Milky Way Galaxy

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.

1. Would you describe the distribution of clusters on the plot as random, or is there a pattern (explain your answer)?
2. 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, let’s 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 earth’s 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, let’s 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.

1. 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

1. 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

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

Diameter of disk of Galaxy = ___________ ± ___________ kpc

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

Diameter of Halo of Galaxy = ___________ ± ___________ kpc

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

Distance to Disk of the Galaxy = ___________ ± ___________ kpc

Let’s 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. 

1. Determine the position of the Galactic Center from Diagram 3. 

RA = ____________________ ± ____________________

Dec = ____________________ ± ____________________

1. 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.
2.  Compare the distributions of globular clusters and novae. Is the bulge equally big (give the numbers behind your answer)?
3. Is the disk equally thick (give the numbers behind your answer)?
4. 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

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

Now that you’ve 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).

1. 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 let’s 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 Sun’s mass = 1

M.  In kilograms 1M equals 2 x 1030 kg.)

1. Convert the mass of the Milky Way Galaxy calculated in Question #2 into solar masses, or M⊙.   (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.

1. How do the two measurements for mass of the Galaxy compare? Identify any sources that would make your calculation inaccurate.
2. 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.

Material

 

Required: 

· Calculator

· Computer and internet access

· Ruler

· Pencils and pens, eraser

· Digital camera and/or scanner

·

Milky Way Globular Clusters/Novae Distribution Graphs (Diagrams

1

&

3

)

(please print out)

·

X-Z Plot (Diagram

2

)

 (please print out)

Time Required: approximately 3 hours

Pre-exploration Study and Information

Introduction

Our understanding of the size and shape of our galaxy the Milky Way is hindered by the simple fact that our view of it is from inside itself. Therefore, determining the shape and size of our own Galaxy is quite a challenge. In this lab, you will analyze some of the original data of two astronomers Shapley and Curtis. Both of them came up with some right and some wrong conclusions about the size and shape of our galaxy, the Milky Way. Their results led to new philosophical interpretations about our location, role, and importance in the universe.

The appropriate location of the center of the Milky Way was discovered by Harlow Shapley seventy years ago. Today, we have now located the center more precisely by observing at infrared and radio wavelengths (these can penetrate dust, thus eliminating the problem of stellar extinction). Shapley’s method is interesting nonetheless, and it is an example of how some thought and educated guessing can lead to a correct result. Shapley used globular clusters to define the “skeleton” of the Milky Way.

Globular clusters are compact groups of stars, which are roughly spherical in shape. A globular cluster may contain a million stars and is therefore much brighter than a single star. Thus ,they can be spotted at large distances. Shapley reasoned that since globular clusters can be identified at great distances, he might be able to determine the edge and the center of our Galaxy.

Part 1: The Milky way according to Shapely

In this exercise, you will repeat Shapley’s study using the data on globular clusters given in Table 1. Your task is to make two plots, one of RA versus Dec (Diagram 1) and another one with x versus z (Diagram 2). We must start, however, with a discussion of the coordinates involved. Skip the next text section if you are already familiar with right ascension and declination.

Sky Coordinates

The latitude of an object in the sky is called the declination. An object (like Polaris) whose position is over the Earth’s north pole has a declination of +

9

0 degrees; an object over the south pole is at -90 degrees. Objects on the celestial equator have a declination of zero.

Longitude on the sky is called right ascension. On Earth, the line dening zero degrees longitude is fairly arbitrary. It is the circle that goes from the north pole to the south pole which passes through Greenwich, England. Right ascension (RA) on the sky also has an arbitrary zero point. It is a circle from the north celestial pole (Polaris) to the south celestial pole that passes through one of the points where the ecliptic crosses the celestial equator (in the constellation of Pisces). The only difference between longitude on earth and right ascension on the sky is that longitude is usually measured in degrees (from 0 to 3

6

0 degrees), while right ascension is measured in hours (from 0 to 2

4

hours).

The point to the above discussion is that the coordinate system astronomers have for objects in the sky is similar to the coordinate system that map makers have on earth. Just as every location on earth has a longitude and latitude, every object in the sky has a right ascension and declination.

+4.1

-0.7

+7.0

-2.3

+4.8

Table 1. Global Cluster Positions
Name	RA	DEC	x(kpc)	z(kpc)
1	NGC 2 8 08	09h 10 .1m	-64o 39’	+2.0	+1.8
2	NGC 4 14 7	12 h 07.6m	+ 18 o 49’	-1.4	+18.2
3	NGC 5 024	13 h 10.5m	+18o 26’	+3.1	+ 19 .7
4	NGC 5139	13h 23.7m	-47o 03’	+3.2	+1.3
5	M 5	15 h 16 .0m	+02o 16’	+5.5	+5.9
6	M 80	16h 14.1m	– 22 o 52’	+ 11 .9		+4.1
7	M 13	16h 39.9m	+36o 33’	+2.4
8	M 19	16h 59.5m	26o 11’		+7.0	+1.2
9	NGC 6293	17 h 07.1m	-26o 30’	+9.7	+1.4
10	M 9	17h 16.2m	-18o 28’	+12.6	+2.2
11	NGC 6366	17h 25.1m	-05o 02’	+16.0		+4.8
12	M 14	17h 35.0m	-03o 15’	+13.1	+3.5
13	NGC6397	17h 36.8m	-63o 39’	+2.7	-0.6
14	NGC 6441	17h 46.8m	-37o 02’	+7.5	-1.1
15	NGC 6522	18h 00.4m	-30o 02’	+8.5		-0.7
16	NGC 6541	18h 04.4m	-43o 44’	+3.9
17	M 28	18h 21 .5m	-24o 54’	+4.7	-0.5
18	M 22	18h 33.3m	-23o 58’	+2.9	-0.4
19	NGC 6723	18h 56.2m	-36o 42’		-2.3
20	NGC 6752	19h 06.4m	-60o 02’	+4.3
21	M 56	19h 14.6m	+30o 04’	+1.7
22	M 75	20h 03.2m	-22o 05’	+29.6	-15.3

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.

1. Would you describe the distribution of clusters on the plot as random, or is there a pattern (explain your answer)?

0. 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, let’s 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 earth’s 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, let’s 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.

 

0. 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

0. 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

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

Diameter of disk of Galaxy = ___________ ± ___________ kpc

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

Diameter of Halo of Galaxy = ___________ ± ___________ kpc

0. Look at your answers in questions 3, 4, 5, 6 and 7. Which of those quantities have the largest uncertainty, which one’s the least? Explain your answer.

8. How far above or below the Disk of The Galaxy would you place our Solar System?

Distance to Disk of the Galaxy = ___________ ± ___________ kpc

Let’s 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. 

9. Determine the position of the Galactic Center from Diagram 3. 

 

RA = ____________________ ± ____________________

Dec = ____________________ ± ____________________

 

10. 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.

11.  Compare the distributions of globular clusters and novae. Is the bulge equally big (give the numbers behind your answer)?

12. Is the disk equally thick (give the numbers behind your answer)?

13. 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

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

Now that you’ve 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).

2. 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 let’s 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 Sun’s mass = 1

M.  In kilograms 1M equals 2 x 1030 kg.)

3. Convert the mass of the Milky Way Galaxy calculated in Question #2 into solar masses, or M⊙.   (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.

4. How do the two measurements for mass of the Galaxy compare? Identify any sources that would make your calculation inaccurate.

5. 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.

Submit