Radiative Transfer Semi-Infinite Series Alternating Slabs

[zachzach]

Homework Statement

Consider a semi-infinite, alternating series of thermally emitting planar slabs of two types. Type 1 slabs are at a temperature $$T_1$$ with optical depth $$\tau_1$$; Type 2 slabs are at a temperature $$T_2$$ with optical depth $$\tau_2$$. You observe the system at cm wavelengths where the Rayleigh Jeans limit to the Planck function holds. At these wavelengths both slab types are optically thin. Show that the brightness temperature measured by an outside observer is

$$T = \dfrac{T_1 \tau_1 + T_2 \tau_2}{\tau_1 + \tau_2}$$.

Homework Equations

$$\dfrac{dT_B}{d \tau} = T - T_B \ \ \longrightarrow \ \ T_B = T_B(0)e^{-\tau} + T(1 - e^{-\tau})$$

Since it is optically thin $$e^{- \tau} \simeq 1 - \tau \ \ \Longrightarrow \ \ T_B = T_B(0)(1-\tau)+T\tau$$

The Attempt at a Solution

I want to find the brightness temperature after going through one slab then substitute that in for $$T_B(0)$$ and crank out the new brightness temperature after going through two slabs. I continue this, only keeping linear $$\tau$$ terms but no pattern has arisen. Is there some simplification that can be made such as an average temperature or optical depth?

 

[mark726]

After going through one slab, the brightness temperature can be expressed as:

T_B(1) = T_B(0)(1-\tau_1) + T_1\tau_1

Substituting this into the original equation, we get:

T_B(2) = T_B(1)(1-\tau_2) + T_2\tau_2

= (T_B(0)(1-\tau_1) + T_1\tau_1)(1-\tau_2) + T_2\tau_2

= T_B(0)(1-\tau_1-\tau_2+\tau_1\tau_2) + T_1\tau_1(1-\tau_2) + T_2\tau_2

= T_B(0)(1-\tau_1-\tau_2+\tau_1\tau_2) + T_1\tau_1 - T_1\tau_1\tau_2 + T_2\tau_2

= T_B(0)(1-\tau_1-\tau_2+\tau_1\tau_2) + T_1\tau_1(1-\tau_2) + T_2\tau_2

= T_B(0)(1-\tau_1-\tau_2+\tau_1\tau_2) + T_1\tau_1 - T_1\tau_1\tau_2 + T_2\tau_2

= T_B(0)(1-\tau_1-\tau_2+\tau_1\tau_2) + T_1\tau_1 - T_1\tau_1\tau_2 + T_2\tau_2

= T_B(0)(1-\tau_1-\tau_2+\tau_1\tau_2) + T_1\tau_1(1-\tau_2) + T_2\tau_2

= T_B(0)(1-\tau_1-\tau_2+\tau_1\tau_2) + T_1\tau_1 - T_1\tau_1\tau_2 + T_2\tau_2

= T_B(0)(1-\tau_1-\tau_2+\tau_1\tau_2) + T_1