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Radiation Transport
1 I
 nˆ  I     I
c t
Current astrophysical research
using radiation transport
• Interstellar medium:
–
–
–
–
PN evolution
Photoevaporation of cometary knots
Evolution of proplyds
HII formation and evolution
• Intergalactic medium
NOAO/AURA (Abell 39)
– Photoevaporation of cosmological mini-haloes
– Cosmological reionization
Nakamoto 2001
STScI-PRC1994-24b
STScI-PRC1995-45a
STScI-PRC1996-13a
Radiation transport equation:
The specific intensity
angle.
I v is the energy flux per unit frequency per unit solid
dE  I v dt dAdd
The transport equation is then expressed in terms of the specific intensity:
1 I
 nˆ  I     I
c t
Where v is the volume emissivity and  v is the absorption coefficient.
In the absence of emitters or absorbers, the radiation equation becomes:
I
DI v
 cnˆ  I  0 
0
t
Dt
The specific intensity is constant along a given ray or characteristic
More generally though, along a given characteristic we have:
dI
1 I I

    I  v  S v  I v
c t
s
d
Where we have used the quasi-static approximation and S v 
v
and d   v ds

Source function
• If the source function is known then the transport equation can be

solved:
  0 
   
I v    I v  0 e
  Sv  e
d 
0
• Typically the source function (and the optical depth) depend on the
chemistry of the material (temperature, degree of ionization) as well
as the distribution of the material (density). And the chemistry is
usually dependent on the radiation field. However in thermal
absorption and emission, the source function is just the Planck
function.
Sv  Bv T 
• More typically in astrophysical plasmas, we have something closer
to pure isotropic scattering in which the source function is just the
angle averaged specific intensity. S  J  1 I d
v
v
v
4 
• But now the transport equation becomes an integro-differential
dI v
1
equation:
d

I d  I

4
v
v
Specific intensity

I r , nˆ, t 
Difficulties with calculating specific intensity
• Specific intensity is a function of position, time, and direction as well
as frequency
• Generally has a non-local nature (hard to calculate in parallel)
• Feedback between radiation field and material
Simplifying factors
• Specific intensity often dominated by point sources in which the
specific intensity is mono-directional.
• Often only need to calculate the specific intensity at a few
frequencies around ionization potentials of HI, HeI…
• Diffuse field tends to be nearly isotropic
• Radiation field quickly adjusts to changes in source terms – quasistatic approximation
Computational Methods
•
•
•
•
•
•
Lambda iteration
Frequency decomposition
Moment methods
Short Characteristics
Hybrid Characteristics
Multiple Wave Fronts
Λ Iteration
dI v
1

I v d  I v

d 4
• Integro-differential equation can be solved by first
calculating I v using the previous value of J v and
then using this to calculate the new value of J v and
then repeating this process until convergence.
• Typically, scattering will be isotropic, but rarely will it be
coherent.
–
In HII regions, absorptions are always from the ground state,
but recombinations can occur to any excited state followed by a
radiative cascade back to the ground state. As a result, the
source function for each frequency will depend not only on the
mean intensity at that frequency, but on the mean intensity over
a wide range of frequencies.
Local averaging of narrow frequency bands
• In most astrophysical plasmas we are only interested in modeling
the interaction of the radiation field with hydrogen and helium. As a
result we are only interested in calculating the specific intensity at
frequencies capable of ionizing HI, HeI, and HeII.
• Typically the incident radiation field can be integrated over the cross
sections so that the optical depth and radiation field at the ionization
potentials of HI, HeI, and HeII enables accurate calculations of
ionization rates.
• Typically the scattered field will contain contributions capable of
further ionization from the recombinations of HI, HeI, and HeII. As
the temperatures in diffuse astrophysical plasmas are often ~2eV,
the scattered photons from recombination will typically have
energies within a few eV of the ionization potentials – so the
scattered field will contain rather narrow bands.
• Three other bound-bound transitions of He are capable of further
ionization, but these are also confined to a narrow frequency band
with a small doppler width.
Moment methods
• For fields that are fairly isotropic (diffuse fields), the
specific intensity can be approximated by its various
moments.
1
1




J

I
d

E

I
d

 
 






4
c


 i 1

1
i
i
i
 I n d  or  F   I n d 
 H 
4
c




 K ij  1  I n i n j d
 P ij  1  I n i n j d



 



4
c


• Then taking the first two moments of the transfer
equation in the quasi-static approximation we arrive at:
4
 i F 
   E
c
ij
i
 j P    F
i
• But now we need some way to close the moment
equations.
ij
ij
– Eddington factor closure: P  f E
– Diffusion approximation:
where f i is given by:
F i  f E
i

 E
f   R R where R 
 E
i
 1 R2
1
1   3  45 for R  1
 R    coth R     
R
R 1 1

for R  1
 R R2
Equations of Radiation Hydro Dynamics
D

   v  0
Dt


Dv
1

 p   F F
Dt
c
De

    c E E  4P B  p  v
Dt   
D E

      F  v : P  4P B  c E E
Dt   


 D F
1
     P   F F
2
c Dt   
c

1
 F     F  d
F0

1
 P     B d
B0

1
 E     E  d
E0
Point Sources
• Due to the linear nature of the specific intensity, the
contribution from point sources can be separated from
that due to diffuse scattering.
• The diffuse field is often fairly isotropic and can be
modeled using some form of moment method.
• Solving for the direct field from the point source(s) then
reduces to solving the transport equation with no source
terms, which reduces to calculating the optical depth
from the point source(s) to any given cell. This can be
done by some form of ray tracing – short or long
characteristics.
Short vs. Long Characteristics
•
When dealing with point sources, we need to
calculate the optical depth to every cell.
Since characteristics from the point source to
any given cell cross many other cells, we
would normally have to add the contributions
from cells close to the point source many
times. This can be avoided by using short
characteristics. And then working our way
outwards by interpolating the optical depth at
the upwind corners and then adding the local
contribution. However, this method is not
suited for parallel computations and the
interpolation introduces additional numerical
diffusion.
Rijkhorst (2006)
Hybrid Characteristics
Rijkhorst (2006)
Hybrid characteristic methods use “short” long characteristics to calculate the
optical depth to each cell within each patch combined with “long” short
characteristics to accumulate the total optical depth from a given point source.
Non-isotropic fields
•
•
In the case of multiple point
sources or non-isotropic
scattering, neither moment
methods nor characteristic
methods are well suited.
However, recent advances in
computational capability have
made five dimensional
calculations possible. These
calculations basically involve
using short characteristics
beginning at the outer boundary
and interpolating across the grid,
however here the calculations can
be done in parallel as there are
many different directions and the
calculations can be done
independently. Multiple Wave
Front method.
Nakomoto (2001)
Comparison project
Iliev (2006)
References
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Nakamoto T., Umemura M., Susa H., 2001, MNRAS, 321, 593
Stone J., Mihalas D., Norman M., 1992, ApJS, 80, 819S
Hayes J., Norman M., 2003, ApJS, 147, 197H
Juvela M., Padoan P., 2005, ApJ, 618, 744J
Rijkhorst E., Plewa T., Dubey A., Mellema G., 2006, A&A, 452, 907R
Henney W., 2006, astro-ph, 2626H
Iliev I. et al, 2006, MNRAS, 371, 1057I