Angles Lab. name: Part I - How tall is Founders Hall? Astronomical Ideas, Fall 2012

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1 Angles Lab Haverford College Astronomical Ideas, Fall 2012 name: Collaborators: With this lab, you will get quantitative experience with the basic relationship between angular size, distance, and physical size that is fundamental to astronomical measurements. You will also think carefully about both the concepts of and calculation of measurement uncertainty, and be required to develop your skills of informed estimation of quantitative parameters. You are encouraged to collaborate with other students on this lab, but be sure that your measurements, calculations, and answers are all your own. Please list all of your collaborators above, and ask me if you have any questions about appropriate collaboration practices. Part I - How tall is Founders Hall? You will use your body to estimate the physical height of Founders Hall from the base of the stairs to the top of the cupola. You will need to go out to Founders Green to make your measurements. A) Draw and label a schematic that includes you, Founders Hall, the distance between you and Founders (d), the physical height of Founders (s), and the angular size of Founders (θ ). Use the small angle approximation to write down an expression that relates these three quantities.

2 B) Stand out on Founder s Green where you can walk in a straight path to the porch stairs. Use your body (and the cloth tape measures, if desired) to estimate the angular size of and distance to Founders Hall. Repeat five times and record your data in the table on the next page. In the space below: i.) Describe how you measured θ and d; ii.) describe one source of random measurement uncertainty; iii) use the measurements themselves to make an approximate calculation of the size of the random measurement uncertainty associated with each of θ and d.

3 C) Calculate the height of Founders Hall. For each of your five calculations, propogate the uncertainty in θ and d to get the uncertainty on height, and record in the table below. Then average the five height measurements, and calculate the uncertainty on the measurement. You don t need to show your work, but write down the equations that you used to calculate the error on each individual height calculation and the error on the average of all five height calculations. Be clear about the definition of each variable. Averaged Height of Founder s Hall Cupola: Include units, use a sensible number of significant digits, and include random error. distance +/- error angular size +/- error height +/- error

4 D) The true height of the cupola is approximately 57 feet. Comment separately on the precision and accuracy of your measurement. Include a brief discussion of systematic uncertainty in your answer.

5 Part II - Angular Sizes of stars In Chapter 7 of Cosmos (pgs 134 and 157), Carl Sagan comments that the distances to stars can be inferred using only knowledge of the physical size of the Sun and imagining that you took the Sun and moved it so far away that it was just a tiny twinkling point of light. In Part II of this lab, we will play with the quantitative side of this statement. A) Use only the physical size of the Sun (look it up in a book or on the internet) and your own estimate of the apparent angular sizes of stars in naked eye observations to estimate the distances to stars that we see in the night sky. Express your answer in units of AU and light years. You do not need to incorporate measurement uncertainty into this estimate.

6 B) What assumptions did your calculation in A depend on? Critically evaluate whether these were good assumptions using the knowledge that you currently have from class and the readings. (Continue answer on next page.) C) Assuming that the assumptions inherent to your calculation in A) holds true, does the estimate give a sense of the typical distances to stars, an upper limit on the distances to stars, or a lower limit on the distances to stars? Briefly explain.

7 D)The closest star (actually a binary star) is Proxima Centauri, an M dwarf star at a distance of 4 light years. The radius of this star is about 0.14 RSun. What angular image resolution on a telescope would be needed to resolve this star? Resolve means to see the disk of the star, rather than it appearing only as a tiny twinkling point of light. E) Same question as D), but for Betelgeuse. Betelgeuse is a nearby, very bright red giant star in the Orion constellation. It is at a distance of 625 light years, and has a radius of ~1000 RSun.