AY202a Galaxies & Dynamics Lecture 6: Galactic Structure, con’t Spirals & Density Waves.

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AY202a Galaxies & Dynamics Lecture 6: Galactic Structure, con’t Spirals & Density Waves

Transcript of AY202a Galaxies & Dynamics Lecture 6: Galactic Structure, con’t Spirals & Density Waves.

Page 1: AY202a Galaxies & Dynamics Lecture 6: Galactic Structure, con’t Spirals & Density Waves.

AY202a Galaxies & Dynamics

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

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A Rotation Pattern with Two Inner LB Resonances

ΩP

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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

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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|

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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

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Note that

K < 0 corresponds to Leading Arms

K > 0 “ “ “ Trailing “

and

i (mφ – ωt) = i m(φ – ΩPt)

ΩP = ω/m

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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

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Density Wave

Models+

Bar Potential

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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

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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

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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)

||

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

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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

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Schmidt-Kennicutt Law vs Elmegreen/Silk

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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

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Normal Disks

Starburst Galaxies

Kennicutt ‘06

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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

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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

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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

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Toomre2

model for

the Antennae

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Toomre2

galaxy.interaction.mpg

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Bar Driven Density Wave

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Self Propagating Star formationMueller & Arnett 1976 Seiden & Gerola

1978, Elmegreens 1980’s+

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

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Seiden & Gerola 1978

Spore

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Galaxy Rotation Curves

MW HI

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D. Clemens 1985

MW Rotation Curve

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Zwicky’s Preface