# 2019年USAAAO决赛第7题

## 英文题目

7. (15 points)

a) Mass-Radius Relation Stellar physics often involves guessing the equation of state for stars, which is typically a relation between the pressure P and the density ρ. A family of such guesses are known as polytopes and go as follows

$$P = K\cdot ρ^γ$$ (1)

where K is a constant and the exponent γ is fixed to match a certain pressure and core temperature of a star. Given this, show that one can obtain a crude power-law scaling between the mass M of a polytopic star and its radius R of the form $$M\propto R^α$$. Find the exponent α for polytopic stars (justify all steps in your argument). Also, indicate the exponent γ for which the mass is independent of the radius R. Bonus: Why is this case interesting?

b) Black Holes as Blackbodies The mass radius relation for ideal non-rotating, uncharged black holes is known from relativity to be

$$R = \dfrac{2GM}{c^2}$$ (2)

Moreover, Stephen Hawking showed that a black hole behaves like a blackbody, where its temperature (known as the Hawking temperature) is given by

$$T =\dfrac{ħc^3}{8πk_BGM}$$ (3)

Given this information, show that the lifetime of a black hole (justify this phrase!) $$t^*$$scales with its mass M as

$$t^* \propto M^β$$ (4)

where you should find the exponent β

c) Minimal Black Holes Using the information of the previous part, and Wien’s displacement law, estimate the smallest possible mass of a black hole. State any possible flaws with this estimate.