FFYS4327 Galactic Dynamics

Spring Term 1999

Lecture 2

Stellar Distances
Distances to stars can be directly measured
via the parallax effect, in which the position of the star changes slightly as
the Earth moves around the Sun.
This is the only direct method to measure the
distance of stars.

Source
http://wwwhip.obspm.fr/hipparcos/SandT/hip-SandT.html

Until the late 1980's only a few
hundred stars had been measured accurately in this way, since the effect is very
small and difficult to measure with ground based telescopes.  The European
Space Agency's Hipparcos
mission revolutionised this field, measuring the parallaxes of some 120,000
stars, some 15,000 with an accuracy of better than 10%.  Distances, D,
allow the absolute magnitude MV of the stars to be determined
using the apparent magnitude V

MV = V - 5 log (D) +
5

The apparent magnitude is a
measure of the flux from the star as seen at the Earth in a particular pass-band
(or filter). The simplest such system for optical light measures the flux in
five filters termed U, B, V, R, I for ultra-violet, blue, visual, red and
infra-red. The colour index  of a star is defined by the logarithmic
ratio of the flux in two filters

B - V = 2.5 log(Flux in B/Flux in
V)

This leads to the
Colour-Magnitude Diagram (or CMD) in which the colour of stars is
plotted against their absolute magnitudes. The CMD
of the Hipparcos stars for which the
distances were measured to better than 15 % is shown below:

The spectral types are (very
roughly) indicated at the top of the diagram.  The broad sweep of stars
from upper-left to lower-right is called the main sequence, and is the
site of Hydrogen burning. After stars on the main sequence exhaust their fuel
they move off the main sequence and become giants (most of the giants
seen in the diagram would have been A-G main sequence stars up until a few
million years ago), later ending up in the stellar graveyard as white
dwarfs.

The Solar Neighbourhood
A
useful concept is that of the solar neighbourhood, which refers to the
properties of the Galaxy near the Sun.  Thus we refer to a volume
which is small relative to the large scale properties of the Galaxy. Since the
global properties of the Galaxy vary on kpc scales or greater, the solar
neighbourhood may refer to with 1 kpc of the Sun (e.g. for bright OB stars) or
just 10 pc from the Sun (e.g. for very faint M dwarfs).

The following table shows the amount
of gas and stars of various types in the solar neighbourhood. The total amount
of non-stellar material, integrated over 1 square parsec above and below the
Sun's position is about 13 Mo. The total amount of stellar
material is about 35 Mo, giving a total of 48 Mo per
square parsec.  The
total mass density of matter near the Sun is about 0.1 M0 per cubic
parsec.

Inventory of the Solar
Neighbourhood

Component

volume
density

vertical
velocity
dispersion

surface
density

Mo
pc-3

km
s-1

Mo
pc-2

Gas

Molecular Hydrogen H
2

0.021

4.0

3.0

Ionised Hydrogen
H I

0.028

8.0

8.0

warm
gas

0.001

40.0

2.0

Stars

giants

0.0006

17.0

0.4

MV <
2.5

0.0031

7.5

0.9

2.5 < MV
< 3.0

0.0015

10.5

0.6

3.0 < MV
< 4.0

0.0020

14.0

1.1

4.0 < MV
< 5.0

0.0024

19.5

2.0

5.0 < MV
< 8.0

0.0074

20.0

6.5

MV >
8.0

0.014

20.0

12.3

white
dwarfs

0.005

20.0

4.4

brown
dwarfs

0.008

20.0

6.2

stellar
halo

0.0001

100.0

0.6

Round numbers to remember :

About 1/3 of the local disk is in
gas and about 2/3 is in stars.

About 1 star in 500 near the Sun
comes from the stellar halo (spheroid), the rest coming from the
disk.

About 5 percent of the local
stars belong to the young disk, the rest to the old disk.

Population
Types
Moving away
from the solar neighbourhood, the properties of stars begin to change. A useful
characterisation of these changes is called the population concept. Stars
are found to broadly belong to two population types.

The concept was proposed by Walter
Baade in 1944. Baade noticed in studies of the nearby spiral galaxy M31 that the
disk stars in M31 were like nearby disk stars in young associations in the
Galaxy, while the bulge stars mere nearly resembled those of the Galaxy's
globular clusters.

Young, metal rich stars rotating
with the disk of the Galaxy are termed Population I, while old, metal
weak stars are termed Population II. The fundamental difference between
these two types is very probably when and how they were formed.

Population
I
The disk of the
Galaxy is thought to be about 10 Gyr old. It presently contains about 20% gas by
mass, and along spiral arms and in giant molecular clouds new stars are forming
at a rate of about a solar mass per year. This rate is just about right to build
up the disk mass over its lifetime of 10 Gyr.

The Young Disk

The very youngest stars are of type
O and B, and are found in associations and clusters, often still containing the
proto-stellar gas. Hot stars of this type light up the gas clouds around them by
ultra-violet photo-ionisation, causing the clouds to glow conspicously in the
light of Hydrogen emission lines --- these are called HII regions. Cold gas
clouds, which are in the process of  collapsing under gravity before stars
begin to form, can be detected via emission from molecules such as
CO.

All of these objects, as well as
much of the gas layer, are concentrated very close to the galactic plane, with a
scale height of about 100 pc. Scale height hz is
defined as the exponential falloff in density (rho) of the stars/gas as a
function of height z above the Galactic plane. This is termed the
young disk.

rho(z) = rho(0)
exp(-|z|/hz)

The amount of heavy elements or
metals in stars relative to the amount in the Sun is called the metallicity
or the abundance and is denoted by the symbol [Fe/H] (note the
square brackets).

[Fe/H] = log(Fe/H)star -
log(Fe/H)Sun

i.e. (Fe/H) is the amount of iron
(Fe) relative to the amount of Hydrogen (H) in the star.  Note that this
defines the solar metallicity as [Fe/H] = 0.0.  A star with a
metallicity of [Fe/H] = -1.0 has 10 times less metals than the Sun relative to
Hydrogen, while a star with twice the metallicity of the Sun has [Fe/H] =
0.3.

The metallicity of young disk
stars was classically thought to be 2 to 3 times that of the Sun ([Fe/H] = 0.3
to 0.5) but this is now much less certain.

Stars in the young disk rotate
around the Galactic center at just about at the same speed as the disk
gas. Stars are born in
gas clouds, which are buffeted by the surrounding gas clouds and other
irregularities, so that the stars rapidly dissolve into the general stellar
field. Stars start in these associations with a very small scatter amongst
their velocities, typically less than a few km/s. The scatter in the
velocities amongst a group of stars is called the velocity
dispersion.

Motions of Stars in the disk. Source  http://astrosun.tn.cornell.edu/courses/astro201/circ_motions.htm

On the movements of
stars

The space velocity of a star
can be determined by measuring it's distance, proper motions and
radial velocity. The space velocity is a star's true motion in space
relative to some system, such as the Cartesian Galactic system of coordinates
(defined above).  The radial velocity is the component of the space
velocity projected along the line of sight from us to the star. The proper
motions are the angular movement of the star projected onto the sky as seen from
the Earth. Radial velocities are measured using the spectral redshift, in which
the spectrum of the star is shifted to the blue if it is moving towards us or
the red if moving away. The proper motions are measured by observing the star's
very small motion across the sky relative to other stars, usually by measuring
it's exact position over many years.The radial velocity of a star is in general
much easier to measure than the proper motions.

The three components of a star's
velocity along the Galactic coordinate system (X, Y, Z) are usually termed (U,
V, W), and are defined by the following time derivatives:

U = dX/dt
V = dY/dt
W = dZ/dt.

In physical terms, U is the velocity
of the star toward the Galactic center, V is it's velocity in the direction of
Galactic rotation, and W is it's velocity out of the Galactic
plane.

The velocity dispersion of a group
of stars is usually measured by the RMS (root-mean-square) scatter in the
velocities. These are usually denoted

where (for example) the velocity
dispersion in V is

Here the sum is over
i for n stars, and Vi are the individual
V-velocities.

In physical terms, the velocity
dispersion is a measure of the kinetic energy of the group of stars, and is
closely related to how the stars will move within the Galactic gravitational
field.

The total velocity dispersion
of the young disk is very small, about 20 km/s.

The Old Disk

The old disk refers to stars
in the disk but older than a few Gyr, which generally means A-type stars or
later (for historical reasons hot, blue stars are called early type while
cool, red stars are called late type).

The old disk rotates at about 200
km/s rather than 220 km/s, like the gas. The total velocity dispersion is about
twice that of the young disk, 40 km/s. Objects of the old disk include, F, G, K
and M stars, white dwarfs, planetary nebulae and some types of variable
star.  Perhaps 80% or 90% of the local material belongs to the old
disk.  The abundances of old disk stars range from about [Fe/H] = -0.5 to
[Fe/H] = 0.3. Most of the stars have abundances in a narrow peak at a value a
bit less than the Sun, [Fe/H] = -0.2.

Histogram of metallicities of K
dwarfs in the solar neighbourhood, showing the sharp peak at about [Fe/H] =
-0.2

Source :
Kotoneva and Flynn, 1999. See http://xxx.uni-augsburg.de/abs/astro-ph/9901159

Population
II
Population II
stars are old.  The physical distribution of Population II is spheroidal,
unlike Population I objects of the disk. Population II therefore includes both
the stellar halo and the bulge. The stellar halo and bulge differ
in their metallicities --- the halo is metal weak ([Fe/H] < -1) while the
bulge is generally metal rich (with a metallicity distribution quite like the
disk).

The bulge luminosity distribution
can be modelled by a de Vaucouleurs' law with Galactocentric radius
R

I(R) = Ie exp(-7.67 (
(R/Re) )-1)

where Re is the projected
half light radius and Ie is the surface brightness at
Re.

The density distribution (rho)
of the stellar halo follows a power-law  with Galactocentric radius,
R

rho(R) =
R-a

where a = 3.5 +/-
0.5.

Both the bulge and halo are
centrally concentrated --- that is most of their mass and light is within the
solar circle (the  orbit of the Sun around the center of the Galaxy
at R = 8 kpc).

Typical stellar halo objects are
globular clusters, subdwarf stars, certain kinds of RR Lyrae stars
and long period variables. Subdwarfs are low mass (< 1.5 M0) metal
weak, stars which typically lie 1 to 2 magnitudes below the main sequence (as
defined by metal rich disk stars --- see
above).

The stellar halo is old, 12 -15
Gyr. This has been determined from the colour magnitude diagrams of globular
clusters and nearby subdwarfs. It would seem that the halo stars were the very
first to form as the Galaxy was collapsing from a gas cloud in the early
universe.  As a consequence they retain   information about this
early time.

Neither the halo or bulge
rotate very much. The halo rotates very little, about 20 +/- 20 km/s, which may
be compared to the disk rotation of 220 km/s.  This means that the halo is
pressure supported rather than rotation supported. Halo stars have
a large velocity dispersion (in all three components) of about 120 km/s and this
is what keeps them in their spheroidal configuration. The bulge rotates slowly
at about 100 km/s and has a velocity dispersion of about 100 km/s. Hence it is
somewhat intermediate between the disk and the halo in these
respects.

Random motions : The stars in star clusters and in the spheroidal components
of galaxies, and the galaxies in clusters of galaxies, have motions that are
random in three dimensions. That is, they have no preferred plane (disk).

Source:  http://astrosun.tn.cornell.edu/courses/astro201/random_motions.htm

Summary of Population
properties

Population I

Population
II

Age

Young disk : < ~1
Gyr
Old
disk : ~1 - 10 x 109 years

12-15 x 109
years

Metallicity
[Fe/H]

-0.5 <~ [Fe/H] <~
0.3

Halo : -3.0 <~ [Fe/H]
<~ -1.0
Bulge : -0.5 <~ [Fe/H] <~ 0.5

Rotation,
km/s

Young disk:
220
Old disk : 180 - 200

Halo : 20 +/-
20
Bulge : circa 100

Velocity Dispersion in
(U,V,W) km/s

Young disk : (40, 30,
10)
Old
disk : (80, 60, 20)

Halo : (150, 100,
100)
Bulge : ~ 130 km/s

Clusters and Stellar
Ages

Ages can be measured for individual
stars which have evolved slightly off the main sequence (where they have spent
most of their lives) but not yet moved up the giant branch. This means F and G
stars in practice --- they are the only stars for which accurate ages can be
obtained individually. Stars in groups however, where a range of stars masses
have all formed at the same time, can be dated as a group. This is the case in
open clusters and globular clusters.

On
the left, the globular cluster M92. On the right, the Galactic open cluster
M67.
Source
http://www.stud.ifi.uio.no/~mikkels/messier.html

Most open clusters are young (less
than 2 Gyr) and metal rich, move in circular orbits around the Galactic center
and have low velocity dispersion --- hence they are Population I. Globular
clusters are mostly distributed around the Galactic disk in a spherical
distribution, have large random velocities, no or very little net rotation
around the Galactic center and much less metal content thatn the Sun (-2.5 <
[Fe/H] < -1). These are archetypical Population II objects. A typical open
cluster contains 100 to 1000 stars in is of order 1 to 10 pc in radius. There
are about 105 open clusters in the Galaxy (a very loose number since
the vast majority are hidden from view in the Galactic disk behind dust clouds,
or lost amongst all the foreground and background stars). A typical globular
cluster contains 104 to 106 stars. These are much easier
to find since they well away from the dust layer. There are about 150 known, and
probably about 200 in total.

Summary of cluster
properties

Globular
Clusters

Open
Clusters

central
density

circa 104
Mo pc-3

circa 100 Mo
pc-3

core radius

1.5 pc

1 pc

median
radius

10 pc

2 pc

tidal radius

50 pc

10 pc

central velocity
dispersion

7 km/s

1 km/s

mass-to-light
raio

2 Mo /
Lo

1 Mo /
Lo

typical
mass

5x105
Mo

250
Mo

lifetime

1010
years

108
years

Adapted from Binney and Tremaine, Table 1-3, page 26

The central density of stars in
globular clusters is very high, about 10,000 solar masses per cubic parsec
compared to only about 0.05 in the Solar neighbourhood. In the table, the
core radius refers to the radius where the surface brightness falls to
hald its central value; the median radius referes to the radius of a
sphere which contains half of the total light, while the tidal radius is
the point at which the gravitational field from the Galaxy is the same as the
field of the cluster -- at this point stars can easily escape.

The key to aging a cluster is the
colour-magnitude diagram. The colour of the bluest stars on the main sequence
(called the main sequence turnoff) is directly related to the age of the
cluster --- the older the cluster, the redder the position of the
turnoff.

Animation of stellar evolution
:  click here
Source
http://www.astro.soton.ac.uk/PH308/Milky_Way/clusters.html

Colour magnitude diagram for M67.
Stars are shown as dots, and sequences of isochrones show the expected
positions of stars in the cluster for a range of masses but the same
age.  The isochrones shown here have ages of (left-to-right) 3, 4, 5
and 6 Gyr (take your pick of which one fits best!)

Source
http://adsbit.harvard.edu/cgi-bin/nph-iarticle_query?1992AJ....103..151D

Original source: Demarque, Guenther & Green 1992,  Astronomical
Journal, 103, 151

Open cluster colour magnitude
diagrams are typically beset with the problems that the clusters are
contaminated with foreground and background stars, there are not many stars in
the cluster anyway,  and the amount of dust obscuration, metallicity and
distance to the cluster have intrinsic uncertainties.  Various ways around
these difficulties have been developed, and most clusters can be dated to within
+/- 30%.

There are much more stars in
globular clusters, and deriving ages from isochrone fits is usually easier than
in open clusters. The colour magnitude diagram for M92 is shown below: Four
isochrones with ages of  12, 15, 18 and 21 Gyr are shown. Globular clusters
are old, typically 12-15 Gyr.

Source
http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1983AJ.....88.1159S

Original Source : Sandage, 1983,
Astronomical Journal, 88, 1159

Source : Steve Majewski,
private communication, 1999

Distribution of ages (MAI - age
index in Gyr) for open and globular clusters, showing that most open clusters
are younger than 10 Gyr while Globular clusters are always at least this old.
The peak in the open clusters at young ages is because open clusters are not
very stable --- they are born and rapidly break up, dissolving into the stellar
field. Globular clusters, being much denser at having halo-like rather than
disk-like orbits are much more difficult to break up.

Age of the Disk
The colour magnitude diagram of nearby stars
shows a mixture of all stellar ages, up to the oldest disk stars at about 10
Gyr. The three sets of isochrones shown on the figure below are for ages of 8,
11, 13 and 15 Gyr (left-to-right), and for three different metallicities [Fe/H]
= -0.5 (green), 0.0 (red) and +0.3 (blue). This plot shows how sensitive
isochrones are to metallicity! If we assume that all stars have solar
metallicity (they don't!) then there are apparently very few stars older than
about 10 Gyr, since these would turn up below and to the right of the red lines
(all the stars are instead above and to the left --- meaning they are 10 Gyr or
younger).

Source
http://adsbit.harvard.edu/cgi-bin/nph-iarticle_query?1998MNRAS.299..515J

Original Source: Jimenez, Flynn and Kotoneva, 1998, Monthly Notices of the
Royal Astronomical Society, 299,515

The Age-Metallicity and Age-Velocity
relations

The galaxy is thought to have
collapsed from a cloud of primordial gas (whose composition was set essentially
by the Big Bang) consisting of about 75% Hydrogen and 25% Helium plus a very
small amount of the other elements (much less than 1%). As this cloud collapsed
generations of stars were born. Each generation consists of many low mass stars
which evolve slowly and are still with us today, and high mass stars which
quickly spent their fuel and have since become white dwarfs, neutron stars or,
possibly, black holes.

The stars that explode as
supernovae at the end of their lives distribute back into the primordial
gas a large amount of heavy elements which have been built up in the star via
nucleosynthesis. As a result the metallicity of the gas from which each
new generation of stars forms rises steadily, which results in correllations
between the kinematics (motions) of stars and their
metallicities.

Source: Rocha-Pinto et. al. Preprint 1999.

The graph above shows the rate at
which stars have been forming in the solar neighbourhood, calculated by finding
and measuring the ages of long-lived (low mass) stars near the Sun. Over the age
range of 0 to 10 Gyr the amount of stars formed is roughly constant, with some
bumps located at A, B and C where the formation rate seems to have been a bit
above average. Beyond 10 Gyr many stars are missing and the determination is not
reliable.

As these stars formed there has been
a generally rising metallicity, althouhg with a great deal of scatter. Older
textbooks will speak of a quite good relationship between metallicity and age
(called the AMR or Age-metallicity relation). Studies within the last decade
have shown that this relationship is much weaker and contains more intrinsic
scatter than previously thought.

Source : Edvardsson
et al 1993
Relationship between
metal abundance [Fe/H] and log Age (in Gyr) for F and G dwarf stars near the
Sun. There is a general trend of rising abundance with decreasing stellar age,
with a very large amount of intrinsic scatter.

Much clearer in general is the
relation between stellar age and kinematics, or velocity of the
stars.

Source : Rocha-Pinto et.al.
Preprint, 1999.

The (U, V, W) velocities of G and K
dwarf stars near the Sun for which an age has been measured. Note that young
stars generally have much smaller velocities than old stars (> approx 1 Gyr).
Young stars have recently been born from gas clouds and are tightly constrained
to a few 100 parsecs of the Galactic plane. Older stars have had time for their
orbits to be changed by encounters with irregularities in the disk of the
Galaxy, such as spiral arms and giant molecular clouds, and show much larger
random motions and a larger scale height. This diagram shows why the dividing
line between the young and the old disk is set at about 1-2 Gyr.

Dark Matter
There is a further aspect to Galactic
dynamics which we have hardly considered yet, the issue of dark matter. It was
discovered in the 1970's that galaxies rotate in a peculiar way, in that the
outer parts of galaxies seem to move too fast. A great deal of work has gone
into this issue over the last 30 years and has led to the conclusion that the
visible parts of galaxies represent only a small part of the mass of the
galaxy.  Most of the mass in galaxies seems to be in a dark form, the
composition of which still remains completely uncertain. Since more than 90
percent and perhaps as much as 99 percent of the Universe might be in this form,
we currently do not seem to know what most of the mass in the Universe
is.

The main evidence for dark
matter  comes from rotation curves of disk galaxies. A rotation
curve measures the rate at which stars and/or gas move in their circular orbits
around the center of the galaxy. In general these are "flat", as shown in the
next figure:

For more information click on the image.

Source
http://astrosun.tn.cornell.edu/courses/astro201/rotation_flat.htm

The rotation curve of the milky way
is much harder to measure than for external galaxies, since the gas and stars
beyond the solar circle are moving mostly transverse to the line-of-sight rather
than along it. However, so many rotation curves have been measured for other
disk galaxies there is no reason to suppose that the Milky Way would be
particularly different, so even if the flattness of the rotation of the outer
Galaxy is poorly measured it is thought to be flat nevertheless. Improved
astrometric space missions within a decade or two will resolve this
observational difficulty.

Source
Merrifield 1992.
http://adsbit.harvard.edu/cgi-bin/nph-iarticle_query?1992AJ....103.1552M

The Sun moves at about 220 km/s in a
circular orbit around the center of the galaxy, like almost all the stars near
the Sun. We can estimate the amount of matter inside the Sun's orbit by assuming
that all the matter is at the Galactic center, a not too bad
approximation.

Let the speed be V0 , the
mass of the Galaxy be MG and the distance of the Sun from the
Galactic center be R0. Then the centrifugal force due to rotational
speed must balance the gravitational force due to the mass of the
Galaxy.

GMG/R02
= V02/R0

where G is the gravitational
constant.

Hence

MG =
V02 R0/G

substituting values of 8 kpc and 220
km/s for the Sun and G = 0.00430 M0 (km/s)2 / pc, we
get

MG = 1011
M0    (i.e. the mass of the Galaxy within a radius of
8 kpc from the center outwards).

This is already quite close to the
mass determination of 6 x 1011 M0 obtained by more
realistic modelling of the disk, showing that the approximation is not too
bad.

However, measurements of the
rotation of the outer edge of the Milky Way show that the stars out there also
rotate at 220 km/s, out to about 20 kpc.

Thus,  within a radius of 20
kpc we get a mass of

MG = 2x1011
M0.

If the light distribution of the
Galaxy were proportional to the mass distribution, then the two mass estimates
above would imply that the amount of light emitted by the 8 kpc region would be
the same as the region from 8 kpc to 20 kpc... whereas measurements show that
the 8 kpc region emits about 10 times more light than the 8 kpc to 20 kpc
region.

The major conclusion is that the
distribution of emitted light is not necessarily the same as the underlying
distribution of matter.