Name Midterm Exam March 4, 2010

Transcription

1 Name Midtem Exam Mach 4, 00 This test consists of thee pats. Fo the fist pat, you may wite you answes diectly on the exam, if you wish. Fo the othe pats, use sepaate sheets of pape. Useful equations can be found on the last page of this test; feel fee to tea it off. Pat I: Multiple Choice [40 points] Fo each question, choose the best answe ( points each). Obsevations indicate that the Sun is cuently moving vetically at about 7 km/s nothwads, away fom the galactic disk. What is the long tem effect of this motion? A) It will move to a positive z-value, and then obit indefinitely at that z-value B) It will oscillate up and down, but neve staying vey fa fom the disk C) It will follow a lage obit which takes it well out into the halo D) It will settle back down into the plane of the galaxy ove time; this is a tempoay phenomenon E) It will leave the galaxy in this diection. Which of the following is not a typical chaacteistic of a thick disk sta, compaed to a thin disk sta? A) It will tend to have a lage adius, and hence is thicke B) It will tend to be olde C) It will have moe andom motions; i.e. less pefectly cicula D) It will tend to have lowe metallicity E) Actually, all of these ae chaacteistics of thick disk stas 3. Most of the mass of ou galaxy is pobably in the fom of A) Black holes in the nucleus B) Stas, gas, and dust in the disk C) Globula clustes out in the halo D) Old stas in the bulge E) Unknown dak matte spead thoughout the halo 4. Although the bulge of ou galaxy is obscued by dust, we ae still confident it is thee because A) We can detect its gavitational contibution to the obits of stas B) Neutinos fom it ae obseved at Eath C) We can see stas fom it that have been expelled to ou neighbohood D) We can see it with infaed light, which penetates the gas and dust E) We simply infe its pesence since othe spial galaxies like ous have bulges

2 5. If the suface tempeatue of a sta is doubled, but its adius emains the same, its luminosity would incease by a facto of A) B) 4 C) 8 D) 6 E) Spectoscopic paallax is a technique fo measuing distance based on the idea that A) Fo main sequence stas, the spectal class moe o less tells you the luminosity (absolute magnitude), and combined with the appaent magnitude you can get the distance B) The spectum of a sta tells you its velocity; combined with othe infomation, this gives you the distance C) Thee is a elationship between the spectum of a vaiable sta and its peiod D) The spectum of a sta tells you its appoximate age; this can be combined with how long it took light to get hee to get the actual distance E) The appaent angle of the spectum as viewed fom diffeent points in Eath s obit tells you the distance to the sta 7. Suppose one had a catalog of the appaent magnitude of all the bightest planetay nebulas in a neaby galaxy. What additional infomation would we need to find the distance to this galaxy? Assume thee is no dust obscuing the galaxy. A) The peiod of at least one Cepheid vaiable sta B) The paallax to at least one of the planetay nebulas in that galaxy C) The physical size of at least one of the planetay nebulas D) The spectal class of one o moe planetay nebulas E) None of the above; you aleady have sufficient infomation to find the distance 8. List the types of electomagnetic adiation below in ode fom longest wavelength to shotest A) X-ays, infaed, visible, ultaviolet, adio B) Radio, ultaviolet, visible, infaed, X-ays C) X-ays, ultaviolet, visible, infaed, adio D) Radio, infaed, visible, ultaviolet, X-ays E) None of the above ae coect 9. Which of the following is most likely to be found vey close to the galactic plane? A) Thin disk stas B) Halo stas C) Cool molecula clouds and sta foming egions D) Hot thin hydogen clouds E) Globula clustes 0. The oldest stas can be found in the A) Thin disk B) Thick disk C) Bulge D) Globula Clustes E) Nucleus

3 . In the moving cluste method, how does one detemine the actual diection that a cluste of stas is moving? A) One calculates the tansvese and adial velocities, then you poject the esulting vecto B) The angula motion of all the stas in the cluste ae extended to the vanishing point, o convegent point, whee they all come togethe; this is the diection of motion C) Diffeential Dopple shifts on diffeent stas thoughout the cluste ae used to detemine the diection of oveall motion D) The ages of the stas is estimated, and based on the location of the molecula cloud fom which it was bon, one can then detemine the diection of motion E) The speed of the stas in the cluste is compaed with the mass and size, and the Viial theoem then tells you the diection of motion. The distance to the cente of ou galaxy is pobably closest to A) 80 pc B) 800 pc C) 8 Mpc D) 80 Mpc E) 8 kpc 3. The abundance of which of the following elements would be included when computing the metallicity of a sta? A) Ion, but not neon B) Ion and neon, but not cabon C) Ion, neon, and cabon, but not helium no hydogen D) Ion, neon, cabon, and helium, but not hydogen E) Ion, neon, cabon, and hydogen, but not helium 4. The hottest spectal type of those listed is A) A B) B C) K D) M E) F 5. The O and B main sequence stas ae the fist to die because A) They have the least mass, and theefoe un out of fuel the fastest B) They bun thei fuel the fastest, and theefoe un out of fuel the fastest C) They ae the coolest, and theefoe they soon lose tempeatues sufficient to sustain fusion D) They have the most mass, and attact moe mass making them unstable E) The question is false; these stas live the longest, not the shotest 6. Which of the following is pobably closest to the speed of the Sun aound the cente of the galaxy? A) 00 km/s B) 600 km/s C),000 km/s D) 6,000 km/s E) 0,000 km/s 7. The mass of the black hole at the cente of ou galaxy was estimated by A) Obseving the Schwazschild adius B) Sending a pobe and measuing the gavitational pull C) Studying the obit of some stas that wee vey close to it D) Measuing the Dopple shift of gas falling into it E) Studying the bending of light fom objects behind it

4 8. In galaxies like ous, how does the appoximate otational velocity depend on distance as you move away fom the cente of the galaxy? A) At lage adii, it falls off as about R B) At lage adii, it falls off as about R C) At lage adii, it ises as about R D) At lage adii, it ises as about R E) It is oughly constant at lage adii 9. The disk of ou galaxy is in some ways moe difficult to study than that of othe galaxies, pimaily because A) Ou galaxy s disk is vey dim, making it difficult to see B) The motion of ou disk is exceptionally chaotic, making pattens had to spot C) We ae in the plane of ou galaxy, whee the dust blocks ou view of much of the disk D) The stas in ou galaxy ae so close they oveexpose in most telescopes E) The stas ae so close that we can t use the usual assumption that the distance to all the stas in the galaxy is identical 0. The spial ams of a galaxy, accoding to cuent theoy, A) Ae self-popagating waves of highe density B) Contains a fixed set of stas and gas that obits togethe C) Is actually a magnetic phenomenon, connected by magnetic field lines D) Repesents a seies of exploding stas, which set off fomation of new stas in a spial patten E) Is actually a giant pinwheel toy made by ultapoweful aliens fo thei amusement Pat II: Shot essays [60 points] Wite appoximately a paagaph o two explaining the following (5 points each). Explain what a Cepheid Vaiable sta is, and how it can be used to detemine the distance to an object containing one. Include all appopiate equations, explaining how they ae elevant. A Cepheid Vaiable is a vaiety of giant sta which is luminous and pulsates with a egula peiod. Thee is a elationship between the ate at which it pulsates and the luminosity, called the peiod-luminosity elationship. It is given among the fomulas below: M.8log P.43 whee the peiod P is in days and M is the absolute magnitude. If we then measue the appaent magnitude m of the sta, we can then get the distance using the standad distance fomula d 0 mm 5 whee the distance d comes out in pasecs. Cepheids ae useful distance indicatos because they ae bight, consistent, and not so ae as to be had to find.

5 . Two clustes have identical metallicity, but one is much olde than the othe. Explain how one might obsevationally detemine which cluste is olde. As clustes age, the main sequence stas will cease being main sequence stas, going on to be giant stages. If we plot the Hetzspung-Russell diagam fo a sta cluste, many of the stas will follow the diagonal path of the main sequence, but towads the top the stas will have tuned off of the main sequence. The O and B stas ae the fist to tun off, then the coole A, F, and eventually G stas (K and M stas ae neve old enough to have left the main sequence). The tun off point, the place whee the stas leave the main sequence, gives you an estimate of the age. As the young, hot, bluish stas leave the main sequence, the cluste will evolve in colo, fist fom a blueish tinge (epesenting the hottest O and B stas) and then tuning moe neutal and ending up with a ed tinge (when only the coole main sequence and a few giant stas dominate the light). Hence the oveall colo can also give some indication of age. 3. Most of the stas in ou neighbohood ae membes of the thin o thick disk, but a vey small numbe ae halo stas. Explain how these stas might noticeably diffe fom odinay stas in ou neighbohood. Give at least two ways they geneally diffe. Halo stas have a vaiety of diffeences. Most noticeable is thei metallicity, which can be one pecent o even less of the sola abundance. Halo stas also move vey fa fom the plane of the galaxy. Neaby halo stas, of couse, ae quite close to the plane, but they will likely have consideable vetical motion. They also don t move paticulaly close to cicula obits. Hence thei velocity elative to the Sun will commonly have significant backwads motion, and pobably have a lot of adial motion as well. Finally, Halo stas ae always old. 4. What is the souce of the cm adio line? What type of gas does it come fom? Why is it impotant in undestanding the otation of the galaxy? The cm line comes fom neutal hydogen atoms. It is poduced when the electon in a hydogen atom flips its spin compaed to the spin of the poton. The magnetic enegy eleased comes out in the fom of electomagnetic adiation. Unlike visible light, the adio waves at cm can penetate the gas and dust in the plane of ou galaxy. We can pee though this muk using this adio wave, and see the neutal hydogen atoms cicling the galaxy. By measuing Dopple shift, we can wok out the elative otation ate at vaious adii compaed to ou own. The ecessional velocity compaed to us is given by V V 0 v R0 sin l R R0 whee V and R ae the otational speed and adius of the obseved cloud, V 0 and R 0 ae the same thing fo us, and l is the galactic longitude, the angle elative to the diection of the cente of the galaxy.

6 Pat III: Calculation: [00 points] Fo each of the following poblems, give the answe, explaining you wok. The value of each potion appeas in squae backets. Some possibly useful equations appea below. [0 points each] Units AU.4960 m 6 pc m 9 ev.600 J 7 y = s 8 R m 30 M.9890 kg 6 L W T 5777 K Bightness/Magnitude W/m 0 m b Moving Cluste v tan d Physical Constant 3 kb.380 J/K 5 kb ev/k W/m /K J s ev s G m /kg/s Planetay Nebula M * A cetain galaxy has an unusual distibution of matte. The density of matte takes the fom A at a distance fom the cente of the galaxy. (a) [6] What is the total amount of mass contained within a adius R fom the cente of the galaxy? We divide the space into thin spheical shells, each having thickness d and aea We then multiply the volume by the density, to yield R Black Body Radiation 4 kt B u 3 5 c 3 T m K max Type Ia SN M max 9.3 A M R d A d A R Gavitational Potential Enegy 3 3 G EP d d 3 E d P Black Hole GM RS c Relative speed, Cicula Obits V V 0 v R0 sin l R R0 Dopple Shift v z v Cepheid Peiod/Luminosity M.8log P.43 0 c c 4.

7 (b)[8] What is the magnitude of the gavitational acceleation g as a function of R? As demonstated with the help of Gauss s law, the gavitational acceleation fom a spheically symmetic mass distibution is simply given by GM R GA g R ˆ ˆ GA ˆ R (c) [6] What is the velocity V fo cicula obits as a function of R? The centipetal acceleation, given by acceleation, so a V R V gr GAR, V GAR., must match the gavitational 6. A distant sta is a ecuent nova. By shee luck, thee is a cicula ing of gas suounding the sta, as sketched at ight, exactly pependicula to ou line of sight, with cm epesenting ac-second. Exactly 30 days afte the nova goes off, the ing is suddenly illuminated. ( d = 86,400 s) (a) [8] What is the adius of the ing, in AU? The light that fom the ing fist went to the ing, then to ou eyes. The distance of the ing and the sta ae equal, and so the time delay fo the light to each us is just the time it took to get to the ing. The adius theefoe is the speed of light c times the time we had to wait, o s AU ct.9980 m/s30 d 500 AU d.4960 m (b) [] What is the distance to the nova/ing, in pc? The ing as dawn is 4.8 cm wide, o.4 cm in adius, so this means the angula size is.4. The distance is elated to the appaent size by R d, so we have R 500 AU pc d 00 pc.4 AU ad This places it well within the galaxy.

8 7. A type Ia supenova goes off in a distant galaxy! The appaent magnitude as a function of time is sketched at ight. (a) [5] What is the appoximate distance to this galaxy? The bightest point fo this supenova occus at about m = 9.5. But accoding to ou simplified fomulas, it should have an absolute magnitude of about M max 9.3. We theefoe use the distance fomula to find m time (days) mm pc 575 Mpc d (b) [7] With a good telescope, one can obseve planetay nebulas that ae as dim as about m = 30. Would planetay nebulas be a good way to check the distance to this galaxy? Why o why not? We need to figue out how dim a planetay nebula would be at this distance. Using an altenate vesion of the distance fomula, we have mm d 8 5log 5 5log Actually, we could have gotten this moe quickly by noting that this combination will be identical fo the supenova and fo the planetay nebulae. The planetay nebulae have absolute magnitudes no bighte than about M = -4.47, so the bightest appaent magnitude they can have is about m M * This is about fou magnitudes too dim to use the planetay nebula method on. (c) [8] A study of the spectum fom this galaxy shows that the Lyman-alpha line, nomally at a wavelength of.57 nm, has been shifted to a wavelength of 53.8 nm. What is the speed of this galaxy, and is it towads us o away fom us? You may leave you answe as a faction of c if you wish. We use the Dopple shift fomula and solve fo the velocity: v c , v c , v c v c v c v c , v c v.5897 Since the esult came out positive, it is moving away fom us o m/s

9 8. A cluste of stas has total mass M and adius R. The mass distibution is such that the gavitational potential enegy is given by GM EP R (a) [8] Accoding to the Viial theoem, what is the kinetic enegy of this cluste? What is the total enegy of this cluste? The Viial theoem says that the E E 0, so the kinetic enegy is about P K and the total enegy is E ET EK EP GM R K GM EP R (b) [8] Due to an amazing coincidence, half of the stas suddenly explode, causing them to completely expel thei mass fom the cluste. Hence the mass suddenly is educed, so M M. The speed and position of all emaining stas ae unchanged. Immediately afte the explosion, what is the new potential, kinetic, and total enegy of the cluste? The potential enegy is easy to calculate. We simply substitute the new fomula fo the mass, and we find G M GM E P R R Howeve, the kinetic enegy takes some thought. Since the pocess happens instantaneously, the system will no longe satisfy the Viial theoem. Now, the kinetic enegy of each sta is given by E mv, so if we simply eliminate some faction of the mass, but keep the velocities the i i i same, the kinetic enegy will decease popotional to the mass. Hence the new kinetic enegy is down by a facto of, and the esulting kinetic enegy is GM E K EK R The new total enegy is the sum of these two, which is E E E 0. T P K (c) [4] Does the cluste now have enough gavity to hold it togethe? O is it too little enegy? O is it ight at the bounday? The cluste has enough enegy to hold togethe if its total enegy is negative; if it s positive, the cluste dissolves. We e ight at the bounday, so it will spead out, slowly and gadually. Effectively, the cluste will dissolve.

10 9. A sta at 30 is tacked ove the couse of fou yeas. Plotted at ight is the position of the sta in mas, with x denoting the motion y paallel to the plane of the Eath s obit aound the Sun, and y denoting the appaent motion pependicula to that obit. (a) [7] Explain qualitatively the two types of motion seen; i.e., what is causing the oscillatoy motion, and what is causing the gadual dift? x The position of the sta seems to oscillate, difting back and foth ove time, o moe pecisely, moving in an appaent oval. This does not epesent eal motion of the sta, but athe motion of the Eath. As we obit the Sun, the appaent diection of the sta will vay cyclically. The appaent position of the sta also shows some eal motion, at least in the y diection. This epesents actual motion of the sta, o at least, motion of the sta elative to the Sun. (b) [5] What is the angula velocity x and y fo this sta? Ignoing the oscillatoy motion, it is appaent that it is difting downwads in the y- diection, though thee is no obvious motion in the x-diection. I have dawn in some lines to guide the eye, and we estimate that the y-cuve has dopped fom 00 to 40 in fou yeas, so we have 0, 40 mas/y. x y (c) [8] What is the appoximate paallax p and distance d in pasecs fo this sta? The paallax is the amount it moves in the x-diection due to the obit of the Eath. Reading fom the gaph, it is moving fom a position of 00 mas up and down to 50 mas and 50 mas espectively, fo a paallax of about p 50 mas We can also deduce it fom the y- motion, but this motion is educed by a facto of sin This motion appeas to be fom y = 00 mas up and down to 5 mas and 75 mas, which is a motion of 5 mas, but we need to divide by sin 0.50 to get the paallax, which again is p 50 mas So it looks like it s woking out. We then find the distance fom d 0.0 pc p