Measurements of Velocities and Distances

Step 3: Finding the velocity of each galaxy

The velocity is relatively easy for us to measure using the Doppler effect. An object in motion (in this case, being carried along by the expansion of space itself) will have its radiation (light) shifted in wavelength. For velocities much smaller than the speed of light, we can use the regular Doppler formula:

The quantity on the left side of this equation is usually called the redshift, and is denoted by the letter z.
The formula for redshift should remind you of the process where you calculated your percentage error: [(your value) - (true value)] / (true value). Thus, we can view the redshift (at least for those galaxies with a recessional velocity much less than the speed of light) as a "percentage" wavelength shift. It is a measure of the ratio between the velocity of the galaxy and the speed of light.
For this lab, all wavelengths will be measured in Ångstroms (Å), and we will approximate the speed of light at 300,000 km/sec. Thus, we can determine the velocity of a galaxy from its spectrum: we simply measure the (shifted) wavelength of a known absorption line and solve the equation v = z * c.

For Example: A certain absorption line that is found at 5000Å in the lab (rest wavelength) is found at 5050Å when analyzing the spectrum of a particular galaxy. We first calculate z:

redshift = [(measured wavelength) - (rest wavelength)] / (rest wavelength)

We find that z = 50/5000 = 0.001 and conclude that this galaxy is moving with a velocity v = 0.001 * c = 3000 km/sec away from us.

Measuring the Spectral Lines

Here is the listing of the files that contain the real data for the 27 galaxies.

Step 4: Finding the distance to each galaxy

A trickier task is to determine the distances to galaxies. For nearby galaxies, we can use standard candles such as Cepheid variables or Type I supernovae. But, for very distant galaxies, we must rely on more indirect methods. The key assumption for this lab is that galaxies of similar Hubble type are, in fact, of similar actual size, no matter how far away they are. This is known as "the standard ruler" assumption. We must first calibrate the actual size by using a galaxy to which we know the true distance. We are looking for galaxies in the sample that are Sb galaxies, as we would use the nearby Sb galaxy, M31 the Andromeda galaxy, to calibrate the distances. We know the distance to the Andromeda galaxy through observations of the Cepheid variables in it. Then, to determine the distance to more distant, similar galaxies, one would only need to measure their apparent (angular) sizes, and use the following approximation for small angles:

a = s / d
d = s / a

where a is the measured angular size (in radians), s is the galaxy's true size (diameter), and d is the distance to the galaxy.

Measuring the Galaxies

Here is the listing of the files containing the real data for the 27 galaxies.

Checking Your Data

It would be a good idea to have your instructor look at your data now, before you do a ton of calculations. You wouldn't want to spend hours of your time only to discover that you made mistakes in steps 3 and 4.

Initial Calculations

If you feel confident of your data, then you are ready for the preliminary calculations:

Velocity Determination
For each measured line calculate the (redshift z), and enter this value in the box under the measured wavelength. Then take the average redshift of the measured lines for each galaxy, and enter it on the appropriate column. Finally, use this average redshift to calculate the velocity of the galaxy using the modified Doppler-shift formula:

v = c * z

Distance Determination
Determine the distance (in Mpc) to each galaxy using the following, revised version of the small angle formula. Recall, we have had to make an important assumption: all of these galaxies are about the same actual size. Once you have the angular diameter in mrad (and with some adjusting of units), just take the actual size of each galaxy -- 22 kpc -- and divide it by the measured angular diameter. For example, if one of the galaxies had a measured angular diameter of 0.50 mrad, 22 / 0.50 = 44 Mpc.

Details for the manipulation of the units to come out with the correct distances
From calibrations, we know that galaxies of the type used in this lab are about 22 kpc (1 kiloparsec = 1000 pc) across. We may then find the distance to the galaxies:
distance (kpc) = size (kpc) / a (rad)

or equivalently, upon multiplying the left side by 1000 and dividing the right side by 0.001 (which is exactly the same thing):
distance (Mpc) = size (kpc) / a (mrad)
Note that we now have the equation in a form where we can simply substitute the size in kpc (22) and divide it by the angle returned by our measurements (already in mrad).

Go on to Steps 5 and 6

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