Here is one HST image of a
beautiful edge-on dust disk surrounding Beta Pictoris
(WFPC2 image obtained by Chris Burrows's group):
There are also newer images showing the disk even better:
the STIS image by Sally Heap's group
The vertical dust density profile.
We use the following formula for d(tau)/ds, optical thickness per
unit length:
d(tau)/ds = const* tau(r)/w *exp(-(|z-z_c|/w)^p)
where w=width of the profile in vertical (z) direction,
z_c = the coordinate of the maximum dust density, p=power law
determining the shape of the vertical profile, and const=constant chosen
so that the vertical disk opacity is indeed tau(r).
We used p=0.7, which corresponds to a central sharp core of density
distribution (super-exponential). This is also a departure from previous
models, the ancient ones (A+B+P 1989 had p=2, Kalas & Jewitt 1995 had p=1).
Beta Pic disk has a dense and thin midplane region (w=1 in the
linear plot below):
The disk width changes with radius in the following manner:
w(x) = 0.055 r_m * x^0.75
which means that the disk is thicker outside than inside but
not flaring, and has thickness/radius ratio~0.06 at 120 AU, which is similar to
that of the solar nebula scale height (at Jupiter distance).
Rather thin.
The disk is inclined by a small angle (we used here i=1.3 degrees) to the plane including the line of sight. (The line of nodes nearly perpendicular to the l.o.s.) The inclination gives the values of z_c throughout the model disk.
Scattering phase function of the dust.
We use an analytical
approximation to empirical f(theta) for cometary dust compiled from
measurements of solar
system comets. The formula:
f(theta)=0.3*(.2+.5*theta)^(-3.)+1.4*(theta/3.3)^4+0.2.
Importantly, alternative phase functions, most notably the
Zodiacal Light empirical phase function and some theoretical Mie
scatterring functions will be INAPPROPRIATE due to too large a peak
in the forward scattering direction. [ZL function from
Lamy Perrin 1991 Orig&EvoInterplDust, 163) makes isophotes climb higher
instead of going more horizontally
near the vertical axis. Other ZL function are less peaked.
Of course, we know that comet-like planetesimals are good candidates for
parent bodies of the dust in Beta Pic, so this is a nice consistent result.
(Notice that
Halley comet dust gives the best fit to the 10 micron silicate emission
feature of the Beta Pic disk).
are reasonably well reproduced by our simple model
throughout the image (coloring worked so-so, suggesting larger
differences in normalization than they really are):
This isophotal shape agreement
is much harder to get than just the agreement of the "midplane
brightness" profiles along the spine of the disk:
Notice the power-law break at 100 AU mentioned already by A+B+P (1989)
but first solidly proven by Golimowski, Durance & Clampin (1993)
based on adaptive-optics ground-based coronography.
Model results:
The isophotes in the STIS image (512 AU by 160 AU):
The solid line is the NE extension of the disk (l.h.s. on images)
and the dotted line is the SW side, and the dash-dotted lines are the
theoretical model.
This modeling was done by trial and error. In the future we will try to
automate it and join with:
Work ahead: modeling of disk asymmetries...
last modified: April 1998