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Given equations:
\frac{d T^4}{dr} = - \frac{3 \kappa \rho}{a c} F_{rad}
where a = 4 \sigma_{B} / c with \sigma_{B} being the Boltzmann constant.
Also, define the optical depth \tau = \int \kappa \rho dr
Optical depth measured at ground level is \tau_{g}
Where \tau_{g} = \int^\infty_{ground} \kappa \rho dr and the optical depth at the photosphere equals 2/3
Also 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}
Now, I have to use the first equation given to find:
T^4_{g} = T^4_{p} [1 + \frac{3}{4} (\tau_{g} - \frac {2}{3})]
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) \frac{d T^4}{dr} = - \frac{3 \kappa \rho}{a c} F_{rad}
2) \int^{photosphere}_{ground} d T^4 = - \frac{3}{a c} F_{rad} \int^{photosphere}_{ground} \kappa \rho dr (because F_rad is constant)
3) (T^4_{g} - T^4_{p}) = - \frac{3}{a c} F_{rad} ( \int^\infty_{ground} \kappa \rho dr - \int^\infty_{photosphere} \kappa \rho dr)
4) (T^4_{g} - T^4_{p}) = - \frac{3}{a c} F_{rad} (\tau_g - \frac{2}{3})
5) (T^4_{g} - T^4_{p}) = - \frac{3}{4} T_{g}^{4} (\tau_g - \frac{2}{3})
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.
Thanks!
\frac{d T^4}{dr} = - \frac{3 \kappa \rho}{a c} F_{rad}
where a = 4 \sigma_{B} / c with \sigma_{B} being the Boltzmann constant.
Also, define the optical depth \tau = \int \kappa \rho dr
Optical depth measured at ground level is \tau_{g}
Where \tau_{g} = \int^\infty_{ground} \kappa \rho dr and the optical depth at the photosphere equals 2/3
Also 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}
Now, I have to use the first equation given to find:
T^4_{g} = T^4_{p} [1 + \frac{3}{4} (\tau_{g} - \frac {2}{3})]
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) \frac{d T^4}{dr} = - \frac{3 \kappa \rho}{a c} F_{rad}
2) \int^{photosphere}_{ground} d T^4 = - \frac{3}{a c} F_{rad} \int^{photosphere}_{ground} \kappa \rho dr (because F_rad is constant)
3) (T^4_{g} - T^4_{p}) = - \frac{3}{a c} F_{rad} ( \int^\infty_{ground} \kappa \rho dr - \int^\infty_{photosphere} \kappa \rho dr)
4) (T^4_{g} - T^4_{p}) = - \frac{3}{a c} F_{rad} (\tau_g - \frac{2}{3})
5) (T^4_{g} - T^4_{p}) = - \frac{3}{4} T_{g}^{4} (\tau_g - \frac{2}{3})
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.
Thanks!
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