac{2}{3})[/tex]
    I know I'm close and it's not complete and I'm stuck... Did I make a mistake anywhere or what? If anyone could help me out that'd be great.
    Last edited: Nov 14, 2005
  2. jcsd
  3. Nov 15, 2005 #2


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Could you be more explicit about what you're trying to show?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook