Introduction to Stellar Astrophysics

\fbox{ \parbox{1.5in}{\small PAPER CODE NO.\\ {\bf PHYS251} }}


THE UNIVERSITY
of LIVERPOOL





JANUARY 2000 EXAMINATIONS

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

INTRODUCTION TO STELLAR ASTROPHYSICS




TIME ALLOWED : Two Hours

INSTRUCTION TO CANDIDATES
Answer all Questions.
Question 1 carries 50% of the total marks.
Questions 2 and 3 each carry 25% of the total marks.
The marks allocated to each part of a question are indicated in square brackets.
In the event of a student answering both parts of an either/or question and not clearly crossing out one answer, only the answer to part (a) of the question will be marked.
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.

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Section A

(a) Explain how the mass of a star may be observationally measured, and include in your explanation problems with the method. [10]

(b) Using the equation of hydrostatic equilibrium, and the mass of a shell dm, estimate the central pressure of the Sun. [10]

(c) From the equation of hydrostatic equilibrium and the ideal gas equation of state, show that the temperature scales as $T \sim M / R$. Use this to show that the luminosity of very hot stars scales as $L \sim M^3$. [8]

(d) Describe how both low mass and high mass stars lose mass whilst on the main-sequence. State the rate at which mass is lost in both cases. [8]

(e) Two stars are measured to have the same effective temperature of 30,000K and luminosities of 105$L_{\odot}$and 0.1$L_{\odot}$. One of the stars has a radius of 15$R_{\odot}$. What is the radius of the other, and what sort of star is it? [7]

(f) Describe the chemical evolution of the central regions of a 1$M_{\odot}$ star from its formation to the end of its life. [7]

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Section B

Answer either (a) or (b)

(a)

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

(ii) 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$M_{\odot}$, a radius of 5$R_{\odot}$, and a luminosity of 2x104$L_{\odot}$. [9]

(iii) Why are stars assumed to be in hydrostatic equilibrium during their main sequence lives? [2]

(iv) Calculate the efficiency of the energy source (ie. the fraction of the rest-mass energy liberated) for the two stars described in (ii) 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). [9]

(v) The fraction of rest mass energy liberated for chemical reactions, fission reactions and fusion reactions is 10-10, 5 x 10-3, and 10-2 respectively. From this data determine the energy source of the stars (give reasons for your answer). [3]

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(b)

(i) Low mass stars ( $M \raisebox{-0.6ex}{$\,\stackrel {\raisebox{-.2ex}{$\textstyle <$}}{\sim}\,$}2.5\mbox{$M_{\odot}$}$) undergo a phase in their evolution called ``helium flash''. Explain what the helium flash is, how it starts and finishes, and state how long it lasts. [9]

(ii) Photons travelling though the stellar interior do so by continually being absorbed and re-emitted in arbitrary directions by the stellar interior.

Calculate the mean free path of a photon $\lambda_p$ in the interior of a star, given that the radiative cross-section per unit mass $\kappa \approx 1$cm2g-1, and the mean density of stellar material $\bar{\rho} \approx 1.4\times 10^3$kg m-3. [3]

(iii) The distance travelled by a photon being continually absorbed and re-emitted $\Delta R$ is

\begin{displaymath} \Delta R = \sqrt{n} \lambda_p \end{displaymath}

where n is the number of absorption and re-emission processes. Assuming that each absorption/re-emission process takes about 10-8s, calculate the time taken (in years) for a photon to travel from the core of a 1$M_{\odot}$ star to the atmosphere (assume that the star has a radius of 1$R_{\odot}$). [5]

(iv) Why is this time similar to the thermal timescale for the star? [3]

(v) Calculate how long it would take for a photon to travel from the centre of a 1$M_{\odot}$ star to the surface, assuming that it could escape freely. [2]

(iv) With reference to the previous parts of this question, why do we not observe stars brightening dramatically whilst they undergo helium flash?

[3]

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Answer either (a) or (b)

(a)

(i) By using a suitable order of magnitude approach, show that a sphere of isothermal gas in hydrostatic equilibrium will be gravitationally unstable to collapse if it has a radius larger than the Jeans length RJ:

\begin{displaymath}R_J = \frac{c_s}{[(4/3)\pi G \rho]^{1/2}}\end{displaymath}

where cs is the isothermal sound speed ( $c_s^2 = dP/d\rho$), G is the gravitational constant, and $\rho$ is the density. [10]

(ii) Hence, obtain an expression for the Jeans Mass MJ. [2]

(iii) Calculate the sound speed cs in a cloud with a temperature of 100K, and a density of 10-21kg m-3, and using this calculate the Jeans mass (in $M_{\odot}$) for the cloud. [3]

(iv) Contrast your value of the Jeans mass with the observed stellar mass range, and show how fragmentation of the cloud may occur as the cloud collapses. [6]

(v) How will the effects of rotation and magnetic fields affect the collapsing cloud? [4]

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(b)

Describe the structure, and the central energy source of a of a 1$M_{\odot}$ star evolving from the zero-age main-sequence including: (i) the main-sequence phase [4]

(ii) evolution to helium flash [11]

(iii) evolution to the He burning main-sequence [3]

(iv) evolution to planetary nebula and white dwarf [7]

Include in your answer a Hertzsprung-Russell diagram for each phase clearly indicating the evolutionary track of the star.

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