Draft version May 28, 2007
Preprint typeset using LATEX style emulateapj v. 2/19/04
NOTES ON THE KROLIK’S MODEL OF RADIATION PRESSURE IN AGNS
D.V. & M.E.
—
Draft version May 28, 2007
ABSTRACT
Dejan’s notes (02 April, 2007): I found a mistake in the Krolik’s model: there is one additional
equation describing a boundary condition for the gas specific angular momentum jin at the inner
radius rin .
Subject headings: —
1. DEJAN’S NOTES (02 APRIL 2007)
1.1. Error in the Krolik’s model
All the equations in the model are actually correct
(down to the used approximations), but there is one
equation missing: the boundary condition at rin describing the balance between gravity, rotation and radiation
pressure. Take equation 1 from the Krolik’s model and
write it for the point (r,z)=(rin ,0):
κ
2
),
(1)
F (Rin , 0) = rin Ω2 (rin )(1 − jin
c
where F (Rin , 0) is the total radial net flux at (rin ,0).
Radiative transfer in optically thick dust clouds dictates
balance between the flux entering radially the cloud at
(rin ,0) and the diffuse flux escaping radially out. The
resulting net flux F (Rin , 0) is zero, which means that
jin = 1.
(2)
This leads to rmax = rin in equation 10, j(r) = jin =
const. in equation 9, dE/dr = 0 in equation 19 and
E(r) = Ein in equation 20. That is, the whole model
collapsed into a point.
Krolik ignored the fact that dust radiative transfer dictates the inner dust tours radius. He talks a lot in §2.3
about various physical process that will influence this radius, but by ignoring this boundary condition he got a
freedom to set an arbitrary jin < 1. Indeed, this yields a
large net flux flowing through the tours, which also creates radiation pressure, but at the same time this yields
dust temperature at rin : T (rin , 0) À Tsublimation .
1.2. Radiation pressure on the cloud surface
These assumptions are valid within the cloud, at optical depths where the majority of incoming radiation from
the nucleus is absorbed. On the very surface of the cloud
(defined as τ ∼ layer), however, the diffuse approximation is not valid any more. In this surface at (rin ,0) the
radiation pressure force consists of two opposite forces:
outward force due to UV/optical radiation from the nucleus and inward force due to diffuse infrared radiation.
If the outward flux is F+ and inward F− , where F+ = F−
then the net force is:
κIR
κI R
κV
κV
F+ −
F− =
F+ (
− 1).
(3)
P =
c
c
c
κIR
This force vanishes for big grains where κV /κIR ∼ 1.
Small grains have κV /κIR > 1 and we can make an estimate if the radiation pressure force is bigger than gravity.
Combining the above equation with the nucleus luminosity LBH :
LBH
F+ =
(4)
2 ,
4πrin
Eddington luminosty LE :
LE = 4πcGMBH /κT ,
and gravity:
Fg =
GMBH
,
2
rin
(5)
(6)
yields the radiation pressure force:
P =
κIR κV
LBH
(
− 1)
Fg .
κT κIR
LE
(7)
According to numbers mentioned by Krolik,
κIR /κT ∼ 10 and LBH /LE ∼ 0.1. Since for small
grains κV /κIR > 1, the radiation pressure force is bigger
than gravity P > Fg in the cloud surface. This means
that only big grains can survive in the surface, where
”big” and ”small” is dictated by the spectral shape of
the nucleus radiation (I used index V in κV , but this
actually represents the peak wavelength of the nucleus
spectrum absorbed by the dust).
2. DEJAN’S NOTES (04 MAY 2007)
I looked in more detail at the Krolik’s model and concluded that all the approximations he introduced are
valid. For example, the real mid-IR dust opacity can
vary by a factor of 10, but in the limit of large optical
depths this does not change the Korlik’s results based on
the first order approximation of constant opacity.
His model is valid as long as the total net flux flowing through the dust cloud is a free parameter. And
this is the key mistake in his model! In §1.1 I already
talked about that, but here I elaborate it further. Here
we will work with a dusty slab. The non-radial flux flowing through the dusty tours cannot exceed this limiting
case.
2.1. Gray slab
In the limit of large optical depths we can use the diffusion approximation. This is what Krolik is also using.
If we want only first order approximations then we can
assume a constant dust opacity in the wavelength range
of interest (mid-IR). This is also what Korlik is doing.
But this means we are using the grey approximation and
2
V.D., M.E.
for that we can solve the problem analytically. The textbook solution (e.g. Mihalas “Stellar Atmospheres”) for
angular averaged intensity J(τ ) at the optical depth τ is
(notice that Krolik is using energy density E = 4πJ/c)
3
J(τ ) = J(0) − FT τ
(8)
4
where FT and J(0) are constants. Under these approximations the diffuse flux in two opposite directions is
F (τ ) = F+ (τ ) − F− (τ ) = J(τ )/3
3. OPACITY AVERAGES
Planck average of dust cross section:
Z
π
σP (T ) =
σλ Bλ (T )dλ
σSB T 4
Rosseland average:
σR (T ) = R
∂B/∂T
=R
∂Bλ
1
σλ · ∂T dλ
4σSB 3
π T
∂Bλ
1
σλ · ∂T
(12)
dλ
(13)
(9)
and
σSB 4
T (τ )
(10)
π
where σSB is the Stefan-Boltzmann’s constant and T is
temperature.
We know that the temperature range is from ∼ 1000K
on the illuminated side to ∼ 100K on the dark side. This
means that J(0) À J(τT ) in a slab of total optical depth
τT À 1. From these we see that the diffuse flux on the
illuminated side is F (τ ¿ τT ) À F (τT ), where F (τT ) is
the flux at the dark side. To total flux is equal to F (τT )
because τT À 1 ensures that no incoming radiation Fin
entering the dust cloud can emerge through. This means
that the net flux balance on the illuminated side is
J(τ ) =
Fin − F− (0) = F (τT )
(11)
where the diffuse back-radiation is F− (0) = F (0)/2.
Since F (τ ¿ τT ) À F (τT ), this yields the proof that
Fin À F (τT ).
Krolik’s equation 1 sets the absolute value for his net
flux. In his model the net flux F (τT ) is a free parameter and he hides it by introducing jin < 1 as a free
parameter. Without any justification he uses jin = 0.5
(i.e. Fin NOTÀ F (τT )) and he can do that because he
ignores the above described solution.
2.2. Dust temperature at the inner tours rim
Notice that in the above description the critical information was the temperature range (from 1000K to 100K)
which set the limit on the net flux. An alternate approach is to repeat what DDN do when they derive the
inner disk radius from dust sublimation Tsub boundary
condition. This radius is not a free parameter and it is set
by the balance between the emitted blackbody flux from
4
the inner dust edge σSB Tsub
and the flux from the black
2
hole LBH /4πrin . This is also ignored by Krolik. Notice
that in the most general terms the boundary condition is
NOT the dust sublimation, but flux balance at the inner
rim. We can use whatever temperature we want at that
radius and the Krolik’s model would still fail. From the
physical properties of AGN dust torii we know that this
temperature is dust sublimation, but sublimation is not
the key argument.
2.3. Conclusion
The key mistake by Krolik was to ignore the basic property of optically thick dust clouds: the net flux is small
compared to the incoming flux from the black hole! This
can be shown either by using the analytic solution to
the approximate model of gray dust or by simply setting
the flux balance at the inner torus radius where the dust
evaporates.
4. MOSHE (MAY 26, 2007)
4.1. Boundary Condition
Krolik assumes that the net bolometric flux at the dust
2
. However, Fe is
illuminated boundary is Fe = L/4πrin
the illuminating flux, and the net flux is smaller than that
because of the dust emission in the opposite direction.
The diffuse flux vanishes only in the case of a spherical
dust shell, and that’s the only case where the net flux is
Fe , thanks to symmetry: at any point P on the shell’s
inner surface the intensity on every incoming ray is equal
to that from across the inner cavity by the opposite ray
along the same path (both rays are at the same angle
from the local radius vector; see figure 1). This symmetry is broken in every other geometry. In particular, the
inner cavity of the AGN torus is bisected by the accretion disk, which cuts communication between diametrical
segments. In that case the net flux at the boundary is
the difference between the external illuminating flux Fe
and the local diffuse flux from the torus. For simplicity, consider gray opacity, as assumed in K07, and an
illuminated dusty slab with optical depth τ (figure 1).
The transmitted flux is then Fe e−τ . When τ >> 1, the
diffuse fluxes emanating from the bright and dark sides
are related to the corresponding dust temperatures via
Fdark /Fbright = (Tdark /Tbright )4 . Flux conservation,
Fe − Fbright = Fe e−τ + Fdark
(14)
implies that
1 − e−τ
Fbright
=
(15)
Fe
1 + (Tdark /Tbright )4
so that the net flux is only a fraction of the illuminating
flux:
e−τ + (Tdark /Tbright )4
Fe − Fbright
=
. 10−4 (16)
f=
Fe
1 + (Tdark /Tbright )4
NOTES: Krolik - AGN - radiation pressure
P
Fbright
Fdark
Fe
Fe e-τ
Fe
Fig. 1.— Left: Symmetry in spherical geometry. Right: Slab
boundaries
3
same effect also reduces the radiation pressure asymmetry between radial and vertical directions, which is
the foundation of the K07 model. We should produce a
LELUYA figure similar to the one for IRC+10011. The
numerical calculation should involve a simple geometry
similar to what Krolik starts with. It would be good to
produce a curve that shows along some directions the radial and vertical components of Prad normalized to the
value Krolik claims on the surface, to show how much
smaller they (hopefully) are, in addition to a contour
map that shows how small the actual asymmetry is.
2
REFERENCES
Krolik, J. H. (2007) astro-ph/0702396
1
.0
1
-0
- 0.07
- 0.10
-0
.04
Switching from gray opacity to the optical properties
of actual dust modifies these results, but the conclusion
that f ¿ 1 remainsless
valid
whenfew
the percents
slab optical
is
than
of depth
the radial
large. We have run exact radiative transfer calculations
with the code DUSTY for slabs illuminated by typical
AGN spectrum to a temperature Tbright = 1500 K, corresponding to dust sublimation. The slab opacity corresponds to standard Galactic interstellar dust and its
optical depth is τV at visual. At τV = 100 the net flux
fraction is only f = 0.1 (Tdark is 390 K). This fraction decreases further to f = 0.07 at τV = 200 (Tdark = 350 K)
and f = 0.03 at τV = 500 (Tdark = 290 K).
The reason f is smaller than for gray opacity is that
the slab becomes optically
thin
long wavelengths,
a map of
theatratio
of tangentialand
to raradiation “leakage” at those wavelengths reduces the contrast between the bright and dark sides. But this very
Z
4.2. Realistic Dust
- 0.04
0.02
The last inequality is for the K07 estimates Tbright ∼
1000 K and Tdark ∼ 100 K, and holds as long as τ ≥ 10.
diminishes
Instead, K07 assumed
f = 1. fast with radial distance,
Fθ / Fr
01
- 0.
0
0
1
X
2