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\title{\Huge{Stars and {\bf Galaxies}}}
\author{Tom Theuns\\
			Office 307\\
        Institute for Computational Cosmology\\
			Ogden Centre for Fundamental Physics\\
			Durham University\\
         and\\
         University of Antwerp\\
	      Campus Drie Eiken\\
			tom.theuns@durham.ac.uk}
\date{2007-2008}
\maketitle
% title page
{\Large\bf
\begin{enumerate}
\item[$\star$] Chapter 1: Introduction
\item[$\star$] Chapter 2: The discovery of the Milky Way and of other galaxies
\item[$\star$] Chapter 3: The modern view of the Milky Way
\item[$\star$] Chapter 4: The Interstellar Medium
\item[$\star$] Chapter 5: Dynamics of galactic disks
\item[$\star$] Chapter 6: The dark halo
\item[$\star$] Chapter 7: Elliptical galaxies I
\item[$\star$] Chapter 8: Elliptical galaxies II
\item[$\star$] Chapter 9: Groups and Clusters of galaxies
\item[$\star$] Chapter 10: Galaxy statistics
\item[$\star$] Chapter 11: Active Galactic Nuclei
\item[$\star$] Chapter 12: Gravitational lensing
\end{enumerate}
}
\chapter*{Aim}
The {\em Hubble Deep Field} shows how even a tiny patch of sky
contains thousands of galaxies of different sizes and shapes,
lighting-up the Universe with a dazzling display of colours. When did
they form? Which physical processes shaped them? How do they evolve?
Which types are there, and whence the huge variety? How does the Milky
Way galaxy fit in?

12 lectures are too few to answer any of these fundamental questions
in any detail. In addition, galaxy formation and evolution require
a cosmological setting, see the later lecture
courses on cosmology (L3) and Galaxy Formation (L4). The more modest
aim of these lectures is to give an overview of the properties of
galaxies (galaxy types, properties of spiral galaxies in general and
the Milky Way galaxy in particular, properties of elliptical
galaxies), look at some of the more spectacular phenomena (quasars,
gravitational lensing), and investigate the extent to which all this
can be described with simple physics. These lectures will {\em use}
the physics you've learnt in other lectures (in particular classical
mechanics, electricity and magnetism, quantum mechanics, and of course
the earlier parts Observational Techniques and Stars) to galaxies,
rather than teach new physics.

The beauty of galaxies is for most people enough to warrant their
study, but of course astronomy and cosmology enable one to study
physics in situations which cannot be realised in a
laboratory. Examples of fundamental physics made possible by astronomy
and cosmology are tests of general relativity (Mercury's orbit,
dynamics of pulsars), investigations whether the fine-structure
constant evolves (from quasar absorption spectra), the discovery of
dark matter (from the motions of galaxies), and more recently the
discovery of dark energy (from the expansion of the Universe). Several
of these fundamental discoveries rely on a basic understanding of
galaxies, the building blocks of the Universe.

\noindent {\bf Learning outcomes:}\\

\begin{itemize}
\item ability to classify galaxies and state defining properties of spiral, elliptical and irregular galaxies
\item explain observational basis for discovery of the Milky Way
\item explain observational basis of the modern view of the Milky Way
\item apply classical mechanics to dynamics of galactic disks, including arguments for the presence of dark matter
\item apply classical mechanics and other basic physics to the properties of the interstellar medium in spirals (properties of gas and dust, 21-cm radiation, concept and application of Jeans mass)
\item apply mechanics and hydrostatics to describe elliptical galaxies and clusters of galaxies
\item understanding of galaxy scaling relations and statistical properties of galaxies
\item observational manifestations of active galactic nuclei, and the connection to central supermassive black holes
\item basics of gravitational lensing, and its applications
\end{itemize}

\noindent {\bf Additional information}\\
There are many excellent sites with images of galaxies, and
explanations of some of the issues discussed below.  The latest
version of {\em Google Earth} has the option to look at the night sky,
with images of galaxies taken from the {\em Sloan Digital Sky Survey}
and the {\em Hubble Space Telescope}. 

Most of the images shown during the lectures come from one of

\begin{itemize}
\item http://www.seds.org/
\item http://www.aao.gov.au/images.html/
\item http://hubblesite.org/newscenter/archive/
\item http://teacherlink.ed.usu.edu/tlnasa/pictures/picture.html
\item http://astro.estec.esa.nl
\item http://space.gsfc.nasa.gov/astro/cobe/cobe\_home.html
\item http://astro.estec.esa.nl/Hipparcos/
\item http://astro.estec.esa.nl/GAIA/
\end{itemize}

Additional explanations can be found in

\begin{enumerate}
\item[1] Carroll \& Ostlie, {\it Modern Astrophysics}, Addison-Wesley,
1996
\item[2] Zeilik \& Gregory, {\it Astronomy \& Astrophysics}, Saunders College Publishing, 1998
\item[3] Binney \& Merrifield, {\it Galactic Astronomy}, Princeton,
1998
\item[4] Binney \& Tremaine, {\it Galactic Dynamics}, Princeton, 1987
\end{enumerate}

(These are referred to as CO, ZG, BM, and BT in these notes) The first two are 
general and basic texts on astronomy, which, together with the web pages, should be your first 
source for more information. [3] is a much more advanced text on galaxies,
and [4] is a superb and detailed discussion of the dynamics of galaxies.

\hbox{http://burro.astr.cwru.edu/Academics/Astr222/index.html} is a truly excellent
web site where you'll find additional material on what is discussed in these lectures, and
several images in these notes  originate from there.

\newpage

\chapter*{Summary}

\section*{\em Chapters 1-6: spiral galaxies}
Galaxies come in a range of colours and sizes. In {\bf spiral}
galaxies such as the Milky Way, most stars are in a thin disk.  The
disk of the Milky Way has a radius of about 15~\kpc, and contains
$\sim 6\times 10^{10}\msun$ of stars. The disk stars rotate around the
centre of the galaxy on nearly circular orbits, with rotation speed
$v_c\sim 220$ km s$^{-1}$ almost independent of radius $r$ (unlike the
motion of planets in the solar system where $v_c\propto
1/r^{1/2}$). Such {\em flat rotation curves} require a very extended
mass distribution, much more extended than the {\em observed} light
distribution, indicating the presence of large amounts of invisible
{\em dark matter}. Curiously, the spiral arms do not rotate with the
same speed as the stars, hence stars may move in and out of spiral
arms. 

Stars are born in the large molecular gas clouds that are concentrated
in spiral arms. Massive short-lived stars light-up their natal gas
with their ionising radiation, and such \lq\HII\rq\ regions of ionised
hydrogen follow the arms as beads on a string. The blue light from
these massive and hot stars affects the colour of the whole galaxy,
one reason why spirals are blue. Remember that the luminosity $L$ of a
star is a strong function of its mass $M$, with approximately
$L\propto M^3$ on the main sequence, hence a single $100\msun$ star
outshines nearly $10^6$ solar-mass stars: the colour of a galaxy may
be affected by how many massive (hence young) stars it harbours.

The stars in the central region of the Milky Way (and most spirals)
are in a nearly spherical {\em bulge}, a stellar system separate from
the spiral disk, with mass $\approx 10^{10}\msun$. Dust in the
interstellar medium of the disk prevents us from seeing the Milky
Way's bulge on the night sky. Disk and bulge are embedded in a much
more extended yet very low-density nearly spherical {\em halo},
consisting mostly of dark matter.  The stellar halo contains a sprinkling of
\lq halo\rq~ stars, but also globular clusters, very dense groups of
$10^5-10^6$ stars of which the Milky Way has $\sim 150$.  The total
mass of the Milky Way is $\sim 10^{12}\msun$, but stars and gas
make-up only $\sim 7\times 10^{10}\msun$ with the rest in dark matter.

\section*{\em Chapters 7-8: elliptical galaxies}
{\bf Elliptical galaxies} are roundish objects which appear yellow.
The redder colour as compared to spiral galaxies is due to two reasons:
firstly the stars in elliptical galaxies tend to have higher
metallicity than spiral galaxies of comparable mass, and the metals in
the stellar atmospheres redden the stars. Secondly,
elliptical galaxies have no current star formation, and hence they do
not contain the massive and short-lived stars that produce the blue
light in spirals.

The stars move at high velocities, and it is these random motions, not
rotation, that provides the support against gravity. This is similar
to how the velocities of atoms in a gas generate a pressure, $p\sim
\rho v^2\sim \rho\,T$.  However, unlike the atoms in a gas, the stellar velocities
are in general {\em no}t isotropic, meaning that also the pressure is not
isotropic. The result is that, whereas stars are round, elliptical
galaxies are in general flattened, {\em i.e.} elliptical in shape.

The observed stellar motions, but also the hot X-ray emitting
gas detected in them, both suggest the presence of dark matter in
elliptical galaxies. 

\section*{\em Chapters 9-10: galaxy statistics}
Roughly half of all stars in the Universe are in spirals, and half are
in ellipticals. Spiral galaxies tend to live in region of low galaxy
density. The Milky Way is part of a \lq local group\rq~ of galaxies,
containing in addition to Andromeda a swarm of smaller
galaxies. Ellipticals in contrast tend to cluster in groups of tens to
hundreds of galaxies, called clusters of galaxies.  Such clusters
contain large amounts of hot, X-ray emitting gas. Together with the
large observed speeds of the galaxies, this hot gas provides evidence
that a large fraction of the cluster's mass is invisible dark matter.

\section*{\em Chapter 11: Active Galactic Nuclei}
A small fraction of galaxies contains an active nucleus, which may
generate as much or even more energy than all the galaxy's stars
combined.  The observational manifestations of such active galactic
nuclei, or AGN, are quite diverse, with energy being
emitted from visible light in quasars, to immensely power full radio
lobes, to X-rays and even gamma-rays. Some fraction of energy may even
be released in the form of a powerful jet.

The energy source is thought to be accretion onto a supermassive black
hole, with masses from $10^6$ to as high as $10^9\msun$.  It is
thought that most, if not all, massive galaxies contain such a
supermassive black hole in their centres. The evidence is particularly
convincing for the presence of a $\sim 10^6\msun$ black hole in the
centre of the Milky Way.

Clearly, if most galaxies contain the engine but have little or no
observational AGN manifestations, it must imply that the majority of
black holes is dormant.

\section*{\em Chapter 12: Gravitational lensing}
A gravitational field bends light. In general this is only a small
effect and our view of the distant universe is not significantly
deformed by the gravitational lensing caused by the intervening
potentials generated by stars, galaxies or clusters of galaxies.  The
phenomenon has been used to test Einstein's theory of relativity, to
constrain the nature of dark matter in the Milky Way, to determine the
masses of galaxies and clusters, and even to enable astronomers to use
clusters of galaxies as truly giant telescopes enabling the detailed study of distant
galaxies.

  

%% Roughly half of all stars in the Universe are in spirals, and half are
%% in ellipticals. Edwin Hubble classified galaxies in a \lq Hubble
%% diagram\rq~ (see Fig.\ref{fig:Hubble} below) which looks like a
%% tuning-fork. Elliptical galaxies are ordered along the stem from round
%% to flattened, and spirals along the vibrating section of the fork,
%% ordered according to how tightly wound their spiral arms are. Spiral
%% galaxies tend to live in low density regions, for example the Milky Way
%% together with Andromeda (M31) and a swarm of smaller satellite
%% galaxies, lives in a low density region called the local
%% group. Elliptical galaxies in contrast tend to occur in regions of high
%% density called galaxy clusters. The biggest clusters contain thousands
%% of galaxies, and are the most massive gravitationally bound structures
%% in the Universe. Most of their $10^{15}\msun$ of mass, as inferred from
%% the galaxy motions, properties of their X-ray emitting gas, or from
%% gravitational lensing, is in the form of dark matter. The observable
%% Universe contains about $10^9$ galaxies with masses equal to, or larger
%% than, the Milky Way.

%% Distances between galaxies are comparable to their sizes hence, unlike
%% most stars, galaxies affect each other significantly. The Magellanic
%% cloud, a satellite of the Milky Way, trails a large stream of stars and
%% gas due to tidal forces from the Milky Way. The Sagitarius dwarf
%% galaxy, discovered only recently, is presently being pulled apart by
%% the Milky Way, and crashing into its disk. In a few billion years,
%% Andromeda and the Milky Way will crash into each other, and the tidal
%% forces will probably destroy both disks, changing the night sky for
%% ever. But don't worry: the expanding outer parts of the Sun as it
%% enters the Red Giant phase, will have made life on earth impossible way
%% before that.  Super nova explosions drive gas from the disk into a
%% galactic fountain, and may even fling gas into intergalactic space
%% during a galaxy wide wind. The violent lives of galaxies is in sharp
%% contrast to the majestic and eternal sight they provide in images.
 
%% Gravity tries to make gas collect into clouds and ultimately into stars, but nuclear energy released in 
%% fusion and during supernova explosions tries to counteract this. 

%% This course on galaxies runs over 14 lectures. The aim is to give you
%% an overview of galaxy properties and of some of the physical processes
%% that we think shape galaxies. In contrast to the beauty of stellar
%% physics, you'll see that galactic physics is often much more murky and
%% complex -- a bit like weather predictions: we (think we) know the
%% physics, but even so it still is difficult to make accurate
%% calculations. And often, we don't know the physics very well -- or even
%% at all. On the bright side, galaxies are beautiful. And we can in
%% principle probe physics on scales which are impossible to reach within
%% the laboratory and so learn about really fundamental physics -- just
%% recall the recent excitement about the possible non-zero value for the
%% cosmological constant. Anyway, I hope you'll enjoy the lectures.

%% As to the organisation of the course, I will basically follow the
%% structure of Simon Morris, i.e. there will be hand-outs after each
%% lecture, but more up-to-date versions will appear on the web in
%% DUO. Look there as well for (most of) the pictures that I show during
%% the lectures. Suggestions on material included in this course, its
%% presentation, and lists of errors in the notes, will be most
%% appreciated.

%% And for those that are only interested in passing their exams: there
%% will be two questions on galaxies. The first consist of 5 very short
%% questions where you're asked to describe {\em very briefly} some
%% concepts that we have discussed. For example, what is the interstellar
%% medium, and how do we observe it. Given five differences between
%% football and basketball. The second type of question usually starts from
%% a derivation that we've done. The end of the question is something you
%% have not seen, but is very similar to what we have done.

\newpage
\chapter{Introduction}

\section{Historical perspective}
Galaxies are extended on the night sky as can be seen even
without using a telescope\footnote{The disks of stars on the sky can only be
  resolved with special techniques, and even then only for very nearby stars.}.
{\it Andromeda} is similar in size to the Milky Way and extends over
several degrees, whereas the {\it Large Magellanic Cloud} or LMC, visible
from the Southern hemisphere on a clear night, extends over six
degrees (diameter of full moon is 1/2 a degree). The Sagittarius dwarf
galaxy, a small galaxy gravitationally bound to the Milky Way, extends
over a large fraction of the sky.

The Milky Way galaxy is a spiral galaxy, in which most of the stars
lie in a thin ($\sim 1/2\kpc$) disk of radius $\sim 15$\,\kpc, with
the Sun at a distance $\sim 8\kpc$ from the centre. The Milky Way's
disk can be seen as a faint trail of light on the night sky, while the
other visible stars are not obviously part of a disk simply because
they are nearby.

Other extra-solar objects which appear extended on the night sky
include planetary nebulae, supernova remnants, and star clusters.
Before intergalactic distances were first measured in the 1920s it was
not realised that galaxies were a separate class, and both Messier's
catalogue (1780, in which Andromeda is M~31) and Dryers (1988) {\em
  New General Catalogue} (in which it is NGC~224) make no distinction
between galaxies and these other \lq nebulae\rq. Emmanuel Kant was one
of the first to suggest that galaxies were other Milky Ways. The study
of galaxies therefore only started in the 20th century, and we now
know that the observable Universe contains $\sim 10^9$ galaxies more
massive than the Milky Way. Surveys of galaxies now regularly
catalogue properties of $10^4-10^6$ galaxies.

\section{Galaxy classification}
The light from normal galaxies is produced predominantly by stars.
Luminosities of stars vary widely: recall from the earlier lectures
that along the main sequence luminosity $L$ depends on mass $M$ as
$L\propto M^\alpha$, with $\alpha\approx 3$, hence a single $100\msun$
star outshines nearly $10^6$ solar-mass stars. For given $M$, $L$
depends strongly on the phase of stellar evolution, with giant and
asymptotic giant branch stars {\em much} more luminous than the
earlier main sequence phase. Therefore the observed luminosity of a
galaxy will be dominated by its giant stars.

\subsection{Observables}
\begin{itemize}
\item {\em Luminosity $L$} Total luminosity of all stars combined.
\item {\em Spectrum} The spectrum of a galaxy is the combined spectrum
  of all of its stars, weighted by luminosity. 

\item {\em Colour} A measure of the fraction of light emitted in long
  versus short wavelengths. {\em True colour} images of
  galaxies\footnote{Sometimes narrow-band filters are used to stress
    the presence of particular emission lines, for example the
    H$\alpha$ emission line produced in star-forming regions.} are made
  by combining photographs of the galaxy taken through standard
  broad-band filters (see observational techniques), and mapping these
  to RGB colours.  Since massive stars are young, hot and blue,
  galaxies which contain massive stars will tend to be bluer than
  those without them.

\item {\em Extent} The angle the galaxy extends on the sky.

\item {\em Flux} Just as for stars, the flux $F$ of a galaxy with
  luminosity $L$ at distance $d$ is $F=L/4\pi d^2$, and is usually
  expressed using a system of magnitudes. Whereas $L$ is an intrinsic
  property of the galaxy, the flux also depends on the distance to the
  observer.

\item {\em Surface brightness and intensity} Approximate a galaxy as a
  slab of stars, with surface density $\sigma$ (in stars per unit
  area), each of identical luminosity $L$.  The total luminosity
  ${d\cal L}$ of an area $dS$ of this galaxy is $d{\cal L}=\sigma
  L\,dS$. The {\em intensity I} is the luminosity per unit area,
  therefore $I=d{\cal L}/dS=\sigma L$ for a slab. It is an intrinsic
  quantity, {\em i.e.} it does not depend on the distance to the
  observer. The flux $dF$ an observer at distance $d$ receives from
  this surface area is
  \begin{eqnarray}
    dF&=&{d{\cal L}\over 4\pi d^2}\nonumber\\
      &=&{\sigma L dS\over 4\pi d^2}\nonumber\\
      &=& {I\over 4\pi}\,d\Omega\,.
   \end{eqnarray}
   Here, $d\Omega=dS/d^2$ is the solid angle the surface area $dS$
   extends on the sky.  The quantity $dF/d\Omega=I/4\pi$, the flux
   received per unit solid angle, is called the {\em surface
     brightness}, 
\begin{equation}
{\hbox{\rm Surface brightness}} = {dF\over d\Omega}={I\over 4\pi}\,.
\end{equation}
Note that it is independent of distance\footnote{This is only true for
  nearby galaxies with redshift $z\ll 1$. When $z$ is not $\ll 1$,
  surface brightness dims with redshift $\propto 1/(1+z)^4$.}. Since
$\sigma$ decreases with radius $r$ from the centre, surface brightness
is higher in the centre than in the outskirts. Surface brightness is
usually expressed in magnitudes per square arc seconds but this is
not correct. What is meant is that one has converted the flux $dF$,
measured in a solid angle of 1 square arc seconds, into magnitudes.

\end{itemize}

\subsection{Galaxy types}
Images of galaxies immediately show there are two types, called {\em
  elliptical} (E, also called early type, or spheroidal) and {\em spiral}
{S, also called disk type, or late type}. Large galaxies that do not
fit into either category have usually undergone a recent violent
collision, and most small galaxies are of type {\em irregular}.

Defining characteristics for E and S galaxies are
\begin{tabbing}
\= \hspace{5cm}\= Elliptical\hspace{5cm}  \= Spiral\hspace{5cm} \\
\opt{lecturer}{
\> Shape     \> spheroidal   \> most stars in a disk \\
\> Colour    \> red \> blue \\
\> Stars     \> old stars  \> old and young stars\\
\> ISM       \> little gas or dust \> gas and dust \\
\> Stellar Dynamics  \> large random motions (hot) \> circular orbits (cold) \\
\> Environment \> dense (clusters)  \> low density (groups and field) \\
}
\opt{student}{
\> Shape     \> \makebox[3truecm]{\hrulefill} \> \makebox[3truecm]{\hrulefill}\\
\> Colour    \>\makebox[3truecm]{\hrulefill} \> \makebox[3truecm]{\hrulefill}\\
\> Stars     \>\makebox[3truecm]{\hrulefill} \> \makebox[3truecm]{\hrulefill}\\
\> ISM       \>\makebox[3truecm]{\hrulefill} \> \makebox[3truecm]{\hrulefill}\\
\> Dynamics  \> \makebox[3truecm]{\hrulefill}\>\makebox[3truecm]{\hrulefill} \\
\> Environment \> \makebox[3truecm]{\hrulefill}\>\makebox[3truecm]{\hrulefill} \\
}
\end{tabbing} 

\noindent {\em Ellipticals} Isophotes (lines of constant surface
brightness) of Es are smooth and elliptical and are further classified
as E$n$, where $n=10(1-b/a)$, where $a$ and $b$ are the major and
minor axis of the isophotes, respectively. A round elliptical ($a=b$)
is E0, whereas the most flattened ellipticals, E7, have $b=0.3a$. There
is little evidence for rotation in elliptical galaxies (except may be
small ellipticals), so their flattening is not due to angular
momentum. The intrinsic (as opposed to projected) shapes of Es are
thought to be triaxial, with iso-density contours $a>b>c$.\\

\noindent {\em Spirals} The very thin flattened disk of spirals
galaxies is due to rotation, and the stars in the disk are on nearly
circular orbits around the galaxy's centre. Gas collects in Giant
Molecular Clouds in the spiral arms of disks, where some fraction
collapses into new stars. The massive, newly formed stars ionise their
natal gas, and such \HII\ regions of ionised hydrogen follow the arms
as beads on a string. Once the cloud is dispersed, the blue light of
these stars contributes to making the whole spiral appear blue, in
contrast to the yellow/red light emitted by the older stars in
Es. Spirals also contain dust which causes dark bands across the disk
as the dust obscures background stars. The presence of dust prevents
us from seeing the Milky Way's central bulge in visible light. The
{\em bulge} is the central spheroidal stellar system, with many
properties in common with elliptical galaxies. Spirals are further
divides as Sa to Sc, where along the sequence the ratio of
bulge-to-disk luminosity decreases, and the spiral arms become more
loosely wound.

Some spirals also contain a {\em bar}, an almost rectangular stellar
system in the disk, sticking-out of the bulge. These are designated as
SB.  So for example, an SBc galaxy is a barred (B) spiral (S), with loosely
wound spiral arms and a small bulge (c), where an Sa has no bar, and a big
bulge. The Milky Way is between types SBb and SBc, Andromeda is type
Sb.\\


\noindent {\em Hubble's classification}

\begin{figure*}
\center\resizebox{.9\textwidth}{!}{\includegraphics{TuningFork.pdf}}
\caption{The Hubble sequence.}
\label{fig:Hubble}
\end{figure*}
Hubble used the above classification ordering galaxies in a tuning
fork diagram called the Hubble Sequence (Fig.~\ref{fig:Hubble}). The commonly used
nomenclature of early types (for Es) and late types (for Ss) comes
from the mistaken belief that Es evolve in Ss.\\

The range in physical scales of Es is huge, from masses as
little\footnote{It has been suggested that globular
  clusters are small ellipticals and should represent the low-mass
  end.} as $10^7\msun$ to as much as $10^{13}\msun$, with linear sizes
ranging from a few tenths of a \kpc\ to hundreds of \kpc. In contrast,
spirals tend to be more homogeneous, with masses
$10^9$--$10^{12}$\msun, and disk diameters from 5 to 100\kpc\ or so.

cDs and S0s are unusual types of Es and Ss, respectively.  cDs are very
large ellipticals, found in the centres of clusters, with a faint but
very large outer halo of stars\footnote{cD refers to properties of the
  spectrum, but think of it as standing for {\bf c}entral {\bf
    D}ominant.}. S0s (or SB0 when barred) or lenticulars are the divide
in Hubble's sequence between E and S, they have disks without gas or
dust, and no recent star formation.

Most small galaxies like {\em e.g.} one of our nearest neighbours, the Small Magellanic Cloud or SMC, have no well
defined disk, nor a spheroidal distribution of stars, and hence are
neither of type E nor S, they are classified as type {\em Irregular}.
The two main types of galaxies are then elliptical and spiral, with most small galaxies being irregulars.

\section{Summary}
After having studied this lecture, you should be able to 

\begin{itemize}
\item Describe the Hubble Sequence
\item Describe the main galaxy types, Es and Ss, and list five
defining characteristics
\item Define surface brightness, and show it to be independent of
distance
\item Derive the relation between the surface density of stars in a
galaxy, and its surface brightness
\end{itemize}

\newpage
\chapter{The discovery of the Milky Way and of other galaxies}
Before distances to galaxies were measured, their nature and that of
the Milky Way itself, was unclear.  One possibility was that nebulae
where stages of stellar evolution, such as {\em e.g.}  proto-planetary
systems. Observations gave contradictory results, with star counts
suggesting the Sun was near the centre of the Milky Way, while globular
clusters suggested it was not. The reason for the confusion is
interesting and discussed below.

\section{The main observables}
\subsection*{\bf Star clusters}
\begin{itemize}
\item {\em Globular clusters} (GCs) are small ($\sim $1\pc) dense
concentrations of $10^5-10^6$ stars of low-mass, old stars.  Most
galaxies contain 10s to 1000s of GCs, spherically distributed around
their centres; the MW galaxy has about 150 GCs.\\

\item {\em Open clusters} contain typically $10^4$ stars, are much
  less dense than GCs, and are restricted to the galactic plane. They
  are gravitationally bound systems left-over after star formation has
  ceased in a Giant Molecular Cloud, and the remaining gas has been
  dispersed. They are of crucial importance for testing theories of
  stellar evolution: since their stars are thought to be coeval, any
  differences in properties should be a consequence of differences in
  initial mass.\\
\end{itemize}

\subsection*{\bf Star counts}
Star counts can probe the number density of stars in a stellar system,
as a function of position.  William Herschel and his sister counted
the numbers of stars as function of their magnitude in many (700!!)
directions in the sky.  Kapteyn and his collaborators used
photographic plates for counting stars.

Suppose for simplicity that all stars have the same luminosity,
$L$. The flux of a star, $F=L/4\pi r^2$, then only depends on distance
$r$, and $dF/F=-2dr/r$. If the number density of stars at distance $r$
in direction $\hat\Omega$ is $n(r,\hat\Omega)$, the number of stars
with flux between $F$ and $F+dF$ in that direction, is
\begin{eqnarray}
dN(F,\hat\Omega) &=& n(r,\hat\Omega) r^2 dr\,d\Omega\nonumber\\
\opt{lecturer}{
                 &=& -{1\over 2}\,n(F,\hat\Omega)\,({L\over 4\pi})^{3/2}\,F^{-5/2}\,dF\,d\Omega\,.
}
\opt{student}{
                 &=& \makebox[8truecm]{\hrulefill}\,dF\,d\Omega\,.
}
\end{eqnarray}
Therefore counting $dN(F)$ allows one to infer the density $n(r)$.

\subsection*{\bf Parallax}
Nearby stars appear to move with respect to more distant stars as 
Earth moves around the Sun. The extent of the excursion $\theta$
depends on the Earth-Sun distance  and the distance to the star. Measuring
$\theta$  gives a {\em parallax} distance to the star. A
star at 1\pc\ distance has by definition a yearly parallax of 2~\arcsec . The name derives  from {\bf par}allax of one {\bf sec}ond of arc.

\subsection*{\bf Standard candles}
Standard candles are objects with a known absolute property, for
example a known size, or known luminosity. If the intrinsic size $D$
is know, the distance can be determined by measuring angular
extent. If the luminosity is known, distance can be determined by
measuring flux. 

{\bf Cepheid variables} are an important example. Henrietta Leavitt
studied variable stars in the Magellanic Cloud in 1912. She found a
relation between the period $P$ of the periodic variation $\Delta m$,
and the magnitude $m$ of these Cepheids. Since all these stars are at
(nearly) the same distance, this must mean it is actually a relation
between the {\em absolute} magnitude $M$ and $P$. Calibrating this relation makes Cepheids standard candles. In addition,
Cepheids have a very characteristic saw-tooth shaped $m(t)$, making it
easy to recognise them. Leavitt's discovery was therefore of great
importance.

Parallax and Cepheid variables are the first two steps in the {\em
  distance ladder}, which use one method (e.g. parallax) to calibrate
another distance measure (e.g. Cepheids), that then can be used to
calibrate another distance indicator, and so on to ever greater
distances. {\em Hubble Space Telescope} recently measured Cepheids in
the Virgo cluster (at a distance of $\sim\,17\Mpc$), thereby providing
the first accurate distance to that cluster of galaxies, and getting
an accurate measurement of Hubble's constant in the process.


\section{Discovery of the structure of the Milky Way}
\subsection*{Star counts} 
{\em Jacobus Kapteyn} confirmed in the beginning of the 20th century
using photographic plates the star count results first obtained by
Herschel. The MW is a flattened elliptical system, with stellar
density $n$ decreasing away from the centre. $n$ drops to half its
central value at 150\pc\ perpendicular to the MW plane, and 800\pc\ in
the galactic plane. The Sun is at 650\pc\ from the centre. Although
the star counts are correct, the interpretation to the shape of the MW
and the position of the Sun are wrong.

The conversion from $F$ to $r$, for given $L$, assumes $F\propto
1/r^2$, which neglects possible absorption of the light on route.  But
how can one test whether absorption is important? A plausible source
of absorption is light scattering off atoms or molecules along the line of
sight (Rayleigh scattering). The strength of scattering is colour dependent (stronger for
shorter wavelengths): scattering of Sun light off molecules in the
atmosphere makes the sky appear blue. Therefore if this were
important, distant (hence fainter) stars should appear redder than
more nearby stars. Kapteyn did measure the predicted reddening, but
the amount was too small to be important.

However the reddening is not due to scattering off atoms, but due to
scattering off  {\em dust} (Tyndall  scattering). The
colour dependence of dust scattering is much less, so the small amount
of reddening actually implies that absorption is quite important.


\subsection*{Globular clusters}
{\em Harlow Shapley} estimated the distances to the brightest globular
clusters using their RR Lyrae variables, and found they were not
centred around the Sun, but around a point 15\kpc\ away in the
direction of Sagittarius. This implied a much bigger MW than Kapteyn's,
and relegated the Sun to the MW outskirts. But who was right?

\subsection*{Other nebulae}
Another famous Dutch astronomer, {\em van Maanen}, observed galaxies
over several years, and decided he could see them move with respect to
the stars. This must mean they are relatively nearby, certainly within
the MW. 

But another astronomer Slipher measured large (1000s \kms) velocities
for some Nebulae, and finds evidence for rotation. He also claims the
light is produced by stars, not by gas. This suggests the nebulae are
galaxies, and outside of the MW, in conflict with van Maanen'sand Kapteyn's  MW picture.


\subsection*{Hubble's discovery}
Hubble used the 100-inch Hooker Telescope on Mount Wilson in 1919 to
identify Cepheid variables in other nebulae, including Andromeda.
He inferred\footnote{The modern value is $\sim 0.7\Mpc$.} a distance of $\sim 0.3~\Mpc$, much larger than even Shapleys size for the MW, and final prove that the nebulae were other galaxies outside
of the Milky Way. Hubble went on to discover that the more distant
galaxies move away from us at ever increasing speed. It is difficult
to underestimate the importance of this discovery. In one great swoop, the size of
the Universe increased dramatically. The Sun got relegated to the
outskirts of the MW, and the MW itself was found to be just one out of
billions of other galaxies. And the Universe was found to be
expanding, making Einstein's attempts to build a static cosmological
model out of his theory of relatively irrelevant.

\subsection*{Epilogue}
{\em Trumpler} found in the 1930s what had been
wrong with Kapteyn's interpretation, by studying MW Open Clusters.  He
assumed the size of Open Clusters to be a standard candle, and hence
assumed you could infer their distance from their angular size.  Using
this distance indicator, he found that the stars in the more distant
clusters (as inferred from cluster size) were invariable much fainter
than stars in more nearby clusters. He concluded that the light from
distant stars was attenuated much more than expected from scattering
off atoms. It had to be absorption by dust, and Kapteyn's neglect of
this lead to his error in interpretation.


\subsection*{Time-line}
\begin{enumerate}
\item[1610] Galileo resolves the MW into stars
\item[1750] Immanual Kant suggests that some of the other Nebulae are
other galaxies, similar to the MW.
\item[end of 1700s] Messier and Herschel catalogue hundreds of
Nebulae. Herschel counts stars, and deduces that the Sun lies near the
centre of an elliptical distribution with axes ratio 5:5:1
\item[1900-1920] Kapteyn counts stars, decides wrongly that extinction
is unimportant, and deduces the MW to be $5\kpc\times5\kpc\times1\kpc$
big, with the Sun at 650\pc\ from the centre.
\item[1912] Leavitt discovers the $P(L)$ relation for Cepheids.
\item[1914] Slipher measures large (1000s \kms) velocities for some
Nebulae, and finds evidence for rotation. The spectra he takes suggests
presence of stars, not of gas. A clear indication these are not
proto-planetary structures in the MW, but other galaxies.
\item[1915] Shapley finds the centre of the MW's globular cluster
system to be far away from Kapteyn's MW centre.
\item[1920] van Maanen claims (erroneously) that some spiral nebulae
have a large proper motion, suggesting they are within the MW.
\item[1920] Shapley and Curtis debate publicly over the size of the
MW, but the matter is not settled.
\item[1923] Hubble resolves M31 (Andromeda) into stars, using the newly
commissioned 100-inch telescope. Given the large inferred distance
means that M31 must be outside the MW. He also discovers Cepheids, and
the distance to M31 is estimated at 300\kpc. So Andromeda is indeed
another galaxy.
\item[1926] Lindblad computes that Kapteyn's MW is so small, it cannot
gravitationally bind its Globular Clusters. But Shapley's much bigger MW could.
\item[1927] Jan Oort shows that several aspects of the local motion of
stars can nicely be explained if the Sun (and the other nearby stars),
is on a nearly circular motion around a position 12\kpc\ away in the
direction of Sagittarius. Nearly the same position as found by Shapley,
and implying a much larger MW than Kapteyn's.
\item[1927] larger MW picture, where many of the nebulae are
extra-galactic MWs, gains general acceptance.
\item[1929] Hubble discovers his expansion law. His derived value is a
factor of 10 too large!
\item[1930] Trumpler uses open clusters to show the importance of
extinction, and explains why Kapteyn's measurement were faulty
\item[1930-35] Hubble's new data confirm the modern picture of
galaxies, and demonstrates van Maanen's measurements must have been
wrong.
\end{enumerate}

\section{Absorption and reddening}
Consider a light ray of intensity $I$ traversing a space containing
very large dust grains (bricks). The bricks will absorb a fraction of
the incoming light, and the intensity of the ray will decrease as

\begin{equation}
{dI\over dr}=-A\,I\,,
\end{equation}
which expresses the fact that each distance $dr$ will absorb a
constant {\em fraction} $dI/I=-A\,dr$ of the light. The constant $A$
depends on the number density of bricks, and their size.

The solution to this equation is

\opt{lecturer}{
\begin{equation}
I(r) = I_0\,\exp(-Ar)\,.
\end{equation}
}
\opt{student}{
\begin{equation}
I(r) = \makebox[8truecm]{\hrulefill}\,
\end{equation}
}

In terms of magnitudes, $\Delta m=-2.5\log(I/I_0)=\hat A\,r$, where
the relation between $\hat A$ and $A$ is left to you. $\hat A$
therefore has units of magnitude per unit length. Absorption therefore
changes the usual relation $(m-M)=5\log(r)-5$ between apparent and
absolute magnitude to $(m-M)=5\log(r)-5+\hat A\,r$.

If the size of the particles is of order of the wavelength $\lambda$
of the light, then the value of $A$ will be wavelength dependent.
This leads to reddening of the star, since smaller wavelengths (blue
light) will be absorbed more strongly than longer wavelengths (red
light). 

If we apply this reasoning to light in the $B$ versus $V$ band, for example, we obtain
\begin{eqnarray}
(m-M)_\B&=&5\log(r)-5+A_\B\,r\,\nonumber\\
(m-M)_\V&=&5\log(r)-5+A_\V\,r\,\nonumber\\
E_{\B-\V}&\equiv&(m_\B-m_\V)-(M_\B-M_\V)=(A_\B-A_\V)\,r\,.
\label{eq:colour}
\end{eqnarray}
The quantity $E_{\B-\V}$ is called the {\em colour excess}, note that it is
the difference between the observed and intrinsic colour of the star,
\begin{equation}
E_{\B-\V}=(\B-\V)_{\rm observed}-(\B-\V)_{\rm intrinsic}\,.
\end{equation}

Trumpler's measurements, and also laboratory measurements, show that
for interstellar dust grains
\begin{equation}
E_{\B-\V}\approx {1\over 3}A_\V\,r\,.
\end{equation}

This is a crucial result. Reddening and hence $E_{\B-\V}$, is easy to
measure, and so if we do this for stars of known distance, we
find\footnote{The amount of dust is not the same everywhere: there are
regions where the absorption is much stronger, not surprisingly called
dark clouds, and some directions along which the absorption is much
less, a well known direction is called Baade's window.}  $A_\V\approx
1{\rm mag}\,\kpc^{-1}$. If we now measure $E_{\B-\V}$ for another star
of {\em known} colour (from stellar evolutionary models say) we can
estimate $r$.

\pagebreak
\section{Summary}
After having studied this lecture, you should be able to

\begin{itemize}
\item Describe what are Open and Globular clusters
\item Explain and apply the relation between star counts and the density of stars
\item Explain what a standard candle is.
\item Explain why Cepheids are good standard candles.
\item Explain and apply parallax measurements
\item Explain how parallax and Cepheids can be used to walk-up the
distance ladder.
\item Explain how Hubble's observations revolutionised our view of the
MW and the realm of the Nebulae.
\item Derive and apply the effect of scattering by dust on the apparent magnitude
and colour of distant stars.
\end{itemize}

\newpage

\chapter{The modern view of the Milky Way}
Our current picture of the Milky Way is based on a observations using a range of different techniques, often  developed for military purposes ({\it e.g.} radar).

\section{New technologies}

\subsection{Radio-astronomy}
Radio-waves are EM-radiation with long wavelength $\lambda$ and therefore are not susceptible to absorption by interstellar dust. Therefore they can probe the dense, dusty regions where stars are formed.
Earth's atmosphere is nearly transparent for $1~{\rm cm}< \lambda< 10~{\rm m}$. Radio telescopes usually consist of a (large) single dish (for example at Jodrell bank; the Phjysics department has a small dish on its roof) or an {\em interferometer} 
with many inter-connected dishes, for example the {\em Giant Meter Radio-telescope} in India. Radio-observatories separated by 100-1000s of kilometers sometimes combine their signal to obtain very high angular resolution\footnote{Recall that the angular diffraction limit depends on telescope diameter $D$ and observed wavelength $\lambda$ as $\theta\propto \lambda/D$. A radio-interferometer with $D=10^3$~km and $\lambda=1$~mm has $\theta\approx 10^{-9}$, whereas an optical telescope with $D\sim 10$~m and $\lambda\sim 5\times 10^{-5}$~m has  $\theta\approx 5\times 10^{-8}$.} Radio-waves are usually produced by one of:

\begin{enumerate}
\item Roto-vibrational transitions in molecules
\item Thermal radiation from (cold) dust
\item Synchrotron radiation from electrons moving in a magnetic field
\item Hyperfine transitions, {\em e.g.} the 21-cm line in hydrogen (see Ch~4)
\end{enumerate}

\noindent {\em Roto-vibrational} transitions correspond to the rotational or vibrational transitions in molecules. For example energy can be stored in the vibration of the C and O atoms in a CO molecule, whereby the distance between C and O varies. A quantum transition whereby the amount of vibration in the molecule  decreases is associated with the emission of a photon. A diatomic molecule such as CO can also store energy in rotation along one of the two axes perpendicular to the C-O molecular bond. A decrease in the amount of rotation again results in the emission of a photon. The associated energies $\Delta E$ are in general much less than of {\em electronic transitions} hence  the associated wavelengths $\lambda=hc/\Delta E$ are longer (IR or radio-waves, as opposed to optical or UV-radiation). \\

\noindent {\em Thermal radiation} Dust heated by nearby stars cools by radiating radio/infrared waves. If the temperature are low, the blackbody may peak in  micro/mm wavelengths\footnote{Recall Wien's displacement law, relating the peak emission wavelength $\lambda$ and temperature $T$ as
\opt{lecturer}{$\lambda=b/T$, with $b=3\times 10^{-3}$~m~K}
\opt{student}{$\lambda=$\makebox[3truecm]{\hrulefill}}.}\\

\noindent {\em Synchrotron radiation} Electrons moving in a magnetic field may emit radiation at radio-wavelengths depending on their speed (just as in a terrestrial synchrotron). Astronomical examples include  supernovae remnants.\\

\subsection{Infrared astronomy}
IR photons are not significantly absorbed by dust. In fact, much of the visible and UV-light absorbed by
dust grains, is re-emitted in the IR and sub mm, and so IR observations can look deep inside star forming regions. The DIRBE instrument on the COBE (Cosmic Background Explorer) satellite provided us with
one of the best views of the the Milky Way, because it could see through the dust clouds that obscure large parts of the MW in the optical. Unfortunately earth's atmosphere absorbs most IR radiation, except in some narrow bands. IR observations therefore require balloon, rocket or satellites, or are limited to narrow regions of EM radiation in between the atmosphere's absorption bands.

\subsection{Star counts}
The Hipparcos satellite used diffraction limited observations above the atmosphere to obtain superbly accurate positions of stars on the night sky. By repeating measurements over several years, the satellite 
measured parallaxes of many 1000s of stars, and in addition proper motions for some stars ({\em i.e.} the velocity of some stars in the plane of the sky). Parallax distances are the crucial first step to the distance-ladder: parallax distances to Cepheids are required to calibrate the period-luminsity relation.

GAIA\footnote{http://astro.estec.esa.nl/GAIA/}, the successor to HIPPARCOS,  is due for launch in $\sim $ 2012. It will measure positions of stars up to 15-th magnitude with a resolution of 10$^{-5}\arcsec$. Gaia's expected scientific harvest is \lq of almost inconceivable extent and
implication\rq\, (according to their modest web page). In total, about $10^9$
MW stars will be measured (as compared to $10^4$ measured by
Hipparcos). Gaia will be so accurate that it can even measure proper
motions of some of the nearest globular clusters and galaxies. It will revolutionize the study of the MW.

\section{The components of the Milky Way}
The techniques above have shaped our view of the MW: it consist of three well-defined separate stellar systems: {\em disk},  {\em bulge} and {\em halo}. A summary of the properties of these is in Tables~\ref{table:disk} and \ref{table:halo}.

\subsection{Disk}
The disk is a round, thin distribution of stars. The Sun is part of the MWs disk.
\begin{itemize}
\item luminosity $L\sim  2\times 10^{10}\lsun$ in the B-band, $\approx 70$  per cent of the MWs total B-band luminosity.
\item radius\footnote{The distance from Sun to the galactic centre has recently been determined with remarkable accuracy to be $R_0=7.94\pm 0.42 ~\kpc$, from the observed motions of stars around the galactic centre (see lecture 11 for more details).} $R\sim 15\kpc$. 
\item Define the thickness $t$ of the disk as the ratio $t\equiv \rho/\sigma$ of the volume density of stars, $\rho$, at the galactic plane, and the surface density $\sigma$. The thickness $t$  depends on the type of star, and is $\sim 200\pc$ for young stars, $\sim 700\pc$ for stars like the Sun. 
\end{itemize}

The disk also contains gas and dust (see lecture 4 on the interstellar medium). On top of the
smooth disk are spiral arms, traced by young stars, molecular clouds,
and ionised gas. The disk stars are in (nearly) circular motion around
the centre, with speeds $\sim 220\kms$. The oldest disk White Dwarfs are $\sim 10-12\times 10^9$ years old, but these could have formed {\em before} the disk.

The density distribution of stars in disk in cylindrical coordinates $(R,\phi,z)$ can be written as
\begin{equation}
n(R,\phi,z) \propto \exp{({-R\over R_h})}\,\exp{({-|z|\over z_h})}\,,
\end{equation}
{\em i.e.} independent of $\phi$ since it is cylindrically symmetric, and falling exponentially both in radius $R$ and height $z$ above the disk, with {\em scale-length} $R_h\approx 3.5~\kpc$ and {\em scale-height} $z_h\approx 0.3~\kpc$.

Note that there isn't really an edge to the disk, either in radius or height. It can be traced to a
distance of around 30\kpc. With a height of 0.3\kpc, this is a ratio
100:1, which is thinner than a compact disk!\\

\noindent {\bf The thick disk} About 4 per cent of the MW's stars belong
to a thicker disk, aligned with the (thin) disk, but with a larger 
scale height of $z_h\approx 1\kpc$.  Stars in this thick disk differ from the thin disk stars discussed above both in composition (having a lower metal content) and kinematically.

\subsection{Bulge}
The bulge is a spheroidal distribution of stars in the centre of the MW (and of most spirals).

\begin{itemize}
\item central spheroidal stellar system with radius of $\sim 1\kpc$
\item luminosity $\approx 30$ per cent of total MW luminosity
\item luminosity profile is a de Vaucouleurs or \lq $r^{1/4}$\rq\, profile, defined as 
\begin{equation}
I(r)=I_e \exp[-7.67(r/r_e)^{1/4}-1]\,,
\label{eq:r4}
\end{equation}
with {\em effective radius} $r_e\approx 0.7\kpc$. 
\end{itemize}
The bulge stars are generally older and more metal poor than disk stars, suggesting the bulge formed before the disk, and before stellar evolution had managed to pollute the gas from which stars form with metals. In contrast to the disk, there is little or no star formation in the bulge.

The MW's bulge is elongated with axis ratio 5:3, with strong evidence for a bar. Whereas the disk stars are rotating in ordered fashion around the MW centre, the bulge has little net rotation, but the stars have large random velocities.  All these properties: no star formation, large random stellar motions, spheroidal geometry,  are reminiscent of the properties of an  elliptical galaxy: it is as if there is
a small elliptical galaxy at the centre of each spiral. 

The bulge is bright but obscured in the visible due to dust: we need IR observations to make it visible.

\subsection{Halo}
A very small fraction ($\le 1~$ per cent) of the MWs stars are contained in a large, spheroidal, extremely tenuous
stellar system called the halo, mostly (99\%) made-up out of single \lq field\rq\ stars with a sprinkling of Globular Clusters. The halo does not appear to rotate, and the halo stars have very low metallicities, typically 1/10 to 1/100 times the solar metallicity. When a halo star occasionally plunges through the disk, it is spotted because of its very high velocity, which is partly due to its intrinsic high random velocity, partly due to the fact the the Sun (and other disk stars) races with 220 km s$^{-1}$ around the MW centre, but the halo star does not partake in this rotation. The halo contains $\approx 150$ Globular Clusters (GCs), although there is some evidence that younger GCs may be associated with the disk.

The density of halo stars, and of GCs, falls spherically as
\begin{equation}
n(r)=n_0 ({r\over r_0})^{-3.5}\,,
\end{equation}
and extremely distant field stars have been detected out to more than
$\approx 50~\kpc$. 

\subsection{The dark matter halo}
See lecture 5 for evidence that as much as 90 per cent of the mass in the MW is invisible, and consists of some unknown type of matter.

\section{Metallicity of stars, and stellar populations}
Nuclear-synthesis during the first three minutes after the Big Bang produced mostly Hydrogen and Helium, and trace-amounts of other elements such as Li and B\footnote{See L3 and L4 lectures.}. By mass, a fraction $X\approx 0.76$ was Hydrogen, with most of the remaining mass fraction $Y\approx 1-X\sim 0.24$ in Helium.  All other elements were synthesised in stars and flung into space either 
due to winds (in AGB stars), during a Planetary Nebula phase, or during supernova explosions\footnote{Elements heavier than $^{56}$Fe
are endothermic, meaning energy is required to synthesise them, as opposed to elements that release energy during synthesis,
and are almost exclusively produced during SNe explosions.}. These elements are usually (but erroneously) referred to as \lq metals\rq\, in astronomy, and their mass fraction denoted as $Z\equiv 1-X-Y$.

A star formed out of gas already enriched in metals by (a) previous generation(s) of stars, will have a higher metal fraction $Z$ than stars formed from more pristine gas. Since most of the stellar burning converts Hydrogen into Helium, such a star will have also tend to have $X<0.76$ and hence $Y>0.24$. For example the Sun has $X\sim 0.7$, $Y\sim 0.28$ leaving a total fraction of $Z=1-X-Y\approx 0.02$ in \lq metals\rq. The composition of a star therefore contains a wealth of information on the properties of earlier generations of stars. It is quite an awesome realisation that the metals in the Sun (and also in you and me) were produced by 1000s of stars and SNe that have long since perished.

Metallicity  is often expressed relative to that of the Sun on a logarithmic scale, denoted (for example for Fe) as

\begin{equation}
\opt{lecturer}{
{\rm [Fe/H]}\equiv \log_{10}\left[{M_{\rm Fe}/M_{\rm H}\over (M_{\rm Fe}/M_{\rm H})_\odot}\right]\,,
}
\opt{student}{
{\rm [Fe/H]}\equiv \makebox[7truecm]{\hrulefill} \,,
}
\end{equation}
that is (the logarithm of ) the ratio of Fe-to-H mass, divided by that ratio for the Sun.
A star with  [Fe/H]=0 has the same Fe abundance as the Sun, a star with  [Fe/H]=-1 is ten times more Fe poor, a star with  [Fe/H]=1 ten times more Fe rich.\\

\noindent Disk stars have $-0.5\le $ [Fe/H] $\le +0.3$, with a clear trend of increasing [Fe/H] toward  the centre of the disk, hence the MWs material has been processed more vigourously in its interior than towards its outskirts. The disk population of relatively metal rich stars, still undergoing star formation is called {\em population I} stars. The true nature of the thick disk stars is not completely clear, but thick disk stars tend to be more metal poor, [Fe/H]$\approx -0.6$\\

\noindent  Bulge stars have a wide range of $-3\le $ [Fe/H] $\le 0.3$, with no significant ongoing star formation. This population of relatively metal rich stars without star formation is called {\em population II}\footnote{The stellar population of elliptical galaxies is similar}.\\

\noindent Halo stars have much lower abundances  $-3<$ [Fe/H]$ < -1$. A population with such extremely low abundance, composed of old stars, is called {\em extreme population II}.

\label{sect:simulation}
\begin{table}
 \hrule
 \vskip0.1truecm
 \hrule
 \centering \begin{minipage}{120mm} \caption{Disk parameters}
 \vskip0.1truecm
 \hrule
 \vskip0.05truecm
 \begin{tabular}{llll} 
        & Neutral Gas    & Thin Disk           & Thick Disk \\[1pt]
$M/10^{10}\msun$ & 0.5	& 6	& 0.2 to 0.4\\
$L_B/10^{10}\lsun$ &	& 1.8 & 0.02\\
$M/L_\B$ (\msun/\lsun) & 	& 	3 &\\
Diameter (\kpc) &	50	&	50	& 50\\
Distribution &	$\exp(-z/0.16\kpc)$ &	$\exp(-z/0.325\kpc)$ & $\exp(-z/1.4\kpc)$ \\
${\rm [Fe/H]}$ & $>0.1$ &	-0.5 -- 0.3 & -1.6 -- -0.4\\
Age (Gyr) &	0 -- 17 & $<12$ & 14 -- 17\\
\label{table:disk}
\end{tabular}
 \vskip0.05truecm
 \hrule
\end{minipage}
\end{table}


\label{sect:simulation}
\begin{table}
 \hrule
 \vskip0.1truecm
 \hrule
 \centering \begin{minipage}{120mm} \caption{Spheroid parameters}
 \vskip0.1truecm
 \hrule
 \vskip0.05truecm
 \begin{tabular}{llll} 
        & Central Bulge    & Stellar Halo        & Dark Matter Halo \\[1pt]
$M/10^{10}\msun$ & 1	& 0.1	& 55\\
$L_B/10^{10}\lsun$ &0.3	& 0.1 & 0\\
$M/L_\B$ (\msun/\lsun) &3	& 	$\sim 1$ & -\\
Diameter (\kpc) &	2	&	100	& $>200$\\
Distribution &	bar &	$r^{-3.5}$ & $(a^2+r^2)^{-1}$ \\
${\rm[Fe/H}]$ & -3 -- 0.3 &	-4.5 -- -0.5 & \\
Age (Gyr) &	10 -- 17 & 14 -- 17 & 17\\
\label{table:halo}
\end{tabular}
 \vskip0.05truecm
 \hrule
\end{minipage}
\end{table}

\begin{figure*}
\center\resizebox{.7\textwidth}{!}{\includegraphics{galactic_coordinates.pdf}}
\caption{Diagram illustrating galactic longitude $l$ and galactic latitude $b$. From {\em thinkastronomy.com}.}
\label{fig:galactic_coordinates}
\end{figure*}

\subsection{Galactic Coordinates}
The position of an object on the sky as seen from the Sun can be characterised by two angles (see Fig.~\ref{fig:galactic_coordinates}) :
(1) {\em galactic latitude}: the angle $b$ above the Milky Way plane and (2) {\em galactic longitude}: the angle $l$ between the direction Sun-Galactic centre, and the projection Sun-star onto the Milky Way disk.

\pagebreak
\section{Summary}
After having studied this lecture, you should be able to

\begin{itemize}
\item Explain which processes generate observable radio and IR radiation, and why this radiation was important to clarify the MW structure.
\item Explain how the Hipparcos satellite was a major step in setting
the scale of the MW, by performing accurate parallax and proper motions
measurements, and fixing a reference frame with respect to distant
objects.
\item Describe the three main stellar components of the MW, and give two characteristic properties of each
\item Explain what is meant by the metallicity of a star, and how it is expressed
\item Explain how the Galactic Coordinate system $(l,b)$ is defined
\end{itemize}


\chapter{The Interstellar Medium}
The interstellar medium (ISM) is the {\em stuff} between the stars. It is a rich but complex physical system, with gas collecting in large molecular clouds ({\em Giant Molecular Clouds}), which form stars that then destroy the clouds, polluting the gas with metals. Stellar winds and super nova explosions stir the gas, any may even expell some gas from the MW into the surrounding intergalactic medium. Dust, magnetic fields and cosmic rays\footnote{Cosmic rays are energetic particles (photons, protons, nuclei of heavier elements), with energies up to orders of magnitude higher than what can presently be achieved in labs such as SLAC or Cern. With Profs Rochester and Wolfendale, Durham has a rich history in cosmic ray physics, and continues to do so with its involvement in the HESS telescope in Namibia.} play an important but poorly understood r$\hat{\rm o}$le. The L4 course on the ISM treats some of the issues in more detail. This chapter concentrates on the composition of the ISM, how the different components can be observed, how stars and gas interact, and the important concept of Jeans mass, relevant for how stars form in clouds.

\section{Interstellar dust}
Interstellar dust is composed of metals ({\em e.g.} Si), affects the chemistry of the ISM, and also the propagation of light.
Dust particles interact with light both through {\em scattering} and {\em absorption}. Sunlight passing through the Earth's atmosphere scatters off molecules (mostly N and O molecules), with blue light scattered more strongly than red light. Away from the Sun, we see the scattered Sun light on the sky, which is hence much bluer than the Sun itself: this is why the sky is blue. Conversely, looking at the Sun itself, we see a larger fraction of red light: the Sun reddens at sunset or sunrise. Sunlight can also scatter on dust (for example following a volcanic eruption or a dessert storm): this makes the sun appear even redder at sunset/rise. Figure~\ref{fig:blue_sky} illustrates this phenomena:


\begin{figure*}
\center\resizebox{.7\textwidth}{!}{\includegraphics{blue_sky.pdf}}
\caption{Scattering of Sunlight in the Earth's atmosphere, from The Physics and Relativity FAQ.}
\label{fig:blue_sky}
\end{figure*}

The scattered light is also {\em polarised}. Sun glasses are sometimes polarised to block sunlight scattered from the Ocean, or from road surfaces. Light may also be absorbed by clouds our dust, darkening the day. The same process of scattering and absorption occur in the ISM: distant stars appear redder, the light detected from reflection nebulae is blue and polarized, all due to the effects of dust grains.\\

{\em Scattering}: changes the direction of the incoming photon, but not its energy. EM radiation with wavelength much longer than the size of a grain scatters must less efficiently, hence red light, and in particular IR and radio-waves, is hardly affected by dust.\\

{\em Absorption}:  a photon interacting with a dust grain may break a molecular bond, or simply heat the dust grain.
This destroys the photon, which is lost to the observer. The grain may cool by emitting EM radiation, with black-body spectrum corresponding to its temperature: the typical energies of this cooling radiation is much less than those of the incoming photon. The net effect is that the dust grain has converted short wavelength (blue) photons to long wavelength (red or IR) ones. Dust in the surroundings of star forming regions for example reprocesses the blue light emitted by the hot stars into IR radiation. This obscures the star forming region in the visible, but makes it glow in the IR.\\

Although in general the amount of scattering/absorption increases with decreasing wavelength $\lambda$, some $\lambda$s correspond to resonance transitions in the molecules or dust grains, allowing determination of their composition\footnote{A typical example is the detection of the Si-O resonance in dust grains, showing they contain both Si and O.}.


\section{Interstellar gas}
Gas in the ISM can be in molecular, neutral, or ionised form. What determines which phase the gas is in? How are the different phases  observed? The processes discussed below are illustrated in Fig.~\ref{fig:ISM}.\\

\begin{figure*}
\center\resizebox{1.4\textwidth}{!}{\includegraphics{ISM.pdf}}
\caption{Important physical processes in the ISM.}
\label{fig:ISM}
\end{figure*}


\noindent {\em Notation} Astronomers use the (confusing) notation XIII to denote the {\em doubly} ionised state of element X, for example CIII$\equiv$C$^{2+}$, 
HI is neutral Hydrogen, HII ionised Hydrogen, and OVI is five-times ionised O.\\


\subsection{Collisional processes}
\noindent {\em Kinetic theory of gases} The typical velocity $v$ of particles with mass $m$ in a gas with temperature $T$ is $v^2\sim kT/m$.
A collision between such particles will have typical kinetic energy $E\sim mv^2\sim kT$, and will be elastic (energy conserving) when $E$ is much less than any excitation energies of the particle. Rotational and vibrational excitation energies are small, and so even at relatively low temperatures $T\sim 100$~K, the kinetic and roto-vibrational energy levels are in equipartition (\lq thermal equilibrium\rq). Because of such \lq internal degrees of freedom\rq\, a diatomic gas ({\em e.g.} H$_2$) does not behave the same as a mono-atomic gas ({\em e.g.} HI)\footnote{Recall that a diatomic gas has adiabatic exponent $\gamma=7/5$ in contrast to the ideal (mono-atomic gas) case $\gamma=5/3$.}.\\

At higher $T$, the kinetic energy may be high enough for collisions to break the molecular bond in the particles: collisions are no longer elastic, but destroy molecular bonds. The molecular binding energy in H$_2$ is $\sim 11$eV\footnote{An energy of 1eV corresponds to a temperature  T=1eV/\kB$\approx 10^4\K$.}, so gas at temperatures of $T\ge 10^3\K$ will cease to be molecular. At even higher $T$, $E$ is enough to ionise the atoms. For example the binding energy of the Hydrogen atom is 13.6eV, hence  for $T\sim 10^4\K$, Hydrogen will tend to be ionised. This suggests that {\bf increasing $T$ leads to initially molecular gas to become first neutral, then ionised.}\\ 

How about the reverse processes? Ionised gas can become neutral through recombinations. This process produces a photon which could escape from the gas, taking energy with it. This energy loss means the gas is cooling: it is called {\em radiative cooling}.  Neutral gas can become molecular through gas-phase reactions, but in the ISM mostly because of reactions on dust grains.\\

The ISM has regions where the gas is cold, warm , and hot . Because we expect the {\em pressure} in these regions to be  similar (since otherwise gas would flow from high to low pressure regions, equilibrating the pressure), the cold regions tend to be dense and molecular (called molecular clouds), the warm regions tend to be neutral, whereas the hot regions are at low density and contain ionised gas.\\

\subsection{Photo-ionisation and HII regions}
\begin{figure*}
\center\resizebox{0.5\textwidth}{!}{\includegraphics{hydrogen_spectrum.pdf}}
\caption{Electronic transitions in the Hydrogen atom, and nomenclature of the corresponding emission lines. Figure taken from the internet encyclopledia.}
\label{fig:hydrogen-spectrum}
\end{figure*}

When a photon impinging on an atom has enough energy, it may ionise the atom in a process called {\em photo-ionisation}. The ionisation energy for HI is 13.6eV, and the corresponding wavelength of the photon is in the UV. The ionised Hydrogen nucleus may catch a free electron and recombine: it is the inverse process. On its way to the $n=1$ ground-state, the electron will pass through one or more allowed electronic energy levels, emitting photons with characteristic energies 13.6eV ($1/n_1^2-1/n_2^2$) for a transition $n_2\rightarrow n_1$.  The series of emission lines which end in the ground state ($n_1=1$) are called the {\em Lyman-series}, with Lyman-$\alpha$ corresponding to $n_2=2$, Lyman-$\beta$ to $n_2=3$, {\em etc}. Transitions with $n_1=2$ are called the Balmer series, with Balmer-$\alpha$ or H$\alpha$ corresponding to $n_2=3$, H$\beta$ to $n_2=4$, {\em etc}. Fig \ref{fig:hydrogen-spectrum} illustrates these transitions.\\

Only massive stars are hot enough to produce Hydrogen ionising  photons in abundance. Since massive stars have such short lifetimes,
they are almost exclusively found in the star forming region where they were born. As the hot stars ionise the gas, most of Hydrogen gas is ionised and the region is therefore called an {\bf HII region}.  Recombinations make HII regions emit the very characteristic pattern of recombination lines discussed above. It is the H$\alpha$ emission lines which is used to trace HII regions in spiral galaxies\footnote{Other lines are too far into the UV to be easily observable from the ground}. Such HII regions follow spiral arms as beads on a string: this is because the spiral structure sweeps gas into dense clouds, which can then form stars, which subsequently ionise the gas making it radiate in recombination lines and be observable. All this stellar feedback destroys the cloud, the massive stars die, and so there are no HII regions nor massive stars, in between spiral arms. The Rosetta Nebula and Orion are well known HII regions in the Milky Way.\\



\subsection{Str\"omgren spheres} 
Suppose a hot star starts at time $t=0$ to emit ionising photons at a rate $\dot N$ into a gas cloud,
which initially contains neutral Hydrogen gas with uniform density $n$. The radius $R$ of the sphere within which the gas is ionised will increase in time. We can find $R(t)$ by requiring that the number of photons emitted by the source in time $t$, $\dot N\,t$, equals the number $N$ of (initially) neutral Hydrogen atoms within $R(t)$, $N=(4\pi/3)\,n\,R^3(t)$, hence
\begin{equation}
\dot N\,t = {4\pi\over 3}\,n\,R^3(t)\,.
\end{equation}

This simply expresses the fact that the source needs to emit one ionising photon per neutral atom. Take the time derivative to obtain the speed of the {\em ionisation front},

\begin{equation}
\opt{lecturer}{
{dR(t)\over dt} = {\dot N\over 4\pi n R^2(t)}\,.
}
\opt{student}{
{dR\over dt} = \makebox[8truecm]{\hrulefill}\,.
}
\label{eq:Rt}
\end{equation}


Initially when $R$ is small the speed of the front is high, then decreases $\propto 1/R^2$. 
Equation (\ref{eq:Rt}) predicts the speed becomes arbitrarily large for small $R$, but this cannot be right: it must be limited by the speed of light, so our treatment cannot be applied at very early times\\

As the ionisation front runs into the cloud, some of the Hydrogen ions behind the front will start to recombine. Each neutral atom within the HII region will absorb an ionising photon, which then can no longer ionise a neutral atom outside of the HII region: recombinations will
slow the ionisation front. Eventually, there may be so many recombinations occuring within the HII region that the speed of the front decrease to zero: the front has reached its {\bf Str\"omgren radius} $R_s$. The rate $d n_{\rm HII}/dt$ at which Hydrogen ions recombine, is given by

\begin{equation}
{\dot n_{\rm HII}\over dt}=- \alpha n_e n_{\rm HII}\,.
\end{equation}

Figure \ref{fig:ISM} illustrates the recombination of a Hydrogen ion with an electron. Since both a proton and an electron should be present for a recombination, the recombination rate is proportional to the product of the densities $n_e n_{\rm HII}$ of electrons and protons, respectively. The proportionality constant is denoted $\alpha$\footnote{Determine the units of $\alpha$ as an exercise.}\\

When Hydrogen gas is highly ionised, $n_e=n_{\rm HII}\approx n$, and the Str\"omgren radius $R_s$ is found by equating the rate of recombinations in a sphere of radius $R$, $\dot N_r\equiv d n_{\rm HII}/dt\,\times (4\pi/3)\,R^3$, with the rate at which the source produces ionising photons, $\dot N$:

\begin{eqnarray}
\dot N &=& {dn_{\rm HII}\over dt}\,{4\pi\over 3}\,R_s^3\nonumber\\
R_s &=&  
\opt{lecturer}{({\dot N\over (4\pi/3)\,\alpha\,n^2})^{1/3}\,.}
\opt{student}{\makebox[8truecm]{\hrulefill}\,.}
\end{eqnarray}

Using typical values $n_{\rm H}=5\times 10^3{\rm cm}^{-3}$ for the
density of a cloud, $\dot N=10^{49}{\rm s}^{-1}$ for the
ionisation rate of a massive star, and using $\alpha\approx 3.1\times 10^{-13}$ cm$^{3}$ s$^{-1}$ (valid for temperatures $\sim 10^4$~K in HII regions) gives a Str\"omgren radius $R_S\approx 0.21\pc$.


\subsection{21-cm radiation}
\begin{figure*}
\center\resizebox{.8\textwidth}{!}{\includegraphics{21cm_radiation.pdf}}
\caption{Emission of  21-cm radiation in Hydrogen due to a hyperfine transition, taken from Schombert.}
\label{fig:21-cm}
\end{figure*}

The Lyman and Balmer series discussed earlier corresponded to transitions of the electron between different {\em orbital} energy levels, but there are other possible transitions for an electron in an atom, which result from its {\bf spin}. Spin is a purely quantum mechanical degree of freedom, with some similarities to angular momentum\footnote{See later lectures on quantum mechanics}. 
A spinning electric charge generates a dipole moment, and spin is also associated with a magnetic dipole. However, the electron's magnetic dipole ${\bf S}$ is {\em quantised}: along any axis, the dipole is either up or down: ${\bf S}=\pm 1/2$. The electron has a series of energy levels resulting from the interaction of this dipole moment with its orbital angular momentum, ${\bf L}$, and with the proton spin, ${\bf S_p}$. This is similar to the interaction between two bar magnets: when they are aligned, they will repel each other, since like-poles repel. When they are anti-alingned, they will attract each other.\\

\noindent {\em fine-structure lines} result from coupling between ${\bf S}$ and ${\bf L}$: for given ${\bf L}$, the electron's energy is slightly higher when ${\bf S}$ and ${\bf L}$ are parallel, than when they are anti-parallel.\\

\noindent {\em hyperfine structure lines} result from coupling between the electron spin ${\bf S}$ and the {\em proton} spin ${\bf S_p}$: the energy of the electron is higher for  ${\bf S}$ and ${\bf S_p}$ parallel than when they are anti-parallel, see Fig.\ref{fig:21-cm}.\\

The wavelength of the hyperfine transition in the Hydrogen atom is $\lambda\approx 21.1$~cm and it is therefore called the {\bf Hydrogen 21-cm line}. The transition probability is very low, which is why it is very hard to observe it in the laboratory, because collisions between the particles
will de-excite the aligned spin well before there has been a spontaneous transition. But in the ISM, densities are far lower than can be generated in the lab, and {\bf this line is of great observational importance} because it can be used to detect neutral gas. The long wavelength also means it is not affected by dust\footnote{The radio-dish on top of the physics department can detect 21-cm radiation from neutral gas in the MW}.\\

An external magnetic field shifts the energy levels of the electron in a Hydrogen atom, the \lq Zeeman
effect\rq, and the 21-cm radiation can be used to estimate the strength of
the magnetic field in the ISM.\\

\subsection{Other radio-wavelengths}
The 21-cm line cannot be used to study gas in dense clouds, because the gas will tend to be molecular instead of atomic. But radio-telescopes can be used to detect roto-vibrational transitions of these molecules, enabling the study of molecular clouds.\\

The MW contains molecular clouds with a wide range in masses, up to 
{\em Giant Molecular Clouds}, (GMCs) with masses up to $10^7\msun$. These enormous complexes of gas and dust are almost exclusively found in spiral arms, and are the sites of star formation in the MW: most, if not all, stars are thought to form in GMCs. 

\subsection{The Jeans mass}
The masses of GMCs, $\sim 10^6\msun$, are very much higher than those of the stars that form in them: why?\\

The {\bf Jeans} mass $M_J$ in a gas with uniform density, $\rho$, and temperature, $T$, is the characteristic mass for which the thermal energy, $K$, and gravitational energy, $U$, in a sphere are in virial equilibrium,
$2K=U$. When $M>M_J$, gravity dominates, and the sphere will tend to collapse; when $M<M_J$, pressure dominates, and the sphere is stable to collapse.\\

The thermal energy $K$ in a sphere with mass $M$ is

\begin{eqnarray}
K &=& Mu\nonumber\\
u&=& {\boltz T\over (\gamma-1)\,\mu\mH}\,,
\end{eqnarray}

where $\gamma=5/3$ is the adiabatic exponent, $\boltz$ Boltzmann's constant, and $\mu\mH$ the mean molecular weight per particle. The gravitational potential energy $U$ of the sphere is

\begin{eqnarray}
U &=& {3\over 5}{GM^2\over R}\nonumber\\
M&=& {4\pi\over 3}\rho\,R^3\,.
\end{eqnarray}

In virial equilibrium, $2K=U$, and the mass of  the sphere is the Jeans mass\footnote{You may find expressions which differ by factors of order unity in other texts}:

\begin{equation}
M_J =\left({10\boltz T\over 3(\gamma-1)\mu\mH G}\right)^{3/2}\, \left({3\over
4\pi\rho}\right)^{1/2}\,.
\end{equation}


\noindent{\em Fragmentation} Consider the fate of a cloud
with mass $M=M_J$ that starts to collapse. In general, both $T$ and $\rho$ will change, and hence $M_J$ will change as well. If the gas behaves  {\em adiabatically}, $\rho\propto T^{3/2}$, 
and hence $M_J\propto \rho^{1/2}$. Hence $M_J$ {\em increases} at the cloud starts to collapse, pressure will increase faster than gravity, and the cloud will bounce back. However if the gas is  {\em isothermally}, $T=$constant, then $M_J\propto 1/\rho^{1/2}$, and the Jeans mass will {\em decrease}. Now smaller spheres will have $M=M_J$ and hence the cloud may fragment.

\pagebreak
\section{Summary}
After having studied this lecture, you should be able to 

\begin{itemize}
\item Describe how we know the properties of interstellar dust from
scattering and absorption of star light.
\item Explain why we find different ionisation states of interstellar
gas, depending on density, temperature, and ionising background.
\item Compute the speed of an ionisation front.
\item Derive the Str\"omgren radius for an HII region, and explain the
concept.
\item Explain the origin of the hydrogen 21-cm line, and explain its
importance in understanding the structure of the MW.
\item Explain the concept of Jeans mass, and its relation to
fragmentation of clouds.
\end{itemize}


\chapter{Dynamics of galactic disks}
The stars in the Milky Way disk are on (almost) circular orbits, with
gravity balancing centripetal acceleration. Given that most of the
light of the disk comes from its central parts, we would expect the
circular velocity in the outer parts of the disk to fall with distance
as appropriate for Keplerian motion. We should also be able to compute
how velocities of stars in the solar neighbourhood depend on
direction. Observations do not follow these expectations at all, which
leads to the startling conclusion that most of the mass in the Milky
Way is invisible.

\section{Differential rotation}
\subsection{Keplerian rotation}
\label{sect:KR}
The velocity of a test mass $m$ in circular motion around a mass $M$ at distance $R$
is 

\begin{equation}
{V_c^2\over R} = {GM(<R)\over R^2}\,,
\label{eq:vcirc}
\end{equation}
independent of $m$ along as $m<<M$. Mass $M$ need not be a point mass: Newton's theorem guarantees this equation is also correct for an extended spherical mass distribution, as long as 
$M$ is the mass enclosed\footnote{We'll use the notation $M(<R)$ for the mass enclosed in a sphere of radius $R$.} by the orbit $R$:  mass outside $R$ does not contribute to the gravitational force. This equation describes for example the motion of planets in the solar system, with $M$ the mass of the Sun.\\

This equation also describes the circular motion of stars around the MW's centre:
Newton's theorem also applies for motion in a disk\footnote{Demonstrate this as an exercise.}
In the outskirts of the disk, for example at the position\footnote{$R_\odot\sim 8\kpc$ denote here the distance of the Sun to the centre of the MW.} $R_\odot$  of the Sun, we observe that most of MW's light is at $R\le R_\odot$. This suggests that also most of the mass is within $R_\odot$, hence $M(<R)\approx M_{\rm MW}$, the total mass of the MW. For $M(<R)=M_{\rm MW}$ constant, Eq.~(\ref{eq:vcirc}) gives
$V_c\propto R^{-1/2}$. The function $V_c(R)$ (circular velocity as function of radius) is called the {\bf rotation curve}, and we expect  the disk to be in {\em differential rotation}, with rotation curve
$V_c\propto R^{-1/2}$. We can test this assertion by studying the motion of stars in the solar neighbourhood.\\

\subsection{Oort's constants}
\begin{figure*}
\center\resizebox{.8\textwidth}{!}{\includegraphics{oort.pdf}}
\caption{(Taken from http://www.csupomona.edu/\~{}jis/1999/kong.pdf)
The observer is moving on a circular orbit with velocity $\Theta_0$ and
radius $R_0$, and observes a star at distance $r$, and galactic
longitude $l$, on circular orbit with velocity $\Theta$ and radius
$R$. In the text, $r$ is denoted as $d$ and the circular velocities are
$V_c$ and $V_{c,0}$ for the star and the Sun, respectively.}
\label{fig:Oort}
\end{figure*}

Assume all disk stars are on circular orbits, with circular velocity $V_c(R)$ for a star at distance $R$ from the MW centre. The observer (at radius $R_0$) is moving with circular velocity $V_{c,0}\equiv V_c(R=R_0)$ measures the following radial, $V_r$ and tangential, $V_t$ of a star at
distance $d$ and radius $R$, which has galactic coordinates $(b=0,l)$ (see Fig.\ref{fig:Oort})

\begin{eqnarray}
V_r & = & V_c\cos(\alpha) - V_{c,0}\sin(l)\nonumber\\
V_t & = & V_c\sin(\alpha) - V_{c,0}\cos(l)\,.
\label{eq:oort1}
\end{eqnarray}

In the indicated right-angled triangle

\begin{eqnarray}
d+R\sin(\alpha)&=&R_0\cos(l)\nonumber\\
R\cos(\alpha) &=& R_0\sin(l)\nonumber\\
R_0&=& d\cos(l) + R\cos(\beta)\approx d\cos(l) +
R\,{\hbox{\rm when $d\ll R_0$}}\,,
\end{eqnarray}

where $\beta$ is the angle Sun-MW Centre-Star. Combining these
equations gives

\begin{eqnarray}
V_r&=&(\Omega-\Omega_0) R_0\sin(l)\nonumber\\
V_t&=&(\Omega-\Omega_0) R_0\cos(l)-\Omega\, d\,,
\label{eq:oort}
\end{eqnarray}

where $\Omega_0\equiv V_{c,0}/R_0$ is the {\em angular velocity} of the observer,
and $\Omega\equiv V_c/R$ the angular velocity of the star. For {\em nearby stars},
$R\approx R_0$ and $\Omega(R)\approx\Omega_0+(d\Omega/dR)(R-R_0)$.
If we now define {\bf Oort's constants} $A$ and $B$ as
\begin{eqnarray}
A &\equiv & -{1\over 2} \left[ {dV_c\over dR}|_{R_0} -
{V_{c,0}\over R_0} \right]\approx 14.4\pm 1.2\km\,\s^{-1}\,\kpc^{-1}\nonumber\\
B &\equiv & -{1\over 2} \left[ {dV_c\over dR}|_{R_0} +
{V_{c,0}\over R_0} \right]\approx -12.0\pm
2.8\km\,\s^{-1}\,\kpc^{-1}\,.
\label{eq:oortct}
\end{eqnarray}

then 

\begin{eqnarray}
V_r &=& A\,d\,\sin(2l)\nonumber\\
V_t &=& A\,d\,\cos(2l)+Bd\,.
\label{eq:oort}
\end{eqnarray}

For example a star toward the galactic centre (or anti-centre; $l=0$ and $l=180^o$ respectively), has 
$V_r=0$ and $V_t=(A+B)d$.\\

Jan Oort\footnote{Note that these are values appropriate for the Sun.} measured ($V_r,V_t$) for stars as function of $l$ and $d$, and inferred 
$A\approx 14.4\pm 1.2\km\,\s^{-1}\,\kpc^{-1}$ and $B=-12.0\pm 2.8\km\,\s^{-1}\,\kpc^{-1}$.\\

Using the results from section \ref{sect:KR} we find that for a Keplerian disk, $V_c\equiv V_{c,0} (R_0/R)^{1/2}$, hence $dV_c/dR=-(1/2)V_c/R$.  At the position of the Sun,  $V_{c,0}\approx 220\kms\,$, with $R_0\approx 8\kpc$, hence we expect the values of Oort's constants to be
\begin{eqnarray}
A_{\rm Kepler} &=&  {3\over 4} {V_{c,0}\over R_0}= 19.4\kms\,\kpc^{-1}\nonumber\\
B_{\rm Kepler} &=& -{1\over 4} {V_{c,0}\over
R_0}=-6.5\kms\,\kpc^{-1}\,.
\end{eqnarray}

Our expected values are inconsistent with Oort's measured values. Assuming we measured
 $V_c/R$ correctly, the measured and expected  values for $dV_c/dR$ are:

\begin{eqnarray}
{dV_c\over dR}|_{\rm measured}  &=& -(A+B) = -2.4\pm 3.0 \kms\,\kpc^{-1}\nonumber\\
{dV_c\over dR}|_{\rm Keplerian} &=& -{1\over 2}{V_c\over R} = -13.0\kms\,\kpc^{-1}\,.
\end{eqnarray}

As expected, the Keplerian circular velocity drops $\propto 1/R^{1/2}$, hence ${dV_c\over dR}|<0$, but the {\em measured} value is consistent with zero: {\bf the Mily Way's rotation curve is flat, {\em i.e.} $V_c(R)\approx $ constant}.\\

Of course in a real galaxy stars are not {\em exactly} on circular orbits, and each star has a small {\em peculiar} velocity with respect to the perfect circular motion $V_c(R)$. A standard of rest that moves on an exact circular orbit is called the \lq local standard of rest\rq\,\footnote{Note this is not an inertial system}: the speed of the Sun is $\sim 16$ km s$^{-1}$ with respect to its local standard of rest.\\

Oort also discovered a small number of stars with very large deviations from the expectation given by Eqs.~(\ref{eq:oort}), which he called {\em high velocity stars}. He correctly identified these with stars belonging to the halo: the high velocity is because the halo does not rotate, whereas the disk, and the Sun with it, rotates with a speed $\sim 220$ km s$^{-1}$.

\subsection{Rotation curves measured from HI~21-cm emission}
The HI 21-cm emission line can be used to measure the velocity of the HI from its Doppler shift.
This can be used to determine the rotation curve of the HI gas\footnote{You can attempt this in L4, using the radio dish on the physics building}, if it were possible to determine $R$.  Eqs.~(\ref{eq:oort1}) show that $V_r$ has a
{\em maximum}

\be
V_{r,{\rm max}}=V_c-V_{c,0}\sin(l)\,,
\ee

for any $l$ with $|l|<\pi/2$, which occurs for $\alpha=0$, at which
point $d=R_0\cos(l)$.
Therefore
\begin{equation}
{dV_{r,{\rm max}}(l)\over dl} = {dV_c(R)\over dR}\,{dR\over dl} - V_{c,0}\,\cos(l)\,
\end{equation}
hence
\begin{equation}
{dV_c(R)\over dR}={dV_{r,{\rm max}}(l)\over dl}\,/R_0\,\cos(l)+V_{c,0}/R_0\,.
\end{equation}
The 21-cm analysis confirmed Oort's measurements: the MW's rotation curve near the Sun is essentially flat.\\

It is much easier though to measure rotation curves in other spiral galaxies, by simply measuring the Doppler shift
as function of distance to the centre.  Such measurements are not available for tens of spirals, and they all show flat rotation curves in their outskirts. The MW is definitely not unusual in this respect.

\section{Rotation curves and dark matter}
\label{sect:rotcurve}
A flat rotation curve, instead of a $1/R^{1/2}$ falling rotation curve, implies the mass is not all concentrated within the solar circle, but is more extended. To find the shape of the density distribution that gives rise to a flat rotation curve,  take the derivative\footnote{Show that $dM/dR=4\pi\rho(R)\,R^2$ in a spherically symmetric density distribution.} with respect to $R$ of

\begin{equation}
V_c^2\,R=GM\,,
\end{equation}
for $V_c$ is constant:
\begin{eqnarray}
V_c^2 = G{dM\over dR}\nonumber\\
\rho(R) ={V_c^2\over 4\pi GR^2}\,.
\end{eqnarray}

Hence a spherical distribution of mass, with $\rho(R)\propto 1/R^2$, gives rise to a flat rotation curve. But the observed light distribution in the MW is very different from this. This suggests three equally astonishing alternatives, 

\begin{enumerate}
\item The mass-to-light ratio of stars in spirals conspires such that, although the light is very much centrally concentrated, the mass in stars is not. But stellar populations do not seem to vary significantly between the centre and the outskirts. However there may be unseen gas for example in the outskirts of the Milky Way providing the required $\rho(R)\propto 1/R^2$ density.
\item The Milky Way contains invisible matter, which does not emit, nor absorb light.
\item Gravity does not behave as $1/R^2$ on galactic scales. If this were true, then our reasoning above is simply not valid.
The theory of {\bf Mo}dified {\bf N}ewtonian {\bf D}ynamics (MOND) is able to provide very good fits to
measured rotation curves with a small modification of gravity that
cannot be probed in other regimes.
\end{enumerate}

The currently favoured option is (2), namely that the MW, and other galaxies, contains invisible dark matter. More evidence for this later,
including the fact that this matter cannot be baryonic\footnote{Baryons are subatomic particles made out of three quarks, such as protons and neutrons. The dark matter has to be composed of something else, hence cannot be in the form of faint stars, planets, rocks, or past exam papers.} in nature.  

%\section{The Oort limit}
%Jan Oort performed another interesting measurement: count stars as
%function of their height above the disk. From this he was able to
%estimate how much mass was in the plane of the MW. By counting stars in
%the plane of the disk as well, he computed a mass-to-light ratio, which
%was also a bit on the high side, suggesting the presence of dark matter
%in the disk, even at the position of the Sun.

%Here is a simplified way how to do this. First let me prove that for a
%uniform disk, the force on a star is {\em independent} of the distance
%to the disk.

%As an intermediate step, let's compute the gravitational acceleration
%on a star, at distance $d$ from the centre of a ring of matter with
%radius $\omega$ and thickness $d\omega$, and of surface density
%$\Sigma$. From symmetry, the acceleration is toward the middle of the ring,
%and it has magnitude

%\begin{equation}
%da(d,\omega) = -2\pi\,G\,\Sigma {\cos(\alpha)\omega\,d\omega\over
%\omega^2+d^2}\,,
%\end{equation}

%where $\alpha$ is the angle at the position of the star, between the
%vertical and the ring, so $\tan(\alpha)=\omega/d$. Now the acceleration
%due to the whole disk is found by summing over all such rings, hence

%\begin{equation}
%a(d) =- 2\pi\,G\,\Sigma\int_0^\infty {\cos(\alpha)\omega\,d\omega\over
%\omega^2+d^2} = -
%2\pi\,G\,\Sigma\,\int_0^{\pi/2}\,\sin(\alpha)d\alpha=-2\pi\,G\,\Sigma\,,
%\end{equation}

%independent of $d$! So the equation of motion for the star is

%\begin{equation}
%{d^2z\over dt^2} = -2\pi\,G\,\Sigma\equiv -f\,.
%\end{equation}

%This is just the same as that for a ball thrown vertically on the
%earth's surface, so the solution is $z=z_0+\dot z_0
%t-(1/2)f\,t^2$. From this, one can easily obtain the fraction of stars
%you expect that have $z$ between $[z_1,z_1+\delta]$ and between
%$[z_2,z_2+\delta]$, which depends only on $f$. So, if you measure this from
%your data, you can determine $\Sigma$. The value Oort found is called
%the \lq Oort limit\rq\, since the MW disk needs to have at least this
%surface density.

\section{Spiral arms}
Before leaving the subject of spiral galaxies, we should address what
is in fact probably the first question you'd like to know about them:
{\em why} do they have spiral arms? So let's start by reviewing what we
know about spiral arms.

Firstly, spiral arms are really evident when we look at massive stars
and HII regions -- these really delineate the spiral structure. But
there is more: we also detect the arms in HI, and even better in
molecular gas. In fact, GMCs are probably the best tracers of the
pattern. So may be they have something to do with star formation?

However, they are also present in the \KB\, band where we are probing
{\em old} stars. So first conclusion: it's really the whole galaxy that
takes part in the spiral pattern. The {\em density
contrast}\footnote{the relative difference in density at given $r$ from
the centre, within vs outside of the arm} is not very large, probably
less than a factor of two. The contrast in {\em surface brightness} can
be quite a bit higher though, since the young stars in the arms are
massive and hence bright.\\

{\em The winding problem} Given that the disk is in differential
rotation, a spiral arm cannot be a material structure (meaning that it
is always the same stuff in the arm going round and round) -- because
of it were, we would have a {\em winding problem}. To realise why this
is so, consider the following thought experiment.

Suppose you paint all stars along a radius in the disk green, and
follow how those green stars move (see figure~(\ref{fig:winding}). If
the circular velocity is a constant, $V_c$, then the period of an orbit
is proportional to its radius, $P\propto R$. Now concentrate on two
green stars, one at distance $R$ (star 1), the other at distance $2R$
(star 2), having periods $P_1=P$ and $P_2=2\,P$. After a time $P$, star 1 will
be at the original position, whereas star 2 will be at the other side
of the galaxy. After time $2P$, both stars will be back at their
original position, and so the green line of stars in between them will
have circled the galaxy, i.e. {\em the initially straight line of green
stars now encircles the galaxy}. Applying this to stars originally in a
spiral arm, it is clear now that very quickly, the spiral arm will wind
tighter and tighter. Considering how loosely wound some spiral arms
are, this would imply they are all very recently formed, placing us at
a peculiar instant in time where spiral structure forms.

In cases where we were are able to determine the sense of rotation of
the pattern, it turned out that the spiral arms are {\em trailing},
that is, at least they are rotating in the same sense as the spiral
pattern of our green stars.

\begin{figure*}
\center\resizebox{.9\textwidth}{!}{\includegraphics{winds.pdf}}
\caption{A spiral pattern made out of stars will rapidly wind-up in a
disk undergoing differential rotation. }
\label{fig:winding}
\end{figure*}

So what's the solution? May be it's good to realise that there is quite
a variety in spiral patterns. {\rm Grand Design} spirals have two
really well defined relatively open arms. Some spirals have {\em
flocculent} arms, which are not very long (i.e. you can't trace them
for very long), but often there are many in the disk. So may be we're
looking at more than one possible origin.

The theory of {\em spiral density waves} suggest that a density wave
rotates with a constant pattern speed in the disk. You can think of
such a wave as a natural mode of oscillation of the disk, much as you
have sound waves in air, or harmonic waves in a violin string. If the
rotation period of a star is shorter than of the pattern, then the star
will occasionally overtake the wave. The gravitational force of the
matter in the wave will make the star linger a bit near the pattern,
before it leaves the arm again. An analogy often made is that of a slow
lorry on the motor way causing a pile-up of faster cars behind
it. Although most cars go fast, the speed with which {\em the pile-up}
moves is determined by the slow lorry.

Another process which we {\em know} to occur, is the triggering of
spiral arms by {\em galaxy encounters}. When two galaxies come close
together, they will exert tidal forces on each other.\footnote{The
tidal force exerted by the moon causes the tides in the earth's
oceans.} This effect can be convincingly demonstrated with numerical
simulations. And so, although these arms may quickly disappear when
left on their own (winding problem), the satellite galaxy may
continually re-excite the spiral arms. Another hint comes from the fact
that in barred galaxies, the spiral arms emanate from the ends of the
bar (have a look at NGC~1365 if you don't believe me). It's is thought
that Grand Design spirals are either triggered by a tidal encounter, or
by the rotating bar.

Goldreich and Lynden-Bell said about flocculent spirals: \lq a swirling
hoch-potch of spiral arms is a reasonably apt description\rq\ . So here
we're probably faced by some kind of gas-dynamical instability in the
disk.

So the conclusion about spiral arms is that there may be several
processes that give rise to spiral arms, and depending on the
particular spiral galaxy, one of these may dominate. So tidal
encounters are probably responsible for the beautiful Grand Design
spiral arms. The more messy pattern of flocculent spirals is probably
due to a gas-dynamical instability in the disk. And once excited, a
spiral pattern may live for a long time due to the spiral-density
wave.


\pagebreak
\section{Summary}
After having studied this lecture, you should be able to

\begin{itemize}
\item Derive the rotation curve for a Keplerian disk
\item Derive the equations for the radial and tangential velocity of stars
on circular orbits in a disk in differential rotation, and derive
expressions for Oort's constants.
\item Compute Oort's constants $A$ and $B$ for a Keplerian
disk. Explain how $A$ and $B$ are measured in the MW.
\item Explain how the 21-cm emission line can be used to estimate the
rotation curve of the MW.
\item Explain why both Oort's constants, and the rotation curve
measured from 21-cm emission, suggest the presence of dark matter in
the outer parts of the MW.
\item Explain how Oort derived a limit on the amount of dark matter in
the MW disk from the vertical motion of stars.
\item Describe why spiral arms cannot all be material structures by
explaining the winding problem.
\item Discuss solutions to the winding problem.
\item Explain the density wave theory of spiral arms.

\end{itemize}

\chapter{The Dark Halo}

Measurements of the rotation curve from HI 21-cm emission, analysis of
the motions of stars in the solar neighbourhood with Oort's constants,
and the Oort limit, all suggest the presence of a large amount of
invisible \lq dark matter\rq\ in the MW. Given such a startling
conclusion, it may be a good idea to look for other evidence for dark
matter in galaxy haloes.

\section{High velocity stars}
A number of high-velocity stars near the Sun have measured
velocities\footnote{The quoted velocity is wrt to the centre of mass
velocity of the MW. Do not confuse these with Oort's high velocity
stars, which are typically low mass, low metallicity stars in the {\em
Galactic Halo}. The velocities of Oort's stars are of order
200\kms. The present high velocity stars are typically $A$-type stars,
presumably born in the disk, that have acquired their high velocity
following a super nova explosion.}  up to $v_\star\approx
500\kms$. Their existence provides us with a probe of the galaxy's
mass, if we assume that these stars are still bound to the MW: it means
that {\em the escape speed\footnote{Stars travelling at the local
escape speed are just able to leave the potential well and escape to
infinite.} at the solar circle, $v_e>v_\star$}.

\subsection{Point mass model}
If we assume there is no dark halo, then it is easy to find a relation
between the circular velocity $V_c$ and the escape speed $v_e$. If we
assume all mass $M$ of the MW is interior to the solar radius $R_\odot$
\footnote{$R_\odot$ is the radius of the circular orbit around the MW,
at the position of the Sun -- the solar galacto-centric radius.}, then
$V_c^2=GM/R_\odot$, and the potential $\Phi=-GM/R_\odot=-V_c^2$. The
velocity of a star just able to escape from the gravitational well is
the {\em escape speed}, $v_e$. So since that star has zero total
energy,

\begin{equation}
0=E={1\over 2}v_e^2+\Phi={1\over 2} v_e^2-V_c^2\,.
\label{eq:ve}
\end{equation}

So in this case, $v_e=2^{1/2}\,V_c\approx 311\kms$ (Using
$V_c=220\kms$.) So for a point mass model of the MW, these high
velocity stars cannot be bound to the MW -- suggesting once more the
presence of more mass in the outskirts of the MW than expected, given
the rapid decrease in the distribution of light.

\subsection{Parameters of dark halo}
Given the failure of the point mass MW model, let's assume
there to be a dark halo. In addition, let us assume the MW to be
spherical, with a constant circular speed $V_c$ out to some radius
$R_\star$. Since $V_c^2=GM/R={\rm constant}$, we have that

\begin{eqnarray}
M(R) &=& {V_c^2\,R\over G}       \,\,\,\,\,\,\,\,\,\hbox{\rm when $R<R_\star$}\nonumber\\
     &=& {V_c^2\,R_\star\over G} \,\,\,\,\,\,\,\,\,\hbox{\rm when $R\ge R_\star$}\,.
\end{eqnarray}

Now the equation for the potential $\Phi$ is

\begin{equation}
{d\Phi\over dR} = {GM\over R^2} = {V_c^2\over R}\,,
\end{equation}

for $R\le R_\star$. Integration gives $\Phi={\rm
constant}-V_c^2\ln(R_\star/R)$. We can determine the value of the
constant at $R=R_\star$, since then $\Phi=-G\,M/R_\star=-V_c^2$. Hence

\begin{equation}
\Phi(R) = -V_c^2\,\left[1+\ln(R_\star/R)\right]\,.
\end{equation}

Using Eq.(\ref{eq:ve}) for the escape speed, we obtain

\begin{equation}
v_e^2 = 2\,V_c^2\left[1+\ln(R_\star/R)\right]\,.
\end{equation}

For the Sun, $R\approx 8.5\kpc$, $V_c=220\kms$, $v_e\ge 500\kms$,
implying $R_\star\ge 41\kpc$, hence

\begin{equation}
M(R=R\star)\ge 4.6\times 10^{11}\msun\,.
\label{eq:mwmass1}
\end{equation}

Since the total luminosity of the MW is of order $1.4\times
10^{10}\lsun$ (in the \V\ -band), it means\footnote{Remember: we've
only derived a lower-limit to the MW mass, hence a lower-limit to the
mass-to-light ratio.} that the mass-to-light ratio is at least
30. Given that the mass-to-light ratio of the MW {\em stars} is around
3 or so, our reasoning again suggests that a dominant fraction of the
MW mass is not in stars.

Another estimate of the MW mass comes from the motion of neighbouring
galaxies in the {\em Local Group}.

\section{The Local Group}
\label{sect:local_group}
The MW is located in a rather average part of the Universe, away from
any dense concentrations of galaxies. The \lq Local Group\rq\ consist
of the MW, Andromeda (M31), and about 40 small, irregular galaxies,
gravitationally bound to each other. 

\subsection{Galaxy population}
The MW has a couple of small galaxies gravitationally bound to it, for
example the Large and Small Magellanic Clouds. The Magellanic stream is
a stream of gas which originates from the Large Magellanic Cloud, which
can be traced over a significant fraction of the whole sky. It may
result from the tidal forces exerted by the MW.

The Sagittarius Dwarf galaxy is the neighbouring galaxy closest to the
MW. It was discovered as recently as 1994, having eluded detection
since it happens to be at the other side of the MW bulge and therefore
is hiding behind a large amount of foreground stars. Sagittarius has
ventured so close to the MW that the tidal forces will probably tear it
apart in the near future. Its discovery was actually hampered by its
close proximity, since the large area on the sky ($\sim 8^o\times 2^o$)
made it difficult to pick out against the bright MW foreground. Even
away from the galactic plane (where dust obscuration limits our view),
new members of the LG continue to be identified. The study of these
small galaxies is exciting, since they allow us to constrain models for
galaxy formation and evolution.

The Andromeda galaxy M31 is about twice as bright as the MW, and is
the brightest member of the Local Group (LG). It is very similar to the
MW, and so also has its own set of small galaxies orbiting around
it. These galaxies, together with another 20 small galaxies, form the
LG. M31 and the MW are by far the most massive galaxies in the LG.

The LG is almost certainly gravitationally bound to other nearby
groups, and so does not really have a well defined edge. Galaxies in
the outskirts of the LG may in fact belong to other groups. 

The distribution of galaxies in the LG is rather untidy and lacks any
obvious symmetry. Most of the galaxies are found either close to the
MW, or close to M31. There is a third, smaller condensation of
galaxies, hovering around NGC~3109.

The motion M31 can be used to estimate the mass of the Local Group,
using the Local Group {\em timing argument}.

\subsection{Local Group timing argument}
The dynamics of M31 and the MW can be used to estimate the total mass
in the LG. From the Doppler shifts of spectral lines, one can measure
the radial velocity of M31 with respect to the MW\footnote{What one
measures is the radial velocity wrt to the {\em Sun}. Since the Sun is
on a (nearly) circular orbit around the MW, one needs to correct the
measured heliocentric velocity to obtain the radial velocity of
Andromeda wrt the MW.},

\begin{equation}
v=-118\kms\,.
\end{equation}

The negative sign means that Andromeda is moving {\em toward} the
MW. This may be surprising, given that most galaxies are moving apart
with the general Hubble flow. The fact that Andromeda is moving toward
the MW is presumably because their mutual gravitational attraction has
halted, and eventually reversed their initial velocities. Kahn and
Woltjer pointed out in 1952 that this leads to an estimate of the
masses involved.

Since M31 and the MW are by far the most luminous members of the LG we
can neglect in the first instance the others, and treat the two
galaxies as an isolated system of two point masses. Since M31 is about
twice as bright as the MW, and given that they are so similar, it is
presumably also about twice as massive. If we further assume the orbit
to be radial, then Newton's law gives for the equation of motion

\begin{equation}
{d^2r\over dt^2} = -{GM_{\rm total}\over r^2}\,,
\end{equation}

where $M_{\rm total}$ is the sum of the two masses. Initially, at $t=0$, we can
take $r=0$ (since the galaxies were close together at the Big Bang).

The solution can be written in the well known parametric form

\begin{eqnarray}
r &=& {R_{\rm max}\over 2} (1-\cos\theta)\nonumber\\
t &=& \left({R^3_{\rm max}\over 8\,G\,M_{\rm total}}\right)^{1/2}\,
(\theta-\sin\theta)\,.
\end{eqnarray}

The distance $r$ increases from $0$ (for $\theta=0$) to some maximum
value $R_{\rm max}$ (for $\theta=\pi$), and then decreases again. The
relative velocity

\begin{equation}
v = {dr\over dt} = {dr\over d\theta}/{dt\over d\theta} =
\left({2\,G\,M_{\rm total}\over R_{\rm max}}\right)^{1/2} \left({\sin\theta\over
1-\cos\theta}\right)\,.
\end{equation}

The last three equations can be combined to eliminate $R_{\rm max}$, $G$
and $M_{\rm total}$, to give

\begin{equation}
{v\,t\over r} = {\sin\theta\,(\theta-\sin\theta)\over
(1-\cos\theta)^2}\,.
\label{eq:lgt}
\end{equation}

$v$ can be measured from Doppler shifts, and $r\approx 710\kpc$ from
Cepheid variables. For $t$ we can take the age of the
Universe. Current estimates of $t$ are quite accurate\footnote{From
properties of the micro-wave background radiation.}, but even using
ages of the oldest MW stars, $t\sim 15\Gyr$ gives an interesting
result. Using these numbers, we can solve the previous equation
(numerically) to find $\theta=4.32$ radians, {\em assuming M31 is on
its first approach to the MW\footnote{Equation~(\ref{eq:lgt}) has no
unique solution for $\theta$, since it describes motion in a periodic
orbit. On its first approach, $\theta$ should be the {\em smallest}
solution to the equation.}.}

Substituting yields $M_{\rm total}\approx 3.66\times 10^{12}\msun$, and
hence for the MW mass, $M\approx M_{\rm total}/3$
\begin{equation}
M\approx 1.2\times 10^{12}\msun\,,
\end{equation}
comfortably higher than our lower limit Eq.(\ref{eq:mwmass1}).

Since the luminosity of the MW (in the \V\ band) is $1.4\times
10^{10}\lsun$, the corresponding mass-to-light ratio for the MW is
around $\Gamma=100$. Furthermore, the estimate of $M$ is increased if
the orbit is not radial, or M31 and the MW have already had one (or
more) pericenter passages since the Big Bang.

If all stars in the MW and M31 were solar mass stars, we would expect
$\Gamma=1$. Now even in the solar neighbourhood, most stars are less
massive than the sun, and so the mass-to-light ratio for the {\em
stars} is about 3 or so\footnote{Recall that the luminosity of stars
scales with their mass quite steeply, $L\propto M^\alpha$, with
$\alpha\approx 5$, and hence $M/L\propto M^{1-\alpha}$. See your notes
on stars, p.~34. }. So the very large mass inferred from the LG
dynamics strongly corroborates the evidence from rotation curves and
Oort's constants, that most of the mass in the MW (and presumably also
in M31) is dark.

From these numbers, we can also estimate the extent $R_\star$ of such a putative
dark halo. If the the circular velocity $V_c=220\kms$ out to $R_\star$, then
from $V_c^2=GM/R_\star$ we find

\begin{equation}
R_\star={GM\over V_c^2} \approx {G\,10^{12}\msun\over (220\kms)^2} \approx
100\kpc\,.
\end{equation}

If, as is more likely, the rotation speed eventually drops below
220\kms, then $R$ is even bigger. Hence the extent of the dark matter
halo around the MW and M31 is truly enormous. Recall that the size of
the stellar disk is of order 20\kpc\ or so, and the distance to M31
$\sim 700\kpc$. So the dark matter haloes of the MW and M31 may almost overlap.

\pagebreak
\section{Summary}
After having studied this lecture, you should be able to

\begin{itemize}
\item Show that in a point mass model of the MW, the high velocity
stars are not bound.
\item Estimate the parameters of a dark halo, assuming the high
velocity stars are bound to the MW.
\item Describe the properties of the Local Group in terms of the
galactic content.
\item Estimate the mass and extent of the dark halo of the MW from the
Local Group timing argument.
\end{itemize}

\chapter{Elliptical galaxies. I }
Elliptical galaxies are spheroidal stellar systems with smooth
luminosity profiles which contain old, metal rich stars. Unlike spirals, they have a large range in sizes, and rotation is generally unimportant.  Their interstellar medium is very different from that of spirals, consisting of very tenuous,  hot, X-ray emitting gas. Observations of this gas, and stellar dynamics, both suggest the presence of significant amounts of dark matter.

\section{Luminosity profile}
\label{sect:ellipticals}
Elliptical galaxies have smooth surface brightness (SB) profiles, which in projection are ellipsoidal in shape (Fig.\ref{fig:2D}). Note that the isophotes near the centre are well defined, becoming more noisy in the outskirts.

\begin{figure*}
\center\resizebox{.8\textwidth}{!}{\includegraphics{ellipticals_2D.pdf}}
\caption{(Taken from astro-ph/0206097) Lines of constant surface
brightness (isophotes) in the \KB-band for four elliptical
galaxies. In successive isophotes, the surface brightness increases by
0.25 magnitudes.}
\label{fig:2D}
\end{figure*}

Average intensities as function of radius for these galaxies are shown in Fig.~(\ref{fig:1D}). The drawn
line which fits the data well, is the de Vaucouleurs or \lq$r^{1/4}$\rq\ fit introduced previously in Eq.~(\ref{eq:r4}) for the MW's bulge:

\begin{equation}
I(r)=I_e \exp[-7.67(r/r_e)^{1/4}-1]\,.
\end{equation}

Here, $I(r)$ is the intensity at projected distance $r$ from the centre. When surface brightness, $\mu=-2.5\log(I)+{\rm const}$,  is plotted as function of $r^{1/4}$ this profile is a straight line as in Figure~(\ref{fig:3379}).  The intensity profiles of most elliptical galaxies can be fit with just the two parameters $I_e$ and $r_e$.\\

\begin{figure*}
\center\resizebox{\textwidth}{!}{\includegraphics{ellipticals_1D.pdf}}
\vskip0.5truecm
\caption{(Taken from astro-ph/0206097) Surface brightness as function
of distance to the centre for the galaxies from Fig.~\ref{fig:2D}. The drawn line is the best $r^{1/4}$ fit.}
\label{fig:1D}
\end{figure*}

\begin{figure*}
\center\resizebox{.8\textwidth}{!}{\includegraphics{3379.pdf}}
\caption{Luminosity profile for galaxy NGC 3379 (open symbols) and
$r^{1/4}$ fit (drawn line). Taken from de Vaucoleurs \& Capaccioli,
ApJS 40, 1979.}
\label{fig:3379}
\end{figure*}

\begin{figure*}
\center\resizebox{0.8\textwidth}{!}{\includegraphics{1399.pdf}}
\caption{SB profile for NGC~1399 from Schombert (filled symbols; ApJS,
1989). The outer parts of the profile do not follow the $r^{1/4}$ fit
(bottom panel), but are nearly a straight line in a SB-$\log(r)$
(i.e. a $\log-\log$) plot (top panel), meaning the profile is close to
a power-law.}
\label{fig:1399}
\end{figure*}

\begin{figure*}
\center\resizebox{1.2\textwidth}{!}{\includegraphics{1399_GCS_crop.pdf}}
\caption{Image of NGC~1399 with smooth, best fit $r^{1/4}$ profile
subtracted (Bridges et al., AJ 101, 469 (1991). The centre of NGC~1399
is at the left, the object toward the top right is another galaxy in
the galaxy cluster. The other extended objects in the image are other
galaxies. Clearly seen are hundreds of high SB unresolved objects,
which are GCs in the halo of NGC~1399.}
\label{fig:1399GCs}
\end{figure*}

Some of the giant elliptical galaxies that are invariably found in the
centres of big clusters of galaxies have an extended stellar halo not
fit well by the de Vaucouleurs profile. Such galaxies are called cD
galaxies. An example is NGC~1399, the cD galaxy in the nearby Fornax cluster of galaxies
(Fig.~\ref{fig:1399}). The extent of such a halo can be truly enormous and be traced to around a \Mpc\,, making the size of NGC~1399 on the night sky about the size of the moon, even though Fornax is at a distance of $\sim 20$~Mpc.\\

Ellipticals also have many globular clusters. Figure~(\ref{fig:1399GCs}) is an
image of NGC~1399, where an $r^{1/4}$ fit to the SB-profile of the
galaxy has been subtracted. Clearly visible are 100s of high surface brightness
objects, indistinguishable on this plate from foreground stars, which
are in fact GCs in the halo of the galaxy. \\

What is the origin of the $r^{1/4}$ profile? Numerical simulations of  of collisions between two spiral galaxies produce objects with surface brightness profiles resembling the  $r^{1/4}$ profile. Since elliptical galaxies are found in dense environments where such collisions are frequent, it seems plausible that elliptical galaxies form from collisions between spirals galaxies.
However the stellar populations of ellipticals and spirals differ: ellipticals contain mostly old and metal rich stars (see below), whereas spirals contain younger and more metal poor stars. So ellipticals may still have formed from collisions, but not collisions between {\em todays} spiral galaxies.

\section{Stellar populations and ISM}
\noindent The redder colours of stars in elliptical galaxies is due to
\begin{enumerate}
\item the absence of star formation: the blue light in spirals is partly due to young and massive stars, but these massive stars are not present in ellipticals
\item the higher metallicity of stars in ellipticals. The presence of heavy elements (\lq metals\rq) in stars leads to redder colours, both because of changes in the stellar structure, and because the metals in the stellar atmosphere redden the emitted light.
\end{enumerate}

The fact that a stellar population is blue either because it is young (and the blue light is emitted by massive and hence young stars), or because it is metal poor, or both, is called the {\em age-metallicity degeneracy}: the colour of the population is not enough to tell a young but metal rich population, from an old metal poor one. Metallicity measurements of stars in elliptical galaxies have shown that both effects, absence of star formation, and high metallicities, cause the redder colours of ellipticals.\\


\noindent{\em Dust and gas} Whereas the ISM in spirals consists of dust and cold gas, these are generally absent in ellipticals. The beautiful Sombrero galaxy (Fig.~\ref{fig:sombrero}) is an exception: a recent merger with a smaller system has left a strong dust lane strung across this elliptical. Such recent mergers may be quite common: a number of ellipticals have faint rings of stars around them (see for example Fig.~\ref{fig:shellgalaxy}), probably a result of such  \lq galactic cannibalism\rq{}. 

\begin{figure*}
\center\resizebox{1.2\textwidth}{!}{\includegraphics{sombrero.pdf}}
\vskip-5truecm
\caption{Image of the \lq Sombrero\rq\ galaxy, with its striking dust lane.}

\label{fig:sombrero}
\end{figure*}

\begin{figure*}
\center\resizebox{1.2\textwidth}{!}{\includegraphics{shellgalaxy.pdf}}
\vskip-5truecm
\caption{Image of NGC~3923 taken by David Malin. Several faint shells
of stars appear as ripples in the outer parts of the galaxy, probably resulting from the merger of the central galaxy with a much smaller system. (see
http://www.ast.cam.ac.uk/AAO/images/general/ngc3923.html)}
\label{fig:shellgalaxy}
\end{figure*}

\section{X-rays}
\label{sect:xray}
X-rays are absorbed by the Earth's atmosphere hence are observed from rockets or satellites. They are focussed onto a detector by placing a set of nearly parallel plates in the beam: the X-rays ricochet off these plates onto the detector\footnote{See additional images on the web-page or http://chandra.harvard.edu/xray\_astro/.}.\\

The emission of X-rays from elliptical galaxies is due to two unrelated sources: hot gas in their interstellar medium (thermal bremsstrahlung), and binary stars. Binary stars may emit X-rays due to mass transfer (see your notes on stars). Subtracting such point sources from the X-ray image of the galaxy reveals extended X-ray emission from {\em thermal bremsstrahlung}.\\

Thermal bremsstrahlung\footnote{German for \lq braking radiation\rq.} is the process whereby high energy radiation (X-rays) is produced in a plasma of ions and electrons,  due to the electromagnetic deflection of a high-speed (thermal) hot electrons passing close to a positively charged ion. The deflection of the electron means it is accelerated, and an accelerating
charge emits electromagnetic dipole radiation: this is the radiation that we observe\footnote{Note that the detected radiation is {\bf not} blackbody radiation from the hot gas: the spectrum of thermal bremsstrahlung is not the same as that of a black body. Also other ions undergo electromagnetic deflections, but since they are much more massive than the electrons, their contribution to the emission is negligible.} . The plasma is highly ionised because it is so hot, and the electrons are not bound to any particular ion. For this reason, thermal bremsstrahlung is sometimes called \lq free-free\rq\ radiation. \\

The {\em emissivity} $\epsilon$ (the amount of energy radiated per unit volume) is

\begin{equation}
\epsilon  \propto n_e\,n_i\,T^{1/2}\,,
\end{equation}

where $n_e$ and $n_i$ are the electron and ion number densities respectively, and $T$ is the temperature of the plasma.\\

%\subsubsection{Radiated power\footnote{This section is not required reading}}
%Let's try to obtain an expression for the radiated power. The aim of
%this derivation is to show that the power emitted is $\propto
%n_e\,n_i\,T^{1/2}$, where $n_e$ and $n_i$ are the electron and ion
%densities, and $T$ is the temperature. You could have guessed the first
%bit, since we're talking about an encounter between {\em two}
%particles. So the exercise is how to get the $T^{1/2}\propto v$
%dependence.

%If the deflection angle is small, we can simple neglect it, and say
%that the electron moves at constant speed $v$ along a straight line
%with impact parameter $b$. The acceleration is then

%\begin{equation}
%\bfa(t) = {Ze^2\over m_e d^3}\bfd = {Ze^2\over
%m_e(b^2+v^2t^2)}{\bfd/d}\,,
%\end{equation}

%where $Z$ is the charge of the ion in units of the electron charge $e$,
%and $\bfd$ is the distance between electron and ion. $t=0$ is the time
%of closest approach, and so the encounter lasts from $t\rightarrow
%-\infty$ to $t\rightarrow +\infty$.

%The dipole electric field at distance $r$ from the electron is

%\begin{equation}
%|{\bf E}(t)| = {e\,a\over c^2\,r}\sin(\theta)={Z\,e^3\sin(\theta)\over
% m_e\,c^2\,r\,(b^2+v^2t^2)}\,.
%\end{equation}

%This is the electric field as function of time $t$, but we want it as a
%function of frequency $\nu$. The trick to obtain $E(\nu)$ is by Fourier
%transforming $E(t)$,

%\begin{eqnarray}
%E(\nu)&\equiv&\int_{-\infty}^{\infty} E(t)\exp(2\pi\,i\nu\,t)
%dt\nonumber\\
%&=&{Ze^3\sin(\theta)\over m_ec^2\,r}\int_{-\infty}^{\infty}
%{\exp(2\pi\,i\nu\,t)\over b^2+v^2t^2} dt \nonumber\\
%&=& {Ze^3\sin(\theta)\over m_ec^2\,r}{1\over b\,v}
%\int_{-\infty}^{\infty} {\exp(2\pi\,ix(\nu\,b/v))\over (1+x^2)}\,dx\nonumber\\
%&=& {Ze^3\sin(\theta)\over m_ec^3b^2v^2}\,{\pi\over bv}\,\exp(-2\pi\nu\,b/v)\,.
%\end{eqnarray}

%Along the way, I've introduced the variable $x=vt/b$. The integral
%$\int\exp(2\pi x\alpha)/(1+x^2)\,dx=$ some function of $\alpha$ is a
%little tricky to perform, and I don't expect you to be able to do
%it. What is important is to note that this change of variables gives
%rise to a factor $1/bv$ .

%Now we're done, because it means that the power radiated by this single
%electron per unit frequency is

%\begin{equation}
%{dW\over d\nu}\propto \int E^2(\nu)\,d\Omega\propto {1\over
%(b\,v)^2}\exp(-4\pi\nu\,b/v)\,.
%\end{equation}

%So the spectrum is independent of $\nu$ for low frequencies $\nu\ll
%v/b$, and cuts off exponentially for large $\nu$. Now the number of
%encounters with impact parameter between $b$ and $b+db$, in time $dt$
%is $n_e 2\pi b\,db vdt$, and so this gives the power as

%\begin{equation}
%{dW\over d\nu\,dt\,dV}\propto n_e\,n_i\,v^{-1}\,\exp(-4\pi\nu\,b/v)\propto
%n_e\,n_i\,T^{-1/2}\,\exp(-4\pi\nu\,b/v)\,.
%\end{equation}

%So the origin of the $1/v$ is that the energy radiated per encounter
%$\propto 1/v^2$ whereas as the rate of encounters is $\propto v$.  

%To obtain the final result, we still need to integrate over the impact
%parameter $b$, and average over all impact velocities $v$ (for example
%by assuming a Maxwell-Boltzmann distribution for $v$). The result is
%that the total power emitted is
%\begin{equation}
%{\hbox{\rm power }} \propto n_e\,n_i\,T^{1/2}\,.
%\end{equation}

%Notice that the total power is much more dependent on the density of
%the gas ($\propto $ density {\em squared}), than on its temperature
%($\propto T^{1/2}$)

Thermal bremsstrahlung produces a power-law spectrum up to some cut-off frequency, but the observed X-ray spectra of elliptical galaxies also show {\em absorption lines}, 
where electronic transitions in some highly ionised elements present in the hot plasma absorb the thermal bremsstrahlung.
This makes it possible to measure {\em abundances} of elements in the ISM.\\

It is often possible  to detect several different transitions of Iron. 
The relative strength of these transitions depends only on the temperature $T$ of the gas, and not on the abundance of Iron: this makes it possible to measure $T$ from the X-ray spectrum.

\section{Evidence for dark matter from X-rays}
It is possible to constrain the density distribution of the elliptical galaxy by assuming the hot X-ray emitting gas is
in {\em hydrostatic equilibrium}, much in the same way as one can derive the equations for stellar structure.
Gravity pulls the gas inward, whereas the pressure of the gas pushes it outward. In hydrostatic equilibrium these two forces balance. We can determine the temperature $T$ of the gas from its X-ray spectrum, its density $\rho$ from the emissivity, $\epsilon\propto n_e\,n_i\,T^{1/2}$ and hence infer its pressure, $p$, as function of radius. Hydrostatic equilibrium then allows us to measure the gravitational potential and hence the total density distribution. Assume for simplicity that the galaxy is spherical.\\

Let $M(<r)$ be the mass inside the shell of radius $r$ (consisting of
stars, gas and dark matter: $M=M_\star+M_{\rm gas}+M_{\rm dm}$), and
$dr$ its thickness. The gravitational force $F_g$ on the shell is due to this total enclosed mass
\begin{equation}
F_g(r)= {GM(<r)M_s\over r^2}\,,
\label{he:1}
\end{equation}
where $M_s =4\pi r^2\rho_{\rm gas}(r) dr$ is the mass of the
shell.\\

Let $p(r)$ be the pressure in the hot gas. The pressure force on a
surface is $p(r)$ times the surface area, $4\pi r^2$. The net outward
force is the difference between the outward and inward pressures:
\begin{equation}
F_p(r) = 4\pi r^2 (p(r)-p(r+dr)) = -4\pi r^2 {dp\over dr}\,dr\,,
\label{he:3}
\end{equation}
where the pressure was expanded in a Taylor series, assuming $dr<<r$. Expressing hydrostatic equilibrium, $F_g=F_p$, yields

\begin{equation}
{GM(<r)\over r^2} 4\pi r^2 \rho_{\rm gas}(r) dr = -4\pi r^2 {dp\over
dr} dr\,,
\end{equation}
hence
\begin{equation}
{GM(<r)\over r^2} = -{1\over \rho_{\rm gas}(r)} {dp\over dr}\,,
\label{eq:HE0}
\end{equation}
or in terms of the gravitational potential $\Phi$

\begin{equation}
\nabla\Phi=-{\nabla p\over \rho_{\rm gas}}\,.
\label{eq:HE}
\end{equation}

This should look familiar from hydrostatic equilibrium in stars!\\

The X-ray observations can be used to determine the right-hand-side: the X-ray spectrum yields $T$, and the emissivity $\rho$.
Application of Eq.~(\ref{eq:HE0}) then yields $M(<r)$.\\

The temperature $T$ is observed to remain approximately constant, in
which case we can obtain an approximate expression for the density
profile. The pressure $p$ in an ideal gas depends on $T$ and $\rho$ as 

\begin{equation}
p={\boltz\,T\over \mu\,\mp}\,\rho_{\rm gas}\,,
\end{equation}

where $\mu\,\mp$ is the mean molecular weight per particle. Substituting this into Eq.~(\ref{eq:HE0}) and assuming $T$ to be constant yields
\begin{equation}
G M(<r) = -{\boltz T\over \mu\,\mp} r^2 {d\ln\rho_{\rm gas}\over dr}\,.
\label{eq:HE3}
\end{equation}

Finally taking the derivative of both sides with respect to $r$, gives

\begin{equation}
G 4\pi r^2 \rho_{\rm tot} = -{\boltz T\over \mu\,\mp} {d\over dr} (r^2
{d\ln\rho_{\rm gas}\over dr})\,,
\label{eq:HE2}
\end{equation}

where $\rho_{\rm tot}$ is the {\em total mass density}, {\em i.e.} due to stars, gas and dark matter.\\

{\bf If} the total density and gas density were proportional, $\rho_{\rm gas}\propto \rho_{\rm total}$, we could solve Eq.~(\ref{eq:HE3}) by trying a solution of the form $\rho_{\rm total}=\rho_0\,(r_0/r)^\beta$. Substitution shows this is indeed a solution provided $\beta=2$, and yields
\begin{equation}
\rho_{\rm tot}(r) = {\boltz\,T\over 2\pi\,G\mu\mp}\,r^{-2}\,.
\end{equation}

Even if total and gas density do not trace each other it is possible to infer $\rho_{\rm total}$ from the X-ray data.
We can then compare the inferred total density $\rho_{\rm total}$ with that from stars, $\rho_\star$, and gas, $\rho_{rm gas}$ combined: this analysis demonstrates that elliptical galaxies also contain dark matter, since $\rho_{\rm total}>\rho_\star+\rho_{\rm gas}$.

\pagebreak
\section{Summary}
After having studied this lecture, you should be able to 

\begin{itemize}
\item describe the surface brightness profiles of Es in terms of the de Vaucouleurs profile.
\item explain why we think that this profile results from galaxy encounters
\item recall that Es have typically hundreds of GCs.
\item explain why dust lanes in ellipticals and shells of stars around them are
thought to be evidence for galactic cannibalism.
\item contrast the stellar population and ISM of ellipticals with those of spirals
\item describe how X-rays are detected, and explain the process with
which X-rays are produced in the hot gas in ellipticals
\item explain how X-ray observations can be used to infer the
gravitational potential and derive the equation for hydrostatic equilibrium, relating gas properties to the total density
\end{itemize}


\chapter{Elliptical galaxies. II}
Stars in a spiral disk are on nearly circular orbits. Although some of the smaller elliptical galaxies sometimes rotate as well, the
angular momentum of their stars is not sufficient to balance gravity. It is the {\bf large random motions} of stars in an elliptical which balances its gravity. This is in fact similar to hydrostatic equilibrium in a star, where it is the pressure,  $p\propto \rho\,T$, which  balances gravity.\\

The random velocities $v$ of the gas atoms generate the temperature, $T\propto v^2$.
Similarly the high velocities of stars in an elliptical generate something akin to pressure, $p\propto \rho\,v^2$, where $\rho$ is the density of stars, and $v^2$ their velocity dispersion. However there is a significant difference: the velocities of gas particles are isotropic, $\langle v_x^2\rangle=\langle v_y^2\rangle=\langle v_z^2\rangle$, whereas in general those of stars in an elliptical are not. This leads to a significant difference: stars are spherical\footnote{Except for effects due to rotation and tidal or magnetic fields.} but elliptical galaxies are not. This chapter explores why this is.\\

The Jeans's equations are the stellar dynamical analogue of the gas equations, and they can be used to relate stellar density and velocity distribution.  We will not derive these equations here, but simply make them plausible. But first we need to show that a description in terms of fluid variables makes sense for a stellar system, by introducing the concept of {\em relaxation time}.

%$\sigma_{ij}^2=\langle (v_i-\langle v_i\rangle)(v_j-\langle
%v_j\rangle)\rangle$ with the gravitational potential of the system, and
%are the stellar analogue of the equation of hydrostatic equilibrium
%$\nabla\Phi=-\nabla p/\rho$ in a star. They do not describe the
%properties of the orbits of a single star, but rather assume one can
%use averaged properties. To investigate whether such an approach makes
%sense, I first need to introduce the concept of relaxation time.

\section{Relaxation in stellar systems}
\label{sect:relax}
A star moving through a stellar system feels the gravitational pull of all other stars.
Is the motion of this star mainly determined by the average gravitational force of all other stars
combined, or is it mostly sensitive to the force due to near stars?\\

To see that this would make a huge difference, consider the case where
a single star (a test particle) moves through a smooth density
distribution with spherically symmetric density distribution $\rho(r)$,
where $r=|\bfr|$. In such a time-independent potential both the angular
momentum, ${\bf L}=m\,{\bf r}\times{\bf v}$, and energy,
$E=v^2/2+\Phi$, are conserved along the stellar orbit.\\

Now consider the case where the smooth stellar density $\rho$ is made-up of stars each with mass $m$.
When the test star comes close to another star, it will be deflected from its orbit,  in the process exchanging both energy and angular momentum\footnote{Such \lq gravitational slingshots\rq\ are exploited in interplanetary travels, for example the Voyager space craft gained enough speed to escape the solar system by passing very close to other planets.}.
Therefore in this case neither $E$ nor $L$ are conserved. But which description applies to stars in an elliptical galaxy?\\

Clearly, if the test star has many close encounters with other stars, its energy $E$ will start to differ considerably from its initial value.  {\em The relaxation time $T_E$ is a measure of how long the star
remembers its initial energy}. If $T_E$ is very long, the energy of each star in (nearly) conserved, and the fluid description applies. Conversely if $T_E$ is short, the energy of the star changes rapidly, and the second description applies: systems with short $T_E$ cannot be described with fluid equations.\\



We will see that the more stars are
there in the galaxy, the longer is the relaxation time. You might have
expected this, since for an infinite number of stars, the density is
smooth, ${\bf L}$ and $E$ are constant, and hence $T_E\rightarrow
\infty$.\\

To estimate $T_E$,  compute the change in energy $\Delta E$ of a
star due to a single encounter with another star, as function of the
impact parameter $b$ and impact velocity $v$ (the velocity at infinity,
before the encounter started), and then sum over encounters. This
derivation was first done by Chandrasekhar, the derivation here is
slightly different from BT (p.~188).\\

Assume that the change in velocity $\delta\bfv$ as the test star passes another star is small, and therefore approximate the orbit as a straight line, traversed with constant velocity $v$. The force perpendicular to the orbit is

\begin{equation}
\bfF_\perp = {Gm^2\over b^2+(vt)^2}\,\cos(\theta)\approx {Gm^2\over
b^2}\,\left[1+({vt\over b})^2\right]^{-3/2}\,,
\end{equation}

assuming equal mass stars. The approximation
$\cos(\theta)=b/r\approx b/vt$ is valid for $b$ small. Combined with Newton's law


\be m{\bf {\dot v}}_\perp = \bfF_\perp\,,  \ee

yields the net change $|\delta\bfv_\perp|$ over a single encounter:

\be
|\delta\bfv_\perp| \approx {Gm\over bv} \int_{-\infty}^{\infty}
(1+s^2)^{-3/2}\,ds\,={2Gm\over b\,v}\,.
\label{eq:dv}
\ee

Thus the change is roughly equal to the force at closest approach,
$Gm/b^2$, times the time this force acts, $\Delta t\sim b/v$. Because
of symmetry, encounters are equally likely to change the velocity in
the positive direction, as in the negative direction. Therefore the
mean change $\langle \delta\bfv_\perp\rangle=0$. We will therefore
consider the rate of change of $\delta\bfv_\perp^2$.\\

To compute the rate of change of $\delta\bfv_\perp^2$, due to many
encounters, proceed as follows. Suppose the test star moves with
velocity $v$ through a stellar system with radius $R$, and (uniform)
number density of stars, $n$.  When travelling for a time $\delta t$,
the number of encounters with impact parameter between $b$ and $b+db$,
is simply the volume of the cylindrical shell, $dV=2\pi\,b\,db\,\times
v\delta t$, times the number density of stars, $n$. Here, $v\delta t$
is the length of the cylinder, $2\pi\,b\,db$ is the surface area of the
shell. Since each of these encounters leads to a change in velocity
squared by an amount $\delta\bfv_\perp^2$, we find for the rate of
change

\begin{equation}
{d^2 \delta\bfv_\perp^2\over dt\,db} = n\,v\,2\pi\,b\,\times ({2Gm\over
b\,v})^2\,.
\end{equation}

This is the product of the {\em rate} of encounters, $n\,v\,2\pi\,b db$, with the
velocity change {\rm per} encounter. Now integrate over impact
parameter to obtain for the rate of change in velocity

\begin{equation}
{d\delta\bfv_\perp^2\over dt} = \int_0^\infty  n\,v\,2\pi\,b\,({2Gm\over
b\,v})^2\,db=2\pi\,nv\,({2Gm\over v})^2\,\int_0^\infty {db\over b}\,.
\end{equation}

Unfortunately this integral diverges, both at small impact parameters
(which are few in number, but cause large velocity changes), and at
large impact parameters (which cause small velocity changes yet are
very numerous). How to overcome this divergence?\\

A real stellar system has finite size, therefore the impact parameter should be limited to the radius 
$R$ of the system: this avoids the divergence at infinity. Our derivation so far assumed the changes in velocity of the star during a single encounter to be small, but this is not true for $b\rightarrow 0$ which causes the divergence at small $b$.
Therefore we want to replace the lower limit of integration by $b_{\rm min}$, which is that impact parameter
for which the {\em change} in velocity is comparable to the {\em initial} velocity, $b_{\rm min}=Gm/v^2$
Making these somewhat suspicious changes we obtain


\begin{equation}
{d\delta\bfv_\perp^2\over dt} \approx 2\pi\,nv\,({2Gm\over
v})^2\,\int_{b_{\rm min}}^R {db\over b}=2\pi\,nv\,({2Gm\over
v})^2\,\ln({R\over b_{\rm min}})\,.
\label{eq:TE1}
\end{equation}

When the stellar system is in {\bf virial equilibrium} its kinetic energy $K\sim N\,m\,v^2/2$ is half its the potential energy, $U\sim
G\,(N\,m)^2/R$, hence 

\begin{equation}
v^2\approx {GNm\over R}\,.
\label{Eq:VE}
\end{equation}

Combining this with the expression for $b_{\rm min}$ it is easy to show
that $R/b_{\rm min}=N$, the total number of stars.\\

So far we have computed the {\em rate} of change of the velocity. To
judge whether the change of velocity is large or small, we can multiply
the rate with the {\em crossing time} of the system, $t_{\rm cr}\equiv
R/v$. It is a measure of how long it takes to cross the stellar system,
on average. The velocity change during a single crossing of the stellar
system is 

\begin{eqnarray}
\delta\bfv_\perp^2\sim {d\delta\bfv_\perp^2\over dt}\,t_{\rm cr} &=& 2\pi\,nv\,({2Gm\over
v})^2\,\ln(N)\times {R\over v}\nonumber\\
 &=& {6\ln(N)\over N}\, v^2\,.
\end{eqnarray}

The first line combines Eq.(\ref{eq:TE1}) with our finding $R/b_{\rm
min}=N$ and the definition of the crossing time. The second step
assumes virial equilibrium, Eq.~(\ref{Eq:VE}).\\

We now define the {\em relaxation time} $T_{\rm relax}$, as the
time it takes for encounters to change the velocity of a test star by order of itself,
hence

\begin{equation}
{d\delta\bfv_\perp^2\over dt}\, T_{\rm relax}\equiv v^2\,.
\end{equation}

Combing the last two equations provides a relation between the relaxation time and the crossing time of a stellar system:

\begin{equation}
T_{\rm relax} = {N\over 6\ln(N)}\,t_{\rm cr}\,.
\end{equation}


Systems of the same mass and size, have the same crossing time $t_{\rm cr}\propto R^{3/2}/M^{1/2}$
However their relaxation time depends on the {\em number} of stars, with $T_{\rm relax}\propto N/\ln(N)$.
Put differently, the effect of encounters decreases with increasing particle numbers as $N/\ln(N)$.\\

A\footnote{A handy approximate relation is $1\km\,\s^{-1}\approx 1\pc\,\Myr^{-1}$.} typical globular cluster has $N=10^5$ stars,  radius $R=10\pc$, and velocity dispersion $v=10\kms$, and hence $t_{\rm cr}\approx 10^6$ years but $T_{\rm relax}\approx 10^9$ years.
The relaxation time of such a system is therefore much less than its age, and encounters between stars have significantly changed the stellar orbits. It would be wrong to describe globular clusters with fluid equations.\\

However, a big elliptical galaxy may have $N=10^{11}$ stars, radius $R=20\kpc$, and velocity dispersion $v=200\kms$, therefore $t_{\rm cr}\approx 10^8\yr$, and $T_{\rm relax}\sim 10^{19}\yr$. The relaxation time is {\bf much} longer than the age of the Universe ($\sim 10^{10}\yr$), and encounters between stars have had negligible effect on stellar orbits. 
Such a system is called {\em collisionless} and can be described using a set of fluid-like equations called the Jeans equations.

\section{Jeans equations}

%\subsection{A continuity equation}
%We can derive fluid-like equation for the gas of stars in an elliptical
%galaxy, using the concept of a {\em continuity equation}. A continuity
%equation expresses conservation of a quantity. This is a general
%concept, let's do a simple example first: conservation of number of
%cars on a motor way.

%Suppose you're standing on a hill above a tunnel on a motor way, and
%you count the number of cars entering the tunnel on one side and the
%number that leaves the tunnel on the other side.

%If $n$ is the number density of cars in the tunnel (in cars per km
%say), and $\Delta l$ is the length of the tunnel, then the number of
%cars in the tunnel is obviously $N=n\,\Delta l$. The change in $N$
%between two times is

%\be
%\Delta N=[n(l,t+\Delta t)-n(l,t)]\,\Delta l\approx {\partial
%n(l,t)\over\partial t}\,\Delta l\,\Delta t\,.
%\ee

%Assuming there is no other exit, this is also the change between the
%number of cars entering and leaving the tunnel,

%\be
%\Delta N=[n(l,t)\dot l(l,t)-n(l+\Delta l,t)\dot l(l+\Delta l,t)]\,\Delta
%t\approx -{\partial n(l,t)\,\dot l(l,t)\over \partial l}\,\Delta l\,\Delta
%t\,.
%\ee

%The first term $n(l,t)\dot l(l,t)\,\Delta t$ is the number of cars entering
%the tunnel in time $\Delta t$, the second term is how many are leaving
%in that time. $\dot l(l,t)$ obviously denotes the speed of the cars.

%Combining these two equation, we get the following continuity equation:

%\be
%{\partial n(l,t)\over \partial t} + {\partial n(l,t)\,\dot l(l,t)\over \partial l}
%= 0\,.
%\label{eq:ce}
%\ee

%A little bit of thought shows that, in a N-dimensional case (spaceships
%through a worm-hole?), the more general equation would be

%\be
%{\partial n(\bfl,t)\over \partial t} + {\partial n(\bfl,t)\,\dot\bfl(\bfl,t)\over \partial \bfl}
%= 0\,,
%\label{eq:ce}
%\ee

%so there is a partial derivative $\partial/\partial l_i$ for every
%Cartesian coordinate $l_i$, and $n$ is multiplied by $\dot l_i$ in the
%second term, which is the velocity of the spaceships in the
%$i$-direction.

%\subsection{Boltzmann's equation}
%We're interested in finding an equation that describes the distribution
%of stars in a galaxy. So we want to know is how many stars there are in
%a small volume $d\bfx$ around a position $\bfx$, that have velocity in
%a small interval $d\dot\bfx$ around a given velocity $\dot\bfx$. This density
%of stars, $f(\bfx,\dot\bfx,t)$, is called the {\em distribution
%function}. If we don't care about the velocity of stars, then we can
%integrate $f$ over all velocities, to obtain the density of stars,

%\be
%n(\bfx,t)=\int f(\bfx,\dot\bfx,t)\,d\dot\bfx\,.
%\ee

%Now we only have to realise that stars acts as our cars on the motor
%way, and hence $f$ should satisfy the continuity equation
%(\ref{eq:ce}). The way to do it is to substitute for the $l$
%coordinates in that equation the positions and velocities of the stars,

%\begin{eqnarray}
%\bfl     &\equiv & (\bfx,\bfv)\nonumber\\
%\dot\bfl &\equiv & (\bfv,\dot\bfv) = (\bfv,-\nabla\Phi)\,,
%\end{eqnarray}

%where I've written $-\nabla\Phi$ for the gravitational acceleration
%$\dot\bfv$ of the star.

%So this $\bfl$ vector has 6 coordinates, $l_\alpha=\bfx$ for
%$\alpha=1\rightarrow 3$, and $l_\alpha=\bfv$ for $\alpha=4\rightarrow
%6$. Now these coordinates are very special, since

%\begin{eqnarray}
%\sum_{\alpha=1}^6 {\partial\dot l_\alpha\over \partial l_\alpha} 
%&=&\sum_{i=1}^3 \left({\partial v_i\over\partial x_i}+{\partial
%\dot v_i\over\partial v_i}\right)\nonumber\\
%&=&\sum_{i=1}^3 -{\partial\over\partial v_i}\left({\partial\Phi\over
%\partial x_i}\right)\nonumber\\
%&=&0\,.
%\end{eqnarray}

%Here, $\partial v_i/\partial x_i=0$ since the positions and velocities
%of the stars are independent variables, and the second term is zero
%because the gravitational acceleration $\partial\Phi/\partial x_i$ does
%not depend on velocity.

%We can use this equation to simplify the continuity equation, and
%to obtain

%\be
%{\partial f\over\partial t}+\sum_{\alpha=1}^6\dot l_\alpha{\partial
%f\over\partial l_\alpha}=0\,.
%\ee

%Rewriting in terms of positions and velocities, we obtain the {\bf
%collisionless Boltzmann equation},

%\be
%{\partial f\over\partial t} + \sum_{i=1}^3\left(v_i{\partial
%f\over\partial x_i}-{\partial\Phi\over\partial x_i}{\partial
%f\over\partial v_i}\right)=0\,.
%\label{eq:ce1}
%\ee

%or in vector notation
%\be
%{\partial f\over\partial t}+\bfv\cdot{\partial f\over\partial \bfr}-\nabla\Phi\cdot {\partial
%f\over \partial\bfv}=0\,.
%\label{eq:ce2}
%\ee

%\subsection{Moments of the Boltzmann equation}
%Although the collisionless Boltzmann equation (CBE for short) provides
%us with a complete description of the stellar dynamics problem, it is
%rather unwieldy for practical applications. The Jeans equations, which
%give a clearer physical picture of what is going on, are {\em moments}
%of the Boltzmann equation. Here is how to obtain them.

%Suppose we integrate Eq.~(\ref{eq:ce1}) over all velocities. The first
%term will become

%\be
%\int {\partial f(\bfr,\bfv,t)\over \partial t}\,d\bfv = {\partial\over\partial t}\int
%f(\bfr,\bfv,t)\,d\bfv\equiv {\partial n(\bfr,t)\over\partial t}\,.
%\ee

%The integral of the distribution function over velocities, $ n\equiv
%\int\,f\,d\bfv$ is just the density of stars. And we're allowed to
%interchange the integral over velocities, with the derivative wrt time,
%since the range of velocities over which we integrate does not depend
%on $t$.

%Integrating the second term, gives

%\be
%\int v_i{\partial f\over\partial x_i}\,d\bfv = {\partial\over\partial
%x_i}\,
%\int v_i\,f\,d\bfv\equiv {\partial( n\bar v_i)\over\partial x_i}\,,
%\ee

%where I've introduced the mean velocity $\bar v_i$ as

%\be
% n\,\bar v_i=\int v_i\,f\,d\bfv\,.
%\ee

%Now the last term is zero, since

%\be
%\int_{-\infty}^\infty {\partial f\over\partial v_i}
%dv_i=f|_{v_i=-\infty}^{v_i=+\infty}\,,
%\ee

%since there are no stars with infinite velocity.

%And so the first moment of the CBE is a continuity equation for the
%density,

%\be
%{\partial n\over\partial t}+{\partial ( n\bar v_i)\over\partial x_i} = 0\,.
%\ee

%Now the second moment is what we're after. The trick is to {\em first} multiply the
%CBE with $v_j$, and {\em then} integrate over all velocities. The
%calculation proceeds exactly as for the density, and the result is
%after a bit of algebra:

%\be
% n\,{\partial\bar v_j\over\partial t} +  n\,\bar v_i\,{\partial \bar
%v_j\over\partial x_i} = - n\,{\partial\Phi\over\partial x_j} - {\partial
%n\,\sigma^2_{ij}\over \partial x_i}\,.
%\ee

%where

%\be
%\sigma^2_{ij}\equiv  \overline{v_i v_j}-\bar v_i\bar v_j\,.
%\ee

%This was a lot of mathematics to come to a simple equation: suppose the
%system is in a steady state, $\partial/\partial t$=0, and there are no
%streaming motions, so that $\bar v_i=0$. Then the previous equation
%simplifies to

%\be
%{\partial\Phi\over\partial x_j} = -{1\over n}{\partial
% n\sigma^2_{ij}\over \partial x_i}\,.
%\ee

The Jeans equations are the stellar dynamical analogues of fluid equations. Here we only consider the stellar dynamical analogue of the equation for hydrostatic equilibrium in an ideal gas

\begin{eqnarray}
\nabla\phi &=& -{1\over \rho}\nabla p\,\nonumber\\
p    &=& n\,\boltz T\,,
\end{eqnarray}

where $\phi$ is the gravitational potential, $\rho$ the density, $T$ the temperature, $n=\rho/m$ the particle density, and $p$ the pressure. Temperature and velocity dispersion of the particles are related by\footnote{These relations are derive from the kinetic gas theory.}

\begin{equation}
{1\over 2} mv^2={3\over 2}\,\boltz T\,,
\end{equation}

hence $p=\rho v^2/3$. Therefore the equation for hydrostatic equilibrium in an ideal gas can be written as

\begin{equation}
\nabla\phi = -{1\over\rho}\,\nabla\,{\rho v^2\over 3}\,,
\label{eq:jeans}
\end{equation}

or in Cartesian Coordinates 

\begin{eqnarray}
{\partial\phi\over\partial x}  &=&-{1\over\rho}{\partial\over\partial x}\,\rho v^2_x\nonumber\\
{\partial\phi\over\partial y}  &=&-{1\over\rho}{\partial\over\partial y}\,\rho v^2_y\nonumber\\
{\partial\phi\over\partial z}  &=&-{1\over\rho}{\partial\over\partial z}\,\rho v^2_z\,.
\label{eq:J1}
\end{eqnarray}



Collisions between atoms in a gas guarantee that velocities are isotropic, $\langle v_x^2\rangle=\langle v_y^2\rangle=\langle v_z^2\rangle$, therefore the right-hand-sides of these equations are the same, hence also ${\partial\phi\over\partial x}={\partial\phi\over\partial y}={\partial\phi\over\partial z}$ and the density distribution for the {\em star} is spherical.\\

However, encounters between stars in an elliptical galaxy are so infrequent that this is not true, $\langle v_x^2\rangle\ne\langle v_y^2\rangle\ne\langle v_z^2\rangle$, and the right-hand-sides are {\bf not} equal, and the density distribution can be ellipsoidal: it need not be spherical. It is the anisotropy of velocities of the stars in an elliptical galaxy that allow it to be ellipsoidal in shape\footnote{Careful readers may have noticed that the semi-major axes of the ellipsoid  are along our chosen Cartesian axes. In general this is of course not the case, and the correct equations are not Eq.~(\ref{eq:J1}). Fortunately it is always possible to rotate the system of axes so that they do turn into this set.}.\\

In conclusion: the behaviour of {\rm collisionless} stellar systems is
similar to that of self-gravitating gas spheres (stars). The role of
temperature is taken over by that of the stellar velocity dispersion.
It is the high velocity dispersion of the stars in an elliptical (and
also in the bulge of spirals), which balances gravity. Since the required velocities are so high, such systems are
called \lq hot stellar systems\rq. Recall that in disk galaxies, it is
the ordered motion of the stars -- i.e. the rotation of the disk --
which supports the system against gravity. Such systems are dynamically
cold, i.e. the stellar velocity dispersion is small compared to
the streaming (circular) motions\footnote{Compare for example the large random velocities of shoppers in a shopping mall  
during the sales (hot system) versus the streaming motion of commuters leaving Durham station (cold system). The speeds of people may be the same in both cases, but the commuters all move in the {\em same} direction, hence their relative velocities are much smaller than those of the shoppers.}\\


%For disks, the \lq velocity dispersion\rq\
%refers to the small velocities that stars have with respect to their
%\lq local standard of rest\rq\ (10s of \kms), versus the rotational
%velocity of the disk, 200\kms.

In a gas the particle velocities are isotropic and hence stars are spherical.  In
contrast in collisionless stellar systems velocities are not isotropic and the system is elliptical.
\pagebreak
\section{Summary}
After having studied this lecture, you should be able to 

\begin{itemize}
\item interpret the equation for the stellar relaxation time, and discuss its relevance for being able to describe a stellar system as collisionless (a galaxy) or collisional (a globular cluster)
\item explain why we can describe the dynamics of elliptical galaxies
in terms of fluid equations.
\item explain the difference in dynamics between an elliptical (hot
stellar system) and a spiral (cold, rotating stellar system)
\item explain why the ellipticity of stellar systems is related to the underlying orbits of the stars.
\end{itemize}


\chapter{Groups and clusters of galaxies}

Galaxies are not distributed homogeneously in the Universe. Galaxies such as the Milky Way huddle together in small groups (the Local Group), voids are very large regions of space where galaxies are very rare, and {\em clusters} of galaxies are regions where the galaxy density is very high.
Galaxy groups and clusters themselves are clustered in super
clusters (i.e. clusters of clusters). But there are no clusters of super clusters: on sufficiently large scales the Universe is not a
fractal but becomes homogeneous.

\section{Introduction}
The Milky Way, Andromeda, and some 40 small irregular galaxies within
2\Mpc\ or so from the MW, are part of a gravitationally bound system,
called the Local Group (see Sect. \ref{sect:local_group}). Most spiral galaxies are found in such loose groups.\\

{\bf Galaxy clusters} are gravitationally bound systems of 10s-100s of elliptical galaxies and 1000s of smaller dwarf galaxies. Fornax and Virgo are two clusters of galaxies within $\sim 20\,\Mpc\,$ of the Milky Way.
The galaxy M~87, which has a super massive black hole (see Sect 11) with mass $\sim 10^9\,\msun$ lurking in its centre, is the central object in the nearby  {\em Virgo} cluster of galaxies. The Local Group, including the Milky Way, is falling toward Virgo. The large cD galaxy NGC~1399 (see Sect.\ref{sect:ellipticals}) is at the centre of the {\em Fornax} cluster of galaxies. The {\em Coma} cluster of galaxies, at a distance of 90\Mpc, has a mass as large as $10^{15}\msun$. Clusters such as Coma contain the most massive galaxies known.  \\

George Abell and collaborators eye balled photographic plates in the 1950s to identify
galaxy clusters as over dense concentrations of galaxies.  He discovered some 4500 clusters, ranking them
from poor to rich, according to the number of galaxies they had. Distant clusters are still identified with the number Abelll allocated.\\


Clusters of galaxies are the most massive virialised structures in the
Universe. Their crossing time\footnote{If $v$ is the typical velocity
of a galaxy in a cluster of galaxies, and $R$ is the radius of the
cluster, then the crossing time $T_{\rm cr}=R/v$, i.e. it is the
typical time a galaxy takes to cross the cluster once.} is sufficiently
small that each galaxy has had the opportunity to cross the system
several times, and hence these systems are thought to be in dynamical equilibrium.\\

\subsection{Evidence for dark matter in clusters from galaxy motions}
The Swiss astronomer Fritz Zwicky measured Doppler velocities of galaxies to infer the dynamical mass of galaxy clusters in the 1930s. Inferring the mass in stars from the observed luminosity, he decided that clusters must contain dark matter: this was the first time that the need for dark matter arose. Zwicky's mass measurement was based on the following argument.\\

Assume all cluster galaxies have the same mass
$m$, and that the cluster contains $N$  galaxies.
The kinetic energy of the system  is

\begin{equation}
K={1\over 2}\sum m\,v^2={1\over 2}\,M\,\sigma^2\,,
\end{equation}

where $M=N\,m$ is the total mass of the cluster galaxies, and the
velocity dispersion $\sigma^2$ is defined as

\begin{equation}
\sigma^2 = {\sum m\,v^2\over M}\,.
\end{equation}

Note that these velocities are measured with respect to the {\em mean} velocity of the cluster. The potential energy in the system is of order

\begin{equation}
U=-{G\,M^2\over R}\,,
\end{equation}

where $R$ is a measure of the size of the system. When the system is in
{\em virial equilibrium}, $2K=|U|$ hence $M$ can be determined from measuring $\sigma$ and $R$ from

\begin{equation}
\opt{lecturer}{
M={R\,\sigma^2\over G}\,.
}
\opt{student}{
M=\makebox[8truecm]{\hrulefill}\,.
}
\label{eq:cluster1}
\end{equation}

This allowed Zwicky to estimate the dynamical mass, $M$, of the cluster, from the observed velocities and size of the system.\\

Note that the velocity inferred from the Doppler shift, $\lambda=\lambda_0\,(1+v_r/c)$, is only the {\em radial component} of the velocity, but the kinetic energy employes the {\em total} velocity, $v^2=v_r^2+v_t^2$, the sum of  radial and tangential components. Since we cannot measure $v_t$, Zwicky assumed\footnote{The extra factor  2 arises because the tangential velocity has two Cartesian components. To see this, choose a Cartesian system of axes with $z$ along the radial direction, then $v_r^2=v_z^2$, whereas the tangential velocity $v_t^2=v_x^2+v_y^2$.
Assuming isotropy, $v_x^2=v_y^2=v_z^2$, hence $v_t^2=2v_r^2$.}
that $v_t^2=2v_r^2$, since we (the observers) are presumably not in a special position.\\

Zwicky estimated the masses of the galaxies as inferred from their luminosities, $M_L=\Gamma\,L$, 
where $\Gamma_L\sim 3\,\msun/\lsun$ is the stellar mass-to-light ratio, 
and compared the result with the dynamical mass, $M$. He found that, not only was the dynamical mass much larger, $M\gg M_L$, 
the velocities of the cluster galaxies was so high that, given the
amount of mass inferred to be in the galaxies, the cluster was not even
a gravitationally bound system: $K+U>0$. Left on its own, it would cease to be a
high density region of galaxies on a short time scale because the
galaxies would fly apart very rapidly. So either clusters were just
chance superpositions of galaxies, or there was much more mass in them
than was inferred from just the galaxies. Zwicky concluded that most of
the mass in clusters must be invisible, hence introducing {\em dark matter}.

\section{Evidence for dark matter from X-rays observations}
The hot gas\footnote{Note that the emission is extended, and not just due to the elliptical galaxies themselves: most of the hot gas is between the galaxies, not associated with any particular galaxy.} in clusters emits X-rays due to {\em thermal bremsstrahlung}, which can be used to determine their mass assuming the hot gas is in hydrostatic equilibrium. We did the same for elliptical galaxies in Section~\ref{sect:xray}.
The (total) mass of the cluster in a sphere of radius $r$, relates to gas density,  $\rho_{\rm gas}$,  and pressure, $p$, as (see Eq.~\ref{eq:HE0})

\begin{equation}
{GM(<r)\over r^2} = -{1\over \rho_{\rm gas}(r)} {dp\over dr}\,.
\end{equation}

As for ellipticals, the X-ray data can be used to measure the right hand side, allowing a determination of the encloded mass, $M(<r)$. Comparing this to the mass in stars and gas, $M_\star=\Gamma\,L$, and $M_{\rm gas}$ inferred from the X-ray emissivity, yields $M\gg M_\star+M_{\rm gas}$: X-rays strongly indicate the presence of dark matter in clusters.\\

What is the origin of the gas, and why is it so hot? A high-mass cluster will suck gas in from its surroundings due to its large gravitational pull. The accreting gas slams into the gas already there, and the rapid compression of the gas converts kinetic energy into thermal energy in an  {\em an accretion shock}. \\

Assume a parcel of gas starts at infinity has velocity $v=0$, and is accreted by a cluster with mass $M$. It slams into the gas already present at distance $R$ (the radius of the cluster). Expressing energy conservation yields

\begin{equation}
E=0={1\over 2}v^2 - {G\,M\over R}\,,
\end{equation}

since the energy of the parcel at infinity is zero. When the gas converts its kinetic energy $(1/2)\,v^2$, into thermal energy, it gets heated to temperature $T$ given by

\begin{equation}
{3\over 2}{\boltz T\over \mu\,m_p}={1\over 2}\,v^2\,.
\end{equation}

Here, $\mu$ is the mean molecular weight of the gas. These equations relate the cluster's mass and radius to the {\bf virial temperature} $T$ of the gas: 

\begin{equation}
\opt{lecturer}
{
{G\,M\over R} = {3\over 2}\,{\boltz T\over \mu\mp}\,.
}
\opt{student}{
{G\,M\over R} = \makebox[8truecm]{\hrulefill}\,.
}
\end{equation}

Clusters have such high values of $T$ that all the hydrogen atoms are fully ionised, hence\footnote{A more accurate expression for $\mu$ takes into account the presence of Helium and other elements, giving $\mu\approx 0.6$.} $\mu=1/2$.

\section{Metallicity of the X-ray emitting gas.}
The X-ray spectra of  clusters consists of a power-law component due to thermal bremsstrahlung,
with additional emission lines mostly from inner-shell transitions of the Fe ion, see
Figure~\ref{fig:xrayspec}.  Recall that the ratio of emission lines can be used to determine $T$, whereas the emissivity $\epsilon\propto \rho_{\rm gas}^2\,T$ then determine $\rho_{\rm gas}$. The first surprise is that most of the baryonic mass in the cluster is in the X-ray gas: $M_\star\ll M_{\rm gas}$. Although we identified galaxy clusters as regions
with a high density of galaxies, most of the baryons in the cluster have not actually collapsed to form stars.\\

Once $T$ and $\rho_{\rm gas}$ are known, the strength of the lines allows one to determine the {\em amount} of Fe in this intra-cluster medium. Comparison with the total amount of gas gives a mean metallicity $M_{\rm Fe}/M_{\rm gas}\approx {1/3}$ of the solar value\footnote{Recall that the Sun contains a mass fraction of 0.02 of metals.}.\\

\begin{figure*}
\center\resizebox{1.0\textwidth}{!}{\includegraphics{isothermal_plasma.pdf}}
\caption{Theoretical emission spectrum of a hot plasma. Emission lines due to a range of elements (indicated) appear on top of a smooth continuum caused by thermal bremsstrahlung.}
\label{fig:xrayspec}
\end{figure*}

The high metallicity of the cluster {\em gas}, and the enormous amount of metals outside of galaxies, is an even bigger surprise.  Note that metal abundance is higher than that of {\em stars} in most  globular clusters and dwarf galaxies. Presumably these metals were produced in stars,  and subsequently ejected from their parent galaxy into the intra-cluster gas.  This demonstrates that galaxies are far from closed boxes: a large fraction of the products of stellar evolution are blown  out of the galaxy. Current models suggest that this is due to the combined effect of many super nova explosions.\\

\section{The dark matter density of the Universe}
\begin{figure*}
\center\resizebox{0.9\textwidth}{!}{\includegraphics{big-bang.pdf}}
\caption{The production of various elements during Big Bang
nucleosynthesis as a function of the baryon density. ({\em Nature} {\bf 415}, p. 27, 2002) }
\label{fig:bbn}
\end{figure*}

\label{sect:dmU}
Clusters can be used to determine the dark matter density of the Universe as a whole.
Assume the Universe starts-out smooth, with a (nearly) constant ratio
$\omega$ of dark matter to baryons:

\begin{equation}
\omega = {\rho_{\hbox{\rm dm}}\over \rho_{\hbox{\rm b}}}\,.
\end{equation}

The forming cluster grows through accretion of dark matter and baryons, and so the density of
both components increases. As the potential well of the cluster deepens, it is probably a good approximation to assume that eventually neither dark matter, nor baryons, can ever escape from the cluster, meaning the ratio of dark matter to baryons, {\em in the cluster as a whole}, $M_{\rm dm}/M_{\rm b}$, equals the primordial value:


\begin{equation}
\omega\approx {M_{\hbox {\rm dm}}\over M_{\hbox{\rm b}}} \,.
\end{equation}

Therefore we can determine $\omega$ by measuring the dark matter mass of a cluster, $M_{\rm dm}=M-M_{\rm b}$, and the baryonic mass, $M_{\rm b}$. Recall that we could determine $M$ from either galaxy motions (Eq.~\ref{eq:cluster1}) or X-ray observations, with $M_{\rm b}=M_\star+M_{\rm gas}$ from combining the stellar mass (from the observed stellar luminosity) and the gas mass from the X-ray emissivity. Doing so for a range of clusters yields $\omega\approx 6$.\\

An estimate for the mean baryon density, $\rho_{\rm b}$, follows from the  {\em abundance} of Deuterium\footnote{Recall that the nucleus of a Deuterium atom differs from that of ordinary Hydrogen in that it has a proton and a neutron.} relative to normal Hydrogen, $\rho_{\rm D}/\rho_{\rm H}$. Deuterium is produced during nucleosynthesis in the first three minutes after the Big Bang, and destroyed during nuclear burning in stars. Therefore 
any Deuterium found\footnote{Deuterium abundance measurements use quasar spectra (see Chapter~11 for a discussion of quasars) to probe nearly primordial gas in between the galaxies.} is primordial and left over from Big Bang. Crucially, {\em the Deuterium abundance depends on the baryon density} (Figure~\ref{fig:bbn}).
The inferred mean baryon density is $\rho_{\rm b} \approx 2.2\times 10^{-7}$ hydrogen atoms per cm$^{-3}$.
This is approximately 30 {\rm orders of magnitude} lower than the density of the paper that these notes are printed on (or the density of the screen if you read them on your computer).\\

Combining the mean baryon density with $\omega=6$ yields $\rho_{\rm dm}\approx 2.2\times 10^{-30}$ g cm$^{-3}$ for the mean dark matter density of the Universe.

\section{Summary}
After having studied this lecture, you should be able to 

\begin{itemize}
\item Define what is a cluster and a group of galaxies by listing some
of their properties.
\item Explain how galaxy motions suggest the presence of dark matter in clusters.
\item Explain how X-ray observations suggest the presence of dark
matter in clusters.
\item Explain the origin of the high temperature of the cluster gas.
\item Explain how we know that most of the baryons in a cluster are in
hot gas, and not in stars.
\item Explain why the high observed metallicity of the cluster gas
suggests that a large fraction of the products of stellar evolution are blown out of galaxies
\item Explain how clusters have been used to estimate the mean dark
matter density of the Universe.
\end{itemize}


\chapter{Galaxy statistics}
%% Having studied some properties of spiral and elliptical galaxies in
%% detail, we now take a step back to examine the statistics of the
%% different types. The two degree-field galaxy redshift survey (2dF), and
%% the Sloan Digital Sky survey (SDSS), are two very recent {\em galaxy
%% redshift} surveys. In a galaxy redshift survey, one first identifies
%% galaxies in images of the sky, and then takes a spectrum for each of
%% them. Because the Universe is expanding, spectral lines of stars in a
%% distant galaxy are shifted from the laboratory toward longer
%% wavelengths: they are \lq redshifted\rq\,. There is a relation between
%% the distance to the galaxy and its redshift, and so a galaxy redshift
%% survey provides us with the 3D position of a set of galaxies, about
%% $2\times 10^5$ for the 2dF, about $10^6$ for the SDSS. So what kind of
%% general trends do they uncover? We will discuss some of these here.\\

%% The {\em distribution} of galaxies probably traces to some extent the
%% distribution of dark matter in the Universe. For example, we've seen
%% that there is evidence for a lot of dark matter in the Universe as a
%% whole (from the cluster argument), but also in individual galaxies,
%% groups and clusters. And so, since we find that galaxies are
%% distributed very inhomogeneous in groups and clusters, presumably
%% also the dark matter is distributed in an inhomogeneous fashion. How?
%% And what process determines this in the first place?\\

%% We have a good idea of how the dark matter is distributed on a
%% cosmological scale, from some general considerations, backed-up by
%% observations of the micro-wave background (i.e. the sea of photons
%% left-over from when the Universe was still hot). Given this input, we
%% can use numerical simulations to predict how dark matter is clustered
%% on large scales. The nice part of the story is that this theory is
%% highly predictive, and does predict structures in the galaxy
%% distribution very similar to what we observe: a filamentary pattern
%% delineating large regions of space devoid of galaxies, called \lq
%% voids\rq\,, with regions of high galaxy density, clusters of galaxies,
%% at the intersection of the filaments.\\
{\em Galaxies surveys} select objects on the night sky based on size, luminosity and colour to distinguish galaxies from stars or other foregrounds.
Distances are inferred to relative nearby galaxies using standard candles such as Cepheid variables, or assuming the globular cluster luminosity function to be universal, or using colours or redshifts for more distant galaxies\footnote{Recall the L1 lab on the Hubble constant}. The survey then provides the three dimensional position of each galaxy (two angles and a distance), as well as a large sample of object for which the {\em physical} (as opposed to apparent) properties are known. The recent {\bf 2dF}  (\lq two degree redshift survey\rq; an Anglo-Australian collaboration with an extensive Durham contribution; see the movie!) collected a spectroscopic sample of 250k galaxies, the more recent and larger {\bf Sloan redshift survey} used a dedicated telescope to both image the sky and obtain spectra, and collected about a million galaxies.\\

Inferences about the galaxy population need to take into account observational {\em biases}: whether or not a galaxy makes it past the selection criteria and ends-up in the survey sample depends on external factors, such as its distance (since surveys collect galaxies with flux brighter than some minimum magnitude), its surface brightness, and its position on the sky (nearby bright stars, foreground extinction due to the Milky Way, and other effects may affect the selection). The galaxy's intrinsic properties themselves may bias the sample: for example it is much easier to measure the redshift of an emission line galaxy ({\em e.g.} a spiral with recent star formation) than of a galaxy without emission line (an old elliptical galaxy, say).
Consequently emission line galaxies may be over-represented with respect to galaxies without emission lines.\\

The visible Universe contains around $10^9$ galaxies more massive than the Milky Way, with approximately half of all stars in spirals, half in ellipticals. Galaxies are not homogeneously distributed, with groups lining-up along {\em filaments}, and rich clusters of galaxies at the intersection of filaments. The filaments themselves delineate large under dense regions (nearly) devoid of galaxies called {\em voids}.  Cosmological numerical simulations that follow the gravitational amplification of small fluctuations, as seen in the Cosmic Microwave Background, are able to produce a pattern very similar to this \lq cosmic web\rq~(see movies). This chapter discusses the statistical properties of galaxies, as inferred form such surveys.\\

\section{The density-morphology relation}
Once galaxies are identified it may be possible to classify them according to the Hubble sequence\footnote{This is only possible for relatively nearby galaxies: at high redshifts, galaxies are so small on the sky that a morphological classification becomes impractical.}.
When Hubble type is plotted against galaxy density ({\em i.e.} the density of galaxies per unit volume), it appears that {\em low density regions ({\em e.g. groups}) are mostly populated by spiral galaxies, whereas high density regions ({\em e.g. clusters}) are mostly populated by S0s and Es (see Fig.\ref{fig:densmorp})}: this is called the {\bf density-morphology relation}.\\

%In the previous chapter we saw how some galaxies are in low density
%environments, some are in higher density {\em groups}, and others in
%dense {\em clusters} containing several hundred
%galaxies. Figure~\ref{fig:densmorp} shows that the {\em fraction} of
%elliptical and S0 galaxies increases rapidly with increasing galaxy
%density. In other words, in a region of high galaxy density, such as a
%dense group or cluster of galaxies, most galaxies tend to be S0 or E,
%and very few are of type S. Vice versa, in regions of low galaxy
%density, most galaxies tend to be of the type S.  This is called {\bf
%the density-morphology relation}, since it {\bf relates the density of
%galaxies in a given region with their typical morphology} (i.e. S, E,
%S0). Clearly this is telling us something on the nature of galaxy
%formation and evolution.\\
%Recall from the movie I showed of the motion of galaxies in a cluster,
%that encounters between galaxies are very frequent in a region of high
%galaxy density. Such tidal encounters could be responsible for
%converting S-type galaxies into S0 and E, but other processes might
%operate as well. Recall that we already discussed that this cannot be
%the whole story, since the stellar populations of Ss and Es tend to
%differ as well.

The origin of this relation has long been debated, but it seems plausible that galaxies in high-density regions suffer more tidal encounters with other galaxies, and this may destroy the fragile disk  (recall the movie of a collision between two spirals), hence converting spirals into S0s or Es. Another possibility is that the hot gas associated with clusters (Chapter~9) removes the gas from spiral galaxies that fall into a cluster (by {\em ram-pressure stripping}), thereby suffocating the galaxy (descriptively termed {\em strangulation}).

\begin{figure*}
\center\resizebox{1.2\textwidth}{!}{\includegraphics{density-morphology.pdf}}
\caption{The fraction of Elliptical (E), S0, and Spiral + Irregular
(S+I) galaxies as function of the logarithm of the projected density( $\rho_{\rm proj}$,
in galaxies \Mpc$^{-2}$). At low density (small $\rho_{\rm proj}$), most galaxies are type S or Irr, whereas at high densities, most galaxies are S0 or E: there appears to be a relation between density of galaxies, and galaxy type: this is the \lq density-morphology\rq\ relation. From Dressler, ApJ 236, p. 351 (1980).}
\label{fig:densmorp}
\end{figure*}

%% \section{The density-morphology relation}
%% In the previous chapter we saw how some galaxies are in low density
%% environments, some are in higher density {\em groups}, and others in
%% dense {\em clusters} containing several hundred
%% galaxies. Figure~\ref{fig:densmorp} shows that the {\em fraction} of
%% elliptical and S0 galaxies increases rapidly with increasing galaxy
%% density. In other words, in a region of high galaxy density, such as a
%% dense group or cluster of galaxies, most galaxies tend to be S0 or E,
%% and very few are of type S. Vice versa, in regions of low galaxy
%% density, most galaxies tend to be of the type S.  This is called {\bf
%% the density-morphology relation}, since it {\bf relates the density of
%% galaxies in a given region with their typical morphology} (i.e. S, E,
%% S0). Clearly this is telling us something on the nature of galaxy
%% formation and evolution.\\

%% Recall from the movie I showed of the motion of galaxies in a cluster,
%% that encounters between galaxies are very frequent in a region of high
%% galaxy density. Such tidal encounters could be responsible for
%% converting S-type galaxies into S0 and E, but other processes might
%% operate as well. Recall that we already discussed that this cannot be
%% the whole story, since the stellar populations of Ss and Es tend to
%% differ as well.

%% \begin{figure*}
%% \center\resizebox{.9\textwidth}{!}{\includegraphics{density-morphology.pdf}}
%% \caption{The fraction of Elliptical (E), S0, and Spiral + Irregular
%% (S+I) galaxies as function of the logarithm of the projected density,
%% in galaxies \Mpc$^{-2}$. The data shown are for all cluster galaxies in
%% the sample and for the field. Also shown is an estimated scale of true
%% space density in galaxies \Mpc$^{-3}$. The upper histogram show the
%% number distribution of the galaxies over the bins of projected
%% density. Es and S0s tend to populated regions of high galaxy density,
%% and Ss regions of low galaxy density. From Dressler, ApJ 236, p. 351
%% (1980).}
%% \label{fig:densmorp}
%% \end{figure*}

\section{Galaxy scaling relations}

\subsection{The Tully-Fisher relation in spirals}
Spiral galaxies have flat rotation curves: the circular velocity is independent of radius away from the centre.
The {\em Tully-Fisher} relation $L\propto V_c^4$ relates the luminosity $L$ of the spiral to its  circular velocity $V_c$: bright spirals (high $L$) spin faster (higher $V_c$), see Fig~\ref{fig:tf}.\\

To measure this relation we need to determine $L$, and the circular velocity\footnote{The circular velocity of spirals rises from the centre, then remains nearly constant. This maximum of the  \lq circular velocity\rq\ is denoted $V_c$ below. Note also that the {\em measured} value depends on the inclination of the disk: $V_c$ below is assumed to have been corrected for this.} $V_c$. $L$ is inferred from the apparent magnitude, once the distance to the galaxy is known. $V_c$ is inferred from measuring velocities of stars or gas in the interstellar medium, for example using H$\alpha$ as emission line, or 
more usually using the HI 21-cm line.\\

Consider a galaxy at distance $r$, which is edge-on (so we need not correct for inclination). On average, such a galaxy will be moving away from us at speed $v=H_0\,r$ due to the expansion of the Universe, where $H_0\approx 72\,\km\,{\rm s}^{-1}\,\Mpc^{-1}$ is the Hubble constant.
This will redshift the measured wavelength of the HI-21cm line from its laboratory value, $\lambda_0=21.106\,\cm$, to $\lambda=\lambda_0\,(1+v/c)$. If $V_c$ is the rotation speed of this galaxy,  the gas on one side (the right-hand side, say) will move with speed $v+V_c$,  while on the left-hand side the speed is $v-V_c$. Consequently the measured wavelength right and left are $\lambda_{\rm r}=\lambda_0\,(1+(v+V_c)/c)$ and  $\lambda_{\rm l}=\lambda_0\,(1+(v-V_c)/c)$, respectively. The {\em difference} $(\lambda_{\rm r}-\lambda_{\rm l})/\lambda_0=2V_c/c$ yields the rotation speed.\\

%The circular velocity as function of radius -- the rotation curve -- in
%spirals tends to be flat, i.e. the rotation velocity is independent of
%radius sufficiently far out in the disk. We argued in section
%\ref{sect:rotcurve} that this was evidence for the presence of dark
%matter in spiral disks, since it implies lots of mass in the outer
%parts, whereas we see very little light there. Figure\,\ref{fig:tf}
%compares the measured rotation velocity $V_c$ (written $W$ in the
%figure) with the total absolute magnitude $M=-2.5\log(L)$+constant, for
%a sample of spiral galaxies. {\bf Brighter spirals} (more negative $M$)
%{\bf rotate faster} (larger $W$). Since the figure shows there to be a
%linear relation between $M$ and $V_c$, it implies a power-law relation
%between the intrinsic luminosity $L$ and the circular velocity $V_c$,
%called the {\bf Tully-Fisher relation}:
%
%\begin{equation}
%L\propto V_c^\alpha\,,\,\,\,\,\,\,\,\alpha\approx 4\,.
%\label{eq:tf}
%\end{equation}

%The slope $\alpha$ of this relation depends somewhat on which colour is
%used (i.e., does $L$ refer to a \V\ -band or \I\ -band luminosity).\\


\begin{figure*}
\center\resizebox{.9\textwidth}{!}{\includegraphics{tfr.pdf}}
\caption{R-band absolute magnitude $R=-2.5\log(I_R)$ of a set of spiral galaxies as function
of their circular velocity $W\equiv V_c$ (in \kms). {\em Brighter}
spiral galaxies (more negative $R$) rotate {\em
faster} (higher $W$).}
\label{fig:tf}
\end{figure*}

What is the origin of this relation? The circular velocity depends on enclosed mass
$M(<R)$ as

\begin{equation}
V_c^2 = {{\rm G}\,M(<R)\over R}\,.
\end{equation}

Let 

\begin{equation}
\Gamma \equiv {M(<R)\over L}\,,
\end{equation}

denote the global mass-to-light ratio,  then 

\begin{equation}
V_c^2 = {\rm G}\Gamma\, {L\over R}\,.
\end{equation}.

The mass-to-light ratio $\Gamma$ will depend on the type of stars in the
galaxy (which determines $L$), and the amount of dark matter (which
mainly determines $M$). The intensity $I$ is the luminosity per unit
area, $I=L/(\pi\,R^2)$, or solving for $R$

\begin{equation}
R = \left({L\over \pi\,I}\right)^{1/2}\,.
\end{equation}

Therefore

\begin{equation}
L = {1\over {\rm G}^2\,\Gamma^2 \pi\,I} \,V_c^4\,.
\label{eq:tf}
\end{equation}

This reproduces the Tully-Fisher relation {\em assuming} intensity $I$ and mass-to-light ratio $
\Gamma$ are {\em independent} of
the luminosity of the galaxy. Notice that this is not a {\em
derivation} of the Tully-Fisher relation: we are just trying to
understand what is required for spirals to follow this relation. The
fact that $\Gamma$ and $I$ are independent of $L$ implies that somehow
stars and dark matter are closely linked, or in other words: the star
formation history is largely determined by the mass of the dark halo of
the galaxy.\\

The exponent $\alpha$ in the {\em observed} Tully-Fisher relation, $L\propto V_c^\alpha$, depends on colour ({\em i.e.} the broad-band filter through which $L$ is measured: clearly our argument is not complete.

\subsection{The Faber-Jackson relation in ellipticals}
The Faber-Jackson law, Figure~\ref{fig:fj}, relates the velocity
dispersion $\sigma$ of the stars in an elliptical galaxy with its luminosity $L$,

\begin{equation}
L\propto \sigma^4\,.
\label{eq:FJ}
\end{equation}

This is similar to the Tully-Fisher, but replaces $V_c$ by $\sigma$. Measuring this relation again requires determining distances to a sample of elliptical galaxies, and measuring $\sigma$ from line-widths in spectra.\\


Comparison of Figure \ref{fig:fj} with Figure \ref{fig:tf} shows that
the {\em scatter} in the Faber-Jackson relation is significantly larger that that in the Tully-Fisher relation: spirals follow the TF relation much better than ellipticals follow the FJ relation. Following a similar reasoning that lead to the TF relation, we can improve on FJ.

\begin{figure*}
\center\resizebox{.9\textwidth}{!}{\includegraphics{faberjackson.pdf}}
\caption{B-band absolute magnitude $M_B$ for a sample of elliptical
galaxies plotted versus their stellar velocity dispersion $\sigma$ (in
\kms). {\em Brighter} ellipticals galaxies (more negative $M_{\rm B}$)
have a {\em larger} velocity dispersion: the {\bf Faber-Jackson} relation}
\label{fig:fj}
\end{figure*}

Suppose an elliptical galaxy has luminosity $L$, stellar velocity dispersion $\sigma$, radius $R$, intensity $I$ and mass-to-light ratio $\Gamma$.
Assuming virial equilibrium,
\begin{equation}
\sigma^2 =  {{\rm G} M\over R}={{\rm G}\,\Gamma\,L\over R}\,.
\end{equation}

Now {\em assume} $\Gamma$ depends on $L$ as 

\begin{equation}
\Gamma = \Gamma_\star\,(L/L_\star)^\alpha\,,
\end{equation}

for some value of $\Gamma_\star$, $L_\star$ and $\alpha$ to be determined. Using $I=L/\pi R^2$ to eliminate $L$ yields

\begin{equation}
\sigma^2 \propto {L^{1+\alpha}\over R} \propto {(R^2\,I)^{1+\alpha}\over R} \propto I^{1+\alpha}\,R^{1+2\alpha}\,,
\end{equation}
or
\begin{equation}
R \propto \sigma^{2/(1+2\alpha)}\,I^{-(1+\alpha)/(1+2\alpha)}\,.
\label{eq:fp}
\end{equation}

%From all the parameters we can measure for an elliptical, the total
%luminosity $L$, the velocity dispersion $\sigma$, the effective radius
%$R_e$ and the intensity $I_e$ (which both enter in the de Vaucouleurs
%profile, Eq.~(\ref{eq:r4})), only three are independent. So we can try
%to obtain a tighter relation by introducing a second parameter in
%Eq.~(\ref{eq:FJ}), for example $R_e$. This is how it goes: suppose that
%the galaxy is in virial equilibrium, then its kinetic energy
%$M\,\sigma^2$ will be proportional to its potential energy $M^2/R_e$,
%and so we expect

%
%Introducing again the mass-to-light ratio, \mol\, this can be written
%as

%\begin{equation}
%\sigma^2 \propto \mol {L\over R_e} \propto {L^{1+\alpha}\over R_e}\,,
%\end{equation}

%where I've assumed \mol\ to depend on $L$ as
%\begin{equation}
%\mol\propto L^\alpha\,.
%\label{eq:mol}
%\end{equation}

%The intensity $I_e\propto L/R_e^2$, hence
%\begin{equation}
%R_e\propto \sigma^{2/(1+2\alpha)}\, I_e^{-(1+\alpha)/(1+2\alpha)}\,.
%\label{eq:fp1}
%\end{equation}

\begin{figure*}
\center\resizebox{.9\textwidth}{!}{\includegraphics{fp.pdf}}
\caption{{\bf The Fundamental plane relation} for elliptical galaxies between radius $r_e$, 
(central) intensity $I_e$ and velocity dispersion $\sigma$, $r_e\propto \sigma^{1.24}\,I_e^{-0.82}$ (drawn line). Data points refer to the
same sample of Es plotted in Figure~\ref{fig:fj}. The galaxies follow
this relation with significantly less scatter than in the
Faber-Jackson relation in Fig.\ref{fig:fj}.}
\label{fig:fp}
\end{figure*}

Figure~\ref{fig:fp} shows that the same galaxies plotted in Figure~\ref{fig:fj} follow the relation

\begin{equation}
R\propto \sigma^{1.24}\,I^{-0.82}\,,
\label{eq:fp2}
\end{equation}

very well. Choosing $\alpha=0.25$ in Eq.~(\ref{eq:fp}) yields
$R\propto \sigma^{1.33}\,I_e^{-0.82}$, close to what is observed. Put differently, {\em if}
the mass-to-light ratio of ellipticals depends on luminosity as
$\Gamma\propto L^{0.25}$, then we can understand why ellipticals follow
the relation plotted in Figure~\ref{fig:fp} so tightly.\\

Relation Eq.~(\ref{eq:fp2}) is called the {\bf fundamental plane}: in the 4D space with axes $L$,  $R$, $\sigma$ and $I$, galaxies are close to the plane defined by this relation, hence \lq fundamental plane\rq. Note that once more we have not {\em derived} this equation: we used simple arguments to understand what is needed for galaxies to follow the observed relation. In contrast to spiral galaxies, we had to assume that $\Gamma$ depends on $L$: somehow stars and dark matter in an elliptical galaxy conspire to have a mass-to-light ratio that depends on total luminosity as
$M/L\propto L^{0.25}$.

\section{Tully-Fisher and Fundamental plane relations as standard
candles}
The Tully-Fisher relation in spirals , $L\propto V_c^4$ (Eq.\ref{eq:tf}), and the fundamental plane relation,
$R\propto \sigma^{1.24}\,I^{-0.82}$ (Eq.~\ref{eq:fp2}), both relate quantities independent of distance ($V_c$, respectively, $\sigma$ and $I$) 
to intrinsic quantities (luminosity $L$ and size $R$, respectively). This makes both relations useful as {\em standard candles}.\\

Suppose we were to determine the proportionality constant $A$ in the Tully-Fisher relation, $L=A\,V_c^4$, by inferring
luminosity $L$ from flux $F$ and distance $r$ from Cepheid measurements for local spirals (using $F=L/4\pi r^2$). For spirals too distant for Cepheid measurements, we can determine their distance from measurements of $V_c$ (distance independent) and flux, by applying the Tully-Fisher relation.\\

Similarly we can use local ellipticals to determine the proportionality constant $B$ in the fundamental plane relation, $R=B\,\sigma^{1.24}\,I^{-0.82}$.
For more distant ellipticals we can determine the distance independent quantities $\sigma$ and $I$, apply the fundamental plane relation to infer $R$, and get the galaxy distance by measuring its angular size $\theta=R/r$. \\

%

%
%relates the luminosity $L$
%of a galaxy to its circular velocity. Now suppose we were able to
%measure the proportionality constant, by determining the luminosity for
%galaxies with known distance. Then by just measuring the circular
%velocity of a spiral galaxy, we could infer its luminosity hence
%absolute magnitude $M$, and consequently from its apparent magnitude
%$m$ obtain the distance $r$, since

%\begin{equation}
%m-M=5\log(r)-5\,.
%\end{equation}

%Therefore, the Tully-Fisher relation can be used as a standard candle,
%since it allows us to determine the luminosity from the distance
%independent quantity $V_c$. Note also that $V_c$ is relatively easy to
%determine from spectra, and so we have found a good way of measuring
%distances to distant galaxies!\\

%The fundamental plane relation plotted in Figure~\ref{fig:fp} can also
%be used as a standard candle: all we need to do is measure the velocity
%dispersion of the stars $\sigma$ from a spectrum, and determine the
%intensity $I_e$ (recall that the surface brightness and hence also
%$I_e$ is distance independent). By putting the E galaxy onto
%Figure~\ref{fig:fp}, we then find $R_e$. And so from the apparent size
%of the galaxy, we can estimate its distance.\\

Both methods are widely used, since measuring velocities is
relatively easy from a good quality spectrum, and can be done even for
faint and/or distant galaxies. And since the scatter in the relations
is small, we can obtain a relatively accurate distance estimate.

\section{Galaxy luminosity function}
The galaxy luminosity function is simply a histogram of the number of galaxies per unit volume, binned as function of their luminosity (Fig.~\ref{fig:sf}).
This requires knowing  distances $r$ to the galaxies, for example using the Tully-Fisher or fundamental plane relation, or simply assuming $r=v/H_0$. The shape of the luminosity function is well captured by the {\bf Schechter} function

\begin{equation}
{dN\over dL/L_\star} = N_\star\,\left({L\over
L_\star}\right)^\alpha\,\exp(-L/L_\star)\,,
\end{equation}

%
%From a big galaxy redshift survey, we can make a histogram of the
%number of galaxies with absolute luminosity between $L$ and $L+dL$,
%where $L$ is either the total (or bolometric) luminosity, or the
%luminosity in some band, e.g. the B-band, per unit
%volume. Figure~\ref{fig:sf} shows that the number of faint galaxies
%increases like a power law, whereas there is an abrupt cut-off toward
%the brighter galaxies. This behaviour is well captured by the {\bf
%Schechter} function

%\begin{equation}
%{dN\over dL/L_\star} = N_\star\,\left({L\over
%L_\star}\right)^\alpha\,\exp(-L/L_\star)\,.
%\end{equation}

which has three parameters: $L_\star$, $\alpha\sim -1$ and
$N_\star$. At low luminosities $L\ll L_\star$,  the number of galaxies increases $\propto L^\alpha\sim
1/L$. At high luminosities $L\ge\ L_star$, the number of galaxies drops exponentially.  Finally, $N_\star$ is a normalisation constant, which determines the mean number density of galaxies.\\

Observed values are $N_\star\approx 0.007$\Mpc$^{-3}$,  $-1.3\le \alpha \le -0.8$, and $L_\star\approx 2\times 10^{10}\lsun$ in the B-band. From
Table \ref{table:MW}, we find that the total B-band luminosity of the MW
is about $1.8\times 10^{10}\lsun$, so the MW is just slightly fainter
than an $L_\star$ galaxy.\\

When the faint-end slope is steeper than $\alpha=-1$ the total number of galaxies per unit volume, $N=\int_0^\infty (dN/dL)\, dL$,
diverges, because there are so many faint galaxies. In reality, this
does not happen of course: there must be a {\em finite} number of
small galaxies. The mean luminosity follows from  integrating

\begin{equation}
\langle L\rangle=\int_0^\infty L\, {dN\over dL}\, dL/N\,.
\end{equation}

$N\,\langle L\rangle$ is the mean luminosity per unit volume. Combining this with the
mean density of the universe as obtained in section \ref{sect:dmU}, we
can estimate the mass-to-light ratio for the Universe as a whole!\\

\begin{figure*}
\center\resizebox{.9\textwidth}{!}{\includegraphics{schechter.pdf}}
\caption{Galaxy luminosity function (symbols) together with best-fit Schechter function (drawn line).}
\label{fig:sf}
\end{figure*}


Finally, Figure\,\ref{fig:lf} shows how different galaxy types contribute to the 
total luminosity function, and how this depends on environment.


%Finally, Figure\,\ref{fig:lf} shows how the Schechter luminosity function
%is composed of contributions from the various galaxy types, E, S0 and
%the different types of spirals, and irregular galaxies. Top panel is
%for a sample of galaxies away from big clusters, bottom panel refers to
%the Virgo cluster. Here we recognise the density-morphology once more:
%most of the brighter galaxies in the top panel are Spirals, whereas in
%the cluster sample, E an S0s play a much bigger role. 

\begin{figure*}
\center\resizebox{.99\textwidth}{!}{\includegraphics{galaxy-luminosty-function.pdf}}
\caption{Contribution of different galaxy types to the total galaxy luminosity function, $\Phi\equiv dN/dL$.
The different panels illustrate how the luminosity function depends on environment: note the predominance of types E and S0 in the Coma galaxy cluster (top left), how spirals contribute more in groups, and the almost complete absence of E and S0 types in low density regions (\lq extreme field\rq). Figure taken from Dr Helmut Jerjen at the  Australian National University}
\label{fig:lf}
\end{figure*}

\section{Epilogue}
The classification of galaxies that we have used so far, in terms of
spirals and ellipticals, is based largely on how these galaxies appear
in pictures. With several $10^5$ galaxies in modern surveys, it is not
really possible to eye-ball all of them in order to classify them. In
addition, it is not even necessarily the best way, and certainly not
the only way, of classifying galaxies and identifying different
types. With so many spectra and images available, new statistical
techniques are presently being developed, where computer programs which
do not suffer from human prejudices, such as principle component
analysis and neural networks, decide how galaxies should be grouped and
classified.

\section{Summary}
After having studied this lecture, you should be able to 

\begin{itemize}
\item describe the density-morphology relation for galaxies

\item describe the Tully-Fisher relation, and explain how it is used as a distance indicator

\item describe the Faber-Jackson and fundamental plane relations, and explain how they are used as a distance indicator

\item explain what the luminosity function is, and sketch it.
\end{itemize}

\chapter{Active Galactic Nuclei}
Quasars are the most luminous objects in the known Universe, with the most extreme objects outshining the combined star light of a bright galaxy by a factor $10^5$ or more. They are one manifestation of an {\em active galactic nucleus}: a powerful source of radiation lurking in the centres of a small fraction $\sim 1$\% of large galaxies. They were originally discovered by the powerful radio-radiation that some of them emit. 
It is thought they are powered by accretion of matter onto a super-massive black hole. 
%

%Some galaxies harbour immensely powerful sources of radiation in their
%centres, detected from the longest (radio) to the shortest wavelengths
%(X-rays): these are called Active Galactic Nuclei (or AGN for
%short). We think they are powered by the accretion of matter onto super
%massive black holes (SMBH), the same process that powers some galactic
%X-ray binaries. However, the SMBH have masses ranging from $10^6$ to
%$10^9\msun$, and so, unlike galactic black holes, cannot be stellar
%remnants.  We think that {\em most} or even all reasonably massive
%galaxies have a SMBH lurking in their centres, but only a small
%fraction of these are \lq active\rq\, at any given time. The evidence
%that the MW has its own SMBH is particularly impressive.\\

%Where do these SMBHs come from? Are they really black holes
%(i.e. objects with an event horizon) or are they just very massive,
%extremely dense objects (MDO)? Why are some associated with strong
%radio-sources (radio-loud AGN), but others are not? How do radio-loud
%AGN power the immense radio-lobes we observe? What is the effect of the
%AGN on the galaxy itself? Most of these questions remain without a
%clear answer, even today.\\

%I will start by discussing some of the observations of AGN. Then I'll
%take you through the evidence that these really are black holes, but
%leave it up to you to decide whether you believe it or not. My own view
%is that the evidence for super massive and very dense objects is very
%convincing -- but I have not seen evidence for an event horizon (yet).

\section{Discovery and observational properties}
Carl Seyfert reported in 1943 that a small fraction of galaxies have a
very bright unresolved nucleus, emitting radiation characterised by emission lines
produced by atoms in a wide range of ionisation stages. These are the most feeble members of the AGN family.\\

The first {\em radio surveys} in the 1950s uncovered sources not obviously associated with any bright galaxy.
The third brightest radio source on the sky, Cygnus A, appeared to be associated with a peculiar-looking
cD galaxy, whose centre is encircled by a ring of dust. The spectral lines in the optical (galaxy) spectrum of Cygnus A are redshifted
to $\Delta\lambda/\lambda_0=0.057$, implying a Doppler shift of 16000 \kms and hence a distance $r=v/H_0\sim 220\,\Mpc$.
Knowing the distance and the measured radio {\em flux} allows one to compute the radio luminosity: about ten times the bolometric\footnote{The bolometric luminosity is the total energy output, {\em i.e.} integrated over the whole EM-spectrum.} luminosity of the Milky Way.\\

In 1960, Thomas Matthews and Alan Sandage attempted to identify the optical counterpart to the radio source 3C~48\footnote{The C stands for Cambridge -- this is the third Cambridge radio-survey.}.  It turned out to be a faint 16th magnitude star like object, with a very curious optical spectrum featuring broad emission lines that they could not identify with any known element or molecule. Sandage put it
very scientifically: \lq This thing is exceedingly weird\rq. Another such weird object was found in 1963, associated with the radio-source 3C~273.
Given these objects looked like stars, they were called quasi-stellar objects, or QSOs.\\

In the same year, the Dutch astronomer Maarten Schmidt recognised that
the pattern of lines in 3C~273 was similar to the Balmer
series\footnote{The Balmer series is the series of electronic
transitions which end at $n=2$.} of
the Hydrogen atom -- but only if they were redshifted to the
(improbably large ?) value of $\Delta\lambda/\lambda=0.16$, implying a
recession velocity of $0.16c$ and a distance of $\sim$ 660\Mpc. The distance
to 3C~48 was even greater: a recession velocity of $0.367c$ and a
distance of 1800\Mpc.\\

Such large distances also implies these sources are exceedingly bright: a distance of $r=660\Mpc$ implies a distance modulus of $m-M=-5\log(r/10)\approx 39$, therefore the measured apparent magnitude of $m=12$ of 3C~273yields an absolute magnitude of $M=-26$, compared to $M=5$ for the Sun. The luminosity of this source is therefore $L=10^(31/2.5)\,L_\odot=2.5\times 10^{12}L_\odot$, or about 100 times the luminosity of the entire Milky Way. What kind of object can be so small yet so tremendously powerful?\\

Current quasar surveys have uncovered 1000s of sources, with the current redshift record held by a source\footnote{When interpreted as a Doppler shift, the velocity of this source $v=(1+z)\,c=7.4\,c$. A proper description applies the Friedmann-Robertson-Walker metric to the Universe, and the redshift is not interpreted as a velocity: the photons from the quasar always move with just the speed of light. Note that it is incorrect to attempt to \lq correct\rq\ this seemingly superluminal motion by using special relativity.} at $z=6.4$.  The luminosity of the brightest quasars can range up to $10^5$ times the luminosity of the MW.\\

In general, AGN emit radiation over a large part of of the EM-spectrum, from radio, over optical and UV, to X and gamma-rays, and the most energetic cosmic rays are also suspected to be due to AGN. However not all AGN are visible over this entire wavelength range. QSOs by definition are bright in the optical, but some powerful AGN are only visible in the radio, or in X-rays.\\

The shear luminosity of QSOs alone already suggests we are looking at
something unusual. In addition, some of these QSOs have radio-lobes
which are tens of times the size of a galaxy. What is the engine that
powers them?

\section{The central engine and unification schemes}
AGN certainly are very bright. But are they {\em unusually} bright?
Yes. The argument is neat, and based on their {\em variability}. The luminosity of AGN varies on a range of time-scales -- hours,
days, and months, depending on the wavelength at which you observe
them. But variations in the X-ray properties tend to have very short
time-scales, of order of minutes. Now this implies that the X-rays come
from a small region -- of the order of light minutes. To come to this
conclusion, all we need is to say that the information about the
variation has to be able to travel across the object that is producing
the emission\footnote{Think of it in terms of a {\em Mexican wave} in a sports stadium: spectators need to be able to see the wave approaching if they want to take part.} , and since the fastest  information can is $c$ it implies the AGN engine is only light minutes in size. This is comparable to the size of the Solar system. Therefore an engine the size of the solar system or smaller produces as much energy as $10^5$ galaxies.\\


The most efficient way of producing energy is by mass accretion onto a compact object\footnote{Actually, annihilation of
matter--anti-matter is even better!}. This has an efficiency\footnote{This numbers comes from the fact that there are no stable circular orbits around 
a central object for sufficiently small radii due to general relativistic effects.} from  $\epsilon= L/\dot Mc^2\sim 10$ \%, much better than that of nuclear fusion that power stars, which only manages $\epsilon\approx 0.7$ \%.\\

The steady-state luminosity of an object accreting mass spherically is limited by the {\em Eddington luminosity} $L_{\rm Edd}$, which follows from equating the radiation pressure of the source to its gravitational pull. Therefore we can estimate the minimum mass of a QSO so that it does not exceed $L_{\rm Edd}$: this implies masses $\sim 10^8\,M_\odot$. Given the small inferred size together with this huge mass and extreme luminosity, Donald Lynden-Bell and Martin Rees suggested that accretion onto a supermassive black hole must be the energy source powering QSOs.\\

However, how can we reconcile the immense variety in emission spectra of AGN with the extreme simplicity of a black hole, which is
characterised by just two numbers, spin and mass? The gas is thought to accrete onto the black hole by passing through an
accretion disk, where it loses angular momentum and spirals in. It is thought that it is the accretion disk that governs the appearance of the AGN, although the details of how this happens are still unclear. For example it could be that a fortuitous orientation of this disk lets us see almost straight onto the black hole and see the quasar, whereas an edge-on disk might hide the optical source, and the AGN would not be classified as a QSO.\\

\section{Evidence for a super massive black hole}
The standard model suggests that AGN are powered by accretion onto a super massive black hole (SMBH). Recent evidence suggests that most large galaxies contain such a SMBH at their centres, yet only a small fraction of these is active at any given time. The evidence for the presence of a SMBH at the Milky Way centre is especially convincing, yet the Milky Way does not harbour an AGN. Recall that only a small fraction, $\sim 1$ \% of galaxies show AGN activity. How can we reconcile the fact that most galaxies harbour a SMBH, yet do not show AGN activity? The suspicion is that in most galaxies, the SMBH is starved of fuel (infalling gas), and therefore lies dormant until gas drains to the galaxies centre, for example following a galaxy collision.\\

But what\footnote{This section follows a nice review paper by Luis Ho, see\\
http://nedwww.ipac.caltech.edu/level5/March01/Ho/Ho.html for the full
text.} evidence is there that that the central object really is a black hole ({\em i.e.} an object with an event horizon) as opposed to simply a massive compact object, or even an entirely different engine? According to Luis Ho, \lq our confidence that
SMBHs (super massive black holes) must power AGN largely rests on the implausibility of alternative
explanations\rq. Most of the arguments in the next sections just
suggest the presence of a very massive, dense object (MDO) in the
centre of galaxies, but not really require the MDO to be a {\em black hole}.


\subsection{Stars}
\subsection{Central cusps}
The Jeans equations relate the potential well in a stellar system, with the density $\rho$ and velocity dispersion $\sigma$ of stars (see Eq.~\ref{eq:jeans}):

\begin{equation}
{G\,M(<r)\over r^2}=-{\nabla \rho\sigma^2\over\rho}\,.
\label{eq:jeans2}
\end{equation}

Away from the SMBH where $M$ is dominated by many stars and dark matter, $M$ is a slowly increasing function of $r$. However near a SMBH, the mass is dominated by the SMBH, and the left-hand-side increases rapidly with {\em decreasing} $R$, $M/r^2\approx M_{\rm SMBH}/r^2\propto r^{-2}$. The stars can react  by a rapid increase in either density $\rho$ or velocity dispersion, $\sigma$, towards the centre (or both).\\

A sharp increase of stellar density $\rho$ towards the centre would be seen as a cusp in the surface brightness.
Unfortunately the expected size of such a cusp is small: of order $r\approx {GM_{\rm BH}/ \sigma^2}$.
For  a QSO at distance $R$, the corresponding angular extent is $\approx 1\arcsec
(M_{\rm BH}/2\times 10^8\msun)\,(\sigma/200\kms)^{-2}\,(R/5\Mpc)$. Such
a small extent is hard to see from the ground (due to atmospheric seeing), and even Hubble Space Telescope (HST)  can resolve such a core only relatively nearby. Some galaxies do indeed show a dramatic increase in brightness toward the centre, consistent with the presence of very massive central object.\\

It is also possible to measure a steep increase of $\sigma$ towards the centre in some galaxies, also consistent with the presence of very massive central object. \\

Unfortunately neither measurements are ultimately conclusive: Eq.~(\ref{eq:jeans2}) assumes isotropic orbits, but the velocity dispersion of stars need not be isotropic. Anisotropy in stellar velocities can hide, but also mimic the presence of a central massive object.

\subsection{Stellar kinematics in the MW centre}
The Milky Way centre has a weak AGN in its centre (the radio source Sagittarius A). 
Genzel and collaborators measured the motion of stars near the MW centre 
using near-IR observations\footnote{Recall the need to go to the IR: there is a lot of
absorption by dust toward the MW centre but IR-light undergoes far less
absorption. See the movie!} They also determined very accurate positions of the
stars. By doing so over several consecutive years, they were able to
follow the full orbit of several stars as they swung by the centre of the MW at speeds of several 1000 \kms, 
and hence measure {\em accelerations}.\\

 Using Kepler's laws\footnote{Since they measured the velocities of stars both in the plane of the sky ({\em i.e.} the proper-motion)
 and along the line of sight (from radial velocities), they could construct the 3D orbit. By relating the projected orbit to the true 3D orbit they could measure the distance to the Milky Way centre very accurately.} they were able to put very stringent limits on the mass of the central
concentration, $M_{\rm DO}\approx 2.6\times 10^6\msun$, in a region
smaller than 0.006\pc, implying an enormous mass density higher than
$2\times 10^{12}\msun\,\pc^{-3}$. This leaves {\em almost} no room to
escape the conclusion that the MDO is indeed a SMBH, at least in the MW. The presence of the large mass is also supported by the presence of
other very high velocity stars. From that, and from the fact that the
putative centre, i.e. the radio source Sagittarius A itself does not
seem to move, Genzel derived a lower limit to the density of $3\times
10^{20}\msun\,\pc^{-3}$. 



\subsection{Methods based on gas kinematics}
Unlike the situation for stars where velocity anisotropy clouds the
picture, gas kinematics is in general much easier to interpret. However 
there is a caveat in that there may be non-gravitational forces acting on the gas, magnetic fields or radiation pressure for example.\\

{\bf Optical emission lines} HST found several examples of nuclear gas disks,
some in Keplerian rotation with rotation speed up to 1000 \kms\ at a distance
of 19 \pc\ from the centre\footnote{Don't get confused with evidence for
dark matter from {\em flat} rotation curves. Here we are looking close
to the centre, whereas for rotation curves we were looking at large
distances, $\sim 20\kpc$ from the centre.}. This implies a MDO with
mass $\approx 2.4\times 10^9\msun$. Similar evidence for a MDO now
exists for other galaxies as well.\\

{\bf Maser emission} Some AGN have luminous maser\footnote{A {\em
maser} is a laser that operates in the mm regime.} sources close to the
centre. Radio observations of these maser lines show that the masers
are in nearly Keplerian rotation, and from the speed one can place
tight constraints on the mass density of the MDO.\\

{\bf Reverberation mapping} An elegant way to determine the extent
of an AGN is \lq reverberation mapping\rq. Many of the emission lines
seen from an AGN are due to the light from the central source being
reprocessed by the surrounding material. As the central source varies
in luminosity, the emission lines vary as well, but with a time-lag
that corresponds to the light-travel time between the central source
and the line-emitting gas. So by measuring the time-lag, we can
estimate the size of the light-emitting region.

\subsection{Profile of X-ray lines}
The Fe K$\alpha$ line at 6.4keV is a common feature in the X-ray
spectra of AGN. It is thought to arise from fluorescence in the colder 
material in the surroundings of the source, irradiated by the X-rays from the source.\\

This line is produced very close to the SMBH, and it exhibits Doppler
motions approaching relativistic speeds, $\sim 10^5\kms\approx
0.3c$. Now crucially, the line-profile displays asymmetries consistent
with a {\em gravitational redshift}. The photons lose energy as they
climb out of the deep potential well near the black hole. In doing so,
their wavelengths become longer -- a \lq gravitational redshift\rq.\\

For the AGN galaxy MCG-6-30-15, the best fitting accretion disk has an
inner radius of 6 Schwarzschild radii, i.e. {\em very} close to the
event horizon. A similar analysis has now been performed for other AGN
as well. {\bf The shape of the line probably provides the best evidence
to date for the existence of a SMBH.} However, other mechanisms for
generating the line profile are possible, but implausible. With better
data, detailed modelling of the profile even has the potential to
determine the spin of the SMBH, by measuring another general relativistic effect called frame-dragging.\\

\section{Discussion}
A small fraction of galaxies contains an immensely luminous source of radiation in their centres: these are called active galactic nuclei.
These source display a large variety of observational manifestations, and are thought  to be powered by accretion onto a black hole.\\

We have evidence that probably most, if not all, galaxies with masses of order the MW mass or higher, have a SMBH in
their centres, although the evidence that this is really a black hole (as opposed to a very massive dense object) is weaker.
Most of these SMBHs are starved of fuel: this explains why most galaxies have a SMBH yet are not AGN.\\

And finally: a star venturing too close to a SMBH may be
torn apart as a result of the tidal forces. This could
lead to a very bright flare lasting for of order months. The
realisation that SMBHs may be quite common provides fresh motivation to
search for such flares. Upcoming missions designed to detect SNe will
find these flares -- if they exist.\\

\section{Summary}
After having studied this lecture, you should be able to 

\begin{itemize}
\item describe observational manifestations of AGN.
\item describe briefly the current paradigm for energy generation in
AGN (mass accretion through a disk onto a SMBH)
\item describe three arguments that suggest the presence of SMBHs in
galaxies.
\end{itemize}


\chapter{Gravitational lensing}
A massive object distorts space-time around it, as described by General Relativity.
The distortion due to the presence of the Sun makes the planets orbit around it on ellipses. 
When light travels through a curved space-time due to the presence of mass it also gets deflected. The deflection is in general small, therefore our view of the near and distant Universe is generally not greatly affected. \\

The gravitational field generated by a mass
distribution, for example due to stars, galaxies, or clusters of galaxies, distorts the image from background objects.
A good analogy is how hot air (above a road in summer, say), distorts the view behind
it, because the index of refraction of air depends on temperature.
Similarly, the gravitational distortion of space-time acts as if the index of refraction varies in space. This effect is called \lq gravitational lensing\rq, although \lq gravitational miraging\rq\ might be a better description since the
distortion is more \lq blurring\rq\ (like the hot air) than focusing
(like a proper lens).\\

A very good introduction to the subject is the recent review by Joachim
Wambsganss\footnote{see
http://www.livingreviews.org/Articles/Volume1/1998-12wamb}, who claims
that a German physicist in 1804 was the first to suggest
that gravitational lensing might be observable, for example in terms of
the deflection of star light passing close to the Sun.\\

Just as with any lens, there are several possible observable effects of gravitational lensing:
\begin{enumerate}
\item {\em deflection}: the image position is displaced from the source
position,
\item {\em amplification}: the image may be become brighter or
fainter,
\item {\em distortion}: if the source is extended (for example lensing
of a galaxy by a galaxy cluster), then lensing may distort the image,
\item {\em multiple images}: in general a single source will be imaged in
multiple images.
\end{enumerate}
All these effects have been observed.\\

Even though gravitational lensing is generally weak, it has become a powerful tool in Astronomy: this chapter discusses some of the applications.

%
%In general, the deflection and amplification in real systems is usually
%very small, therefore our view of the Universe is not greatly distorted by
%gravitational lensing. But occasionally, some objects are lensed quite
%strongly. We'll start by deriving a very simple -- but unfortunately
%not quite correct -- expression for the deflection angle, but also
%state the correct result. Then we'll be armed with enough understanding
%to look at some applications of lensing, and you'll realise the
%tremendous potential of this technique, on a really wide variety of
%fronts, from constraining the nature of dark matter, via measuring the
%masses of galaxy clusters, to detecting planets and constraining models
%of stars.

\section{The lens equation}
\subsection{Bending of light}
Suppose a particle with mass $m$ and velocity $v$ flies with impact
parameter $b$ past an object with mass $M$. The gravitational pull of
$M$ will deflect $m$, and give its velocity a component $v_\perp$ perpendicular
to its initial $v$. For the case $m=M$, this is in fact the situation
we envisaged when deriving the relaxation time in
Section~\ref{sect:relax}. There we found that (see Eq.~\ref{eq:dv})
when the deflection angle $\alpha$ is small,

\begin{equation}
v_\perp = {2GM\over b\,v}\,.
\end{equation}

Convince yourself that this is the correct result also in case $m\ne
M$. Finally, since $\tan(\alpha)=v_\perp/v$ and when $\alpha$ is small
so that $\tan(\alpha)\approx \alpha$,

\begin{equation}
\alpha = {2GM\over b\,v^2}\,.
\label{eq:lens0}
\end{equation}

We use a trick to infer the deflection due to gravitational lensing:  the expression for $\alpha$ does not depend on
the mass $m$ of the particle being reflected, only on the mass $M$ of
the deflector. So we might be tempted to put $m=0$, $v=c$ and claim
this is the angle by which light will be deflected as well.\\

Applying this to starlight grazing the surface of the Sun, we can use
$M=\msun$, $R=\rsun=7\times 10^5\km$,  which yields $\alpha=0.83\arcsec$ --
so $\alpha\ll 1$ is certainly satisfied. This was actually the value
Einstein found in 1911, though {\em Johann Soldner} already obtained it
almost a century earlier.\\

Curiously, the answer is wrong! Once Einstein formulated his theory of
General Relativity, he did the problem again, and found that the
deflection angle is {\em twice} our \lq Newtonian\rq\ value. So the
correct deflection angle for light is

\begin{equation}
\alpha = {4GM\over b\,c^2}\,,
\label{eq:lens1}
\end{equation}

and it was great vindication for GR when Eddington verified this during a solar eclipse in 1920. Note that the deflection
angle increases with $M$ and decreases with $b$ -- the closer you pass
to the more massive an object, the bigger is the deflection.\\

%In the case of the Sun, the deflection is so small that you might think
%it is of little use in Astronomy. This was actually true for a long
%time, but recently it has really taken off as a new way of looking at
%the Universe. Before I show you some applications, we need to do a
%little bit more maths first.


\subsection{Point-like lens and source}
\begin{figure*}
\center\resizebox{.7\textwidth}{!}{\includegraphics{evafig1-crop.pdf}}
\caption{Gravitational lensing when source (S), lens (L) and observer (O) are all
aligned. Light from S is deflected by an angle $\alpha$ toward
O. Because of symmetry, O sees the source S as ring of radius $R_E$
centred around L.}
\label{fig:er}
\end{figure*}

\begin{figure*}
\center\resizebox{.7\textwidth}{!}{\includegraphics{evafig2-crop.pdf}}
\caption{Gravitational lensing of point sources when source (S), lens (L) and observer (O) are not
aligned. O sees two images of S, one of which is depicted here while
the other one lies on the other side of L.}
\label{fig:er2}
\end{figure*}

Suppose light source (S), lens  (L) and observer (O) are
exactly aligned, as in Fig.\ref{fig:er}. Also assume they are 
point like (i.e. not extended like for example a galaxy). Light rays
from S are deflected by L over an angle $\alpha$ toward O. Because of
symmetry of the situation, O sees a ring of light around L: an Einstein
ring\footnote{The figure is not to scale: $\alpha$ should be small if
we want to apply the lensing equation.}.\\

Introducing the distances between observer and lens $D_{OL}$, observer
and source $D_{OS}$, and lens and source $D_{LS}$, we can compute $R_E$
and the angle $\theta_E$ under which O sees the ring. Since
$\psi+\theta_E=\alpha$ we get, using the lensing equation and the figure,

\begin{eqnarray}
\theta_E &=& {R_E\over D_{OL}}\nonumber\\
\psi     &=& {R_E\over D_{SL}}\nonumber\\
\alpha   &=& {4GM\over R_E\,c^2}\,,
\end{eqnarray}

and hence
\begin{equation}
R_E^2 = {4GM\over c^2}\,{D_{OL} (D_{OS}-D_{OL})\over D_{OS}}\,.
\end{equation}

If O, L and S are not aligned, as in Fig.\ref{fig:er2}, O does not
see a ring, but {\em two} images. Denote the angular position of S from the line OL by
$\theta_S$, and the angle between S and its image I by $\theta_I$. Then
$\alpha(D_{OS}-D_{OL})=D_{OS}(\theta_I-\theta_S)$. A bit of juggling
and using the lensing equation yields:

\begin{equation}
\theta_I^2-\theta_I\theta_S = \theta_E^2\,.
\end{equation}

This is a quadratic equation for $\theta_I$ for a given $\theta_S$ and
so there will be {\em two} images, at positions $\theta_{I,\pm}$.\\

Often the change in position is to small to be unobservable (an in fact, we don't usually {\em know} the unlensed position!). But in addition to being lensed, images will also be {\em
magnified} (or de-magnified). So when the displacement is too small to
see two images, you'll only see one but magnified by total
magnification $A=A_++A_-$ of each image separately. Interestingly, for
a small fraction of peculiar alignments $A$ formally {\em diverges}, in practice yielding
potentially very large amplifications.

\subsection{More realistic lensing}
An extended lens yields an odd number of multiple images (3,5,7,...), depending on impact parameter and density
distribution. If the {\em source} is extended as well, then different parts of the source may be deflected and amplified in various ways, and so the image may be highly deformed. We are now in a position
to look for applications. \\

Figure \ref{fig:EinsteinRing} shows an example of an {\em Einstein ring}: a distant spiral galaxy gets lensed by an intervening elliptical, with a fortuitous alignment resulting in an almost perfect ring.\\

Figure \ref{fig:EinsteinCross} shows how a nearby spiral galaxy has lensed a very distant background quasar 
into multiple images, in this case five images.\\

\begin{figure*}
\center\resizebox{0.7\textwidth}{!}{\includegraphics{gravring-crop.pdf}}
\caption{Hubble Space Telescope image of an Einstein ring, where a
fortuitous alignment has caused an intervening elliptical galaxy (the
central spot) to lens a background spiral galaxy into a ring.}
\label{fig:EinsteinRing}
\end{figure*}

\begin{figure*}
\center\resizebox{0.7\textwidth}{!}{\includegraphics{EinsteinCross-crop.pdf}}
\caption{Hubble Space Telescope image of the \lq Einstein cross\rq,
where a distant quasar has been lensed into multiple images by an
intervening spiral galaxy.}
\label{fig:EinsteinCross}
\end{figure*}

\section{Micro-lensing}
\begin{figure*}
\center\resizebox{.7\textwidth}{!}{\includegraphics{microlensing-crop.pdf}}
\caption{Luminosity of a star in the LMC in the blue (top) and red
(bottom) as a function of time. Lensing causes it to brighten around
day 420.}
\label{fig:lmc_lensing}
\end{figure*}
A distant star may be lensed by an intervening star\footnote{I took this part, including the figures, from W Evans' 2003
review, see astro-ph/0304252}, a situation where we may apply the results for point source and point lens. We expect very small deflections, but potentially large magnifications for chance alignments.\\

Fortuitous alignment of a foreground Milky Way star with a star in the bulge, or the LMC, will brighten that star by a relatively large amount (up to a factor of 30), during the few days where the alignment is very good. This phenomena
is called {\em micro-lensing} (because the deflection angle is so small, $\sim 10^{-6}\arcsec$). The alignment needs to be so good that only a very small fraction, around 1 in $10^{7}$, of bulge or LMC stars, is brightened significantly at any time. Movements of the stars breaks the good alignment, which is why the duration of the lensing event is so short.\\

How can we be sure that the variation in flux is due to lensing, and not simply because we found a variable star?
The deflection angle is independent of wavelength, as so is the amount of amplification. This is quite crucial, since luminosity variations in variable stars are usually colour dependent. In addition, the {\em shape} of the light curve is well defined, and because lensing is so unusual, a star brightened by gravitational lensing is expected to be lensed only once.
If the flux detected from a star varies only once, with amplitude independent of wavelength, and with amplitude as function of time as well-fit by micro-lensiing, then we can be confident we have indeed detected lensing.
Figure~\ref{fig:lmc_lensing} shows an example of micro-lensing
of a star in the LMC, which brightens considerably around day 420,
equally much in the blue as in the red.\\

Although we so far only considered lensing by stars, in fact any
massive object will produce lensing. This has been exploited by the
{\bf MACHO collaboration}\footnote{http://wwwmacho.mcmaster.ca/} to
constrain the nature of the dark matter in the halo of the Milky
Way. Indeed, suppose dark matter consisted of some type of massive
objects, like black holes, or Jupiter-like objects, or any other type of
{\bf MA}ssive {\bf C}ompact {\bf H}alo {\bf O}bject. Since
we know that the outer parts of the Milky Way halo are dominated by dark
matter, there should be many such objects between the Sun and the
LMC. If they exist, then we should be able to detect them by their lensing signal.\\

The MACHO collaboration imaged millions of stars in the
LMC every night for four years, and indeed found several lensing events like the
one pictured in Fig.\ref{fig:lmc_lensing}. However, it is now believed
that in each case, the lens is actually a normal star in the LMC, and
{\em not}  a dark matter clump in the Milky Way's halo. As a consequence, this experiment has
now ruled-out a wide range of dark matter candidates. If the dark
matter is some type of elementary particle, it will not lens LMC stars
(well, it will, but $M\rightarrow 0$!), so that is still an option.\\

The future of micro-lensing is very bright. Suppose for example that
the lens is a binary star -- then the light curve of the lensed object
may be much more complicated, as the source may be lensed by each
component of the binary in turn. Such events have already been detected
toward the bulge. This is also a very promising way for detecting {\em
planets} around distant stars, and planned satellite missions will
undoubtedly detect planets in this way.

\section{Strong lensing}
\begin{figure*}
\center\resizebox{0.7\textwidth}{!}{\includegraphics{abell2218-crop.pdf}}
\caption{Hubble Space Telescope image of the rich cluster of galaxies
Abell 2218. The large galaxies in the image are all massive elliptical
galaxies in this galaxy cluster. The tens of arcs are images of
{\em background} galaxies that have been lensed and distorted by the cluster's gravitational potential.}
\label{fig:abel2218}
\end{figure*}
The effect of lensing increases with the mass of the lens. So can we
use galaxy clusters with masses $\sim 10^{15}\msun$ as lens? Yes we
can, and we do get large deflections of several tens of arc seconds up
to several arc minutes.\\

Figure~\ref{fig:abel2218} is an HST image of a rich cluster of galaxies, containing many red ellipticals as expected.
Because of the superb image quality of HST you also notice tens of {\em tangential arcs}, centred on the cluster centre\footnote{Lack of resolution in ground-based observations makes it impossible to see these arcs.}
These arcs are the distorted images of background galaxies lensed by the gravitational potential of the cluster. Such lensing can be used to determine the mass of the cluster, and also to study distant galaxies.\\

The locations and amplification factors of the lensed galaxies depend on the shape of the gravitational potential of the cluster, and hence lensing can be used to constrain the density distribution and also the total {\em mass} of the cluster. 
Note that the signal depends on the {\em total mass}, sum of baryonic and dark matter mass. In addition, the signal does not depend on the dynamical state of this mass. This makes gravitational lensing ideal for mass measurements\footnote{Previously we used the motion of galaxies, {\em assuming virial equilibrium}, or the properties of X-ray gas, {\em assuming} hydrostatic equilibrium, to determine the mass of clusters. Gravitational lensing does not need any such assumptions: the signal depends only on the mass.}. Fortunately different mass measurements of clusters (dynamics, X-ray gas, gravitational lensing) agree reasonably well. All result in a mass considerably higher than can accounted for in stars or gas, hence suggesting large amounts of dark matter. Recent observations of the {\em bullet cluster} even suggest that the centre of mass, and the centre of the X-ray emission, do not coincide, even more suggestive for the presence of dark matter.\\

Several lensed galaxies are {\em multiply imaged}, i.e. you see the
same lensed galaxy at various positions in the cluster. In addition, some
have been magnified by a lot, sometimes as much as a factor of 30, and their angular size dramatically increased. 
The cluster acts as a giant telescope, allowing the study of faint and distant galaxies. The intervening lens makes 
distant small and faint galaxies appear bright and big, and so they can be studied in much more detail\footnote{The cluster also distorts the images of the galaxies: we need a good lens model to infer the intrinsic shape of these galaxies.}. Astronomers deliberately target clusters in order to study better the faint galaxies behind it.\\

\section{Weak lensing}
The arclets seen in clusters arise because the image of the background
galaxy is strongly deformed. Equation~(\ref{eq:lens1}) shows that as we
look at galaxies further from the projected centre of the cluster --
hence with increasing impact parameter $b$  -- the distortion will
become smaller and smaller. Eventually, we won't be able to detect the
deformation for a single galaxy any more, but we could look at many
galaxies in a given small patch of sky, and try to determine whether
perhaps they are all slightly elongated in the same direction, as would
be the case if there were a lot of mass nearby. This is called {\em
weak lensing}: you try to identify the presence of a large mass in a
given direction, from the fact that galaxies nearby in projection, tend
to be elongated due to their distortion, in the {\em same} direction. 
This makes it possible to directly trace the mass distribution, without prejudice of requiring there to be light associated with the mass.\\

\pagebreak
\section{Summary}
After having studied this lecture, you should be able to 

\begin{itemize}
\item derive the Newtonian lensing equation (\ref{eq:lens0})
\item explain why alignment produces an Einstein ring, and derive its
radius.
\item explain how micro-lensing works and has been used to search for
massive compact halo objects in the Milky Way halo
\item explain how gravitational lensing has been used to estimate the
mass of galaxy clusters, and hence infer that they contain dark matter.
\end{itemize}

\tableofcontents

\end{document}
