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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 21031149 .1 radian : 2 x 105 arcseconds
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WMa : VpMp zzAPp/P.=AA/Ao= startmm (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 Ghezet 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
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calculate the Schwarzshﬂd radius of the black hole. (1 * i 1 deﬂate 6m. W'fbf’é’ 2006 #55")
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(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
postNewtonian 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 postNewtonian phenomena
have had time to make a noticable difference. 0 r  I PMCegs‘M”. 01% feﬁ'agiéroﬁ t’ﬂe ﬂair (cape/v
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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 pestNewtonian approximation. Note that Newtonian physics predicts no redshift due to gravity at all. b) Using the postNewtonian 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 (996) between 1996 and 2006.
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(is? ﬂag“ 0) Is the use of the post—Newtonian formula here justiﬁed? Why or why not?
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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
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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.
 Spring '08
 MarkSaltzman

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