Consider a spherical black asteroid (made of rock) which has...

Consider a spherical black asteroid (made of rock) which has been ejected from the solar system, so that the radiation from the sun no longer has a significant effect on the temperature of the asteroid. Radioactive elements produce heat uniformly inside the asteroid at a rate of $\dot{q}=3 \times 10^{-14}$ $\mathrm{cal} / \mathrm{g} \cdot \mathrm{sec}$. The density of the rock is $\rho=3.5 \mathrm{~g} / \mathrm{cm}^3$, and the thermal conductivity is $k=5 \times 10^{-3} \mathrm{cal} / \mathrm{deg} \cdot \mathrm{cm} \cdot \mathrm{sec}$. The radius of the asteroid is $R=100 \mathrm{~km}$. Determine the central temperature $T_c$ and the surface temperature $T_n$, of the asteroid assuming that a steady state has been achieved.

Key Concepts

Steady-State Heat Conduction

This concept describes a thermal system in which the temperature distribution does not change over time. In the context of the asteroid, the steady state implies that the internal heat generated by radioactive decay is balanced by the heat conducted to the surface, resulting in a constant temperature profile during the period of interest.

Internal Heat Generation

Internal heat generation refers to the production of heat uniformly throughout a material due to processes such as radioactive decay. In this problem, the heat is produced at a specific rate per unit mass, and when combined with the density of the material, it results in a volumetric heat generation that drives the temperature gradients within the sphere.

Thermal Conductivity

Thermal conductivity is a material property that quantifies the ability of a material to conduct heat. It governs the rate at which heat flows from regions of high temperature (the interior) to regions of lower temperature (the surface) and is crucial in determining the temperature distribution in the asteroid.

Spherical Geometry and Symmetry

The radial symmetry inherent in a spherical object simplifies the mathematical treatment of heat conduction. By assuming spherical symmetry, the heat conduction equation can be expressed in spherical coordinates, which facilitates the integration and analysis of how heat generated internally is transported to the surface.

Boundary Conditions in Heat Transfer

Boundary conditions play a critical role in solving the heat conduction equation. In this problem, the conditions at the center (requiring symmetry) and at the surface (where the temperature is determined by the balance between conduction and radiative losses) are essential in determining the specific temperature distribution from the center to the surface of the asteroid.

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