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The Big Bang and The Expansion of the Universe CLEA Exercise: The Hubble Redshift-Distance Relationship

From Wikipedia for free use, downloaded from http://en.wikiversity.org/wiki/Educational_Media_Awareness_Campaign/Astronomy/POTD_3. In this exercise you will be measuring the reshifts and distances of increasingly distant galaxies and then discuss possible interpretations. In fact, you will walk in the steps of Hubble, who basically made his discoveries about galaxies in the universe in a similar manner.

The computer program was written and designed by Department of Physics Gettysburg College Gettysburg, PA 17325 Telephone: (717) 337-6028 email: [email protected]

The lab instructions are written and edited by Esther L. Zirbel Department of Engineering Science and Physics College of Staten Island Staten Island, NY 10314 Email: [email protected]

CLEA Hubble Redshift  Lab 13  1

Magnitude Distributions and Distances of Galaxies in Clusters

(a) Virgo Cluster

(b) Hercules Cluster

(c) Coma Cluster

(d) Centaurus Cluster

(e) Perseus Cluster

(f) Hydra Cluster

Fig 2: Images from Kitt Peak National Observatory at http://www.noao.edu/image_gallery/galaxies.html 2  Lab 13  CLEA Hubble Redshift

Part I – Galaxy Magnitudes 1) Which of the clusters in Figure 2 are the most and least distant ones? Most distant: _________________________________________________________________________ Least distant: _________________________________________________________________________ 2) Compare the brightest galaxy in the Coma cluster to the fifth brightest one in the Virgo cluster. Which one is brighter, which one is more luminous, and which one is more distant? Brightest ____________________________________________________________________________ More luminous _______________________________________________________________________ More distant _________________________________________________________________________ Fig 3a: The Images of group members of some of the clusters are displayed on the left. It seems that, on average, the fainter the galaxy, the farther away it is. However since galaxies have a range in luminosities (they may be 100,000 times fainter to 100 brighter than our own galaxy) this general rule is not always correct. However, if we refer to the brightest or second brightest galaxy in a cluster we can pretty much assume that these have comparable luminosities.

Fig 3b: Corresponding spectra of the galaxies are shown in the middle of each spectrum (red is at the left, blue at the right). The top and bottom spectra (the vertical lines) are comparison spectra of a known gas (these are needed for calibration purposes). Look at the H and K lines; relative to the comparison spectra, the spectra of the galaxies are shifted towards the redder part of the spectrum. For more distant galaxies, this shift in wavelength (the arrow) is larger. In this lab you will be measuring this shift in wavelength, and from it determine the redshift and the velocity of the galaxies.

Original image in J. Silk, The Big Bang, 2nd Ed., and from http://www.fas.org/irp/imint/docs/rst/Sect20/A9.html.

3) Which of the galaxies in Figures 3a and b are the most and least distant ones? Most distant: _________________________________________________________________________ Least distant: _________________________________________________________________________ CLEA Hubble Redshift  Lab 13  3

Part II – Using the CLEA Software As you have read in the Pre-Lab, Edwin Hubble discovered a relation between a galaxy’s distance and its redshift — a relation that showed that our universe is expanding. In this exercise, you’ll make your own measurements using a computer simulation and a virtual telescope. Well, have fun with it.

1. As in previous labs turn on the computer, click on “Hubble” and log in. This takes you to a starting

page where you can select ‘start’ to begin. Youu’ll see a control panel on the left, and what looks like the inside of a closed telescope dome in the center. Open the dome by clicking the ‘dome’ button. Voila, the sky — with galaxies and stars. This view through the ‘finder’ — a CCD camera that shows you a wide portion of the sky to help you find your objects. (Below the ‘monitor’ button is some text that tells you what mode the camera is in. Currently it is set to ‘finder.’) Notice that the galaxies and stars appear to move, because the Earth is turning underneath the sky. Click ‘tracking’ to turn on the motor that counteracts the Earth’s rotation.

2. Your mission is to use the spectrometer to measure the wavelength of the calcium K and H line of 10 galaxies (two per field). Most galaxies are dominated by sunlike stars (F, G, K, M stars) which have relatively strong H and K lines. These lines are often used to determine galaxy redshifts. If you were looking at a galaxy with active star formation, which lines would you expect to see?

a) Use the left mouse button to center the galaxy (press and hold N, S, E, or W buttons). b) Click on the ‘monitor’ button and the mode will switch to ‘spectrometer’ and the view will now show the slit of a spectrometer. Fine-tune your centering to put the brightest part of the galaxy on the slit. c) Press the ‘take reading’ button at upper right. Press ‘start’ to start recording a spectrum. After a while, you’ll see that there are two absorption features in the spectrum. When you see that “Signal/Noise” is at least 20, stop the measurement. Click the left mouse button exactly at the bottom of the H (the right line) and the K line (the left line). d) Record the wavelengths AND the apparent magnitude of the galaxy in Table 1. e) Click ‘return’ to get back to the telescope view, switch the mode to ‘finder’, and then at the top press ‘change field.’ f) Select a different field and then repeat steps (a) - (e) until you have collected data on (at least) two galaxies from EACH of the 5 fields. Note that the galaxies in the field Ursa Major II are very faint! You might have to wait until you are in spectrometer mode to see them. Record your data in the Table. 4  Lab 13  CLEA Hubble Redshift

Galaxy

λmeas ZK ZH

zH + zK 2

Zmean

_________________

m

m-M

d [Mpc]

⎛ m − M − 25 ⎞ ⎜ ⎟ ⎝ ⎠ 5

⎛ m − M − 25⎞ ⎜ ⎟ ⎝ ⎠ 5

(be mindful of significant figures!)

this is 10 to the power of

Ho

v c

⎡ km sec ⎤ ⎢ ⎥ ⎣ Mpc ⎦

Redshift correlates to velocity by: z =

v [km/sec]

km sec Mpc

Distance in mega-parsecs: d = 10

Mean redshift: z mean =

λmeas-λrest

H line (right line) λmeas

The mean Ho for all galaxies from this table is:

v Hubble constant: H o = ; v = 3.0 x 108 m/sec d

λmeas − λrest λrest

λmeas-λrest

The redshift is defined by: z =

Cluster

K line (left line)

Table 1: Using the Program determine λH, λK, and the apparent magnitude of the galaxies (yellow columns). Instructions are on next page. (Use scientific notation and be mindful of significant figures. Consult the Toolkit if necessary.)

Part III – Determining the Hubble Constant (Complete and Analyze Table 1) 1. Did you notice that fainter galaxies had longer wavelength K and H lines? For each galaxy calculate the shift in wavelength, the redshift, and the recessional velocity, and record values in Table 1.

If you are uncertain how to do it, here is a list of individual points; a) Calculate the shift in Wavelength; i.e., λmeas -λrest. The rest frame wavelengths of the H and K lines are λrest = 3968.85 Å and λrest = 3933.67 Å b) Then calculate the redshift for each line of each galaxy. The redshift is defined by: z =

λmeas − λrest λrest

c) For each galaxy you have two values of the redshift. One determined from the H line, and one from the K-line. Ideally you’d have even more lines. Take the mean of those lines for a more accurate redshift. d) Mean redshift: z mean =

zH + zK 2

e) Next calculate the velocity. Redshift correlates to velocity by: z =

v c

Velocity of light: v = 3.0 x 108 m/sec

2. Next, you’ll use each galaxy’s apparent magnitude to calculate its distance from Earth. For simplicity we are going to assume that all galaxies are equally bright, and the apparent magnitude depends on the distance of the galaxy from us. Look at Figure 2 again and comment on whether or not this is a “good” assumption.

If we had more clusters, a better method might be to look only at the “brightest cluster member” (BCM) and use those galaxies are ‘standard candles’ (the magnitudes of BCM’s are all within 0.3 magnitudes of the mean magnitude). This was method was introduced by Hubble, and is still used today, though to much larger distances.

3. Let’s assume all your galaxies are equally bright (with absolute magnitudes of M = -21.0). Then calculate the distance for your galaxies. You have done the math before in the Spectroscopy-Lab:

a) For all galaxies calculate the distance modulus, i.e., m – M. b) Then calculate the distance using m − M = 5 log d − 5 First solve for the distance: d = 10

⎛ m − M − 25 ⎞ ⎜ ⎟ ⎝ ⎠ 5

(This is “10 to the power of

m − M − 25 5

”!)

Then plug in your numbers, and you will get your answer in units of Mega-parsecs.

4. Remember that Hubble’s Law says that a galaxy’s velocity and distance is related by H o =

v ; d

a) Calculate Ho for each galaxy and write these values into your Table. b) Determine the average value of Ho for the 10 galaxies. Use the statistical functions of your calculator, type in all values of Ho, and then determine the mean and the standard deviation (check your Toolkit for the definitions of “mean” and “standard deviation”) c) Be mindful of significant figures! (Rusty? Check the Toolkit)

The mean Ho from the Table is: ____________ ± ________

The standard deviation goes in here

km sec Mpc

5. Since the velocity is measured in km/sec, and the distances to the galaxies in Mpc, this means that the units of the Hubble constant are Ho =

km sec velocity in km sec , i.e., the units are (which is often written as “km·s-1·Mpc-1”) dis tan ce in Mpc Mpc

Note that the units of Ho are strange, since both km and Mpc are distances, but measured in different units! So if the units were to be converted to the same kind (i.e., km to Mpc, or Mpc to km) then they would cancel out (also check the Appendix). Now think of what the units would be for Ho.

The units of Ho are

km sec and _____________ Mpc

CLEA Hubble Redshift  Lab 13  7

6. Analyze the Hubble Redshift-Distance Relation

a) Plot of the points of all galaxies. b) Then draw a straight line that best fits all the points and goes through the origin at (0,0). Use a RULER. c) Next, determine the value of Ho directly from your plot. Measure the slope of your line by picking two points on the line and dividing velocity by distance. slope = H o =

v 2 − v1 d 2 − d1

Ho from the Slope of the Graph is: _____________

km sec Mpc

7. Estimate the accuracy of your value of Ho from your Plot. Take a pen and draw a line of the steepest possible slope and another of the shallowest possible slope that barely fits your data. Then calculate the Hubble constant for both values.

The steepest slope is ____________ The shallowest slope is ____________

Your value of Ho is: ____________ ± ________

km sec Mpc

8. Which method of determining the Hubble constant do you prefer. Determining the way you did in section 6, or calculating it from the above plot. Explain your answer.

8  Lab 13  CLEA Hubble Redshift

The Hubble Redshift-Distance Relation 50000

Velocity in km/sec

40000

30000

20000

10000

0 0

100

200

300

400

Distance in Mpc

CLEA Hubble Redshift  Lab 13  9

500

Part IV – Interpretation 1. What can you say about the velocities of more and more distant galaxies?

2. Explain how this implies that the universe is expanding.

3. If everything is moving away from us, does that put us into the center of the universe? Explain.

10  Lab 13  CLEA Hubble Redshift

4. Explain how this implies that there was a “Big Bang,” i.e., there was a “point” in space and time.

5. There are also alternate theories that explain these observations. Make some suggestions.

6. Now we know that the universe is expanding and that there was a “Big Bang”. But – when did this happen? Since we know the relationship between distance and velocity, we can calculate that.

Write down the relationship between velocity, distance and time.

________________________

Write down Hubble’s law.

________________________

Now combine both equations and solve for time. This is the age of the universe.

Translate your answer into words.

CLEA Hubble Redshift  Lab 13  11

7. The age of the universe

Answer from Part III, question 5

a) The units of Ho are km/sec/Mpc or __________. The age, being the inverse Ho, then has units of (km/sec/Mpc)-1 and __________. b) So, you will have to convert the units and change Mpc to km as illustrated in the Appendix. For a Hubble constant of 75 the age of the universe is 13 billion years. Your Hubble constant is probably slightly larger than 75 km/sec/Mpc. Would a larger Hubble constant make the age of the universe longer or shorter? Explain in words.

c) Calculate the age of the universe for your value of Ho. [There is an easy and a long & more difficult method. The easy method involves using the information given in the Appendix and requires you to think in ratios, rather than recalculating everything. This will save you lots time and cut down the amount of algebra.]

d) If you knew a galaxy was 800 Mpc away, what would you measure for its velocity? [Hint: Remember that there is a correlation between distance, velocity and the Hubble constant. First write down this relationship, then insert your value of Ho and the distance of 800 Mpc…]

e) Hubble’s original measurement of the Hubble constant was 530 km/sec/Mpc. By what factor (calculate a ratio) would this change the age of the universe? Calculate that age and comment on whether nor not this could be a reasonable number.

12  Lab 13  CLEA Hubble Redshift

Lab Report 1. Objective of the Lab.

2. Describe the experimental procedure. [What did you measure? How did you measure that? How did you reduce your data? How did you arrive at your final answer?]

3. What did you learn about scientific methodology and deduction?

4. What do you think about this lab? Please comment.

CLEA Hubble Redshift  Lab 13  13

Appendix — Sample Calculation for H = 75 o

The calculation itself is easy — you just take the inverse of the Hubble constant, and you got the age. For example, for Ho = 75 km/sec/Mpc, the age of the universe would be 0.0133 Mpc/(km/sec). Now that is a pretty meaningless result. So therefore the whole problem lies in converting the units. Here is an example of how to do this. 1 ⎡ km ⎤ ⎢ ⎥ 75⎢ sec ⎥ ⎢ Mpc ⎥ ⎣ ⎦ To solve this we need to change the units from Mpc to km. t empty =

1 = Ho

Use 1 pc = 3.09 × 1016 m and 1m = 10 −3 km m km 1Mpc = 10 6 pc = 10 6 pc × 3.09 × 1016 = 3.09 × 10 22 m = 3.09 × 10 22 m × 10 − 3 = 3.09 × 1019 km pc m

t empty =

1 = Ho

1 1 1 = = 4.12 × 1017 sec = 1 ⎡ km ⎤ ⎡ km ⎤ 2.43 × 10 −18 75⎢ ⎥ ⎢ sec ⎥ sec ⎣ sec ⎦ 75⎢ ⎥ 19 ⎢ Mpc ⎥ 3.09 × 10 km ⎢⎣ ⎥⎦

So the age of the universe for a Hubble constant of 75 km/sec/Mpc is 4.12 x 1017 seconds. To most people this number is meaningless, therefore, let’s convert seconds to years. One year has 365 days, each day has 24 hours, each hour 60 minutes and each minute has 60 seconds. Therefore, one year has 365 x 24 x 60 x 60 = 3.16 x 107 seconds. So, converting seconds to years we get: t empty = 4.12 × 1017 sec = 4.12 × 1017 sec ×

1 yr = 1.31 × 1010 yrs 3.16 × 10 7 sec

The age of the universe is roughly 13 billion years. There are two methods of calculating the age of the universe for a different Ho. a) Replace the value of “75” with your value of Ho and repeat the same calculation as above. b) Using a little trick. You have the answer for Ho = 75 km/sec/Mpc. For Ho = 50 km/sec/Mpc, you would have to take the inverse of 50 km/sec/Mpc. Or you could the previous for the inverse of Ho = 75 km/sec/Mpc, multiply it by 75, and then divide it by 50. The age is then: tempty =

1 = Ho

1 75 75 × = 1.3 × 1010 yrs × = 2.0 × 1010 yrs km 50 ⎡ ⎤ 50 ⎢ sec ⎥ 75⎢ ⎥ ⎢ Mpc ⎥ ⎣⎢ ⎦⎥

14  Lab 13  CLEA Hubble Redshift