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

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

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Page 1: Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

Planet Formation

Topic:

Viscous accretion disks

Lecture by: C.P. Dullemond

Page 2: Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

shock

shock

The formation of a disk

• 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:

Page 3: Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

The formation of a disk

3-D Radiation-Hydro simulations of disk formation

Yorke, Bodenheimer & Laughlin 1993

Page 4: Planet Formation Topic: Viscous accretion disks 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

Page 5: Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

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!

Page 6: Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

Outward angular momentum transport

A B

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.

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.).

Page 7: Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

Molecular viscosity? No!

Problem: molecular viscosity is virtually zero

Reynolds number

Typical disk (at 1 AU): N=1x1014 cm-3, T=500 K, L=0.01AU

Assume (extremely simplified) H2 (1Ang)2.

L = length scale<u>= typical velocity = viscosity

Molecular viscosity:lfree = m.f.p. of molecule<uT>= velo of molecule

Page 8: Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

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:

Page 9: Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

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).

Page 10: Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

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)

Page 11: Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

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

Page 12: Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

Magneto-rotational instability (MRI)

Johansen & Klahr (2005); Brandenburg et al.

Page 13: Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

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)

Page 14: Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

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

Equation of hydrostatic equilibrium:

Page 15: Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

Shakura & Sunyaev model

Vertical structure

Because of our assumption (!) that T=const. we can write:

This has the solution:

with

A Gaussian!

Page 16: Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

Shakura & Sunyaev model

Define the surface density:

Radial structure

Integrate continuity equation over z:

(1)

Integrate radial momentum equation over z:

(2)

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...

Integrate tangential momentum equation over z:

(3)

Page 17: Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

Shakura & Sunyaev model

(2)

Let’s first look closer at the radial momentum equation:

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

Page 18: Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

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

Page 19: Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

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)

Page 20: Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

Shakura & Sunyaev model

Further on alpha-viscosity:

Here the vertical structure comes back into the radial structure equations!

So we obtain for the viscosity:

Page 21: Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

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.

Page 22: Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

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 theaccretion rate of the disk.

Page 23: Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

Shakura & Sunyaev model

Suppose we know that is a given power-law:

Ansatz: surface density is also a powerlaw:

Stationary continuity equation:

from which follows:

The radial velocity then becomes:

Proportionality constants are straightforward from here on...

Page 24: Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

Shakura & Sunyaev model

Examples:

Page 25: Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

Shakura & Sunyaev model

Examples:

Page 26: Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

Shakura & Sunyaev model

Examples:

Page 27: Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

Shakura & Sunyaev model

Examples:

Page 28: Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

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)

Page 29: Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

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!

Page 30: Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

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:

Page 31: Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

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:

Page 32: Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

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:

Page 33: Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

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:

Page 34: Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

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.

Page 35: Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

Shakura & Sunyaev model

Example: Dust power-law opacity

Remember from previous page:

Remember relation between and :

Remember from power-law solutions:

Page 36: Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

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

Page 37: Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

Non-stationary (spreading) disksGiven 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:

Page 38: Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

Non-stationary (spreading) disks

Time steps of 2x105 year

Lynden-Bell & Pringle (1974), Hartmann et al. (1998)

Page 39: Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

Formation & viscous spreading of disk

Page 40: Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

Formation & viscous spreading of disk

Page 41: Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

Formation & viscous spreading of disk

Page 42: Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

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

Page 43: Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

Formation & viscous spreading of diskA numerical model

Page 44: Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

Formation & viscous spreading of diskA numerical model

Page 45: Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

Formation & viscous spreading of diskA numerical model

Page 46: Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

Formation & viscous spreading of diskA numerical model

Page 47: Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

Formation & viscous spreading of diskA numerical model

Page 48: Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

Formation & viscous spreading of disk

Hueso & Guillot (2005)

Page 49: Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

Disk dispersal

Haisch et al. 2001

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?

Page 50: Planet Formation Topic: Viscous accretion disks Lecture by: C.P. Dullemond

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: