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?

M51 & Hubble Heritage NGC 1300

• Some spirals are clearly tidally driven, others may be bar-driven – clear driving mechanism

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

Lindblad diagram • Jacobi integral

conserved

IJ = E - pLz

• so E = pLz

• 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, Vc = 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 activity

rot = 50 in these units

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

Lz J & JR

• Angles

w & wR

A true instability of the perturbed disk

• Restart N=50M case from time 1400 with reshuffled phases

– green: w only

– blue: both wR 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 R0 and V0 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 JR)

– azimuth is 2w – wR

• 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 = 0.6R (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

Other work

• Quinn et al. (1993) worried that disks thickened without an obvious mechanism

• McMillan & Dehnen (2007) also worried – scrambling test suppressed collisions too

– concluded (wrongly) that vertical heating was caused by spirals

• In fact, every thin disk simulation with N < 2M thickens by relaxation – House et al. (2012) compared their model with

MW data – not really meaningful

Conclusions • Spiral patterns are short-lived (a few rotations)

unstable modes

– superposition can lead to apparently continuously shearing patterns

– low-mass disks manifest multi-arm patterns that cause slower heating

– radial migration is alive and well

– gas still needed to prolong activity

• Non-linear effects cause the recurring cycle

– larger amplitude delayed with increasing N

– supporting evidence for the recurrence mechanism from nearby stars