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## Q. 12.6

A spherical balloon of radius R is in orbit around the earth and enters the earth’s shadow. The balloon has a perfectly transparent wall and is filled with a gray gas with constant absorption coefficient κ, such that $κR \ll 1.$ Neglecting radiant exchange with the earth, derive a relation for the initial rate of radiant energy loss from the balloon if the initial temperature of the gas is $T_0.$

## Verified Solution

Using the emission approximation, Equation 12.16,

$I(S)=\int_{S^*=0}^{S}{\left[\int_{\lambda =0}^{\infty }{k_\lambda } (S^*)I_{\lambda b}(S^*)d\lambda \right] }dS^*$                        (12.16)

Figure 12.1b shows that the intensity at the surface for a typical path S is $I(\theta )=(κσT_0^4/\pi )S=(κσT_0^4/\pi )2R\cos \theta$ The radiative flux leaving the surface is

$q=2\pi \int_{\theta =0}^{\pi /2}{I(\theta )\cos \theta \sin \theta d\theta } =4k\sigma T_0^4R\int_{\theta =0}^{\pi /2}{\cos ^2\theta \sin \theta d\theta }=\frac{4}{3}k\sigma T_0^4R$

The initial rate of energy loss from the entire sphere is then $Q = 4/3 κσ T_0^4R(4 \pi R^2)=4κσ T_0^4V_s,$ where $V_s$ is the sphere volume. This is what is expected, as it was found in Section 1.6.4 that any isothermal gas volume with negligible internal absorption radiates according to this relation.