# Teacher’s Guide to Specific Sections

## Introduction

Let students read through the Introduction on their own. Ask them why the discovery that the universe was expanding was so unexpected. Point out that its discovery required three separate threads to come together: Einstein’s General Relativity, Slipher’s measurements of redshifts, and the building of the new 100″ telescope on Mount Wilson. Ask them how the three other lines of evidence support the big bang theory. Get them thinking about how different lines of evidence add up to support a scientific theory. Ask: what makes a good theory?

## The Simple Diagram

This section is designed to give students some exposure to how astronomers know that the universe is expanding. This section, up to an including Exercise 4, can stand alone as a shorter project.

Make sure students understand the difference between absolute and relative distance. Have them explain why the Hubble diagram requires only relative distance. Have them study the picture above Exercise 1. Some students may have difficulty understanding the logarithmic scale of magnitude. Explain that other scales are also logarithmic: in the Richter Scale, a magnitude 7 earthquake is 10 times stronger than a magnitude 6. In the decibel scale, a 100 decibel sound is 10 times louder than a 90 decibel sound. Likewise, a magnitude 6 star or galaxy is 2.51 times brighter than a magnitude 7 star/galaxy. Tell students that each wavelength of the SDSS survey (u,g,r,i,z) has its own magnitude. In Exercise 1, they may choose any wavelength, but they should be consistent with their choice.

The key to doing Exercise 1 is using the Navigation Tool. In the Navigation Mosaic window, students should click “Select by ObjID,” then carefully type the appropriate ID into the window. Click on the red Zoom button to switch to the zoom window; the galaxy will be circled, and its five magnitudes will appear in the table at the top right. For more information about the galaxy, students should click the “More” link. To save the galaxy in their notebook, students should click the “Save” link.

Be sure students understand the concept of redshift before discussing spectra. Remind them that looking at spectra is a way to measure redshift: spectral lines serve as markers for how redshifted a galaxy is.

Students will need to use a graphing program. A tutorial for Excel can be found on the SkyServer web site.

A “model” is a theoretical explanation for the relationship seen in the data. In the case of the Hubble diagram, scientists use a linear model for redshift as a function of distance – they assume that the underlying astrophysics is such that redshift will be linearly dependent on distance. Note that the mathematical model, by itself, says nothing about the underlying astrophysics; it says only that the laws should be such that they will produce a linear relationship between distance and redshift.

To test whether a model is a true description of data, scientists test the “fit” of the model. They find a trendline – the line that comes closest to fitting through each data point. Then, they calculate the distance the line falls from each actual data point. The value r^{2} gives a quantitative measure of how close the line comes to actually describing the data. A value of r^{2} = 0 means that a line can’t describe the data at all; a value of r^{2} = 1 means that a line fits perfectly. In Exercise 4, students use Excel to find the r^{2} value of their simple Hubble diagrams. They do not need to know how to calculate r^{2} themselves, but if they are interested, refer them to an introductory statistics book.

The goal of finding r^{2} is to quantitatively determine how well the model describes the data. What value is considered a “good fit” varies among different sciences and research topics. In astronomy and other physical sciences, r^{2} = 90% is considered a good fit.

Exercises 5 and 6 are designed as “teasers.” Students repeat the analysis they did in Exercises 1-4 with six other galaxies, and discover that the Hubble diagram they get looks terrible. Do these exercises only if you intend to have students do Part II of the project.

The fit of the new diagram is less than 10% in all wavelengths. Why does the new diagram, which was made in exactly the same way as the old diagram, look so different? The answer is that the six galaxies in Exercises 5-6 have vastly different properties, so students cannot simply assume that any observed differences are due to different distances from Earth. Ask: what could some of those differences be? Have students study the six galaxies with the Navigation Tool to generate some ideas.

## Distances

Exercise 7 should give students practice in finding relative distances to galaxies from their magnitudes. Exercise 8 is important, because students must be able to calculate actual relative distances to make their Hubble diagrams. Be sure all students can solve Exercise 8 before moving on. The paragraph below the question tells students why finding relative distances to galaxies is not enough, and gives them a solution: look at galaxy clusters.

Teach your students the idea of “populations.” When astronomers find the distance to a galaxy, they need to control for galaxy variations by comparing “average” galaxies. This is like taking a survey to find the average family income in a town. If you ask only one family, you have no idea if their income is anywhere near the town’s average. If you ask several families, the average of their incomes will be closer to the true average income for the town (which is what you are trying to find). The more families you ask, the more confident you can be that you are finding the true average. Similarly, the more galaxies astronomers look at, the more confident they can be that the average properties of galaxies in their sample are to the average properties of galaxies in general. They know that galaxies in a cluster are at roughly the same distance, so they can be sure that the ratios between observed properties of galaxies in the same cluster is the same as the ratios between the true properties of those galaxies.

Exercise 9 is a difficult question. If your students have not learned trigonometry, tell them to skip it. The key to the question is recognizing that the largest possible distance between two galaxies in a cluster is equal to the diameter of the cluster, which is equal to the cluster’s distance times the tangent of the cluster’s angular size (see the answer key for an explanation and a diagram).

The analogy to cities and towns should get students thinking about how to recognize similarities and differences based only on what they can see. Ask: what kinds of buildings do towns and cities have? How big are the buildings? Where do the buildings sit in relation to one another? Once they have explored the analogy, ask what is similar between buildings and galaxies? What is different? How might you apply the analogy to understanding galaxies? Once students have generated a list of ideas for how to recognize galaxies in clusters, let them practice by doing Exercise 10.

The paragraph after Exercise 10 tells students that to compare galaxies in different clusters, they must be sure to compare similar galaxies. To know whether two galaxy clusters are truly at different distances, you must look at two galaxies of the same type, size, and intrinsic brightness.

In Exercise 11, students should pull together all that they learned from this section to examine galaxies in three clusters in the same area of the sky. In Exercises 12 and 13, they look at these galaxies with the Navigation Tool. Encourage them to come up with creative ways to determine the galaxies’ relative distances. These exercises will give them the distance data they will need to make a Hubble diagram.

## Redshifts

Make introductory physics and astronomy textbooks available so that students can look up terms and concepts as they work through this section. Tell students that spectra are one of the most important tools that astronomers use. In addition to redshift, spectra can tell astronomers the temperatures and compositions of stars and galaxies.

If you have a hydrogen discharge tube, have students measure the wavelengths of the Balmer lines for themselves. If not, tell them to use the table provided on the site. In Exercise 14, point out that the lines are marked on the spectrum. Ask: why are the Hg and Hd lines valleys instead of peaks? Have them use the formula provided to calculate redshift from the spectral lines.

The fact that redshift can be interpreted in two ways is a subtle but important point. When objects are close to Earth, their redshifts should be interpreted as coming from Doppler shifts due to relative motion. When objects are far from Earth, their redshifts should be interpreted as coming from the cosmological stretching of space. Be sure that students understand the concept of the stretching of space, because they will need it to understand the big bang in the next section.

In Exercise 15, explain the idea of a template – an expected, baseline curve to which experimental curves should be compared. Explain that not all galaxy spectra will match one of the SDSS’s nine templates, but most will. All the galaxies in Exercise 15 will match one template or another.

Show students how to use the redshift application, then let them find the galaxy redshifts on their own. Remind students that they do not need to get the curves to match perfectly, but they should find the template and redshift that gets the best match to the major emission and absorption lines.

In Exercise 16, students will pull together what they have learned in this section to find the redshifts of ten galaxies in the three overlapping clusters from the last section (SDSS does not take spectra for every object, so these are the only ten galaxies in this region with redshifts available). Students should find the redshifts using the Get Spectra tool; however, if they are especially interested in the definition of redshift, they can calculate redshift as they did in Exercise 14 using the same table the SDSS software uses.

Exercise 17 asks students to find the average redshift to galaxies in the SDSS database using the Get Plates tool. They should examine at least 20 spectra to find the average. Quasars are distant, bright objects that shine when matter falls into a black hole, heating up and glowing as it falls. Quasars are the brightest objects in the universe, and astronomers can see them at much greater distances than other objects. The most distant object ever seen, a quasar at z = 6.28, was found in 2000 by astronomers using SDSS data.

## Conclusion

The text describes the two observations Hubble made to establish that the universe is expanding. Ask students why his observation that our location in the cosmos is not special is so important. How can astronomers assume that all other galaxies would see the same thing?

Have students think about the differences between the explosion model and the big bang model. How could you tell, based only on what you can see and measure from Earth, which model correctly describes our universe? Remind students that the process of deciding between models based on observable evidence is at the heart of the scientific method.

Remind students of the difference between relative and absolute distance. Astronomers do know absolute distances to some galaxies, based mainly on looking at apparent magnitudes for known standard candles (such as Cephid variables or Type Ia Supernova).. From these absolute distances, they can use the equation to find a numeric value for H_{0}. Finding the value of H_{0} has been an important project in the past decade.

In Exercise 19, students pull together everything they have learned to make an improved Hubble diagram. They made this diagram following almost exactly the same steps that Edwin Hubble followed in 1929. Exercise 20 asks them to throw out one data point to improve the fit of the line. Teach them the concept of an “outlier” – a data point significantly different from other points. Get them thinking about when it is appropriate to throw out outliers, and when they should keep all their data (this is not always an easy decision!)

In the last question, you may wish to have students make a written outline, or even a formal proof, of the logic behind the big bang. Have them think about deductive logic: what does it mean to argue that something is true? How do you get from one logical point to another? Lastly, comment on how Hubble and Humason returned to collect more data. How much data would you need to collect to prove something is true? What is required to convince someone of an argument?

Exercise 21 is the final challenge. This exercise should not be done in the classroom. It is a completely open-ended exercise: students choose their own galaxies, all over the sky, and find relative distances and redshifts to each. Encourage students to complete this exercise on their own, for fun. You may wish to give extra credit to students who do it.