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An exactly soluble problem of radiative transfer without redistribution in frequency in an inhomogeneous atmosphere
J. Quanr.
Specrrosc.
&dim.
Trmsfer.
Vol.
9, pp.
1017-1024.
Pergamon Press 1969. Printed in Great Briain
AN EXACTLY SOLUBLE PROBLEM OF RADIATIVE TRANSFER WITHOUT REDISTRIBUTION IN FREQUENCY IN AN INHOMOGENEOUS ATMOSPHERE M. LECAR Smithsonian Astrophysical Observatory and Harvard College Observatory. Cambridge, Massachusetts (Received
Abstract-The
2
1 November
1968)
transfer of radiation by coherent scattering and gray absorption
is described by the integral
equation 2 S(T) = (1-E)
dr’K()r - r’))S(r’)+ &B(r). I II For the case when the medium is one dimensional, so that the kernel K is an exponential, and homogeneous (E constant), the resolvent was given by Mime. The resolvents are given here for a class of prescribed variations of E with r. 1. INTRODUCTION
of radiative transfer in a plane-parallel atmosphere with gray absorption (K) and coherent scattering (b) is THE EQUATION
aI,
pax
= -(K+o)z,+d,+KB,
(using standard notation ; e.g. KOURGANOF@))
pz=
I,-(1
or -&)J,-&EB,
(1)
with dr = -(k+c)dx
and
E = IC/(K+O).
The equation of transfer for a two-level atom is
4WvW dv-&(d (2) 1. -cc The physical implications of this equation and its various approximate forms have been extensively discussed by R. N. Thomas, J. T. Jefferies, and their collaborators. References to their work and a summary of their ideas are found in JEFFERIES.‘~) Computational methods 1017
1018
M.
LECAR
that have been developed to solve this equation were recently reviewed by HUMMERand RYBICKI(3) whose notation I have used. If the profile function 4(v) is taken to be a narrow rectangle (e.g., 4(v) = l/A when Jv-v,-J < A/2 and 4(v) = 0 otherwise), an integration over frequency results in equation (1). Since J,(z) =
3
s
dr’E,(]z - z’])&(r’),
(3)
0
and
S”(7)= (1- 4J”b) + ah),
(4)
equation (1) can also be written as the integral equation cc
S,(r) = (1 -&
dz’E,(lz-z’l)S,(z’)+&B,(~).
I
(5)
0
The difficulties in obtaining numerical solutions to equation (1) or equation (5) occur when E is small, as is typically the case in line-transfer problems. Accordingly, in order to allow maximum flexibility in the s-dependence, the angular dependence is simplified via the Eddington approximation. Introducing a two-point quadrature in p, we have +1
s
.fb)dp =
.f-( -$)+f(+fj):
-1
the exponential integral is approximated
E,(x)
=
by
s@$f
z
J3 e-
J(3b.
In this approximation, J,(r) = i[Z\’ ‘(r) + I:- ‘(r)]
(6)
where f1
Z\+‘(z) =
II-)(z)
s
=
s 0
J3 dr’ e-
J3(,‘-~)~~(~~)+~~+)(~~),-J3(r1-r)
,/3 dr’ e- J3’r-r’)Sy(r’).
(74
(7b)
Radiative transfer without redistribution
Differentiating
in frequency in an inhomogeneous
atmosphere
1019
equations (7), we obtain
I:+‘(4 = ZI-
‘(z)=
1 dJ,W
J,W +J7
J,(z)
-
dT
,
1 dJd4
J3
dz
(84
WI
and
T
=3[J&) - S,(z)].
Making use of the definition of S,(r) (equation (4)), we obtain d* J,(T) = 3s[J,(r) - B,(r)]. dr*
(9)
It will prove convenient to work in the depth variable dt = J(3s) dr, in terms of which equation (9) is written d*J
$+2/l(t)%
= J,-B,
(10)
where
This is the equation we shall work with, subject to the boundary conditions Z;-‘(O) = J,(O)-&?
= 0
(114
and Z:+‘(tl) = J,(tl)+Jedi
dJ,(t,)
(lib)
When /? = 0, the general solution of d*J 2 = J,-Z3, dt* t Jdt) = 3
(12)
fl
ds e-“-“‘B,(s)+~
$
dse-(“-‘)B~s)-Rae-‘+fDe’,
(134
1020
M. LECAR
where A = K[(l-~)Z~-(l-J~)~e~~“Z,+2(1-~/~)e~~’Z~+~(t,)],
(13b)
D = K[(~-JE)~~-~“Z,-(~-E)~-~‘~Z,+~(~+JE)~-’~Z~+’(~,)],
(13c)
K = [(~+JE)~--(~-~JE)~~-~‘~]-‘,
(13d)
fl IO =
s
ds e-‘&(s),
(13e)
ds e”&(s).
(13f)
0
fl I,
=
I
0
(MILNE’~))
For example, when tI = co, the solution t Jv(t) = 9
s
is m
dse- “-“‘B&)+~
0
s
ds e-@‘)B,(s)
f
In a gray atmosphere the spectrum of the emergent flux is
f&(O)=
JE
1 3
1
m dt e-‘&(t),
I o
+JE
where the dimensionless frequency u = hv/kTe. Figure 1 shows H, vs. u for B, (the Planck function) and (T/Te)4 = J3/4[1 +(t/J.$], which is the value it takes in radiative equilibrium. The frequency u,,_,.~at which H, peaks varies as C1’* as would be predicted from the simple argument that u,,, cc T(t = 1) CC (I JE)~‘~. 2. SOLUTION
FOR
c--a
STEP
FUNCTION
A useful variant of Milne’s problem, considered first in another context by is obtained by letting E = constant when t < tl whent 2 t,.
&=l Since E = 1 for t 2 t,
, m
Z’+)(t,)=
J3 dr e-J3(r-rl)B(r), s 71
SCHUSTER,‘~)
Radiative transfer without redistribution in frequency in an inhomogeneous atmosphere
0
1021
1022
M. LECAR
where we have dropped the frequency subscript. The specification of I(+)(t,) together with equations (13) completely specifies the solution. It is of interest to look at two examples of B(r). The choice B(r) = 1 +b,/(3) r p rovides a useful check on our algebra, for when b = 1, the solution is always J = B, independent of E. We note that when E < 1, if t, S I (i.e., r1 9 l/J&)
J(o)J+JE = JE'
(144
1+
while if tI < 1, J(0) z +(l + b)
1 +(bll +b)
JVh
1+fJ(3h
1
(14b)
As a second example, we let B(r) = 1 -a e-““. To the lowest order in a, when r1 B l/ JE, if J >> l/ JE, then J(0) z JE(~ -a); while if I < l/ JE, then J(0) g JE-ad. On the other hand, when 1 < z1 + l/J&, if 1 $ rr, then J(0) E
&p-4>
while if 1 < zl, then
We note that to lowest order, 1 does not appear in J(0) unless 1 4 l/ JE; i.e., features on a smaller scale than l/ JJEare not seen. Further, when z1 + l/ JE, J(3)7, just replaces JE in the formulas for J(0).
3. A SOLUTION
FOR
p #O
If in equation (lo), we let J = E- ‘j4j, we obtain
-$+( l+~‘+~)j If 1+ 8’ + d/?/dt = y2 = constant, the introduction written as
= .s1/4B.
(15)
of do = y dt allows equation (15) to be
where
(16)
Radiative transfer without redistribution in frequency in an inhomogeneous atmosphere
1023
This is again equation (12), and its solution is contained in equations (13) except that in the new variables, if we specify that E = 1 at o = o1 , the boundary conditions are
Z’-)(O)
= 0 = ~~1~4{(l+&/~o)jo-y~~o~}
(17b)
where &g= &(O= O),. . . In this case,
A = ~{~[~-J~o~y-8,~1~~+~~~,~~~"'+~~-J~~~~-B~ll~~+~~-B~~l~~ (18)
-[~-JE~(Y-B~~I[~-(Y+~~)I~~ e-2W'Ij D = K{2[1+J~,(y+~,)]1~~~(0,)e~“~+[1-~~,(y-~,)1[1-(y+~,)]l~e~*”~
-[~+JE~(Y+B~~I[~-(Y+B~)Z~I~-*“‘},
(19)
with 01 ’
I, =
I
doe-“b(w),
0
WI
I, =
s
dw e%(o),
0
and
The allowable forms for e(r) when
dB = constant y* E 1 +/I* +% are E = co[ 1 - 2&/(3c,)r] - *
(20)
&= &o[l-~oJ(3&o)r]-4
(21)
y* > 1,po = 0:
&= &o[l-(y2-
(22)
y2 < l,/Io = 0:
&= EO[l+ (1 - y2)3Eor2]- *.
B = constant: y2 = 1:
1)3&,2*]-*
(23)
With the exception of equation (23) which has E decreasing with z, the forms are similar
1024
M. LECAR
and we investigate in detail the case when p = constant. Then 6(t) = c0 e4@, J(3)N
(24) - e-2ot),
= &(I &=l
when
o=o,
(25) =Lln48
1 &cl
or J(3)Z = J(3)z,
= g. Eo
(26)
As an example, we find for J(0) when B(z) = 1, and E 6 1, when p 6 1, J(0) E JEO
(lga)
while when
To lowest order in JE and l/z,, these results agree with the step function model (equations (14), b = 0). In general, when E(T)and B(z) vary slowly on the scale of the “thermalization length,” physical reasoning based on mean-free-path arguments seems to be reliable (RYBICKI and HUMMER(@). On the other hand, the situation when E and B vary rapidly is not as well understood, and analytic examples are still useful in providing insight. In this ccnnection, the step function model (dating back to SCHUS-~JZR~~)) seems particularly simple to formulate, solve, and interpret. Acknowledgements-Helpful suggestions from Dr. R. N. Thomas, who read an earlier version of this paper. and Dr. G. Rybicki are gratefully acknowledged.
REFERENCES I. V. KOURCANOFF. Basic Method~y in Transfer Problems, Clarendon Press. Oxford (1952). 2. J. T. JEFFERIES, In Temperature-Its Measurement and Control, Vol. 3, p. 703, Reinhold Publishing Corp.. New York (1962). 3. D. G. HUMMERand G. RYBICKI,Methods qf Computational Physics, Vol. 7, p. 53. Academic Press, New York (1967). 4. E. A. MILNE,Hdb. Astrophys. 3, 166 (1930). 5. A. SCHUSTER.Astrophvs. J. 21, 1 (1905). 6. G. RYBICKIand D. G. HUMMER,Thermalization lengths and their distribution function in line transfer problems, MNRAS, 143 (in press).
Specrrosc.
&dim.
Trmsfer.
Vol.
9, pp.
1017-1024.
Pergamon Press 1969. Printed in Great Briain
AN EXACTLY SOLUBLE PROBLEM OF RADIATIVE TRANSFER WITHOUT REDISTRIBUTION IN FREQUENCY IN AN INHOMOGENEOUS ATMOSPHERE M. LECAR Smithsonian Astrophysical Observatory and Harvard College Observatory. Cambridge, Massachusetts (Received
Abstract-The
2
1 November
1968)
transfer of radiation by coherent scattering and gray absorption
is described by the integral
equation 2 S(T) = (1-E)
dr’K()r - r’))S(r’)+ &B(r). I II For the case when the medium is one dimensional, so that the kernel K is an exponential, and homogeneous (E constant), the resolvent was given by Mime. The resolvents are given here for a class of prescribed variations of E with r. 1. INTRODUCTION
of radiative transfer in a plane-parallel atmosphere with gray absorption (K) and coherent scattering (b) is THE EQUATION
aI,
pax
= -(K+o)z,+d,+KB,
(using standard notation ; e.g. KOURGANOF@))
pz=
I,-(1
or -&)J,-&EB,
(1)
with dr = -(k+c)dx
and
E = IC/(K+O).
The equation of transfer for a two-level atom is
4WvW dv-&(d (2) 1. -cc The physical implications of this equation and its various approximate forms have been extensively discussed by R. N. Thomas, J. T. Jefferies, and their collaborators. References to their work and a summary of their ideas are found in JEFFERIES.‘~) Computational methods 1017
1018
M.
LECAR
that have been developed to solve this equation were recently reviewed by HUMMERand RYBICKI(3) whose notation I have used. If the profile function 4(v) is taken to be a narrow rectangle (e.g., 4(v) = l/A when Jv-v,-J < A/2 and 4(v) = 0 otherwise), an integration over frequency results in equation (1). Since J,(z) =
3
s
dr’E,(]z - z’])&(r’),
(3)
0
and
S”(7)= (1- 4J”b) + ah),
(4)
equation (1) can also be written as the integral equation cc
S,(r) = (1 -&
dz’E,(lz-z’l)S,(z’)+&B,(~).
I
(5)
0
The difficulties in obtaining numerical solutions to equation (1) or equation (5) occur when E is small, as is typically the case in line-transfer problems. Accordingly, in order to allow maximum flexibility in the s-dependence, the angular dependence is simplified via the Eddington approximation. Introducing a two-point quadrature in p, we have +1
s
.fb)dp =
.f-( -$)+f(+fj):
-1
the exponential integral is approximated
E,(x)
=
by
s@$f
z
J3 e-
J(3b.
In this approximation, J,(r) = i[Z\’ ‘(r) + I:- ‘(r)]
(6)
where f1
Z\+‘(z) =
II-)(z)
s
=
s 0
J3 dr’ e-
J3(,‘-~)~~(~~)+~~+)(~~),-J3(r1-r)
,/3 dr’ e- J3’r-r’)Sy(r’).
(74
(7b)
Radiative transfer without redistribution
Differentiating
in frequency in an inhomogeneous
atmosphere
1019
equations (7), we obtain
I:+‘(4 = ZI-
‘(z)=
1 dJ,W
J,W +J7
J,(z)
-
dT
,
1 dJd4
J3
dz
(84
WI
and
T
=3[J&) - S,(z)].
Making use of the definition of S,(r) (equation (4)), we obtain d* J,(T) = 3s[J,(r) - B,(r)]. dr*
(9)
It will prove convenient to work in the depth variable dt = J(3s) dr, in terms of which equation (9) is written d*J
$+2/l(t)%
= J,-B,
(10)
where
This is the equation we shall work with, subject to the boundary conditions Z;-‘(O) = J,(O)-&?
= 0
(114
and Z:+‘(tl) = J,(tl)+Jedi
dJ,(t,)
(lib)
When /? = 0, the general solution of d*J 2 = J,-Z3, dt* t Jdt) = 3
(12)
fl
ds e-“-“‘B,(s)+~
$
dse-(“-‘)B~s)-Rae-‘+fDe’,
(134
1020
M. LECAR
where A = K[(l-~)Z~-(l-J~)~e~~“Z,+2(1-~/~)e~~’Z~+~(t,)],
(13b)
D = K[(~-JE)~~-~“Z,-(~-E)~-~‘~Z,+~(~+JE)~-’~Z~+’(~,)],
(13c)
K = [(~+JE)~--(~-~JE)~~-~‘~]-‘,
(13d)
fl IO =
s
ds e-‘&(s),
(13e)
ds e”&(s).
(13f)
0
fl I,
=
I
0
(MILNE’~))
For example, when tI = co, the solution t Jv(t) = 9
s
is m
dse- “-“‘B&)+~
0
s
ds e-@‘)B,(s)
f
In a gray atmosphere the spectrum of the emergent flux is
f&(O)=
JE
1 3
1
m dt e-‘&(t),
I o
+JE
where the dimensionless frequency u = hv/kTe. Figure 1 shows H, vs. u for B, (the Planck function) and (T/Te)4 = J3/4[1 +(t/J.$], which is the value it takes in radiative equilibrium. The frequency u,,_,.~at which H, peaks varies as C1’* as would be predicted from the simple argument that u,,, cc T(t = 1) CC (I JE)~‘~. 2. SOLUTION
FOR
c--a
STEP
FUNCTION
A useful variant of Milne’s problem, considered first in another context by is obtained by letting E = constant when t < tl whent 2 t,.
&=l Since E = 1 for t 2 t,
, m
Z’+)(t,)=
J3 dr e-J3(r-rl)B(r), s 71
SCHUSTER,‘~)
Radiative transfer without redistribution in frequency in an inhomogeneous atmosphere
0
1021
1022
M. LECAR
where we have dropped the frequency subscript. The specification of I(+)(t,) together with equations (13) completely specifies the solution. It is of interest to look at two examples of B(r). The choice B(r) = 1 +b,/(3) r p rovides a useful check on our algebra, for when b = 1, the solution is always J = B, independent of E. We note that when E < 1, if t, S I (i.e., r1 9 l/J&)
J(o)J+JE = JE'
(144
1+
while if tI < 1, J(0) z +(l + b)
1 +(bll +b)
JVh
1+fJ(3h
1
(14b)
As a second example, we let B(r) = 1 -a e-““. To the lowest order in a, when r1 B l/ JE, if J >> l/ JE, then J(0) z JE(~ -a); while if I < l/ JE, then J(0) g JE-ad. On the other hand, when 1 < z1 + l/J&, if 1 $ rr, then J(0) E
&p-4>
while if 1 < zl, then
We note that to lowest order, 1 does not appear in J(0) unless 1 4 l/ JE; i.e., features on a smaller scale than l/ JJEare not seen. Further, when z1 + l/ JE, J(3)7, just replaces JE in the formulas for J(0).
3. A SOLUTION
FOR
p #O
If in equation (lo), we let J = E- ‘j4j, we obtain
-$+( l+~‘+~)j If 1+ 8’ + d/?/dt = y2 = constant, the introduction written as
= .s1/4B.
(15)
of do = y dt allows equation (15) to be
where
(16)
Radiative transfer without redistribution in frequency in an inhomogeneous atmosphere
1023
This is again equation (12), and its solution is contained in equations (13) except that in the new variables, if we specify that E = 1 at o = o1 , the boundary conditions are
Z’-)(O)
= 0 = ~~1~4{(l+&/~o)jo-y~~o~}
(17b)
where &g= &(O= O),. . . In this case,
A = ~{~[~-J~o~y-8,~1~~+~~~,~~~"'+~~-J~~~~-B~ll~~+~~-B~~l~~ (18)
-[~-JE~(Y-B~~I[~-(Y+~~)I~~ e-2W'Ij D = K{2[1+J~,(y+~,)]1~~~(0,)e~“~+[1-~~,(y-~,)1[1-(y+~,)]l~e~*”~
-[~+JE~(Y+B~~I[~-(Y+B~)Z~I~-*“‘},
(19)
with 01 ’
I, =
I
doe-“b(w),
0
WI
I, =
s
dw e%(o),
0
and
The allowable forms for e(r) when
dB = constant y* E 1 +/I* +% are E = co[ 1 - 2&/(3c,)r] - *
(20)
&= &o[l-~oJ(3&o)r]-4
(21)
y* > 1,po = 0:
&= &o[l-(y2-
(22)
y2 < l,/Io = 0:
&= EO[l+ (1 - y2)3Eor2]- *.
B = constant: y2 = 1:
1)3&,2*]-*
(23)
With the exception of equation (23) which has E decreasing with z, the forms are similar
1024
M. LECAR
and we investigate in detail the case when p = constant. Then 6(t) = c0 e4@, J(3)N
(24) - e-2ot),
= &(I &=l
when
o=o,
(25) =Lln48
1 &cl
or J(3)Z = J(3)z,
= g. Eo
(26)
As an example, we find for J(0) when B(z) = 1, and E 6 1, when p 6 1, J(0) E JEO
(lga)
while when
To lowest order in JE and l/z,, these results agree with the step function model (equations (14), b = 0). In general, when E(T)and B(z) vary slowly on the scale of the “thermalization length,” physical reasoning based on mean-free-path arguments seems to be reliable (RYBICKI and HUMMER(@). On the other hand, the situation when E and B vary rapidly is not as well understood, and analytic examples are still useful in providing insight. In this ccnnection, the step function model (dating back to SCHUS-~JZR~~)) seems particularly simple to formulate, solve, and interpret. Acknowledgements-Helpful suggestions from Dr. R. N. Thomas, who read an earlier version of this paper. and Dr. G. Rybicki are gratefully acknowledged.
REFERENCES I. V. KOURCANOFF. Basic Method~y in Transfer Problems, Clarendon Press. Oxford (1952). 2. J. T. JEFFERIES, In Temperature-Its Measurement and Control, Vol. 3, p. 703, Reinhold Publishing Corp.. New York (1962). 3. D. G. HUMMERand G. RYBICKI,Methods qf Computational Physics, Vol. 7, p. 53. Academic Press, New York (1967). 4. E. A. MILNE,Hdb. Astrophys. 3, 166 (1930). 5. A. SCHUSTER.Astrophvs. J. 21, 1 (1905). 6. G. RYBICKIand D. G. HUMMER,Thermalization lengths and their distribution function in line transfer problems, MNRAS, 143 (in press).