Galaxies

Liverpool John Moores University
SCHOOL OF ENGINEERING

Semester 1 Examinations, 1999/2000

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: TJTM Subject leader: PAJ

1.

(a)

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

$\begin{displaymath} \phi(M) = \frac{dN}{dM} = K\left(\frac{M}{\mbox{\,$\rm M_{\odot}$}}\right)^{-2.5} \end{displaymath}$

between stellar mass lower and upper limits M_L and M_U respectively. K is a constant, and no stars are formed outside the mass limits.

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 immediately after the star formation burst is dominated by high-mass stars. [5]

(b)

Taking `standard' values for M_L and M_U of 0.1 and 100 $\rm M_{\odot}$ respectively, show quantitatively the effects on the mass-to-light ratio of the young galaxy of each of the following changes:

i) Removing all the low mass stars by setting M_L to 1 $\rm M_{\odot}$
ii) Removing all the high mass stars by setting M_U to 50 $\rm M_{\odot}$

Comment on your findings in both cases. [5]

(c)

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

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

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

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

Using this result, how would you expect the red luminosity of an elliptical galaxy to vary with its age? [10]

(d)

Describe what is meant by the term `age-metallicity degeneracy' in the analysis of stellar populations in galaxies. [5]

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2.

(a)

A dwarf irregular galaxy has a half-light radius of 1 kpc and a velocity dispersion of 50 kms^-1. Calculate the `dynamical' or `crossing' time t_d for this galaxy. [4]

(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, which you can take as 10⁹. Estimate t_r for the dwarf irregular galaxy, and comment on the answer. [4]

(c)

Name and describe the process thought to be responsible for giving the smooth, relaxed structures of elliptical galaxies. What is the timescale for this process, and how does it differ physically from the 2-body relaxation process described in part (b)? Describe the characteristics of a system produced by this process. [9]

(d)

The old stellar population in the irregular galaxy is estimated to have an age of 5 x 10⁹ years, and these same red stars are observed to have a highly clumpy, irregular distribution. Explain how this might arise, given your answers to parts (a)-(c). [8]

3.

(a)

A galaxy is modelled as a uniform slab of stars, thickness T metres and volume emissivity E Wm^-3. The dust distribution within this slab is modelled by

(i) a very thin ( $\ll$ T) dust layer down the centre of the slab, and
(ii) dust evenly mixed with stars throughout the thickness T.

For dust distribution (i), the surface brightness is given by

$\begin{displaymath} I_{total} = \frac{ET}{2cos(i)}\left[1+exp-\left({\frac{\tau}{cos(i)}}\right)\right]. \end{displaymath}$

Derive the corresponding expression for the surface brightness of the galaxy in Wm^-2, as a function of the total dust optical depth $\tau$ normal to the slab, and the viewing angle i, for dust distribution (ii). [8]

(b)

Comment on the effectiveness of the surface brightness versus inclination test for determining $\tau$ in the limits $\tau \ll 1$ and $\tau \gg 1$ , for both dust distributions. [8]

(c)

Explain what is meant by the Freeman Law for spiral galaxy disks, and specify carefully the observational parameter which was used in defining this law. Describe how observations since Freeman's initial work have modified our understanding of this law. [9]

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4.

(a)

Explain what is meant by the Fundamental Plane of elliptical galaxies. Describe the physical meaning of the three parameters which define the axes within which this plane lies, and say how they might be determined observationally. [6]

(b)

The D_n- $\sigma$ relation is a projection of the Fundamental Plane. Define the two parameters involved, and use a simple physical argument to show that one would expect a relation of the form

$\begin{displaymath} log(D_n)=2 log\sigma + C \end{displaymath}$

where C is a constant. You may simplify the analysis by assuming circular stellar orbits. Comment on how realistic the predicted form of the relation is. [8]

(c)

Using a similar argument to that in part (b), show that the expected form of the Tully Fisher relation, between absolute magnitude M and rotation velocity V_rot in spiral galaxies, is of the form

M = -10 x log(V_rot) + const.

State explicitly the 3 assumptions that are necessary for this derivation. [11]

5.

(a): Describe the techniques which have been used to search for high-redshift galaxies using optical and near-infrared imaging. What properties of young galaxies do they exploit, how successful have they been, and why? [10]
(b): Describe four physical mechanisms which have been used to explain the evolution of spiral and elliptical galaxies. Mention the observational evidence, if any, for each of these mechanisms. [8]
(c): Define morphological segregation, give specific examples of this effect for different galaxy types, and list those mechanisms, mentioned in part (b), which might contribute to this effect. [7]

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