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Spiral density waves
Spiral density waves
Dispersion relation
Toomre Q parameter
Torque formula
gap opening criterion
planet migration
History
• Galactic spiral arms first
considered by Lynden-Bell as
density waves
• Lin-Shu hypothesis and modal
view
• Toomre view on stability,
amplification, transients
• Applications to ring dynamics by
Goldreich and Tremaine, Torque
formula
• Gap opening criterion, Lin,
Papaloizou?
• Planet Migration: Ward
Euler’s equation
• In cylindrical coordinates
• To first order and assuming all terms
• Φ1 Contains both forced terms (e.g. from a planet)
and that due to self-gravity in the disk.
Where
• Parameter dependent on distance to Inner or
Outer Lindblad resonances
• B Oort constant
• κ, Ω: Epicyclic frequency and angular rotation
rate (functions of r)
• c sound speed
• h enthalpy
WKB approximation
•
•
•
•
Tight winding limit
Perturbations go as
or
with α or kr >1
Only terms with r derivatives kept
Use Poisson’s equation to relate density to potential
perturbation for self-gravitating structures
• Thin sheet
everywhere but in plane
• Use Gaus’ law in a pill box
to show that b=k and
Mass conservation
• To first order
• Everything using WKB (dropping all radial terms that
don’t have a k in them)
Relate potential to
density using Poisson’s
equation
Relate enthalpy to density
with speed of sound
Then we get a dispersion
relation relating k to ω
Dispersion relation
• Pattern speed
• At large k we have sound waves
• We get very open waves (small k) near Lindblad
resonances invalidating the WKB approximation
• Waves can reflect off or be absorbed at
boundaries such as the Lindblad resonances
• Waves can be driven at Lindblad resonances
• Waves carry angular momentum
• Angular momentum is a second order quantity
Toomre Q and Stability
• For m=0
• Stability ω2>0 no negative imaginary or growing solutions
• Stability when
• For tightly wound axisymmetric disturbances but this turns
out to be okay for non-axisymmetric disturbances too
• For a stellar or non-collisional disk short wavelength waves
are damped via Landau damping
• Q parameter is approximately the same as for a gaseous
disk but with sound speed replaced by velocity dispersion
Toomre Q
• Ratio of Kinetic energy to potential energy
• Used to set up N-body simulations so exhibit bar
instability and spiral arms
• Modified if there is a multiphase disk, gas+stars
• A small amount of very cold gas can allow somewhat
stable stellar disk to display spiral density waves
• Recipes for star formation can involve feedback setting
Q -- attempts to explain Kennicutt-Schmidt star
formation law.
Gas + Planetesimal disk
• To order of magnitude
• An extremely cold or massive disk required for
instability
• Gas in a circumstellar system is in thermal equilibrium
with the starlight
• However cold planetesimals are damped
• Goldreich + Ward proposed that settling solids could
form an unstable disk leading to clumping and growth
of planetesimals
• Larger bodies are less affected by gas pressure and so
rotate at Keplerian velocity
• Gas feels pressure and so rotates more slowly.
Headwinds and turbulence
• Bodies feel a headwind which can slow them
down
• Sheer between gaseous upper layers and midplane containing planetesimals. Sheer can cause
turbulence which can prevent Q instability
New instabilities?
• Recently proposed gas/planetesimal streaming
instabilities may allow clumps to grow (Johanson,
Youdin and collaborators).
• Clumps naturally form in disk because of selfgravity
• Back reaction onto the gas. Gas is slowed down
by a clump
• Pressure drop in clump, which allows other
planetesimals to collect in same region and
maintains clump
• Decrease in planetesimal formation timescale
Spiral like structure observed in
circumstellar disks
• Spiral like structure is observed in some 107 year old
circumstellar A star disks (e.g. HD100546)
• Many alternative explanations ruled out leaving instabilities
left to be exploited to understand underlying structure
HD100546
Spiral Structure in
Galaxies
• If boundaries reflect and absorb waves then some
particular waves might grow better than others and some
may last a long time. Some dissipation required so that
trailing waves primarily present
 Modal view – Lin Shu
• Noise can be amplified. Leading waves
turn into amplified trailing waves
 Transient spiral structure view
Toomre
• One pattern likely to fit only a moderate range
in radius
• Multiple Fourier components present
(supports transient view; Elmegreen)
• Trends in opening angle with Hubble type
support modal view?
• Kinks suggesting resonances such as 4:1
important --- second order term makes steady
state pattern impossible to support past the
4:1 (Contopoulos, Patsis)
• Feathering of dust show that swing
amplification takes place (Ostriker + Kim)
• As sound speed is very much below rotation
speed shocks in gas happen as gas passes over
the density peak.
• Dust tells you whether an arm is leading or
trailing. Note: If a bar is present would be
leading. We expect trailing for spiral waves.
• Dust CO IR and star clusters present near
spiral arm peaks
• Q parameters for both gas and stars close to 1
What is seen in simulations?
• Bar’s slow down,
spiral structure
exterior to bar often
with differing pattern
speed
• Possibility of nonlinear mode coupling
• Multiple waves
present can cause
appearance of
transient spirals
Same simulation but in polar coordinates
Waves driven at resonance
• Now consider a disk perturbed by a planet
• Response is large near a resonance
• But now potential perturbations from planet rather
than self-gravity. Gravitational potential terms
for planet in a circular orbit
In Saturn’s rings
• Waves are only driven at resonances. Otherwise
perturbations don’t add in phase with motions of disk
• Waves become more open and stronger the closer to
resonance
• Waves are rapidly damped away from resonance
Waves driven at resonance
• Insert velocity perturbation expressions into that for
conservation of mass (don’t yet use WKB)
• with
• Expand in terms of distance x=(r-rL)/rL from
resonance
Forced wave equation
• Result is a wave equation like this
Where the regular wave equation
has spiral density wave solutions (both leading and
trailing) and
Perturbation is an inhomogenous term
• Analogy is a string with a sinusoidal perturbation at
x=0
• Solved with a Green’s function and by asserting that
travelling waves leave in both directions from x=0
Forced string
Analogy
• Solutions to homogenous equation
• Construct a function that satisfies homogenous equation
everywhere but at x=0
• Different solution for x>0 and x<0
• Is continuous at x=0 but has a discontinuity in first derivative
at x=0
• Travels to left for x<0 and to right at x>0
solution A=B=1
Spiral density waves driven at a
resonance
• Insist that only trailing waves are present
distant from the resonance
• This allows you to relate inhomogenous part
of equation to traveling wave solution
• Wave equation is not easily solved near
resonance as equation involves terms
proportional to 1/x and WKB approximation is
not necessarily justified
Torque
angular momentum per
unit area
• Momentum flux through a radius r
-- is constant away from resonance
• Should be consistent with integral of r x∇Φ
• Because torque is dependent on u,v it is second order in ψGT
and mass of planet
• To order of magnitude
Only one derivative of Laplace integral because the torque is integrated
over dθ and we need cross terms in u,v in order to get terms in phase
Torque formula
• Angular momentum flux at m-th Lindblad
resonance
• Evaluated at resonance
• Remarkably robust and independent of source
of dissipation (e.g., Artymowicz 93,94)
Torque between planet and disk
• Wave launched at resonance
• The act of launching the wave pushes the planet.
Torque formula gives torque between planet and disk
• If the wave is damped locally then the planet pushes
the disk away from the planet near the planet and you
get a gap
• Solution of wave equations near disk done with
boundary condition assuming propagation of wave,
however torque on disk is estimated from the
approximation that the waves are damped rapidly (see
Takeuchi et al. )
Torque cutoff
• To find total torque between planet and disk must sum
torques at each resonance
• The closer to the planet the stronger the torque as Wm
depend on Laplace coefficients which diverge
• Pressure in disk moves locations of resonances
• Resonance location normally at
• new resonance condition
Ω0 is Keplerian rotation rate and so is set by radius
Torque cutoff
distance to resonance
• For large m
• Thus the resonances stop getting closer and
closer to the planet
• Distances between planet and resonance
approaches h for large m
• Only resonances with m<r/h effectively drive
waves (Atrymowicz 93,94)
Gap opening
• Viscous torque from accretion disk
• Close to planet
where x is distance to resonance
• Torque ∝ψGT2/D,r and D ∝ x-1 Torque ∝q2/x4
Summing all resonances to x=Δ
• If we set Δ=rroche then
• Set T to Tν (balance torques) and solve for q
• Gap opening criterion is q > 40 Re-1
with Re= (r2Ω/ν)
Lin & Papaloizou, Bryden, later Crida
Disk edges: a balance between inward diffusion
and outward mass flux due to spiral density
waves driven by the planet.
2D simulation of a Jupiter mass planet
opening a gap in a low viscosity disk
Simulated planet disk interactions
Adding up many waves, they all cancel except at one angle -> one armed pattern
If gap is very wide then high m are not driven -> two or 3 beating patterns
Waves
• Simulations show waves travelling far but they are assumed
to be damped quickly in gap opening criterion
• Viscously damped vs shock dissipation – or perhaps both,
shock dissipated near planet and then viscously damped
further away (see discussion by Takeuchi et al. 94)
• Torque formula and gap opening estimate seems pretty
accurate nevertheless
• Waves driven by planets detectable in circumstellar disks?
No or extremely difficult. Gap opening: Rapidly winding
up and so difficult to resolve. Open armed detected
structure is not driven by planets. Non-gap opening, more
open but fainter and smaller. Detectable perhaps via
thermal or chemical signatures.
Planet migration
Torque from waves generated on both sides of planet. So
involves a difference. (Bill Ward)
• Type I migration : Planet is embedded in disk of smooth
surface density. Torque is proportional to square of planet
mass Mp so migration rate da/dt ∝Mp Can be fast,
particularly for Earth sized objects. Rate independent of
viscosity but proportional to disk surface density
• Type II : Planet opens a gap, not flat surface density.
Migration rate set by viscous timescale of disk. Planet
keeps up with disk. Rate independent of planet mass or
disk surface density though if disk is too diffuse migration
cannot take place until enough mass piles up next to the
planet.
• Type III: with or without a gap but possibly faster than
predicted via Type I or II because of torques associated with
corotation resonances. Disk is asymmetric.
Planet migration rates
Type I migration
• Torque depends on Mp2 Σ
• Angular momentum of planet depends on Mp
Type II migration
• independent of everything but disk viscosity
terms coined by Bill Ward
Gaps and clearings
• Gaps undetectable in SEDs
• Large clearings are seen in about 5% of few Myr old
stars.
• Clearing timescale depends on host of factors, stellar
mass, cluster environment …..
• Extremely empty clearings could be due to binaries
(e.g., CoKuTau4)
• Clearings with inner disks perhaps more likely to be
hosting planets
• Disk edges held up by planets unlikely to be extremely
steep so some gas can pass the gap, moving either
onto a planet or to an inner circumstellar disk
Eccentricity evolution of disk and
planet
• Regular Lindblad resonances tend to damp
eccentricity of planet
• Corotation and higher order ones can increase
eccentricity of either disk or planet
• Usually rapid eccentricity damping seen in
simulations of both disk and planet
• We have seen low mass inner disks become
eccentric – possibly related to a class of
oscillation periods seen in binaries
Explaining hot Jupiters
• Planet migration seems necessary as massive
planets need condensed material from which
to form and this requires cold temperatures
• However eccentricity distribution just outside
tidal circularization region implies close
planets have been scattered
• When does migration stop? Magnetic effects?
When are spiral density waves NOT
driven?
• Collision timescale in disk must be shorter
than libration timescale in resonance
• Is there a qualitative way this makes sense?
• Resonance causes perturbations on this
timescale. They are only in phase if collisions
erase memory between oscillations.
• Not intuitive as libration timescale did not
enter at all in torque estimate.
Low density disks
• Implies that spiral density waves are not
driven in disks with collision times greater
than a few hundred orbits (that covers most
debris disks)
• Gap clearing involving ejection of particles
because of close approaches by planet
• Only work done in diffusive approximation,
simulations with better treatment of collisions
could improve understanding
To truncate a disk a
planet must have
mass above
 n 
log10   6  0.43log10 
3 
5

10


 u / vK 
 1.95 

0.07


(here related to
observables)
Log Planet mass
Using the numerical measured fit to
diffusive disk simulations
Log Velocity dispersion
A lower mass object can
truncate a disk if spiral
density waves are not driven
Gap opening applied to accretion disks
• Accretion rate, mass surface
density Σ
• Viscosity, ν with α
prescription
– h=scale height
– cs=sound speed
• hydrostatic equilibrium
Ω is angular rotation rate
Hydrostatic equilibrium
Gap opening and accretion
• Gap open criterion depends on disk viscosity
• We need to know temperature to find scale
height and sound speed and so viscosity
• Heat sources
– star light
– accretion
Thermal structure
• Dissipation due to accretion
radiated away
• T is that at surface, however disk could be optically thick and
so interior temperature exceeds that at surface
• You need an opacity law for τ to relate surface temperature to
disk interior. This law may also depend on temperature. Also
emissivity ε likely to depend on temperature
• Optically thin and thick limits possible each giving different
temperature profiles
• Can consider 2 temperature sandwiches or complex thermal
structure (most recently Garaud & Lin)
Irradiated disk
• Flux absorbed = Flux radiated away thermally
• Amount of light absorbed by disk depends on
grazing angle
• β is albedo, L* luminosity of star
• For either energy source (irradiation or
accretion) you can solve for T(r) and then
estimate the viscosity
Minimum Gap Opening Planet In
an Accretion Disk
accretion,
optically
thick
Gapless disks
lack planets
qmin  M 0.48 0.8 M *0.42 L*0.08
Edgar et al. 07
Reading
• Binney & Tremaine, Galactic Dynamics
• Artymowicz 1993, ApJ, 419, 155, On the Wave Exitation and
Generalized Torque Formula for Lindblad Resonances
excited by an External Potential
• Bryden, G. et al. 1999, ApJ 514, 344, Tidally Induced Gap
Formation
• Goldreich, P., & Tremaine, S. 1979, ApJ, 233, 857, Excitation
of Density waves ..
• Takeuchi, T., Miyama, S. M. & Lin, D. N. C. 1996, ApJ, 460,
832, Gap Formation in Protoplanetary Disks
• Tanaka, H., & Ward, W. R. 2004, ApJ, 602, 388, Threedimensional Interaction between a Planet and an
Isothermal Gaseous Disk. II. Eccentricity Waves and
Bending Waves (handy formulas in this paper for
eccentricity damping and migration rates)