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The Milne problem for partially plane-polarized light
« Quant Spectrosc Radlat Transfer. Vol 14, pp 661-677 Per8amon Press 1974 Prmted m Great Bntam
THE MILNE PROBLEM FOR PARTIALLY PLANE-POLARIZED LIGHT S. A. MOURAD Division of Science and Mathematlcs, Fordham Unlverslty, Lincoln Center Campus, New York, New York 10023, U.S A
(Recewed25 September 1973) Abstraet--An accurate numerical method has been devised to solve half-range problems for the scattenng of azlmuth-mdependent polar~zed radmtlon m stellar atmospheres The use of the method is fllustrated by solving the Mllne problem for the radmtlon mtens~ty at any opUcal depth, for the extrapolaUon &stance, and for the law of darkenlng.
1. INTRODUCTION UPON scattering, light generally becomes polarized. Polarization greatly affects the intensity of hght, espe¢lally in planetary atmospheres and in those stellar atmospheres where free electrons are predominant. In pra¢ti¢e, parallel plane atmospheres are orten consldered sufficiently general sinee the transmission layer around a star or planet is optieally thin ¢ompared to the opti¢al radius o f the star or planet. The polarized light field is eharactenzed by four parameters known as the STOKESparameters. ¢1) If the in¢ident light is perpendi¢ular to the straUfied layers of the atmosphere, the system possesses azlmuth independen¢e and two of the parameters are always zero. The remalning two are the mtensitles of the light in different stares of polarization. The equation of transfer governing these lntensltles ¢an be found on¢e a suitable scatterlng law is estabhshed. For atmospheres of free electrons, atoms and molecules, Rayleigh's law is used. However, the presence of anisotropic partlcles causes depoladzatlon. Thus, the scattedng o f planepolarized light ean be descnbed by Raylelgh- and isotrople-s¢attering eomponents. The model just descrlbed also applies to resonance line phenomena where the two parameters represent the mtenslties o f photons in the ground and exclted states. The equaUons of the transfer for these problems were formulated by CHANDRASEKHAR.(2) In addition to the academie lmportan¢e and the mathematical aesthetic, the exact solution o f these equations would lncrease our knowledge of the stars, planets, and lllumination of the sunht sky. In the first part o f thls century, Sobolev and Chandrasekhar were the pdn¢lpal investigators mterested in the solutlon o f the aforementloned equations. Both scientists eoncerned themselves with the hmitmg case of Raylelgh scattering type. SOBOLEVt3) enumerated the different atmospheres where these equat10ns apply and gave the appropriate boundary conditlons for each. CHANDRASEKHARt2) mvestlgated the problem more thoroughly. Recently, an exact and more elegant solution of thls problem was obtained by SmWERT and FRALEY(4) who made use o f Case's method. They proved full-range and half-range •ompleteness and solved for the extrapolatlon length 661
QSRT VoL 14 No 8 - - A
662
S. A MOLr~D
Atmospheres in which amsotropic particles exist have not been glven much attentlon. Chandrasekhar is credited wlth formulatmg the correspondlng equaUons but he dld not attempt to solve them. It is the purpose of the present paper to find exact solutions for these equations and to use them m solving the Mllne problem for such quantmes as the exat distribuUons, the extrapolation distance and the degree of polarization. Due to the elegance and computatlonal merits of the method, Case's techmque is used. In Section 2 the equatmns of transfer are presented, thelr soluUons are dlscussed, and the parameters pertinent to the Milne problem are defined. The apphcatmn of the boundary conditions yields two singular equations which are reduced to one equation due to some particular properties of the problem. This reductlon is shown in Sectlon 3. The equation is then regularized in Section 4 by convertmg it lnto a Fredholm type. The ob•ective of SecUon 5 is to apply the method, developed in the previous sections, to the Milne problem. All calculatlons are carrled out numerlcally. The results are shown m Sectmn 6. 2 DEFINITION
OF THE
PROBLEM
The scattering of polanzed light m stellar atmospheres is described by equaUon (1) whieh has been formulated by CnANDRASEKHAR.(2) Thus, #
I(z,#) + I(z,p) = 2 -1 [c~K(#, #') + C2 E]I(~, #') d#'
(1)
with K(#,#')
]
#,2
1
1
Here # is the direction COslne (as measured from the mward normal to the free surface) of the directed radiation. The scattering kernel for th~s model conslsts of two parts. Orte part, welghed by c~, is of the Raylelgh type, and the second, weighed by c2, is isotroplc. The latter part causes depolanzation. Since the me&um is conservative, et and c2 sum to unity. We seek a non-triviat solution of equation (1) obeying the boundary conditlons of the Mllne problem: I(0,/# = 0 for tL ~ (0, 1) (3a) and hm ~b-q(% #) = 0. (3b) f--# 0o
A general solutlon to equation (1) was constructed m a paper by MOURADand SmWERT.(5) They used Case's method of normal modes and found +t
+1
l(z,/~) -- A+ ~+ + A_(z,/00+ + ~_f, Al(r/)~t(t/,/~)e-mdt/+
f~ A201)0201, tOe-'/" dt/, t
(4) where A +, A_, At(~/) and A2(r/) are arbltrary coefficients to be determined from the boundary conditions, and the eigenvectors are given by 0+= O,(r/, #) =
11 1'
(5a)
½(1 - r/2)ct ~~ r / - tL
, - c 2 ~(n - tL)
I
(5b)
The Milne problem for partially plane-polarizedhght
663
and _
~2)
] {r/(1
o2(r/, /~) =
[{q(1
P n -
+
~«(~) ~(n -
u)
_ 1~)
u
P r/2) ~-~ + ,;L2(q)ô(r/
,
(5c)
where 2,=(-1)'+3(1-q2)20(q)
for z = l o r 2
(5d)
and 20 = I -
t/tanh-l (q).
(5e)
W e need to apply the boundary con&tions given by equations (3) to determine the arbitrary coefficlents.By virture of equation (3b), t/is not allowed to assume values in the interval (0, - I); equation (4) thus becomes
föAl(n)~dq, 1
I(, #) = A+ @ + + A_(z, #)0+ +
l
l.Oe- m dtl + föA2(t/)O2(q ]~)e-'/~dr/. (6)
N o w we use the boundary condition given by equation (3a)to obtain 1
0 =A+*+
1
- v A - * + +föAiQl)Oi(q,l#dq+föA2(q)*2(ti, l~)dq,
#~(0, 1).
(7)
Choosing the normalize the solution by setting A_ equal to unity and transposing the second term to the left hand side, we obtain 1
1
/~(ID+ = A+ O+ + fö Al(t/)@l(q,/0 dr/+ fö A2(r/)O2(q, U) dq,
/~ 6 (0, 1).
(8)
The solution of thm equation for the arbitrary constants will be dealt with in Section 3. Once these coeflicmnts are known and introduced into equation (6), we can determine the üght intensity at any depth • in the atmosphere. In the remainder of this section, the parameters related to the MiMe problem will be defined.
(a) The extrapolation distance An important parameter assooiated with the Milne problem is the extrapolation distance %. The total intenmy, in other words the zero moment of the sum of the two angular intensities, is obtained from
-zf£
l(z) a _l t t 11 I(*,/~) de.
(9)
Ifwe introduce the asymptotic part of I ( , #) as given by equation (4), with At(t/) = A2(q) = 0 and A_ = 1, equatlon (9) yields I.,(,) = A+
+,
(I0)
The extrapolation length is the distance from the boundary x = 0, for which I~(,) = 0; thus Zo = .4+. (11)
664
S A. MOURAD
(b) The emergent radiatwn Of partlcular interest here lS the dmtribution of radiation at the boundary of the semlinfimte half-space. Tlus special oase is more commonly known as the exlt distnbution and is glven by equation (4) with • = 0 and la e ( - 1, 0), thus the integrals are no longer singular. We find
It(O, -la) = A+ + la + fö {cxr/(1
-
r/2)2Al(n) r / dr/ + la + :ó ½r/(1-- r/2)A2(r/) 71dq +-""~,
-
la ~ (0, 1), (12a)
and /r(O, --la) = A+ "{- la "4"
dr/ 371(1 -- F]2)A2(~) 71 -']- la,
la ~ (0, 1).
(12b)
(c) The degree of poIarization In general, the two &stnbutions Il(O, - # ) and I,(0, -la) are not the same. The degree of polarlzation is a measure of the difference in the two components. Let us denote the degree of polarizaüon by P(la); by definition, we can write I,(O,
-
la) -- I,(O,
-
la)
P(la) = Ic(O, - u ) + I,(O, - l a ) '
la ~ (0, 1).
(13)
la e (0, 1),
(14)
la E (0, 1).
(15)
Using equation (12), P(la) takes the torm c 1 A(la) , P(la) = 2I,(0, --la) + ct A(la) where A(la) =
fl
9r/(1 __r/2)2
A, (r/)
dr/,
r/+la
It is obvious from the previous definltions that all quantities pertinent to the Milne problem are expressed in terms of A+, At(r/) and A2(r/). In SecUons 3 and 4 we will develop a method to enable us to calculate these coefficients numerically. 3 THE SINGULAR EQUATION If instead of the Mitne problem another half-space problem is studled, we would obtain equation (8) with a different functlon, say F(#), on the left-hand side. Thus for half-space problems, in general, we must solve the equation 1 1
F(ù) = A+®+ + f0Al(r/)®,(r/, la)dr/+ f~a2(r/)®2(r/, la)dr/,
la~(0, 1).
(16)
To show that thm expansion m posslble, for any F(la), is eqmvalent to provlng half-range completeness. When such a theorem is proved, we can calculate the arbitrary coefficients by using the results of the proof or by constructing adjoint vectors or eren by resorting to numerical analysis. The proof of the half-fange completeness theorem for the parUcular case cx -- 1 has been achieved by SmWERTand FRALEY.(4) When the medium scatters isotropically (Cl = 0), the problem is slmilar to the one-speed neutron equation; the half-range theorems for this case have been proven by CASE.(6)
The Mflne problem for pamally plane-polanzed hght
665
In mathematleal construetion, two-group neutron problems are similar to the polarized light scattering equaüons; e.ach group is equivalent to one stare of polarization. A brief review of the completeness theorem in the neutron problem is thus helpful for the current study m polarized light. Although LEONAgD and FERZIGER(7) claim to have proven half-range completen¢ss their results are not ' constructive' in the sense that they do not y~eld expllc~t expressions for the coefficients and also they are not amenable to numerical analys~s. SHIEH and SIEWERT(s) mtroduced Chandrasekhar's S-function method to solve two-group neutron problems, but they did not prove the exlstence of a solution to the mtegral equatmns obtained. It should be mentioned, however, that the S-matrix approach has the merit of making the problem amenable to numerical analysxs Numerical solutlons to the two-group Milne problem have been suggested by METCALF and ZWEIFEL.<9) Thls paper was based on the eigenfunction method. The most direct approach to prowng half-space eompleteness is to follow the method used by CASE~6) for one-speed neutron problems. However, due to a dffficulty that remains unc~rcumvented, analytlcal results do not result, and thus we resort to a different method. Some inherent properties of the problem considered hefe allow us to reduce the half-range application to proving the existenee of a solutmn to a Fredholm equation. We will make use of the completeness of the set ~ + , O~(r/, #) and O2(r/, #) on the half-fange, for the case where c~ = 0, to reduce the coupled singular equations of expansion (16) to a singular equation, then the proof of half-fange completeness is established. In the remainder of this seetion, we will denve the equation. In Section 4 we will then attempt to solve ~t and show the possibility ofproving its ex~stence without actually solving it. Notice that O~(r/, #) can be wntten as
01(r/, I~) = c~OR(r/, #) + c2 ~P~(r/, #),
(17)
where OR(r/, #) and Of(r/, #) are the llmlting forms of Ol(r/, #) when c~ = 0 and cx = 0, respectively, i e. ®x(r/, #) = ô ( r / - #)
I
1, _ 1
(18a)
~R(r/, #) = ] ~b(r/, #) , I 0
and
q~(r/, #) = ¼(1 - r/2)
(18b)
(1 - #2) ~r / _ # + ~l(r/) 6(r/- u) •
(18c)
EquaUon (16) thus ean be wntten m the form
1
1
1
F(p) = A+ O+ + fo ¢lA'(r/)~I~R(r/' fl) dr/--~ fo ¢2 Al(r/)~l)/(r/,/A) dr/4- fó A2(r/)~[~2(?~,/./)dr/, (19) Transposing the term eontaining Os(r/, #) to the left hand side, we obtain
1
1
GR0~) = A+ ~ + + fó c, ~l(r/)®,(r/, u) dr/+ f~ A~(r/)®,(r/, U) dr/. where we have defined
(20)
1
GR(#) __4F(#) - f ctA1(r/)Oa(r/, #) dr/. Jo
(21)
666
S A. MOURAD
Smce O+ and O20/, la) are independent of cl and c2, equation (20) represents the expansion of G(#) m terms of the set of functlons O + , Ot(q, la) and O2(q, #) which has been shown to be complete by lncorporating the exlstlng proof of one-speed neutron theory. For this proof, we make use of the orthogonality relations for c 1 = 0. It is obvious that we shall obtain A+ and A2(r/) as functions of A~(~/), in addition to a singular integral equation for
Ad,0. Denotmg the vector adjolnt to Oi( q, la) by ¢~t?(q, la), we find S(r/) [1 , 0It(r/' la) = " 7 " OI(?/' la) -]'- "~O(la)O"(la) 1
(22a)
where S(q)
=
(22b)
~o(~/)Ao+(r/)Ao-(r/),
and r/
P
(22c)
O~(#) = -2 r / - l a + 2o(n) ~(n - la).
Here O~(la) lS Case's contmuum elgenfunctmn for the conservatwe one-speed neutron problem, A o + (z), ~o(~) and ?0(la) are the functlons for the one-speed ¢onservatlve problem. Multiplying equation (20) by ~1t(~/, #) and mtegrating over # from 0 to 1, we obtam, since OI?(q, #) is orthogonal to ~ + and O2(~/, #),
1 1 Go(r/) = fó A,(v)¼(1 - v2)c, fo ¢(v, p), [--~IS(t/) 6(q
] -- #) + ?o(#)~ù(#)
dp dv
+ c2 S(rl)Al(~l),
r/« (0, 1),
(23)
where we have defined 1
G°(rl)~-föflll?(rl' la)F(la)d#.
(24)
If we introduce y(q)
=
¼(1 -- ~/2)
(25a)
.4(,7) =
y(,0.41(,0
(25b)
and
in equatlon (23), we find Go(,0 =
Sòc [~ 1,4(v)
~(v, ,0 +
sò
]
~ o D ) ~ ( v , ~)~~0~) du dv "['- C2 S(~)A(?])/y(l~),
q ~ (0, 1).
(26)
We notice that th(v, #) can be wntten as 3v th(v, #) = 3(1 -- v2)Ov(#) + - ~ (v + la) - 6(v - la).
(27)
The Milne problem for paxtially plane-polaxlzed light
667
Introducing equation (27) into equation (26) we obtain the integral equation
S(r/)
rs(r/)
6o(r/) = c, L-T- ,h(r/) + - T
- ~o(r/)&(r/)] a(r/)
+ c2 S(r/)A(r/) + cly(r/)S(r/) A(v) y(r/)
v dv v -
,1
- fó1c~A(v)r0(v) ~r~r/_P v d r - ¼clr/fòvA(v)dv.
(28)
This equation can be written m the form 6o(r/) =/~(r/),4(r/) +
sò
R(r/, v)A(v) dv +
So"
S(v, r/) ~ _ ~ A(v) dr,
(29)
where c2 s(~)
B(r/) = - y(~) -
+ [2y(r/)s(r/) -
yo(r/)]Ct2o(r/),
= - ¼c~vr/,
R(v, r/)
(30a) (30b)
and
s(v,
,1) = v c l y ( r / ) s ( r / )
cd/
+ Œ
~0(v).
(30c)
Equatlon (29) is a singular lntegral equation for A(r/), r/e (0, 1). We, then, illustrate how the remaining expansion coeffieients A+ and A2(r/), r/e (0, 1), are obtained once A(r/), r/e (0, 1), is established. We multiply equation (20)alternately by Wo(~)~~(p)lll and Wo(~)l|l, the adjoints of 0 + and 020/) for ct = 0, and integrate over/z from 0 to 1 ; the following expressions for A2(r/) and A+ will be obtamed once relations previously developed by Ku~dER°°) and the orthogonality theorem are invoked. Au(r/) =
Sò~o(/~)~n(#) IT[ 1 F(#)dlz+ctAl(r/)-clr/
(3la)
and 1 IlIF(bt) + - 3cl A+ = fóTo~) ~ - fjol [3r/ I-~-Q/- z o ) - ro(r/)]Al(r/) dr/,
(31b)
where zo, the Milne problem extrapolation distance for the conservaüve one-speed netttron problem, is » .1
Zo = jo#7O(#) d/~.
(32)
In thIs section we have used known properties for the extreme cases ct = 0 to develop singular equations for A(r/). Furthermore, expressions for Az(r/) and A+ have been constructed by employmg orthogonality relations relevant to this limiting case. For this set, the singular equation plus the associated expressions for A+ and Az(r/), we attempt to prove half-range completeness m the followmg manner: the continuum coefficient A~(r/) is a solution of a singular equation. If this equation admits a s o l u ü o n , then
668
S.A. Mo~
thls solution, along with expressions for A2(r/) and A +, allow us to conclude that the expansion given by equation (16) is unique. In other words, we can then conclude that the set O+, O1(r/, #) and O2(r/, #) is complete. The main problem ~s to investlgate the existence of the solution to the smgular equation. This problem is treated in the next Section 4. 4. R E G U L A R I Z A T I O N
OF THE
SINGULAR
EQUATION
We believe that the reduction of equatlon (29) to a non-slngular form would put us m a better position to decide about the exlstence of a unique solution. Thls reduction is attalned either by regularizing the singular kernel or by subtractlng the singularity. According to POC,ORZ~LSKI,(u) the first approach is preferred by Hilbert and Poincaré, the founders of the theory of singular lntegral equatlons; the second method is favored by more recent workers. We prefer this last approach since the resultmg equat~on is of the Fredholm type. The later approach, often referred to as the Carlemann-Vekua method, is presented in a rlgorous and ¢lear manner by GAKHOVO2) and also by POGORZELSKI.tu) We rewrite equatmn (29) in the form 6o(t/) = #(r/),4(r/) + s(r/, r/)
fo1A(v)
P
v - t~
dv +
fò s(v' r/)v - s(r/' t/) •(v) t~
+
fo~R(v, r/)A(v)dv. (33)
Defining 1
1
- ~l s(t/, t/) A(v) dv - fo R(v, t/)A(v) dr, Fo(r/) a_ Go(t/) - fo s(v, ,7)v ---
(34)
we observe that the integrals involving A(r/) are non-slngular since S(v, t/) satisfies the Hölder condition (GAKHOV(x2)) and R(v, r/) is, by definition [see equation (30b)], a regular function. Using the defmiuon given by equation (34), we find that equation (33) takes the form:
Fo(r/) = fl(r/)A(r/) + S(t/, t/)
A(v)
V--r~
dr.
(35)
We consider now a function N(z) analytic m the complex plane cut from 0 to 1 along the real axis and defined by 1
dr/, N(z) = K~ fò r~A(r/) -
(36a)
with boundary values from above ( + ) and below ( - ) the cut given by Ne(t/) = ~
A(t/')
_
dt/' + ~ A(t/).
(36b)
Introducing equation (36b) into equaüon (35), we obtain Fo(r/) = Tt(t/)N+(r/) - T3(t/)N-(t/),
(37)
where Te(t/) = il(r/) + (i)«27rS(r/, t/),
« = 1 or 3.
(38)
The Milne problem for partially plane-polarized light
669
For Ta(r/) ~ 0, we can write
Fo(rl)
G(rl)N+ (rl) _ N-(q),
=
(39)
where
(40)
a(u) = Tl(u) r3(u----)
We would like to construct a function X(z) analytic and non-vamshing in the plane cut from 0 to 1, in such a manner that the boundaries X±(U) on the out are related by
x+(u)
G(#) = X-(U)"
(41)
The solution of equatlon (39) would then be
N(z) = 1
1 fä Y(U)Fo(#)
2~--iX(--~
# ~
z
du,
(42)
where 7(#) =
x-(u)
(43)
T3(U)
This solution for N(z) should have the same properties as those of the initial definition #ven by equation (36). In particular, when z tends to infinity, N(z) vanishes at least as fast as l/z. This is possible if X(z) behaves as z~, n > 0 when z approaches infinity. In Appendix 7.7 of the author's thesis, MOURAI),(13) it has been shown that the nonvanishing solution to equation (41) exists. This solution is given by
1 f~ In G(U) X(z) = exp ~-i v # -~ z du. However, we do not need this explicit form of the soluüon since in the following analysis we will use y(n) rather than X(z). A regular non-linear integral equation for y(r/) has been developed in Appendix 7.8 of MOURAD(13) and should be useful for numerical calculation. Now we take the difference in boundary values, equation (36b), to obtain A ( 0 = N + ( 0 - N-(~).
(44)
If we evaluate the boundary values of N(z) as given by equation (42), and enter them into equation (44), we find
1 x P du+ 1 1 1 ù"':~ [Z,, x-,}So,O,,o~ù,_, ~[~+~],<,o,,
<,,,,
Since equation (43) can be put into the form X+(~)
X-(0
~(0 = #(~) + i~s(~, ,1) = 13(0 - i~s(~, ~)'
(463
equation (45) can be written as [/~2(~)+ n2s2(n ' ~)]A(O = s(n, n)
SO~(u) n -P u Fo(U) du + fl(g~01)Fo(q).
(47)
670
S . A . MoLrgAD
Substitutmg for Fo0/) from equation (34), we find [/~2(tl) + ~ 2 s 2 ( t l , tl)]A(tl) = S(tl, tl)
fó ~(~) ~ P- ,7 Go(P) d/t
+ S(tl, tl) fó1 fr(#_)tl d# fó S(IZ,v)v- IzS(#'#) A(v) dv
+ s(tl, tl)fó ;(#)tl_d~, föR(ù,,)d, +,(,O,(tl)Go(~) löS(tl,/~) - ~(tl)~(tl)
(tl, tl) A(Ig) dbt
-
~
tl
- B(tl)?(tl) föR(tl, z)~4(z)dl,.
(48)
We now use the Poincaré-Bertrand formula, #-tlv-/~
=
_
v-tl
+ ~2 ~ ( ~ _ tl) 6 ( ~ - v),
p-
v
(49)
to express equation (48) in the form 1
A(tl)
=
(50)
F(tl) + fo h(tl, #)A(#) d#,
where [fl2(tl) + ~z2S2(tl, tl)]F(tl) = S(tl, tl)
f
l o
p y(#)
#--tl
Go(#) d# + fl(tl)y(tl)Go(tl)
(5la)
and
[fl2(tl) + n2S2(tl, tl)]h(tl, v) = S(tl, tl) L(
- L(tl) + tl
?(#)
P
R(/g, v) dir
#--tl
_ ~(tl)?(tl){s(tl,v)_-v_tls(tl,tl) + R(ù,~)}; (51b) here we have lntroduced the notatlon L(x)
=
fl ~,(x') S(x', x)x ' -- S(x', x') dx'. x
(52)
Equatlon (50) is the Fredholm eqmvalent to the singular equation (29); thus we will be able to decide about the solvability of equatlon (29) by mvestigatmg equation (50). This equation has a unique soluUon lf, and only if, 1
1
f0 dtl fó dtl'[ h(r/, tl')l 2 < 1,
(53)
(see Mlra-ILIN(~4)). The complexlty of the kernel h(tl, tl') prohlblts us from analyfically performing the test given by equation (53). But an accurate numerical determination of this expression would at least give an indication of the uniqueness of the solution to equation (50) and hence of that of equation (29).
The Mflne problem for partiatly plane-polarized light
671
5. NUMERICAL ANALYSIS In tl'as section we wdl first demonstrate numencally that equation (29) with F(#) =/zO+ has a solution; then we will describe the numerical method for solving this equation. The verification of the solution is an lmportant step before proceedmg to the calculatlon of the different quantities, such as the exit distnbutlons, and the degree of polarization. Programming was carried out in F O R T R A N IV and executed on an IBM 360 Model 75 computer.
(a) Existence and uniqueness Here we seek to demonstrate the existence and uniquertess of the solution to equation (29) by reduclng the equatlon to a Fredholm integral equation by calculating numencally the norm of the Fredholm kernel. Though this procedure does not prove rigorously the required existence and uniqueness theorems, it does lend evldence to that suppositlon. Equatlon (50) is equivalent to the singular equation (29). A solution to the Fredholm equatlon exists and is unique if the condition given by equation (53) is satisfied. The function h(r/, v) may be written as h(r/, v) = a(r/, v) - b(q, v),
(54a)
where a(r/, v) and b(r/, v) are always positive for t/and v ~ (0, 1),
b(n, v) =/~(n)r(~)R(n, v),
(54b)
a(r/, v) = h(r/, v) + b(r/, v).
(54c)
Ih(,1, v)[ 2 _ la(q, v) l 2 + [b(rt, v) l 2
(55)
and
It follows then that
and, thus, the convergence criterion, equation (55), can be established numencally by verifying V(cx) < 1,
(56)
where 1
1
V(c,) ~ fó fo {I a(t/, v) l 2 + [ b(t/, v) l 2} dr/dv.
(57a)
The calculated values of V(cl) and 1
1
h(cx)~-fo fo Ih(•,
v)l 2 «vati
(57b)
are given in Table 1 for the values of the parameter % We note that, in all cases, the convergence cnterion is clearly satisfied numerically. Although the numerical verification of equaUon (53) is not a rigorous proof, we can eonclude that any numerical inaeeuracies reported in Table 1 are not sufficiently large to alter the conclusion that a solution to equation (29) exlsts and is unique. We proceed then to solve numerieally the singular integral equation equivalent to equation (50). Equation (29) is solv¢d by an iterative proeess; first, however, we remove the singularity in the following manner.
672
S.A. MOUI~.D Table 1. Check on the existence and umqueness of the solutlon to equatlon (50)
cl
h(cl)
V(cl)
0.0 0.1 02 03 04 0"5 0-6 0"7 0"8 0.9 1-0
0-000 0.041 0 083 0"125 0-167 0.209 0"248 0"284 0-313 0"328 0.321
0 000 0.044 0-089 0.135 0-180 0 225 0.269 0.308 0"339 0.356 0 348
(b) The singular equation Consider the integral
F(y) = f l f ( x ) dx, x-y which may be rewritten as
F(y) =
fô1 f ( xx) -- - fy( y ) F fó x~)- y dx.
Clearly the second integral in tbas last equatlon can be easily evaluated. The first term is not singular; however, when x = y, the mtegrand tends to the derivative of f (x). Ifwe divide the mterval (0, 1) into N ordinates and use a scheme of integration with weights w, we can write F(y) in the form N
F(y) = ~. wj Fj - F, + w, dF(y) x=x~ + F, In -1- -, x, Xj-
X~
~
Xt
(58)
where F, ~- F(xt). Expresslons s~mllar to equaUon (58) are used to evaluate the singular mtegrals in equatlons (29) and (3la). The integration is performed by an 81-point improved Gausslan quadrature scheme. At the end of every iteration, A + is calculated from equation (3 lb). The criterion o f convergence lsapphed to A + rather than every A I(~/), since A + is the quantity initially of interest. The lteratlons are cons~dered converged when e, the relative difference between two successive values o f A +, is less than 10-15. When they converge, equation (31 a) is used to evaluate
A~(,I). Contrary to most iteratave processes, this program requires an interpolation at the beginning of every cycle, instead of an extrapolaüon. Before we proceed to the evaluation of the outgoing dlstnbutions and the degree o f polarization, we need to show that the coeff~~ients obtained will yield a solution satisfymg equatlon (7).
The Mflnc problem for partlally planc-polanzcd hght
673
(c) Numerical verification of the solution The venfication consists of introducing the calculated values for the coefficients A+, Al(r/) and A2(v/) toto equation (29) and illustrating that to a certain degree of accuracy, the right-hand side is equal to the left-hand slde. IdeaUy we should like to venfy equation (29) polntwise, i.e. for every value of/Z ~ (0, 1). Such a verlfication, however, would necessltate the numencal evaluation of pnnc~pal value mtegrals, which obviously would lntroduce further errors. Rather than verifying the equation pomtwise, we prefer to compare various moments of that equatlon. Before taking moments, we dlgress and develop a recursmn relatlon for the moments of the function ¢¢(/Z). First we define M,(tl) as t h e / t h moment of ¢ù(/Z), viz. 1
M,(~/) & fö/z'?o(/Z)¢s(/z) d/z,
(59)
wherc ~s(g) is given by cquatmn (22c) and 70(/z) is the ?-function for the onc-speed conservat~vc case. Thus M,(t/) = t/fä1 ?o(/Z)/Zt- 10.(#)
~/ 2
(60a)
and M~(~/) = ~/M,_ ~ - -~/. 2
(5Ob)
But, we know from McCORMICK and KU~~ER,~~») that Mo(t/) = 0
(6la)
M~(T/) = --~/.
(61b)
and Thus all M,(~/) can be evaluated. We notice also that OR0/, #) and O20/, #) can be writtcn as ~R(t/, # ) = [¼(1 - ~/2)[3(1 - ~/2)~s(/z) + ¼ t / ( t / + / z )
I
- 6(~/-
#)][
0
and ~ 2 0 / , U) = 3(1 -
~2,o.,~,
11
1 + 60/--
(62a)
I #)
1-11
~02b~
1 "
I f w e now multiply cach component of equation (29) by/ZS?o(/Z), make use of equations (59), (60) and (61), and integrate over/z from 0 to + 1, we find that A+~o~ = ),o~+1 - ~q + r/M~_ t(r/) + -~ [rD'oN + 7oN+I] + 3(1 _~/2) _ 2 + MN_t _ r/N?0(T/)(C2 _ Ct)
(63a)
and A + ~ 0 N = ~ 0 N + ~ - - ?/N~o(?/)(C I - - C 2 ) "4-
[~
3(1 -- ~/2) _ 2 + MN-10/)
]
,
(63b)
674
S. A MOURAD
where 1
y0N - ~o#NY0(#)d#.
(64)
Letting Li N (i = 1 and 2) denote the left hand side of equatmns (63a) and (63b), and lettmg R, N denote the nght hand side of these same equatlons, we introduce out first verification by evaluating the followmg errors: eN ZX1
Rf L N'
i = 1 or 2,
(65a)
and e N &-- 1
R1N + R2N L1N + L2 N •
(65b)
For each cx consldered in this work, e, N, t = 1 and 2, and eNhave been calculated numencally for the first 10 values of N: N = 1, 2 . . . . . 10. Such calculations ylelded values for the various e's of less than 10 -6. The range of values is consldered satisfactory and help to establish confidence in the obtamed results for the numencal solutlon of the singular integral equation, equation (29). Another measure of the accuracy of the solutlon of equation (29) is to calculate the exit current whlch has been normahzed to unlty. The current is glven by J ~ f
+1g II
-12
1
I(z,/a) d#.
(66)
For every cl for wtuch the program was run, J is very close to umty The error, ] 1 - J I , does not exceed 10 -7. It is normal to obtaln better verlficatmn this way than by taklng moments because the evaluatlon o f the current does not lnvolve singular integral calculaUon. 6 RESULTS In thls sectlon we present graphs and some tables of most of quantltles related to the Mllne problem, the extrapolation dlstance Zo, the degree of polarlzatlon P(a), and the exit &stributions. It is clear from the definitlons given in Sectlon 2 that all quantltles related to the Mllne problem are expressed in terms of A + , Al(r/) and A2(r/). In partlcular, these coefficlents are used in equatlon (6) which gwes the mtenslty of the partlally polarized hght in the two states of polanzatlon. The coetficlent AI(~/) is the solutlon o f the singular lntegral equation (29), which has been solved numerlcally. The two other coefficients are expressed m terms of Al(r/) and are given by equations (3la) and (31b). The two continuum coefficlent A~(r/) and A2(r/) have been calculated for values of c~ ranging from 0 to 1 In steps of 0.1. Plots of these two furrctlons are shown in Flgs. 1 and 2. We noUce that, as cl increases, the resonance in Al(r/) is sharper and shifts towards ~/= 1. Table 2 gwes the exact values for the extrapolation distance, To, which we found to be equal to the dlscrete coeffioent A + In Sectlon 2, we have derlved expresslons for the ex~t distnbutlons and the degree of polarizauon Equatlons (12a), (12b) and (13) were calculated numerically for values of # ranglng from 0 to 1. The two components It(0, - # ) and I,(0, -/a), and P(#) have been rounded to s~x declmals and are presented m Table 3. We notlce that, for c~ = 0, the degree
The Mdne problem for partlally plane-polarized hght
-A1(,7)
~,«o /f~,o,
-0.20
-0.15
-0.10
-005
OI
0.2
0.3 0.4
0.5 OB 0.7 0.8 0.9
It
Flg, 1. The contmuum coefficlent At07).
-O'ZO['Aa(q)
0.1 O.z 0.3 0 4 0.5 0.60.T
0.8
Fig. 2. The contmuum cocfl~c~ent ,'lz('q).
Table 2 The exact results for extrapolat~on chstancc C1
"r o
00 0.1 0"2 03 0.4 05 0.6 07 08 0"9 1.0
0.710446 0-710539 0-710641 0.710754 0-710878 0-711017 0.711174 0.711353 0.711561 0.711808 0.712110
I0
675
s. A MOUSAD
676
Table 3. The exlt dlstr~butlons and the degree of polarlzaUon Cl = 1 B
Il
c I = 0.5
I,
P
I~
ct = 1 0
I,
P
Il
I,
P
0.0 01 0.2 03 04 0.5 0.6 0.7 0-8 0.9
0.3439 0.4290 0.4988 0.5649 0.6291 0.6922 0.7546 0.8164 0 8779 0.9391
0.3439 0.4290 0.4988 0-5649 0.6291 0.6922 0.7546 0-8164 0.8779 0-9391
0.0 0.0 0.0 0.0 0.0 0.0 00 0.0 0.0 0-0
0.3204 0.4111 0.4840 0.5525 0.6188 0-6838 0-7480 0.8116 0.8747 0.9375
0-3532 0.4349 0.5032 0 5683 0-6318 0.6943 0.7561 0 8175 0 8786 0 9394
00488 0.0281 0.0195 0 0142 0.0104 0.0076 0 0054 0.0036 0.0022 0-0010
0.2882 0.3820 0.4578 0.5294 0.5989 0.6671 0-7346 0.8015 0.8679 0-9341
0-3647 0.4435 0.5113 0 5740 0.6364 0-6979 0-7589 0.8195 0.8799 0-9400
0 1171 0.0745 0.0541 0 0404 0 0303 0.0225 0 0163 00111 0 0068 0-0032
10
1.0000
1.0000
0.0
1.0000
1 0000
0-0000
1 0000
1 0000
0-0000
P* 0.1171 0-0745 0.0541 0.0404 0 0303 0.0225 0.0162 0.0111 0.0068 0.0032 0-0000
* Chandrasekhar's results. o f p o l a n z a U o n is ldenUcally zero f o r all #. This result ~s expected since, for tlus e x t r e m e oase, the scattering is purely l s o t r o p i c ; t h a t is, each mtenslty c o m p o n e n t scatters equally in stares o f polarizaUon. F o r any value o f the p a r a m e t e r c I different t h a n zero, the degree o f polari z a ü o n is m o r e p r o n o u n c e d w h e n the scatterlng is at 90 °. In Fl g 3, the degree o f p o l a r l z a t l o n P ( # ) is p l o t t e d vs cl for the three values o f p. F o r any o f these curves, it is o b v l o u s that the p o l a r l z a t l o n effect b e c o m e s m o r e p r o n o u n c e d as cl tends t o 1 012 011 010 ó
009
~
0 OB
«
'ö
O.OZ o o6 0 05
0 04 t-
oo~
oo~/ «,
«~
o,
«,
oo
«o
o.,
«,
«,
,o
CI
Fig. 3. The degree of polarizaUon for three values of I.L.
1. 2. 3 4.
REFERENCES G. G. STOKES,Trans. Camb. Phd Soc. 9, 399 (1852) S. C'HANDRJ~g.tt~R, Astrophys. J 103, 351 (1946). V.V. SOBOX~v,A Treatbe on Radlatwe Transfer. Van Nostrand, Prmceton (1963) C E SEWERTand S. K FRALEY,Ann. Phys. 43(2), 338 (1967)
The Milne problem for partially plane-polarized light 5. 6. 7. 8 9. 10 11. 12 13. 14. 15.
S A. MOURADand C. E. SIEWEltT,Astrophys.J. 155, 555 (1969) K. M. CASE,Atm. Phys. 9(1), 1 (1960). A. LEONARDand J. H. FERTZIGER,Nucl. Sci. Engng 26(2), 170 (1966). P.S. SH~H and C E. SWWeRT,Astrophys. J 155, 265 (1969). D B. M~rCALFand P F. ZWEn~L, NucL Scl. Engng 33(3), 318 (1968). I KUSCER,N. J. McCoRMICK and G. C. SUMS~R~£D, Atm. Phys. 30(3), 411 (1964). W PORGO~ZELSKLlntegral Equatlons and Thezr Apphcations. Pergamon Press, Poland (1969). F . D . GAKHOV,Boundary Value Problems. Addlson-Wesley, Reading, Mass. (1966). S. A MOURAD,PhD Thesis, North Carolina State Universlty, Raleigh, N.C. (1970). S. G. MIKHLIN,IntegralEquattons. Macmdlan, New York, N.Y. (1964). N. J. McCoRMICK and I. Kus¢~R, J. math. Phys. 6(12), 1939 (1965)
QSRT Vol 14 No 8--B
677
THE MILNE PROBLEM FOR PARTIALLY PLANE-POLARIZED LIGHT S. A. MOURAD Division of Science and Mathematlcs, Fordham Unlverslty, Lincoln Center Campus, New York, New York 10023, U.S A
(Recewed25 September 1973) Abstraet--An accurate numerical method has been devised to solve half-range problems for the scattenng of azlmuth-mdependent polar~zed radmtlon m stellar atmospheres The use of the method is fllustrated by solving the Mllne problem for the radmtlon mtens~ty at any opUcal depth, for the extrapolaUon &stance, and for the law of darkenlng.
1. INTRODUCTION UPON scattering, light generally becomes polarized. Polarization greatly affects the intensity of hght, espe¢lally in planetary atmospheres and in those stellar atmospheres where free electrons are predominant. In pra¢ti¢e, parallel plane atmospheres are orten consldered sufficiently general sinee the transmission layer around a star or planet is optieally thin ¢ompared to the opti¢al radius o f the star or planet. The polarized light field is eharactenzed by four parameters known as the STOKESparameters. ¢1) If the in¢ident light is perpendi¢ular to the straUfied layers of the atmosphere, the system possesses azlmuth independen¢e and two of the parameters are always zero. The remalning two are the mtensitles of the light in different stares of polarization. The equation of transfer governing these lntensltles ¢an be found on¢e a suitable scatterlng law is estabhshed. For atmospheres of free electrons, atoms and molecules, Rayleigh's law is used. However, the presence of anisotropic partlcles causes depoladzatlon. Thus, the scattedng o f planepolarized light ean be descnbed by Raylelgh- and isotrople-s¢attering eomponents. The model just descrlbed also applies to resonance line phenomena where the two parameters represent the mtenslties o f photons in the ground and exclted states. The equaUons of the transfer for these problems were formulated by CHANDRASEKHAR.(2) In addition to the academie lmportan¢e and the mathematical aesthetic, the exact solution o f these equations would lncrease our knowledge of the stars, planets, and lllumination of the sunht sky. In the first part o f thls century, Sobolev and Chandrasekhar were the pdn¢lpal investigators mterested in the solutlon o f the aforementloned equations. Both scientists eoncerned themselves with the hmitmg case of Raylelgh scattering type. SOBOLEVt3) enumerated the different atmospheres where these equat10ns apply and gave the appropriate boundary conditlons for each. CHANDRASEKHARt2) mvestlgated the problem more thoroughly. Recently, an exact and more elegant solution of thls problem was obtained by SmWERT and FRALEY(4) who made use o f Case's method. They proved full-range and half-range •ompleteness and solved for the extrapolatlon length 661
QSRT VoL 14 No 8 - - A
662
S. A MOLr~D
Atmospheres in which amsotropic particles exist have not been glven much attentlon. Chandrasekhar is credited wlth formulatmg the correspondlng equaUons but he dld not attempt to solve them. It is the purpose of the present paper to find exact solutions for these equations and to use them m solving the Mllne problem for such quantmes as the exat distribuUons, the extrapolation distance and the degree of polarization. Due to the elegance and computatlonal merits of the method, Case's techmque is used. In Section 2 the equatmns of transfer are presented, thelr soluUons are dlscussed, and the parameters pertinent to the Milne problem are defined. The apphcatmn of the boundary conditions yields two singular equations which are reduced to one equation due to some particular properties of the problem. This reductlon is shown in Sectlon 3. The equation is then regularized in Section 4 by convertmg it lnto a Fredholm type. The ob•ective of SecUon 5 is to apply the method, developed in the previous sections, to the Milne problem. All calculatlons are carrled out numerlcally. The results are shown m Sectmn 6. 2 DEFINITION
OF THE
PROBLEM
The scattering of polanzed light m stellar atmospheres is described by equaUon (1) whieh has been formulated by CnANDRASEKHAR.(2) Thus, #
I(z,#) + I(z,p) = 2 -1 [c~K(#, #') + C2 E]I(~, #') d#'
(1)
with K(#,#')
]
#,2
1
1
Here # is the direction COslne (as measured from the mward normal to the free surface) of the directed radiation. The scattering kernel for th~s model conslsts of two parts. Orte part, welghed by c~, is of the Raylelgh type, and the second, weighed by c2, is isotroplc. The latter part causes depolanzation. Since the me&um is conservative, et and c2 sum to unity. We seek a non-triviat solution of equation (1) obeying the boundary conditlons of the Mllne problem: I(0,/# = 0 for tL ~ (0, 1) (3a) and hm ~b-q(% #) = 0. (3b) f--# 0o
A general solutlon to equation (1) was constructed m a paper by MOURADand SmWERT.(5) They used Case's method of normal modes and found +t
+1
l(z,/~) -- A+ ~+ + A_(z,/00+ + ~_f, Al(r/)~t(t/,/~)e-mdt/+
f~ A201)0201, tOe-'/" dt/, t
(4) where A +, A_, At(~/) and A2(r/) are arbltrary coefficients to be determined from the boundary conditions, and the eigenvectors are given by 0+= O,(r/, #) =
11 1'
(5a)
½(1 - r/2)ct ~~ r / - tL
, - c 2 ~(n - tL)
I
(5b)
The Milne problem for partially plane-polarizedhght
663
and _
~2)
] {r/(1
o2(r/, /~) =
[{q(1
P n -
+
~«(~) ~(n -
u)
_ 1~)
u
P r/2) ~-~ + ,;L2(q)ô(r/
,
(5c)
where 2,=(-1)'+3(1-q2)20(q)
for z = l o r 2
(5d)
and 20 = I -
t/tanh-l (q).
(5e)
W e need to apply the boundary con&tions given by equations (3) to determine the arbitrary coefficlents.By virture of equation (3b), t/is not allowed to assume values in the interval (0, - I); equation (4) thus becomes
föAl(n)~dq, 1
I(, #) = A+ @ + + A_(z, #)0+ +
l
l.Oe- m dtl + föA2(t/)O2(q ]~)e-'/~dr/. (6)
N o w we use the boundary condition given by equation (3a)to obtain 1
0 =A+*+
1
- v A - * + +föAiQl)Oi(q,l#dq+föA2(q)*2(ti, l~)dq,
#~(0, 1).
(7)
Choosing the normalize the solution by setting A_ equal to unity and transposing the second term to the left hand side, we obtain 1
1
/~(ID+ = A+ O+ + fö Al(t/)@l(q,/0 dr/+ fö A2(r/)O2(q, U) dq,
/~ 6 (0, 1).
(8)
The solution of thm equation for the arbitrary constants will be dealt with in Section 3. Once these coeflicmnts are known and introduced into equation (6), we can determine the üght intensity at any depth • in the atmosphere. In the remainder of this section, the parameters related to the MiMe problem will be defined.
(a) The extrapolation distance An important parameter assooiated with the Milne problem is the extrapolation distance %. The total intenmy, in other words the zero moment of the sum of the two angular intensities, is obtained from
-zf£
l(z) a _l t t 11 I(*,/~) de.
(9)
Ifwe introduce the asymptotic part of I ( , #) as given by equation (4), with At(t/) = A2(q) = 0 and A_ = 1, equatlon (9) yields I.,(,) = A+
+,
(I0)
The extrapolation length is the distance from the boundary x = 0, for which I~(,) = 0; thus Zo = .4+. (11)
664
S A. MOURAD
(b) The emergent radiatwn Of partlcular interest here lS the dmtribution of radiation at the boundary of the semlinfimte half-space. Tlus special oase is more commonly known as the exlt distnbution and is glven by equation (4) with • = 0 and la e ( - 1, 0), thus the integrals are no longer singular. We find
It(O, -la) = A+ + la + fö {cxr/(1
-
r/2)2Al(n) r / dr/ + la + :ó ½r/(1-- r/2)A2(r/) 71dq +-""~,
-
la ~ (0, 1), (12a)
and /r(O, --la) = A+ "{- la "4"
dr/ 371(1 -- F]2)A2(~) 71 -']- la,
la ~ (0, 1).
(12b)
(c) The degree of poIarization In general, the two &stnbutions Il(O, - # ) and I,(0, -la) are not the same. The degree of polarlzation is a measure of the difference in the two components. Let us denote the degree of polarizaüon by P(la); by definition, we can write I,(O,
-
la) -- I,(O,
-
la)
P(la) = Ic(O, - u ) + I,(O, - l a ) '
la ~ (0, 1).
(13)
la e (0, 1),
(14)
la E (0, 1).
(15)
Using equation (12), P(la) takes the torm c 1 A(la) , P(la) = 2I,(0, --la) + ct A(la) where A(la) =
fl
9r/(1 __r/2)2
A, (r/)
dr/,
r/+la
It is obvious from the previous definltions that all quantities pertinent to the Milne problem are expressed in terms of A+, At(r/) and A2(r/). In SecUons 3 and 4 we will develop a method to enable us to calculate these coefficients numerically. 3 THE SINGULAR EQUATION If instead of the Mitne problem another half-space problem is studled, we would obtain equation (8) with a different functlon, say F(#), on the left-hand side. Thus for half-space problems, in general, we must solve the equation 1 1
F(ù) = A+®+ + f0Al(r/)®,(r/, la)dr/+ f~a2(r/)®2(r/, la)dr/,
la~(0, 1).
(16)
To show that thm expansion m posslble, for any F(la), is eqmvalent to provlng half-range completeness. When such a theorem is proved, we can calculate the arbitrary coefficients by using the results of the proof or by constructing adjoint vectors or eren by resorting to numerical analysis. The proof of the half-fange completeness theorem for the parUcular case cx -- 1 has been achieved by SmWERTand FRALEY.(4) When the medium scatters isotropically (Cl = 0), the problem is slmilar to the one-speed neutron equation; the half-range theorems for this case have been proven by CASE.(6)
The Mflne problem for pamally plane-polanzed hght
665
In mathematleal construetion, two-group neutron problems are similar to the polarized light scattering equaüons; e.ach group is equivalent to one stare of polarization. A brief review of the completeness theorem in the neutron problem is thus helpful for the current study m polarized light. Although LEONAgD and FERZIGER(7) claim to have proven half-range completen¢ss their results are not ' constructive' in the sense that they do not y~eld expllc~t expressions for the coefficients and also they are not amenable to numerical analys~s. SHIEH and SIEWERT(s) mtroduced Chandrasekhar's S-function method to solve two-group neutron problems, but they did not prove the exlstence of a solution to the mtegral equatmns obtained. It should be mentioned, however, that the S-matrix approach has the merit of making the problem amenable to numerical analysxs Numerical solutlons to the two-group Milne problem have been suggested by METCALF and ZWEIFEL.<9) Thls paper was based on the eigenfunction method. The most direct approach to prowng half-space eompleteness is to follow the method used by CASE~6) for one-speed neutron problems. However, due to a dffficulty that remains unc~rcumvented, analytlcal results do not result, and thus we resort to a different method. Some inherent properties of the problem considered hefe allow us to reduce the half-range application to proving the existenee of a solutmn to a Fredholm equation. We will make use of the completeness of the set ~ + , O~(r/, #) and O2(r/, #) on the half-fange, for the case where c~ = 0, to reduce the coupled singular equations of expansion (16) to a singular equation, then the proof of half-fange completeness is established. In the remainder of this seetion, we will denve the equation. In Section 4 we will then attempt to solve ~t and show the possibility ofproving its ex~stence without actually solving it. Notice that O~(r/, #) can be wntten as
01(r/, I~) = c~OR(r/, #) + c2 ~P~(r/, #),
(17)
where OR(r/, #) and Of(r/, #) are the llmlting forms of Ol(r/, #) when c~ = 0 and cx = 0, respectively, i e. ®x(r/, #) = ô ( r / - #)
I
1, _ 1
(18a)
~R(r/, #) = ] ~b(r/, #) , I 0
and
q~(r/, #) = ¼(1 - r/2)
(18b)
(1 - #2) ~r / _ # + ~l(r/) 6(r/- u) •
(18c)
EquaUon (16) thus ean be wntten m the form
1
1
1
F(p) = A+ O+ + fo ¢lA'(r/)~I~R(r/' fl) dr/--~ fo ¢2 Al(r/)~l)/(r/,/A) dr/4- fó A2(r/)~[~2(?~,/./)dr/, (19) Transposing the term eontaining Os(r/, #) to the left hand side, we obtain
1
1
GR0~) = A+ ~ + + fó c, ~l(r/)®,(r/, u) dr/+ f~ A~(r/)®,(r/, U) dr/. where we have defined
(20)
1
GR(#) __4F(#) - f ctA1(r/)Oa(r/, #) dr/. Jo
(21)
666
S A. MOURAD
Smce O+ and O20/, la) are independent of cl and c2, equation (20) represents the expansion of G(#) m terms of the set of functlons O + , Ot(q, la) and O2(q, #) which has been shown to be complete by lncorporating the exlstlng proof of one-speed neutron theory. For this proof, we make use of the orthogonality relations for c 1 = 0. It is obvious that we shall obtain A+ and A2(r/) as functions of A~(~/), in addition to a singular integral equation for
Ad,0. Denotmg the vector adjolnt to Oi( q, la) by ¢~t?(q, la), we find S(r/) [1 , 0It(r/' la) = " 7 " OI(?/' la) -]'- "~O(la)O"(la) 1
(22a)
where S(q)
=
(22b)
~o(~/)Ao+(r/)Ao-(r/),
and r/
P
(22c)
O~(#) = -2 r / - l a + 2o(n) ~(n - la).
Here O~(la) lS Case's contmuum elgenfunctmn for the conservatwe one-speed neutron problem, A o + (z), ~o(~) and ?0(la) are the functlons for the one-speed ¢onservatlve problem. Multiplying equation (20) by ~1t(~/, #) and mtegrating over # from 0 to 1, we obtam, since OI?(q, #) is orthogonal to ~ + and O2(~/, #),
1 1 Go(r/) = fó A,(v)¼(1 - v2)c, fo ¢(v, p), [--~IS(t/) 6(q
] -- #) + ?o(#)~ù(#)
dp dv
+ c2 S(rl)Al(~l),
r/« (0, 1),
(23)
where we have defined 1
G°(rl)~-föflll?(rl' la)F(la)d#.
(24)
If we introduce y(q)
=
¼(1 -- ~/2)
(25a)
.4(,7) =
y(,0.41(,0
(25b)
and
in equatlon (23), we find Go(,0 =
Sòc [~ 1,4(v)
~(v, ,0 +
sò
]
~ o D ) ~ ( v , ~)~~0~) du dv "['- C2 S(~)A(?])/y(l~),
q ~ (0, 1).
(26)
We notice that th(v, #) can be wntten as 3v th(v, #) = 3(1 -- v2)Ov(#) + - ~ (v + la) - 6(v - la).
(27)
The Milne problem for paxtially plane-polaxlzed light
667
Introducing equation (27) into equation (26) we obtain the integral equation
S(r/)
rs(r/)
6o(r/) = c, L-T- ,h(r/) + - T
- ~o(r/)&(r/)] a(r/)
+ c2 S(r/)A(r/) + cly(r/)S(r/) A(v) y(r/)
v dv v -
,1
- fó1c~A(v)r0(v) ~r~r/_P v d r - ¼clr/fòvA(v)dv.
(28)
This equation can be written m the form 6o(r/) =/~(r/),4(r/) +
sò
R(r/, v)A(v) dv +
So"
S(v, r/) ~ _ ~ A(v) dr,
(29)
where c2 s(~)
B(r/) = - y(~) -
+ [2y(r/)s(r/) -
yo(r/)]Ct2o(r/),
= - ¼c~vr/,
R(v, r/)
(30a) (30b)
and
s(v,
,1) = v c l y ( r / ) s ( r / )
cd/
+ Œ
~0(v).
(30c)
Equatlon (29) is a singular lntegral equation for A(r/), r/e (0, 1). We, then, illustrate how the remaining expansion coeffieients A+ and A2(r/), r/e (0, 1), are obtained once A(r/), r/e (0, 1), is established. We multiply equation (20)alternately by Wo(~)~~(p)lll and Wo(~)l|l, the adjoints of 0 + and 020/) for ct = 0, and integrate over/z from 0 to 1 ; the following expressions for A2(r/) and A+ will be obtamed once relations previously developed by Ku~dER°°) and the orthogonality theorem are invoked. Au(r/) =
Sò~o(/~)~n(#) IT[ 1 F(#)dlz+ctAl(r/)-clr/
(3la)
and 1 IlIF(bt) + - 3cl A+ = fóTo~) ~ - fjol [3r/ I-~-Q/- z o ) - ro(r/)]Al(r/) dr/,
(31b)
where zo, the Milne problem extrapolation distance for the conservaüve one-speed netttron problem, is » .1
Zo = jo#7O(#) d/~.
(32)
In thIs section we have used known properties for the extreme cases ct = 0 to develop singular equations for A(r/). Furthermore, expressions for Az(r/) and A+ have been constructed by employmg orthogonality relations relevant to this limiting case. For this set, the singular equation plus the associated expressions for A+ and Az(r/), we attempt to prove half-range completeness m the followmg manner: the continuum coefficient A~(r/) is a solution of a singular equation. If this equation admits a s o l u ü o n , then
668
S.A. Mo~
thls solution, along with expressions for A2(r/) and A +, allow us to conclude that the expansion given by equation (16) is unique. In other words, we can then conclude that the set O+, O1(r/, #) and O2(r/, #) is complete. The main problem ~s to investlgate the existence of the solution to the smgular equation. This problem is treated in the next Section 4. 4. R E G U L A R I Z A T I O N
OF THE
SINGULAR
EQUATION
We believe that the reduction of equatlon (29) to a non-slngular form would put us m a better position to decide about the exlstence of a unique solution. Thls reduction is attalned either by regularizing the singular kernel or by subtractlng the singularity. According to POC,ORZ~LSKI,(u) the first approach is preferred by Hilbert and Poincaré, the founders of the theory of singular lntegral equatlons; the second method is favored by more recent workers. We prefer this last approach since the resultmg equat~on is of the Fredholm type. The later approach, often referred to as the Carlemann-Vekua method, is presented in a rlgorous and ¢lear manner by GAKHOVO2) and also by POGORZELSKI.tu) We rewrite equatmn (29) in the form 6o(t/) = #(r/),4(r/) + s(r/, r/)
fo1A(v)
P
v - t~
dv +
fò s(v' r/)v - s(r/' t/) •(v) t~
+
fo~R(v, r/)A(v)dv. (33)
Defining 1
1
- ~l s(t/, t/) A(v) dv - fo R(v, t/)A(v) dr, Fo(r/) a_ Go(t/) - fo s(v, ,7)v ---
(34)
we observe that the integrals involving A(r/) are non-slngular since S(v, t/) satisfies the Hölder condition (GAKHOV(x2)) and R(v, r/) is, by definition [see equation (30b)], a regular function. Using the defmiuon given by equation (34), we find that equation (33) takes the form:
Fo(r/) = fl(r/)A(r/) + S(t/, t/)
A(v)
V--r~
dr.
(35)
We consider now a function N(z) analytic m the complex plane cut from 0 to 1 along the real axis and defined by 1
dr/, N(z) = K~ fò r~A(r/) -
(36a)
with boundary values from above ( + ) and below ( - ) the cut given by Ne(t/) = ~
A(t/')
_
dt/' + ~ A(t/).
(36b)
Introducing equation (36b) into equaüon (35), we obtain Fo(r/) = Tt(t/)N+(r/) - T3(t/)N-(t/),
(37)
where Te(t/) = il(r/) + (i)«27rS(r/, t/),
« = 1 or 3.
(38)
The Milne problem for partially plane-polarized light
669
For Ta(r/) ~ 0, we can write
Fo(rl)
G(rl)N+ (rl) _ N-(q),
=
(39)
where
(40)
a(u) = Tl(u) r3(u----)
We would like to construct a function X(z) analytic and non-vamshing in the plane cut from 0 to 1, in such a manner that the boundaries X±(U) on the out are related by
x+(u)
G(#) = X-(U)"
(41)
The solution of equatlon (39) would then be
N(z) = 1
1 fä Y(U)Fo(#)
2~--iX(--~
# ~
z
du,
(42)
where 7(#) =
x-(u)
(43)
T3(U)
This solution for N(z) should have the same properties as those of the initial definition #ven by equation (36). In particular, when z tends to infinity, N(z) vanishes at least as fast as l/z. This is possible if X(z) behaves as z~, n > 0 when z approaches infinity. In Appendix 7.7 of the author's thesis, MOURAI),(13) it has been shown that the nonvanishing solution to equation (41) exists. This solution is given by
1 f~ In G(U) X(z) = exp ~-i v # -~ z du. However, we do not need this explicit form of the soluüon since in the following analysis we will use y(n) rather than X(z). A regular non-linear integral equation for y(r/) has been developed in Appendix 7.8 of MOURAD(13) and should be useful for numerical calculation. Now we take the difference in boundary values, equation (36b), to obtain A ( 0 = N + ( 0 - N-(~).
(44)
If we evaluate the boundary values of N(z) as given by equation (42), and enter them into equation (44), we find
1 x P du+ 1 1 1 ù"':~ [Z,, x-,}So,O,,o~ù,_, ~[~+~],<,o,,
<,,,,
Since equation (43) can be put into the form X+(~)
X-(0
~(0 = #(~) + i~s(~, ,1) = 13(0 - i~s(~, ~)'
(463
equation (45) can be written as [/~2(~)+ n2s2(n ' ~)]A(O = s(n, n)
SO~(u) n -P u Fo(U) du + fl(g~01)Fo(q).
(47)
670
S . A . MoLrgAD
Substitutmg for Fo0/) from equation (34), we find [/~2(tl) + ~ 2 s 2 ( t l , tl)]A(tl) = S(tl, tl)
fó ~(~) ~ P- ,7 Go(P) d/t
+ S(tl, tl) fó1 fr(#_)tl d# fó S(IZ,v)v- IzS(#'#) A(v) dv
+ s(tl, tl)fó ;(#)tl_d~, föR(ù,,)d, +,(,O,(tl)Go(~) löS(tl,/~) - ~(tl)~(tl)
(tl, tl) A(Ig) dbt
-
~
tl
- B(tl)?(tl) föR(tl, z)~4(z)dl,.
(48)
We now use the Poincaré-Bertrand formula, #-tlv-/~
=
_
v-tl
+ ~2 ~ ( ~ _ tl) 6 ( ~ - v),
p-
v
(49)
to express equation (48) in the form 1
A(tl)
=
(50)
F(tl) + fo h(tl, #)A(#) d#,
where [fl2(tl) + ~z2S2(tl, tl)]F(tl) = S(tl, tl)
f
l o
p y(#)
#--tl
Go(#) d# + fl(tl)y(tl)Go(tl)
(5la)
and
[fl2(tl) + n2S2(tl, tl)]h(tl, v) = S(tl, tl) L(
- L(tl) + tl
?(#)
P
R(/g, v) dir
#--tl
_ ~(tl)?(tl){s(tl,v)_-v_tls(tl,tl) + R(ù,~)}; (51b) here we have lntroduced the notatlon L(x)
=
fl ~,(x') S(x', x)x ' -- S(x', x') dx'. x
(52)
Equatlon (50) is the Fredholm eqmvalent to the singular equation (29); thus we will be able to decide about the solvability of equatlon (29) by mvestigatmg equation (50). This equation has a unique soluUon lf, and only if, 1
1
f0 dtl fó dtl'[ h(r/, tl')l 2 < 1,
(53)
(see Mlra-ILIN(~4)). The complexlty of the kernel h(tl, tl') prohlblts us from analyfically performing the test given by equation (53). But an accurate numerical determination of this expression would at least give an indication of the uniqueness of the solution to equation (50) and hence of that of equation (29).
The Mflne problem for partiatly plane-polarized light
671
5. NUMERICAL ANALYSIS In tl'as section we wdl first demonstrate numencally that equation (29) with F(#) =/zO+ has a solution; then we will describe the numerical method for solving this equation. The verification of the solution is an lmportant step before proceedmg to the calculatlon of the different quantities, such as the exit distnbutlons, and the degree of polarization. Programming was carried out in F O R T R A N IV and executed on an IBM 360 Model 75 computer.
(a) Existence and uniqueness Here we seek to demonstrate the existence and uniquertess of the solution to equation (29) by reduclng the equatlon to a Fredholm integral equation by calculating numencally the norm of the Fredholm kernel. Though this procedure does not prove rigorously the required existence and uniqueness theorems, it does lend evldence to that suppositlon. Equatlon (50) is equivalent to the singular equation (29). A solution to the Fredholm equatlon exists and is unique if the condition given by equation (53) is satisfied. The function h(r/, v) may be written as h(r/, v) = a(r/, v) - b(q, v),
(54a)
where a(r/, v) and b(r/, v) are always positive for t/and v ~ (0, 1),
b(n, v) =/~(n)r(~)R(n, v),
(54b)
a(r/, v) = h(r/, v) + b(r/, v).
(54c)
Ih(,1, v)[ 2 _ la(q, v) l 2 + [b(rt, v) l 2
(55)
and
It follows then that
and, thus, the convergence criterion, equation (55), can be established numencally by verifying V(cx) < 1,
(56)
where 1
1
V(c,) ~ fó fo {I a(t/, v) l 2 + [ b(t/, v) l 2} dr/dv.
(57a)
The calculated values of V(cl) and 1
1
h(cx)~-fo fo Ih(•,
v)l 2 «vati
(57b)
are given in Table 1 for the values of the parameter % We note that, in all cases, the convergence cnterion is clearly satisfied numerically. Although the numerical verification of equaUon (53) is not a rigorous proof, we can eonclude that any numerical inaeeuracies reported in Table 1 are not sufficiently large to alter the conclusion that a solution to equation (29) exlsts and is unique. We proceed then to solve numerieally the singular integral equation equivalent to equation (50). Equation (29) is solv¢d by an iterative proeess; first, however, we remove the singularity in the following manner.
672
S.A. MOUI~.D Table 1. Check on the existence and umqueness of the solutlon to equatlon (50)
cl
h(cl)
V(cl)
0.0 0.1 02 03 04 0"5 0-6 0"7 0"8 0.9 1-0
0-000 0.041 0 083 0"125 0-167 0.209 0"248 0"284 0-313 0"328 0.321
0 000 0.044 0-089 0.135 0-180 0 225 0.269 0.308 0"339 0.356 0 348
(b) The singular equation Consider the integral
F(y) = f l f ( x ) dx, x-y which may be rewritten as
F(y) =
fô1 f ( xx) -- - fy( y ) F fó x~)- y dx.
Clearly the second integral in tbas last equatlon can be easily evaluated. The first term is not singular; however, when x = y, the mtegrand tends to the derivative of f (x). Ifwe divide the mterval (0, 1) into N ordinates and use a scheme of integration with weights w, we can write F(y) in the form N
F(y) = ~. wj Fj - F, + w, dF(y) x=x~ + F, In -1- -, x, Xj-
X~
~
Xt
(58)
where F, ~- F(xt). Expresslons s~mllar to equaUon (58) are used to evaluate the singular mtegrals in equatlons (29) and (3la). The integration is performed by an 81-point improved Gausslan quadrature scheme. At the end of every iteration, A + is calculated from equation (3 lb). The criterion o f convergence lsapphed to A + rather than every A I(~/), since A + is the quantity initially of interest. The lteratlons are cons~dered converged when e, the relative difference between two successive values o f A +, is less than 10-15. When they converge, equation (31 a) is used to evaluate
A~(,I). Contrary to most iteratave processes, this program requires an interpolation at the beginning of every cycle, instead of an extrapolaüon. Before we proceed to the evaluation of the outgoing dlstnbutions and the degree o f polarization, we need to show that the coeff~~ients obtained will yield a solution satisfymg equatlon (7).
The Mflnc problem for partlally planc-polanzcd hght
673
(c) Numerical verification of the solution The venfication consists of introducing the calculated values for the coefficients A+, Al(r/) and A2(v/) toto equation (29) and illustrating that to a certain degree of accuracy, the right-hand side is equal to the left-hand slde. IdeaUy we should like to venfy equation (29) polntwise, i.e. for every value of/Z ~ (0, 1). Such a verlfication, however, would necessltate the numencal evaluation of pnnc~pal value mtegrals, which obviously would lntroduce further errors. Rather than verifying the equation pomtwise, we prefer to compare various moments of that equatlon. Before taking moments, we dlgress and develop a recursmn relatlon for the moments of the function ¢¢(/Z). First we define M,(tl) as t h e / t h moment of ¢ù(/Z), viz. 1
M,(~/) & fö/z'?o(/Z)¢s(/z) d/z,
(59)
wherc ~s(g) is given by cquatmn (22c) and 70(/z) is the ?-function for the onc-speed conservat~vc case. Thus M,(t/) = t/fä1 ?o(/Z)/Zt- 10.(#)
~/ 2
(60a)
and M~(~/) = ~/M,_ ~ - -~/. 2
(5Ob)
But, we know from McCORMICK and KU~~ER,~~») that Mo(t/) = 0
(6la)
M~(T/) = --~/.
(61b)
and Thus all M,(~/) can be evaluated. We notice also that OR0/, #) and O20/, #) can be writtcn as ~R(t/, # ) = [¼(1 - ~/2)[3(1 - ~/2)~s(/z) + ¼ t / ( t / + / z )
I
- 6(~/-
#)][
0
and ~ 2 0 / , U) = 3(1 -
~2,o.,~,
11
1 + 60/--
(62a)
I #)
1-11
~02b~
1 "
I f w e now multiply cach component of equation (29) by/ZS?o(/Z), make use of equations (59), (60) and (61), and integrate over/z from 0 to + 1, we find that A+~o~ = ),o~+1 - ~q + r/M~_ t(r/) + -~ [rD'oN + 7oN+I] + 3(1 _~/2) _ 2 + MN_t _ r/N?0(T/)(C2 _ Ct)
(63a)
and A + ~ 0 N = ~ 0 N + ~ - - ?/N~o(?/)(C I - - C 2 ) "4-
[~
3(1 -- ~/2) _ 2 + MN-10/)
]
,
(63b)
674
S. A MOURAD
where 1
y0N - ~o#NY0(#)d#.
(64)
Letting Li N (i = 1 and 2) denote the left hand side of equatmns (63a) and (63b), and lettmg R, N denote the nght hand side of these same equatlons, we introduce out first verification by evaluating the followmg errors: eN ZX1
Rf L N'
i = 1 or 2,
(65a)
and e N &-- 1
R1N + R2N L1N + L2 N •
(65b)
For each cx consldered in this work, e, N, t = 1 and 2, and eNhave been calculated numencally for the first 10 values of N: N = 1, 2 . . . . . 10. Such calculations ylelded values for the various e's of less than 10 -6. The range of values is consldered satisfactory and help to establish confidence in the obtamed results for the numencal solutlon of the singular integral equation, equation (29). Another measure of the accuracy of the solutlon of equation (29) is to calculate the exit current whlch has been normahzed to unlty. The current is glven by J ~ f
+1g II
-12
1
I(z,/a) d#.
(66)
For every cl for wtuch the program was run, J is very close to umty The error, ] 1 - J I , does not exceed 10 -7. It is normal to obtaln better verlficatmn this way than by taklng moments because the evaluatlon o f the current does not lnvolve singular integral calculaUon. 6 RESULTS In thls sectlon we present graphs and some tables of most of quantltles related to the Mllne problem, the extrapolation dlstance Zo, the degree of polarlzatlon P(a), and the exit &stributions. It is clear from the definitlons given in Sectlon 2 that all quantltles related to the Mllne problem are expressed in terms of A + , Al(r/) and A2(r/). In partlcular, these coefficlents are used in equatlon (6) which gwes the mtenslty of the partlally polarized hght in the two states of polanzatlon. The coetficlent AI(~/) is the solutlon o f the singular lntegral equation (29), which has been solved numerlcally. The two other coefficients are expressed m terms of Al(r/) and are given by equations (3la) and (31b). The two continuum coefficlent A~(r/) and A2(r/) have been calculated for values of c~ ranging from 0 to 1 In steps of 0.1. Plots of these two furrctlons are shown in Flgs. 1 and 2. We noUce that, as cl increases, the resonance in Al(r/) is sharper and shifts towards ~/= 1. Table 2 gwes the exact values for the extrapolation distance, To, which we found to be equal to the dlscrete coeffioent A + In Sectlon 2, we have derlved expresslons for the ex~t distnbutlons and the degree of polarizauon Equatlons (12a), (12b) and (13) were calculated numerically for values of # ranglng from 0 to 1. The two components It(0, - # ) and I,(0, -/a), and P(#) have been rounded to s~x declmals and are presented m Table 3. We notlce that, for c~ = 0, the degree
The Mdne problem for partlally plane-polarized hght
-A1(,7)
~,«o /f~,o,
-0.20
-0.15
-0.10
-005
OI
0.2
0.3 0.4
0.5 OB 0.7 0.8 0.9
It
Flg, 1. The contmuum coefficlent At07).
-O'ZO['Aa(q)
0.1 O.z 0.3 0 4 0.5 0.60.T
0.8
Fig. 2. The contmuum cocfl~c~ent ,'lz('q).
Table 2 The exact results for extrapolat~on chstancc C1
"r o
00 0.1 0"2 03 0.4 05 0.6 07 08 0"9 1.0
0.710446 0-710539 0-710641 0.710754 0-710878 0-711017 0.711174 0.711353 0.711561 0.711808 0.712110
I0
675
s. A MOUSAD
676
Table 3. The exlt dlstr~butlons and the degree of polarlzaUon Cl = 1 B
Il
c I = 0.5
I,
P
I~
ct = 1 0
I,
P
Il
I,
P
0.0 01 0.2 03 04 0.5 0.6 0.7 0-8 0.9
0.3439 0.4290 0.4988 0.5649 0.6291 0.6922 0.7546 0.8164 0 8779 0.9391
0.3439 0.4290 0.4988 0-5649 0.6291 0.6922 0.7546 0-8164 0.8779 0-9391
0.0 0.0 0.0 0.0 0.0 0.0 00 0.0 0.0 0-0
0.3204 0.4111 0.4840 0.5525 0.6188 0-6838 0-7480 0.8116 0.8747 0.9375
0-3532 0.4349 0.5032 0 5683 0-6318 0.6943 0.7561 0 8175 0 8786 0 9394
00488 0.0281 0.0195 0 0142 0.0104 0.0076 0 0054 0.0036 0.0022 0-0010
0.2882 0.3820 0.4578 0.5294 0.5989 0.6671 0-7346 0.8015 0.8679 0-9341
0-3647 0.4435 0.5113 0 5740 0.6364 0-6979 0-7589 0.8195 0.8799 0-9400
0 1171 0.0745 0.0541 0 0404 0 0303 0.0225 0 0163 00111 0 0068 0-0032
10
1.0000
1.0000
0.0
1.0000
1 0000
0-0000
1 0000
1 0000
0-0000
P* 0.1171 0-0745 0.0541 0.0404 0 0303 0.0225 0.0162 0.0111 0.0068 0.0032 0-0000
* Chandrasekhar's results. o f p o l a n z a U o n is ldenUcally zero f o r all #. This result ~s expected since, for tlus e x t r e m e oase, the scattering is purely l s o t r o p i c ; t h a t is, each mtenslty c o m p o n e n t scatters equally in stares o f polarizaUon. F o r any value o f the p a r a m e t e r c I different t h a n zero, the degree o f polari z a ü o n is m o r e p r o n o u n c e d w h e n the scatterlng is at 90 °. In Fl g 3, the degree o f p o l a r l z a t l o n P ( # ) is p l o t t e d vs cl for the three values o f p. F o r any o f these curves, it is o b v l o u s that the p o l a r l z a t l o n effect b e c o m e s m o r e p r o n o u n c e d as cl tends t o 1 012 011 010 ó
009
~
0 OB
«
'ö
O.OZ o o6 0 05
0 04 t-
oo~
oo~/ «,
«~
o,
«,
oo
«o
o.,
«,
«,
,o
CI
Fig. 3. The degree of polarizaUon for three values of I.L.
1. 2. 3 4.
REFERENCES G. G. STOKES,Trans. Camb. Phd Soc. 9, 399 (1852) S. C'HANDRJ~g.tt~R, Astrophys. J 103, 351 (1946). V.V. SOBOX~v,A Treatbe on Radlatwe Transfer. Van Nostrand, Prmceton (1963) C E SEWERTand S. K FRALEY,Ann. Phys. 43(2), 338 (1967)
The Milne problem for partially plane-polarized light 5. 6. 7. 8 9. 10 11. 12 13. 14. 15.
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