Further Stellar Astrophysics

Liverpool John Moores University
SCHOOL OF ENGINEERING

Semester 2 Examinations, 1999/2000

ENGAS3056 Further Stellar Astrophysics

Duration 2 hours

Instructions to candidates

Do not open this question paper until you have been told to do so by the invigilator.

There are five questions.
Answer three questions only.
A figure in [ ] denotes the number of marks available for that question or part of question.
Each question is worth 25 marks. The total number of marks available is 75.

Examiner: MS Moderator: JMP Subject leader: TJTM

1.

(a)

Describe briefly the central burning phases of a typical Type II supernova progenitor. [5]

(b)

What is the difference between the prompt explosion and the delayed explosion mechanism in Type II supernovae? [6]

(c)

Explain why in neutron stars free neutrons are stable while in laboratory experiments they decay after $\approx$ 15 minutes. [7]

(d)

Let's assume that the light curve of a Type II supernova is controlled by the energy released in the radioactive decay of an isotope i with decay constant $\lambda$ ; the luminosity at a given instant is proportional to the number of atoms of the species i remaining in the sample. Demonstrate that the slope of the bolometric light curve is given by:

(dM_bol/dt) = 1.085 $\lambda$ [7]

2.

(a)

What determines the average colour and colour extension of the Horizontal Branch in a given globular cluster? [6]

(b)

What feature in the colour-magnitude diagram of a globular cluster is the most suitable age indicator? What is the corresponding evolutionary phase? [6]

(c)

Two globular clusters GC1 and GC2 have the same age and two different metallicities, Z1 and Z2 respectively, with Z1>Z2. Describe, explaining your reasoning, the differences in their Horizontal Branch and Red Giant Branch location. [8]

(d)

The scale-length of convective motions in the framework of the mixing-length theory is proportional to the pressure scale height H_P defined as

1/H_P= - (1/P) (dP/dr)

Derive the value of H_P (in solar radii) in the envelope of a Main Sequence globular cluster star assuming log(g)=2.5, log(P)=11.0 and log( $\rho$ )=1.7, where g, P and $\rho$ are, respectively, the local acceleration of gravity, pressure and density expressed in S.I. units. [5]

3.

(a): Why, when looking at the Hertzsprung-Russell (H-R) diagram of a stellar cluster, can one deduce that stars are not chemically homogeneous after the Main Sequence phase? Include a diagram to illustrate your answer. [9]
(b): Explain what is the Schönberg-Chandrasekhar limit and how its effect on the post-Main Sequence evolution is displayed in the H-R diagrams of stellar clusters. [8]
(c): What class of stellar objects evolves at constant radius, right after the pre-MS phase? Why? [8]

ENGAS3056/MS Page 1 of 2Semester 2 1999/2000

4.

(a)

State the evolutionary phase and mass range of Cepheid stars. [5]

(b)

The period of oscillation $\Pi$ of a pulsating star is approximately equal to

$\Pi$ = 2 R/ c_s

where R is the stellar total radius and c_s an average value for the sound speed.

By considering a pulsating star of total radius R, made of a completely ionized perfect Hydrogen gas, with average temperature T and average density $\rho$ , use the virial theorem to show that

$\Pi \propto$ 1/( $\rho^{1/2}$ ) [11]

(c)

What is the physical reason for the red and blue edge of the instability strip? [9]

5.

(a)

Consider a star with density $\rho$ and total radius R. Using the equation of hydrostatic equilibrium in the assumption of spherical symmetry, derive the relation between the pressure P and the radial coordinate r all along the stellar structure. Assume $\rho$ =constant throughout the star and P(R)=0. [6]

(b)

If the equation of state of the stellar matter is well described by a perfect gas completely ionized with an initial chemical composition X=0.70, Y=0.28 and Z=0.02, what is the central temperature of the star? Assume R=7.0 10⁸ m and $\rho$ = 2 10³ Kg m^-3 [9]

(c)

In the approximation of constant $\rho$ and R, discuss if the center of the star can become electron degenerate at the end of the Main Sequence phase and if it can ignite helium burning. Remember that the electron Fermi energy for a chemical composition made by one element i with ratio $\frac{Z_{i}}{A_{i}}$ between its atomic number and atomic weight is given by:

$\varepsilon _{F}$ = $\frac {h^{2}}{8 m_{e} \pi^{2}}$ $\lgroup 3 \pi^{2} (\frac{Z_{i}}{A_{i}}) \frac{\rho}{H} \rgroup ^{2/3}$

where h is the Planck constant, m_e the electron mass, H the mass of the Hydrogen nucleus. [10]

ENGAS3056/MS Page 2 of 2Semester 2 1999/2000