Laboratory 8 - Hubble's Law Name: Background In 1929 the astronomer Edwin Hubble measured the distances to many galaxies. He also obtained spectra for these galaxies. From the spectra he calculated the velocities of the galaxies using the Doppler shifts of spectral lines. He found that all of the galaxies (except for a couple of the closest) displayed edshifts and are therefore moving away from us. Hubble then correlated the velocities of the galaxies with their distances and found a linear relation: the greater the distance to a galaxy, the faster it is moving away from us. Hubble's data is shown in Figure 1. Velocity-Distance Relation among Extra-Galactic Nebulae. FIGURE 1 Figure 1 The original graph found in Hubble's 1929 paper. Each circle in Hubble's graph is an individual galaxy. We have better data today, but even with Hubble's data we can clearly see there is a linear relationship. You may recall the equation of a line is y=mx+b. where m is the slope of the line and b is the y-intercept. For Hubble's line, we can replace y with velocity and x with distance. We'll give the slope a special letter so that m becomes . Also, we can see that the y- intercept is O. With these substitutions the equation of the line becomes: where is the recessional velocity, is the distance to the galaxy, and is the slope. We call the Hubble constant. The above equation is known as Hubble's Law.Historically, there has been considerable debate over the Hubble constant due to uncertainty in the galaxy data. Galaxies that are close to us have peculiar motions (motions about the center of gravity of their particular group of galaxies) which can be a large fraction of their recessional velocities due to the expansion of the universe. While galaxies that are very far away, often have rather uncertain distances. Thus, there have been "spirited" debates over the value of . In 2019 astronomers used Hubble Space telescope data to determine a Hubble constant of 69.8 km/(s:Mac). (An Mac is equal to a million parsecs. Hubble's Law is evidence that the universe is expanding. Space itself is expanding! The distance between all galaxies in the universe has been getting bigger with time, like the distance between raisins in a rising loaf of bread. An observer on any galaxy, not just our own, would see all the other galaxies traveling away, with the furthest galaxies traveling the fastest. Using Hubble's Law we can calculate the age of the universe. If all the galaxies in the universe are moving away from one another as space expands, then it stands to reason that all of these galaxies were closer together in the past. Let's assume that at the beginning of the universe (the Big Bang), everything was located in a single point. If the universe has been expanding ever since the Big Bang, then the time it took the universe to expand from a single point to the size it is now is the age of the universe In this laboratory you will calculate the age of the universe using Hubble's Law and data from Stellarium. PART A We define a quantity called the redshift (z). The redshift is a measure of how much a spectrum is redshifted. The larger the redshift, the faster the galaxy is moving away from us. The redshift can be determined from the following equation: Here is the wavelength of an absorption line at rest (what you measure in the laboratory) and is the wavelength of an absorption line that you observe in the galaxy. If the galaxy is moving away from us will be longer, or redshifted. If the galaxy is toward us will be shorter, or blueshifted The velocity of the galaxy can then be calculated from the redshift: As before, y is the velocity of the galaxy, and c is the speed of light. Figure 2 below is a theoretical spectrum of a galaxy that is "at rest". The galaxy is not moving toward or away from us. There are two absorption lines that are very close to one another. These lines originate from calcium. They are named the calcium K and H lines. The other line you can see is from hydrogen.The rest wavelength () for each line is given in Table 1. Spectrum For a Galaxy "At Rest" 0.9- 0.8- 0.7 Relative Intensity 0.3 0.2 0.1- 380 390 400 410 420 430 440 450 Wavelength Figure 2 Absorption line spectrum for a theoretical galaxy that is not moving toward or away. Table 1 Absorption Line (nm Calcium K 393.4 Calcium H 396.9 Hydrogen y 434.1 Using the information in Table 1, circle the absorption lines in Figure 2 and label them (Calcium K, Calcium H, and Hydrogen y) and then answer the questions below. Imagine a galaxy that is moving toward us. Where would you expect the calcium and hydrogen absorption lines to appear in the spectrum compared to spectrum for a galaxy at rest? Imagine a galaxy that is moving away from us. Where would you expect the calcium and hydrogen absorption lines to appear in the spectrum compared to spectrum for a galaxy at rest?Astronomers use the measured absorption lines in galaxies to determine the redshift and how fast the galaxy is moving away from us. This is the recessional velocity, or vr from Equation 3. PART B Let's first see if the universe is homogenous and isotropic. Start Stellarium and open the Sky and Viewing Options window. Turn off the atmosphere and the ground with the A and G keys. In the DSO ("Deep Sky Objects") tab, lets first choose to display only galaxies. To do this, in the Display Objects from Catalogs section, put check marks next to M. IC and NGC only. Then check the Filter By Type box and put check marks next to only the first four boxes - the galaxy ones. Finally, check the box next to Labels and Markers, and the drag the slider next to Hints all the way to the right. This will show a small red oval wherever there's a galaxy in the sky. Close the Sky and Viewing Options window. As you can see there a LOT of galaxies in the sky! Do the galaxies look evenly spread around the sky? You should see a "gap" in the distribution of galaxies. To see where this gap is in the sky, press the "C" and "V" buttons to display the constellations and their names and look back at the Milky Way assignment! For another hint, open the open the Sky and Viewing Options window. In the Markings tab put a check mark next to Galactic Equator and then close the window. What does the "gap" in galaxies correspond to in the sky? Is this gap due to the fact that there actually aren't any galaxies in that section of the sky, or that there's something blocking our view of galaxies in the section of the sky. Which do you think it is? If we ignore this gap, are the galaxies evenly distributed in the sky? If so, we say they are isotropic. Just as you did for the Milky Way assignment, pick 10 random spots in the sky away from the "gap" in the distribution of galaxies, and zoom in until the Field of View is about 5 degrees. Count the number of galaxies (red ovals) in the field and record the number in Table 2 below. Table 2 Random spot | Number of galaxies10 If the galaxy distribution is isotropic, these numbers should be all roughly the same. Are they? By "roughly", I mean do the counts differ by more than a factor of 5-10?) Are the galaxies distributed isotropically? To be homogenous, the galaxies would have to be evenly distributed in distance as well, but it's hard to get that information from Stellorium, so you'll just have to take our word for the fact that they are! PART C In this section we will use data from Stellarium to make a graph like Hubble's and see if we can reproduce the famous discovery that the universe is expanding! Let's measure the distances and velocities of a number of galaxies and plot them on a graph to see if the velocity of a galaxy away from us depends on its distance. Click on the Configuration icon on the left-hand side of the screen, and in the Extras tab, in the Additional Buttons section, check the DSS survey button. Then close this window and click on the DSS icon at the bottom of the screen. This will show real-life images of any deep sky object you click on. Pick twenty galaxies at random from the hundreds of little red ovals on the screen and click on them one by one. After you click on them, press the forward slash key to zoom in, and let the Deep Sky Survey image load to see what the galaxy actually looks like! For each galaxy, record its distance and redshift, , in the Table 3 below. I've done one galaxy for you. If no distance or redshift are listed for the galaxy, just pick another one. Then calculate the velocity of each galaxy by multiplying the redshift by the speed of light. Rearranging Equation (3) from above, we have: where z is the redshift, and c is the speed of light (). Almost all of these galaxies we observe are flying away from us (they have positive values of z), and, as Hubble discovered, the speed with which they'removing increases as they get farther way - in other words, the universe is expanding! Table 3 Galaxy Name Redshift (2) Distance (Mpc) Velocity ) km/s IC 4884 0.014922 170.6 14477 In Excel or on a separate sheet of graph paper, plot the velocity vs. distance for the galaxies in Table 3. (Velocity should be on the y-axis, and distance on the x-axis.) If you did everything correctly, you should have a straight line. Calculate the slope (either by hand or with Excel). The slope of the line is the Hubble constant Ha- What is your value for H.? How does your value compare with the currently accepted value (67.8 km-s/Mpc)? PART D Finally, we can calculate the age of the universe! The amount of time since the universe began expanding can be determined from taking the inverse of the Hubble constant. This is the age of the universe:ASTR101 LABORATORY B The only problem is our Hubble constant, H., is in units of (km/s per Mpc). We would like the units to be (1/years). That way, when we take the inverse of Ho. we'll have an age of the universe in years. Follow the steps on the next page very carefully! Please only keep three significant figures and show your units! First divide your H, by the number of kilometers in a Mpc: . This gets H, in units of 1/seconds. Write your value below. SHOW YOUR UNITS! Now multiply the number you just calculated (your H, in 1/s) by the number of seconds in a year: . This gets H, in units of 1/years. Write your value below. SHOW YOUR UNITS! Finally, take the inverse of the number you just calculated (your H, in 1/years) to determine the age of the universe in years. Write your answer below. SHOW YOUR UNITS! The currently accepted value for the age of the universe is between 13 and 14 billion years. (A billion is 10" years. Therefore, in scientific notation 13 billion years looks like ) How does your value compare with the currently accepted value? (If your value is much different explain why.)