Structure and evolution of protoplanetary disks Part I Accretion disk theory Lecture by: C.P. Dullemond

Keplerian rotation (a reminder) Disk material is almost (!) 100% supported against gravity by its rotation. Gas pressure plays only a minor role. Therefore it is a good approximation to say that the tangential velocity of the gas in the disk is: Kepler frequency

The angular momentum problem Angular momentum of 1 M in 10 AU disk: 3x1053 cm2/s Angular momentum of 1 M in 1 R  star: <<6x1051 cm2/s (=breakup-rotation-speed) Original angular momentum of disk = 50x higher than maximum allowed for a star Angular momentum is strictly conserved! Two possible solutions: Torque against external medium (via magnetic fields?) Very outer disk absorbs all angular momentum by moving outward, while rest moves inward. Need friction through viscosity!

Outward angular momentum transport Ring A moves faster than ring B. Friction between the two will try to slow down A and speed up B. This means: angular momentum is transferred from A to B. A B Specific angular momentum for a Keplerian disk: So if ring A looses angular momentum, but is forced to remain on a Kepler orbit, it must move inward! Ring B moves outward, unless it, too, has friction (with a ring C, which has friction with D, etc.).

Molecular viscosity? No! Problem: molecular viscosity is virtually zero L = length scale <u>= typical velocity  = viscosity Reynolds number Molecular viscosity: lfree = m.f.p. of molecule <uT>= velo of molecule Typical disk (at 1 AU): N=1x1014 cm-3, T=500 K, L=0.01AU Assume (extremely simplified) H2 (1Ang)2.

Turbulent ‘viscosity’: Reynolds stress The momentum equation for hydrodynamics is: Now consider this gas to be turbulent. We want to know the motion of average quantities. Assume that turbulence leaves  unaffected. Split v into average and perturbation (turbulence): The momentum equation then becomes:

Turbulent ‘viscosity’: Reynolds stress Now average over many eddy turnover-times, and use: (tensor!) and but Then one obtains: The additional term is the Reynolds stress. It has a trace (=turbulent pressure) and off-diagonal elements (=turbulent ‘viscous’ stress).

Turbulent ‘viscosity’: Reynolds stress Problem with turbulence as origin of viscosity in disks is: most stability analyses of disks show that the Keplerian rotation stabilizes the disk: no turbulence! Debate has reopened in the last decade: Non-linear instabilities Baroclynic instability? (Klahr et al.) But most people believe that turbulence in disks can have only one origin: Magneto-rotational instability (MRI)

Magneto-rotational instability (MRI) (Also often called Balbus-Hawley instability) Highly simplified pictographic explanation: If a (weak) pull exists between two gas-parcels A and B on adjacent orbits, the effect is that A moves inward and B moves outward: a pull causes them to move apart! A B The lower orbit of A causes an increase in its velocity, while B decelerates. This enhances their velocity difference! This is positive feedback: an instability. A B Causes turbulence in the disk

Magneto-rotational instability (MRI) Johansen & Klahr (2005); Brandenburg et al.

Shakura & Sunyaev model (Originally: model for X-ray binary disks) Assume the disk is geometrically thin: h(r)<<r Vertical sound-crossing time much shorter than radial drift of gas Vertical structure is therefore in quasi-static equilibrium compared to time scales of radial motion Split problem into: Vertical structure (equilibrium reached on short time scale) Radial structure (evolves over much longer time scale; at each time step vertical structure assumed to be in equilibrium)

Shakura & Sunyaev model Vertical structure Equation of hydrostatic equilibrium: Equation for temperature gradient is complex: it involves an expression for the viscous energy dissipation (see later) radiative transfer, convection etc. Here we will assume that the disk is isothermal up to the very surface layer, where the temperature will drop to the effective temperature

Shakura & Sunyaev model Vertical structure Because of our assumption (!) that T=const. we can write: This has the solution: with A Gaussian!

Shakura & Sunyaev model Difficulty: re-express equations in cylindrical coordinates. Complex due to covariant derivatives of tensors... You’ll have to simply believe the Equations I write here... Radial structure Define the surface density: Integrate continuity equation over z: (1) Integrate radial momentum equation over z: (2) Integrate tangential momentum equation over z: (3)

Shakura & Sunyaev model Let’s first look closer at the radial momentum equation: (2) Let us take the Ansatz (which one can later verify to be true) that vr << cs << v. That means: from the radial momentum equation follows the tangential velocity Conclusion: the disk is Keplerian

Shakura & Sunyaev model Let’s now look closer at the tangential momentum equation: (3) Now use continuity equation The derivatives of the Kepler frequency can be worked out: That means: from the tangential momentum equation follows the radial velocity

Shakura & Sunyaev model Radial structure Our radial structure equations have now reduced to: with Missing piece: what is the value of ? It is not really known what value  has. This depends on the details of the source of viscosity. But from dimensional analysis it must be something like: Alpha-viscosity (Shakura & Sunyaev 1973)

Shakura & Sunyaev model Further on alpha-viscosity: Here the vertical structure comes back into the radial structure equations! So we obtain for the viscosity:

Shakura & Sunyaev model Summary of radial structure equations: If we know the temperature everywhere, we can readily solve these equations (time-dependent or stationary, whatever we like). If we don’t know the temperature a-priori, then we need to solve the above 3 equations simultaneously with energy equation.

Shakura & Sunyaev model Suppose we know that is a given power-law: Ansatz: surface density is also a powerlaw: The radial velocity then becomes: Stationary continuity equation: from which follows: Proportionality constants are straightforward from here on...

Shakura & Sunyaev model Examples:

Shakura & Sunyaev model Examples:

Shakura & Sunyaev model Examples:

Shakura & Sunyaev model Examples:

Shakura & Sunyaev model Formulation in terms of accretion rate Accretion rate is amount of matter per second that moves radially inward throught he disk. Working this out with previous formulae: We finally obtain: (but see later for more correct formula with inner BC satisfied)

Shakura & Sunyaev model Effect of inner boundary condition: Powerlaw does not go all the way to the star. At inner edge (for instance the stellar surface) there is an abrupt deviation from Keplerian rotation. This affects the structure of the disk out to many stellar radii: Keep this in mind when applying the theory!

Shakura & Sunyaev model How do we determine the temperature? We must go back to the vertical structure......and the energy equation..... First the energy equation: heat production through friction: For power-law solution: use equation of previous page: The viscous heat production becomes:

Shakura & Sunyaev model How do we determine the temperature? Define ‘accretion rate’ (amount of matter flowing through the disk per second): End-result for the viscous heat production:

Shakura & Sunyaev model How do we determine the temperature? Now, this heat must be radiated away. Disk has two sides, each assumed to radiate as black body: One obtains:

Shakura & Sunyaev model How do we determine the temperature? Are we there now? Almost.... This is just the surface temperature. The midplane temperature depends also on the optical depth (which is assumed to be >>1): The optical depth is defined as: with the Rosseland mean opacity. We finally obtain:

Viscous heating or irradiation? T Tauri star

Viscous heating or irradiation? Herbig Ae star

Flat irradiated disks Irradiation flux: Cooling flux:  Irradiation flux: Cooling flux: Similar to active accretion disk, but flux is fixed. Similar problem with at least a large fraction of HAe and T Tauri star SEDs.

Flared disks flaring irradiation heating vs cooling vertical structure Kenyon & Hartmann 1987 Calvet et al. 1991; Malbet & Bertout 1991 Bell et al. 1997; D'Alessio et al. 1998, 1999 Chiang & Goldreich 1997, 1999; Lachaume et al. 2003

Flared disks: Chiang & Goldreich model The flaring angle: Irradiation flux: Cooling flux: Express surface height in terms of pressure scale height:

Flared disks: Chiang & Goldreich model Remember formula for pressure scale height: We obtain

Flared disks: Chiang & Goldreich model We therefore have: with Flaring geometry: Remark: in general  is not a constant (it decreases with r). The flaring is typically <9/7

Non-stationary (spreading) disks Given a viscosity power-law function , one can solve the Shakura-Sunyaev equations analytically in a time-dependent manner. Without derivation, the resulting solution is: Lynden-Bell & Pringle (1974), Hartmann et al. (1998) where we have defined with r1 a scaling radius and ts the viscous scaling time:

Non-stationary (spreading) disks Time steps of 2x105 year Lynden-Bell & Pringle (1974), Hartmann et al. (1998)

Formation & viscous spreading of disk From the rotating collapsing cloud model we know: Initially the disk spreads faster than the centrifugal radius. Later the centrifugal radius increases faster than disk spreading

Disk formation and spreading Molecular cloud core:

Disk formation and spreading Molecular cloud core:

Disk formation and spreading Molecular cloud core:

Disk formation and spreading Molecular cloud core:

Disk formation and spreading Molecular cloud core:

The formation of a disk shock Infalling matter collides with matter from the other side Forms a shock Free-fall kinetic energy is converted into heat Heat is radiated away, matter cools, sediments to midplane Disk is formed At 10 AU from 1M star:

The formation of a disk 3-D Radiation-Hydro simulations of disk formation Yorke, Bodenheimer & Laughlin 1993

Formation & viscous spreading of disk A numerical model

Formation & viscous spreading of disk A numerical model

Formation & viscous spreading of disk A numerical model

Formation & viscous spreading of disk A numerical model

Formation & viscous spreading of disk A numerical model

Formation & viscous spreading of disk Hueso & Guillot (2005)

Coupling to 2D disk structure models t=0.1 Myr Dullemond et al. in prep.

Coupling to 2D disk structure models t=0.2 Myr Dullemond et al. in prep.

Coupling to 2D disk structure models t=0.4 Myr Dullemond et al. in prep.

Coupling to 2D disk structure models t=0.8 Myr Dullemond et al. in prep.

Coupling to 2D disk structure models t=1.6 Myr Dullemond et al. in prep.

Coupling to 2D disk structure models t=3.2 Myr Dullemond et al. in prep.

Coupling to 2D disk structure models t=6.4 Myr Dullemond et al. in prep.

Disk dispersal It is known that disks vanish on a few Myr time scale. But it is not yet established by which mechanism. Just viscous accretion is too slow. - Photoevaporation? - Gas capture . by planet? Haisch et al. 2001

Geometry: only gravity vs pressure dP/dz Kz

Geometry: adding vertical B-field

Flux freezing: MHD in a nutshell Strong field: matter can only move along given field lines (beads on a string): Weak field: field lines are forced to move along with the gas:

Geometry: adding vertical B-field dP/dz Kz

Geometry: adding vertical B-field Slingshot effect. Blandford & Payne (1982) (courtesy: C. Fendt) Use cylindrical coordinates r,z Gravitational potential: Effective gravitational potential along field line (incl. sling-shot effect):

Geometry: adding vertical B-field Blandford & Payne (1982) Critical angle: 60 degrees with disk plane. Beyond that: outflow of matter. Gas will bend field lines

Photoevaporation of disks (Very brief) Ionization of disk surface creates surface layer of hot gas. If this temperature exceeds escape velocity, then surface layer evaporates. Evaporation proceeds for radii beyond:

Extreme-UV Photoevaporation Hollenbach 1994; Clarke et al. 2001 Alexander, Clarke & Pringle 2006

EUV versus FUV Photoevaporation Strong, but Works > 50 AU EUV Photoevap: Weak, but Works around 1 AU 1 AU 50 AU Hollenbach et al. 1994 Gorti & Hollenbach 2007

Far-UV Photoevaporation Solve full 2-D (=1+1D) structure of disk on-the-fly while doing disk evolution and compute from this the mass loss at each radius. Gorti & Hollenbach (2007) Gorti, Dullemond & Hollenbach (in prep)

Far-UV Photoevap: First Results

Far-UV Photoevap: First Results

Far-UV Photoevap: First Results

Far-UV Photoevap: First Results

‘Dead zone’ MRI can only work if the disk is sufficiently ionized. Cold outer disk (T<900K) is too cold to have MRI Cosmic rays can ionize disk a tiny bit, sufficient to drive MRI Cosmic rays penetrate only down to about 100 g/cm2. full penetration of cosmic rays partial penetration of cosmic rays

‘Dead zone’ Hot enough to ionize gas Only surface layer is ionized by cosmic rays Tenuous enough for cosmic rays Above dead zone: live zone of fixed  = 100 g/cm2. Only this layer has viscosity and can accrete.

Accumulation of mass in ‘dead zone’ Remember: Stationary continuity equation (for active layer only): For >0 we have mass loss from active layer (into dead zone)

Gravitational (in)stability If disk surface density exceeds a certain limit, then disk becomes gravitationally unstable. Toomre Q-parameter: For Q>2 the disk is stable For Q<2 the disk is gravitationally unstable Unstable disk: spiral waves, angular momentum transport, strong accretion!!

Gravitational (in)stability Spiral waves act as `viscosity’ Rice & Armitage