2020年3月7日 (六) 09:50 Jingsong Guo

2020-03-07T09:50:41Z

Jingsong Guo：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…”

2020-03-07T09:24:04Z

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…”

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==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*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 M9Rα. 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 = 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 = ħc^3/8πkGM (3)

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

t* is proportional to Mβ (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.

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 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*ρ^γ (1)
+$$P = K\cdot ρ^γ$$ (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 M9Rα. 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?
+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 = 2GM/c^2 (2)
+$$R = \dfrac{2GM}{c^2}$$ (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 = ħc^3/8πkGM (3)
+$$T =\dfrac{ħc^3}{8πk_BGM}$$ (3)
-Given this information, show that the lifetime of a black hole (justify this phrase!) t ˚ scales with its mass M as
+Given this information, show that the lifetime of a black hole (justify this phrase!) $$t^*$$scales with its mass M as
-t* is proportional to Mβ (4)
+$$t^* \propto M^β$$ (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.
+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.

2019年USAAAO决赛第7题 - 版本历史

2020年3月7日 (六) 09:50 Jingsong Guo

Jingsong Guo：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…”