Name: |

Nature lives in motion. |

-- James Hutton |

- Explore the Tully-Fisher relation for spiral galaxies.
- Use this relation to estimate the Hubble constant.
- See how mass to light ratios give evidence of dark matter.

Why doesn't the Moon fall down to Earth? Galileo and Newton
realized that this is not quite the right question to ask;
instead, we should ask why the Moon does not *fly away* from
the Earth, because the natural tendency is for objects to move in
a straight line, not a circle. Newton showed that the Moon is *constantly*
falling to Earth, because to move in a circle the Moon must
accelerate toward Earth at the center. It is **gravity**
that gives this acceleration and causes the Moon to keep orbiting
the Earth.

We know that the Sun is moving in a circle around the center
of the Milky Way. What keeps the Sun moving in this circle
instead of flying away? Gravity, of course. We notice likewise
that all other spiral galaxies appear to have spinning disks:
when we point the spectrograph at one side of the disk, the
spectral lines are all **Doppler-shifted** towards
the red, meaning that the stars are receding from us. On the
other side of the disk, the stars seem to be moving towards us,
so we conclude that the disk is spinning. Gravity is once again
at work, keeping the stars in these disks from flying off. In
this lab you will use the Doppler shift to measure how fast
several spiral galaxies are spinning. The faster the disk
spins, the stronger gravity must be. By measuring this gravity, we are actually measuring how much
**mass** there is in the spiral galaxy, since
Newton's Law of Universal Gravitation shows that gravity is
proportional to mass. Therefore, by measuring the spin rate of the galaxy
we can determine how much matter is in the galaxy. We
can also measure how much light is coming from the galaxy, and
since we know about how much light a star puts out, we can
estimate how many stars are in each spiral galaxy. Then we can
ask whether there is more mass in the galaxy than can be
accounted for by stars (i.e. what is the **mass-to-light
ratio**?). As we shall see, there must be some kind of **dark
matter** whose gravity keeps the spiral galaxy together,
but which does not appear to shine like stars do.

Along the way to discovering dark matter, you'll notice an
amazing correlation between how fast a spiral galaxy *spins*,
and how *bright* it is. This relation became famous in the
70s and is called the **Tully-Fisher Relation**
after the first people to use it. This lets us use spiral
galaxies as standard candles, similar to the Cepheid stars but
millions of times brighter. By comparing your spiral galaxies to
our next-door neighbor spiral galaxy (the Andromeda galaxy),
you'll be able to figure out how far away they are, and measure
the Hubble constant and the age of the Universe!

Your instructor will give you a set of rotation curves for six
very distant spiral galaxies along with CCD photographs of these
six galaxies. These data were taken by Prof. Bernstein while he was at Michigan
who actually used the data in essentially the same way you will
in this lab. Look at the pictures of the six galaxies. You can't
see the spiral arms because the galaxies are nearly edge-on, and
because the pictures are over-exposed. Notice that some of them
are bigger than others, and some are brighter than others. The **apparent
magnitudes**, **m**, of these galaxies have
already been determined for you and are given in Table 1.

Next look at the rotation curves. These plots are of the
observed redshift for the **H-alpha** line at
various positions along the disk of each galaxy. Remember that
H-alpha normally has a wavelength of **656.3 nm**,
so these galaxies must all be redshifted considerably since the
observed values are greater than this for each one. But also, the
wavelength for one side of the galaxy is apparently moving away
from us *faster* than the other side, since the wavelength
for one half of the curve is greater than that for the other
half. The most obvious explanation for this is that these
galaxies are *spinning*, just like our own Milky Way.

You can use the Doppler shift formula to calculate how fast
each half of the galaxy is moving away from us. You should recall
that the Doppler shift formula gives us the **recessional
velocity** as:

As usual *c* is the speed of light, 300,000 km/s, and
λ_{o} = 656.3 nm.

For each galaxy, determine the typical wavelength for the Hydrogen line on both sides of the disk, and enter them into Table 1. Then use the Doppler shift formula to change these wavelengths into speeds (in km/s).

Even the "approaching" side of the disk appears to
be moving away from Earth at several thousand km/s. This is
because of the expansion of the Universe, which of course makes
the entire galaxy appear to be rushing away at some expansion (or
recessional) velocity, v_{exp}. If the **rotation
speed** is v_{rot}, then the
"approaching" side will show a Doppler recession
velocity of v_{exp} - v_{rot}, and the receding
side will be v_{exp} + v_{rot.} The difference in
these values is 2 * v_{rot}. So in the column labeled **v**_{rot},
you should enter *half* the difference between the left and
the right side. Then calculate the log_{10} of each of
these numbers on your calculator for the last column.

Plot the **m** and **log**_{10}**(v**_{rot}**)**
values for each of the six spiral galaxies on the graph. Draw a
line across this graph that best fits the six points you've
drawn. This line relates the rotation speed of a galaxy to its
magnitude, and is the so-called **Tully-Fisher Relation**.

The six galaxies you have been studying are all (roughly) the *same*
distance away. We know this because the rotation curves all
straddle the same wavelength, (roughly) 671.5 nm. This means they
are all moving away from us at approximately the same speed.
According to the Hubble Law, the recessional velocity is
proportional to a galaxy's distance, so these galaxies must all
be around the same distance from us.

Now you are going to figure out how far away this is using the
Tully-Fisher relation that you plotted. The plot you made shows a
pretty good correlation between a galaxy's rotational velocity
and its apparent magnitude. But apparent magnitude changes with
distance, so if we use a galaxy at a different distance, then
this relation tells us the magnitude it *would* have if it
were as far away as these galaxies. So if we know the distance to
some spiral galaxy, and we measure its rotational speed and
apparent magnitude, then we can figure out how far away these
other six galaxies are.

Lucky for us, there is a spiral galaxy, M31, also called the **Andromeda
galaxy**, which is very close by so that we can measure
its distance using Cepheid variable stars. M31 is 770 kpc away
from us, it has an apparent magnitude of 1.04, and its rotational
speed has been observed to be 275 km/s.

First, you should determine how bright M31 would be if it were
at the distance of the other six galaxies. To do this, you need
to calculate the logarithm of its velocity (275 km/s) and see
where this falls on your graph to get the corresponding apparent
magnitude. We'll call this magnitude **m**_{d}
since it is the magnitude M31 would have at the as yet
undetermined distance, **d**.

Of course M31 is much closer to us than the other six, and its apparent magnitude is in fact seen to be m=1.04. The difference in these two numbers tells us something about the distance. Specifically, every 5 magnitudes corresponds to a factor of 10 in distance. Therefore, you can calculate d from:

where Δm = m_{d} - m_{actual}
and d_{M31} is the observed distance to M31.

Now you can figure out the Hubble constant, which is the H_{0}
appearing in the Hubble Law:

**v**_{exp}**
= H**_{o}**
D**

You just calculated d, so you just need a value for v_{exp}.
For each galaxy, this is just the average of the left and right
velocities in Table 1. So an average value for the six galaxies
would be the average of all 12 numbers.

Now you can estimate the total mass of your galaxies. Consider a star of mass m that is near the edge of a spiral galaxy (distance R), going around in circles at speed, v. Since it is orbiting the center of the galaxy, it obeys Newton's form of Kepler's 3^{rd} law, and since it is moving in a roughly circular orbit, it obeys the laws of circular motion.

Newton's Law: Law of Circular motion (centripetal force):

Since the force of gravity is the centripetal force, we can set the two equations equal to each other and solve for the mass, which give us

where G is the gravitational constant, 6.67×10^{-11}
m^{3} s^{-2} kg^{-1}. This equation required that you use "physics" units of kg, m, s, etc. However, astronomers prefer units like parsecs and solar masses, in which case the equation becomes (you may be asked to show this):

Using this equation, you can find the mass of an entire spiral galaxy if you know the rotation speed, v_{rot} (we know
it already!), and its size, R; so all you need now is the radius,
R, of the spiral galaxy. That's not too hard -- you can do that
from the pictures you have.

For each galaxy, measure the distance from one end to the
other (in mm). This will correspond to the **diameter**
of the galaxy. Next measure the length of the calibration line
shown under the image for galaxy 5. This gives you the scale for
the images. Multiply the **scale** (in arcsec/mm) by
your diameter measurements to get the angular diameter, θ,
in *arcsec*.
You can now infer the real size based on the galaxy's distance
from us, which you already calculated. To do this, use the small
angle formula, which in this case would be:

Remember that if you measured d in kpc, you now have R in kpc
also. The 1/2 comes from the fact that what you measured was the *diameter*,
and you really want the *radius*. Now you have everything
you need to determine the mass of the galaxy. Just make sure you
convert R to parsecs, pc, for the above equation to find M_{g}.

To investigate the possibility of dark matter in these
galaxies, you need to know how much light all this mass is
producing. To figure out how many Sun's worth of light they are
emitting, you need to determine how much brighter the galaxies
are than the sun. To do this, you need the **absolute
magnitudes**, **M**, of these galaxies. Use
the formula (from your text), **M = m - 5 * log d + 5**.
This should be a negative number, since these galaxies are much
brighter than individual stars. Remember to use d in *parsecs*
(not kpc or Mpc).

The absolute magnitude of the Sun is 4.8. Now you can figure out how many Sun's worth of light your galaxy is putting out. The **luminosity**, **L**, measured in
solar luminosities is: where ΔM is the difference between the sun's absolute
magnitude and the galaxy's. Enter this number into Table 2. It
tells you how many times more light energy is emitted by your
spiral galaxy than by the Sun.

Finally, with the mass and luminosity of each galaxy,
calculate the so-called **mass-to-light ratio**, **M/L**.
For a typical set of stars, this ratio should be around 1. If a
galaxy were made up entirely of stars like the sun, this ratio
would be exactly 1. (However, this doesn't mean you will get M/L
values near 1 for you galaxies . . .)

Last Modified: 5/22/06 by sam