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\chapter*{Prologue}
\section{Before the beginning}
This course on galaxies runs over 14 lectures. The aim is to give you
an overview of the types of galaxies we know of, and to try to
understand why galaxies look like they do, based on the physics we
think that applies. In contrast to the beauty of stellar physics,
you'll see that the galactic physics is often much more murky and
complex -- a bit like weather prediction: 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 excitementq 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.

My name is Tom Theuns (tom.theuns@durham.ac.uk, tel 8275), and my
office is room 307 in the Ogden Centre. This is the first time I teach
this course, and in case I go too fast (or too slow), I count on you to
tell me! Suggestions, and lists of errors, will be most appreciated.

And for those that are only interested in passing their exams: there
will be two questions on galaxies.

\section{My source of knowledge}
Although this field is continually evolving, much of the basics is of
course text-book stuff. The web provides a very valuable source of
information, not just for pictures. Often, satellite mission web-pages
have extremely well designed pages. Some links you might want to look
at (and were I got many of the pictures) are:

\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/GAIA/
\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}

and the books I've used are

\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}

(I'll refer to these as CO, ZG, BM, and BT.) The first two are clearly
more general, [3] provides a nice overview on galaxies in much more
detail than I'll do, and [4] of course concentrates on the dynamcis of
galaxies. [1] contains everything you {\em need} to know, in [2] you
can often find the basics formulated in a very clear way, and [3] and
[4] are where you could start looking if you want to know more.


\chapter{Introduction}
\section{Historical perspective}
Galaxies were discovered in the 20th century. This is a bit surprising,
since many of them you can see at night with the naked eye. They differ
from stars, in that, even with the naked eye, you can see that they are
extended. For example, Andromeda has an extent of several degrees
(compare to the diameter of full moon, which is 0.5 degrees). And the
Large and Small Magellanic clouds, are even bigger (but you can only
seem them in the Southern hemisphere). And the nearest galaxy (which
was only discoverd in the last years, extends over a significant
fraction of the whole sky! To distinguish them from stars, they were
called {\it nebulae}. However, there are other extra-solar objects
which are also extended, like Planetary Nebulae, star clusters, and
super nova remnants, for example. Since, before the 1920s, we didn't
know the distance to those objects, it wasn't clear what was the
difference between all these types. And so, many of the catalogues of
nebuale, like Messier's catalogue (objects denoted by an \lq M\rq~,
like M31, which is Andromeda), or NGC (New General Catalogue, Dreyer
1888), contain a wide variety of different types of objects.

Immanual Kant, the German philosopher, already figured out that we see
the faint band of stars accross the night sky -- usually called the
Milky Way (MW for short) -- because, on a larger scale, most stars
around us are distributed in a disk, and the sun is located in that
disk. And so, i you look out at night in the plane of the disk, you see
a faint band of light, wich Galileo already had resolved into stars!
And most of these other nebulae you see, were just other
galaxies. Though this was a correct hypothesis of Kant's, it took until
the 1920s before the matter was settled. The historial introduction in
BM is well worth reading.

As a final point: why do we study galaxies? Are they in some sense the
most important building blocks in the Universe? I like BMs reasoning:
stars are huddled together in galaxies. And since (not surprisingly if
you think about it) we (humans) can {\em see} stars, we are naturally
inclined to study them. But, for example, although a cluster of
galaxies contains thousands of galaxies, they are by no means the main
constituent. Most of the {\em mass} is in dark matter -- which we
cannot see at all. But most of the baryons is in hot gas, which we can
{\em observe} with X-ray sattelites, but we cannot {\em see}. And so
the choice to study galaxies is a bit human centred: our eyes can see
them.

\section{Bringing order to the zoo of galaxies}
Before we start doing any calculations, it might be a good idea to look
around us and make an inventory of what types of galaxies are out
there. There are many good web pages to start, for example I like
http://www.taas.org/gnto/astropic.html, because it shows pictures made
with a small telescope -- so those are galaxies as you might have seen
them yourself, if your an amateur
astronomer. http://www.aao.gov.au/images.html/ is very nice as well, it
shows professional images. If you look at the more professional pages
(such as the HST archive for example), beware of an amusing bias:
professional astronomers tend to observe (and make picture of) unusual
galaxies.

\subsection{Vocabulary}
{\bf Colours.} These images are usually made through a broad-band
filter, typically \V, \B or \R, and so should be just black and
white. The blackness being a measure of how much, or how little of the
light in that wavelength band fell on that part of the
photograph/CCD. One often combines images in different filters to
produce a colour image, which tries to mimick the colour your eye would
see. But often, one of the filters is a narrow-band filter, e.g. in
H$\alpha$ or O[III]. Combing with other filters, we get a false colour
images, i.e. the colour of the object is {\em not} how your eye would
see it, but the colour coding has been chosen to bring out a particular
feature. Those really pretty colour pictures of Planetary Nebulae are
false colours.

{\bf Surface brightness.} The brigthness depends, apart from the size
and quality of the optics of the camera, the integration time, and the
transparancy of the atmosphere (all of which we should be able to
calibrate-out), on the apparant brightness of the galaxy. So this is
what we want to characterise. Since galaxies are extended, the
brightness varies over the image, and so what we want to measure is the
surface brightness (SB). The SB measures how much light we receive from
the galaxy, per unit solid angle (usually per \arcsecsq) so it should
have units of flux (e.g. in W), per unit wavelength range, per unit
solid angle on the sky. So for example in units $L_{\odot\V}\arcsecsq$,
in units of the solar \V luminosity. Usually, SB is expressed in
magnitudes, by taking $-2.5\log_{10}$ of the previous number -- we say
(incorrectly, or at least confusingly) that we measure SB in magnitudes
per square arc seconds. (It's confusing, because this suggests we
divide the magnitude by \arcsecsq, but no: we first divide by the solid
angle, and then take the logarithm). As is usual with magitudes, we add
a constant to it.

The number that characterises the {\rm intrinsic} brightness
distribution of the galaxy (as opposed to apparent) is how much light
the galaxy emits per unit {\rm surface area}, so in for example units
of $L_{\odot\V}\kpc^2$. Astronomers do not use separate names for
these two types of SB, so check the units!

{\bf Star counts} To quantify this a bit more, assume you observe a
distribution of stars, all of the same absolute luminosity $L$, and
with uniform number density $n$ (in stars per \pc$^3$, say). The flux
you receive from a single star, when at distance $r$, is $L/4\pi r^2$
(in W\,m$^{-2}$). The number of stars at distance between $r$ and
$r+dr$, and within a solid angle $d\Omega$ on the sky, is
$dN(r,d\Omega)=nr^2drd\Omega$. For the whole sky, $d\Omega=4\pi$, and
so the number is $dN(r)=4\pi nr^2dr$, which is the product of the volume
of the shell, $4\pi r^2dr$, with the density $n$ of stars. Combining
these two, the luminosity of all stars within $r$ and $r+dr$, and
within $d\Omega$, is
\begin{equation}
dL = LdN(r,d\Omega)={L n \over 4\pi}d\Omega dr\,.
\end{equation}

The surface brightness is $\propto dL/d\Omega=L ndr/4\pi$.  For an
external galaxy, you could integrate $\int ndr$ along the line of sight
through the galaxy, to get the surface density of stars, $\int
ndr=\Sigma$ of stars. The SB is then $L\Sigma$. The important thing to
notice is that it is {\em independent} of the distance $r$.  Going back
through the calculation, you can see what has happened: the luminosity
from a single star at distance $r$ {\em decreases} $\propto 1/r^2$, but
the number of stars within $d\Omega$ {\em increases} $\propto r^2$. And
since the SB is proportional to the product of these two, it is
independent of $r$\footnote{In a cosmological setting, this argument
does not hold, and the SB of very distant galaxies decreases with
distance, an effect called surface brightness dimming.}. (You might
object saying we've assumed all stars to have the same luminosity. So
just redo the calculation with the {\em average} luminosity then!)


When looking-out into a stellar distribution -- for example when
counting stars in the Milky Way -- we should be able to infer $n(r)$ by
counting how many stars we see as function of their luminosity. Here is
how to do it: the number of stars within a solid angle $d\Omega$, and
with distance between $r$ and $r+dr$, $dN(r,d\Omega)=nr^2drd\Omega$, as
we had before. Now again assume them to have all the same intrinsic
luminosity $L$. The trick is to change variables from distance $r$ to
apparent magnitude $m$, using
\begin{equation}
m=-2.5\log(L/r^2)=M+5\log(r)\,,
\end{equation}
up to a constant. Recall where this came from: since the observed
luminosity $\propto L/r^2$, the apparent magnitude
$m=-2.5\log(L/r^2)=-2.5\log(L)-5\log(r)=M-5\log(r)$, up to a
constant. (Also recall that, by convention, $m=M$ for a distance of
$r=10\pc$. And hence the proper equation is $m-M=5\log(r)-5$, when $r$
is expressed in \pc. $m-M$ is called the {\em distance modulus}.) Now
we're almost done, since rearraging
\begin{equation}
r\propto 10^{0.2(m-M)}\,,\nonumber
\ee
and by differentiating
\be
dr/dm\propto 10^{0.2(m-M)}\propto r\,.\nonumber
\ee

Substituting this in the expression for $dN$ gives
\be
dN(m,d\Omega)\propto n(m) 10^{0.6(m-M)} dm\,.
\ee

For example, if the number density is constant (and hence also $n(m)$
is constant), then, $dN(m+1,d\Omega)/dN(m,d\Omega)=10^{0.6}\approx
4$. In other words: by going one magnitude fainter, we get around 4
times as many stars. If we get fewer, then either the more distant stars
are intrinsically fainter, or light has been lost on the way, or the
density of stars drops (or some combination).

As a last exercise: we found that the luminosity of all stars in a
shell, within a given solid angle, was independent of the radius of the
shell. So, integrating over all radii, we get an infinite
luminosity. Which clearly conflicts with the fact that the night sky is
dark. This is called Olber's paradox. What is wrong with this
reasoning?

\subsection{Galaxy properties}
Looking at the different images of galaxies, it becomes immediately
clear that there are two basic types, elliptical and spiral galaxies.

Ellipticals (also called early type galaxies)
\begin{itemize}
\item have smooth light distributions, where the isophotos (lines
of constant SB) have (nearly) elliptical shapes. They are denoted as
E$n$, where $n=1-b/a$, and $a$ and $b$ are the semi-major and
semi-minor axis of the ellipse. So, an E0 is round, and an E7 is
flattened, $n=7$ is about how flattened E's get. A 3D ellipsoidal
shape, with semi-axes $a$, $b$ and $c$ (ordered as $a\ge b\ge c$), is
called oblate when $a=b>c$ (a pancake), prolate when $a>b=c$ (a cigar),
and tri-axial when $a>b>c$ (a squashed American football). We think
that E's are probably tri-axial.
\item have no current star formation
\item have old stars
\item have little gas and dust
\item occur preferentially in groups and clusters of many Es
\item have little evidence for rotation, so the stars must have large
random motions to keep the system from collapsing (also called a hot
stellar system)
\end{itemize}

Spirals (or late type galaxies)
\begin{itemize}
\item have most of the light coming from a thin disk. Seen face on, it
is nearly circular, seen edge on, it is nearly a line. In their centre,
they often have a nearly elliptical stellar system, similar in shape to
a small elliptical, called a bulge. In addition, some have a bar as
well.
\item have light profiles with a characteristic spiral pattern
\item have star formation going on, especially in their spiral arms
\item have both old and young stars
\item have gas and dust
\item seem to avoid dense groups, and are very rare in clusters
\item have large rotational veleocities, enough to keep their stars on
circular orbits. 
\end{itemize}

We will see later that the SB as a function of radius, for these two
types, are quite different.

These are the archetypical properties of such systems. For example,
some ellipticals do have a lot of dust, as well as evidence for star
formation. Some even have a small disk. Given that some spirals have a
very large bulge, and some elliptical have a disk, the distinction
between the two types is not always as clear as you might
think. Clearly it would be interesting to know what processes are
responsible for forming an elliptical vs a spiral.

Both types are seen to have hundreds and sometimes thousands of {\em
globular clusters} (GCs for short). These are very dense, spherical,
gravitationally bound systems of typically $10^5$ to $10^6$ low mass
stars, and they are usually found in a spherical distribution around
their parent galaxy. The Milky Way has about 150 of them, but some of
the larger Es have several tens of thousands.

There are many schemes for classifying galaxies, based on their
morphology. What do you expect from a good classification? Ideally, it
should be unambigous, and have some physical basis behind it. Also,
since for most galaxies you don't know the distance, it should be
independent of distance. The {\em Hubble sequence} is still very
popular. It divides galaxies in ellipticals and spirals. Spirals are
divided in two strands, according to whether they are barred (SB), or
not (S).  The sequence has the shape of a tuning fork -- the Hubble
tuning fork. Along the handle of the tuning fork, the range goes from
E0 to E7. The sequence then forks, towards the top are unbarred spirals
-- denoted S, along the bottom are barred spirals, denoted SB. Both
strands are further divided along the sequence (away from the handle),
as Sa towards Sc, and analogously SBa to SBc, depending on the relative
strength of bulge to disk, and, correlated with this, how
tightly/loosely wound the arms are. Sa's and SBa's have a large bulge
and tightly wound arms, Sc's and SBc's have a much smaller bulge, and
more loosely wound arms. Finally, any galaxy that does not fit into the
sequence, is called irregular (Irr).

The physical scale of galaxies is huge: E's can have masses from as
little as $10^7\msun$ to $10^{13}\msun$, 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.

Hubble thought (incorrectly) that this sequence was due to evolution,
where galaxies evolve from being E0s, towards Sa or Sba, and later
evolving in the more loosely wound Sc's or SBc's. Therefore these types
are also called early type and late type. So E's are also called early
type, and Ss are called late type. And an Sc is a late-type spiral, and
Sa an early type spiral. We know that this is not the way galaxies
evolve, yet the names are still widely used.

There are other reasons why this is not a particularly good scheme. For
example, galaxies may look quite different, depending on which filter
you use.

Another type you'll often come accross are cD galaxies, which stands
for {\bf c}entral {\bf D}ominant: these are very large ellipticals, found
in the centres of clusters, with a faint but very large outer halo of
stars.

Another type are called S0 (or SB0 when barred) or lenticulars: they
are the divide in Hubble's sequence between E and S. They have disks
without gas or dust, and no recent star formation. 

Galaxies can fail to fit in the sequence, for example because they
recently had a tidal encounter with another galaxy. This typically
leads to spectacular tidal arms, like the case of e.g. the Antennae.

Now, most smaller galaxies are of the Irreglar type: no disk, no nice
spherical distribution of stars, but more a loose aggregate of stars,
like for example the Magellanic Clouds. Since there are far more small
than large galaxies, most galaxies are in fact of type Irr. The aim of
these lectures is to understand which types of observation can be used
to describe these different types a bit better (the \lq biology bit of
the lectures\rq), and also try to understand the physical processes
behind them. May be you'll be surprised to learn that, for many of the
most basic questions you might think of, such as -- why are some
galaxies of type E, and others of type S?, or, why do Ss have spiral
arms? -- we are only just beginning to understand the underlying
physics, and we have by no means a well defined, generally accepted
textbook answer. May be something for you to work on, when you start
doing astronomy research yourself?



\chapter{The discovery of the Milky Way and of other galaxies}
Around the 1920s, a great debate was raging about the nature of the
nebulae, with two camps. According to one camp, the nebulae were
structures within the Milky Way, probably proto-planetary systems or
something similar, and the Sun was near the centre of MW. The other
camp claimed that many of the nebulae were galaxies outside of the MW,
and the Sun was at large distance from the centre of the MW. The reason
they couldn't decide was mainly because they didn't know how to
determine distances to the nebulae. The reason the first group got it
wrong was not because they were stupid or anything like that: their
mistake was quite interesting. And so we'll go through it, hoping we
can learn from their mistakes. Before we do this, let's describe the
main observables, and observing techniques.

\section{The main observables}
\noindent{\em Star clusters} Two types of star cluster were (and are)
known. We already described {\em Globular clusters}, which are
spherically distributed around the MW. Stars form in the MW disk out of
large molecular clouds. Sometimes, after the gas has been dispersed,
the newly formed stars are gravitationally bound to each other, in what
is called an {\em Open clusters}. They typically contain of order
$10^4$ stars. They are of great importance for studies of stellar
evolution, since we know all the stars in it have (more or less) the
same age, distance and initial composition.

\noindent{\em Star counts} Previously we described how star counts can
be used to probed the number density of stars, as a function of
position. In the early days, William Herschel and his sister, counted
the numbers of stars as function of their magnitude in many (700!!)
directions in the sky, by spending (most?) nights gazing through his
telescope! Later, photographic plates were used for the same
purpose. These confirmed that the stars around the Sun are distributed
in a disk.

\noindent{\em Standard candles} are objects with a known absolute
property, for example a known size, or known luminosity. So measuring
their angular size size, or apparent luminosity, you can determine
their distance. A most important example is that of {\bf Cepheid
variables}. Henrietta Leavitt studied variable stars in the Magellanic
Cloud in 1912. She found a relation between the period $P$ of the
variation, and the luminosity of Cepheid. Since all these stars are at
the same distance, this must mean it is a relation between the {\em absolute}
luminosity and $P$. There are many types of variable stars, but
Cepheids have many advantages (a) the shape of their light curve is
a very characteristic sawtooth pattern, (b) they are luminous - and so
can be seen to large distances and (c) the $P(L)$ relation has little
scatter. Unfortunately, they are also relatively rare. {\bf RR Lyrae}
are similar to Cepheids, but occur in a different type of star. They
are also used as standard candles. Of course, to get an absolute
distance, we need somehow to find the distance to some Cepeids or RR
Lyrae using another method, for example using the parallax. 

\noindent{\em Parallax} Stretch your arm, point your index finger
upward, and look with your left eye alone towards a distant wall. Now,
look with your right eye alone: your finger seems to move with respect
to the background. You've done (your first?) parallax measurement. If
the wall is sufficiently distant, there is a relation between how much
your finger appeared moved, the length of your arm, and the distance
between your eyes.

In astronomy, you can increase the distance between your eyes, to twice
the distance earth-Sun, by looking at the same object half a year
apart. Since one can relatively easily measure angles to a fraction of
an \arcsec\ (a 60th of an arc minute, or 1/3600th of a degree, now we
can do much better!), we have a new distance unit: the \pc. An object
at 1\pc distance, has a parallax of 2\arcsec. Let's compute how much
this really is. Consider an equi-lateral triangle, with two sides of
lenght 1\pc, third side 1\AU. By definition of \pc\, 1\pc/1\AU=1\arcsec
in radians, or 1/206265 radians (approximately). (We've used the fact
that this angle is very small). Hence \be 1\pc={180\times 3600\over
\pi}\AU\approx 206265\AU\,.  \ee

Unfortunately, this litterally doesn't get use very far: the distance to
even the nearest stars is of this order. We'll come back to the
distance scale later.

\section{The main players, their measurements, and their mistakes}
The famous Dutch astronomer, {\em Jacobus Kapteyn}, and his
collaborators, started a major project of counting stars around the
1900s, to determine once and for all the structure of the MW. So what
he did is count stars on photographic plates as function of their
luminosity, in several directions.

His analysis basically confirmed Herschel's picture: 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. As we'll discuss later, this is
wrong!

In our derivation of counts of stars as function of distance $r$, we
assumed that the luminosity drops as $1/r^2$. One reason this could be
wrong, is because of absorption or scattering of the stellar light. For
Rayleigh scattering off atoms, there is a colour dependence. Of course,
not taking absorption into account would give you the wrong answer, and
Kapteyn was fully aware of this. If absorption were important, one
would expect fainter (more distant stars) to be on average
redder. Kapteyn found this indeed to be the case, yet the amount of
reddening was very small, and so absorption was no thought to be an
issue. The mistake was that the absorption is due to {\em dust}, not
Rayleigh scattering. Dust absorption is much less colour dependent,
and so the small reddening measured by Kapteyn actually implied a much
greater dimming of the distant stars.

The American astronomer {\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 was in conflict with the
Kapteyn universe -- it seemed difficult to reconcile the Sun to be near
the centre of the MW, yet the centre of the globular cluster system not
to coincide with the centre of MW?

Meanwhile, another famous Dutch astronomer, {\em van Maanen}, found
that some of the nebulae had proper motions, implying they were nearby,
within the MW. (He compared the positions of the nebulae on plates
taken several years apart). It was generally accepted these nebulae
were probably proto-planetary systems, or something like that. {\em
Edwin Hubble} later redid the proper motion measurements with better
data (using a better telescope), and failed to reproduce the proper
motions. van Maanen probably just made a mistake! In the 1930s, Hubble
made his even more famous discovery that the spectral lines of atoms in
other galaxies are shifted to the red, by an amount proportional to the
distance to the galaxy. The universe is expanding!

So it was a combination of mistakes (van Maanen), and not taking into
account the importance of dust (Kapteyn), that led to the confusion.
In the end, Hubble's better data clinched the matter. He resolved
Cepheid's in Andromeda, one of the bigger nebulae, and found a distance
of 300\kpc\, and so clearly outside of the MW. And so the size of the
universe all of a sudden increased dramatically.

It was not until the 1930s that {\em Trumpler} found what had been
wrong with Kapteyn's results. He studied Open Clusters in the MW
plane. He assumed them to be all of about the same size, and so got an
independent relative distance for them. He used another estimate of
their distance, based on \lq spectroscopic parallaxes\rq\, by observing
the spectral type of some stars in the cluster, and using a calibration
between spectral type and absolute magnitude. He showed that the more
distant clusters (as judged from their angular size), appeared to have
unusually faint stars in them. Or equivalently, that the more distant
clusters (as judged from their photometric parallax) were
larger. Clearly, something was wrong here. He correctly interpreted
this, as due to the fact that the light from these more distant stars
was strongly attenuated. By comparing the reddening with the
attenuation, it was clear that a given amount of reddening corresponded
to much more dimming than you'd expect for Rayleigh scattering. And
hence Kapteyn's scales were wrong.

Before I summarize this in a time-line, just think of the fact that
Einstein formulated his theory of general relativity, on which the
current cosmological models are still based, before this puzzle was
resolved. In fact, our view of the Universe around Einstein's time, was
just plain wrong. I guess he rightly paid little attention to what the
astronomers of his days told him!

\subsection{Time-line}
\begin{enumerate}
\item[1610] Galileo resolves the MW into stars
\item[1750] Immanual Kant suggest some of 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, axes ratio 5/1
\item[1900-1920] Kapteyn counts stars, decides wrongly that extinction
is unimportant, and deduces the MW is $5\kpc\times1\kpc$ big, with the
Sun at 650\kpc\ from the centre.
\item[1912] Leavitt discovers the $P(L)$ relation for Cepheids.
\item[1914] Slipher measures large (1000s\kms) velocities for some
Nebuale, and finds evidence for rotation. Spectra suggest presence of
stars, not of gas.
\item[1915] Shapley find centre of globular clusters far away from
Kapteyn's MW centre.
\item[1920] Van Maanen claims (erroneously) that some spiral nebuale
have a large proper motion, suggesting they are within the MW.
\item[1920] Shapley and Curtis debate publically over the size of the
MW, but the matter is not settled.
\item[1923] Hubble resolves M31 (Andromeda) into stars, using the newly
commisioned100-inch telescope. So M31 must be outside the MW. He also
discovers Cepheid's, and the distance to M31 is estimated at 300\kpc.
\item[1926] Lindblad computes that Kapteyn's MW is so small, it cannot
gravitaionally 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 Sagitarius. 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}

And so in a very short time indeed, three sweeping changes in our view
of the visible universe took place:
\begin{itemize}
\item an 8-fold increase in the size of the MW
\item the acceptance that many of nebulae were external galaxies
\item the universe is expanding
\end{itemize}

\section{Absorption, scattering and reddening}
What is the effect of dust on the light from distant stars? Suppose the
wavelenght of the light is much smaller than the size of the dust
grains (bricks, say) -- then you could compute the dimming just from
the geometric cross-section of the bricks along the line of sight. Now
consider a ray of light, with itensity $I$. The amount of light of this
ray, absorbed per unit lenght, will satisfy
\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. This is what we want since,
if the light source were twice as bright, then the bricks will stop
twice as many photons. $A$ is some constant, which will depend on the
size of the bricks, and their number density. This equation is easy to
integrate, and the solution is $I=I_0\exp(Ar)$. Taking the $\log$ of
both sides, we find that the effect on the magnitude will be $\Delta m=A\,r$.
So the relation between apparent ($m$) and absolute magnitude ($M$) will
change from $m-M=5\log(r)-5$ to
\begin{equation}
m-M=5\log(r)-5+A\,r\,,
\label{eq:dm}
\end{equation}
due to the effect of absorption. So the unit of $A$ is magnitudes per
\kpc\,. (Careful readers will notice that I have slightly changed the
definition of $A$ along the way.)

If the size of the particles is of order of the wavelength $\lambda$ of
the light, then it is not a good approximation to use the geometric
cross-section of the dust grains. The amount of absorption (the value
of $A$) will then depend on $\lambda$, and so $A_\V\ne A_B$: absorption
in the \B\ band will differ from that in the \V\ band, and so
absorption will also lead to a colour change (reddening, since the
longer wavelenghts get {\em less} absorbed than the shorter ones).

Appling Eq.(\ref{eq:dm}) to two bands, we find
\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\,.
\end{eqnarray}
The quantity $E_{\B-\V}$ is called the 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
$A_\V\approx 1{\rm mag}\kpc^{-1}$\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}. If we now measure $E_{\B-\V}$ for another star,
we can estimate $r$! (Well, we need to know its intrinsic colour from
models.

\chapter{The modern view of the Milky Way}
Hipparcos/GAIA
COBE/DIRBE
Radio observations

Our observational view of the MW has improved due to steady hard work,
but occasionally in great leaps, when new technology was applied to old
problems. Let's quickly describe some of these.

\section{New technology}
\subsection{Radio-astronomy}
Radio astronomy used the new technology developped during the war (for
other purposes ...), and really
flourished. Roto-vibrational\footnote{Most transitions you know, which
occur in the visible and UV-part of the spectrum, are {\em electronic}
transitions: in emission, they occur because an electron changes from a
higher energy state to a lower one, e.g. from $n=2\rightarrow 1$ for
the hydrogen Lyman-$\alpha$ transition, and from $n=3\rightarrow 2$ for
the hydrogen H$\alpha$ transition. An in absorption, they go the other
way around. But melocules also have excited levels, due to their
rotation or vibration. Since these are also quantum-mechanical, these
energy levels are quantized. And transitions between them correspond to
rotational or vibrational transitions. Since the energy-levels are
lower, these correspond to longer wavelengths -- typically mm and cm:
in the radio.} transitions in molecules have low exitation levels,
which are ideally suited to probe the low tempertatures in interstellar
clouds. Because the wavelenghts of these transitions are long, one can
build very large dishes to observe them, and also it is more
straigthforward to build interferomoters.\footnote{One way to think of
this is to realize that for a good \lq mirror\rq\, it's departure from
the ideal shape should not be larger than some small fraction of the
wavelenght of the radiation you plan to observe. So for optical
telescopes, it should be of the order of 5000\AA say, but for radio,
fractions of a mm, and up to a m. There is a down-side. The resolution
you get, when making diffraction limited observations, is of order
$\lambda/D$, where $\lambda$ is the wavelength, and $D$ the diameter of
the telescope. And so, radio-telescopes have to be much bigger to
obtain a similar resolution. In an interferometer, instead of building
one giant dish -- which would be very expensive, and difficult to move
-- one builds many smaller dishes, put far apart, and combines their
output. The result is that the effective $D$ is not the size of a dish,
but rather the biggest distance between them. What you loose, is that
the collecting area is only the sum of the sizes of all dishes, not $D^2$.}

Today, one regularly combines data from radio-observatories accross the
earth to get Very Long Baseline Interferometry, which gives superb
angular resolution ($10^{-6}\arcsec$!). Another great advantage of
radio over optical observations, is that radio waves are not absorbed
by dust. (By now you know why: it's because the wavelenghts are much
larger than the size of the dust grains.) And so, yo can use
radio-waves to probe into very dense regions, for example where star
formation occurs. One of the bigger problems facing radio-astronomy
today, is mobile phones ...

\subsection{Infrared observations}
IR astronomy took-off relatively recently, with the advent of rockets
and satellites to get the instruments above the atmosphere. The
construction of the camera for near IR observations is
challenging. They need to be cooled, since otherwise the main source of
signal is thermal radiation from the camera itself. I suspect that,
again, a major driver behind their construction was not really
astronomical, but over the past 10 years or so, it seems that much of
the technology has becomes available to the astronomers.

IR-observations share with radio the great advantage not to be too
bothered about dust. In fact, much of the visible and UV-light absorbed
by dust grains, is re-emitted in the IR and submm. And so we can use IR
to probe the dust and look deep inside star forming regions. We thank
the DIRBE instrument on the COBE (Cosmic Background Explorer) satellite
for one of the best views of the Milky Way.

\subsection{Star-counts, again}
Although counting stars seems a bit boring, the Hipparcos satellite
mission\footnote{http://astro.estec.esa.nl/Hipparcos/} was really a
tremendous revolution, due to the incredible accurary of its
measurements. This satellite was launched in 1989 and specifically
designed to measure very accurately the positions of stars, several
times during the satellite's lifetime. For distant stars (some of which
were in fact very bright, extra-galactic sources called quasars), this
made it possible to construct an absolute reference system, that can be
used define the positions of extra-solar objects. But for more nearby
stars, Hipparcos could not only measure their parallax and hence their
distance, but also, by comparing the positions at two different times,
the {\em propoer motion} of the stars with respect to this \lq
fixed\rq\ reference system.

The parallax is the ultimate step in creating a distance scale. We
first need to determine the parallax to standard candles, like
e.g. Cepheids, before we can use them to look further. 

The proper motion, i.e. the dance of nearby stars around us, tells us
about the distribution of mass. For example, the oscillation of stars
perpendicular to the disk of the MW tells us how much mass there is
locally in the disk.

GAIA\footnote{http://astro.estec.esa.nl/GAIA/} is a planned mission to
be launched before 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 of almost inconceivable extent and implication
(according to the webpage). 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.

\section{The components of the Milky Way}
We've already briefly discussed what the major constituents of the MW
are: stars, gas and dust. Now we'll describe how they're
distributed. We think the MW is a pretty much ordinary spiral galaxy,
of which we know several million others, but we know this one in a bit
more detail. Even so, bare in mind that some of the details are
uncertain. We will discuss the {\em disk} (there is a thin and a thick
disk), {\em bulge} and {\em halo}.

\subsection{The disk}
Most of the MW's stars are in a thin disk, which represents about 70 per
cent of the total star light of the MW. The centre of the disk is in
the direction of Sagittarius, $\alpha=17^h\,42^m\,29.3^s$,
$\delta=-29^{o}\,59'$, at a distance of $R_0$. The latter is called the
solar galactocentric distance, and there is quite some discussion to
how large it really is. $R_0=8.5\kpc$ is usually quoted, but some
studies find $R_0=8.0\kpc$. These studies try to determine the centre
of a set of objects, for example globular clusters, variable stars such
as RR Lyrae or Mira variables, or O and B stars. An $H_2O$ maser source
is located near the centre of the MW. The maser emission is thought to
come from a spherical shell, and one can measure both the radial and
transverse speed of the shell. If the shell is expanding spherically,
then one can determine its distance.

Apart from stars, the disk also contains gas and dust. On top of the
smooth disk are spiral arms, traced by young stars, molecular clouds,
and ionized gas. The disk stars are in (nearly) circular motion around
the centre. For the sun, the circular velocity is $\approx 220\kms$.

How old is the disk? We don't really know. But we can put limits on the
ages of the {\em stars} in the disk, but have to keep in mind that the
stars could possibly be older than the disk (i.e., they have formed
somewhere else). One neat age estimator are White Dwarfs. As you
recall, these donnot undergo nuclear fusion anymore, but are presently
just cooling down. So, by measuring their current temperatures, and
cooling rates, we can compute how old they are. Some of the older White
Dwarfs are thought to be $\sim 10--12\times 10^9$ years old.

Let's describe the position of a star in the MW using cylindrical
coordinates $(R,\varphi,z)$, where $R$ is the distance to the centre,
$z$ the distance to the plane, and $\varphi$ an angle. The density of
stars in the disk goes approximately like $n(R,\varphi,z)\propto
\exp(-R/R_h)\exp(-z/z_h)$. Since the disk is (nearly) axisymmetric,
$n(R,\varphi,z)$ is independent of $\varphi$. Away from the centre, $n$
decreases exponentially, and at distance $R_h\approx 3.5\kpc$, it has
fallen to $1/e$ of it's central value. Also perpendicular to the disk,
the fall-off is exponential, with $z_h\approx 0.3\kpc$. $R_h$ and $z_h$
are called the {\em scale length} and {\em scale height} of the disk,
respectively. These scale lengths really depend on the type of star
you're using. Young stars are born out of gas, which is much more
concentrated towards $z=0$. And so the scale height of young stars is
much smaller than of older stars. 

Note that there isn't really an edge to the disk. 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 disc!

Given the exponential distribution in $n$, the luminosity profile (in
$L_\odot$ per unit volume) of the disk is often modelled as

\begin{equation}
L(R,z)=L_0 \exp(-R/R_h) {2\over \exp(z/z_h)+\exp(-z/z_h)}\,,
\label{eq:diskL}
\end{equation}
where $L_0$ is some normalisation constant. In the \B-band, the total
luminosity is $\sim 2\times 10^{10}\lsun$.

Because of the component I'll describe next, the disk is also sometimes
called the thin disk.

\subsection{The thick disk} About 4 per cent of the MW's stars belong
to a thicker disk, which is aligned with the (thin) disk, but has a
scale height of $z_h\approx 1\kpc$. It's existence has been discovered
only recently, in the 1980s. The difference with the thin disk is not
just the kinematics of it's stars: it turns out that thick disk stars
are usually older than the stars of the thin disk, and have a lower
metallicity [Fe/H]$\sim $-0.5, i.e. they have less Fe (and other metals
as well, but we can usually measure Fe accurately) for a given amount
of hydrogen (we'll see later what this notation means).


\subsection{The bulge} The bulge is a spheroidal stellar system with radius of
$\sim 1\kpc$, located at the centre of the MW. In fact, it's properties
are quite similar to that of an elliptical galaxy: it is as if there is
a small elliptical galaxy at the centre of each spiral. The surface
brightness profile (in $L_\odot$ per unit area) of this component goes like

\begin{equation}
I(r)=I_e \exp[-7.6(r/r_e)^{1/4}]\,.
\label{eq:r4}
\end{equation}

This is a famous fit to the SB-profile of elliptical galaxies,
introduced by {\em de Vaucouleurs} in 1948, and called the \lq de
Vaucouleurs\rq\ or \lq $r^{1/4}$\rq (r-to-the-one-quarter) profile.
($r=(R^2+z^2)^{1/2}$ is the 3D distance to the centre.)

$r_e$ is called the effective radius. Note that $I(r=r_e)=I_e$. The
factor 7.6 is chosen such that half of the light comes from stars
within $r_e$. For the MW, $r_e\approx 0.7\kpc$. 

Although the bulge is bright, we can't see much of it, due to the large
amount of obscuring dust (in particular, you can't see it with the
naked eye!). The total amount of extinction towards the centre is abut
28 magnitudes in the visual! In other spirals, we can see the bulge very well (and
recall that the classification of spirals is based on the bulge/disk
ratio), but for the MW we need to go the the IR to get a nice picture.
of it. The MW's bulge is rather elongated with axis ratio 5:3, with
strong evidence for a bar.

The stellar content of the bulge is a bit of a mixed bag, with both
very old, low metallicity stars, but also young, metal rich stars. The
mean meallicity is about {Fe/H]$\sim 0.3$, i.e. twice the solar value. 
The mass of the bulge is $\sim 10^{10}\msun$, with a \B-band luminosity
of $\sim 3\times 10^9$\lsun. 

\subsection{The stellar halo}
The final luminous component of the MW is the stellar halo, composed of
globular clusters (GCs), and \lq field stars\rq. These stars, just like
the stars in the globular clusters, are typically of very low
metallicity, [Fe/H]$\sim -0.8$. The current record holder of low
metallicity has [Fe/H]$\approx -5$.

Unlike the disk, the halo rotates very little, if at all. As a
consequence, halo stars all appear to move at high velocity (in fact,
rather it is the sun that is moving at $\sim 220\kms$), which is how
they were discovered initially.

We know about 150 GCs in the halo. There is some evidence that there
are two types, distinguished according to age, metallicity and
position. The younger, more metal rich ones have a relatively flattened
distribution, and may in fact be associated with the disk. The older,
more metal poor ones, are spherically distributed.

There is quite some discussion on the current ages of GCs, with
estimates for the oldest ones varying between 12 and 17Gyears
(1Gyear$\equiv 10^9$ years). For a long time, these estimates were
rather embarrassing for cosmologists, given that this was older than
the best estimates for the age of the {\em Universe}. This discrepancy
is now resolved, since the Universe is thought to be older now, and the
ages of the clusters have come down a bit.

If the absolute age of the GCs is difficult to estimate, the relative
ages are more secure. And so it is rather more secure to say that there
is a relatively large spread in the ages of the MW's GCs. Presumably,
this tells us something about how the MW formed.
 
The density of halo stars (and of GCs as well) falls spherically as
\begin{equation}
n(r)=n_0 (r/r_0)^{-3.5}\,,
\end{equation}
and extremely distant field stars have been detected out to
$r=50\kpc$. The total mass is actually quite low, $M\sim 10^9\msun$ and
luminosity $\sim 10^9\lsun$, or only 10 per cent of that of the disk.

\subsection{The dark matter halo}
We'll discuss evidence for dark matter later, but it appears to be the
case that most, may be as much as 90 per cent, of the mass in galaxies
is actually invisible. We don't know what it is made of, but this
matter does not emit or absorb any light -- hence {\em dark} matter,
yet there seems to be a lot of it.

\section{Metallicity of stars}
Astronomers call all elements more massive than Helium\footnote{The Big
Bang produced hydrogen, helium, and trace amounts of more massive
elements} \lq metals\rq, and denote them by $Z$ ($X$ and $Y$ being the
hydrogen and helium abundance by mass, respectively). These are
produced in stars, but for some elements, like e.g. Carbon, we don't
really know which stars are the dominant source. A lot of metals are
produced during super novae (SNe) explosions. There are two basic types
of SNe. SNe of type II are explosions of very massive stars. Nuclear
fusion during the explosion is responsible for many elements of the
$\alpha$-type, like Oxygen and Silicon. SNe of type I, on the other
hand, are thought to occur during mass-transfer in binary stars --
these are the dominant producers of Iron, Fe. The chemical mix of the
stars is therefore a fossil relict of the relative fractions of type I
and type II SNe. Typically though, one measures just one metallicity,
for example the Fe abundance since that is often the easiest to
measure, and then assumes that the other elements scale with Fe. But we
know for a fact that some stellar systems have a rather different
relative abundance pattern of the elements.

For the sun, the total amount of metals by mass, is about
$Z_\odot=0.02$, or 2 per cent of the mass in the solar system is {\em
not} hydrogen or helium. This is inferred not just from observations of
the sun, but also from the composition of comets.

For other stars, one usually compares the metallicity in units of the
solar value, on a logarithmic scale,
\begin{equation}
[Fe/H]\equiv log_{10}\left[(Fe/H)/(Fe/H)_\odot\right]\,.
\end{equation}

So if a star has [Fe/H]=0, it has the same Iron abundance as the Sun,
and for [Fe/H]=-1, it has one tenth the solar value.

One might expect the metallicity of a star to be related to its
age. Indeed, very old stars formed before there had been many
generations of stars\footnote{The Sun is thought to be a third
generation star.}, and hence before many SNe exploded, and so would
have been formed from gas containing mostly hydrogen and helium. But
stars forming now, will contract from gas that has already been
polluted by SNe, and hence will be more metal rich. Within the MW, this
seems to be born-out, at least to some extent. Stars in old GCs, for
example, typically have [Fe/H]$\sim -1$, so quit a bit below the solar
value. And most of the old stars in the halo also have low
metallicity. These are called {\em population II} stars. In contrast,
stars in the disk usually have higher metallicity, and are called {\em
population I}. There is no strict divide between those, some disk stars
also have low $Z$ for example.

Recently, there has been quite a bit of interest in the very first
generation of stars that formed after the Big Bang. Those would have
$Z=0$! They are called {\em population III}. The halo star with then
lowest metallicity currently known, has [Fe/H]$\approx -5$, and might
well be one of the first stars to form in the MW.

The evolution of $Z$ within the MW, or within galaxies in general, is
called their chemical evolution (which is a misnomer, since the
elements are produced in nuclear reactions which do not involve
chemistry).


Table of numbers for the MW

\subsection{Galactic Coordinates}
When describing the position of a star within the MW, it is useful to
use Galactic Coordinates $(l,b)$, $0^o\le l\le 360^o$, and $-90^o\le
b\le 90^o$. From the vantage point of the Sun, the direction to the
galactic centre is taken to be $l=0$, $l=180^o$ is the ante-centre, and
$b=0$ corresponds to the galactic plane. $b=\pm 90^o$ to the galactic
poles. Knowing these direction in terms of right ascension $\alpha$ and
declination $\delta$, one uses spherical trigonometry to convert
between them. Since the plane of the MW is inclined with respect to the
celestial equator, the planes $\delta=0$ and $b=0$ do not coincide.


\chapter{The Interstellar Medium}

\end{document}
