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# Rotation of Spiral Galaxies

 Nature lives in motion. -- James Hutton

## Overview

• 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.

## Introduction

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!

## Step One: Measuring the Galaxies' Rotations

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 λo = 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, 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. Then calculate the log10 of each of these numbers on your calculator for the last column.

Plot the m and log10(vrot) 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

## Step Two: Determine the Distance and Measure the Hubble Constant

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 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.

## Step 3: Estimate the Mass of a Galaxy and Check for Dark Matter

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 3rd 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 m3 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, vrot (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 Mg.

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 . . .)