Galaxies

THE UNIVERSITY
of LIVERPOOL

JANUARY 2001 EXAMINATIONS

Degree of Bachelor of Science : Year 3
Degree of Master of Physics : Year 3

GALAXIES

TIME ALLOWED : Three 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.
Note that this division of marks is different from that of other Year 3 examinations.

The marks allotted 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.

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Section A

1. (a) The total luminosity emitted by quasars during the `quasar epoch' at z $\sim$ 2 has a mass equivalent of 10¹⁸ M $_{\odot}$ . Explain the consequences for the existence of black holes in galaxies, and for their likely masses. [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:

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

State any assumptions or approximations you need to make.
[8]

(c) Sketch the general form of the observed rotation curves of spiral galaxies. Mark the characteristic `turnover' radius on your plot. Calculate how M(r), the total galactic mass 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. [10]

(d) What is meant by the ``Disk-Halo'' conspiracy for spiral galaxies? [5]

(e) 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. The minimum radius of a region that can be stabilised by differential rotation in a thin stellar disk with second Oort constant B is

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

Combine these expressions for M_J and R_rot to derive the Toomre stability criterion in terms of a limiting disk velocity dispersion and explain in words what this criterion means. [10]

(f) Define what is meant by ``morphological segregation'', and give 3 examples of observations which demonstrate this effect. Briefly describe 3 physical mechanisms which have been used to explain morphological segregation. [9]

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Section B

2. Answer either (a) or (b)

(a)

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

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

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]

(ii) Taking `standard' values for M_L and M_U of 0.1 and 100 M $_{\odot}$ respectively, show quantitatively the effects on the mass-to-light ratio of the young galaxy of each of the following changes:
Removing all the low mass stars by setting M_L to 1 M $_{\odot}$ , or
Removing all the high mass stars by setting M_U to 50 M $_{\odot}$
Interpret your findings in terms of the types of stars which dominate mass and light. [5]

(iii) 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]

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

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(b)

(i) The D_n- $\sigma$ relation between galaxy diameter and velocity dispersion is a projection of the Fundamental Plane of elliptical galaxies. Use a simple physical argument to show that one would expect a relation of the form

$\begin{displaymath} D_n \propto I_e^{-0.07} \sigma^{1.4}, \end{displaymath}$

given the equation of the Fundamental Plane

$\begin{displaymath} R_e \propto I_e^{-0.9} \sigma^{1.4}, \end{displaymath}$

and elliptical galaxy light profiles of the form

I(R) = I_e (R/R_e)^-1.2.

(I(R) is the surface brightness of the galaxy at radius R, and I_e is the value of surface brightness measured at the effective radius R_e.) Why does the D_n- $\sigma$ relation not suffer from the scale-size dependence which affects the Faber-Jackson relation? [10]

(ii) 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. [8]

(iii) The visual B-band Tully-Fisher relation has a lower slope, of approximately 8. Suggest a reason for this. [3]

(iv) Would you expect the optical B-band or near-infrared H-band Tully-Fisher relation to have the smaller scatter? Give two reasons for your answer. [4]

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3. Answer either (a) or (b)

(a)

(i) 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 EITHER

A very thin ( $\ll$ T) dust layer down the centre of the slab, OR
Dust evenly mixed with stars throughout the thickness T.

For the first dust distribution, the surface brightness is given by

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

For the second, evenly-mixed dust distribution, 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.
[8]

(ii) 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]

(iii) 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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(b)

(i) Define the galaxy luminosity function $\phi(L)$ . [4]

(ii) The equation for the luminosity function defined by Schechter is

$\begin{displaymath} \phi(L) = N_0 \left(\frac{L}{L_{\star}}\right)^{\alpha} exp\left(\frac{-L}{L_{\star}}\right)\frac{dL}{L_{\star}} \end{displaymath}$

with a best-fitting value of $\alpha$ being -1.5.
Sketch the shape of this function, plotted as $log\phi$ vs absolute magnitude. Mark the location of the characteristic luminosity $L_{\star}$ . Sketch the approximate distribution of luminosities for spiral, elliptical and dwarf galaxies on the same diagram. [6]

(iii) If galaxies are distributed homogeneously in space with luminosities described by the Schechter function above, what would be the median luminosity of galaxies in a magnitude-limited survey? [6]

(iv) For the faintest dwarf galaxies, $L<<L_{\star}$ . Derive a simplified form of the Schechter function that is applicable in this regime. [3]

(v) Use your answer to part (iv) to demonstrate the importance of the index $\alpha$ on the total luminosity, and hence mass, contribution provided by dwarf galaxies, by considering values of $\alpha$ of -1, -2 and -3. (Assume constant mass-to-light ratio for all galaxies.) Discuss why it is particularly important to determine whether $\alpha$ is greater than or less than -2.0. [6]

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