課程名稱︰恆星天文物理
課程性質︰必修
課程教師︰闕志鴻
開課學院：理學院
開課系所︰天文物理所
考試日期（年月日）︰2018年1月12日
考試時限（分鐘）：150分鐘（15:30 - 18:00）
試題 :
Stellar Astrophysics Final Exam (01/12/2018)
(1) When the energy transport in a star is entirely radiative, the stars of different
masses are self-similar (or homologus) if the specific energy generation rate
ε= aρT^{n}, where a and n are constants. ρ is the mass density and T is the
temperature. In this case you can let the Bremsstrahlung be the main
mechanism for radiative diffusion, where the inverse of the photon mean free
path, or the Rossland mean opacity <α> = bρ/T^{3.5}, where b is also a constant.
(a) Please write down the set of governing differential equations that describe
the spherically symmetric solution of a star in terms of relevant variables,
such as the mass density, the temperature and the luminosity L. (10%)
(b) From these differential equations, show that solutions of different stellar
masses can be self-similar. (10%)
(c) Obtain the luminosity-temperature relation from the self-similar scaling
relation. (10%)
(2) If a fully ionized, charge neutral plasma is composed of electrons and hydrogen
H+, what is the average mass per particle? (5%) The ratio of the average mass to
the hydrogen mass is called the mean molecular weight, μ, which is a
dimensionless quantity. Please express the pressure P in term of the mass
density ρ, the temperature T and the mean molecular weight, μ. (5%) Now, in a
fully ionized, charge neutral star consisting of electrons, H+ and He++, where the
composition ratio of H+ and He++ are 4:1 in mass density, what is μ? (10%)
(3) Show that a shell burning star has an isothermal core. (10%) An isothermal core
can become unstable leading to the helium flash when the core mass exceeds the
Schonberg-Chandrasekhar mass limit, which depends on the molecular weight μ
and stellar mass M*. Please derive the Schonberg-Chandrasekhar mass limit. (15%)
(4) The protostar follows the Hayashi track before it becomes the main-sequence star.
Please explain the different phases in the Hayashi track. (25%)