Homework 9due Nov. 26 (after Thanksgiving)• [CO 17.6 parts (a), (b)] [16.6 1sted., parts (a), (b)] Derive the deflection of the light ray passing a massive object. Note that your answer will come out a factor of two smaller than Eq. 28.20 [26.15 1sted.] because you are only asked to consider the term affecting the space-like part of the Schwarzschild metric. The time-like part is also affected (this is the answer to part c of this problem.Special offer: I have placed copious hints for how to solve this problem on the course web site. You’re in luck… I don’t know how to set up a secure web site requiring credit card payments, so these hints are absolutely free of charge. But give the problem a try without the hints as a starter.• [CO 28.15] [26.16 1sted.] Pay attention to the hint given with the problem. And remember the following oldies but goodies:• [CO 28.16] [26.17 1sted.] and the law of sines:Dark Matterso farMass/Luminosity• Local stellar luminosity function: M/L = 0.67• Our Galaxy, at larger scales:• Oort limit: M/L ~ 2.7• Slice through disk (Bahcall & Soniera) ~ 5• Rotation curve > 30• Escape speed > 30• Pop II dynamics (glob. clusters, etc.) ~ 27• Magellanic stream > 80 • Local Group timing 100 • X-ray halo of M87 > 750 • Groups of galaxies 200hVirial Theorem for Clusters • Galaxy clusters – “fair samples” of the universe.• Coma is closest relaxed cluster• Original mass measurement was by Zwicky (1933).• Measure n(r) , σv(r)n(r) = # of galaxies, σv(r) = vel. Dispersion• Fit to models based on collisionless Boltzmann eq.~ isothermal, non-spherical.• Coma: M = 2x1015 M☼M/L = 360h (+0, -180h)• Perseus: M/L = 600hDetermining membershipX-ray emitting gas in clustersHydra A - ChandraHydra A - OpticalX-ray emitting gas in clusters• Same method as used for x-rays from individual galaxies==> M/L ~ 180h• gas is important mass component of cluster• emission by thermal bremsstrahlung (free-free).• LX~ 1043-1045erg/s (5x1044erg/s for Coma)• Mgas= (4/3) πR3nemH= 3x1014M☼• Mstars= (M/L)LocalLV= 2x1013M☼Hydra A - ChandraHydra A - OpticalHz-1,Tamplitudefreq.distr.= hν[CO fig. 27.17]Gravitational Lensing• Foreground cluster distorts images of numerous background galaxies.• Use to determine total mass of foreground cluster.• Shows that 85% of mass is Dark Matter.Gravitational Lensing[CO 17.28]R-W metricGravitational Lenses1938+666HSTradioThe “Einstein Cross”Galaxy at center causes 4 images of same quasar.SourceObserverReflective, stretchy membraneLensing MassGravitational Lens SimulatorBlandford & Narayan 1986 ApJ, 310, 568Gravitational Lensing by a Point Mass(28.20) (28.21) (28.24) [CO Sect. 28.4]Effect of Lensingon Fluxβ/θE F1 /FN F2 /FNFT /FN β= θEClose alignmentNot alignedF = FNSee Refsdal (1964) MNRAS 128, 295Max amplification when β = 0~ θE/αSPoint mass forms two images(or ring)βCaustics & CatastrophesLensing by a Transparent Mass DistributionCaustic Surfaces:Number of images changes by 2 each time a caustic is crossedÎalways an odd number(if lens is transparent).Rays:Wave fronts:θ >> θEθ < θE1938+666HSTradioThe “Einstein Cross”Weak (and not-so-weak) LensingAbell 2218• Foreground cluster distorts images of numerous background galaxies.• Use to determine total mass of foreground cluster.• Shows that 85% of mass is Dark Matter.Using caustics to search for high-redshift background galaxiesThe Remarkable Case of CL0024+1654• Single distant blue galaxy • z ~ 1.2 – 1.8• Lensed by foreground cluster • z = 0.39• 8 different grav. images of blue galaxy.• Allows detailed analysis of mass distribution in cluster.• 83% of mass is non-luminous Dark Matter.• M/L = 270h(390h after allowing for stellar evolution to z = 0)The Remarkable Case of CL0024+1654• Single distant blue galaxy • z ~ 1.2 – 1.8• Lensed by foreground cluster • z = 0.39• 8 different grav. images of blue galaxy.• Allows detailed analysis of mass distribution in cluster.• 83% of mass is non-luminous Dark Matter.• M/L = 270h(390h after allowing for stellar evolution to z = 0)The Remarkable Case of CL0024+1654• Single distant blue galaxy.• Lensed by foreground cluster.• 8 different images.• Allows detailed analysis of mass distribution in cluster.• 83% of mass is non-luminous Dark Matter.Dark MatterGalaxy clusters, typical values• 85% Dark Matter• 15% Normal Matter• 14% hot intergalactic gas• ~1% stars in galaxiesMass/Luminosity• Local stellar luminosity function: M/L = 0.67• Our Galaxy, at larger scales:• Oort limit: M/L ~ 2.7• Slice through disk (Bahcall & Soniera) ~ 5• Rotation curve > 30• Escape speed > 30• Pop II dynamics (glob. clusters, etc.) ~ 27• Magellanic stream > 80 • Local Group timing 100 • X-ray halo of M87 > 750 • Groups of galaxies 200h• Clusters of galaxies 400hAverage for universe:85% dark matter15% normal