Galaxies

Liverpool John Moores University
SCHOOL OF ENGINEERING

Module Examination, Semester 1 1998/99

ENGAS3053 Galaxies

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 5 questions. Answer 3 questions only.

Questions carry 25 marks each. The total number of marks available is 75.

Examiner: PAJ Moderator: DC Subject leader: PAJ

1.

(a)

It has been estimated that the total luminosity emitted by quasars during the `quasar epoch' at z $\sim$ 2 has a mass equivalent of 10¹⁸ $\rm M_{\odot}$ . Explain the consequences of this observation for the probability of black holes existing in galaxies, and for their likely masses. State explicitly any assumptions or approximations you need to make. [8]

(b)

Starting from the virial theorem, derive the following expression for the mass-to-light ratio of a galaxy nucleus as a function of stellar velocity dispersion $\sigma$ , core radius R_c and surface brightness I:

$\begin{displaymath} \frac{M}{L} = \frac{\sigma^2}{G\times I\times R_c} \end{displaymath}$

State explicitly any assumptions or approximations you need to make.

[9]

(c)

The bulge of a spiral galaxy has a stellar velocity dispersion of 100 kms^-1, and has a 10⁸ $M_\odot$ black hole at its centre. Estimate the maximum distance in Mpc at which this galaxy could be situated if the kinematical signature of the black hole is to be detected, using both ground based observations (point source image diameter 1 $^{\prime\prime}$ ) and HST (point source image diameter 0.05 $^{\prime\prime}$ ). Comment on the importance of good seeing quality when making observations of this type. [8]

2.

(a)

An elliptical galaxy has a half-light radius of 4000 pc and a velocity dispersion of 250 kms^-1, whereas for a globular cluster the corresponding values are 10 pc and 7 kms^-1 respectively. Calculate the `dynamical' or `crossing' timescale t_d for both of these systems. [5]

(b)

The timescale for 2-body relaxation, t_r is given by

$t_r = t_d\times \frac{0.06N}{ln(0.15N)}$

where N is the number of stars in the system. Estimate t_r for elliptical galaxies and globular clusters, and comment on the physical effects underlying the dependence of this timescale on N. [4]

(c)

Compare these answers with the likely ages of an elliptical or a globular cluster, and say what the observational consequences are likely to be for the structure of the two systems. [6]

(d)

What process is thought to be responsible for giving the smooth, relaxed structures of elliptical galaxies?What is the timescale for this process? [3]

(e)

Describe briefly the experiment carried out by Eggen, Lynden-Bell and Sandage. What did they infer from their results about the formation of the Galaxy? How does their model of the halo differ from that of Searle and Zinn? [7]

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3.

(a)

Sketch the general form of the observed rotation curves of spiral galaxies, as measured from HI 21 cm emission. Mark the position of the characteristic `turnover' radius on your plot. [4]

(b)

Calculate the rate at which the total galactic mass M(r) included within a radius r must vary with r, to give the two forms for the rotation curve seen within and outside the `turnover' radius. State any assumptions you need to make. [6]

(c)

What is the evidence for a dark matter component in most or all spiral galaxies? Describe the process used to estimate the mass of the dark matter component in a given galaxy. [6]

(d)

Explain what is meant by the ``Disk-Halo'' conspiracy for spiral galaxies. [5]

(e)

The Tully-Fisher relation is the most frequently used distance indicator for spiral galaxies. State the observational parameters used in this correlation, and discuss very briefly why one would expect them to be correlated. [4]

4.

(a)

Explain what is meant by the Jeans mass as applied to a system of self-gravitating stars and gas. [4]

(b)

The free-fall timescale for a system of stars of total mass M and radius R under its own self-gravity is given by

$\begin{displaymath} T_{ff} = \pi \left(\frac{R^3}{8GM}\right)^{\frac{1}{2}}. \end{displaymath}$

Show that the Jeans mass for a region of galaxy disk of radius R and mass surface density $\mu$ is given by

$\begin{displaymath} M_J = \frac{\pi^3\sigma^4}{64G^2\mu} \end{displaymath}$

where $\sigma$ is the velocity dispersion. You should neglect the effects of the non-spherical mass distribution. [6]

(c)

If a region of galaxy disk shrinks from R₀ to R, it acquires an angular momentum

$\begin{displaymath} \Omega = \left(\frac{R_0}{R}\right)^2\times B \end{displaymath}$

where B is the 2nd Oort constant. Show that the minimum radius of a region that can be stabilised by differential rotation in an infinitely thin stellar disk is

$\begin{displaymath} R_{rot} = \left(\frac{2\pi G \mu}{3B^2}\right). \end{displaymath}$

(Again you can assume spherical symmetry when calculating the gravitational forces.) [8]

(d)

Combine the results from (b) and (c) to derive the Toomre stability criterion for a disk and explain in words what this criterion means. [7]

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5.

(a)

A young galaxy forms stars in an instantaneous burst, with an initial mass function (IMF) given by

$\begin{displaymath} \phi(M) = \frac{dN}{dM} = KM^{-(1+x)} \end{displaymath}$

where M is stellar mass, and K and x are constants.

Assuming a stellar mass-luminosity relation of the form

L(M) = CM^3.5

over the entire mass range, where C is a constant, show that the stellar luminosity in the galaxy shortly immediately after the star formation burst is dominated by high-mass stars, if x has a sufficiently high value. [6]

(b)

What type of stars dominate the red light emitted from an old stellar population, like that found in elliptical galaxies? [4]

(c)

Making suitable approximations, which you should state, derive the following expression for the evolution in luminosity of an elliptical galaxy with time:

$\begin{displaymath} L = 0.4 \times E_{GB}(M_{GB}) \times K \times \left(\frac{M_{GB}}{M_\odot}\right)^{2.5-x} \left(\frac{M_\odot}{10Gyr}\right) \end{displaymath}$

using the form of the IMF given above. E_GB is the total giant branch energy emitted by a star of mass M_GB, and you should approximate the main sequence lifetime using

$\begin{displaymath} t_{ms} = 10Gyr \left(\frac{M}{M_\odot}\right)^{-2.5} \end{displaymath}$

[10]

(d)

Given that E_GB depends only weakly on M_GB, and taking x=1.5, find how the luminosity of a passively evolving elliptical depends on M_GB. [5]

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