Introduction to Stellar Astrophysics

THE UNIVERSITY
of LIVERPOOL

JANUARY 2001 EXAMINATIONS

Degree of Master of Physics : Year 2
Degree of Bachelor of Science : Year 2

INTRODUCTION TO STELLAR ASTROPHYSICS

TIME ALLOWED : Two Hours

$\begin{displaymath} \frac{dP}{dr} = -\frac{Gm\rho}{r^2}\;\;\;,\;\;\; \frac{dT}{dr} = -\frac{3\kappa \rho L}{16\pi ac r^2 T^3}, \end{displaymath}$

$\begin{displaymath} \frac{dm}{dr} = 4 \pi r^2 \rho\;\;\;,\;\;\; \frac{dL}{dr} = 4 \pi r^2 \rho \epsilon \end{displaymath}$

The Sun's mass is $\mbox{$M_{\odot}$}~=~1.99\times 10^{30}$ kg, its radius is $\mbox{$R_{\odot}$}~=~6.97\times 10^8$ m, and its luminosity is $\mbox{$L_{\odot}$}~=~3.86\times 10^{26}$ J s^-1.

THE UNIVERSITY
of LIVERPOOL
Section A

(a) Derive an expression for the dynamic, or free-fall, timescale of a star. What is the dynamic timescale for the Sun? With reference to your answer, state why the Sun is assumed to be in hydrostatic equilibrium? [10]

(b) How may eclipsing binary stars be used to measure stellar radii? Mention the major positive attribute of the method, and a possible problem. [8]

(c) Two stars with spectral type M (T $_{\rm eff} = 2700$ K) are observed to have luminosities of 0.1 $L_{\odot}$ and 10⁶ $L_{\odot}$ . If the bright star has a radius of 10³ $R_{\odot}$ , what is the radius of the other? Name what sort of stars these are, and what phase of evolution they are going through. [10]

(d) Name the energy sources in the pre-main sequence and main sequence phases of stellar evolution for low and high mass stars. What is the energy source of the horizontal branch stars? [7]

(e) Use the equation of hydrostatic equilibrium to show that the central pressure of a star can be estimated to be $P_c \approx A G\mbox{$M_{\ast}$}^2/\mbox{$R_{\ast}$}^4$ , where A is a constant of order unity. [8]

(f) Hot main-sequence stars of mass $M_{\ast}$ have radii of $\mbox{$R_{\ast}$}= (\mbox{$M_{\ast}$}/\mbox{$M_{\odot}$})^{1/2} \mbox{$R_{\odot}$}$ . Using the estimate of central pressure given in 1(e) above, derive the central temperature of a hot star supported solely by radiation pressure ( $P_{\rm rad}~=~\frac{1}{3}~a~T^4$ ). [7]

THE UNIVERSITY
of LIVERPOOL
Section B

(i) The contraction of a star to the main sequence contains two main phases, initially the star is isothermal and fully convective, and then it becomes non-isothermal and radiative.

Use a suitable scaling analysis to show how the luminosity L of a hot star changes as the star contracts for both the isothermal, convective stage, and the non-isothermal, radiative stage (for the second part you may assume that pressure P scales with mass M and radius R as $P \sim M^2/R^4$ )

Sketch the evolutionary track you have derived on a Hertzsprung-Russell diagram. [15]

(ii) One feature in pre-main sequence evolution is that the energy transport mechanism changes from convective to radiative. This change is governed by the Schwarzschild Instability Criterion. State the physical criterion for convection to occur as a blob of gas is displaced adiabatically from one point in a star to another.

Hence, show that convection will occur if the pressure P and density $\rho$ are related by

$\begin{displaymath} \frac{\partial {\rm log}_eP}{\partial {\rm log}_e\rho} > \gamma \end{displaymath}$

where $\gamma$ is the ratio of specific heats. [10]

(i) Briefly describe the current structure of the Sun, from the core to the atmosphere, highlighting the energy generation and transport mechanisms. [6]

(ii) What are the main differences between the Sun's structure and a 50 $M_{\odot}$ main-sequence star's structure. [3]

(iii) Estimate the main-sequence lifetime in years of the Sun and a 50 $M_{\odot}$ star if the energy liberated in the fusion of 4H $\rightarrow$ He is E=6.3 x 10¹⁴J kg^-1. [7]

(iv) During the main sequence, stars burn hydrogen to helium. By assuming the perfect gas law, and that the pressure and density remain constant, show that the temperature in the core increases (assume that the core gas is completely ionized). How is this reflected in the Hertzsprung-Russell diagram? [9]

The equation of transfer of radiation in one dimension is

$\begin{displaymath} \frac{dI}{d\tau} = I - S \end{displaymath}$

(i) What is the physical meaning of the quantities $\tau$ , I, S in this equation? [5]

(ii) Now consider a slab of gas with no incident radiation, and a constant source function. If the cloud has optical depth t, show that the emergent intensity $I_{\rm em}$ is

$\begin{displaymath}I_{\rm em} = S (1 - {\rm e}^{-t} ). \end{displaymath}$

Show that in the optically thick case, the emergent intensity is equal to the source function, and that in the optically thin case that the emergent intensity is proportional to the optical depth. [10]

(iii) With reference to the optically thick case for the example above, outline the overall appearence of a stellar spectrum. [4]

(iv) What are bound-bound, bound-free, and free-free processes? What sort of features do these impose on a stellar spectrum? [6]

The equation of state for non-relativistic degenerate electrons is $P = K \rho^{5/3}$ , where K is a constant. There are two phases in a 1 $M_{\odot}$ star's lifetime where non-relativistic electron degeneracy pressure is dominant. The first is immediately before and during the helium flash.

(i) Describe what the helium flash is, how long it lasts, how it starts and terminates, and the composition of the core of the star during this phase. [9]

Why do high mass stars not undergo a helium flash? [3]

(ii) The second stage where electron degeneracy pressure is important in the life of a 1 $M_{\odot}$ star is in the white dwarf phase. What is the chemical composition and temperature structure of a white dwarf? [2]

(iii) Assuming that the pressure P scales with radius R and mass M of a star as $P \sim M^2/R^4$ , use the non-relativistic equation of state above to show that the radius of a star solely supported by non-relativistic degenerate electrons is related to its mass by

$\begin{displaymath} R \sim M^{-1/3} \end{displaymath}$

[6]

(iv) Calculate the mass-radius relationship for an isothermal object, using the perfect gas equation of state, and contrast it with the above. [5]

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