Aspects of Spiral Structure Theory • J. A. Sellwood • and R. Carlberg • Seoul National university, October 21, 2013
Why spirals matter • Present structure of a galaxy is not simply the consequence of its formation • Spirals are major drivers of secular evolution: – angular momentum transport (esp. in the gas) – age-velocity dispersion relation – radial mixing reduces abundance gradients – smoothing rotation curves – galactic dynamos – etc . • How do they work?
• Some spirals are clearly tidally driven, others may be bar-driven – clear driving mechanism M51 & Hubble Heritage NGC 1300
Self-excited patterns • Spirals are ubiquitous in galaxies with gas • They also appear spontaneously in simulations of cool, isolated, unbarred galaxies • Argues that many spiral patterns in galaxies are self-excited
As SC84 but 2M particles • rot 25 or 250 Myr
Heating rate? • Test of N-dependence in 2D
Heating rate? • Test of N-dependence in 2D • Shift in time – 20K heats rather more rapidly, but N=200K seems OK – except for an increasingly delayed heating – later • SC84 essentially correct – spirals still fade quickly
What about 3D? • Weak dependence on N • Without cooling the disk heats and spirals fade – except again heating is increasingly delayed as N rises
Why do others disagree? • 2M particles in both cases • A lower active mass fraction causes less heating – lower Q • Spirals are more multi-armed
Low- mass disk Spiral activity lasts much longer
• Jacobi integral Lindblad diagram conserved I J = E - p L z • so E = p L z • Slope is parallel to circular orbit curve at CR • Random motion created at LRs • less when they are close to CR, as for high m
Low-mass disk • Most power is at m >4 • LRs are close to CR Slow heating
Disk twice as massive • Little power for m >4 • LRs farther from CR More rapid heating
Two superposed steady waves • inner wave has the higher pattern speed
Heavy disk model • Lifetime of each pattern is several galactic rotations • Inner disk heats first and patterns fade
• Thus apparent shearing transient spiral patterns result from the superposition of a small number of coherent, longer lived waves – quite a few recent papers have stressed the apparent shear instead • They have also suggested that spirals corotate with the stars everywhere so that radial migration is affected – Not so!
Disk churning by spirals (SB02) • Changes at CR of a single spiral and multiple transient spirals
Low-mass disk • Multi-arm patterns just as others find • Radial migration still works well • Characteristic pattern of scattering at corotation by waves of fixed frequency
Multiple waves • Responses away from resonances can be calculated by perturbation theory (BT08) • Each pattern causes an independent response at least in linear theory – calculations by Comparetta & Quillen needlessly complicated • The response at each resonance is also independent – except where they overlap
Transient spiral modes • The underlying waves seem to be genuine modes – i.e . standing-wave oscillations that have fixed pattern speed and shape – unstable modes also grow exponentially • Each pattern lasts a few (5-10) galaxy rotations – each mode grows, saturates, then decays
Modes • Standing wave oscillations familiar from guitar strings, organ pipes, etc .
Swing amplification • NOT a mode – shape changes over time – non-constant growth rate • Vigorous response to a perturbation
A linearly stable disk • “Mestel” disk: 1/ r , V c = const • Toomre & Zang introduced central cutout and an outer taper in active density – both replaced by rigid mass • Carried through a global stability analysis of warm disks with a smooth DF – confirmed independently (Evans & Read, S & Evans) • Halve the active mass, in order to suppress a lop-sided instability, and set Q = 1.5 • They proved this disk is globally stable
Simulations of the ½-mass Mestel disk • Linear theory predicts it should be stable • Peak = / from m = 2 with different N • Amplified shot noise at first • Always runaway growth of spiral rot = 50 in these units activity
Simulations of the ½-mass Mestel disk • Rapid growth is more and more delayed as N is increased • Surges once max > 2% – non-linear effect • Since real galaxies are not as smooth as N = 500M, non-linear behavior must happen all the time
No single coherent wave • Several separate frequencies as the amplitude rises – i.e . not a single mode
Action-angle variables • Rosette orbit – uniform angular speed – plus a retrograde epicycle • Actions are L z J & J R • Angles w & w R
A true instability of the perturbed disk • Restart N =50M case from time 1400 with reshuffled phases – green: w only – blue: both w R and w • No visible spiral by t=1400 • Yet the model now possesses a vigorous instability
A true instability of the perturbed disk • Vigorously growing mode – fixed shape and frequency • Best fit shape – peak near corotation – extends to LRs • Decays after it saturates – CR peak disperses – “wave action” drains to LRs
Lindblad diagram again • Notice that stars scattered at ILR stay close to resonance – allows large changes to build up • Does not happen at OLR
Recurrence mechanism • Each coherent wave scatters stars at the resonances (S12) – especially strong at the inner Lindblad resonance • Scattering changes the dynamical structure of the disk • and creates the conditions for a new instability
Feature put in “by hand” • Demonstrates that ILR scattering really does provoke the new instability – Mode is vigorous – probably of cavity-type with a “hard” reflection near the ILR
Emerging picture • Spiral patterns are unstable modes that grow rapidly, saturate, and decay on time scales of several (5-10) galactic rotations • New instabilities develop in rapid succession that are neither a) long-lived quasi-steady modes (Bertin & Lin), nor b) responses to noise (Toomre, Kalnajs, ...) • Does this happen in nature?
Geneva-Copenhagen survey (Nordström et al . 2004) • Known distances, full space motions and ages of 13,240 local F &G dwarfs • DF not at all smooth (Dehnen 98) – Not dissolved clusters (Famaey et al .; Bensby et al .; Bovy & Hogg; Pompéia et al .) • Hard to interpret the structure in velocity space
Project into action space • Scaled by R 0 and V 0 assuming a locally flat RC – Lower boundary: selection effect – L-R bias: asymmetric drift • One strong feature – (bootstrap analysis) • Scattering or trapping?
Phases of these stars • Action-angle variables – radius shows epicycle size – ( 2 J R ) – azimuth is 2 w – w R • Concentration of stars at one phase – m > 2 disturbances are also supported, – suggests an ILR • Exactly the stars (red) in scattering tongue
Resonant stars • S10 – red stars have been scattered by an ILR • Resonant stars are the “Hyades” stream • Far more than just the Hyades cluster • Distributed pretty uniformly around the sky • Hyades cluster (age ~ 650 Myr) is in this resonance
Implications • Evidence for an LR – probably an ILR of an m = 4 spiral • Support for the picture I have been developing from the simulations – spirals are transient – decay of one pattern seeds the growth of another – each is true instability of a non-smooth DF
Gas seems to be essential for spirals • NGC 1533 – Hubble image • Misled the community for many years
Recurrent transients • Random motion rises and patterns fade
Recurrent transients • Random motion rises and patterns fade • Add “gas dissipation” and patterns recur “indefinitely” (SC84) • A natural explanation for the importance of gas
Galaxy formation simulations • Agertz et al . (2010) – barely a remark!
Age-vely dispersion relation • Data from Holmberg et al . (2009) – cloud scattering is inadequate – transient spirals required ( Binney & Lacey, Hänninen & Flynn )
Aumer & Binney • Ages disputed – use color as a proxy for age • Hipparcos data
3D shape of ellipsoid • Ida et al . (1993) showed that cloud scattering sets the ellipsoid flattening: z = 0.6 R ( V =const) • But GMCs redirect peculiar velocities more efficiently than they increase them • Spirals increase in-plane motions only and most 3D simulations do not include GMCs Any thickening people report is due to 2-body relaxation!
2-body relaxation in 3D disks • Demonstrates spirals cause in-plane heating only • Thickening (and segregation) by relaxation
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