AY202a Galaxies & Dynamics

Lecture 6: Galactic Structure, con’tSpirals & Density Waves

A Rotation Pattern with Two Inner LB Resonances

ΩP

Lindblad first noted that for n=1, m=2

(Ω – κ/2) is constant over a large range of radii such that ΩP = Ω – κ/2 and that a

pattern could exist and be moderately stable.

C.C. Lin computed the response of stars & gas:

Assume that the gravitational potential is a superposition of plane waves in the disk:

Φ (r,φ,t) = eiK(r,t)(r-r0)2πGμ

|K|uniformly rotating sheet

Where K = wave number = 2π/λ

and μ = surface density

Now find a dispersion relation

if μ(r,φ,t) = H(r,t) ei(mφ + f(r,t))

then

Φ(r,φ,t) = H(r,t) e-i(mφ + f(r,t)) -2πG

|K|

Differentiate and find

μ(r,φ,t) = Φ(r,φ,t)

These equations have solutions with a spiral like family of curves

m(φ – φ0) = Φ(r) – Φ(r0)

e.g.

μ = μa(r) ei(mφ - ωt)

iK2G

ddr

Note that

K < 0 corresponds to Leading Arms

K > 0 “ “ “ Trailing “

and

i (mφ – ωt) = i m(φ – ΩPt)

ΩP = ω/m

Response of the motions of stars or gas to non-axisymetric forces F1.

F1 is assumed to

be periodic in time and angle.

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N. Cretton

Density Wave

Models+

Bar Potential

With a gas law:

a2 = dP/dρ ≈ dP/dμ

We calculate a dispersion relation for the gas

(ω – mΩ)2 = κ2 - 2πGμ|K| + K2a2

ω2 = κ2 + K2a2 - 2πG|K|μ

Sound speed ~ velocity dispersion of the gas in equilibrium

F. Shu solved the special case of a flat rotation curve, rΩ(r) = constant = v0

Mass Model μ = v02/2πGr

= √2 Ω and the wavenumber

|K| = [ 1 ± (1 – r/r0)]

where r0 is the co-rotation radius

Inner and outer Lindblad resonances are at

r = ( 1 ± √2/m) r0

m√2

4r

For m = 2, LR are at 0.293r0 + 1.707r0

m= 1, There is no inner LR

Response of the Gas depends on a

μ/μ0

t or φ

5

1128

32

8

a = sound speed

in km/s

(Shu etal 1973)

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NB For an adiabatic shock, max μ/μ0 = 4 for =5/3

How does over density relate to SFR?

Schmidt-Kennicutt Law

ΣSFR = (2.5 ±0.7)x10-4 ( ) M☼/yr kpc-2

an exponent of ~1.5 is expected for self gravitating disks if SRF scales as the ratio of gas density to free fall time which

is proportional to ρ-0.5. This lead Elmegreen and separately Silk to argue for an SFR law where the SFR is related to the gas density over the average orbital timescale:

ΣSFR = 0.017 ΣGas ΩG

There also appears to be a cutoff at low surface mass gas density:

ΣGas

1 M☼ pc-2

1.4 ± 0.15

Schmidt-Kennicutt Law vs Elmegreen/Silk

Disk Stability

Toomre (1964) analyzed the stability of gas (and stars) in disks to local gravitational instabilities. Simply, gravitational collapse occurs if Q < 1.

For Gas Q = κ CS / (π G Σ)

For Stars Q = κ σR / (3.36 G Σ)

where Σ is again the local surface mass density,

κ is the local epicyclic frequency,

σR is the local stellar velocity dispersion,

and CS is the local sound speed

Normal Disks

Starburst Galaxies

Kennicutt ‘06

Kennicutt (1989) rephrased the Toomre argument in terms of a critical surface density, ΣC where

ΣC = α κ C / (π G)

Q = ΣC / ΣG

Where α is a dimensionless constant and

C is the velocity disperison of the gas, and

ΣG is the gas mass surface density.

For this definition of the Q parameter, as before, star formation is also suppressed in regions where Q >> 1 and is vigorous in regions where Q << 1

Some facts about spirals1. Density waves are found between the ILR

and OLR2. Stellar Rings form at Co-rotation and OLR3. Bars inside CR, probably rotate at pattern

speed4. Gas rings at ILRFor the MW ILR ~ 3 kpc, CR ~ 14 kpc, OLR ~ 20 kpc

Interaction induced Spiral Structure = Tides

Based on Strong Empirical Evidence for star formation induced by galaxy interactions (Larson & Tinsley 1978)

Models now “abundant” --- Toomre2 1970’s, Barnes et al 1980’s, many more today.

Bars also act as drivers of density waves

Toomre2

model for

the Antennae

Toomre2

galaxy.interaction.mpg

Bar Driven Density Wave

Self Propagating Star formationMueller & Arnett 1976 Seiden & Gerola

1978, Elmegreens 1980’s+

based on galactic SF observations (e.g. Lada)

Seiden & Gerola 1978

Spore

Galaxy Rotation Curves

MW HI

D. Clemens 1985

MW Rotation Curve

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Zwicky’s Preface