ASTR 680MidtermTuesday, March 13Constants and EquationsSpeed of light in vacuum: c = 2.9979 × 1010cm s−1Gravitational constant: G = 6.673 × 10−8g−1cm3s−2Planck’s constant: h = 6.63 × 10−27erg s, ¯h = h/2π = 1.05 × 10−27erg s.Energy conversion: 1 eV=1.6 × 10−12ergThomson cross section: σT= 6.65 × 10−25cm2Electron mass: me= 9.11 × 10−28g=511 keV/c2Proton mass: mp= 1.6726 × 10−24g=938.27 MeV/c2Neutron mass: mn= 1.6749 × 10−24g=939.57 MeV/c2Solar luminosity: L⊙= 3.8 × 1033erg s−1Solar mass: M⊙= 2 × 1033gMinkowski line element: ds2= −dτ2= ηˆαˆβdxˆαdxˆβ= −dˆt2+dˆr2+dˆθ2+dˆφ2Schwarzschild line element: ds2= −dτ2= gαβdxαdxβ= −(1 − 2M/r)dt2+ dr2/(1 − 2M/r) + r2(dθ2+ sin2θdφ2)Schwarzschild conserved quantities: u2, uφ, utTransformation matrices: ηˆαˆβ= eµˆαeνˆβgµν, gµν= eˆαµeˆβνηˆαˆβSchwarzschild radius: Rs= 2GM/c2= 2.94(M/M⊙) kmOverall hint: All of these problems are designed to be straightforward if done properly.If you think you need to write an enormous amount or go through a very long derivation,you aren’t doing it correctly and should probably spend a few minutes thinking about amore efficient method.There are four problems; see the back page for the fourth.1. Terms and concepts. Give short (one-sentence) descriptions of the following (1 pointeach):(a) Opacity (give cgs units as well).(b) Eddington luminosity (I don’t need the value or equation, just the physical definition).(c) Invariant interval.(d) Frame-dragging.2. Particles and radiation (4 points).Some blazars emit photons up to 1013eV. The universe has a large infrared background atenergies of ∼ 10−1eV, meaning that in the center of momentum frame of the IR-gammaray interaction, the product of the energies is up to 1012(eV)2. The rest mass-energy of anelectron is 5.11 × 105eV. Use these facts to argue qualitatively that there should be a sharpcutoff in the gamma-ray spectra of sufficiently distant blazars (and indicate qualitativelywhat criterion would define “sufficiently distant”; I don’t need the actual distance, just anindication of what it means).3. General relativityConsider a particle with nonzero rest mass, so that u2= −1. It starts with zero speed at alarge distance and is dropp ed radially into a Schwarzschild black hole with no energy loss.Therefore, the only nonzero space component of the velocity is the radial component. Withthis information, derive the following (here r and t are the usual Schwarzschild coordinates;see the equation sheet for line elements and the definition of transformation matrices):(a) (1 pt) The proper radial velocity ur= dr/dτ.(b) (1 pt) The radial velocity as seen at infinity, dr/dt.(c) (1 pt) The radial velocity as seen by a lo cal static observer, dˆr/dˆt.(1 pt) In each of these cases, add a sentence at the end to indicate if your answer makesphysical sense as r → 2M. For example, “This answer makes physical sense b ecause...” or“This answer does not make physical sense because...” if you feel your answer is wrong.4. Black holes (4 points)Dr. Sane has been banned from future Nature submissions, so he is taking his discoveriesdirectly to the popular press. Ron Cowen, astronomy writer for Science News, has askedyou to investigate Dr. Sane’s latest claims. In particular:(a) (2 pts) Dr. Sane believes that in some protoplanetary disks, a Jupiter-like planet(mass M ≈ 2 × 1030g, moment of inertia I ≈ 6 × 1049g cm2, and angular velocityΩ ≈ 2 × 10−4rad s−1) can collapse directly into a black hole on a free-fall timescale. Hethinks this is because of special interactions in the extended atmosph ere of the planet.Hint: dimensional analysis is your friend here!(b) (2 pts) Dr. Sane also thinks that the Sun is actually powered by accretion onto a centralMBH= 0.001 M⊙black hole (produced in the early universe), instead of by nuclear fusion.He thinks this happens by Bondi accretion (accretion rate˙M = ρσv, where ρ ≈1 g cm−3isthe average density of the Sun, v = 4 × 107cm s−1is a typical speed, and the cross sectionis appropriate to the Bondi radius rBondi= GMBH/v2).In both cases, give the most concise and robust possible argument showing that Dr. Sane isonce again out of his
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