Exercise 1. Exercise 2.

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1 Exercise. Magnitue Galaxy ID Ultraviolet Green Re Infrare A Infrare B Question. As objects get farther away, their light appears immer, just as a canle across the room looks immer than a canle hel at arm s length. If we assume that all galaxies output the same amount of light, then any ifference in the galaxy s apparent brightness will be ue to its istance from Earth. Magnitue measures the apparent brightness of a galaxy. Therefore, ifferent magnitues correspon to ifferent istances from Earth. Exercise 2. Galaxy ID Reshift

2 Exercise 3. Ultraviolet Magnitue Green 2 20 Magnitue

3 Re.5.5 Magnitue Infrare (i).5 Magnitue

4 Infrare (z).5.5 Magnitue Exercise 4. Wavelength Fit Ultraviolet 95.0% Green 99.9% Re 99.0% Infrare A 98.% Infrare B 95.5% No matter which wavelength stuents chose to look at, a straight line is a goo fit to their ata. Exercise 5. Galaxy ID Ultraviolet Green Re Infrare Infrare Reshift A B

5 Exercise 6. Ultraviolet Magnitue Green Magnitue

6 Re Magnitue Infrare (i) Magnitue

7 Infrare (z) 9.5 Magnitue Wavelength Fit Ultraviolet 8.8% Green 0.5% Re 2.8% Infrare A.9% Infrare B 2.4% No matter which wavelength stuents chose to look at, a straight line is not a goo fit to their ata. Exercise 7. The answers will epen on which wavelength stuents chose to examine to get the magnitues. In any wavelength, they shoul calculate istances from the magnitues m with the equation = 2.5 m, then create ratios where the nearest galaxy has = an farther objects have istances in the same proportions that they calculate: 2 =. x Stuents shoul solve for x, the normalize relative istance, then repeat for the other galaxies.

8 The answers below are for the green wavelength. Other wavelengths shoul give similar relative istances. Galaxy ID Magnitue (green light) Relative istance Exercise 8. From magnitue, = 2.5 m. The quantity 2.5 -m is the raiant flux, the amount of the galaxy s light that reaches a given area of Earth in a given time. Because the area an the time are constant, the raiant flux epens only on the amount of light that reaches Earth: the galaxy s apparent intensity. The inverse-square law of light says that the apparent intensity of a light source ecreases proportionally to the square of the istance to the source. This is because the amount of light the galaxy puts out the galaxy s true intensity is constant, but the light spreas out as it leaves the galaxy. It spreas into the surface of a sphere with surface area A = 4πR 2, where R is the raius of the sphere. To fin the apparent intensity from Earth, think about a sphere whose center is at the galaxy an whose surface intersects the Earth. This sphere s raius is equal to the istance from the galaxy to Earth. So the apparent intensity I is proportional to the istance to Earth square, an the raiant flux F is proportional to I: F = k I = k 2 (4π 2 ). Since stuents only have relative istances, they have no information on the values of the constants k an 4πk 2, an they can only fin relative fluxes. If k an 4πk 2 can be anything, so they might as well be. If the constants are set to, the equation becomes: F = 2, an solving for gives

9 =. F Lastly, substituting for raiant flux from the efinition of magnitue gives = 2.5 m. From apparent size, =, s app where s is the istance across the galaxy s image. The galaxy has some true size; suppose that the istance across the galaxy s true size is r. The galaxy s apparent size r app, epens on its angular size θ when viewe from Earth. The sketch below shows the relationship between r, θ, an, istance from Earth for galaxies at two istances, an 2 : θ θ 2 r 2 r The iagrams show that as increases, θ gets smaller. By the laws of trigonometry, r tan θ =. But galaxies are so far away from Earth that r is always much, much less than. When tan θ <<, they can use the approximation tan θ θ. Using this approximation, the equation becomes r θ =. Solving for gives

10 r =. θ The galaxy s apparent size, r app, is proportional to the angle θ with a constant of proportionality that epens on the galaxy s true size. Since stuents on t know the true size, an since they are only looking for relative istances anyway, they can let the constant equal. For the same reason, they can let the galaxy s true size r (which they on t know an on t nee to know to fin relative istance) equal. Therefore, =. s app Question 2: If the istance to a galaxy foun from magnitues o not agree with its istance foun with apparent sizes, then the galaxy is unusual in some way. Either it is brighter/fainter than a typical galaxy of its size, or it is larger/smaller than a typical galaxy of its brightness. Question 3: Although clusters are large objects, the universe is a very big place. Just as stars in one galaxy are much closer to each other than the galaxy is to another galaxy, galaxies in one cluster are much closer to each other than the cluster is to another cluster. The istance between two stars in a galaxy is always much, much less than the istance between two galaxies; similarly, the istance between two clusters is always much, much less than the istance between two clusters. Therefore, one can say that a single cluster is at a efinite istance from Earth, an each galaxy in that cluster is at approximately that istance. Exercise 9. The fractional error in the assumption that galaxies are at the same istance is equal to the longest istance between two galaxies in the cluster ivie by the assume istance to the cluster from Earth. Clusters are approximately spherical; suppose a cluster has iameter s an is locate at istance from Earth. The sketch below shows such a cluster: Earth θ s The angular size of the cluster measure from Earth is represente by θ. All the galaxies in the cluster are within the circle of iameter s. Therefore, the galaxy nearest to Earth is at istance - s, an the galaxy farthest from Earth is at istance + s. Subtracting these, the error in istance is s, so the fractional error is s.

11 The laws of trigonometry say that tan θ =. s But since galaxy clusters are so far from Earth, is much, much, much greater than s, so <<<<. Therefore, one can use the small angle approximation tan θ θ. Using this s approximation, θ =. Therefore, the fractional error in the assumption that the entire s cluster is at istance is equal to θ, the angular size of the cluster measure from Earth. Question 4. There are many possible answers to this question. Large cities have more builings in the same area than small towns o. Usually, they also have taller builings than small towns. They may have certain types of builings, such as hospitals or airports, that small towns lack. Similarly, large clusters have many more galaxies in the same area than small groups o. Unlike in the towns example, there is no correlation between galaxy size an cluster size larger clusters o not necessarily have larger galaxies. However, large clusters are more likely to have rare galaxies, like the rare builings hospitals an airports in large cities. This is because clusters have larger numbers of galaxies, an by looking at larger numbers of galaxies, one is more likely to fin a rare galaxy. Exercise 0. Most galaxies within the same cluster have similar colors, sizes, an brightnesses. Clusters can have ifferent types of galaxies, although more istant clusters ten to have more elliptical galaxies. Question 5. If Hubble an Humason saw that one approach gave them a smaller ata scatter that is, the reshift-istance ata they foun came closer to graphing a straight line they woul know that that particular approach was a better way to measure relative istance. Exercise. The largest, whitest galaxies belong to the nearest cluster. The smaller, more bluish galaxies belong to the meium cluster. The small, faint, reish galaxies belong to the most istant cluster. Exercise 2. Stuents answers will vary epening on which galaxies they selecte an what measures of relative istance they chose. Because they will nee galaxies with reshifts available, they shoul have selecte at least the ten galaxies in the table below. The table lists magnitues in the green wavelength an the relative istances calculate from those magnitues. Using green-wavelength magnitues of these ten galaxies is a

12 simple way to fin relative istance, but it prouces large ata scatter. If they use a ifferent metho, they will probably fin better estimates of relative istance. Galaxy ID Right Ascension Declination Green Magnitue Relative Distance Exercise 3. Stuents shoul use a ifferent metho to fin relative istances. Their istances shoul be roughly similar to the istances they foun in exercise 2. Exercise 4: Stuents shoul fin the entries in the table in the spectrum. The correct spectral lines are marke H α, H β, H γ, an H δ. Question 6. Using the Doppler shift interpretation, a reshift of z > woul correspon to a velocity greater than the spee of light. Traveling faster than the spee of light is impossible because of Einstein s Special Theory of Relativity. But using the ( 0 ) cosmological interpretation, a reshift greater than one simply means that > 2 ; in other wors, at the time corresponing to reshift z, the separation between two galaxies was less than half of its current separation. There is no conceptual problem with galaxies being twice as close to one another in the past. Therefore, there is no conceptual problem with z >. Exercise 5. Stuents shoul fin the following reshifts: Spectrum Number Reshift ( z)

13 Exercise. Results from Exercise 5 shoul be similar to the values given by Get Spectra. Exercise. The average reshift for the galaxies in Exercise 5 is The average reshift of all the galaxies in the SDSS atabase is about 0.. Galaxies range from a reshift of about 0.0 to about 0.5. Quasars have an average reshift of about.5. Their reshifts range from about 0. to about 6. Exercise. The inverse of H 0 = 70 km/sec/mpc suggests that the universe is 4 billion years ol. With the 0% uncertainty, the universe coul be between 2.6 billion an 5.4 billion years ol. This is generally consistent with stellar ages of to 3 billion years, although there is some small overlap. Exercise 9. Using the values of relative istance obtaine from the green wavelength magnitues gives the following Hubble iagram: Hubble DIagram Relative Distance A trenline through this ata has a fit of 90.8%, meaning a straight line is a goo fit to the ata. If stuents use a ifferent metho to fin the relative istance, their iagram woul look slightly ifferent.

14 Exercise 20. Galaxy ID is unusually im for a galaxy at its istance. Because it is im, using its relative magnitue to measure istance makes the galaxy appear farther from Earth than it actually is. Here is the Hubble iagram with galaxy ID remove: Hubble DIagram Relative Distance The fit has improve slightly, to 9.8%. Question 7. The Hubble iagram inicates that all galaxies are moving away from us, an that the farther a galaxy is, the faster it moves. Stuents shoul realize that to get from this observation to the expaning universe picture, they nee to assume that the universe is fairly uniform on a large scale that is, galaxies can be foun in roughly equal amounts in all irections. They also nee to establish that galaxies are reshifte in all irections of the sky. If Earth were moving through the universe, then behin us, we woul see reshifts, an ahea of us, we woul see blueshifts. If the universe as a whole were expaning, we woul see reshifts wherever we looke. Stuents coul test both these assumptions with Skyserver. They coul make a Hubble iagram in many ifferent parts of the sky to verify that reshifts appear everywhere. They coul also count galaxies in ifferent parts of the sky to verify that galaxies are roughly uniformly istribute.