1Problem 24.36: “Point mass M0at center of MW + mass distributed with density ρ(r) ∝1/r2. (a) Show that Mr= kr+M0.”Correct:Wrong: anything that does not show that you realized that you need to integrate over GvRMesc220=022 RGmMmvesc=GvRMesc20=Problem 24.15: “Assuming that the highest velocity stars are near the escape speed, estimate the mass of the M.W.”Correct: vesc= vcirc+ max vpec= 220 + 65 ~ 300 km s-1. K.E. = Potential Energy ÎÎWrong: follow example 24.3.1 and calculate mass required to hold star in circular orbit with v = 300 km s-1rCMrdrrCMrdvolrMMrrrππρ∫∫+=′′′+=′′+=002200044)()()()( rdvolr′′ρBlack HoleEllipticalsHuge mass range:• Dwarf spheroidals: 107-108M• Blue compact dwarfs: ~109M• Dwarf ellipticals: 107-109M• Normal (giant) ellipticals: 108-1013M• cD galaxies in cluster centers: 1013-1014McD (NGC 3311)Giant E (NGC 1407)Dwarf spheroidal(Leo I)Dwarf ellipticals M32, NGC 2052HST images• Hubble’s law• Modified Hubble’s law•R1/4law usually fits radial surface brightness distribution• + others−−=133.34/110)(eRReIRI21)(+=ooRRIRI21)(+=ooRRIRI2/321)(+=ooRRIRIDiverges, but at least is projection of simple 3D distribution: GoodBetterBestcDr1/4EllipticalsTrue shapes requires statistical analysis• Oblate = pancakes• Prolate = footballs3True shapes requires statistical analysis• Lower luminosity Î rotationally supported •(Vrot/ σ) ~ ε/(1-ε)•Higher L Î pressure supported•(Vrot/ σ) << 1From Binney & Tremaine, Galactic Dynamics= low L ellipticalsx = spiral bulges high L ellipticalsdEgECurve expected for galaxies that are flattened by rotation (i.e. have isotropic random velocity dispersions)CO pgs. 988-989(V/σ)*= 0.7Rotationally SupportedStatistics of ε= (1- b/a)• Oblate, prolate spheroids can’t fit the observed distribution.• Summing over wide range of true values of εwould fill in the dip at εobs= 0.• Triaxial spheroids can fit.• Nearly oblate triaxial spheroids seem best.32100 0.2 0.4 0.6 0.8 1.00 0.2 0.4 0.6 0.8 1.043210P(εobs)P(εobs)Oblate, true ε= 0.7Prolate, true ε= 0.5ε observedTriaxial, Axis ratios 1:0.8:0.3From Binney & Merrifield, Galactic Astronomy4Other evidence for triaxial systems• Isophotal twists• Kinematics (star motions)From Binney & Merrifield, Galactic AstronomyOrbits in E galaxies• Some families of non-closed orbits in a mildly triaxialpotential.From Binney & Tremaine, Galactic Dynamics5The “Tuning Fork” Diagram“Early” “Late”E galaxies are transparent, but 40% still have some dust lanes• Even if complete star formation at t=0, stars must subsequently have lost gas. • Detected by:• X-rays (Brehmsstrahlung): 108-1010M• H I emission lines: 107-109M• H II emission lines: 104-105M• But gas can be lost by• Supernova-driven winds• Ram pressure
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