The main objective of the research program is twofold: (1) to develop a computer code to solve the problem of scattering, absorption and emission of photons by dust grains in a dusty medium with 2 dimensional disk geometry, and (2) to study the various physical and geometrical effects of 2 dimensional radiation transport on the thermal structure and radiation field. These tasks were accomplished and are briefly summarized. The method for solving the radiation transport problem in disk geometry is a generalization of the quasi-diffusion method (QDM) previously developed by the author

Chun, Ming Leung

English

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RADIATION TRANSPORT IN Dcar IN DISK GEXMETRY 
FINAL TECHNICAL REPORT 
(bas&-CR-180995)  P A D I A T I C 6  I E A I S E O E T  III 887-24 157 
rLST Ill DISK GECBETBY €inal ‘Iechnical 
b€Fort, 15 Apr. 1984 - 16 Cct, 1986 
(6ensselaer P c l g t e c h n i c  I n s t . )  9 p Avail: Unclas  
YlIS HC AOZ/EF 401 CSCL 20C G3/70 0076743 
Principal Investigator: Chun Ming Leung 
Period Covered: 4/15/1984 - 10/16/1986 
NASA Grant Number: NAGW-577 
. 
t 
Rensselaer Polytechnic Institute 
Troy, New York 12180-3590 
1 
https://ntrs.nasa.gov/search.jsp?R=19870014724 2020-03-20T10:58:46+00:00Z
The min objective of the research program is twofold: 
a) to develop a computer code to solve the problem of scattering. absorption 
and emission of photons by dust grains in a dusty medium with 
two-dimensional (2-D) disk geometry, and 
b) to study the various physical and geometrical effects of 2-D radiation 
transport on the thermal structure and radiation field. 
As indicated below these tasks have been accomplished. The method for solving 
the radiation transport problem in disk geometry is a generalization of the 
quasi-diffusion method (QDM) previously developed by the Principal 
Investigator. 
In this report we shall only give a brief summary of the major results 
of the research. Detailed discussions can be found in publications resulted 
from this project (see listing at the end of this report). These results can 
be divided into four areas as outlined below 
(1) Radiation Transport Code for Disk Geometry 
To check the feasibility of applying the QDM of Leung (1975, 1976) to 
2-D disk geometries. we first applied the QDM to one-dimensional (1-D) 
cylindrical geometry which differs from the 2-D disk geometry only in having 
no transport in the z-direction. Both geometries involve 2 angles. Results 
indicate that the QDM in this case is stable and rapidly convergent. The 
result of this effort is a computer code which can solve the 1-D radiation 
transport problem in three geometries: planar, spherical and cylindrical. The 
documentation for this code, a generalization of the previous work by Spagna 
and Leung (1983). is being written up for publication (Egan. Leung and Spagna 
1987). Next we generalized the QDM to 2-D geometry and developed a computer 
code to solve the problem of radiation transport in a dusty medium with disk 
geometry (Spagna and Leung 1987a). The code can handle arbitrary dust density 
distribution and other specified local energy source of grain heating (e.g.. 
viscosity, collision with gas component). 
In 2-1) disk geometry, the basic idea of the QDM. a differential equation 
method, is to treat the radiative transfer problem as a boundary value 
problem. The equation of radiation transport is cast into a quasi-diffusion 
form by combining its zeroth and first angular moment equations through the 
introduction of several anisotropy factors (components of the anisotropy 
tensor) which describes the anisotropy of the radiation field. By first 
assuming the radiation field to be isotropic (diffusion approximation), the 
system of nonlinear moment and energy-balance equations are linearized and 
solved to determine the mean intensity and the grain temperature distribution, 
i.e.. the source function. From the source function, we then solve for the 
angular distribution of the radiation field using a ray-tracing method, to 
update the anisotropy factors. The coupled nonlinear moment and energy 
equations are solved by the Newton-Raphson method. Physically, the success of 
the QDM rests upon the fact that the temperature distribution depends strongly 
on the radiation energy density and only weakly on the anisotropy of the 
radiation field. This weak coupling between the thermal structure and 
radiation field anisotropy allows the problem to be separated into two 
2 
distinct steps: solution of the moment and energy-balance equations for a 
given radiation field anisotropy, and solution of the ray equations to 
determine the angular distribution of the radiation field for a given source 
function. 
We have used this computer code to study the physics of 2-D radiative 
transfer in various simple problems (Spagna 1986; Spagna and Leung 1985, 1986. 
1987b). as outlined below. 
(2) Effects of Source Geometry 
Under the assumption of gray (frequency-independent) opacity, we have 
compared results from 2-D disk models with those of 1-D geometries (e.g.. 
slab. sphere and infinite cylinder), and have established the range of 
conditions under which approximation by l-D geometries may be valid (Spagna 
and Leung 1984): 
i) thin-disk approximation by l-D slab geometry when Z << R; 
ii) long-cylinder approximation by l-D cylindrical geometry when 2 >> R; 
iii) thick-disk approximation by l-D spherical geometry when Z 2: R. 
Results indicate that in general the dominant geometry is the one describing 
the surface closest in optical depth. Here R is the disk radius and 2 is the 
half-thickness of the disk. Identifying these conditions allows us to utilize 
existing codes for 1-D geometries in future calculations when these conditions 
are met. In addition. we have investigated the feasibility of the following 
hybrid approach which is intermediate between an exact solution and one based 
on the diffusion approximation. Instead of solving the ray equations to 
determine the anisotropy factors self-consistently. we simply use some 
analytic expressions for these factors, the choice of expressions being guided 
by results from exact solutions. This drastically reduces the computational 
effort since for 2-D disk geometry the requirement of computer time is 
governed by the solution of the ray equations. For some applications, such as 
those discussed below, this approximate approach gives results which are 
fairly accurate (to within 20% of the exact solution). 
(3) Application to Externally Heated Dust Clouds 
We have studied the various physical and geometrical effects of 2-D 
radiative transfer on the thermal structure and radiation field anisotropy. In 
particular, we have computed a set of nongray models (using realistic grain 
opacities) for interstellar dust clouds which are heated externally by the 
ambient interstellar radiation field. To illustrate one interesting result of 
this study, here we show results from models with spherical and disk 
geometries. Both models have the same radius (taken to be 1 parsec) and 
central optical depth (equal to 100 at 0.55 p), the disk model having a 1:l 
aspect ratio (ratio of radius to half-thickness. R:Z). While the dust 
temperature distributions in the two cases are very similar as shown in Figure 
1. the excess flux for the disk model depends sensitively on the viewing angle 
0 (see Figure 2). The excess f l u  is defined as the emergent flux minus the 
flux from the ambient radiation as measured by an observer when the cloud is 
not resolved. The results in Figure 2 are somewhat unexpected. For clouds 
3 
which are unresolved. one would naively expect. since the thermal emission is 
isotropic in the neighborhood of an emitting grain and the emission (in the 
far infrared) is optically thin. that the observed flux spectrum should be 
characteristic only of the dust temperature and independent of viewing angle. 
However, because of the lack of spherical symmetry in disk geometry and the 
resulting radiation anisotropy, this is not the case for disk geometry, as 
shown in Figure 2. 
There are essentially two contributions to the variation in observed 
flux with viewing angle. The first is a purely geometric projection effect. 
For a disk of 
brightness I. 
or 2) depends 
radius R. half-thickness Z. and having a uniform surface 
the observed f l u  as seen by an observer at distance D (D 
on 8 as 
2 I = (T R case + 4 RZ sine) - . 
D2 Fdi sk 
This implies that for a disk with a 1:l aspect ratio, the observed flux 
larger seen edge-on than face-on and it reaches a m i m u m  when 8 'v 520. 
>> R 
is 
Compared to a sphere with the same uniform surface brightness and radius, the 
ratio of observed fluxes for the two geometries is just given by the ratio of 
their projected areas as seen by the distant observer 
/F = (T R" case + 4 RZ sine) T R ~ .  Fdisk sphere 
which indicates that for R = Z. Fdisk 2 Fsphere for all viewing angles. The 
second contribution to the variation comes from radiative transfer effects of 
disk geometry, which affect the radiation field anisotropy, the dust 
temperature distributions, and hence the surface brightness. For example. for 
l-D spherical geometry, energy transport occurs only along a radial direction 
so that the condition for flux conservation takes a simple form: r%(r) = 
constant, where F is the frequency integrated net flux. For 2-D disk geometry, 
the flux conservation equation is more complicated since there is energy 
transport in both r and z directions and in general F(r,z) depends on both r 
and z. This will significantly affect the radiation field anisotropy. The 
relative importance of these two contributions depends on such factors as the 
degree of disk flattening, the cloud optical depth, and the dust density 
distribution. The importance of the second contribution is supported by the 
fact that the same trend (though less drastic) as indicated in Figure 2 still 
persists even when the projection effect is removed by defining an '*equivalent 
flux" (which. in the limit of uniform surface brightness, is independent of 
viewing angle). 
An important consequence of this variation in excess flux with viewing 
angle is that it implies a large uncertainty in estimating the radiating dust 
mass for disk-shaped clouds which are unresolved. In particular, if spherical, 
geometry is incorrectly assumed, the estimated dust mass may easiLy be off by 
ouer an order of magnitude. Compared to externally heated clouds, the 
radiation field in clouds with embedded heat source is expected to be more 
4 
a 
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c3 
w 
1 
1 
.. 
I- 
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l- 
5 
anisotropic. This implies that the effects of disk geometry on the emergent 
flux as a function of viewing angle will be even more drastic. Other physical 
effects of radiation transport in disk geometry and further observational 
implications are discussed in Spa- and Leung (1986, 1987b). 
(4) Limb-Brightening Effects in Dark Globules 
We have studied the problem of infrared emission from dark globules 
(e.g.. B335, B227) which are nearby dense interstellar clouds that show up as 
dark patches of obscuration against the general background of stars. A recent 
review on this subject is given by Leung (1985). If a dark globule contains no 
embedded heat source and is only heated externally by the interstellar 
radiation field, it should exhibit limb-brightening in the far infrared 
(Spencer and Leung 1978). On the other hand, a globule which is internally 
heated (e.g.. by an embedded protostellar object) should exhibit 
limb-darkening. If the embedded heat source is not strong enough. there may 
still be limb-brightening. By determining the ratio of intensities at the limb 
and core, i.e., the limb-brightening ratio (LBR), one can estimate the 
luminosity of the embedded source. Using realistic cloud models and assuming 
spherical geometry, we determined these criteria more quantitatively and 
arrived at the following conclusions: 
a) Limb-brightening effects in the far infrared could be used to set an upper 
limit on the luminosity of a central heat source embedded in a dark 
globule. 
which LBR exceeds 1.2) is related linearly to the luminosity of the central 
heat source. 
c) In the absence of a central heat source of significant luminosity, the LBR 
is related to the optical depth T by a power-law relation 
b) The threshold wavelength for limb-brightening (the longest wavelength for 
k Z 0.3 k LBR a T ,  
in the wavelength region 120 p ( h ( 240 p where thermal emission from 
grains peaks (Figure 3). 
Thus limb-brightening effect in globules can be used as a diagnostic tool for 
probing these objects and any undetected embedded sources. In particular, the 
last conclusion implies that we now have an observational probe for 
determining the far infrared properties of dust grains in dark globules, i.e.. 
by observing the LBR at various wavelengths within the specified spectral 
range, the power-la relation can be exploited to determine the emissivity lau, 
of the dust grains in the fur infrared. However, to determine the LBR 
accurately, observation at high angular resolution is required. Detailed 
discussions and observational implications can be found in Leung. Dubisch and 
O'Brien (1987). 
6 
L 
9 
L 
B 
R 
0 
100 pm 
0 120 pm 
140 pm 
0 l5Opm 
A 180 pm 
A 190 pm 
X 200 pm 
235 pm 
- 300 pm 
400 pm 
0.6 T 
e. 
X 
I 0 q. . me. ; 
' e 0 '  I I I I I 
-1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 
Log(TAU) 
FIGURE 3 Limb-brightening ratio (LBR) vs optical depth (TAU) at 
various wavelengths for a model of dark globule. For wavelengths 
(120 - 2 4 0  pm) at which thermal emission from grains mostly 
occurs, the quantities LBR and TAU satisfy a power-law relation 
which can be exploited to determine observationally the 
emissivity law (opacity vs wavelength) in the far infrared of 
dust grains in dark globules. 
7 
0 
Egan. M. P.. Leung. C. M.. and Spagna. G. F.. Jr. 1987, in preparation, to be 
submitted to Comp. Phys. Comm. 
Leung. C. M. 1975, A p .  I.. 199. 340. 
1976. I. Quant. Spectrosc. Rad. Tranf . .  16. 559. 
1985, Protostars and Planets. 11. ed.. D. C. Black and 
M. S. Matthews, Univ. of Arizona Press. 
submitted to A p .  1. 
Leung. C. M.. Dubisch. R., and O'Brien, E. 1987, in preparation, to be 
Spagna. G. F.. Jr. 1986. Ph. D. Thesis. RPI. 
Spagna, G. F., Jr.. and Leung. C. M. 1983. Comp. Phys. Corn. ,  28, 337 
1984, Bull. A. A .  S.. 16, 961 
1985, Icarus. 61. 27. 
1986. poster paper presented at the "Summer School on Interstellar 
Processes'' in Wyoming, July 1986. 
1987a. J .  Quant. Spectrosc. Rnd. Tranf., to appear. 
1987b. in preparation. 
Spencer, R. G.. and Leung, C. M. 1978, A p .  1.. 222, 140. 
a 
WORK PARTULLY SWRRTED BY NASA GRANT NAGV-577 
A. 
B. 
C. 
1. 
2. 
3. 
D. 
1. 
2. 
3. 
4. 
5. 
6. 
E. 
Paper Published 
Spagna. G. F.. Jr.. and Leung. C. M.. "Radiative Transfer in Disk 
Geometry: Emergent Intensity from Coo1 Gray Disks.", Icarus. 61. 27 
(1985). 
Paper in Press 
Spagna. G. F.. Jr.. and Leung, C. M.. "Numerical Solution of the 
Radiation Transport Equation in Disk Geometry.", to appear in the 
Journal of QuUntttatiUe Spectroscopy and Radiative l'ransfer. 
Papers in Preparation 
Egan. M. P., Leung. C. M.. and Spagna. G. F., Jr.. "CSDUS3: A Radiation 
Transport Code for a Dusty Medium with 1-D Planar, Spherical or 
Cylindrical Geometry.". to be submitted to Compter Physics 
Communications. 
Leung, C. M., Dubisch. R., and O'Brien. E., "On the Use of 
Limb-Brightening Effects to Probe Isolated Dark Globules.". to be 
submitted to the Astrophysical Journal. 
Spagna. G. F., Jr.. and Leung. C. M., "Radiation Transport in Disk 
Geometry. I. Application to Externally Heated Interstellar Clouds.". to 
be submitted to the Astrophysical Journal. 
Contributed PaperdTalks Resented in MS HeetingdOther 
Gnferences 
Spagna, G. F.. Jr., and Leung. C. M.. "Radiative Transport in Disk 
Geometry: Thermal Structure of Dust Clouds.", W I L L .  Amer. Astron. SOC., 
14. 968 (1982). 
Spgna. 2. F.. Jr.. and Leung. C. M.. "Radiative Transfer in Disk 
Geometry: Emergent Intensity.". BULL. Amer. Astron. SOC., 15. 930 
(1983). 
Spagna, G. F.. Jr.. and Leung. C. M.. "Effects of Source Geometry on 
Continuum Radiation Transport.", Bull. Amer. Astron. SOC., 16, 961 
(1984). 
Spagna. G. F., Jr.. "Transport of Continuum Radiation in Disk 
Geometry.". Bull. Amer. Astron. SOC., 17. 847 (1985). 
Dubisch, R., and Leung, C. M., "Modeling of Infrared Emission from h r k  
Clouds.". talk presented in the ASNY (Astronomical Society of New York) 
meeting in October 1985. 
Spagna. G. F., Jr., and Leung, C. M., "Modeling of Infrared Flux Spectra 
from Disk-Shaped Interstellar Dust Clouds.". poster paper presented in 
the "Summer School on Interstellar Processes.'' held in Wyoming in July 
1986. 
Doctoral Dissertation Completed 
Spagna. G. F., Jr.. "Numerical Calculations of Radiation Transport in a 
Dusty Medium in Disk Geometry.", unpublished Ph. D. Thesis, RPI. May 
1986. 
9 


Radiation transport in dust in disk geometry

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