Dark Matter

Outline

Rotation Curves and Galactic Dark Matter
Evidence for Dark Matter in Clusters
What Can the Dark Matter Be?

Rotation Curves

Here's an example rotation curve for the Solar System:

Note that the velocities obey the equation:

where v is the orbital velocity, r is the orbital radius, M is the Sun's mass and G is Newton's gravitational constant

This is essentially Kepler's Law:

Kepler's law in the original form is:

where T is the orbital period

but we know that
so we can derive
Newton's law of Gravity tells us that the equation is actually
or

This formula allows us to determine M(r), the mass enclosed within a radius r

measure v(r) for a galaxy using the Doppler effect and solve for

Here's an image of the Andromeda galaxy:

and here is the rotation curve for the stars in the disk of the galaxy:

At large r, the rotation curve is flat

i.e. v = a constant (~230 km/sec)
implying
the mass of the galaxy is increasing with distance

We'd like to get the total mass of the galaxy

i.e. see
and get the total mass from
but the rotation curve stops when we run out of stars to look at

Solution - observe neutral Hydrogen at = 21.1 cm with a radio telescope

measure the Doppler shift of this line
most of the Hydrogen lines are in the optical or ultra-violet
but there is a very tiny magnetic energy difference between:

spin of the proton parallel to the spin of the electron
spin of the proton anti-parallel to the spin of the electron
this tiny difference in energy yields photons of = 21.1 cm

the Hydrogen has little total mass, but we can trace its orbit to measure the total mass

Here's a = 21.1 cm radio map superimposed upon a negative optical image of galaxy NGC 3198:

Here's the rotation curve which comes from the Doppler shift measurements of the 21.1 cm line:

The stars in this galaxy extend out only to 10 kpc, but the rotation curve remains flat out to 30 kpc.
There must be something besides the stars dominating the mass of the galaxy.

The curve labeled "disk" indicates the expected rotation curve due to the stars in the galaxy
The curve labeled "halo" indicates the rotation curve due to the "dark matter halo" of the galaxy

the composition of the "dark matter halo" is unknown

The exact amount of the mass associated with the stars isn't known very well

massive stars produce most of the light
low mass stars may dominate the mass but produce little light.
the ratio of low mass stars to high mass stars is uncertain.

Here are two other possible fits to the NGC 3198 data:

maximal disk, minimal halo:

minimal disk, maximal halo:

In either case, dark matter rules!
Here's a summary from Nick Strobel:

Here's what a spiral galaxy really looks like:

or:

Dark Matter in Clusters of Galaxies

Three kinds of evidence:

motions of galaxies in clusters
gravitationally confined hot gas in clusters
gravitational lensing

Motions of Galaxies

is strictly correct only for circular orbits
but it is also correct for the average of lots of orbits
so we can average over lots of galaxy radial velocities in a massive cluster with >1000 galaxies
problem: some galaxies might be in the foreground or in the background and have a large relative velocity due to Hubble's law
The central body of the stick man in the CFA redshift survey slice #1 is the Coma Cluster:

Here's the optical image (almost 0.5 degrees on a side)

Here's an x-ray image from the ROSAT satellite:

This image covers about 2 degrees on the sky.

The cluster is full of hot gas which is gravitationally confined

the temperature of the gas is proportional to v², so we can apply
without dark matter, the gravity of the cluster would be too weak to contain the hot gas and it would evaporate

Here's an image of a group (i.e. a very small cluster) of galaxies superimposed on a x-ray image from the ROSAT satellite.

once again the temperature of the hot x-ray gas can be shown to imply the existence of dark matter.

Gravitational Lensing

Can produce multiple images of a single source as shown in the diagram:

Here's an image of the central regions of cluster 0024+1654:

The cluster galaxies are the yellowish ones

many are quite bright
they have mostly old stars with little radiation in the blue and ultra-violet

The faint blue galaxies are distant high-redshift galaxies that are lensed by the cluster

they have many young stars which radiate in the ultra-violet
this radiation is redshifted to appear blue to us
there is a single lensed galaxy shaped a bit like a squashed ring that appears as at least 4 lensed images

The lensed images are at 4, 8.5, 9 and 10 o'clock from the cluster center

Another view showing the reconstructed galaxy images
HST Press Release describing this image.

Here are some more images of the central regions of clusters with lensed background galaxies distorted into long thin arcs:

An image of Abell 2218 from HST:

An image of cluster CL2244-02 with one long thin arc and some fainter ones:

These gravitationally lensed galaxies provide evidence for dark matter

The mass of stars and hot gas in these clusters is too small to bend the light from the background galaxies so much.
a great concentration of dark matter in the cluster centers is required to give these dramatic lensing events.