Chapter 12, Problem 12.138P

A spherical capsule of 3-m radius is fired from a space platform in earth orbit, such that it travels toward the center of the sun at $16,000\text{ km/s}$ . Assume that the capsule is a lumped capacitance body with a density−specific heat product of $4{\text{ x 10}}^6{\text{ j/m}}^3\cdot K$ and that its surface is black.

1. Derive a differential equation for predicting the capsule temperature as a function of time. Solve this equation to obtain the temperature as a function of time in terms of capsule parameters and its initial temperature $T_i$ .
2. If the capsule begins its journey at ${20}^{\circ }C$ , predict the position of the capsule relative to the sun at which its destruction temperature, ${150}^{\circ }C$, is reached.