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:*>*>■* ((j £S '' ] [ £s ] • *»■:•■;>:*>*>■* A
A
The redistribution function R(v',s'\v The redistribution function R(v',s'\v
,s) is normalised such that ,s) is normalised such that
2^ ,2 <j> dfl'(tdO I di/' I R(i/' ,s' \v,z)&v
<j> dn'cj>dn A
A
'o
J
= 1.
(1.1.10)
o
A Through R(i/' ,A s' \u, s) , we can define both the normalised absorption Through R(v',s';v,s), we can define both the normalised absorption
profile and the normalised emission profile for the process of scattering.
7 Thus if we integrate over all the emitted frequencies and angles, we obtain the probability for absorption from frequency range di/' and solid angle dO'.
This is given by 0(i/')di/'dn74w,
where the
absorption
profile
(v') i-s given by CO
(") = eS(r,i^)/j eS(r,i/)dt/
CXI
= J* R(i/' ,i/)J(r,i/')di/'/f
CO
«K«/')J(r,i>')di/'.
(1.1.20)
9 This implies that the distribution of photons emitted depends on the profile of the incident radiation. In the particular case of frequency independent intensity V>(i/) = P" R(i/' ,i/)dv . J
(1.1.21)
0
If further
R(i/,i/') = R(i/' ,u) ,
(1.1.22)
*(«0 - (")•
(1.1.23)
One example of equality of emission and absorption profiles occurs in the case of thermodynamic equilibrium.
When the scattering is
essentially coherent (i/' = i/) A
A
A
A
R(i/',s';i/,s) = p(s' ,s)(^(i/')S(v - i/'). A
(1.1.24)
A
Here 5 is the Dirac's delta—function and p(s',s) is the phase function.p(s',s) is normalised as
~- I p(s',s)dO' = 1.
(1.1.25)
In the case of isotropic scattering, the phase function A
A
p(s',s) = 1.
(1.1.26)
Another special case is that of Rayleigh's phase function given by p(s',s) = | (1 + cos2 6),
(1.1.27)
8 being the angle between the incident and the scattered pencil. coherent scattering, emissivity or the emission coefficient can be written as
For
10 .28) eS(r,s,i/) - a(r)^(i>)(j> I(r, s',i/)p(s ' , s) j^' ■ ( 1 . 1(1.1.28)
Attenuation or extinction
coefficient
:
The total loss of intensity from a pencil of radiation is usually expressed in terms of an attenuation or extinction coefficient defined as follows :
Fig. 1.1.2
Let the radiant energy pass through an elementary volume dv = dSds. The rate at which the energy from the pencil of incident radiation I(r,s,i/) is removed or attenuated by the volume is proportional to (i)
ds, the length traversed,
(ii)
I(r, s ,i/)dSdf)di/, the rate at which the radiant energy in the frequency interval (v.^+di/) is transmitted across the elementary area dS at r along s within the solid angle dQ.
Then the rate of loss or attenuation of energy from the incident beam in its passage through the volume element dv is
a(r,i/) [I(r,s,i/)d2dfidi/]ds. The factor of proportionality a(r,i/) is called
extinction
coefficient.
(1.1.29) the attenuation
or
11 This rate of loss of energy from the incident beam per unit volume, per unit solid angle and frequency interval may be (a)
due to true absorption by the medium given by k(r,i/)I(r,s,v), where k(r,f) is the volume coefficient of absorption and
(b)
due to scattering from the beam to other beams given by
ff(r,i/)I(r,s,i/) ,
where o(x,v)
is the volume coefficient of scattering. a(v,v)
= k(r,i/) + o(x,i>).
Then
(1.1.30)
Source function : The term source function ?(r,i/) at a particular point r in the medium is the ratio of total emissivity to extinction or attenuation coefficient. If the medium is in local thermodynamic equilibrium we can assign at each point a local temperature T(r).
Then Kirchoff's law
is valid and the source function ?(r,i/) is given by
*9 T^ / ++g(z.*.-P)
d.3.1)
19
Fig. 1.3.1
Let s denote the direction of a pencil of radiation of specific intensity l(z,0,/i')I(r,M')dAl', M - dr . ^
(1.3.6)
1 r^n P°(A»./*') = ■?p(/J,
3. 28) (1 ( 1 3.28)
where the source function ?(r) for a coherent scattering, frequency independent radiation field in a participating medium could be written as . . +1 f(r) - 2£± J I(r,/i')p(/i',M)d^' + BQ(r) + B 1 (r).
(1.3.29)
Here u(r) , the albedo for single scattering = <j(r)/a(r) , 0 < a> < 1.
26 B (r) : Contribution to the source function from radiation reduced o from any incident flux at the bounding surface or surfaces. B 1 (r) : Contribution from internal sources other than scattering. In case the source function is frequency dependent, the distribution function R(i/',i/) defined in (1.1.16) is used [cf. Hummer
].
The radiative transfer equation in this case reads as
4 - Kr./i.i/) + or
1
2 ~ ** |- I(r,/*, s i n 8 s i n
0,
>
(1.4.5)
I+(ro,/i,i/) = B v (T 2 ), for p > 0, where B (T ) and B (T ) are Planck functions.
C)
Opaque, diffusely
emitting
and reflecting
boundaries
:
Let T, and T. be temperatures, e, and e0 , the spectral 2 li> 2i/ hemispherical emissivities and p. , p0 , the spectral hemispherical diffuse reflectivities at r = 0 and T = T
respectively.
Then the
intensity of radiation I_(0,—n) for Q < ft < 1 leaving the surface r = 0 in the negative fi direction -2* -27T |
I > , - ^ ) -
° !
I
J
J
I
= e l l / B v , 0 < jj. < 1, 0 < 7 < 1. Note
7 = 0 = > perfectly absorbing core
and
7 = 1 ==> perfectly reflecting core.
It may, however, be mentioned that the source function in the transfer equation may contain contributions from internal and external sources (other than scattering) in addition to that from diffuse radiation.
(1.4.10) is the boundary condition for diffuse radiation.
The total intensity at any point is calculated taking into account the contribution from all sources.
Boundary conditions
for cylindrical
medium :
The transfer problems in cylindrical geometry are important in the study of heat transfer and neutron transport. For a homogeneous, isotropically scattering, infinite cylindrical medium of radius R, with an incident flux in a specific direction at the boundary surface, the transfer equation for diffuse radiation is written as in (1.3.49).
They are usually solved under the boundary
condition I(R,77,-AO = 0, 0 < n S
1, -1 < rj
?(T) =
eo
T~ J
9(T,) E ( f
l l ~ r ' | ) d T ' + »( T ).
(1.5.16)
with u F.
o I B(r) = B 1 ( 0 + - ~ e
Integral
equation for radiative
—T/U
transfer
' 'o "
in spherical
(1.5.17)
geometry :
The starting point of our consideration is again the general form of integral equation for source function derived in equation (1.5.6).
We express the quantites involved in (1.5.6) in spherical
polar coordinates (r,0,cp).
We assume axial symmetry of the radiation
field unless otherwise mentioned.
Then [cf. Fig. 1.5.1]
41
|r - r'| - |s - s'| = (r2 + r'2 - 2rr'cos B')1,
(1.5.18)
dv' = r'2sin0'dr'd0'd
(1.5.30)
)ds'
r
M„
Furthermore, if o> and a are constants, that is when the medium is homogeneous,
ti>F.
r
V > - -IT
[cf.
p
o
*
R r
,2
exp [-(aBji
o
Leong and SerT
, p.472].
- ar/i ) ] , - 1 are constants), the integral equation for transfer in cylindrical geometry can be written as
4T 7 ?(r')k(r,r')dr' + -[r B(r)
■fr ?(r) = u
(1.5.41)
where
t
1
2
k(r,r') - —
22
(rr')
T -°° f dfl' I K (auy)dy J p,
and K
J
T
(1.5.42)
O
is the modified Bessel function of the second kind.
[cf. ° Heaslet and Warming^(6) , equations (9), (10), (lib); [cf. Heaslet and warmingv
, equations (9), (10), (lib);
Also note that in (1.5.42) k(r,r') = ak(ar,ar') of (lib)]. Now remembering that TTK (ary)I (ar'y), 71
r' < r
K (Quy)dS' Jn
(1.5.43)
° 71 o(ar'y)Io(ary), ^K
r < r'
47 where IQ(x) and K (x) are the zeroth order modified Bessel functions of the first and second kinds respectively, we have 1 2 r°° a (rr') I K (ary)I (ar'y)dy, 2
•
k(r.r')
r' < r (1.5.44)
1 a 2 (rr') 2 J KQ(ar'y)Io(ary)dy,
[cf. Heaslet and Warming^
r < r' .
, equations (14), (15b)].
B (r) can be calculated for cylindrical medium following more or less analogous arguments as in spherical geometry for establishing (1.5.39) and (1.5.41).
B (r) for homogeneous cylindrical medium when
a uniform incident flux TTF. per unit area normal to its direction of incidence is incident at the outer surface in the direction (—6 ,—V> ) o o [cf. Fig. 1.5.4], [cos n and cos i/> = u ] o o o
Fig. 1.5.4 is given by
taF.
V
r)
- -IT [I](
arfi
aR/i
- /*
o exp p * "o '
2 cosh
—
1
Jl - ri
\l
;4 - r,
V-R\ ' - » : •
(1.5.45) i
-l
where the angles cos
y. , cos
reference to Fig. 1.5.2.
■%
/iQ, cos
i
»JQ can be understood with
48 Here
0 < n , fi* < 1, -1 < t) < 1, o o o
(1.5.46)
and
R2(l - n2) o
(1.5.47)
- r2(l - /i*2) , o
and H(r — R.J 1 — /i ) is the Heaviside unit step function. Incidentally it may be mentioned that sometimes it is convenient to use (r,0,2
-
rQ - p,
we have [cf. Heaslet and Warming
If we now set
F(p)/B(p) = 0(r),
]. IT
a> o 0(r) - T - To + -I [ 0(r')E,(IT 2 JQ I I - r'I)dr'. This is the integral equation satisfied by the source function for a plane medium of thickness 2T . Thus we can utilize all the techniques for the plane case.
In particular we can obtain the values of the
source function at the surface of the sphere using Chandrasekhar — Ambarzumian functions. We have [cf. Heaslet and Warming (6).
| r\^x fi(2To)
- fij + | ro(«2 - p2)
+
\ («3 - ^ ) ,
' 'o^ 1
Tt".-',»-r i - id
( Q
i-^ o
WQ
_ 1,
55 where .1 _ a n = | X(/*,2r X ( M , 2 r )/i%, o)M%,
^ B
,1 = | Y(/i, 2r Q )/i%, =
X, Y being the Chandrasekhar — Ambarzumian functions, [cf. Chandrasekhar
] The above analysis was possible as the kernel
of the integral equation took a special form in the homogeneous case. If the attenuation coefficient a(r) is not a constant, then this simple reduction to the plane case is not possible and we have to solve an integral equation of the type W
K.
r£(r) = rB(r) + -^ [ k(r,r')r'5(r')dr'
(2.1.2)
We shall now indicate the different approaches adopted to solve integral equations of this type.
These integral equations belong to
the general class called the Fredholm type of the second kind and take the form y(r) - g(r) + A
k(r,r')y(r')dr'. J
(a)
Neumann series
solution
:
By introducing the m—iterated kernel k (T,T')
k (T.T') m
=
(2.1.3)
a
by the equation
rb k(r,t)k ,(t,T')dt, m > 1 J . m—1 a
k^r.r') = k(T.T'),
the solution of equation (2.1.3) can be written as 00
y(r) - g(r) +
where Am{g(r)) = 1 k^r
Y A m Am{g(r)}. m=l
, r ' )g(r ' )dr ' ,
(2.1.4)
56 .b „b j | |k(r, r' ).| drdr' < 1
provided
[see Miklin (15) , p.13] In our context, we shall show that the kernels satisfy the inequality |AL{X}| < 1 and hence the requirement for the infinite series representation of the solution.
(b)
Degenerate kernels
(Pincherle
Goursat kernels)
:
If the kernel in equation (2.1.3) is an L„—kernel i.e.
. .
rb rb 2
norm k = ||k|| =
a
a
k (r,r')dTdr'
< =>,
then it is known that we can decompose the kernel (in a non—unique way) as follows. k(r,T') = k(r,T') + T(T,T-')
where
k(r,r') =
N V 5C (r)Y, (r') k k=l k
is degenerate and the norm |T| can be made arbitrarily small.
As the
norm of T is small, the Neumann series solution for the kernel T converges fast.
In view of this we first obtain a solution of the
integral equation with kernel T(r,r') and then use standard methods to obtain the solution of the integral equation with degenerate kernels. The details follow. The basic equation (2.1.3) can be written as rb y(r) = G(r) + A J T(r,r')y(r')dr'
(2.1.5)
57 D r
with
G(r) = g(r) + A J
_ k(r,r')y(r')dr'.
a
If r(r.r') is the resolvent kernel of equation (2.1.5) then by definition its solution is b
y ( 0 = G(r) + A
r
r(r,r')G(r')dr', a
= g(r) + J
r(T.T') (g(r') + A J
i.e.
k(r',r")y(r")dr"; df'
* r y ( 0 - g (r) + A J
[
* Xk(r)Yk(r")y(r")dr"
(2.1.6)
with
g (r) = g(r) + A J" r(r,r')g(r')dr', and .b X ^ r ) = J T(r,r')Xk(r')dr'
The solution of the integral equation (2.1.6) with degenerate kernels can be expressed as [cf. Miklin (15) p.19]
£
y(r) - g (r) + [ ^ k \ (r) ' k=l
where £, = A
Y, (r)y(r)dr and satisfies the algebraic equations
*h-\L W k ^ h ' h-1,2,...• ,N k=l
58
with
a^-J
xJ(r>Yh(r>dr.
and b
h = I V r ) g* ( T ) d r a
The resolvent
kernel
and the iterated
kernels
:
If r(r,r') is the resolvent kernel of the integral equation (2.1.3) then it's solution can be written as b r
y(r) = g(r) + A
r(r,r')g(r')dr'.
(2.1.7)
J
a But from equation (2.1.4) we also have But from equation (2.1.4) we also have n
co
y ( 0 = g(r) +
Am J
[
km(r,r')g(r')dr';
-1 a
in'--l
where k (T,T') are the iterated kernels, m Assuming that the series is uniformly convergent, we obtain, by interchanging the order and comparing
T(T.T')
=
A"
f
Li-
m=l
1 -
^ (T,T') m
.
From this we can deduce that the resolvent kernel satisfies the following integral equation :
r(r.r') = k(r.r') + A
rb J
k(r,t)T(t,r')dt,
a
on using the relation on using the relation k (T,T') = m j
k(r,t)k (t.T')dt. m— 1
(2.1.8)
59
(c)
Reduction
to Cauchy initial
value problem [Goldberg
, p.20U] :
Let us denote the solution of the equation (2.1.3) by y(r;A) and its resolvent kernel by r(r,r';A) to emphasize its dependence on the parameter A.
From equation (2.1.8) we know that the resolvent kernel
satisfies the equation
r(r,r';A) = k(T.r') + A
k(r,t)T(t,T';A)dt.
(2.1.9)
Differentiating both sides with respect to A, we have
r (r,T';A) = A
rb J
k(r,t)r(t,r';A)dt
a
+ A
rb
k(r,t)T (t,r';A)dt.
Regarding this as an integral equation for the function T (T,T';A) with the same kernel k(r,t), its solution can be expressed in terms of the resolvent kernel r(r,r';A) as follows :
I\(r,r';A)=J
+A
k(r,t)r(t,T';A)dt
b b I r(r,t;A){[ k(t,t')r(t',T-;A)dt'}dt.
Using equation (2.1.9) this becomes „b r x (r,r',A)-[ r(r,t;A)r(t,T';A)dt.
(2.1.10)
Also from equation (2.1.9) when A = 0, we have r(T,r';0) = k(r,r').
(2.1.11)
60 Thus we obtain a Cauchy integro—differential system to solve for the resolvent kernel.
Following a similar procedure, we can obtain a
Cauchy integro-differential system for y(r;A).
r
We obtain
b
yA(r;A) = j
r(r,t;A)y(t;A)dt,
(2.1.12a)
y(r;0) = g(r).
(2.1.12b)
The above Cauchy systems can be solved numerically using a quadrature formula for the integrals with N quadrature points r.
and weights w..
Thus if r(r.,T.;A) = r..(A) then the Cauchy system for the resolvent kernel becomes N -jf L[r..(A)] = ) r. (A)r . ( A ) W , J dA ljJ L. lm mnJ m m=l T..(0) = k(r.,T.). tj i j The desired solution y(r) can be obtained either from the representation y('"i;A) == g(''i) + A j
r(r1,t;A)g(t)dt a
g(r ) + A l
N V T (A)g(r )w Li, im m m m=l
or by solving the Cauchy system (2.1.12) after computing the resolvent kernel.
(d)
Goldberg's point
method for semi-degenerate
boundary value problem [Goldberg
kernels p.307]
reduction
to two-
:
We assume that the kernel of the integral equation (2.1.3) is of the following degenerate form
61 IN
)
a.(r)b • (s) i
a < s AT) a n d the resulting equation is integrated over (a,b) .
r |
T 4>±(T)
1 [
r
c rf (r) dr - j
g(r)^i(r)dr
i N + A )
„b , J | k(7-,T')4> (r')4. (OdT-dr-
I '.1 I.= I jb
c
i = 1,2
J
a
N.
!
63 These moment equations yield the necessary N algebraic equations for the N unknown constants c .
The solution is clearly approximate as the
equation (2.1.14) is not satisfied completely.
The degree of accuracy
depends not only on the number of terms but also on the choice of the basis function.
A judicious choice will increase the accuracy even
with limited number of terms.
The above two methods belong to the
general class of me methods called projection methods [Goldberg
P-8] for
integral equations.
2.2
BASIC METHODS OF SOLVING THE INTEGRO-DIFFERENTIAL EQUATIONS OF TRANSFER IN SPHERICAL GEOMETRY Let us illustrate the methods by considering a simple model.
The
integro—differential equation of transfer for a spherically symmetric finite homogeneous medium under conservative, Isotropic scattering is [cf. equation (1.3.28) with a = 1, p = 1, B = 0.]
3I(r,/0 ^ l V 3I(r,/0 + ^ ^ ^ + I ( r , M ) 3r
M -
=f J
I(r,M*>)
not
Include the directly transmitted flux ^ Fj exp(-T//iQ) in the direction
HVV The factor l/l* is introduced for securing the symmetry of S and T in the pair of variables (>*,?). (/V'V principle of reciprocity.
aS re(
5 u i r e d by
Helmholtz
's
74 S(T S(T
,i*,tp,(i S((kTrl;fio,<po,n,tp) ,M .?»:•**„ ,
