Physics 321

Astrophysics II:  Lecture
#18

Prof. Dale E. Gary
NJIT

The Milky Way Galaxy -
I

Counting Stars

It is a significant
problem to determine the shape and extent of our galaxy from our point of view
inside it.  On a clear night with a dark sky, the band of stars and dust
that make up the Milky Way can be easily
seen, but when mapped over the whole sky in optical
light, it has a complicated appearance that does not represent its real
shape.

The first attempts to determine
the shape of our galaxy, called the Milky Way
Galaxy were through simply counting stars.  Sir William
Herschel made the first systematic star count in the 1780's, later expanded by
Jacobus Kapteyn the early 1920's.  Both of these attempts ignored
interstellar extinction (the dimming of starlight by dust), and concluded that
the Sun resides near the center of a flat, pancake-shaped distribution of
stars.

However, between 1915 and 1919,
Harlow Shapley estimated the distances to globular clusters, using
RR Lyrae and Pop II Cepheid variables as distance indicators.  He found
that many more clusters where seen in the direction of Sagittarius, and
concluded that globular clusters are distributed uniformly around our
galaxy.  The concentration in the direction of Sagittarius was due to the
fact that our Sun is off-center within the galaxy, and the center is in the
direction of Sagittarius.

We can now obtain images in the
infra-red region of the spectrum, which can penetrate dust easily.  When
we image the entire sky in the infra-red, as was done with the COBE satellite,
we obtain the much clearer picture of the Milky Way Galaxy shown
below.

Image of the Milky Way
Galaxy taken with the COBE
satellite.Note that now we can see through the dust to the central bulge, and the disk appears much more
uniform.  Below are several views of the Milky Way seen in various
wavelengths.
Morphology of the Galaxy

Below is a figure that
gives an overview of the structure of our galaxy.  We will be discussing
different aspects of this figure in the next several lectures, but for now
let's consider only the three main regions.

Most of the stars of the galaxy
reside in a disk, which closely resembles
the disks of spiral galaxies seen elsewhere in the sky.  Also in the
disk are gas clouds, dust, and very young stars (O-B
associations).  The stars tend to have high metallicity (Z >
0.01).

There is a central bulge of stars near the center of the
galaxy that is more spherical in shape.  The stars here are older (and
redder) than the disk population.

Globular
clusters are distributed in an approximately spherical halo above, below, and within the disk.  A
small number of high-velocity stars are also seen
in this halo region.  The stars here are very old, and typically have
low metallicity (Z < 0.001)

Schematic view of the Milky
Way Galaxy.  The top figure shows a "top view" showing the spiral
structure as seen from above.  The bottom figure shows an "edge-on
view" as seen from the side. (The top view image is of the spiral galaxy
M100.  The side view shows the COBE image from the previous
figure.)
In addition to these visible features,
there is indirect evidence of another source of matter, called dark matter, which lies in a large halo extending far
outside the visible confines of the galaxy.  This makes up the dark halo.  The evidence for this enigmatic
matter comes from studies of the kinematics, or motions, of the constituents
of the galaxy.
Kinematics of the Milky Way

As always in astronomy,
the building blocks of great discoveries such as galactic rotation and the
presence of dark matter come from very basic, simple measurements such as the
motions of stars.  As we have seen, a star's speed is measured in two
components:

radial velocity (from doppler
shifts of spectral lines) and

tangential velocity (from
proper motion).

Star catalogs such as the Nearby Stars catalog give these measurements
for each star.

Here, the proper motion
(pm, or m")
is given in arcseconds per year.  In these units, the tangential velocity
of the star, in km/s, is

vq
= 4.74 m"dpc =
4.74 m"/p"
(km/s)
where
p" is the
trigonometric parallax, in arcsec.

Example: For star Gl 4.2A,
above, we have m"= 0.592 and p"= 0.0483, so vq
= 58.1 km/s.  The
radial velocity is shown in the table to be
vr = 2.6 km/s, so the total velocity, or space
velocity is v=
(vr2+
vq2)1/2 = 58.2 km/s.

The quantities in the table,
however, are not so simple to measure.  To convert proper motion to
velocity, one must know the distance, which means either measuring a
trigonometric parallax (which is impossible for all but the nearest stars) or
using a spectroscopic parallax.  In addition, the proper motion must be
measured.  For nearby stars, the proper motion can be quite large, but
for distant stars it can be very small.  Luckily, we can monitor stars
over many decades, after which the proper motion accumulates and can be
measured.

Radial velocity is easier, since
it is determined from spectral line shifts, which can be seen at any distance
(as long as the star is bright enough).  However, for both radial
velocity and proper motion one must subtract other motions such as the Earth
around the Sun (~30 km/s), and the Sun's own motion through space (~19.5
km/s--our text gives 16.5 km/s).

Galactic Coordinates

We learned in the second
lecture that stars are assigned right ascension (a) and declination (d) coordinates depending on
their positions on the celestial sphere--the extension of the Earth's
equator and poles into the sky.  However, it is often more convenient
when discussing the kinematics of the galaxy to change to a new coordinate
system based on the shape and orientation of the galaxy (the Milky Way) in the
sky.

Before we can go further with
stellar motions, we need to define a coordinate system in which to make our
measurements.  Since we expect the galaxy to rotate about its center, and
all of the objects that make up the galaxy should orbit this center in
Keplerian orbits, we could form a coordinate system that is galactocentric, and for some cases that is the
appropriate system to use.  However, since we are observing from our
location in the galaxy, a second coordinate system centered on the Sun will
prove more useful.  This coordinate system is called simply galactic coordinates.  As viewed from the Sun,
coordinates in the galactic plane are at galactic latitude b =
0o, while at the galactic poles (N or S), the
galactic latitude is b = 90o.
Galactic longitude l is measured from the center of the galaxy (the
Sagittarius region) as shown in the figure below.

The transformations between
celestial (a,
d) and galactic
(l, b) coordinates are:

cos b cos (l -
33o) = cos d cos
(a -
282.25o)
cos
b sin (l - 33o) = cos d sin (a -
282.25o) cos 62.6o + sin d sin 62.6o     (from celestial to galactic)

sin b = sin d cos
62.6o- cos d sin (a-
282.25o) sin 62.6o

sin d = cos b sin (l - 33o) sin
62.6o+ sin b
cos 62.6o
cos d sin (a -
282.25o) = cos b sin (l- 33o) cos
62.6o- sin b
sin 62.6o   (from galactic to
celestial)

A graphical representation of these two coordinate systems is shown
below:

Chart for conversion of
celestial coordinates to galactic coordinates.  From Kraus, Radio
Astronomy, Second Edition, Cygnus-Quasar Books,
1986.

We define the kinematics of the
system in terms of the rotational motion, about the galactic center, of all of
the stars in the solar neighborhood.  This allows the Sun to have a
motion with respect to this average rotational motion.  Assumptions that
go into the model are:

Stars share the general
rotation of the galaxy.

Stars have their own
peculiar motions superimposed on this general rotation.

The average velocity (magnitude
and direction) of stars in the local solar neighborhood should be zero with
respect to this general rotation.

The Sun may have a motion
relative to the general rotation.

Thus, if we examine all the motions of
stars in the very local solar neighborhood, and their velocities do not
average to zero, any residual velocity must be due to the Sun's own motion
(due to the fact that the Sun's orbit around the galactic center is not
perfectly circular).  In fact, stars should (statistically) show a
diverging pattern in the direction our Sun is traveling, called the solar apex, and show a converging pattern in the
opposite direction (the solar
antapex).  Once the apex and antapex are found, we determine
the Sun's peculiar velocity by averaging the radial velocities in these
directions.  The solar apex is in the direction toward l =
53o, b = 25o, a point in the constellation Hercules.
Local Standard of Rest

These considerations allow us to
define a Local Standard of Rest (LSR).
Stars moving about the galactic center in perfectly circular orbits would be
expected to have zero velocity in this LSR frame.

The other coordinate system,
centered on the center of the galaxy, has coordinates R, q, and z, corresponding
to radial distance from the center, azimuthal angle, and distance out of the
plane.  The corresponding velocity components are defined as P = dR/dt, Q = R
dq/dt, and Z = dz/dt.  The
rotation rate at the radius of the Sun gives for the LSR velocity:
PLSR = 0, QLSR = Qo, and
ZLSR = 0, where Qo is the rotation rate at the
radius of the Sun.  Peculiar velocities are then defined as

u = P - PLSR = P

v = Q - QLSR = Q
-Qo
w = Z -ZLSR =
Z
Plotting
the peculiar velocities as u vs v, gives the diagram of Figure
22.23 of the text.  Note the different velocity dispersion of
different types of stellar populations.  Young stars show little
dispersion, so are rotating along with the general galactic rotation.
Older stars show a great deal of dispersionn, up to 200-300 km/s
difference.  Note the velocity-metallicity
relation, in which low-metallicity stars have high peculiar
velocities while high-metallicity stars have low peculiar
velocities.
Galactic Rotation Curves

We mentioned before that the
stars should orbit the galaxy in Keplerian orbits.  For Keplerian orbits,
we expect Kepler's Third Law to hold:

P2 = (4p2/GM)R3
where P is the orbital period,
M is the mass interior to the orbit, and R is the radius of the
orbit.  For a circular orbit, the period is related to the speed
by

P = 2pR /
V

so the speed
V is

V = (GM/R)1/2 ~ R-1/2.
(1)

This is the way that velocity
should fall off with radius as long as all of the mass is interior to the
orbits being considered.  In contrast, imagine a spherical distribution
of mass of uniform density, in which particles (stars!) orbit inside
the mass distribution.  The mass interior to the orbit is
then

M(R) = 4pR3r/3

and the equation above
becomes

V =
(GM(R)/R)1/2 =
G(4pR2r/3)1/2 ~
R.
(2)
Clearly, the
way that the mass is distributed in a galaxy will make a big difference in the
velocities of the stars within it.  On the other hand, if we can measure
the velocities of stars as a function of radius (the galactic rotation curve), we can determine the mass
distribution in the galaxy.

If the speed of orbit of stars
depends on radius, then stars far from the solar neighborhood will not be
rotating with the LSR.  If equation (1) is valid, stars interior to the
Sun's orbit will rotate faster, while if (2) is valid they will rotate
slower.