Problem Set 3, AKA First midterm review Astrophysics 4302 Due Date: Sep. 23, 2013

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1 Problem Set 3, AKA First midterm review Astrophysics 4302 Due Date: Sep. 23, δ Cephei is a fundamental distance scale calibrator. It is a Cepheid with a period of 5.4 days. A campaign with the Hubble Space Telescope s Fine Guidance Sensor found atrigonometricparallaxof arcsec,and apropermotionof arcsec/yr. TheaverageV-bandmagnitude during a single period is V =4.07mag. Using infrared measurements, and data in other bands, the extinction to δ Cephei is found to be A V =0.23 mag. a) What is the average absolute magnitude, in the V-band, of δ Cephei? b) By a miracle of fate, you discover a Cepheid variable in the Large Magellanic Cloud with exactly the same period as δ Cephei. This Cepheid has a average apparent V-band magnitude ofv=16.9mag. Recallfrom last week s homework that we found a distance to the Large Magellanic Cloud of 52 kpc. How much V-band extinction, in magnitudes, is there towards the Cepheid in the LMC? Solution a)3points.firstwemustfindthedistancetoδ Cephei via its parallax. d (in pc) = 1/p (in arcsec) = 1./ = 273pc (1) and then we find the absolute magnitude, remembering to correct for extinction d V A V M V =5log 10 pc ; M d V = V A V 5log 10 pc = log = 3.3 mag (2) b) 2 points. We can just use the equation above: d V A V M V =5log 10 pc ;ora d V = V M V 5log 10 pc = log =1.6magofextinction! (3) 2. Galaxy clusters are large aggregations of galaxies which are thought to be gravitationally bound together. AtypicalsizeforagalaxyclusterisR =1Mpc,andhas1000galaxiesinit. Usingmulti-objectspectrographs, it is relatively efficient to measure the radial velocities of 100 s of cluster members in a single night. Let s say a typical galaxy in a cluster has luminosity L=10 10 L,andyoumeasuregalaxiesintheclustertohaveavelocity dispersion of 1000 km/s. Use the virial theorem to find the mass-to-light ratio in the typical galaxy cluster. This type of back of the envelope calculation was used as one of thefirstindicationsthatsomeformofdark matter was necessary. Solution 3points.Thevirialtheoremtellsusthat M = 3rσ2 r G (4) First, lets get big G in happier units then we have 8 cm3 G = gs 2 =4.5 pc M s 2 (5) M = 3106 pc (1000 km/s) 2 1 ( pc km ) pc = M 3 (6) M s 2 The total luminosity of the cluster is L = L.So,themasstolightratiois M/L = M L =70M /L (7) 1

2 This is a big number. Stars have a typical M/L=1-3 M /L.DARKMATTER!! 3. Do problem 4.6 in the book. Make sure to use Eqn for the tidal radius (the book calls it the Jacobi radius), rather than the equation I used in class. Solution 4points. Notethatthebookdoesn tquitegiveyouenoughinformation to complete this problem, namely the luminosity of the Sagittarius dwarf spheroidal. In that case, you only need to to figure out the mass of the dwarf; not its M/L ratio as well. See Solution at the back. 4. Answer this series of questions: a) What is a forbidden atomic transition, in words? b) HI gas has a special transition associated with photons with a wavelength of 21cm. What is this transition called, and what physical process is responsible for it? Can this transition be observed with a ground-based optical telescope? How about an optical space telescope, like the Hubble Space Telescope? As you ll recall, HI gas and the 21cm transition is crucial for measuring the rotation curve of the galaxy. c) Does dust extinction have a larger effect in the ultraviolet, or in the infrared? Very roughly, what does this tell you about the size of the dust grains in interstellar space? d) Recall from class that blobs of Hα emission were seen around the sites of the most recent star formation in the Large Magellanic Cloud. Can you explain why that is the case? e). What observation was made in the last decade that has recently led us to conclude that the Large and Small Magellanic Clouds are making their first passage through the Milky Way system? Solution a)[2points]a forbidden atomictransitionisonewhichislesslikelyaccordingtoquantum selection rules, and is effectively never seen in the laboratory. In particular, allowed transitions often occur spontaneously on 10 8 stimescales,whereas forbidden transitionsoftentake 1 secondtooccur. However, in space, where the density of atoms is so low that atom-atom collisions are nonexistent, forbidden transitions do occur. Studying forbidden lines is a sensitive probe of the density. Forbiddenlinesaredenotedbysquare brackets, e.g. [O III] 5007Å. b) [3 points] This is the hyperfine transition for Hydrogen, and it has to do with the spin alignment of the proton and electron. The high energy state is when their spins areparallel,andthelowenergystateiswhen they are in opposite directions. Since it produces a 21cm photon, this transition is not visible in the optical, but it can be seen in the radio. c) [2 points] Dust extinction is often called reddening, because it takes away more light from the ultraviolet than the infrared, making your spectrum appear redder. Dust grains can only absorb or reflect photons whose wavelength is comparable in size or smaller than the dust grain size. This tells us that dust grains are smaller than 1 micron,which is the traditional cutoffbetween the optical and near-infrared in astronomy. d) [2 points] Recent star formation will produce a handful of hot, O or B type stars. These hot stars produce UV photons that can ionize hydrogen. The ionized hydrogen around sites of star formation does not stay that way permanently. The hydrogen then recombines, producing an array of recombination radiation, including the hydrogen Balmer series which coincide with the optical. TheHα transition is the level 3 to 2 transition and is often indicative of this recombination radiation. e) [2 points] In class we talked about the recent measurements of the proper motion of the LMC and SMC. The values found were much higher than expected, and the velocities of the LMC/SMC are higher than the escape velocity of the Milky Way. After detailed simulations, we now believe that the LMC/SMC are making their first approach near the Milky Way. Proper motion measurements ofthelmc/smcareextremelydifficult. They require precision measurements, over the time span of years, of the positions of thousands of LMC/SMC stars. David Sand 2 HW3

3 Figure 1: This Figure is for problem Check out Figure 1 on the next page. These are spectra of four different stars. List them in order of bluest to reddest. Then list them in order of temperature, from coolest to hottest. What basic radiative process is responsible for the temperature of the stars? Solution [3points]BluesttoReddest: BottomLeft,TopRight,BottomRight,TopLeft. Coolobjects will be redder, as we know from blackbody radiation. Therefore, in order from coolest to hottest, we have Top Left, Bottom Right, Top Right, and Bottom Left. 6. Check out Figure 2. On the left is a spectrum of a spiral galaxy, and on the right is a spectrum of an elliptical galaxy. Which of these has a population of young stars? How can you tell? Hint, there are two reasons. Solution [3points]Thespiralgalaxyhasayoungstellarpopulation. We can tell because the continuum is very blue, and closely resembles that of A-type stars. Second, we see lots of emission lines, including from the Hydrogen Balmer series. This is indicative of recombination radiation; the source of the original ionizing photons is likely a population of hot, young stars like that seen in the bottom left of Figure 1. David Sand 3 HW3

4 Figure 2: This Figure is for problem 6. David Sand 4 HW3

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