EExploring Hubble Constant Data in an Introductory Course
Jeffrey M. Hyde ∗ Department of Physics & Astronomy, Bowdoin College, Brunswick, ME 04011 (Dated: February 7, 2020)
I. INTRODUCTION
Popular accounts of exciting discoveries often draw students to physics and astronomy, but at the introductorylevel it is challenging to connect with these in a meaningful way. The use of real astronomical data in the classroomcan help bridge this gap and build valuable quantitative and scientific reasoning skills [1, 2]. This paper presents astrategy for studying Hubble’s Law and the accelerating expansion of the universe using actual data. Along withunderstanding the physical concepts, an explicit goal is to develop skills for analyzing data in terms of a model.Hubble’s Law is the observation that distant galaxies are receding from us at a rate v proportional to their distance D [3]: v = HD. (1)Knowledge of the Hubble “constant” (or Hubble parameter) H can help answer interesting questions, such as the ageand matter content of the universe. Students may have read about recent debates over its present value (denoted H )[4]. Many excellent resources are devoted to teaching the concept [5–10].In order to incorporate analysis of real data in a meaningful way I use activities that develop two importantskills. First, students use a tangible model of an expanding universe to find a value of H . This builds a workingunderstanding of how the model relates to features that may be observed in data. Second, guided questions for three“case studies” show students how to constructively deal with confusing points in the analysis of data. Before startingthis activity, I have students complete a Lecture-Tutorial on the expansion of the universe [7]. This introduces studentsto basic ideas of Hubble plots and addresses some of the other conceptual difficulties related to the expansion of theuniverse. II. FINDING H IN A MODEL EXPANDING UNIVERSE
To model cosmological expansion, two sheets of paper present snapshots of some part of the universe at differenttimes. Randomly placed dots represent galaxies, and the later snapshot is made by uniformly stretching the entirecoordinate grid by some factor. Students compute the distances from a certain galaxy to several others, and do thesame with the later snapshot.The recession velocity is then computed from the change in distance over the time interval represented by the twosheets. After plotting recession velocity versus distance as seen from the galaxy, the slope will be H . I use the scale1 cm = 1 Mpc (10 parsec), and stretch the distance grid by a multiplicative factor of 1.4 between snapshots 3 Gyrapart, where 1 Gyr = 10 years. This leads to a value of H ≈
130 km/s/Mpc ≈ . − . It’s useful to note thatstretching distances by a factor of m over T Gyr leads to H ≈ m − T × km/s/Mpc, using 1 Mpc/Gyr = 980 km/s ≈ H as observed fromanother galaxy in the example universe, which should give the same value. Follow-up questions build on this tangiblemodel and prepare students for later interpretation of data. Useful questions directly relevant to the next section areWhat would change if H were larger or smaller? and would a larger H correspond to an older or younger universetoday? Observationally-oriented what-ifs are also important. For instance, “are there any factors that the model doesnot take into account? How would these affect the plot?”A few possible points of confusion are worth mentioning. First, connecting recession velocity to the change indistance using v = ∆ x/ ∆ t can be conceptually difficult. Second, since each velocity comes from comparison of twodistances, what is “the” distance? This ambiguity can be minimized by shortening the time interval in betweensnapshots and correspondingly reducing the multiplicative factor m . However, too small of a time interval can make ∗ [email protected] a r X i v : . [ phy s i c s . e d - ph ] F e b (a) (b) FIG. 1: Hubble’s original data [3] (Fig. 1a) and galaxies whose distances were also determined using Cepheidvariables as part of the Hubble Key Project [12] (Fig. 1b).it difficult to reliably measure the smallest distances and therefore obscure the linear relationship we are seeking todemonstrate. Factors such as the increments on rulers used by students and the separation of galaxies on the pageinfluence the best choice of parameters, so it is worthwhile to experiment with some different values.
III. ANALYZING DATA - THREE EXAMPLES
After the hands-on experience described above, students are ready to examine three case studies to sharpen theirthinking about models and their relation to data. It will be important to know that distances are determined bymeasuring the apparent brightness of “standard candles” - stars or supernovae with a known absolute magnitude.Depending on factors such as time constraints, students could plot the data themselves using tables in the citedpapers, or simply be given the plots.
A. Hubble’s Original Data
Hubble’s original data set [3] gives an expansion rate of around 500 km/s/Mpc (Fig. 1a). The data follow a lineartrend, but with quite a bit of scatter - an opportunity for students to brainstorm possible explanations. In this case,all galaxies have their own “peculiar motion” in addition to the Hubble expansion (stretching of the grid), so nearbygalaxies may be moving towards us faster than expansion is taking them away. This is a good place to refer back tothe earlier question of what factors the model did not take into account.Hubble’s value of H implied the age of the universe was around 2 billion years, in contrast with early radiometricdating of rocks on Earth to 3-4 billion years [11]. How could the Earth be older than the universe? This questionprovides good motivation to think broadly about sources of error. We now know that Hubble’s data had a systematicerror where distances were underestimated (largely because two different types of Cepheid variable stars were assumedto be identical standard candles), although I defer this discussion with the class until after the following example. B. Hubble Key Project to measure H The Hubble Key Project compiled a variety of Hubble Space Telescope observations with the goal of decreasinguncertainties in H [12]. Fig. 1a shows the result from their observation of Cepheid variables. This leads to animportant question: is the data set shown in Fig. 1a consistent with the data set shown in Fig. 1b?Students may find it difficult to agree on an answer to “what does it mean for two data sets to be consistent?” Theyoften suggest that if the best-fit lines have any difference at all (no matter how small) then the data sets are totallyinconsistent. This competes with another popular suggestion, that the spread of data points and uncertainty in sloperenders the question impossible to answer. These responses are similar to misconceptions identified in introductoryphysics students’ reasoning about data [13]. In particular, the suggestion that any difference in measurements must (a) (b) FIG. 2: Fig. 2a shows both data sets of Fig. 1 plotted on the same axes. The galaxy NGC 7331 is present in bothsurveys, so it is marked with a star for comparison. Hubble systematically underestimated distances and thereforeoverestimated the slope, or Hubble parameter, H . Fig. 2b shows the Hubble Key Project’s measurement of H using Cepheids (same data plotted in Fig. 2a) and Type Ia supernovae [12].mean they are inconsistent may be thought of as “point paradigm” reasoning that supposes single measurements canrepresent true values with no inherent uncertainty.If students are fitting the data with software, they could consider the quoted uncertainty in slope. A qualitativeapproach is to think about the range of slopes that could plausibly describe each data set. After plotting both on thesame axes (see Fig. 2a), it becomes clear that Hubble Key found a much lower value of H . Students should be ableto conclude that this lower H predicts an older universe than Hubble’s original work, although this is often one ofthe questions that students find more difficult.This is a good time to return to hypotheses about sources of error in Fig. 1a, and bring up some of the historicalcontext (e.g. measuring two populations of Cepheids) mentioned earlier. The galaxy NGC 7331 was part of bothsurveys, and it is interesting to compare its position in each (marked with a star in Fig. 2a). The distance changessignificantly, but the speed has also been corrected for local motions of our solar system, galaxy, and local group.Since different methods may have different systematic errors, it is also worthwhile to discuss results that come froma different standard candle. Fig. 2b shows Type Ia supernovae giving about the same H as Cepheids. C. Accelerated Expansion
Studying accelerated expansion connects students directly with a discovery recognized by the 2011 Nobel Prize inphysics [14, 15], the relevant plots being Fig. 4 or 5 of [14], or Fig. 1 of [15]. The benefits of using a figure from thepublished paper are: (1) the authenticity of the “real thing” is exciting and (2) it has curves showing predictions ofvarious models. Computing such curves is beyond the scope of an introductory class (see for example Chapter 6 of[16], along with the text of the papers cited here), but by this point students will be ready to understand the resultat a qualitative level.Fig. 3 shows a plot of the data from the Cal´an/Tololo Supernova Survey [17] and the Supernova Cosmology Project[15], along with two curves that show predictions based on the contents of the universe. The two predictions shownare for the case where our universe is comprised only of non-relativistic matter, and for the case where our universeis 30% non-relativistic matter and 70% cosmological constant (or dark energy). In Fig. 1 of [15], the curve labeled(Ω M , Ω Λ ) = (1 ,
0) shows a “traditional” case with no cosmological constant/dark energy, and comparison with thedata points and error bars will convince students that the data points tend to fall above that line. These figures plotapparent magnitude versus redshift, so recession velocity is now on the horizontal axis. Guidance from the instructorcan help students determine that the apparent magnitude of a standard candle is a measurement of distance.For interpretation at a qualitative level, it helps to ask questions that point out specific pieces of reasoning. Forinstance, the earlier question about “which galaxies emitted their light further in the past?” can be followed up with“what would the Hubble plot look like if the universe was expanding faster in the past? Slower in the past?” Acommon mistake here is the assumption that time since the Big Bang increases from left to right on any graph.FIG. 3: Standard candles at even greater distances reveal the accelerating expansion. This plot shows data from[15], as well as two curves showing predictions based on the contents of the universe. The solid curve is a predictionif the energy density of the universe is all non-relativistic matter. The dashed curve is a prediction based on thenow-accepted values for the energy density of the universe, where 70% is cosmological constant or dark energy, andonly 30% is matter.
IV. FURTHER EXPLORATION
There are many possibilities for further exploration. For example, students can be asked to find data in the NASAExtragalactic Database (ned.ipac.caltech.edu) and make their own plots. Interesting discussions arise when they haveto choose which of several given measurements is “the” value they will use, or compare results after choosing sampleswith very different distance ranges or numbers of galaxies. Another possibility is to use data from the SternbergSupernova Catalog, where interstellar extinction is seen to dim Type Ia supernovae [18]. Since this dimming increaseswith distance, students will have to determine that the closest supernovae provide the most reliable estimate of H .With the guidance described in this paper, I’ve found this to be an accessible activity that challenges introductorystudents to think hard about Hubble’s Law and how analysis of data is used to evaluate models. ACKNOWLEDGMENTS
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