Exercise 1
Using the data above in Table 1, make a plot of right ascension versus declination on your printed out Milky Way Globular Clusters Distribution Graph (Diagram 1-the top plot). RA is along the x-axis and goes from 0 to 24 hours, Dec is on the y-axis and goes from +90 to 0 to “90 degrees.) Insert the plot into your lab report with your signature and date.
You will type your answers to the below questions in your lab report and then scan/photo your graph(s) and insert them into your lab document. Again, it would be helpful to review the Exploration from Module 1: Math Primer for Astronomy (note this contains link for a free online scientific calculator). There are also good math examples in the Appendix of our eText.

Would you describe the distribution of clusters on the plot as random, or is there a pattern (explain your answer)?
Now look at your plot and point in the direction in which you see most of the globular clusters. This is the general direction of the Galactic Center. Estimate the center of the distribution of the globular clusters. Also estimate (no calculation required ” just an educated estimate) the accuracy of determining this center. You have now determined the rough center of our Galaxy!

RA = ____________________ ± ________________
Dec = ____________________ ± ________________
Shapely was correct in thinking that the distribution of globular clusters could reveal something about the Galaxy as a whole. He went one step further. He used the locations of the globular clusters to determine the distance to the Galactic Center. His result was surprisingly accurate and differed from the modern value by less than 10%. So, lets follow in his footsteps.
The next step is to determine the distance to the clusters. Shapely did this by using RR Lyrae stars. These are variable stars, which have a relatively narrow range of luminosities. From the difference between the apparent magnitudes (measured from his photographic plates) and the absolute magnitudes (calculated from the luminosities), he calculated the distances in parsecs to the star (via: m – M = 5log10(d) + 5). So now we have the distances and the directions of the globular clusters and we can determine the 3-dimensional distributions of the globular clusters relative to us.
However, we will use a different coordinate system that is based on galactic latitude and longitude rather than RA and Dec. The plane of the Galaxy is designated as 0 latitude. Why would we want to do this? RA and Dec is a messy coordinate system that depends on our orientation in space and the earths rotation around its axis. The system based on galactic latitude and longitude is therefore simpler. However, it means that we have to transform the measured RA and DEC positions of the globular clusters and galactic latitude and longitude. To simplify things even further, lets express the galactic latitude and longitude in terms of x, y, and z coordinates. The advantage of this is that x, y, and z have units of parsecs (rather than angles which is the case with galactic latitude and longitude).
So now the z-coordinate tells us how far above or below the galactic plane we are, and the x-coordinate tells us how far away from the origin (in this case from the Galactic Center) we are! The y-coordinate tells us where in the x-y plane (in the Galactic Disk) we would be found. But since we assume that the disk is a round circle (i.e., it is symmetric), we only need to worry about the distance from the center in the disk. Basically, we are only concerned about two quantities: x and z, i.e., how far above and below the Galactic Disk the globular clusters can be found and how far away from the Galactic Center they are.

Using the data given in Table 1 plot x against z on your printed copy of the X-Z Plot (Diagram 2).  In this graph the x-axis points towards the Galactic Center, the z-axis is perpendicular to that, with positive numbers pointing up, and negative numbers pointing down.
On your X-Z Plot identify the disk, the bulge, and the halo of the Galaxy. Clearly label each component. [Remember that this is a two-dimensional drawing: the y-axis is collapsed into the plane of the Galaxy (i.e., the y-axis has been eliminated); you are only looking at the x-z plane].
Assume that the center of the Galaxy is in the center of the distribution of the globular clusters. Figure out where you could draw a line parallel to the z-axis (the vertical axis) such that equal numbers of clusters fall on each side of the line. So then, the z-coordinate of the center should be set to 0. Using a pen of a different color mark the new scale in your plot. Insert the plot into your lab report with your signature and date.
Type out your answers to the below questions in your report.

Most globular clusters are located in a narrow range above and below the galactic plane. Roughly how many kiloparsecs above and below the galactic place are those globular clusters (i.e., how thick is the disk of the Galaxy in kilo-parsecs)? Estimate the uncertainty in that number.

Thickness of Galactic Disk = ___________ ± ___________ kpc

Measure the distance in kiloparsecs from you to the central point in the distribution of the globular clusters. How many kiloparsecs away is the center? Estimate the uncertainty in that number.  (NOTE: this value will also be used for Part 2 of this lab)

Distance to Galactic Center = ___________ ± ___________ kpc

From that plot, what diameter would you infer for the disk of the Galaxy?

Diameter of disk of Galaxy = ___________ ± ___________ kpc

What diameter would you infer for the halo of the Galaxy?

Diameter of Halo of Galaxy = ___________ ± ___________ kpc

Look at your answers in questions 3, 4, 5, 6 and 7. Which of those quantities have the largest uncertainty, which ones the least? Explain your answer.
How far above or below the Disk of The Galaxy would you place our Solar System?

Distance to Disk of the Galaxy = ___________ ± ___________ kpc
Lets compare the data on globular clusters to data on novae. The work has been done for you and the distribution of the novae have been plotted on the Milky Way Novae Distribution (Diagram 3). Your task is to understand and interpret this plot. Compare diagrams 3 and 1 ” the distribution of globular clusters to the distribution of novae – then sketch the Milky Way onto your printed out Milky Way Globular Clusters Distribution Graph (Diagram 3 – the bottom plot), the plot of the novae. 

Determine the position of the Galactic Center from Diagram 3. 

RA = ____________________ ± ____________________
Dec = ____________________ ± ____________________

Diagram 3 seems to have an additional arc of points for right ascensions ranging from 0 to 12 hours. What is this? Is this some type of illusion, or was that omitted in Diagram 1?  Explain your answer.
 Compare the distributions of globular clusters and novae. Is the bulge equally big (give the numbers behind your answer)?
Is the disk equally thick (give the numbers behind your answer)?
Would you expect to derive the same overall shape of the Galaxy from both data? Explain your answer. 

Part 2: Calculating the Mass of the Milky Way

The enclosed equation is the Orbital Velocity Law which allows us to use the orbital speed (v) and radius (r) of an object on a circular orbit around the galaxy to tell us the mass (Mr{“version”:”1.1″,”math”:”$M_r$“}) within the orbit of the object. For our calculations the object will be our sun in its orbit through the Milky Way Galaxy. 
In this formula v equals the velocity of the Sun in its orbit around the galaxy and G is the value of the gravitational constant. Use the following values for your calculations. Show all calculations with your submitted lab.  
r = ____________ kpc (kiloparsecs)  (this value is from Part 1, Question #4 of this lab)
v = 250,000 m/s
G = 6.67 x 10-11 m3 / kg s2
For all measured values of this equation to be equal you r value in kiloparsecs (kilo = 103) must be converted into meters since the distance value for the Gravitational Constant, G,  is given in meters. Use the following conversion value to convert your r value, in kiloparsecs, to an r value in meters:   3.08 x 1019 m / 1 kpc 
Exercise 2

Convert your r value in kiloparsecs to an r value in meters (display your answer in scientific notation)

Now that youve converted this distance to meters all terms are alike for the remainder of the calculation & will cancel out leaving your final value in terms of mass (kilograms or kg).

Now, use the Orbital Velocity Formula to calculate the mass of the Milky Way Galaxy (again, show each step of your work displayed in scientific notation)

Your answer for Question #2 is a very large number that no one has the ability to comprehend so lets try to put it into terms of something we do understand, – our Sun.  The Sun has a solar mass that is signified by M (or the Suns mass = 1
M.  In kilograms 1M equals 2 x 1030 kg.)

Convert the mass of the Milky Way Galaxy calculated in Question #2 into solar masses, or MŠ™.   (again, show each step of your work displayed in scientific notation)

Use this hyperlink, Milky Way Rotational Velocity to find the actual mass of the Milky Way Galaxy and compare your calculation to the actual mass. (you will need to move the shaded red region down to the diameter of the Sun) This is a screenshot of the Milky Way Rotational Velocity Explorer.

How do the two measurements for mass of the Galaxy compare? Identify any sources that would make your calculation inaccurate.
Calculate your percent error in for calculations with the following formula. (show your work)

(m actual – m calculated / m actual)  x 100 = ________
Part 3. Additional Research on Dark Matter.
Find a scientific article that talks about the evidence for the existence of Dark Matter. Write a short paragraph (about 50 words) summarizing the findings of the article.