Question 12.6: A spherical balloon of radius R is in orbit around the earth...
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.
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.