Disk Structure and Evolution
(the so-called a model of disk viscosity)
Ge/Ay 133
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.
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.
CO
Can only “see” Keplerian v
TMB (K)
Dent et al. 2005,
JCMT
vLSR (km/s)
The CO line shape is
sensitive to Rdisk , Mstar,,
Inc.; but the pressure
support is highly subKeplerian & similar to
vdoppler in the outer disk.
M. Simon et al.
2001, PdBI
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?
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
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
What is viscosity?
y
Imagine applying a linear stress field to a
fluid (by placing it between a rotating
and fixed plate, for example):
x
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) ~ 10-3 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.
Viscous Disks
y
If viscosity is characterized as
velocity × a length scale then the
time scale for a fluid to respond
should be
x
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
Plausibly, n~vconvection L ~ 104 cm/s L
or
n ~ 1016 cm2/s
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 BalbusHawley instability (more later when we look into planet migration). For this to
work the disk must be partially ionized.
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?
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).
Hartmann et al. 1998, ApJ 495, 385
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.
Evolution of MMSN (gas) under viscous dissipation:
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
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…