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:
[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 Ho 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 Ho, 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
is the rest or
true wavelength of the radiation,
is the wavelength
as measured at the telescope, making
the fractional
value velocity of the galaxy is of the speed of light.
In this case, wavelengths are measured in Angstroms (Å), an outdated unit equal to 1 ten-billionth of a meter.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 v.
| Example: An absorption line that is found at 5000Å in the lab is found at 5050Å when analyzing the spectrum of a particular galaxy. Therefore this galaxy is moving with a velocity v = (50/5000) * c = 3000 km/sec away from us. |
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 (22,000 parsecs or ~72,000 light years).
To determine the distance to a galaxy one would need to measure only its apparent (angular) size, and use the small angle equation: 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)]
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.
Right-click on this link (hubbles_law1.xls) and open it or choose to save the target on the computer you are using. Ignore the errors 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. Measure the wavelength by clicking at the middle of the spectral line (bottom of an absorption line, top of an emission line) in the galaxy's spectrum.
If using the pre-formatted Excel spread sheet, follow these directions:
The chart now shows the slope of the line in the form: y = mx + b (but here b = 0, so is not given). Round off the slope to 2 significant digits (e.g., 75.4839485 would be 75). The R-squared value is a measure of how well the fit represents the data; if the value was equal to 1, then the fit would be perfect. Anything close to 1 is good here.
If using the method whereby everything is done by hand and calculator, follow these directions:
If the universe has been expanding at a constant speed since its beginning, the universe's age would simply be 1/Ho. (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 |
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The "expansion age" of the universe is t = 1/Ho. 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 = 2/3Ho. Adjust the age of the Universe using this relation by simply multiplying your original age by 2/3.
After you have finished with this exercise, you will need to go to the "CollectIt" website that is part of Catalyst Tools here at the U of W and upload your work.
After uploading your spreadsheet, you should EACH INDIVIDUALLY log onto the web page containing the questions that accompany this exercise, and answer all of them. You need not do so in one sitting; you can choose to save your work and return later to answer additional questions. Although you may confer with your classmates, your answers here must be your own. You can change any answer until you actually submit your work. This assignment is due in one week. It will be counted as late if submitted after 5 pm on that date, and the web site will be unavailable 4 days after that.
Congratulations. You have now earned your first cosmology badge! 