Spiral density waves Dispersion relation Toomre Q parameter

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 • Φ 1 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 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. HD 100546) • Many alternative explanations ruled out leaving instabilities left to be exploited to understand underlying structure HD 100546

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-r. L)/r. L 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 Ω 0 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 ∝ψGT 2/D, r and D ∝ x-1 Torque ∝q 2/x 4 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= (r 2Ω/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. 2 D 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 Mp 2 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. , Co. Ku. Tau 4) • 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 Log Planet mass Using the numerical measured fit to diffusive disk simulations 0 0. 0 α= 1 N -2 0 =1 c = Nc -3 10 Log Velocity dispersion (here related to observables) 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 di a r ir Edgar et al. 07 d ate Gapless disks lack planets

Reading • Binney & Tremaine, Galactic Dynamics • Artymowicz 1993, Ap. J, 419, 155, On the Wave Exitation and Generalized Torque Formula for Lindblad Resonances excited by an External Potential • Bryden, G. et al. 1999, Ap. J 514, 344, Tidally Induced Gap Formation • Goldreich, P. , & Tremaine, S. 1979, Ap. J, 233, 857, Excitation of Density waves. . • Takeuchi, T. , Miyama, S. M. & Lin, D. N. C. 1996, Ap. J, 460, 832, Gap Formation in Protoplanetary Disks • Tanaka, H. , & Ward, W. R. 2004, Ap. J, 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)