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Grey and Eddington Approximations

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Grey and Eddington Approximations ***

restart

 

Problem: Determine the values of Iin and Iout as functions of vertical optical depth (τ), using the Eddington Approximation for a plane-parallel atmosphere. Then find the optical depth at which the radiation field is isotropic to within 5%.

 

Hints:

Begin by equating the first two equations below and then the second two equations below, and substituting.

Find an expression for Iout.

Find an expression for Iin.

 

Data:

r := 0.5e-1

0.5e-1

(1)

Useful Equations:

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Solution:

NULL

 

-Iin*Pi+Iout*Pi = sigma*T^4

-Iin*Pi+Iout*Pi = sigma*T^4

(2)

1/Pi*(-Iin*Pi+Iout*Pi = sigma*T^4)

(-Iin*Pi+Iout*Pi)/Pi = sigma*T^4/Pi

(3)

simplify((-Iin*Pi+Iout*Pi)/Pi = sigma*T^4/Pi)

-Iin+Iout = sigma*T^4/Pi

(4)

(1/2)*Iin+(1/2)*Iout = (3/4)*sigma*T^4*(tau+2/3)/Pi

(1/2)*Iin+(1/2)*Iout = (3/4)*sigma*T^4*(tau+2/3)/Pi

(5)

NULL

2*((1/2)*Iin+(1/2)*Iout = (3/4)*sigma*T^4*(tau+2/3)/Pi)

Iin+Iout = (3/2)*sigma*T^4*(tau+2/3)/Pi

(6)

NULL

Add to find an expression for Iout.

NULL

NULL

Iin+Iout-Iin+Iout = (3/2)*sigma*T^4*(tau+2/3)/Pi+sigma*T^4/Pi

2*Iout = (3/2)*sigma*T^4*(tau+2/3)/Pi+sigma*T^4/Pi

(7)

NULL

1/2*(2*Iout = (3/2)*sigma*T^4*(tau+2/3)/Pi+sigma*T^4/Pi)

Iout = (3/4)*sigma*T^4*(tau+2/3)/Pi+(1/2)*sigma*T^4/Pi

(8)

NULLNULL

Iout := sigma*T^4*(3/4*(tau+2/3)+1/2)/Pi

sigma*T^4*((3/4)*tau+1)/Pi

(9)

NULL

Take half of this expression and subtract from the sum of Iin and Iout to find an expression for Iin.

NULL

1/2*(Iout = sigma*T^4*((3/4)*tau+1)/Pi)

(1/2)*sigma*T^4*((3/4)*tau+1)/Pi = (1/2)*sigma*T^4*((3/4)*tau+1)/Pi

(10)

NULL

NULL

NULLNULL

(1/2)*Iin+(1/2)*Iout = (3/4)*sigma*T^4*(tau+2/3)/Pi

(1/2)*Iin+(1/2)*sigma*T^4*((3/4)*tau+1)/Pi = (3/4)*sigma*T^4*(tau+2/3)/Pi

(11)

NULL

NULL

((1/2)*Iin+(1/2)*Iout = (3/4)*sigma*T^4*(tau+2/3)/Pi)-((1/2)*Iout = (1/2)*sigma*T^4*((3/4)*tau+1)/Pi)

(1/2)*Iin = (3/4)*sigma*T^4*(tau+2/3)/Pi-(1/2)*sigma*T^4*((3/4)*tau+1)/Pi

(12)

NULL

2*((1/2)*Iin = (3/4)*sigma*T^4*(tau+2/3)/Pi-(1/2)*sigma*T^4*((3/4)*tau+1)/Pi)

Iin = (3/2)*sigma*T^4*(tau+2/3)/Pi-sigma*T^4*((3/4)*tau+1)/Pi

(13)

NULL

simplify(Iin = (3/2)*sigma*T^4*(tau+2/3)/Pi-sigma*T^4*((3/4)*tau+1)/Pi)

Iin = (3/4)*sigma*T^4*tau/Pi

(14)

NULL

Iin := (3/4)*sigma*T^4*tau/Pi

(3/4)*sigma*T^4*tau/Pi

(15)

NULL

NULL

For the radiation field to be isotropic to within 5%, the following condition must hold:

 

(Iout-Iin)/(1/2*(Iout+Iin)) = r

(-(3/4)*sigma*T^4*tau/Pi+sigma*T^4*((3/4)*tau+1)/Pi)/((3/8)*sigma*T^4*tau/Pi+(1/2)*sigma*T^4*((3/4)*tau+1)/Pi) = 0.5e-1

(16)

NULL

simplify((-(3/4)*sigma*T^4*tau/Pi+sigma*T^4*((3/4)*tau+1)/Pi)/((1/2)*sigma*T^4*((3/4)*tau+1)/Pi+(3/8)*sigma*T^4*tau/Pi) = r)

4/(3*tau+2) = 0.5e-1

(17)

NULL

Solve for τ.

NULL

solve(4/(3*tau+2) = r, tau)

26.

(18)

NULL

At optical depth τ = 26, the radiation is isotropic to within 5%.

NULL

 

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