Galactic Dynamics IRe’em Sari (Hebrew University, Jerusalem)

Dynamical Friction

Modes in disks

Stability

Lindblad & Corotation resonances

Migration

Dynamical Friction

GM/v2

Estimating Dynamical FrictionFrom conservation of momentum

deflected orbits

From gravity of the wake:contracting spheres

M vb

δb ∼ GM

b2

b

v

2

∼ GM

v2

δM ∼ ρb2δb

dv

dt∼ GδM

b2∼ G2ρM

v2

Independent of b!All scales contributeCoulomb logarithm: counting scales

v⊥ ∼ GM

b2b

v∼ GM

bv

Mdv

dt∼

b2ρv

v||

dv

dt∼ G2Mρ

v2

δv|| =

v2 − v2⊥ − v ∼= −v2⊥v

∼ G2M2

b2v32

(first order)

(second order)

Dynamical Friction - More Accurately Momentum

Vector form:

Where is the velocity relative to the stationary background.

vperp ∼= +∞

−∞

GM

x2 + b2dx

v=

2GM

bv

dv

dt= −

2πbdb ρδv||

M= −4πG2ρ

v2

db

b= −4πG2ρ

v2ln(bmax/bmin)

dv

dt= −4πG2ρ ln

bmax

bmin

v

v3

v

δv|| ∼= −2G2M2

b2v3

M M

M

Dynamical Friction - SummaryA moving object in initially stationary background creates a wake.

Natural scale

Density on that scale higher by order unity, then as b-1.

Vector form:dv

dt= −4πG2ρ ln

bmax

bmin

v

v3

b0

b0 ∼ GM

v2

4b02b0

vM

δρ/ρ ∼ 1/4

δρ/ρ ∼ 1/2δρ/ρ ∼ 1

M

Dynamical Friction - Finite velocity dispersionBody with velocity v, background with velocity distribution

- Like Newton

For isotropic velocity distribution - analogy to Gauss

For small velocity, there is a deceleration timescale independent of v

For large velocity, deceleration timescale = static background

dv

dt= −4πG2ρ ln

bmax

bmin

v − vb|v − vb|3

f(vb)d3vb

f(vb)

f(vb)

Chandrasekhar’sDF formula

1

v

dv

dt= −16π2

3G2ρ ln

bmax

bminf(0)

1

v

dv

dt= −4πG2ρ ln

bmax

bminv−3

dv

dt= −16π2G2ρ ln

bmax

bmin

v

0f(vb)v

2bdvb

v

v3

τ ∝ v3

τ−1 =

τ−1 =

M

M

M

M

Modes in Disks

Axis symmetric modes Dispersion relationStability

Physical argumentsDispersion relation

Spiral wavesDispersion relationExcitations

h/r~10-6

Physical effects

Self Gravity

Shear and rotation

Pressure/velocity dispersion

h/r~10-6

Pressure WavesOrder of magnitude

Sphere of size r, with sound speed cExpansion in time r/c.ω~c/r or ω~kc

Rigorous:

∂ρ

∂t= −∂(ρv)

∂x

∂v

∂t+ v

∂v

∂x= −1

ρ

∂p

∂x

p ∝ ργ

∂δv

∂t= −1

ρ

∂δp

∂x

∂δρ

∂t= −ρ

∂δv

∂x

δp =γp

ρδρ ≡ c2δρ

∂2δρ

∂2t= c2

∂2δρ

∂x2

ω2 = k2c2

Self Gravity - in thin disksOrder of magnitude

Sphere of size r, surface density Σ G (Σ r^2)/r^2 t^2 =r ω2~1/t2~G Σ k

Rigorous:

∂δv

∂t= −∂δφ

∂x∇2δφ = 4πGδΣδ(z)

∂δΣ

∂t= −Σ

∂δv

∂x

δφ = δφ exp [i(kx− ωt)− |kz|]δφ = −2πG

|k| δΣ exp [i(kx− ωt)− |kz|] δg = 2πGδΣ

ω2 = −2πGΣ|k|Unstable

like deep ocean waves

Rotation and Shear - Epicyclic motionGeneral radial acceleration F(r)

Circular orbits at any distancev2/r=F(r)

Non circular orbits

Conservation of angular momentum: Small perturbations around circular orbits:

r = −F (r) + θ2r

r = −F (r) + L2r−3

δr = −F (r)δr − 3L2r−4δr

δr = −(F (r) + 3Ω2)δr

κ2 = Ω2(3 +d lnF

d ln r)

κ2 = Ω2(4 + 2d lnΩ

d ln r)

Keplerian disksPrecessionSolid body rotationUniform velocityConstant LDecreasing L - Instability

κ = Ωω = Ω− κ

κ = 2Ωκ =

√2Ω

κ = 0

Ω

κ2Ω/κ

Aspect Ratio

Axisymmetric Modes - Stability

ω2 = k2c2 − 2πGΣk + κ2

ω2min = κ2 − π2G2Σ2

c2

ω2min = κ2(1−Q−2)

Q < 1Instability:

Most unstable:

Q =κc

πGΣToomre’s Q parameter

k =πGΣ

c2

Q < 1

ω2

k

Stability - Order of magnitudePressure stabilizes short wavelengths:

Thermal energy per unit mass c2

Grav. energy per unit mass GΣrInstability r> c2/GΣ

Rotation and shear:Rotation energy per unit mass (κr)2

Instability r<GΣ/κ2

Together, instability for c2/GΣ <r<GΣ/κ2

Stable disk for c2/GΣ > GΣ/κ2

or Q~ cκ/GΣ >1.

ω2 = k2c2 − 2πGΣk + κ2

Spiral Density Waves

Leading (k>0) & trailing (k<0) wavesm=number of armsTight winding kr>>1Pattern speed

Corotation resonance

Lindblad resonance

At Lindblad k=0Pressure: away from corotationGravity: towards corotation

(ω −mΩ)2 = κ2 − 2πGΣ|k|+ k2c2

exp(imφ+ i

kdr)

Ωp = ω/m

Ωp = Ω

m|Ωp − Ω| = κ

Ω = Ωp ± κ/m

(Find relation between first wavelength and scaleheight H)

Resonances• Out of resonance:

• disk particles are drifting relative to the potential

• torque averages to 0!• Corotation resonance:

• particles move together with the potential• feel a constant force.

• Lindblad resonance:• each disk particle experiences both signs of

force but at different phases of its epicycle• each epicyclic period it drifts one potential

peak.• Inner Lindblad resonance (ILR)• Outer Lindblad resonance (OLR)

Migration in disks• Proper calculations

– involves excitation of density waves.– Inner & outer lindblad resonances.– modes carry all energy and angular

momentum deposited in the disk away.• Dynamical friction??

– which velocity should we take?– what is the Coulomb logarithm?– what is the direction at which it acts

• “One sided” torque:

• Total torque

• Migration time

Σr2(Ωr)2(r/h)3(m/M)2

Σr2(Ωr)2(r/h)2(m/M)2

tdyn(M/m)(h/r)2(M/mdisk)

H

Marginally Stable Galaxies

ω2 = k2c2 − 2πGΣk + κ2

λc =2πGΣ

κ2

Mc =M3

d

M2tot

= δ2Md

δ ≡ Md

Mtot

Q =κc

πGΣ< 1

c

v= δ

Marginally Stable Galaxies

Q < 1ω2

Q > 1Q = 1

ω2 = k2c2 − 2πGΣk + κ2

k

κ2

Slope -2πGΣ

Migration & Bulge Formation

tmig =Mtot

Mc

Mtot

Mdisk

σ

V

2tdyn

δ−3 δ−1 δ2

tmig = δ−2tdynMbulge = Mdisk/tmig

Steady State GalaxiesDekel, Sari & Ceverino

Mbulge = Mdisk/tmig

tmig = δ−2tdyn

δ ≈

tdyntHubble

1/3

Mbulge = Mbulge/tHubble

Steady State Galaxy - Summary

δ ≈

tdyntHubble

1/3

Nc ≈

tdyntHubble

2/3

z=2tdyn = 50Myr

tHubble = 3Gyrδ = 1/4 Nc ≈ 15

Mc ≈

tdyntHubble

Mtot

Mdisk ≈

tdyntHubble

1/3

Mtot

-