Lecture 21 - Galactic Dynamics (4/6/99)
Our Galaxy --- | ---
Galxies and Quasars
Chapter 19, 20 (ZG4)
M51, the Whirlpool galaxy.
||What is the shape and size of our
||Relate the velocities of stars
to galactic properties
- Galactic Rotation and Kinematics
- What is the local standard of rest (LSR)?
- What is differential rotation?
- What are the radial and tangential velocities of nearby stars
as a function of galactic longitude?
- What is a rotation curve?
- What are Keplerian and solid body rotation profiles?
- What is the LSR velocity around the galaxy?
- Galactic Dynamics
- Why do we see only a few kpc in the disk of our galaxy?
- What is the mass of the galaxy within the Sun's orbit?
- What does a flat rotation curve imply for the mass density?
- What are the Oort formulae?
- How do we measure the Oort constants A and B?
- How do we determine the distance to the galactic center using
local star measurements?
- The Galactic Center
- Why do we see only a few kpc in the disk of our galaxy?
- How do globular clusters indicate the position of the galactic
- Is there a bar in the center of our galaxy?
- Why is the galactic center a strong radio emitter?
- Why do we think there is a massive black hole at the center
of the Milky Way?
Our Galaxy in Outline - continued
- The Mass of the Galaxy
- How do we find the mass of our galaxy? We cannot just add
up the luminosity and use some approximate mass-luminosity
relation (that would be incorrect as we see below).
- As usual we use gravity - Kepler's 3rd Law! Note that
since we can ignore the mass of any star in the rotating
disk, Newton's version becomes:
M(<R) = R3 / P2
with R in AU and P in years giving M in Msun as usual.
Note that the mass M corresponds to the mass enclosed
by a sphere of radius R - the mass interior to R.
- Using the Sun's velocity of 220 km/s at R=8.8 kpc, we
get an mass of about 10^11 Msun inside the Sun's orbit
in the disk.
- It is easy to see the relation if we frame it in the form
of Newton's law, with the centripetal velocity:
v2 = G M / R
- Note that if all of the mass in the galaxy were at a point
at the center, like the Sun in the solar system, the
rotation velocities would fall off as the inverse square-root
of the radius (since M is constant). This relation is called
Keplerian rotation, since that is the relation Kepler
found for the solar system.
- However, measurements of the velocity as a function of
radius from the center, v(R), show that after climbing
up from zero in the center to about 200 km/s in the inner
kpc, the rotation curve remains relatively constant
out to the edge of the disk at 15 kpc!
- Thus, it must be that the enclosed mass M(R) is growing
as R, in order to keep v constant in our equation.
Q: What must the mean density within radius R be doing
as a function of R for a constant rotation curve?
- Note that a solid body rotation curve, like a phonograph
record, would have v proportional to R. The galaxy is somewhere
between Keplerian and solid body.
- Since the mass in the galaxy is growing with R in the outer parts
of the galaxy, while the total enclosed luminosity is barely growing
(the central parts are much brighter than the dim outer parts),
most of the outer mass of the galaxy must be made of dark
matter, or at least be made of stuff that has more mass per
unit of luminosity than stuff in the inner galaxy.
- Using the orbits of things in the outermost halo, we find that
the total mass of the galaxy is about 10^12 Msun.
- Thus, the average mass-to-light ratio of the entire galaxy
is about 10^12 Msun / 10^11 Lsun or 10 Msun/Lsun. For comparison,
the mass to light ratio in the solar neighborhood is about
- Stellar Populations (see below)
- In the 1940's, Walter Baade at Mt. Wilson Observatory realised that
there were two different populations of stars.
- Population I stars are stars like our Sun, with similar amounts
of heavy elements, and are found in the disk of the galaxy.
- Population II stars are found in the halo of the galaxy, and
have fewer heavy elements in their spectra than the Sun.
- We use the term metals to denote elements heavier than
helium. Population I stars thus have solar metallicity and
Population II stars are metal poor.
- Pop I stars tend to have low velocities relative to the Sun, while
Pop II stars tend to have high velocities relative to the Sun.
- Pop II stars in the halo tend to be older than their Pop I disk
counterparts. They formed early in the history of the galaxy out
of pristine hydrogen and helium unenriched in metals by supernovae.
- The most recently formed disk stars, like in the Orion Nebula, have
higher metallicity than the Sun.
- Stars in the galactic bulge are old, but metal rich. The bulge formed
very early, but was enriched quickly by many supernovae.
- Evolution in Star Clusters (see below)
- How do we find the age of a distant star? We need some sort
of stellar clock.
- The H-R diagram and stellar evolution gives us this clock. The
main sequence lifetime of a star increases with decreasing mass.
- The mass of the turn off (from L and T and stellar models)
gives us the age of the cluster.
- Plotting the H-R diagram for the cluster gives us the turn off point.
Distance to the cluster is needed to convert apparent magnitudes
- Open star clusters, which contain 10^2 to 10^3 stars and
are in the disk of the galaxy, have ages of 10 billion years or
- Globular clusters, which are dense balls of 10^5 to 10^6
stars and are found in the halo, are 10 to 18 billion years old.
- Clusters in the bulge are older than 10 billion years, but are
- Disk Kinematics
- The galaxy consists of a rotating disk and a halo of
stars with randomly oriented orbits, approximately spherical.
- The disk is relatively thin, with the stars in a thickness
of less than 300 pc.
- The H I gas, as measured by the 21-cm line, is confined to
within 125 pc of the galactic plane.
- The molecular clouds, were stars are being formed, are within
100 pc or less from the plane.
- The disk stars are in nearly circular orbits.
- The rotation curve v(R) is nearly constant at the Sun's orbit
- This means the angular velocity is
proportional to v(R)/R, or 1/R for constant v.
- This causes differential rotation, since the stars inside
the Sun's orbit take a shorter time to make one orbit, and stars
outside the Sun's orbit take a longer time to orbit the center
of the galaxy.
- Stars inside Sun's orbit gain on Sun, outside they lag the Sun.
- Thus, if you like up a structure radially from the center of the
galaxy, it will be wound up into a spiral by the differential
- Spiral Arms
- If you look at some nearby galaxies like M31 the Andromeda galaxy, you
find that they have spiral arms which wind outward from
the center of the galaxy.
- Does our Milky Way galaxy have spiral arms? Its hard to tell since
we are in the disk ourselves and cannot see it from above.
- If you plot out the distances to O and B stars, H II regions,
and molecular clouds, you find them concentrated in bands toward
and away from the galactic center.
- These are believed to coincide with spiral arms in the galaxy.
- There is an overdensity of gas in the spiral arms, causing enhanced
star formation in these regions.
- The leading theory for the formation of spiral arms, at least the
"grand design" two arm spirals, is the density wave theory.
- Density waves are regions of enhanced density, like low-speed high
density "shocks" in traffic patterns found on crowded freeways.
- Clouds move in an out of the arms, but while there they bump into
each other causing star formation.
- These density waves are examples of gravitational instabilities
in the galactic disk.
- Because of the differential rotation, the waves get wound up into
spirals. Eventually they get wound up so much that they disappear.
- What triggers the density wave instability? Most likely a close
encounter with a neighbor galaxy.
- Another sort of strong gravitational instability is the bar in
the center of a galaxy, like our own, and certain other spiral
- Another theory for the formation of spiral patterns is the
propagating star formation model, where the supernovae from
star formation compresses clouds and triggers nearby star formation.
- In this model, weak spiral structures are formed by the differential
rotation. This model works best to explain spirals without strong
two-arm grand design pattern, called flocculent spirals.
- In average galaxies like our Milky Way, both density waves and
propagation star formation are likely to have roles.
- Halo Kinematics
- Halo star orbits are highly elliptical and randomly oriented
through the three-dimensional roughly spherical halo.
- There is low net angular momentum in the halo system, unlike
the high angular momentum disk.
- Unlike the disk, you cannot use the velocity of a halo star
to infer the mass inside its orbit, since for any one star
you do not know where it is in its elliptical orbit.
- Instead, you use the velocity dispersion, or
the root mean square velocity, of a group
of stars at the same distance.
- This can be related to the mass through the energy equation:
E/m = v^2 / 2 - G M / R = - G M / 2 R
and thus the mean square velocity < v^2 > is proportional
to the mass M.
- There is little gas, and thus almost no star formation in the halo.
The halo seems to have formed all its stars a long time ago, and
the leftover gas collapsed to form the disk.
- History of Our Galaxy
- A possible scenario for the formation of our galaxy:
- The first fast collapse of the low angular momentum gas
into the halo and the beginnings of the bulge in the center.
- The globular clusters are formed at this time, before the infalling
gas crosses the center and gets randomized.
- The high density bulge star formation is enriched by supernovae.
- The higher angular momentum gas left over from the halo formation
collapses to the disk through dissipation from cloud-cloud collisions.
- There is on-going star formation in the disk, at the average
rate of 1 solar mass per year or more.
- Note that this rate of star formation would use up all the gas
(10^10 Msun) in the disk in only 10^10 years or less. Since it
seems unlikely that we would just now be on the verge of using
all the disk gas up, it is likely that there is continuous
replenishment by infalling pristine gas from the far halo.
- An infall rate of 1 Msun per year would be enough to support
the star formation rate of the disk.
- This gas may reside in the galactic corona which surrounds
the halo and has a temperature of 10^5 to 10^6 K. Calculations show
that this gas could cool at the rate of 1 Msun/yr and rain down
onto the disk.
- The Center of Our Galaxy
- The galactic center region is heavily obscured by dust
in the intervening spiral arms, and it took Shapley's
study of the globular clusters to show where the center
of the galaxy was located.
- However, dust is transparent to other parts of the electromagnetic
spectrum than visible light. Radio waves, and infrared light,
for example, can pass through dust clouds.
- When radio astronomy began in the years after World War II, one
of the first sources of celestial radio waves found was the
center of our galaxy. This radio source was designated
Sagittarius A (Sgr A) after the constellation in which it was
- The galactic center region is a busy place. There appears to
be rings of hydrogen gas and molecular clouds about 150 to 250
pc from the center, likely stirred by the central bar of our
- Inside these rings are some relatively empty regions (cleared by
the bar), until the inner few parsecs.
- There is a high density of stars in the central region, seen
in the IR. The average spacing in the center is about 1000 AU,
compared with 1.5 pc (300000 AU) near the Sun!
- The stars at a distance of R = 0.3 pc (62000 AU) from the center
seem to be orbiting with a velocity of 200 km/s (P=9000 yrs).
Kepler's law tells us that there must be 3 million solar masses
within this 0.3 pc!!!!
- What could this be? One possibility is a super dense star cluster.
However, even the largest globular clusters with 10^6 Msun are
almost 30 pc in diameter. This is 50 times smaller, or 125000 times
- The most popular explanation is that there is a supermassive
black hole at the center of our galaxy.
- This would explain the large amounts of radio and X-ray energy
being emitted from Sgr A. These sorts of things are also seen
in many other galaxies.
- The presence of some strange object in the nucleus of a galaxy
emitting radio, IR, optical, or Xray energy is called an
active galactic nucleus. We will discuss these in
a more general context later on.
- The presence of a central million solar mass black hole in the
center of our galaxy has no particularly dire consequences. It
makes the stars nearby move fast, occasionally swallowing one
that gets too close, and turns it into energy that it emits in
radio waves or Xrays. No need to worry - it won't swallow up
the rest of our galaxy!
In the early 1940's, during wartime blackouts in Los Angeles, the
astronomer Walter Baade at the Mount Wilson observatory was able to
take many spectra of faint stars without the glare of city lights. What
he discovered was that there were two populations of stars in
our galaxy, distinguishable by the elements seen in their spectra.
The Population I stars are stars like our Sun, with similar amounts
of heavy elements. The term metals is used to denote elements heavier
than helium. Population I stars have solar metallicity, or at least a
similar abundance of metals to the Sun. The Population I stars make up the
disk of the galaxy, and thus have low velocities relative to the Sun, at least
for those stars near the Sun.
The Population II stars are found in the halo of the galaxy. These
stars are metal poor, since their spectra have few lines due to
elements heavier than helium. The Pop II stars tend to have high velocities
with respect to the Sun. This is because they are in the halo, not in the disk
which is rotating along with the Sun. Halo stars have random orbits and thus
cross the disk star orbits with relatively high velocities.
Pop II stars in the halo tend to be older than their Pop I disk
counterparts. They appear to have formed early in the history of the galaxy
out of almost pristine hydrogen and helium unenriched in metals by supernovae.
This is why they are metal poor. There does appear, at least in the halo and
disk, to be a direct correlation between when a star was formed and the
abundance of metals in its photosphere. In fact, the most recently formed
stars in the Orion Nebula are metal rich, and contain as much as twice
the abundance of metals as the Sun!
An anomaly in this age - metallicity relation is the galactic bulge. It
contains some of the oldest stars, and was probably one of the first things
formed in the galaxy, but these stars are also metal rich! The probable
explanation for this is that in the dense gas of the forming bulge, the large
number of supernovae quickly enriched the clouds making the bulge stars,
thus resulting with old stars (now) with solar or higher metallicities.
We have said that Pop I stars are young, while Pop II and bulge stars are
old. How do we know this? How do we find the age of a distant star? We need
some sort of stellar clock. The H-R diagram and stellar evolution gives us
this clock. Remember, the main sequence lifetime of a star increases with
decreasing mass, since M/L is proportional to M^-2.5 measuring the fuel
divided by the rate of burning. Thus, if we had a group of stars of a range
of masses formed at the same time, we could find the point on the main sequence
where the stars more massive than this had evolved off onto the giant branch
- this is called the turn off point. Stellar models can give us the
time it would take a star of this luminosity and temperature, and thus known
mass, to evolve off the main sequence, and thus the age of the cluster.
There are two types of star clusters seen in the galaxy. Open
clusters are loose groups of 100 to 1000 stars in a radius of about 3 to
30 parsecs. The open clusters are in the disk of the galaxy, and are
Population I objects. The turn-off ages for open clusters are found to be
less than 10 billion years.
The globular clusters are dense balls of 100,000 to a million stars
contained in a radius of around 10 to 15 pc. Globular clusters belong to the
Pop II halo The turn-off ages of globular clusters are 10 to 18 billion years,
and thus gives the age of the halo as old.
Some metal rich clusters are found in the bulge and have ages of more than
10 billion years. This gives us the old age for the bulge. Note that since
the most massive stars live less than 1 million years, you can have
significant enrichment in much less than a billion years in a crowded
environment like the bulge. The sparse halo region would take much longer to
enrich since the metals from supernovae would need a long time to travel large
distances to where new stars were being formed. This is a plausible
explanation for the differences in metal content of halo and bulge stars.
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Steven T. Myers