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Disk Topics: Black Hole Disks, Planet Formation 12 May 2003 Astronomy G9001 - Spring 2003 Prof. Mordecai-Mark Mac Low Black Hole Accretion Disks • In protostellar accretion disks, radiation is always efficient, and the assumption Ωr >> cs is good. – thin disk approximation • Now turn to compact objects – deeper potential wells produce higher temperatures – far more energy must be lost to radiation – Some observed supermassive BHs have little radiation (Sag A* is the classic example) – How does accretion proceed? Thin Disk Dissipation • Thin disk approximation • ν = αcs2/Ω (or πrφ = αP) prescription for viscosity • classic radiative disk (Shakura & Sunyaev 1973, Novikov & Thorne 1973) – viscous heating balances radiative cooling – steady mass inflow gives torque (Sellwood) T R M d J R J 0 M d J R M d GMR – dissipation per unit area is then d T d 3GMM d πr R dz 3 dR 2 R dR 4 R – 3 x binding energy, because of viscous dissipation Thin Disk Radiation • if dissipated heat all radiated away, then 3GMM d 4 rad 2 T 4 R3 • this gives temperature distribution T ~ R3/4 • Integrating over the disk gives spectrum 2 • around a BH, energy release is ~ 0.1c M d • Observed luminosities from, e.g. Sag A* appear 4 2 to be as low as 10 c M d • How is BH accreting so much mass without radiating? ADAF/CDAF • Narayan & Yi (1992) and others proposed that the energy is advected into the BH before it can be radiated: advection dominated accretion flow • Numerical models made clear that the extra energy produces a convectively unstable entropy gradient in the radial direction, as well as unbinding some of the gas entirely • convection dominated accretion flow proposed as elaboration of ADAF – outward convective transport balances inward viscous transport, leaving disk marginally stable – analogous to convective zone in stars Problems with ADAF/CDAF • Balbus (2000) points out that convection and MRI cannot be treated as independent forces – instead a single instability criterion must be found – this reduces to the MRI, so no balance exists • Balbus & Hawley (2002) analyze non-radiative MHD flows. – convectively unstable modes overwhelmed by MRI – balanced transport implies that convection recovers energy produced by viscous dissipation, resulting in a dissipation-free flow: but this violates 2nd Law of Thermodynamics! Non-Radiative Accretion Flow • Hawley & Balbus (2002) simulate non-radiative MHD flow numerically, finding outflow and unsteady, slow, accretion And now for something completely different... Ruden 1999 Planet Formation in Disks • Solar planets formed from protoplanetary disk with at least 0.01 M of gas (Minimum Mass Solar Nebula) • Observed disks have comparable masses • Disk evolution Ruden determines initial conditions. Grain Dynamics • Gas moves on slightly sub-Keplerian orbits due to radial pressure gradient • Grains move on Keplerian orbits – grains with a < 1 cm feel drag FD = – (4/3) πa2ρcs(Δv) – coupling time tc = m Δv / FD , so small Ωtc = aρd / Σ means particles drop towards star, large remain. • Vertical settling also depends on Ωtc – – – – vertical gravity gz = (z/r)GM* / R2 = Ω2z settling time ts = z / vz = Ω-1 (Ωtc)-1 = Σ / (aρd Ω) small grains with Ωtc << 1 take many orbits to settle coagulation vital to accumulate mass in midplane Planetesimals • Big enough to ignore gas drag over disk lifetime • How do they accumulate from dust grains? – gravitational instability requires very cold disk with Δv ~ 10 cm s-1 (Goldreich & Ward) – shear with disk enough to disrupt most likely – Collisional coagulation main alternative (Cuzzi et al 93) • Planetesimals collide to form planets – gravitational focussing gives cross-section (Safronov): 2G m1 m2 ve2 a1 a2 1 , where ve 2 a1 a2 V so a planet accreting small planetesimals will have 2 2 v 2 e lV a 1 , with p'mal density l 2 dt V dm p Planet Growth • Orderly growth by planetesimal accretion has long time scale: Ruden 99 • Velocity dispersion Δv must remain low to enhance gravitational focussing. • Dynamical friction transfers energy from large objects to small ones – large objects have lowest velocity dispersion and so largest effective cross sections. – collisions between them lead to runaway growth Final Stages of Solid Accretion • Runaway growth continues until material has been cleared out of orbits within a few Hill radii – Hill radius determined by balance between gravity of planet and tidal force of central star Gmp GM * rH mp 3 3 2 rH r 2 rH r r M* • Protoplanet sizes reach 5–10% of final masses • Final accumulation driven by orbital dynamics of protoplanets – major collisions of planet-sized objects an essential part of final evolution – random events determine details of final configuration of solid planets Gas Accretion • Above critical mass of 10–15 M planetary atmospheres no longer in hydrostatic equilibrium – heating comes from p’mal impacts – increasing heating required to balance radiative cooling of denser gas atmospheres (Mizuno 1980) – collapse of atmosphere occurs until heating from gravitational contraction balances cooling – rapid accretion can occur • Final masses determined either by: – destruction of disk by photoevaporation or tides – gap clearing in gaseous disk Gap Formation & Migration • Giant planets exert tidal torques on surrounding gas, repelling it and forming a gap in disk. • Disk also exerts a torque on the planet, causing radial migration. Gap Formation • Tidal torque on disk with surface density Σ from 2 3 planet at rp r mp 2 4 p Tt f p rp H M* 2 • Viscous torque 2 2 4 H Tv 3r 3 r r • Gap opened if Tt > Tv which means 52 mp 3 H 1/ 2 M* f r • In solar system this is about 75 M or roughly Saturn’s mass. Observations • Disk Observations – spectral energy distributions • density distribution • gaps and inner edges – dust disks (β Pic, Vega) • Poynting-Robertson clears in much less than t* • presence of dust disk indicates colliding planetesimals – Proplyds [Protoplanetary disks], seen in silhouette • Indirect Dynamical Observations – radial velocity searches • need accurate spectroscopy: calibrator (iodine) in optical path – radial distance changes: pulsar timing – astrometry: next generation likely productive (SIM) Observations • Microlensing of planet – superposes spike on stellar amplification curve – can also shift apparent position of star • Direct detections – transits • photometry - eclipse of star (or of planet!) • transmission spectroscopy of atmosphere – direct imaging • adaptive optics • interferometry • coronagraphs (+ AO = Oppenheimer @ AMNH) Search techniques 1. 2. 3. 4. 5. 6. 7. Kepler: space-based transit search COROT: same Doppler: 3m/s ground-based SIM = Space Interferometry Mission FAME = next ESA astrometry mission ground based transit search Lyot = AO + coronagraph (BRO) habitable zone Lyot