1. Alice Quillen University of Rochester Department of Physics and Astronomy May, 2005

2. Motivation—The Galactic Disk Hercules stream Sirius group Pleiades group Hyades stream Coma Berenices group The Milky Way has only rotated about 40 times (at the Sun ’ s Galacto-centric radius).  No time for relaxation! Structure in the motions of the stars can reveal clues about the evolution and formation of the disk. Little is known about the shape of the Galaxy disk We can study our Galaxy star by star. Prospect of radial velocity, proper motion, spectroscopic surveys of hundreds of millions of Galactic stars. Stellar velocity distribution Dehnen 98 Radial velocity Tangential velocity

3. Low Perturbation Strengths Spiral arms give a tangential force perturbation that is only ~5% of the axisymmetric component. Resonances allow a strong affect in only a few rotation periods Jupiter is the Mass of the Sun  resonant effects or long timescales (secular) required Outline of Talk Resonances in the Solar neighborhood Explaining moving groups Chaos in the Solar neighborhood due to resonance overlap Resonant trapping models for peanut shaped bulges Structure in circumstellar disks Disk Edges, CoKuTau/4 Spiral arms: HD141569A, HD100546

4. The amplitude of a pendulum will increase if resonantly forced The planet goes around the sun J times. The asteroid goes around K times. J:K mean motion resonance Perturbations add up only if they are in phase. Even small perturbations can add up over a long period of time.

5. The Galactic Disk– Interpreting the U,V plane Coma Berenices group Orbit described by a guiding radius and an epicyclic amplitude Stellar velocity distribution Dehnen 98 v tangential velocity  u -radial velocity  On the (u,v) plane the epicyclic amplitude is set by a 2 =u 2 /2+v 2 The guiding or mean radius is set by v

6. Orbits associated with Lindblad resonance’s from a bar or spiral mode Figure from Fux (2001) Closer to corotation Location of Lindblad resonances is determined from the mean angular rotation rate  by the guiding or mean radius. On the (u,v) plane, as v changes, we expect to cross Lindblad resonances

7. Simple Hamiltonian systems Harmonic oscillator Pendulum Stable fixed point Libration Oscillation p Separatrix p q I 

8. Weighting by the distance from closed orbits --- similar to making a surface of section but this provides a weight on the u,v plane. Structure set primarily by v Different angle offsets w.r.t the Sun Different pattern speeds 2-armed log spirals VV UU The effect of different spiral waves on the local velocity distribution

9. Each region on the u,v plane corresponds to a different family of closed/periodic orbits Near the 4:1 Lindblad resonance. Orbits excited by resonances can cross into the solar neighborhood

10. A model consistent with Galactic structure Explains structure in the u,v plane Pleiades/Hyades moving groups support the spiral arms. Coma Berenices stars are out of phase. Coma Berenices Pleiades group Hyades group

11. A model consistent with Galactic structure Explains structure in the u,v plane Two dominant stellar arms – consistent with COBE/DIRBE model by Drimmel & Spergel (2001) Excites a 4 armed response locally We are at the 4:1 Inner Lindblad resonance This is a second order perturbation

12. Disk heating and other consequences Nearing corotation  Kink in shape of spiral arms predicted Flocculent structure past Sun In between resonances, the possibility of heating Oort ’ s constant and V_LSR mismeasured

13. Epicyclic motion Higher order terms For discussion on action angle variables Contopoulos 1979, Dehnen 1999, and Lynden-Bell (1979) Zero’th order axi-symmetric Hamiltonian

14. Adding a perturbation from a bar or spiral arm Expand and take the dominant term Perturbation to gravitational potential

15. Hamiltonian including a perturbation This is time independent, and is conserved.

16. In phase space: Bar Mode Increasing radius Closed orbits correspond to fixed points BAR Outside OLR only one type of closed orbit. Inside OLR two types of closed orbits

17. In phase space: Spiral-Mode Increasing radius Closed orbits correspond to fixed points Inside ILR only one type of closed orbit. Outside ILR two types of closed orbits Spiral arm supporting

18. An additional perturbation can cause chaotic dynamics near a separatrix No separatrix Bifurcation of fixed point A separatrix exists

19. Analogy to the forced pendulum Controls center of first resonance and depends on radius Controls spacing between resonances and also depends on radius Strength of first perturbation Strength of second perturbation

20. Spiral structure at the BAR’s Outer Lindblad Resonance Oscillating primarily with spiral structure Perpendicular to spiral structure Oscillating primarily with the bar Perpendicular to the bar Poincare map used to look at stability. Plot every Orbits are either oscillating with both perturbations or are chaotic  heating.

21. Barred galaxies when seen edge-on display boxy/peanut shaped bulges Boxy/peanut bulge From Bureau and Freeman 1997, PASA Bureau et al. (1997) found that all boxy/peanut shaped bulges had evidence of non- circular orbits in their spectra. No counter-examples of: barred galaxies lacking boxy/peanut shaped bulges non-barred galaxies displaying boxy/peanut shaped bulges. NGC 5746

22. Previous Boxy/Peanut bulge formation mechanisms Galaxy accretion (Binney & Petrou 1985) Bar buckling (e.g., Raha et al 1991) also known as the fire-hose instability. Diffusion about orbits associated with the 2:2:1 resonance (banana shaped orbit families) (e.g., Pfenniger & Friedli 1992, Combes et al. 1991) NGC 7582 1.6 μ m Young bar From Quillen et al. 1995

23. A resonant trapping mechanism for lifting stars Resulting Hamiltonian model

24. Vertical resonances with a bar Banana shaped periodic orbits OR 1:1 anomalous orbits Increasing radius Orbits in the plane

25. As the bar grows stars are lifted Resonance trapping Growing bar Extent stars are lifted depends on the radius. A natural explanation for sharp edge to the peanut in boxy-peanut bulges.

26. Starting from a stellar velocity distribution centered about planar circular orbits. Growing the perturbation in 3 rotation periods, resonance traps orbits (even though non- adiabatic growth). Extent of lifting is high enough to theoretically account for peanut thicknesses.

27. Capture into vertical resonances This new model suggests that peanuts grow simultaneously with bars (differing from other models). We don’t know which resonance is dominant, but if we figure it out we may learn about the vertical shapes of galaxy bulges. We used a symmetrical bar, however warp modes may be important during bar formation. Formulism can also be used to address situations where the pattern speeds are changing, but are not well suited towards finding self-consistent solutions.

28. In Summary: Galactic Disks Lindblad Resonances with a two-armed spiral density wave are a possible model for structure in the solar neighborhood velocity distribution. The pattern speed is Uncertainty mostly because of that in Oort ’ s constants. Interplay of different waves can cause localized heating, something to look for in observations. Constraints on properties of waves are possible.

29. In Summary: Galactic Disks Growth of structure can cause resonant trapping. A good way to constrain vertical structure of galaxy bulges... So far no exploration of past history of galaxy! The way spiral waves grow should lead to different heating and capture and so different velocity distributions in different locations in the Galaxy. Better tools coupled with forthcoming large Galactic surveys should tell us about growth and evolution of the Galactic disk.

30. Spiral structure driven by a close passage of the binary HD 141569B,C Quillen, Varniere, Minchev, & Frank 2005 STIS image Clampin et al. 2003 Disk is truncated and spiral structure drawn out as the binary passes pericenter

31. The mass of the perturber affects the amplitude of the spiral pattern and the asymmetry. If the perturber is very low mass, only one arm is driven. The winding of the pattern is dependent on the timescale since the perturber reached pericenter. STIS image of HD 100546 (Grady et al 2001) Time   Flyby Pertruber Mass Spiral structure in HD100546?

32. Flybys and HD100546 Morphology depends on how long since the flyby occurred. However there is no candidate nearby star that could have been in the vicinity of HD100546 in the past few thousand years. Furthermore, the probability that a star passed within a few hundred AU of HD 100546 is currently extremely low, presenting a problem for this scenario. Differences between flybys and a external bound perturber (binary): Both stellar flybys and external planets can produce spiral structure. However external perturbers truncate disks and flybys tend to scatter the outer disk rather than truncate it. Long wavelength SEDs should be sensitive to the difference! Both induce spiral structure that is more open with increasing radius and with increasing amplitude with increasing radius. In contrast to spiral density waves driven by an internal planet which becomes more tightly wound as a function of distance from the planet.

33. Explaining spiral structure in HD100546 with a warped disk If viewed edge on would resemble Beta Pictorus Warps are long lasting – vary on secular timescales rather than rotation timescales Twist caused by precession of an initially tilted disk induced by a planet? Initial tilt caused by an interaction? Disk is too twisted to be explained with a single planet in the inner disk -> could be a Jupiter mass of bodies outside of 50AU