“2019年USAAAO决赛第7题”的版本间的差异
Jingsong Guo（讨论 | 贡献） （创建页面，内容为“==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 bet…”） |
Jingsong Guo（讨论 | 贡献） |
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| 第5行： | 第5行： | ||
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 | 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 | + | $$P = K\cdot ρ^γ$$ (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 | + | 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 | b) Black Holes as Blackbodies The mass radius relation for ideal non-rotating, uncharged black holes is known from relativity to be | ||
| − | R = 2GM | + | $$R = \dfrac{2GM}{c^2}$$ (2) |
Moreover, Stephen Hawking showed that a black hole behaves like a blackbody, where its temperature (known as the Hawking temperature) is given by | Moreover, Stephen Hawking showed that a black hole behaves like a blackbody, where its temperature (known as the Hawking temperature) is given by | ||
| − | T = ħc^3 | + | $$T =\dfrac{ħc^3}{8πk_BGM}$$ (3) |
| − | Given this information, show that the lifetime of a black hole (justify this phrase!) t | + | Given this information, show that the lifetime of a black hole (justify this phrase!) $$t^*$$scales with its mass M as |
| − | t* | + | $$t^* \propto M^β$$ (4) |
where you should find the exponent β | 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. | + | 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. |
2020年3月7日 (六) 17:50的最新版本
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.
