midterm_2_solutions

midterm_2_solutions - NAME: ‘ Midterm Exam #2: Astronomy...

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Unformatted text preview: NAME: ‘ Midterm Exam #2: Astronomy 160b, spring 2007, March 27, 2007 The test relates to some data about the supermassive black hole in the center of the galaxy given below. Note that you do NOT need to know anything particular about supermassive black holes — all the questions can be answered using our standard equations of orbital motion, and information about relativity and so forth that we have have considered in other contexts. The test consists of four problems (worth a total of 30 points) and will last 50 minutes a plan your time accordingly! Please put your name on both pages of the test, and note that we’re using both sides of the test paper. Please do all problems on the test paper — if you need more space, continue on a separate piece of paper labelled with your name and the problem number. Use a different piece of paper for each problem you need ' to continue. The test is open book, but electronic devices such as calculators are not allowed. 1 AU. = 1.5 x 1011111 1M3 =_ 3 x 10””5MG CL] RJ’ 5 G r— 7 X 10“11 in mks units 1 year '2 3 x 107 seconds 1M0 = 2 x 1030 kg A PJ m 11 years c 2: 3 X 108 m/s 1 parsed: 3 X 10mm a3 : Wear/(479) ' V = Zoo/P iMJ 210-31149 .1 radian : 2 x 105 arcseconds O! I D2/D1 WMa : VpMp zzAPp/P.=AA/Ao= start-mm (Hermm Vhsc : R5 3 20.01/62 vtot :(Iul + fog/(1 + Ulla/C2) Zgrav 2 mm w 1 Prof. Andrea Ghez and her collaborators at UCLA have been observing stars near the center of our Galaxy for a number of years. During that time they have seen the stars closest to the galactic center orbiting around an unseen object located at the exact center of the galaxy. From the orbits’of these stars, Ghez and‘her colleagues and competitors have inferred the presence of a massive black hole at the center of the galaxy. Below I have sketched an example of the kind of data they have" obtained (I modified the details a bit for the purposes of this test — for the real thing see Ghez-et al. 2000, Nature vol. 407, p. 349). The points are observations of the position of a particular star over the years (the year of theobservation is noted), the cross marks the center of the galaxy, and the line is the inferred orbit of the star. Note that the orbital period of the star around the galactic center is 20 years. Note also that there is a size scale indicated in astronomical units. It??? I 2000 2.002, 0 - 5 00 1:290 {.090 2m 15770 }00L’ ‘ - 1 ha {4% a on 5' ' Problem 1 (10 points). From the information provided. on the other side of the paper, calculate the Schwarzshfld radius of the black hole. (1 *- i 1 deflate 6m. W'fbf’é’ 2006 #55") "2/ u: 1009 AM g c? 6 4175 643: PZM M: 03) : ,E a 2.9/0 Mr? @ [ lung” 91/”? 6 ‘30 "[3 r; 23.3 S 0'} /_ ,, “so 1’4"“ / / x O /0 : 2*771’”) 7‘sz : E917“ (5,4093 ‘W/O ‘ * $4) L‘M/ Problem 31(8 points). If the star’s motion were perfectly Newtonian, it would follow exactly the same path each orbit, and the shape of the orbit wouldn’t change. However, there are two post—Newtonian effects that will cause a change in the shape of the orbit. List the two post-Newtonian effects, and explain the ways in which they change the shape of the orbit. Then on the diagram below (which shows the current orbit) sketch the orbit as it might appear when re—observed many years from now once these post-Newtonian phenomena have had time to make a noticable difference. 0 r - I PMCegs‘M”. 01% fefi'agiérofi t’fle flair (cape/v \{Feb'7a/fifiCrO/2)‘ 774e, Perm/a b & were We were are a...) s... at, page arid? {the we .91. - I Problem 3 (10 points). Suppose the star is actually a pulsar, and the pulse period was observed to be precisely 3 seconds in 1996. Assume the orbit is'face—on, so that there is no Doppler shift (the motion is neither toward nor away from us at any point in the orbit). Nevertheless, the pulse period will change during the orbit due to changes in the gravitational redshift. - a) Starting from the full relativistic formula for the gravitational redshift given on the first page of the test, derive a pest-Newtonian approximation. Note that Newtonian physics predicts no redshift due to gravity at all. b) Using the post-Newtonian formula derived above and other information provided through— out the test, calculate how much the observed pulse period has increased or decreased 3‘5. ANN ii (9-96) between 1996 and 2006. " i? W“ . (is? flag“ 0) Is the use of the post—Newtonian formula here justified? Why or why not? 3‘ .. ____ F a: ~; (were we ~! 1' RS/r‘ a) . 5) gnhwl fl—v False» L (hp?— W'” 96 L653. ‘Mc'u/p rec/{tr-BL go /Ze c9 (Le/We / (S (L1on © bra/1’)" gar :‘x‘finl 91‘ a 6W0“ (WI/karyzj); \Qi (c9 . “,1 7 0:” "5 ’09.;th 16mm ffl/ ("ea/kg 0d/ Me U!” Adz $P<ié “ APot, @r ’72!” 93 o M .. I ' ix: . gap/m ; anofimwoesnioa / iV." (its wire, 13’ . 3 u h 3 H97 gill?” “’1 1606 0% (we; I? )9'60'4u :* (.7140 sari/a _. 1. (3n 1‘ P ? (i9;th ) g > sum 4,, film. Pea/7L 0. i y than ‘1 r “’0 _ fl” __ Mt Tgfilo’vfi" > r; g M t (9 U is ’ $216; ’ v5 21" 06 {we (care/fair? ‘ ~ ex 36 4 ME?" g ’0 QB : Va £46.; I So C, Hw- W9 -3 Li '27 . ‘ SH 7 _.g- I »5 . : x a" :, 2M0 5 'M . fit“ . _ M ‘WV 1"“;{7 m —-———_......__...Wfi C U594 \‘S ari/(r’v/ é€[4l/oc€ . W (2; : g 1 << / / .m— rig i 25mg, S’wm Problem 4 (2 points). Which of the following observations helped prompt Einstein to develop the general theory of relativity? (Circle all that apply) ‘ Sat/m4. Meg? g / r The constancy of the speed of light for all observers. he equality of the inertial and gravitational mass. Um) c The deflection of light observed from background stars during a solar eclipse. (Am/M d) The discovery of the binary pulsar. "Z, (4’ [CF (OW/7 l (a/L lg?“ L Cum? Sow—re. OM 0ft (5/ h 0m] (1 sllxzf alarm/l} (9F 070 (ii i. 9 we. 74 ...
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This note was uploaded on 02/06/2012 for the course BENG 100 taught by Professor Marksaltzman during the Spring '08 term at Yale.

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midterm_2_solutions - NAME: ‘ Midterm Exam #2: Astronomy...

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