Planet Formation

Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

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

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

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!

A B Specific angular momentum for a Keplerian disk: 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. 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.).

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

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 The momentum equation for hydrodynamics is:

Now average over many eddy turnover-times, and use: and but (tensor!) 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

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 recently: • 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)

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 Magneto-rotational instability (MRI) (Also often called Balbus-Hawley instability) Highly simplified pictographic explanation: 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)

Equation of hydrostatic equilibrium: Shakura & Sunyaev model Vertical structure 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

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

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

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

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: Shakura & Sunyaev model Radial structure Our radial structure equations have now reduced to: with Missing piece: what is the value of ? Alpha-viscosity (Shakura & Sunyaev 1973)

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

If we know the temperature everywhere, we can readily solve these equations (time-dependent or stationary, whatever we like). Shakura & Sunyaev model Summary of radial structure equations: 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 The “local accretion rate” vr(r) Radial velocity can (will) be a function of r. And so is the surface density. Define now the local „accretion rate“ as the amount of gas flowing through a cylinder of radius r: In a steady-state situation this must be independent of r: where is then the accretion rate of the disk.

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

Shakura & Sunyaev model Examples:

Accretion rate is amount of matter per second that moves radially inward throught he disk. We finally obtain: Shakura & Sunyaev model Formulation in terms of accretion rate Working this out with previous formulae: (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!

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

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?

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

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: Shakura & Sunyaev model How do we determine the temperature?

Shakura & Sunyaev model To obtain full solutions, we must first have an expression for the Rosseland mean opacity. We can then express the midplane temperature as a function of the surface density. In the radial structure equations we can then eliminate the temperature in exchange for the surface density. We then have an equation entirely in terms of the surface density. We solve this equation.

Shakura & Sunyaev model Example: Dust power-law opacity Remember from previous page: Remember from power-law solutions: Remember relation between  and :

Non-stationary (spreading) disks • So far we assumed an infinitely large disk • In reality: disk has certain size • As most matter moves inward, some matter must absorb all the angular momentum • This leads to disk spreading: a small amount of outer disk matter moves outward

where we have defined with r1 a scaling radius and ts the viscous scaling time: Lynden-Bell & Pringle (1974), Hartmann et al. (1998) 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:

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

Formation & viscous spreading of disk

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

Formation & viscous spreading of disk A numerical model

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

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

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: Photoevaporation of disks (Very brief)

Planet Formation