|Nature lives in motion.|
|-- James Hutton|
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. Gary 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 based on the observed redshift for the H-alpha line at various positions along the disk of each galaxy. The Doppler shift formula was 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 λo = 656.3 nm.
The velocity curve for one side of the galaxy is apparently moving away from us faster than the other side. The most obvious explanation for this is that these galaxies are spinning, just like our own Milky Way.
Your instructor will assign one galaxy to you. Circle that galaxy's number in column 1 of table 1, then determine the speeds (in km/s) of the left and right sides of the disk. Record these numbers in table 1. Your instructor may also have you put your numbers up on the board and copy everyone else's number.
The velocity on both sides of the disk is positive, implying that 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, vexp. If the rotation speed is vrot, then the "approaching" side will show a Doppler recession velocity of vexp - vrot, and the receding side will be vexp + vrot. The difference in these values is 2 * vrot. So in the column labeled vrot, you should enter half the difference between the left and the right side.
A plot of m and log10(vrot) for each of the six spiral galaxiesis given on the worksheet. 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 velocity (somewhere around 7000 km/s). 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 determine where log(vrot) falls on your graph to get the corresponding apparent magnitude. log(vrot) is given on the worksheet. Mark its location on your best fit line and read off the corresponding magnitude. Record this on the worksheet as md 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 = md - mactual and dM31 is the observed distance to M31.
Now you can figure out the Hubble constant, which is the H0 appearing in the Hubble Law:
vexp = Ho D
You just calculated d, so you just need a value for vexp. 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.
Last Modified: 4/2/07 by sam
Original: G. Bernstein
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