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An imaginary planet is made of a material that sublimates directly (solid to gas). The energy needed for sublimation is supplied by the release of gravitational self-energy. If the heat of sublimation is H joules per kilogram and the planet has uniform density rho, what is the minimum radius for the planet to undergo self-sublimation?

1. Rmin = sqrt(4H / (3 * G * rho * pi))
2. Rmin = sqrt(5H / (4 * G * rho * pi))
3. Rmin = sqrt(H / (G * rho * pi))
4. Rmin = (3/2) * sqrt(H / (G * rho * pi))

Correct answer: Rmin = sqrt(5H / (4 * G * rho * pi))

Solution

Setting the gravitational self-energy per unit mass 3GM/(5R) = H and substituting M = (4/3)*pi*R³*rho gives R = sqrt(5H/(4*pi*G*rho)).

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