MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Earth, Atmospheric, and Planetary Sciences Department. Quiz 2

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1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Departent Earth, Atospheric, and Planetary Sciences Departent Astronoy 8.8J 1.40J April 19, 006 Quiz Nae Solutions (please print) Last First 1. Work any 7 of the 9 probles indicate clearly which 7 you want graded.. Spend 10 or 15 inutes reviewing the probles, and select Iediately after this you have two continuous hours to work the probles. 4. All probles are worth 14 points; everyone gets points just for taking the exa. 5. Closed book exa; you ay use two pages of notes, and a calculator. 6. Wherever possible, try to solve the probles using general analytic expressions. Plug in nubers only as a last step. 7. If you have any questions, e ail the Instructor. 8. Turn exa in during lecture, Friday, April 1. Tie Started: Tie Stopped: Signature Proble Grade Total Grader

2 APPROXIMATE VALUES OF USEFUL CONSTANTS Constant cgs units ks units c (speed of light) G (gravitation constant) k (Boltzann s constant) h (Planck s constant) proton ev (electron Volt) M (solar ass) L (solar luinosity) R (solar radius) σ (Stefan Boltzann cons) Å(Angstro) c/sec dyne c /g erg/k erg sec g erg g erg/sec c erg/c sec K 4 c k (kiloeter) 10 5 c 10 3 pc (parsec) kpc (kiloparsec) Mpc (egaparsec) year day c c c sec sec /sec N /kg J/K J sec kg J kg J/sec J/ sec K 4 sec sec AU 1 (arc inute) /3400 c rad /3400 rad 1 (arc second) 1/00, 000 rad 1/00, 000 rad

3 Proble 1 (Short Answer Questions on Magnitudes) a. A globular cluster has 10 6 stars each of apparent agnitude +8. What is the cobined apparent agnitude of the entire cluster? +8 =.5 log(f/f 0 ) F = F 0 F cluster = F 0 = 630 F 0 cluster =.5 log(630) = 7 b. Find the distance odulus to the Androeda galaxy (M31). Take the distance to Androeda to be 750 kpc. d DM = 5 log = 5 log(75, 000) = pc c. An eclipsing binary consists of two stars of different radii and effective teperatures. Star 1 has radius R 1 and T 1, and Star has R = 0.5R 1 and T = T 1. Find the change in boloetric agnitude of the binary, Δ bol, when the saller star is behind the larger star. (Consider only boloetric agnitudes so you don t have to worry about color differences.) F 1& = 4πσ T 1 4 R1 + T 4 R F eclipse = 4πσT 4 1 R1 F 1& Δ =.5 log F eclipse T 4 R Δ =.5 log 1 + T 1 4 R1 16 Δ =.5 log 1 + = So, the binary is 1.75 agnitudes brighter out of eclipse than when star is behind star 1.

4 Proble (Binary Syste) The orbit of the first accretion powered illisecond X ray pulsar was easured here at M.I.T. using data fro the Rossi X Ray Tiing Explorer Satellite. The Doppler delay curve for the otion of the neutron star is given in the figure below. The aplitude of the sine curve indicates the projected light travel tie across the orbit of the neutron star around the center of ass of the binary. (a) Use the figure to estiate the value of a ns sin(i) for this syste, where i is the orbital inclination angle. Express your answer in c or. We can read off a ns sin i fro the graph = lt sec = c. (b) Assue that the neutron star has a ass of M ns = 1.4 M and that the orbital inclination is i = 30. The orbital period of the binary is P orb = hours. Copute the ass of the copanion star, c. Hint: in order to solve for c you will have to ake the approxiation that c /M ns 1 and, equivalently, M ns / c 1. 4π GM tot G(M ns + M c ) G(M ns + M c ) G(M ns + M c ) = = P a 3 (a ns + a c ) 3 a 3 ns(1 + a ns(1 + M ns /M c ) 3 c /a ns ) 3 a 3 We can now take M ns M c to find: Plugging in nubers, we have: Solve for M c to find 4π GM c 3 GM c 3 sin 3 i = P a 3 M (a ns sin i) 3 M ns ns ns 3 1 GM c 3 4π = 700 ( ) 3 (1.4 M ) M c = M

5 Proble 3 (Hertzsprung Russell Diagras) The figures below show H R diagras for two globular clusters, NGC 1851 and NGC The data were obtained with the Hubble Space Telescope (HST) and reported by Piotto et al. (00). For purposes of this proble you ay consider the HST agnitudes F 439W and F 555W to be the equivalent of the apparent B and V agnitudes. a. Use these HR diagras to find the ratio of the distances to the two clusters. The best defined feature with which to copare agnitudes between the two HR diagras is, perhaps, the horizontal branch. I estiate 16th and 15th agnitudes for NGC 1851 and NGC 5904, respectively. Thus, d 1851 /d d 1851 Δ ag = 5 log 1 d 5904 b. If we take the absolute visual agnitude of a star on the Horizontal Branch to be V = +0.5, find the distance to either cluster. The distance odulus for NGC 1851 is DM = Hence the distance is: distance = /5 pc 1, 600 pc c. Identify as any phases of stellar evolution as you can on one of the globular cluster diagras. You ay ark directly on the figure. Coent on any physical processes that you know are occurring in the stars at these particular phases. The phases and nuclear burning cycles we were looking for included: ain sequence; hydrogen core burning giant branch; hydrogen shell burning horizontal branch; heliu core burning Others that could have been entioned were asyptotic giant branch, ain sequence turnoff, and blue stragglers.

6 Proble 4 (Galaxy Features) The iage below is of the galaxy M101. Assue that this is a typical spiral galaxy seen nearly face on. NASAandESA STSc I PRC0610a a. Give a short discourse on this iage, identifying as any generic features of the galaxy as you can. You ay write on the white space provided and draw arrows to the appropriate places on the figure. Indicate approxiate diensions where appropriate. Coent on the stellar populations in various locations and the corresponding etallicities. Roughly where would the Sun be located if this were the Milky Way? Diensions: kpc in diaeter; perhaps 1 kpc in thickness Pop I concentrated in the spiral ars; Z 0.0; Pop II concentrated in the bulge; low Z ( 0.001) Possible features to point out spiral ars dust lanes galactic bulge star foring regions central black hole globular clusters (hard to discern) dark atter halo

7 Proble 5 (Stellar Atosphere Density Profile) A star of radius, R, and ass, M, has an atosphere that obeys a polytropic equation of state: P = Kρ 5/3, where P is the gas pressure, ρ is the gas density (ass per unit volue), and K is a constant throughout the atosphere. Assue that the atosphere is sufficiently thin (copared to R) that the gravitational acceleration can be taken to be a constant. Use the equation of hydrostatic equilibriu to derive the pressure as a function of height z above the surface of the planet. Take the pressure at the surface to be P 0. Sketch your solution P (z). Start with the equation of hydrostatic equilibriu: dp = gρ dz where g is approxiately constant through the atosphere, and is given by GM/R. We can use the polytropic equation of state to eliinate ρ fro the equation of hydrostatic equilibriu: 3/5 dp P = g dz K Separating variables, we find: 3/5 1 P 3/5 dp = g dz K We then integrate the left hand side fro P 0 to P and the right hand side fro 0 to z to find: 5 P /5 /5 P 0 = gk 3/5 z Solving for P (z) we have: [ ] [ ] 5/ 5/ /5 g P (z) = P 0 gk 3/5 z = P P /5 K z 3/5 0 The pressure therefore, goes to zero at a finite height z ax, where: 5P /5 0 K 3/5 /3 5Kρ 0 5P 0 z ax = = = g g gρ 0

8 Proble 6 (Galaxy Rotation Curve) It has been suggested that our Galaxy has a spherically syetric dark atter halo with a density distribution, ρ dark (r), given by: ( r ) 0 ρ dark (r) = ρ 0, r where ρ 0 and r 0 are constants, and r is the radial distance fro the center of the galaxy. For star orbits far out in the halo you can ignore the gravitational contribution of the ordinary atter in the Galaxy. a. Copute the rotation curve of the Galaxy (at large distances), i.e., find v(r) for circular orbits. GM (< r) v = (fro F = a) r r r ( r ) 0 M (< r) = ρ 0 4πr dr = 4πρ 0 r 0 r 0 r Note that, in general, M = ρ volue! You ust integrate over ρ(r). Fro these expressions we find: v(r) = 4πGρ 0 r 0 = constant b. Find the Oort B ( coefficient ) that would be expected at a radial distance r 0. 1 [Recall: A = r dω 0, and B = A ω 0 ] dr 0 First we find ω(r): Fro above, 4πGρ0 r ω(r) = v 0 = r r dω 4πGρ 0 r = 0 dr r dω 4πGρ0 = dr 0 r 0 1 dω A = r 0 = πgρ 0 dr 0 ω 0 = 4πGρ 0 Thus, B = A ω 0 = πgρ 0 4πGρ 0 = πgρ 0

9 Proble 7 (Diensional Analysis) The objective of this proble is to find a scaling law for how stellar luinosity depends on the ass of a (ain sequence) star. Carry out a diensional analysis of the equation for the teperature gradient (for radiative diffusion): dt dr 3κρL =, 64πσSB r T 3 where L, ρ, and T are the luinosity, density, and teperature at radial distance r, and κ is the radiative opacity (units: cross sectional area per unit ass), and σ SB is the Stefan Boltzann constant. Take the radiative opacity, κ to be: where κ 0 is a constant. ρ κ(ρ, T ) = κ 0 T 7/, Find an expression for L as a function of M and constants, only. Helpful relation: A typical stellar interior teperature is given, also by diensional analysis, to be T GM µ/(kr), where µ is the ean ass per gas particle. A diensional analysis of the above equation first yields: T κρl R c 1 R T 3 where c 1 is a constant constructed fro the constants in the original dt /dr equation. By using the expression given for κ, we find: T ρ L R c R T 13/ where c = c 1 κ 0. The expression for L is then: L c 1 RT 15/ ρ 16c 1 RT 15/ M R 6 where we have approxiated ρ M/(4R 3 ). Next, we ake use of the given relation aong T, M, and R fro the helpful relation. L c 3 M 11/ R 1/ where c 3 = 16c 1 (Gµ/k) 15/. This is as far as one can go with the inforation given in the proble. If you happened to recall that R M for stars on the ain sequence, then you could reach the following proportionality: L M 5

10 Proble 8 (Heliu Burning Star) A pure He star with ass equal to 1 M has a radius of 0. R, a central density, ρ c = 10 4 g c 3 (or 10 7 kg 3 ), and a central teperature of T c = K. It is in theral equilibriu, which eans that the nuclear luinosity generated in the interior equals the luinosity radiated fro the surface. Consider all the He burning to take place in the central 10%, by ass, of the star, and no burning to occur outside this region. Further, assue that the density and teperature throughout the nuclear burning region are constant at ρ c and T c. The energy released per gra per second fro He burning is given by the expression: E(ρ, T ) ρ T 3 exp[ 4.9/T 8 ] ergs g 1 sec 1. 8 For all ks units (including ρ) siply take the leading coefficient to be 40 rather than a. Find the luinosity of this He star. L = E(ρ, T )ρ4πr dr However, there is no need to do an integral since you are told that T and ρ are constant through the burning region. Thus, L = E(ρ, T ) ρ4πr dr E(ρ, T ) 0.1 M Utilizing the expression for E given in the proble, we have: Or, L ρ T 8 3 exp[ 4.9/T 8 ] 0.1 M ergs sec 1. L (10 4 ) (1.3) 3 exp[ 4.9/1.3] 0.1 M ergs sec 1. Note that the definition of T 8 was unfortunately not given in the proble; it actually eans T/10 8 K. No points were taken off if you used the full teperature instead of T 8. b. Find the effective (i.e., surface) teperature of the He star. Use the relation L = 4πσR T 4 to copute T eff : 1/4 L T = 4πσR 117, 000 K c. The reaction that powers this star is 3 6 C 1 He 4 + γ. The atoic ass excess of a He 4 ato is.4 MeV and that of a 6 C 1 ato is 0 MeV. Find the tie for all the heliu in the core of the star (i.e., the central 10% by ass) to be burned to carbon. For each C ato that is fored, 3.4 MeV are released. This corresponds to ergs per gra of burned aterial. The ass to be converted fro He to C is 0.1 M. Thus, a total of ( gras ergs per gra) ergs = ergs of energy will be released over the He core burning phase. At a release rate of L = ergs sec 1, this process will last for seconds, or years.

11 Proble 9 (Short Answer Questions) a. Suppose air olecules have a collision cross section of c. If the (nuber) density of air olecules is c 3, what is the collision ean free path? 1 1 l = = = 10 3 c nσ b. What is the physical echanis by which 1 c radiation is produced? Hyperfine transition in neutral hydrogen atos in the n = 1 state. The interaction is between the intrinsic agnetic oent of the proton and that of the electron. c. What fraction of the rest ass energy is released (in the for of radiation) when a ass ΔM is dropped fro infinity onto the surface of a neutron star with M = 1 M and R = 10 k? GM Δ ΔE = R The fractional rest energy lost is ΔE/Δc, or ΔE GM Δc = 0.15 Rc d. What is the slope of a log N (> F ) vs. log F curve for a hoogeneous distribution of objects, each of luinosity, L, where F is the flux at the observer, and N is the nuber of objects observed per square degree on the sky? The nuber of objects detected goes as the cube of the distance for objects with flux greater than a certain iniu flux. At the sae tie the flux falls off with the inverse square of the distance. Thus, the slope of the log N (> F ) vs. log F curve is 3/.