Astronomy 242: Review Questions #3 Distributed: April 29, 2016 Review the questions below, and be prepared to discuss them in class next week. Modified versions of some of these questions will be used in the final exam on May 13. PART I ESSAY QUESTIONS 1. Hydrogen is by far the most common element in the Universe, and virtually all stars have 70 to 75% hydrogen by mass. However, only a small fraction of stars are seen to have conspicuous lines of Hydrogen in their spectra. Explain this paradox. 2. Interstellar gas is often detected by observing the 21 cm line,which is actually emitted at a wavelength λ e =21.1061 cm. (a) What element or molecule is responsible for this line? What is the physical state (eg, ionized vs. neutral, atomic vs. molecular) of the gas? (b) This line is typically observed in emission. What is the source of the (very small) amount of energy emitted in this line? (c) In rare cases this line is observed as a absorption line. Broadly speaking, what conditions are necessary for this to happen? (d) Suppose we observe this line in emission from disk galaxy seen edge-on, and detect a small range of wavelengths λ centered on an average value λ o λ e.whatcanwededuce from this observation? 3. The Interstellar Medium (ISM) is sometimes said to include everything not in stars. (a) Summarize the key phases of the ISM, and mention the temperature T,particledensity n, andphysicalstateofeachphase. (b) Part (a) is easier to answer if you remember a simple relationship between T and n which most phases of the ISM satisfy. What is this relationship, and what is the one phase which does not satisfy it? (c) Why does the ISM exhibit distinct phases, and not just a smooth range of T and n? There s a simple model which shows that the ISM will develop least two distinct phases. What happens to gas which finds itself at a T and n intermediate between these two phases? 4. Draw a diagram of the Milky Way. Include all the major components, indicate the Sun s position, and provide a scale-bar. Label the population of each component, and summarize the key properties of the stellar populations. Finally, summarize the observed kinematics of stars and gas in the Milky Way. 1
5. Most large galaxies, including the Milky Way, contain super-massive black holes in their centers. These black holes are thought to power active galaxies. What evidence do we have that super-massive black holes are common, and why are they good candidates for the power sources of active galaxies? 6. We believe that visible galaxies make up only a small fraction of the total mass in rich galaxy clusters. This is a remarkable claim, and like all such claims it must be supported by very strong evidence. There are three main lines of evidence. For each one, mention the basic physical principles (with the relevant equations, if possible), the types of observations required, the results of the observations, and the conclusions we draw. 7. The cosmic distance scale enables us to measure distances to remote galaxy in standard units (eg, meters). Assume that the Earth s average distance to the Sun, otherwise known as the AU, is known from radar measurements within the Solar System. Explain how this knowledge can be used to determine distances to nearby stars, how knowing distances to nearby stars enables us to measure distances to more remote stars, and so on to nearby galaxies, distant galaxies, and the measurement of the Hubble parameter. Be sure to pay attention to the relationship between each step and the next. 8. Other galaxies are receding from the Milky Way with velocities proportional to their distances. Describe the observationalbasis for this statement. Then explain how we interpret this observation without assigning the Milky Way a special status in the Universe. 9. We think the early universe was very hot. Describe two different lines of evidence which support this conclusion. 10. Describe the key stages in the evolution of a 1 M star. For each stage, indicate where the star would be plotted on an HR diagram, describe the nature and location of the energy sources which power the star, and explain the forces which support the star against gravity. Finally, explain why and how the star transitions from each stage to the next. 11. What is the Main Sequence? Explain how the Main Sequence is detected observationally. What is the key physical process which occurs in all Main Sequence stars and distinguishes them from stars which are not part of the Main Sequence? What is the fundamental physical parameter of a star which determines where it lives on the Main Sequence? PART II PROBLEMS 12. As seen from Earth, the Sun s apparent diameter is α 0.57,andthebolometric flux we receive is F bol, 1380 W m 2.Usingthisinformation,calculatetheSun seffective temperature T eff,. Show that your answer is independent of the Sun s actual distance. 2
13. The bright star Canopus (αcar) has the following measured properties: bolometric magnitude m bol = 0.72, radial velocity v r =20.5kms 1, proper motion µ =4.8 10 15 rad s 1, parallax π =0.0104 ± 0.0005 arc-sec, effective temperature T = 7350 K. (a) What is the distance d to Canopus? Include an estimate of the uncertainty in d. (b) How fast is Canopus moving with respect to the Sun? (c) What is the absolute bolometric magnitude M bol of Canopus? (d) Given that the Sun has absolute bolometric magnitude M bol, = 4.74, what is the bolometric luminosity of Canopus in units of L bol,? (e) Given that the Sun s effective temperature is T =5780K,whatistheradiusofCanopus in units of R? 14. AtypicalUltra-LuminousInfraredGalaxyemitsL IR 10 12 L,bol of infrared light. This radiation matches a black-body spectrum with a peak at a wavelength of λ peak 5.0 10 5 m. (a) Calculate the temperature T dust of the dust which emits this radiation. (b) Assume this radiation is emitted from a spherical black-body, calculate its radius. 15. In Euclidean (flat) space, a circle s circumference C is related to its radius r by C =2πr. Formally, this can be shown by integrating the infinitesimal distance element ds around the circumference: 2π C = ds = rdθ =2πr, (1) 0 where the second equality holds because the 2-D Euclidean metric is ds 2 = dr 2 + r 2 dθ 2,and dr =0foracircleofconstantradiusr. (a) In a 2-D space of positive curvature (κ = 1), the metric is ds 2 = dr 2 + R 2 sin 2 (r/r)dθ 2, where R is the space s radius of curvature. Set up an integral similar to (1) and calculate the circumference C of a circle of radius r. (b) Using the result for part (a), show that the ratio C/r approaches 2π for very small r, but has a smaller value for a finite r. (c) Using the result for part (a), show that the circumference C has a maximum for a particular (and finite!) value of r. Find this value of r in terms of the curvature radius R, and make a sketch showing why this r gives the maximum C. 16. Given that about 1% of all the hydrogen in the universe has so far been burned in stars, estimate the energy density in starlight. Use u baryon (t o )=0.05u crit (t o ), assume that hydrogen initially makes up 75% of the mass in baryons,andnotethathydrogenburningto helium converts 0.7% mass to energy. Does your result depend on the redshift at which the stars are shining? 3
Figure 1: Left: image of Supernova 2014J and its host galaxy M82. The dashes identify the supernova. Right: apparent magnitude of Supernova 2014J plotted against date of observation. Blue, green, and red represent B, V, and R magnitudes, respectively. 17. Two years ago, an undergraduate class at University College London discovered Supernova 2014J in the nearby spiral galaxy M82 (Fig. 1). This type of supernova results from the thermonuclear explosion of a white dwarf star, and such explosions reach a peak absolute V-band magnitude M V 19.3. At maximum luminosity, these supernovae have colors similar to Vega, with (B V ) 0.0. (a) What (B V )colordoesthissupernovahaveatpeakbrightness? Isthiswhatyou d expect? (b) The galaxy M82 is at a distance d =3.5 ± 0.3Mpc. What apparent V-band magnitude m V would you expect this supernova to have at peak brightness? Is this what we observed? (c) The colors, apparent magnitudes, and even the position of this supernova within its host galaxy all point to a simple interpretation of the observations which explains the results in parts (a) and (b). What is it? (d) At maximum luminosity, this type of supernova has an absolute bolometric magnitude M bol 19.5 andasurfacetemperaturet 10, 000 K. Assuming it radiates like a blackbody, estimate the radius of the supernova when it reaches maximum luminosity, in units of the Sun s radius. 18. Assume that a white dwarf supernova produces 0.6 M of radioactive 56 Ni, which decays to stable 56 Fe via the reactions 56 Ni 56 Co + 2.14 MeV τ Ni =6.10 day, 56 Co 56 Fe + 4.57 MeV τ Co =78.8day, (2) where τ is the half-life of each isotope, and the energy released by the decay is given in MeV ( 1.60 10 13 kg m 2 s 2 ). 4
(a) How much energy is released when 0.6 M of 56 Ni decays to 56 Fe? How does this compare to the total energy of the explosion? (b) Let N Ni (t) andn Co (t) bethenumberof 56 Ni and 56 Co atoms, respectively, present in the supernova remnant at time t (where t = 0 marks the explosion). The rate of 56 Ni decay, in nuclei per unit time, is λ Ni N Ni (t), where λ Ni = ln(2)/τ Ni,andlikewisefor 56 Co. Find two coupled differential equations which describe how N Ni (t) andn Co (t) changewithtime. (c) Show that the solutions to these two differential equations are N Ni (t) =N Ni (0) e λ Nit, N Co (t) = λ Ni λ Ni λ Co N Ni (0) ( e λ Cot e λ Nit ). (3) where N Ni (0) is the number of 56 Ni atoms produced in the explosion. (d) The gas is optically thick to gamma rays, so almost all of the energy released by these radioactive decays goes into heating the supernova remnant. Using the information given here, calculate the heating rates L Ni (t)andl Co (t)duetodecayof 56 Ni and 56 Co, respectively. Sketch these heating rates as functions of time. Which reaction provides the most heat at early times (say, t<10 6 s), and when does L Ni (t) =L Co (t)? (e) After the first two weeks, the supernova is powered almost entirely by radioactive decay, and its bolometric luminosity L(t) closely tracks the total radioactive heating rate. (i)what is the absolute bolometric magnitude of the supernova at t =10 7 s? (ii) Why do we think this luminosity is generated by radioactive decay? (iii) How can we use measurements of supernova luminosity to calculate the total amount of 56 Ni produced in the explosion? 19. Open star clusters are easily disrupted by tidal interactions with giant molecular clouds. This happens so often that few open clusters survive longer than 10 9 yr. Consider an encounter between a star cluster, with mass m cl 10 3 M and half-mass radius r h 2pc, and amolecularcloud,withmassm GMC 10 5 M and radius R GMC 10 pc. (a) Given that half of the cluster s mass lies within radius r h,findtheorbitalvelocityv h and orbital period t h of a star on a circular orbit at radius r h. (You may assume the cluster is spherical.) (b) Suppose the cluster passes the cloud with a relative velocity of v rel 10 4 ms 1 at a minimum distance b. Thedurationofthisencounterist enc 2b/v rel.forthesimplecalculation below to be valid, we must have t enc <t h. What is the maximum value of b which satisfies this condition? (c) If the condition in part (b) is satisfied, we can ignore internal motions within the cluster during the encounter. Focus on the star in part (a). During the encounter, its tidal acceleration, relative to the center of the cluster, is a tid GM GMC b 3 r h. (4) If the encounter lasts time t enc, what is the star s change in velocity v? 5
(d) The cluster is likely to be disrupted if v >v h. Use this condition to find the critical encounter distance b dis at which the cluster is just barely disrupted. 20. Technetium (Tc, atomic number 43) has no stable isotopes. The isotope with the longest half-life τ is 98 Tc (τ 4.2 10 6 yr). Tc does not occur naturally anywhere in the Solar System (except for trace quantities in Uranium deposits, where it is produced by spontaneous fission), but 99 Tc (τ 2.1 10 5 yr) can be produced by exposing Molybdenum (Mo, atomic number 42) to neutrons, forming radioactive 99 Mo which decays to Tc by emitting an electron. (a) If 98 Tc had been present in the Solar Nebula withamass-weightedabundanceof10 9 (ie, 10 9 by mass, similar to elements near Tc in the periodic table), approximately how abundant would it be now? Include only original Tc in your estimate; don t worry about the traces in U ore. (Note: a straight-forward calculation will exceed the 10 ±99 range of most pocket calculators. You ll need to do some math by hand.) (b) Spectral lines of Tc are observed in some asymptotic giant branch (AGB) stars. These stars typically have masses of a few M ;theylivedatleast10 8 yr as main-sequence stars before evolving into AGB stars. Tc is not observed in the spectra of main-sequence stars. What does this prove about the origin of elements? (c) What conditions must exist inside a star for Tc to appear in its atmosphere? Address precursor nuclei, production mechanisms, and transport to the surface. 21. The fusion of 4 protons to form one helium nucleus releases 26.7MeV of energy (where 1MeV 1.60 10 13 kg m 2 s 2 ). Of this energy, 98% heats the Sun s interior, while 2% is carried away by a pair of neutrinos. Solar neutrinos were first detected via the neutrino-induced reaction 37 Cl + ν e 37 Ar + e. (5) Only 0.026% of the neutrinos emitted by the Sun are energetic enough for this reaction. The detector was a tank filled with 6.2 10 5 kg of C 2 Cl 4 ;naturalchlorine,containing 25% 37 Cl and 75% of other Cl isotopes, was used for this experiment. (a) How many neutrinos does the Sun emit per second? (b) What is the average energy carried away by each neutrino? (c) The detector was expected to produce 1.5 atomsof 37 Ar per day. Given this reaction rate, estimate the cross-section of a single 37 Cl nucleus to solar neutrinos. (d) Supernova SN1987A exploded at a distance d 50 kpc from Earth. The explosion released 2.5 10 46 kg m 2 s 1, nearly all of which was carried away by neutrinos with typical energies of 4.2Mev. A total of 11 electron antineutrions were detected by the Kamiokande II detector, primarily via the p + ν e n + e + reaction. The active volume of the detector contained 2.1 10 6 kg of water. What is the cross-section for the p + ν e reaction? 6