Stellar Structure and Evolution

Liverpool John Moores University
SCHOOL OF ENGINEERING

Semester 1 Examinations, 1998/1999

ENGAS2002 Stellar structure and evolution

Duration 2 hours

Instructions to candidates
Do not open this question paper until you have been told to do so by the invigilator.
A figure in [ ] denotes the number of marks available for that question or part of question.
There are 6 questions. Answer 4 questions only.
Questions carry 25 marks each. The total number of marks available is 100.
A Constants Sheet is provided.

You are allowed to quote the following relations without proof:

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

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

where $\kappa,\;\epsilon\;,\;L$ are the local values of opacity, energy generation rate, and luminosity (L = -dE/dt, the rate of energy loss of the star) respectively. The pressure is P, $\rho$ is the density and m is the mass contained within radius r. The constants a and c are the radiation constant, and the speed of light respectively.

Examiner: JMP Moderator: Subject leader: TOB

1.

(a) What is meant by the term hydrostatic equilibrium? [2]

(b) Derive an expression for the timescale over which changes occur if the equilibrium conditions are disturbed. Calculate this timescale for the Sun and also for a star of spectral type B1V which has mass of 10.0 $M_{\odot}$ , a radius of 5 $R_{\odot}$ , and a luminosity of 2x10⁴ $L_{\odot}$ . [9]

(d) Calculate the efficiency of the energy source for the two stars (ie. the fraction of the rest-mass energy liberated) by using Einstein's mass-energy relation E = m c². (Assume that the main sequence lifetime is $t_{MS} = 10^{10}(\mbox{$M_{\ast}$}/\mbox{$M_{\odot}$})^{-2}$ years).

Suggest from the data below the energy source of the stars (give reasons for your answer). [12]

(Fraction of rest mass energy liberated: chemical reactions - 10^-10, fission reactions - 5 x 10^-3, fusion reactions - 10^-2)

2.

(a) If a star is completely supported by radiation pressure, then its equation of state is $P = \frac{1}{3}aT^4$ , where P is the pressure, a is the radiation constant, and T is the temperature.

Show that if the star is in hydrostatic equilibrium, and is completely supported by radiation pressure, then its luminosity (the Eddington luminosity $L_{\rm Ed}$ ) is given by

$\begin{displaymath} L_{\rm Ed} = \frac{4\pi c G \mbox{$M_{\ast}$}}{\kappa}, \end{displaymath}$

where c is the speed of light, G is the gravitational constant, and $\mbox{$M_{\ast}$}$ is the mass of the star. [8]

(b) By considering the equation of hydrostatic equilibrium, what would happen to the star if its luminosity suddenly increased beyond this value? [4]

(c) The mean opacity for the Sun is $\kappa = 0.1$ m²kg^-1, and for a very high mass star ( $M_{\ast}$ = 50 $M_{\odot}$ , $L_{\ast}$ = 6 x 10⁵ $L_{\odot}$ ) $\kappa = 0.04$ m²kg^-1. Calculate how luminous these stars are as a fraction of the Eddington luminosity and comment on your values. [5]

(d) Describe briefly if and how how (i) low mass stars, and (ii) high mass stars lose mass whilst on the main sequence. [8]

ENGAS2002/JMP Page 1 of 4Semester 2 1995/96

3.

(a) Show that the mean molecular mass $\mu$ for a fully ionised gas consisting only of hydrogen (with mass fraction X) and helium is 4/(3+5X). [2]

Consider a main sequence star with a large convective core of constant mass M_c made up of XM_c of hydrogen and YM_c of helium. Nucleosynthesis takes place in the core, liberating a fraction $\epsilon$ of the rest mass of hydrogen.

By calculating the rate that rest mass is converted into energy (via E = mc²), show that the rate of change of the hydrogen mass fraction is

$\begin{displaymath} \frac{dX}{dt} = \frac{-L}{\epsilon c^2 M_c} \end{displaymath}$

where L is the luminosity of the star. [10]

(b) Given that the luminosity varies as $L = A \mu^7 M_c^5$ , (where A is a constant) show that after time t the mean molecular mass is

$\begin{displaymath} \frac{\mu}{\mu_0} = \left[ 1 - \frac{10 \mu_0^8 M_c^4 A}{\epsilon c^2} t \right]^{-1/8}, \end{displaymath}$

where $\mu_0$ is the mean molecular mass at time t = 0 when the star starts burning hydrogen and the value of X = X₀. Hence, show that the luminosity varies with time as

$\begin{displaymath} L = L_0 \left[ 1 - \frac{10 \mu_0^8 M_c^4 A}{\epsilon c^2} t \right]^{-7/8}, \end{displaymath}$

where L₀ is the luminosity at time t = 0. [10]

(c) Illustrate how this affects the star's position on the Hertzsprung-Russell diagram during its main sequence lifetime. [3]

ENGAS2002/JMP Page 2 of 4Semester 2 1995/96

4.

By considering the post-main sequence evolution of stars of (i) low mass, and (ii) high mass,

(a) sketch on a Hertzsprung-Russell diagram the evolutionary tracks of the stars after they have left the main sequence, [5]

(b) describe the main changes in the structure as the stars as they evolve from the main sequence, highlighting the differences in evolution between stars of different mass. [16]

5.

(a) Low to intermediate mass stars ( $M_{\ast}$ $\raisebox{-0.6ex}{$\,\stackrel {\raisebox{-.2ex}{$\textstyle <$}}{\sim}\,$}2.5$ $M_{\odot}$ ) undergo a helium flash. Explain what the helium flash is, how it starts and finishes, and how long it lasts. [10]

(b) The electron pressure for an ideal gas is given by

$\begin{displaymath}P_e = \frac{\rho k T}{2 m_p} (1 + X), \end{displaymath}$

where $\rho$ is the density, k is Boltzmann's constant, T is the temperature, m_p is the mass of a proton, and X is the mass fraction of hydrogen. In degenerate conditions the electron pressure is

$\begin{displaymath}P_e = 2.33\times 10^{-38} \left( \frac{\rho}{2 m_p} (1 + X) \right)^{\frac{5}{3}}. \end{displaymath}$

In a low mass red giant, helium starts to burn at a temperature of 10⁸K. Show that if the density in the core is greater than 4.7 x 10⁷kg m^-3, the burning will commence in degenerate conditions, and the core will undergo helium flash. [8]

(d) Why do high mass stars not undergo helium flashes? [2]

ENGAS2002/JMP Page 3 of 4Semester 2 1995/96

6.

(a) State what the Hayshi forbidden zone is in the Hertzsprung-Russell diagram (include a sketch in your answer). [4]

(b) Consider the HR diagram above of the pre-main sequence evolutionary track of a star of mass $M_{\odot}$ .

(i) Briefly describe the structure (end energy source) of the star as it starts its evolution (from A $\rightarrow$ B). Why does the evolutionary track change from nearly vertical to nearly horizontal at point B? [5]

(ii) From the relevant equations of stellar structure, use a suitable order of magnitude approach (eg. $dP/dr \sim P_c/R_s$ , where subscripts c and s refer to the central and surface values), to show that during the phase B $\rightarrow$ C, the effective temperature is related to the luminosity via

$\begin{displaymath}\mbox{$L_{\ast}$}\sim T_{\rm eff}^{0.8} \end{displaymath}$

given Kramer's law for the opacity $\kappa = \kappa_0 \rho T^{-3.5}$ (where $\kappa_0$ is a constant). [12]

(iii) How does this expression change for high mass (hot) stars? [2]

(c) re-draw the Hertzsprung-Russell diagram and include on it pre-main sequence tracks of both low mass and high mass stars. [2]

ENGAS2002/JMP Page 4 of 4Semester 2 1995/96