Astronomy 1143: Assignment 2, Solutions

Astronomy 1143: Assignment 2, Solutions Part I: Some Short Questions (a) Shorter wavelength photons have higher energy. From highest energy to lowest, the order is: X-ray, ultraviolet, blue visible, orange visible, infrared, radio. (b) The Doppler effect depends on the velocity of the moving object relative to the velocity of the waves. An ambulance is moving much, much slower than the speed of light, so even though its light is Doppler shifted, the fractional change in the color is tiny because v/c is so small. However, the speed of sound is about 300 m/s, and an ambulance in a hurry might be going at 30 m/s, which is a large enough fraction of the speed of sound to have a noticeable effect. (c) It is much easier to measure a galaxy s velocity (from Doppler shift) than to directly measure its distance. Hubble s law helps us to make 3-dimensional maps because we can measure velocity and infer the distance from d = v/h 0. There are inaccuracies in the map because of peculiar velocities, and to know the overall scale of the map we need to measure H 0. (d) I ve known this result for so long that it is hard for me to say whether it would ever have surprised me. It is intuitively difficult to understand how the universe can have a finite age, because it leads to the obvious question of what was there before the universe. But according to Einstein s theory of relativity (which is abundantly confirmed by many precise experiments), space and time are intimately linked, and in at least the simplest versions of the big bang theory the big bang is the origin of space and time. So asking what was there before the big bang? may be a lot like asking what is north of the north pole?. The second of these questions seems to make sense if you are used to thinking of north as the direction that compass needles point, but really it is predicated on an incomplete picture of what north means. Similarly, the first question may be based on an intuitive notion of time based on everyday experience that just doesn t apply at extreme events like the origin of the universe. Once I accept that the universe does have a finite age, I don t feel like I have any intuitive idea for what that age ought to be (outside of my scientific knowledge based on geology and astronomy). Fourteen billion years is much much much longer than a human lifetime, but it seems to me as reasonable a number as any other. There are many other ways (some of which we ll discuss later) to determine the age of the earth, and meteorites, and the sun, which give values of 4.5 billion years, so this implies that the sun was born when the universe was roughly 2/3 of its present age. (e) As we will discuss later in the course, the simplest extrapolation of what we currently know about the universe implies that it will continue to expand forever, at an accelerating rate. The stars within it will gradually die out; the sun will use up its supply of fuel in about 5 billion years, though some stars will last much longer, and some new stars will be formed. However, we expect the universe to gradually become colder and fainter, with matter more and more thinly distributed. There are some ways out of this conclusion because it depends on ideas about the energy content of the universe that are not well tested at present. Indeed, we may never be able to predict the fate of the universe with 100% confidence, though we can develop more confidence in our answers as our observations of the present-day universe improve. For example, when I was a graduate student in the late 1980s, it still seemed quite possible that gravity would cause the universe to recollapse some time in the next 50 billion years. Because of improved observations, that possibility now seems very unlikely. 1

Fossil studies indicate that humans anatomically similar to present-day humans have existed for something like 200,000 years, while more ape-like human species have existed for something like 2 million years. Some species have survived on earth with little change for much longer periods, but in the absence of other information a reasonable guess from our history so far would be that the human species will last in something like its current form for another 100,000 to million years. After that, there might still be species around that are recognizably descendants of humans, but they could be different from us in interesting ways. However, the pace of cultural evolution over the last 10,000 years has been dramatic, and even more so over the last 1,000 years. I can therefore imagine things changing much more rapidly, in either a negative or a positive direction. As a species we now have a huge impact on the environment, most notably by causing global warming, so there is a real possibility that we will drive ourselves extinct (along with the many other species that we have already driven extinct) in mere hundreds of years. We could also wipe out our species via nuclear or biological warfare, though at least the first of these seems less of a threat than it did 30 years ago. On the (maybe) positive side, perhaps we will discover ways to accelerate our own evolution via genetic engineering and change ourselves anatomically in ways we consider desirable. More radically, maybe we will find ways to transfer our consciousness into robots, which would be a very different form of existence. Finally, if humans eventually develop interstellar space travel and spread to planets around other stars, the resulting isolation of communities could lead to evolution into many different species that are adapted to different planetary environments, or perhaps to long-duration interplanetary travel itself. Part II: Measuring Galaxy Redshifts (a) I measure 6608 Angstroms for the peak of this hydrogen emission line.. This is the observed wavelength λ o, while the emitted wavelength is λ e = 6562.8 Angstroms. Using the formula in the assignment, which is derived from our usual Doppler shift formula λ o = λ e (1 + v/c), the velocity is v = c (6608/6562.8 1) = 2066kms 1. This agrees well with the velocity found from the two calcium lines, which is reassuring evidence that I did in fact correctly identify which elements were producing these lines. (b) For NGC 2775, measuring from the bottom of the absorption dips, I get wavelengths of 3955 Angstroms and 3990 Angstroms for the two calcium absorption lines. The implied velocities are v = c (3955/3933.7 1) = 1624kms 1 and v = c (3990/3968.5 1) = 1625kms 1. The good agreement between the two numbers is an indication that I didn t make a mistake in my measurement and that these two dips really are the calcium absorption lines rather than some other features that I misidentified because they were in approximately the right place. For NGC 3368, I measure 3945 Angstroms and 3980 Angstroms for the two lines. The implied velocities are v = c (3945/3933.7 1) = 862kms 1 and v = c (3980/3968.5 1) = 869kms 1. Again the agreement between the two lines is quite good, though I don t think my measurements with a ruler are more accurate than about 2.5 Angstroms, one quarter of the distance between tick marks on the x-axis, so I am somewhat lucky to be getting numbers that match as well as they do. (c) For galaxy velocities, I ll take a value somewhere in the middle of what I obtained from the different lines. For NGC 1382, d = 2050kms 1 /70kms 1 Mpc 1 = 29.3Mpc. For NGC 2775, d = 1625kms 1 /70kms 1 Mpc 1 = 23.2Mpc. For NGC 2775, d = 865kms 1 /70kms 1 Mpc 1 = 12.4Mpc. 2

Part III: Measuring H 0 (a) Start from the equations f = L/4πd 2 and f = L/4πd 2. Multiply both sides by 4πd 2 or 4πd 2 to get L = f 4πd 2 and L = f 4πd 2. Divide the left side of the first equation by the left side of the second and the right side of the first equation by the right side of the second to get L = f 4πd2 L f 4πd 2 = f ( ) 2 d. f d (b) With the formula above, we can just plug in numbers, since the distance in AU is just the same thing as d/d (because d = 1AU). Supernova 1: L/L = 4.4 10 16 (3.1 10 12 ) 2 = 4.3 10 9. Supernova 2: L/L = 1.9 10 16 (4.5 10 12 ) 2 = 3.8 10 9. Supernova 3: L/L = 1.2 10 16 (5.5 10 12 ) 2 = 3.6 10 9. Since these are ratios, there are no units to the answer e.g., for Supernova 2, L/L = 3.8 10 9, or equivalently, L = 3.8 10 9 L. I commented in class that a supernova at its peak can be as luminous as an entire galaxy of several billion stars, and you can see that these numbers are consistent with that statement. The three numbers don t agree perfectly, which may be partly because the distance or flux measurements are imperfect (the distance measurement being the more difficult one), but also because supernovae are not really identical in peak luminosity, just similar. For their analysis, Riess et al. used every galaxy that had a well observed Type Ia supernova and was close enough to a distance with Cepheids, so that they could get the most accurate value possible for the average luminosity of Type Ia supernovae. Even so, that was only eight supernovae. We have done our calculation for three of those eight supernovae, and we could take either the average or the median of our three values as representative. For the next part, I ll take the median value of L = 3.8 10 9 L. (c) Using this value for the supernova luminosity, we now have a calibrated standard candle that we can use to get the distances to galaxies that are too far away for Cepheid measurements. (Cepheids are thousands or tens of thousands of times the luminosity of the sun, not billions of times.) Supernova 4: d/1mpc = 4.85 10 12 3.8 10 9 1/(8.7 10 18 ) = 101. Supernova 5: d/1mpc = 4.85 10 12 3.8 10 9 1/(2.0 10 17 ) = 67. Supernova 6: d/1mpc = 4.85 10 12 3.8 10 9 1/(1.2 10 18 ) = 273. Note the expected result: since we are assuming that the supernovae are standard candles, with the same peak luminosity, a fainter apparent flux corresponds to a larger distance. (d) From Supernova 4 we find H 0 = 7940kms 1 /101Mpc = 79kms 1 Mpc 1. From Supernova 5, H 0 = 5010kms 1 /67Mpc = 75kms 1 Mpc 1. From Supernova 6, H 0 = 17,800kms 1 /273Mpc = 65kms 1 Mpc 1. 3

There is roughly 20% difference between the high and low values. Probably this arises mainly from the range of actual supernova luminosities. Averaging the three values gives H 0 = 73kms 1 Mpc 1, close to the round value of 70kms 1 Mpc 1 I have been using in class. Riess et al. used a much larger number of distant supernovae so that they could average out the supernova-to-supernova variations. However, they only have eight supernovae with Cepheid distances, so their statistical uncertainties are limited by the scatter among those eight. The other main sources of error are the uncertainties in the parallax measurements used to calibrate the Cepheid luminosities and the difficulty of measuring fluxes exactly. Their final result is H 0 = 73.8kms 1 Mpc 1 with an uncertainty of ±2.4kms 1 Mpc 1, making the fractional uncertainty about 3%. Their group and other groups are working on still more accurate measurements to try to bring the uncertainty down to 1% or lower. (e) Because those three galaxies are fairly nearby, peculiar velocities make a significant contribution to their total velocities, so just using H 0 = v/d has a significant error because the total velocity is v = H 0 d + v pec. Specifically, these three galaxies should have expansion velocities H 0 d of 1000 3000kms 1, so a typical peculiar velocity of 300kms 1 would cause a 10-30% error in the inferred value of H 0. The more distant galaxies used in part (d) have velocities of 5000 18,000kms 1, so a typical peculiar velocity makes a much smaller fractional change to the total velocity. For a more complete discussion of the chain used here for measuring H 0 parallaxes of Cepheids to calibrate the period-luminosity relationship, Cepheids to calibrate supernova peak luminosities in relatively nearby galaxies, and supernovae to measure distance to galaxies far enough away that peculiar velocities cause negligible error review the Measuring H 0 section of the course notes. Part IV: Hubble s Law and the Age of the Universe (a) For d = 10Mpc, v = H 0 d = 70kms 1 Mpc 1 10Mpc = 700kms 1. For d = 50Mpc, v = H 0 d = 70kms 1 Mpc 1 50Mpc = 3500kms 1. (b) The time required to go 10Mpc at v = 700kms 1 is t = d/v = 10Mpc (3.09 10 19 km/mpc)/700kms 1 = 4.4 10 17 s. Dividing by 3.16 10 7 s/year gives t = 1.39 10 10 years, or 13.9 billion years. For the second galaxy the numerator and denominator are both larger by a factor of five, so the ratio d/v is the same. (c) If galaxies were moving faster in the past than they are today, then it took them less time to reach their current distance, so the implied age of the universe would be younger. Provided galaxies have been moving at roughly their current velocities for a significant fraction of the age of the universe, the answer for the age shouldn t be radically different from the value d/v = 1/H 0 that you got in part (b), but it wouldn t be surprising for it to be a factor of two bigger or smaller depending on whether galaxies were moving slower or faster in the past. To figure out this effect we will need to think about how gravity affects the expansion of the universe. The answer will turn out to be surprisingly complicated. 4

Extra Credit Let s consider just galaxies D, E, and F. According to Jubble s law, they will have velocities of 70kms 1, 280kms 1, and 630kms 1 away from the Milky Way. Now imagine observing from Galaxy D. The Milky Way will be moving away at 70kms 1 (more precisely, at 70kms 1, since is moving to the left relative to Galaxy D). Galaxy E is moving away at (280kms 1 70kms 1 ) = 210kms 1. Thus, as seen from Galaxy D, the galaxies that are 1 Mpc away on either side are receding with different velocities, only one of which is given by J 0 d 2. Galaxy F, at a distance of 2 Mpc, is moving away at (630kms 1 70kms 1 ) = 560kms 1, also not equal to J 0 d 2. You can look at many other cases, e.g., the velocities of Galaxies A and B as seen from Galaxy E. What you ll find is that in this alternative universe, all observers see that other galaxies are moving away from them, but only observers in the Milky Way see an expansion that is described by the simple mathematical relation v = J 0 d 2. Observers in other galaxies don t even see the same expansion rate towards different directions in the sky. The linear velocity-distance relation discovered by Hubble is special; it is the only relation that will appear the same to observers everywhere in the expanding universe. It is the only expansion law consistent with the idea that the center is everywhere. The result that you found in Part IIId of Assignment 1 is therefore a profound statement about the nature of the universe. 5