------------------------
This is a simulation from Ken Tatematsu of a cluster of protostars at various stages of evolution still embedded in their natal cores. The outer edges of these cores coalesce into the 'atmosphere' of the protocluster core. To estimate e.g. the total mass available in the protocluster cloud for incorporation into stars, and hence the efficiency of star formation within the cloud, one must measure the total flux in this atmosphere. To study how the cloud becomes incorporated into the cores and further into the protostars one must have a good representation of the flux distribution on all scales. The smoothed model flux was measured at IRAM to be 9.4 Jy.
of a modestly bright debris disk at a distance of 12 pc around a sun-like star. The total emission is 10 mJy at this observing frequency, 345 GHz. The disk inner radius is 2 AU; the outer radius is 100 AU. The mass of the disk corresponds to roughly 0.4 lunar masses of dust, or 0.5 Earth masses of gas plus dust assuming a gas-to-dust ratio of 100. The interesting feature about this image for ACA simulation is the weak extended emission carrying the signatures of planetary influence on the dust and gas.
My simulation represents a "typical dusty" disk. One might expect to see of order a dozen of these. Lowering the flux by a factor of 10 would allow for many more detections, but our own solar system is much weaker than that.
Other details of the model are summarized below.
Disk inner and outer radius (in AU) = 2, 100
Dust Temp at 1 au (in Kelvins) = 350
Dust temp power-law index, q = 0.45
Lower limit to allowed temp in disk = 20 K
Surface density power-law index = 1.3
Inclination angle of disk (0=face-on) = 45 deg
Distance to source (in pc) = 12
Position angle of disk (0 = vertical= major axis along dec axis) =
90
Freq of observation in GHz = 345
Location and nature of disk surface features:
1.0 7. 2.0 : dark ring of amp zz, at radius xx AU, width yy AU
2.0 16. 4.0 : dark ring of amp zz, at radius xx AU, width yy AU
1.5 40. 5.0 45. : planet debris of amp zz, at radius xx AU, width yy
AU, PA
3.0 60. 9.0 155. : planet debris of amp zz, at radius xx AU, width
yy AU, PA
checking out the flux density in lee's debris disk. bjb. 2001aug21
the observing and disk geometry parameters:
distance = 12 pc
radius = 100 AU
inclination angle = 45 deg
frequency = 345 GHz
the total flux density in the image is 10.3 mJy.
the flux density in the central pixel is 240.2 uJy. this is almost
exactly what i get for the sun at 12 pc:
F* ~ 6 Teff Rs^2 / Dpc^2 uJy
where Teff is the effective temperature, Rs is the radius in solar
radii, and Dpc is the distance in parsecs. let Teff = 6000 K; Rs = 1;
Dpc = 12, then
F* ~ 250 uJy
so this is right...
assume that temperature, emissivity, and opacity don't vary in
the debris disk. then, the total emission is:
Fd = t e B O
where t = dust opacity, e = dust emissivity, B = dust brightness
O = solid angle of disk. given a disk of radius Rd at inclination
angle p, the solid angle is of course O ~ pi cos(p) Rd^2 / D^2.
the brightness is just the planck equation. the effective opacity
is t = t' / cos(p), where t' is the opacity that would be observed
if it were face-on (note that this causes the inclination angle
factors [cos(p)] to cancel out - as expected...). dust emissivity
is roughly .015 at 345 GHz (very rough, but good enough for this
kind of order-of-magnitude analysis). opacities vary from something
like 1E-3 (beta-pic) to 1E-7 (our own solar system's zodiacal dust).
so, given lee's disk geometry, and a pretty dusty disk (t' = 1E-3),
for a temperature of 50 K (the bulk of the emission comes from the
cold dust, because it occupies the bulk of the solid angle), you get:
F ~ 11.9 mJy
so, 10 mJy in the model is not so far from reality. but it *is*
still a pretty dusty system. a more 'typical' debris disk
dust opacity might be something like 1E-5, and slightly colder
temperatures (20 K), giving a total flux density more like 50 uJy
(factor of 100 less emission).
so, i'd say that this is a good model for a very dusty system. if
you want more typical systems, reduce the flux density numbers by
a factor of 10-100.