Using analyses of images and spectra of 18 spiral galaxies, the students will
In the 1920's, Edwin P. Hubble discovered a relationship that is now known as Hubble's Law. It states that the recessional velocity of a galaxy is proportional to its distance from us:
ν = H_{o}d [Eqn. (1)]
where v is the galaxy's velocity (in km/sec), d is the distance to the galaxy (in megaparsecs; 1 Mpc = 1 million parsecs), and H_{o} is the proportionality constant, called "The Hubble Constant". Hubble's Law states that a galaxy moving away from us twice as fast as another galaxy is twice as far away. The Hubble constant is a hotly contested quantity in astrophysics. In order to precisely determine the value of H_{o}, we must determine the velocities and distances to many galaxies, preferably those extremely far away.
The velocity of a galaxy is measured using the Doppler effect. The radiation coming from a moving object is shifted in wavelength:
[Eqn. (2)]
where λ_{true} is the rest or true wavelength of the radiation, λ_{measured} is the wavelength as measured at the telescope, making the fractional value that the velocity of the galaxy is of the speed of light.
In this case, wavelengths are measured in Ångstroms (Å), an outdated unit equal to 1 tenbillionth of a meter. The speed of light has a constant value of ~300,000 km/sec. The quantity on the left side of equation (2) above is usually called the redshift, and is denoted by the letter z.
We can determine the velocity of a galaxy from its spectrum by measuring the wavelength shift of an absorption or emission line whose wavelength is known and solve for the velocity, v. (If you are unsure about where to click on the spectrum, please review these detailed instructions.)

A trickier task is to determine a galaxy's distance, since we must rely on more indirect methods. One may assume, for instance, that all galaxies of the same type are the same physical size, no matter where in the Universe they are. This is known as "the standard ruler" assumption. To use this assumption, however, we have to know the actual size of the "ruler" and to do that, we need the distances to the galaxies that form our standard ruler. So, since we are working with spiral galaxies, we choose nearby galaxies such as Andromeda, Triangulum, Messier 81, and others to which we have found an accurate distance measure using variable stars or other reliable distance indicator. We find that, on average, the actual size of these standard ruler galaxies is 22 kiloparsecs (22 kpc or 22,000 parsecs or ~72,000 light years).
Theoretically, then, to determine the distance to a galaxy one would need to measure only its angular size and use the small angle formula: a = s / d, where a is the measured angular size (in radians!), s is the galaxy's true size (diameter, 22 kpc in our work here), and d is the distance to the galaxy.
a = s/d or d = s/a [Eqn. (3)]
There is another caveat, however. Each telescope and detector scales the images of celestial objects differently. While the actual angular size of a galaxy does not change, that galaxy might take up 30,000 pixels squared on one detector while on another one (having larger sized pixels) it takes up 5400 pixels squared. Since we may not know anything about the detector ahead of time, we need to figure out how much of the celestial sphere, in radians, is represented by each pixel. We can do this if one of our galaxies has an independent measurement of its angular size. That is the case for NGC 2903, the galaxy that is the closest one in our sample.
Data for NGC 2903 (http://www.messier.obspm.fr/xtra/ngc/n2903.html accessed 9 Feb. 2009):
Right Ascension 9 : 32.2 (h:m);
Declination +21 : 30 (deg:m)
Distance 20,500 (kly);
Visual Brightness 8.9 (mag);
Apparent Dimension 12.6 x 6.6 (arc min)
Taking the apparent dimension along the long axis of 12.6 arc min, and knowing there are approximately 0.00029 radians per arc min, we find that the scale for this telescope and detector is:
12.6 arc min x 0.00029 rads per arc min ÷ 395 pixels = 9.3 x 10^{6} radians per pixel.
Given the distance of 20,500,000 ly or ~6.3 megaparsecs, and using the small angle formula, this gives an actual size for the galaxy of about 23 kpc, close enough to the 22 kpc we've assumed so far.
Print out this worksheet if you are using the preformatted spread sheet to do your calculations (or use the one handed out in class) to record your data (and to use as backup in case something goes wrong!). You may choose to work alone or with a partner on this part of the exercise.
Rightclick Hubble's Law Spreadsheet (Excel format) and open it or choose to save the target on the computer you are using. Ignore and errors (#div by 0, NaN, *****, etc.) shown on the spread sheet as they will all disappear once you start entering your data.
When you click on the galaxy's spectrum link, you will see the full optical spectrum of the galaxy at the top of the spectrum page. Below it are enlarged portions of the same spectrum, in the vicinity of some common spectral features. The small vertical bars near the lower left corners of these spectra indicate the rest wavelengths of the lines.
If using the Hubble's Law Spreadsheet, follow these directions:
The following steps may or may not apply to your work. Check with your instructor!
If using the method whereby everything is done by hand and calculator, follow the directions linked here.
If the universe has been expanding at a constant speed since its beginning, the universe's age would simply be 1/H_{o}. (You may choose to do most of the calculations within the Excel™ spreadsheet.)
EXAMPLE: Your Hubble constant is 75 km/sec/Mpc,
then:
1/75 = 0.0133 = 1.33 x 10^{2} 

The "expansion age" of the universe is t = 1/H_{o}. This is a very simple model for the expansion of the universe. A better model would account for the deceleration caused by gravity. Models like this predict the age of the universe to be: t = 2/3H_{o}. Adjust the age of the Universe using this relation by simply multiplying your original age by 2/3.
1.What are your values for the Hubble constant, maximum age of the Universe, and the age considering deceleration due to gravity?
2. Identify the galaxy with the highest redshift and state its recessional velocity. What fraction of the speed of light is that galaxy receding from us? (There's a remarkably easy way to find this out as you've already calculated it.)
3. Why does the bestfit line to your data need to go through the origin of your graph? Where is this “origin” located in the Universe?
4. Quantitatively compare your maximum age for the Universe to the age of the Sun (5 billion years), and to the age of the oldest stars in the Milky Way (approximately 12.5 billion years). How will this comparison change if you use your value that allows for some deceleration of the Universe?
5. Briefly discuss any discrepancies and comment about your comparisons of the ages as answered in the previous question.
Theoretically, your plot should be a straight line, but it probably isn't. Let's consider a few of the possible sources of error.
6. The formula you used to determine the distances to these galaxies was distance = actual size of the galaxy divided by measured angular size: d = s / a. Is the distance calculated proportional to or inversely proportional to the actual size we assumed for the galaxies? How would an overestimate or an underestimate of the assumed actual diameter of a galaxy affect your estimate of the distance to it?
7. Is the distance calculated proportional to or inversely proportional to the angular size we measured for the galaxies? How would an overestimate or an underestimate of the measured angular diameter of a galaxy affect your estimate of the distance to it?
8. Pertaining to the previous question, what would be the effect on your value for the Hubble constant of your consistently undermeasuring or overmeasuring the angular diameter of the galaxies?
Possible processes for answering question 8: You could combine two equations here, either mathematically or logically, and answer. Mathematically: d = s / a and v = H_{o} x d. Substitute “s / a” for d in the Hubble law, solve for H_{o}, and explain the result. Mentally: If you consistently undermeasured the angular diameter, will the galaxies having a given recessional velocity be calculated to be nearer or farther? If you consistently overmeasured the angular diameter, will the galaxies having a given recessional velocity be calculated to be nearer or farther? Relate how both of these scenarios would individually affect the Hubble's constant. 
9. Another source of error is in the measurements that you make. Quantitatively (give some numbers) how precise do you believe your measurements to be for the wavelengths? For measuring the angular sizes? Give one example of something that might affect the precision in your measurements. (Please note: Stating “human error” is not a correct answer. We don't expect students to make monkey errors, after all!)
10. Another consideration is the fact that galaxies are found in groups or clusters. The motion of these galaxies through space as they orbit their common center of mass is called “peculiar motion.” That is, at a given distance some galaxies will be receding more slowly than others in the cluster while others will be receding more quickly. How does this peculiar motion affect your velocity measurements?
11. Summarize what is meant by “Hubble's Law” and how the relationship was discovered. Include in your discussion here an explanation of the impact that the observations made by Edwin Hubble around 80 years ago and, more recently, by astronomers using the Hubble Space Telescope had on our view of the Universe. What is the currently accepted value for the Hubble constant? How does your value compare? Include any closing comments if you'd like. This is a writing exercise, so please express yourself clearly and concisely. You may wish to draft your work first and copy and paste it here.
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