1. Disk Structure and Evolution (the so-called a model of disk viscosity)
Ge/Ay 133

2. Recapitulation of passive disk structure equations. I.
Equation for hydrostatic equilibrium using only stellar gravity.
For an ideal gas
where c is the sound speed (c2 = RT).
Solving yields the scale height, H,
and mass surface density, S, where the r in the equation above right is taken to mean the density at the disk midplane.

3. Recapitulation of passive disk structure equations. II.
In the previous analysis we did not consider that there is a radial pressure gradient, which exerts an acceleration since pressure = force/unit area. Thus, a(pressure)= dPA/m=dPA/rAdr. Balancing gravity, centripetal acceleration, and pressure gives:
F=PA A dr
Thus, the gas in passive disks moves at slightly sub-Keplerian speeds.

4. CO Can only “see” Keplerian v
Dent et al. 2005,
JCMT
TMB (K)
vLSR (km/s)
The CO line shape is
sensitive to Rdisk , Mstar,, Inc.; but the pressure support is highly sub-Keplerian & similar to vdoppler in the outer disk.
M. Simon et al. 2001, PdBI

5. What do other obs. tell us? Radial structure of the MMSN:
Mass surface density varies as r-3/2 in this model, what do aperture synthesis observations of circumstellar disks have to say?

6. Mm-interferometry observations of disks. I. Overview
If you ASSUME the temperature and mass surface density vary as power laws of radius, and that the mass opacity coefficient varies as a power of the frequency, analytical fits to resolved mm-wave aperture synthesis data can directly constrain the exponents.
Andrews & Williams 2007, ApJ 659, 705

7. Mm-interferometry observations of disks. II. Statistical Results
Mass surface density seems flatter than that from the MMSN, while the temperature distribution (at the midplane) is intermediate between that expected for flat and flared disks. Dust growth and settling? The mass surface density is highly uncertain, both due to materials properties and the unconstrained gas:dust ratio.
Andrews & Williams 2007, ApJ 659, 705

8. What is viscosity? x y
Imagine applying a linear stress field to a fluid (by placing it between a rotating and fixed plate, for example):
Velocity, u
For Newtonian fluids, the shear stress, t, is
t = m(u/y) and n = m/r
where m is the dynamic viscosity, n is the kinematic viscosity and r is the density
Units? m2/s or cm2/s (CGS= 1 Stokes), that is, a velocity × a length scale.
In turbulent fluids, the transport of heat and energy is NOT driven by molecular viscosity, but by eddies. For example, for water:
H2O(1) ~ Pa-s (dynamic viscosity) Ocean ~ 107 Pa-s (from transport)
Similarly, in circumstellar disks the molecular gas viscosity is many orders of magnitude too small to account for the observed transport, some other means of generating viscosity must be found.

9. Viscous Disks
If viscosity is characterized as velocity × a length scale then the time scale for a fluid to respond should be x y
Velocity, u
In circumstellar disks the molecular gas viscosity is many orders of magnitude too small to account for the observed transport, some other means of generating viscosity must be found. What, roughly, is required in a turbulent model?
Time scale ~ 105 – 106 years at a few tens of AU n ~ 1016 cm2/s
Plausibly, n~vconvection L ~ 104 cm/s L or L(eddy) ~ 1012 cm etc.
Source of this viscosity? Opacity from dust? Can work in special circumstances if k is temperature dependent. Most popular culprit is magnetic fields via the Balbus- Hawley instability (more later when we look into planet migration). For this to work the disk must be partially ionized.

10. The so-called alpha disk model:
Assume that the sound speed characterizes the velocity associated with the viscosity and the gas scale height the length scale, multiplied by a dimensionless parameter a.
Values of a near 10-2 are needed to fit the observables (more in just a bit). What does this imply about the radial structure of such disks, and do these properties correlate with the fits derived from mm-interferometry?

11. Radial evolution of a constant a disk:
Time evolution of the mass surface density subjected to a point mass M.
The alpha model requires that the viscosity be a power law function of radius (for constant a).
Solving the eq. for S yields a characteristic time ts, where R1 is the location at which 40% of the mass lies inside R1. Gas within Rt moves inward. Does nRg make sense? Note that the fractional ionization by cosmic rays goes like n-1/2, where n=gas density, and so goes up with R! This should help drive the Balbus- Hawley instability.
Hartmann et al. 1998, ApJ 495, 385

12. Evolution of MMSN (gas) under viscous dissipation:

13. Do a disks correspond to observations? I.
Isella et al. (2009)
The SED and mass surface density profiles for INDIVIDUAL disks can certainly be well fit with the a disk model. However, when statistical samples of disks are studied, the scatter is large!
Andrews & Williams 2007, ApJ 659, 705

14. Do a disks correspond to observations? II.
Andrews & Williams 2007, ApJ 659, 705
The expected trends are likely present, but clearly the actual situation is far more complex than a constant a model predicts. Nevertheless, since we have no analytical physics-based models of disk transport, it is difficult to determine what sort of a(R,z, t) is appropriate.
What about the dust evolution? We’ll start looking into this next time…