On the Approximation of H-Function for Anisotropic Scattering. I.A-Rayleigh’s Phase Function

On the approximation of H-function for anisotropic scattering. I.A-Rayleigh’s phase function

S. Ka b a n ja t a n d S. K . Ba r m a n

Department of Mathematics^ Vniversitif of North Bengal, WeM Bengal, {Beceived 8 December 1972)

One of Karaiijai’s approximate forms for tlio //-function for isotropic scattering has beoji (jxtended to anisotropic; scattering. The pliaso function considered in this paper is the Rayleigh's pliaso function.

Comparison of calcu]at(;d results witli, the exact values given by Chandrasekliar shows good agreement. Results |iav(‘ beem applied to find diffuse reflection according to the Rayleigh’s pha>se function.

Indian J . Phya. 48, 126-132 (1974)

1. Introdttctton

Ih e solution of most of the problems of radiative transfo)’ in semi-infinite plane parallel atmosphere involve Chandrasc^.kJiar's //-function which is solution of the integi'al ecjuation

H { f ,) . . I ,1//'.

0 r-//

( 1 1

and is bmimlcd in the entire half-i)Jane Ji(Z) 0. nhore the cliaraeteristie func­

tion »//{//) is an oven potynoinial in // satisfyiiiR the condition

/ '/'(/0d/< < L ₍2⁾

The case of equality m (

. (2) is the conservative cas(». On rean-anaiiur oq. (1)

H{/i) (3)

To get an exact solution of eq. (3) wo are to find out first, a first approxi­

mation to ^.function and then to substitute it in the right hand side of eq. (3) and to continue the iterative process so long as the difference lietween two successive iterates is negligible.

when llio scattering is isotropic with albwlo w, we Jiavo

= constant = a>/2.

126

(4)

On the approxim ation o f H -fu n ction etc. 127 Thus for isotropic eonsorvativo case

= h 1). ... (5)

Different approximate forms for the £f-function for isotropic scattering have already boon considered by workers in tliis line, e.g. Abu-Shumays (1966a, 19666, 1967), Karanjai (1968a, 19686, 1969), Karanjai & Sou (1970, 1971), Harris (1957), Van de Hulst (1957), ChandrasekJiar & Breen (1947), Chamlborlain & Kuiper (1956), Chamberlain & McEIroy (1966), Sobolev (1949), Savodoff (1952), Stibbs iV: Weir (1959). In the above-referred works everyone considered the case of isotropic scattering. In this series of papers wo like to extend 4iff®i*^ut approxi­

mate forms to the case of anisotropic scattering and tabulatj& the -function and some otluM* related functions witli different phase functions. Since the //-function is a function of both w and /i we denote the function by II{w,/i) instead of i/(//).

Here in this paper w(^ consider the approximate form (Karanjai 1972, Abu-

♦Sliumays 1967)

H{u), fi) — l+a/^+^/i*7 ... (6)

where a and 6 are constants to bo determined, for anisotropic scattering with Rayleigh’s phase functions, according to which //(e^,//), and

are defined in terms of the characteristic functions

^ {1 + 1/819(1- --3 (2 ^ o>)//H I f I

and

respectively.

3a»

8"

3co

... (6a)

... (6l>)

... (6c)

Case / . Whefi the first approximate for the II-function is given hy eq. (6) and ike characteristic function is given by eq. (6a). Determination of the. constants a and b.

The H-function satisfies the following conditions (Chandrasekliar 1960) :

i) / H{w, = 1 - [ 1 - 2 / i/r{/i)d/i]* , ... (7)

-) ^1—2 ^ ^i^(/t)d/t j* / H{w,

Substituting for and from oq. (6) and (6a) into eq. (7) we get 128 S. Karanjai and S. K . Barman

where

K, = PJP,, S _ 2(1-.-t)

A = [^ 1 - 2 / i>(/<)d//y 1 - j ' ,

P, ---

9(1 J 3(2—0)) 1

^ 8 " ‘ 5 3

9(1^^.) 1 3(2-0)) 1

8 6 8 4

9(J-o;) 1 3(2-0)) 1 8 ’ 7 “ S' ' 5

■I-

+

8 3 ’

... (9)

(9a) (9b) (9c)

(9d)

(9o)

• (»f)

• (9g) Substituting for and <,>(/-/) from eq. (H) and (6a) into oq. (8) and using eq. (9) wo got

where

Ti¥+TJ)+Tz = 0, ... (10)

!Ti (w/4)/?/, ... (10a)

n = A(P,-P,K,H(wl2)R,P„ ... (10b)

7*3 = ^(P3 + P3Jf,)-f(w/4)/?i2-P3, ... (10c)

A = Pi+Peifi,. ... (lOd)

J?3 - P^-P,K„ ... (lOo)

p 9(l-<c) 1 3(2-£o) 1 9 1

’ 8 ‘ 8 8 • 6 + 8 ■ 4’ ... (lOf) jj 9(1 oi) 1 3(2—o)) 1 9 1

* 8 9 8 • 7 + 8 - 6 - ... (lOg)

Eq. (9) and (10) give the constants a and b.

O n the a p p ro xim a tion o f H -fuw stion etc.

129

Substituting for H{u}, fi) and from eq. (6) and (6a) into eq. (3) we get the second approximation formula as

1 H(oj,p)

where

= 1- pca [■ 9 2

1

(11)

(12⁾

Case II, When the first approximate for the H~function is given by eq, (6) and the characteristic function is given by eg, (66). Determhudion of the ronstants a and b.

and

Following tlie same procedure as has been done in case I, we get

a ^ K ^ --K 4 > , ... (13)

Tj6 2 + T ,6 + T 3 = = 0 , ... (14)

whore S, and R^ arc given by the equations (9a), (9b), (9c), (lOd) and (lOe), respectively, and

^ = ( * - ? » ) ■ Po = 2/15, P i - 1/12, Pa = 2/36, Ps = 1/24, P , = 2/63, P, = (3a*/16)Pg*,

P* = A(P^-P^Kt)+{^wlS)BiB^, P, « ^(P,+P,iCj)+(3«/16)i?i*-i>g.

Second Approximation formula ;

The second approximation formula is obtained as

where is given by eq. (12).

6

(13a) (13b) (13c) (13d) (14a) (14b) (14c) (14d) (14e)

(15)

130 S. K aranjai and S. K . Barman

Case 111. When the first apirroxmatc for the H-funetion is given by eq.

(6) and the characteristic function is given by cq. (6c). Determination of the constants a and b.

Following the Bamo prociMiuro as has boon doiio in oaso T, wc got

a = ^ K ^ -K .jK ••• (16)

and

Tjb^+T.,b+T^=^(), ... (17)

where, a« ))efoze A\, A'j, S, AV and are, respectively, given by the oq.

(9a), (9b), (9c), (lOd) and (lOe) and

n = 8/15, P, - 1 / 6 , P, = 8/105,

Pa - 1/24, P4 = 8/316, Pi = (3 /64)A2*.

= ^(P,-P3A2)-f-(3o>/32)iJiA2 1\ = J (P ,+ P 3 Ai)+ (3«/6 4)Ai*-P 2 . Second approximation formula :

The second approximation forjnula is obtaiiuMi as 3<u/(

... (16a)

... (16b) ... (16c) ... (16d) ... (17a) ... (I7b) ... (17c) ... (17d) ... (17e)

1 = 1^- • [(A) 2/0-t-/.,) \-a{I^—2I.j-\-I^-\-b{l2—2 /4 + /j)J (18) i/< *'(«,/«)" 32

where i „ is again given by eq. (12).

2. Discussions

Values of the il-function, viz., H(w, y), y), and iP*>(w, y ) have been calculated with the second approximation formulae given, respectively, by the eq. (11), (16) and (18) for w = 1 and /< = 0(0-2)l. In table 1 wo give the results obtained by our calculalions. Table 2 contains those values obtained by Chandrasekhar (1960). Comparison of our results with those o f Chandra­

sekhar shows that maximum error in //(to, y) is less than 0 005 per cent, and those ni f/<‘ >(tt>,/t) and H<^^w,y) are loss than 0 000006 per cent.

On the approxim ation o f H -function etc. 131 Tablo 1. Valuer of ^-functions calciilatod with eq. (11), (15) and (18)

with ci> ~ 1 H(/t)

0.0 1.000000 1.000000 l.OOOOOO

0.2 1.478628 1.013613 1.024467

0.4 1.879236 1.021314 1.032S43

0.6 2.264615 1.026522 1.036W7

0.8 2.642940 1.030274 1.0»9961

1.0 3.017684 1.033116 1.0411^6

3 2, Values of .fl^-functions given by Chandraaokliar (19

m / i )

0.0 1.000000 1.000000 1.000000

0.2 1.48009 1.01362 1.02448

0.4 1.88106 1.02131 1.03236

0.6 2.26660 I .02662 1.03679

0.8 2.64503 1.03028 1.03966

1.0 3.01973 1.03312 1.04170

3. Application

diffuse refloction in accordance with Rayleigh's phase function (con­

servative ease, oi — 1*0) is given by

9>o) ='

I

+ [(l-//2)H '2>(//)ir(l-//,® )»'«(/01onR 2(94„-5»j ... (19) wJiore

= jff(//)(3—c//,), ... (19a)

... (19b) and //(//), are defined in terms o f thc' cliaracteristic functions giv^on by equations (6a), (fib) and (6c) with a> 1.

The constants c and q are related to tJxe moments of the iEf-function by

and

q = 2/3a,

.. (19o) .. (19d)

132 S. Kaxanjai and 8. K. Bannan

For illuBtratioii we consider two caaes only

case (i) /^o ^ and - 0"

case (ii) /<o ~ *^‘8

The diffuse reflection for the above two cases have been calculated and given in table 3.

Table 3. Diffuse reflection for Rayliegh’s phase function

case (i) case (ii)

0.0 0.6377063 0.6675300

0.2 0.7163045 0.6906563

0.4 0.7402489 0,7459697

0.0 0.7583574 0.7839645

0.8 0.7808479 0.8135706

1.0 0.8383860 0.8383860

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Chandrasekhar S. & Breen F. H. 1947 A p .J . 106, 143.

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Chamberlain. ,T. W. & McElroy M. B. 1966 A p .J . 144. 114S.

Harris D. 1957 A p .J . 126. 408.

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