1Spiral Structure[CO 25.3]Grand design (10%) Multi-arm (60%) Flocculent (30%)M51 M101 NGC 2841Inner rings Outer RingNGC 7096 M81 NGC 4340Hwk Set 3Due Wed Sept. 30[CO 25.13, 25.14, 25.16, 25.20]What causes spiral structure?• Winding up of arms• Due to differential rotation• Stochastic, Self-Propagating Star Formation• Chain-reaction star formation• SN shells Î shock fronts Î density enhancements Îstar formation Îmore SN• Differential rotation then winds these regions up into spiral patterns•Density Waves • Wave in gravitational potential• Orbital velocity of stars different than pattern speed• Stars, gas bunch up at position of spiral arms• Causes higher grav. potential• Unclear if self-sustaining or forced.2Retrograde Motion & Ptolemy’s Epicycles• Trying to place Earth at center.• Using only circular motions.• Led to very complicated system.Path of Mars, etc. as seen from EarthPtolemyAlexandria, 140ADEpicycles… the short form.For lurid details, see [CO 25.3 ]Define an effective potential:Taylor series expansion around position of minimum Φeff(circular orbit):Circular symmetry Î independent of φSeparate d2r/dt2into R, φ, z componentsÎ 3 equations.Conservation of specific angular momentumJZ= R2d φ/dtConservation of JZÎ acceleration inφdirection when rchanges.PtolemyAlexandria, 140AD(25.32-25.33)(25.17)(25.19-25.21)U = potential energyΦ = U/M(25.31)ρ= R - Rm3Harmonic oscillation in R, φ, z about circular orbit(Epicycles)Ω = the rotation curve for Milky Way:Rm= R at min. ΦeffΩ = circular ang. vel.Orbits closed if: m (Ω - Ωlp) = nκIn inertial frame:Viewed from frame rotating with Ωlp:Two ways to line up closed elliptical orbits(as seen from frame rotating with Ωlp)2κ−Ω=ΩlpΩ=Ωlp32κ−Ω=Ωlp4κ−Ω=Ωlpn,m“local pattern speed”Angular velocity (Ωlp) ÎFlat rotation curves0xxxxxxΩ=Rvrot4Basic nature of a density waveFrom: Toomre, Annual Review of Astronomy & Astrophysics, 1977 Vol. 15, 437.• At each Rm, stars’positions in epicycles are forced into a specific pattern by gravitational potential of spiral arm.• Sum of positions of stars at this Rmforms an ellipse rotating at pattern speed.• Spiral density pattern is sum of many ellipses, all rotating at same pattern speed.Pendulum exampleRotation is opposite to the example on previous slide:Some Solar System Resonance PhenomenaAsteroid orbitsGaps in Saturn’s rings5Lin & Shu’stheoryHydrodynamicsPerturbed form of collisionlessBoltzmann equation.“quite complicated”Milky WayAngular velocity (Ωlp) ÎInner Lindblad Resonance (ILR)Co-rotation RadiusOuter Lindblad Resonance (OLR)Density wave theory interprets most spirals as 2-armed•4-armed pattern is n / m = 1 / 4• exists over a narrow range of radius.Î less likely to be seen.Important in all disk galaxiesDensity waves cannot propagate across ILR or OLRAngular velocity of ref. frame ÎHwk Set 3Due Wed Sept. 30[CO 25.13, 25.14, 25.16,
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