|The Universe is How Old???
Homework #2, Due Wednesday September 28
**This HW has FIVE parts worth 48 points total. To receive full credit you must turn in each part, with EVERYTHING typed and stapled!
You may complete this homework in groups of 2-3 if you wish. In this case, each member of the group will receive the same grade. Make sure you put the CID # of each group member at the top.
In the first few weeks of class we’ve spent a lot of time talking about light and spectroscopy. Now let's see how your newfound knowledge will allow you to experience the thrill of discovery for yourself!REDSHIFT OF GALAXY SPECTRA: (first figure = slipher's image, second figure = schematic)
As you know (or will learn on Monday), the idea of an expanding universe is generally credited to Edwin Hubble in a famous paper published in 1929. However, Hubble couldn't have come to this conclusion without a lot of ground work by one Vesto Slipher, an astronomer from Flagstaff, AZ. Slipher calculated the velocities for a number of distant fuzzy objects of unknown nature (the “spiral nebulae”) and found that not only were they moving much faster than objects in our own galaxy but they were all moving away from us! Not knowing exactly what these objects were, Hubble measured their distances and much to his surprise discovered that they were actually located beyond the Milky Way. Further, when he combined Slipher’s findings with his own measurements he discovered a very special relationship between the distance of a faraway galaxy and the velocity with which it is receding, leading him to some amazing conclusions. For your assignment, you have the exciting task of exploring this relationship and its implications, but first a bit more background...
How did Hubble & Slipher calculate the velocities of objects (and how do we do this today)? They used the idea of cosmological redshift. Cosmological redshift is an increase in the wavelength of a photon due to the expansion of space. In other words, light from distant galaxies is shifted toward the red end of the electromagnetic spectrum because those galaxies are moving away from us as the Universe expands. Nearly all galaxies display this redshift and there is a direct relationship between the redshift of a galaxy and the velocity with which it is receding. So, if we can measure the redshift, we can determine the velocity. How do we measure redshift? By looking at spectral lines of the galaxies and measuring how much those spectral lines are shifted from where they would be if the galaxy were not moving.
Now, the more the lines are shifted, the higher the velocity of the object. This can be expressed as an equation involving three variables:
1. the speed of light, c = 300000 km/s
2. velocity of the galaxy, v , expressed in km/s
3. redshift, z
where z = (the wavelength of a known spectral line in the object's spectrum - the wavelength of the same spectral line in the Solar spectrum) all divided by the wavelength of the line in the Solar spectrum.
As we discussed in class, z=v/c, or rearranging the expression v=z*c. So, we can use the redshift of spectral lines to calculate a velocity for receding galaxies!
Once Hubble had velocities for his galaxies, he was able to construct a diagram of recession velocity vs. distance. Looking at his diagram, Hubble noticed that the larger the distance to a galaxy, the faster it was receding. Hubble realized that this relationship was consistent with an expanding universe. Further, if everything is moving farther apart as time moves forward then if we go backward in time, everything would be closer together. This is consistent with the Universe originating in an explosion from a single point –THE BIG BANG. The relationship also implies an AGE OF THE UNIVERSE!!
Selecting 10 Galaxies of your choice from your book, you will find their distances, redshifts, & velocities, construct a "yournamehere" diagram and calculate your own age for the Universe!
This assignment has 5 parts:
Part 1. Choose 5 galaxies from p. 300-309 of your text and 5 galaxy clusters from p.318-323 (for a total of 10 objects) with the following stipulation: Do not choose Andromeda, Sagittarius Dwarf, Triangulum, or the Local Group (you’ll see why in part 5).
Set up an Excel (or Numbers if you are a mac user) spreadsheet and create a table containing columns for the galaxy (or cluster) name, distance, redshift, and velocity. If you are not comfortable using Excel, check out the LITS help pages or take their Excel workshop. Make sure your table has column headings that include units where appropriate! Enter the distance to each of your galaxies in ly (light years) or Mly (Million light years) in the distance column.
Now if we had a nice big telescope and spectrograph at our disposal I might be inclined to send you out to observe your galaxies and measure the redshift but luckily, you don’t have to do that - you can benefit from the results of other scientists’ work! This website is a large database of compiled literature on all extragalactic objects with confirmed names, positions, and redshifts. Simply enter the name of your object under input parameters and hit enter (or click submit). You should see a very simple text based table under the first horizontal line and the 7th entry is the redshift:
Record these redshifts in your table. In some cases, the name that comes up may be different - you can always check the “catalog numbers in your book (listed just above the distance) to make sure they are really the same object. You can also do your search using the catalog numbers instead of the name.
By the end of step 1, your table should look something like this (remember redshift is dimensionless):
Part 2. Before we can calculate the age of the universe, we need to get the velocities from the redshifts! Remember, v=c*z, z has no units, and c=300000 km/s so all we have to do is multiply c and z and we’ll get the velocity in km/s!
To do this in excel, you want to use the function capability. Select the first cell in the velocity column (D2 in my case) and hit the function button. This should pop up a window (usually called formula editor) which will allow you to type in a formula. C is my redshift column so to get the velocity for object 2, I just need to enter =C2*300000. Then to repeat the calculation for the other 2 objects (9 in your case) you just need to select the remaining cells in column D and use the “fill down” function (under the insert menu) which should do all of the math for you! (Again, get help from LITS or me or your TAs if you have trouble.)
By the end of step 2, your table should look something like this:
(note, you may choose to enter your distances in Mly instead of ly)
The completed table is what you need to turn in for steps 1 and 2 (20 points)
Part 3. Now you need to plot the distance and velocity to see if the Universe is still expanding! Select the distance column (column B here) and the velocity column (column D here) and go to Insert > Chart which will bring up the Chart Wizard. From the Chart Type list select XY Scatter (you DON'T want lines connecting your points). Once the graph pops up, you should title it "Hubble Diagram" or "[YOURNAMEHERE] Diagram." Also make sure you label your axes with the correct variable and units. Note that by default some versions of Excel will put the first column you select on the x-axis and the scond column on the y-axis so make sure you put them in the right order in your table. If you don’t remember which variable goes on which axis, check your notes from class or read the section on expanding space in the textbook (p42-43).
The completed graph (Hubble Diagram!) is what you need to turn in for step 3 (10 points)
Part 4. You are now ready to calculate your own Hubble constant (Levine constant in my case) and age of the universe. Recall that Hubble’s law is V=Ho*D where V is velocity (units=km/s), Ho is the “your name here” constant (units=km/s/Mpc), and D is distance (units=Mpc) where Mpc stands for Megaparsec. This is the equation of a line - effectively the same line you’ve constructed in your graph! To find your constant, all you have to do is find the slope of your line and multiply by 3,260,000 (if you used ly) or 3.26 (if you used Mly) to put it in the right units since there are one million light years in a Mly and 3.26 light years in one parsec. Excel & Numbers both have built-in slope functions that work pretty much the same way. Select an empty cell and go to your formula editor (or search on the slope function). You then want to type in the parentheses the Y and X ranges of your table within which you want to find the slope. So my example would look something like this:
When I hit enter, I get my answer in my formerly empty cell (E2):
Of course this is my Levine constant in km/s/ly but we want the answer in km/s/Mpc so all we need to do is multiply the number in E2 by 3,260,000 and we’ll have our constant (82.2 km/s/Mpc in my case.)
Finally, to calculate the age of my universe, I just need to find 1/Ho. But hold your horses! If I divide 1/(82.2 km/s/Mpc), I get a number less than 1 in some crazy unit - that doesn’t make sense for the age of the universe! The reason is that I need those pesky distance units in the constant to cancel out AND I also need to convert the seconds in my Levine constant to years. This can be accomplished by dividing and multiplying my answer by the following conversion factors: 3.1 x 10^19 km/Mpc and 3.15 x 10^7 seconds per year, in order to get the age of the universe in years. (note: I'm using x10^XX to indicate scientific notation. Also note that the '*' below implies multiplication)
To summarize this calculation:
yourconstant in km/s/Mpc = slope*3.26 (*1,000,000 if you used ly)
Age of Universe in years = (1/yourconstant km/s/Mpc) * [3.1x10^19 km/ Mpc] / [3.15 x 10^7 s/yr]
For step 4, you need to turn in your value for the slope of your graph, your value for your constant, your value for the age of the universe, and your comments on whether your answer makes sense, using the first few paragraphs of this wikipedia entry for reference.
Make sure to show your work. (10 points)
For part 5 you need to turn in answers to the following questions (2 points each):
1. What type of spectra did Slipher use to find velocities? How can you identify this type of spectrum?
For questions 2-4, go back to the extragalactic database and find the redshift for your choice of the Triangulum, Andromeda, or Sagittarius dwarf galaxies.
2. What galaxy did you choose and what value did you find?
3. What do you notice about this value?
4. What do you think it implies about the motion of this galaxy?
OPTIONAL EXTRA CREDIT - For a maximum of 5 additional points, explain why taking the inverse of the Hubble constant gives an age for the Universe and implies the Big Bang.
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