Alice Quillen University of RochesterDepartment of

Physics and Astronomy

May, 2005

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

Tan

gen

tial velo

cit

y

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

1

1000

Outline of TalkResonances in the Solar neighborhood• Explaining moving groups• Chaos in the Solar neighborhood due to resonance overlap• Resonant trapping models for peanut shaped bulgesStructure in circumstellar disks• Disk Edges, CoKuTau/4• Spiral arms: HD141569A, HD100546

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.

The Galactic Disk– Interpreting the U,V plane

2 22

0

(1 ) ln

2

2

circular orbit epicyclic motionE E E

v uV r

Coma Berenices group

Orbit described by a guiding radius and an epicyclic amplitude

Stellar velocity distribution Dehnen 98

v t

ang

enti

al velo

city

u -radial velocity

On the (u,v) plane the epicyclic amplitude is set by a2=u2/2+v2

The guiding or mean radius is set by v

Orbits associated with Lindblad resonance’s from a bar or spiral mode

Figure from Fux (2001)

Closer to corotationLocation 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

Simple Hamiltonian systems

2( , ) cos( )

2

pH p K

2 2( , ) ( , )

2 2

is constant

0 is conserved

p qH p q H I I

H dI dtH dI

Idt

Harmonic oscillator

Pendulum

Stable fixed point

Libration

Oscillation

p

Separatrix

pq

I

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

Diff

ere

nt

an

gle

off

sets

w.r

.t t

he

Su

n

Different pattern speeds 2-armed log spirals

V U

The effect of different spiral waves on the local velocity distribution

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

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

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

-1 -120km s kpcpattern

Disk heating andother consequences

Neari

ng

coro

tati

on

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

Epicyclic motion

2 22

0 02

1 2 32 21 2 1 2

( , , ; , , ) ( , )2 22

...

r zr z

p pLH p p L r z r z

rI I I

aI bI cI I

1

2

3

radial action ................. =epicyclic frequency (radial osc.)

like angular momentum =angular rotation rate

vertical action ................ =vertical oscillation frequency

I

I

I

Higher order terms

For discussion on action angle variables Contopoulos 1979, Dehnen 1999, and Lynden-Bell (1979)

Zero’th order axi-symmetric Hamiltonian

Adding a perturbation from a bar or spiral arm

1( , , ; , , ) ( , )cos[ ( )]

for a bar mode

cos[ ( ) ln ]

for a logarithmic spiral mode, arms

r z m p

m p

H p L p r z A r z m t

A m t r

m

1 1 2 3 1 2 3 1 1 2

In action angle variables:

( , , ; , , ) cos[ ( )]

near :1 ILR(inner Lindblad resonance)

near :1 OLR(outer Lindblad resonance)

pH I I I I m t

m

m

Expand and take the dominant term

Perturbation to gravitational potential

Hamiltonian including a perturbation

1/ 2

0 1 2 1 2 1 22 21 2 1 2

1 21

0 1 2 2

1 2

( , ; , )

cos[ ( )]

Canonical transformation

( , ; , )

( ) is the resonant anglep

p

H I I I I

aI bI cI I

I m t

H J J

m t

1/ 2

1 2

2 21 2 1 2 1

( )

' ' ' cos[ ]

pJ J

a J b J c J J J

This is time independent, and is conserved.2J

In phase space: Bar Mode

1

angle on the plane

2 distance from origin R I

1/ 221, 1 1 1( ) cos( )H I I I I

Incr

easi

ng r

adiu

s Closed orbits correspond to fixed points

BAR

•Outside OLR only one type of closed orbit.

•Inside OLR two types of closed orbits

In phase space: Spiral-ModeIn

crea

sing

rad

ius

Closed orbits correspond to fixed points

•Inside ILR only one type of closed orbit.

•Outside ILR two types of closed orbits

Spiral arm supporting

An additional perturbation can cause chaotic dynamics near a separatrix

No separatrix

Bifurcation of fixed point

A separatrix exists

Analogy to the forced pendulum

1/ 2 1/ 221 1 1 1cos cos[ ] H I I I I t

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

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.

2t

1/ 2

1/ 2

2 21 2 1 2 1 2

1 21

1 21

cos[ ( )] from spiral

cos[ ( )] from bar

s s

b b

H I I aI bI cI I

I m t

I m t

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

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 μm

Young bar

From Quillen et al. 1995

A resonant trapping mechanismfor lifting stars

1/ 2

20 3, 3 3 3

12

3

23, 3 3 2

3 2

( )

( , ) ( , )cos[ ( )]

' cos[ ( )]

cos2 We chose second order in

so that potential is symmetrical about plane

( ) co

( ) resonant angle2

b

b

b

H I I aI

H x z f r z m t

z m t

I I

H I I I I

m t

s2

Resulting Hamiltonian model

Vertical resonances with a bar2

0 3, 3 3 3( ) ( ) cos2H I I I I t

Banana shaped periodic orbits

OR 1:1 anomalous orbits

Incr

easi

ng r

adiu

s

Orbits in the plane

Orbits in the plane

As the bar grows stars are liftedResonance trapping

Gro

win

g ba

r

Extent stars are lifted depends on the radius.

A natural explanation for sharp edge to the peanut in boxy-peanut bulges.

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.

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.

In Summary: Galactic DisksLindblad 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.

-1 -120km s kpcpattern

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.

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

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

F

lyb

y P

ert

rub

er

Mass

Spiral structure in HD100546?

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.

Explaining spiral structure in HD100546 with a warped disk

If viewed edge on would resemble Beta PictorusWarps are long lasting –vary on secular timescales rather than rotation timescalesTwist 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