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 « 1.

Question

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 [latex]κR \ll 1.[/latex] 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 [latex]T_0.[/latex]

Accepted Answer

Using the emission approximation, Equation 12.16,

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

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

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

The initial rate of energy loss from the entire sphere is then [latex]Q = 4/3 κσ T_0^4R(4 \pi R^2)=4κσ T_0^4V_s,[/latex] where [latex]V_s[/latex] 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.

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

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