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

2. 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

3. 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!

4. 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.).

5. 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.

6. 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:

7. 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).

8. 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)

9. 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

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

11. 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)

12. 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

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

14. 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)

15. 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

16. 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

17. 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)

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

19. 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.

20. 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...

21. Shakura & Sunyaev model
Examples:

22. Shakura & Sunyaev model
Examples:

23. Shakura & Sunyaev model
Examples:

24. Shakura & Sunyaev model
Examples:

25. 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)

26. 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!

27. 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:

28. 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:

29. 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:

30. 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:

31. Viscous heating or irradiation?
T Tauri star

32. Viscous heating or irradiation?
Herbig Ae star

33. 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.

34. 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

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

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

37. 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

38. 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:

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

40. 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

41. Disk formation and spreading
Molecular cloud
core:

42. Disk formation and spreading
Molecular cloud
core:

43. Disk formation and spreading
Molecular cloud
core:

44. Disk formation and spreading
Molecular cloud
core:

45. Disk formation and spreading
Molecular cloud
core:

46. 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:

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

48. Formation & viscous spreading of disk
A numerical model

49. Formation & viscous spreading of disk
A numerical model

50. Formation & viscous spreading of disk
A numerical model

51. Formation & viscous spreading of disk
A numerical model

52. Formation & viscous spreading of disk
A numerical model

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

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

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

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

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

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

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

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

61. 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

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

63. Geometry: adding vertical B-field

64. 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:

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

66. 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):

67. 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

68. 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:

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

70. 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

71. 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)

72. Far-UV Photoevap: First Results

73. Far-UV Photoevap: First Results

74. Far-UV Photoevap: First Results

75. Far-UV Photoevap: First Results

76. ‘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

77. ‘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.

78. 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)

79. 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!!

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