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LUDWIK SILBERSTEIN
example:
216X4th order =X4-18X'+3X2+54X+324X3rd order= +18X3 -54X-6X2nd order = -3X 2+9X-9Xst order = -9XTotal = X4.
Thus, multiplying through by the coefficient E, it is seenthat if Z,o=216E; Z11 ,41 =324E; Z, o=-6E; andZ,,l=-9E; then the term EX4 would be (approximately)computed.
In a similar manner the conditions were determined for
each of the eight terms of the polynomial and combinedinto the formulas shown in Table III.
Derivation of Table VI
The derivation of the formulas in Table VI was simplein plan, but tedious to carry out in detail. The coefficientsof an eighth-degree polynomial through nine anchor pointswere determined in terms of the classical array of dif-
ferences using central differences according to Stirling'sinterpolation formula. The coefficients of -the polynomialwere then eliminated by using the formulas of Table III.
JOURNAL OF THE OPTICAL SOCIETY OF AMERICA VOLUME 33, NUMBER 9 SEPTEMBER, 1943
Simplified Formulae for Scattered and Rescattered Sunlight*
LUDWIK SILBERSTEINEastman Kodak Company, Rochester, New York
(Received May 3, 1943)
THIS subject, under the assumption of a flatT earth or plane-stratified atmosphere, wastreated four years ago by A. Hammad and S.
Chapman.' Their formulae for the reception ofthe primary scattered sunlight are simple enough,but those for the reception of rescattered or"secondary scattered" light are, in their ownwords, "immensely more complicated," so muchso that they are scarcely suitable for a numericalevaluation or even a general estimate of theeffect. To see this, a glance on p. 1091 will suffice.
The purpose of the present paper is merely tosimplify these formulae by certain obvious re-ductions and by discarding unimportant terms,and thus to make them easily applicable. Themerit of the method will be entirely due to thesetwo authors. We shall retain, for the present, all
their assumptions and, on the whole, theirnotation also.
Let p=p(h) be the density of air at a level h
above the ground and M= Jo-pdh, the total massof air over unit area. Then it is convenient tointroduce, instead of h, the variable
1 rhm= - fpdh
M (1)
*Communication No. 929 from the Kodak ResearchLaboratories.
IA. Hammad and S. Chapman, Phil. Mag. 28, 99-1 10(1939), where references to earlier investigations will befound.
or its supplement, m= 1-rm. The former is thefraction of the mass of air below, and the latterthe fraction above the level h.
Now, consider any two points, P at a level hand P' at a level h'. Let r be their distance apartand 0 the angle between their join and thevertical. Then Y-h = r cos 0 or r = (h'-h) sec 0.Now, with a view to the sequel, it is convenient todeal with fpP'pdr as a kind of effective distancebetween these points. By (1), this may be written
pdr = M sec Ofdm = M sec 0 * (m'-m).
The absorption and scattering by any volumeelement of air are assumed to be proportional tothe mass of that element. For a given wave-length, let and a be the absorption and thescattering coefficients of air per unit mass, andlet us put
a=Ma, s=Mo-, and c=a+s.
Then the attenuation exponent of light flux inpassing from P to P', due to the loss by absorp-tion and scattering, will be
(a+'ffpdr=c sec 0(m'-m) =K(m-m),
where K=C sec 0, so that, with the inclusion of
spreading by the inverse square law, the flux
526
FORMULAE FOR SCATTERED SUNLIGHT
received per unit area at the point P'(m') will be
II'exP - KW- m) J' (2)r2
where I is the flux per unit solid angle radiatedfrom the point P(m) and r is the distance from Pto P'. The variable K(m'-m) is thus, in general,a most convenient substitute for geometricaldistance, which is constantly used by Hammadand Chapman. It fails, however, when the twopoints are on the same level. For, in that case,m'-m=O and K= co. We may, therefore, availourselves of the formula (2), keeping in mind,however, that for every horizontal transit(h=const.) the indeterminate product K(m'-m)
is to be replaced by its original meaning,
cp(a+0r)pr=-r,
M
where p is the air density at the contemplatedlevel. This clause will scarcely entail anyinconvenience.
Refraction due to density variations being neg-lected, all light paths are here taken to berectilinear. Polarization, also, is wholly disre-garded. All symbols of intensities or fluxes, aswell as of the optical coefficients, are intended torefer to some narrow spectral region and todenote their amounts per unit wave-lengthinterval.
Let z be the zenith distance (angle) of the sunand Io the flux of sunlight per unit area "at thetop of the atmosphere," that is, before appreci-ably penetrating into it, and let
Z=c sec z.
Then the flux of direct sunlight per unit area at alevel h or m = 1-mn is, by (2), since the variationof the spreading factor is negligible,
I= Io exp [-Zm], (3)
e.g., at the ground, I= Ioe-Z. This formula wouldgive, at sunrise, or sunset, I= 0, which is palpablywrong, the discrepancy being due to the assump-tion of a flat earth. In fine, its application cannotbe pushed just up to z=90 degrees, though itmay be accurate enough for zenith angles butlittle smaller than a right angle.
By the definition of a, the light scattered by theparticle thus illuminated, per unit mass, in alldirections is aI, and thus, in a direction k makingwith the incident beam an angle +,, per unit solidangle, by Rayleigh's law,
El(h, k) =A I(1 + cos2 I),
where A is a numerical constant. Its valueis readily determined. If d is an infinitesimalsolid angle, we have EldC = a-I, and sinceJ(1+cos 2 i)d=47r+(4/3)r, the formula be-comes, by (3),
3a-IoEl(h, k) =- (1 +cos 2 I) exp [-ZT]. (4)
167r
If 0, are the polar coordinates of the direction(unit vector) k, the former counted from thezenith and the latter from the vertical planethrough the zenith and the sun, then
cos VI= cos z cos O+sin z sin cos .
E.g., with the sun in zenith, cos V'= cos .With this meaning of the symbols, formula (4)
gives the emission of primary scattered sunlight inany required direction k(O, 0) from a particleplaced at the level h(mn), per unit mass. Sincecos2 (r - 4) = cos2 , the emission in the oppositedirection (-k) has the same value.
Now, for the reception of the sunlight thusscattered; it will suffice to develop its expressionfor the case in which the receiving element ofarea dS is placed at the ground level. Let k be itsnormal and R(k)d,3dS the primary scatteredlight striking it in directions contained within theelementary solid angle d. All this light will bedue to the air contained within the cone dCo withk as axis and apex P placed at dS. The mass of anelement of this cone, at a distance r from P, beingpr2 drdo, its emission (flux per unit solid angle) isElpr2drd$. By (2), the flux per unit area receivedat dS is Eipdrd&e,-m, so that
RI(k) = pfEie-xmdr = M sec J Eie-,mdm
or, by (4), and since mn= 1-rm,
3s1oR1 (k) = K( +cOS2 V)e-z e(z-K)-dm
167rc
527
LUDWIK SILBERSTEIN
Ultimately, therefore, the primary scatteredradiation received per unit area at the ground levelwithin the cone d&' in any direction or, as we maysay, from any "point" (, ,) of the sky, isR1(O, )dCo in which
3s1O es e-zR1 (9, ') =-K( +COS2 )
167rc Z-K(I)
where cos = cos z cos o+sin z sin a cos q andZ=csecz, K=csecO.
This is Hammad and Chapman's formula,when the receiving point or element is placed ath =0, namely, their formula for Rd 1 (down-goinglight), p. 104, in this special case.
For 0=z, and any 4,
3sIoR1(z, -)= Ze-z(I+cos 2'fa,
167rc
where cos VI= 1-sin 2 z(1 -cosp). Thus the pri-mary scattered radiation received per unit solidangle in the direction straight from the sun
= 0),3sIo
R1(z, 0) = Ze-Z,87rc
while the direct sunlight received is, as above,
Ro(z, 0)=Ioe-Z.
For comparison of the brightness of the sky withthat of the sun, the scattered radiation may becomputed for a cone subtending the same solidangle as the sun (E= 0.000066):
eRi(z, 0) ,3s\= 0.000066 - sec z.
Ro(z, 0) 87r
Since the scattering coefficient s is (for X = 2y) ofthe order of 110, this ratio is of the order of8Xl0- 7 secz, e.g., 1.6X10- 6 for z=60 degreesand about 10-5 for z= 85 degrees. But, for reasonsjust given, the formula can scarcely claim com-plete reliability at such high angles, that is, forsuch low positions of the sun.
If the sun is in zenith, z= 0, Z=c, A=0, and theexpression (I), now of course symmetrical aroundthe sun, reduces to
R 1 (0) 1 +cos 2 0=-* 2c -co I-e-C(see -) (lo)
R1(0) 2c 1-cos 0
where3sIo
RI(0) =- e-c.87r
For small angles, even up to 0= 20 or 25 degrees,
R1(0) 1cos 0
R1(0) 2 cos
A numerical evaluation of these simple for-mulae will be given later on.
Now, for the much more intricate determina-tion of the rescattered or secondary scatteredsunlight.
Since our main purpose, in the present con-nection, is to find out the relative importance ofthe secondary and the primary scattering, it willsuffice to consider the case of the sun in zenith,i.e., z=0, Z=c.
An air particle P placed at the level h (or m)receives primary scattered light from all direc-tions, namely, the amount R1(h, k')d7' fromevery direction k'(O', 0') within the solid angletl)'=sin O'dO'do', of which it rescatters, per unitmass, in any fixed direction k(9, 0),* the fraction(3o/167r)(1 + cos 2 x), where
cos x=kk'=cos 0 cos '+sin 0 sin 9' cos O',
kk' being the scalar product of the two unitvectors. Thus, the complete emission from P(h)of secondary scattered light in the direction k,per unit mass and unit solid angle, is
3 oE2 (h, k) = - J (1+cos2 X)Rl(h, k')dCV.
The expression for R1 is to be constructed asbefore, only with h(m) replacing the ground leveland m', ^6", K written instead of m, 1', K, so that
3sIoR 1 -(1 +cos2 ') sec 9'
1 67r xJ exp [ - Z7m- KJm'- m I]dm',
where m'- m, I stands for the absolute value of
* Since, with the sun in zenith, everything is symmetricalabout the vertical, it is enough to consider k in a singlevertical plane and we may count the longitude 0 from justthis plane.
528
FORMULAE FOR SCATTERED SUNLIGHT
m'-m. Since, in our case, Z=c, ^6'= 6', andm' = 1-rn',
3sI0Ri = (1 +cos2 0') sec O'. e-c
167r
xf exp [cm'-K | m'-m ]dm'.
The required emission of rescattered sunlightthus becomes
9suIo 7/2 27r 1
E2(h, k)= ec r25 67r2 J
Xexp [cm'- ZI'm'-ml] tan 0'(1+cos 2 0')
X (1 +cos2 x)do'dk'dm'.
The angle O' occurs only in cos2 x, as definedabove, and we have
1 rf (1+cos2 X)d4
2+2 cos20 cos2 0'+sin 2
0 sin2 0'.
Next, the integral over m' is
J exp [cm'- K'(m-m')]dm'
+J exp [cm'-K'(m'-m)]dm'
exp -cm]-exp -K'm]=ecK -C
exp E-cm]- exp [- (K'm + c)]+ e'F(O'),say.K' + CJThus,
where
9so-I0E2 (h, k)= Q,
2567r
7/2
Q =f t F(6'){2+2 cos2 0 cos2 0'
+sin 2 sin2 0I} (1 +cos2 ') tan 'dM'
or, with x=sec 6' as integration variable,
Q== Jf (x) 2+sin2
1 1 dx+(4-2 sin2 0)-+(2-3 sin2 0)- -.
X 2 ~X4 X
If we introduce the coefficients
AO=2+sin 2 0, A=4-2 sin2 0,
A 2 =2-3 sin2 0, (6)
depending only on , then
.0 dx dx dxQ=Ao F(x)-+A fF(x)-+A 2 F(x)-.
(7)
It remains to evaluate the three integrals interms of the level (m) of the emitting particle, say
r0 dxLn,(m)f= F(x)-,
f1 Xnn=1, 3, 5, (8)
where
exp -cm] exp -cm]cF(x)= + -
x-1 x+1exp [-cmx] exp [-c(mx+l)]- - *~~~~~~~~~ (9)
x-1 x+1
The integral Li(m) is easily evaluated. Thefirst two terms of F(x) lead to logarithms and thelast two to the exponential integral as defined by
-T e-uduEi(-x) = ,
U
Collecting all terms, we find without difficulty
cL1(m)=exp [-cm][Ei(E)-log e+log 2cm- Ei (- 2cm) -Ei - cm) +e-eEi (- cm),
where e-*O. Now, Ei(e) -log e is Euler's con-stant, C=0.5772-* *, so that, ultimately,
(5) cLj(m) =exp[-cm][C+log 2cm-Ei(-2cm)]-Ei(-cm) +ecEi(-cm). (10)*
The remaining two integrals, L3(m) and L5(m),may be derived from Li(m) by.means of a recur-rence formula given by Hammad and Chapman.
In the first place, however, let us simplify therigorous expression (10) for Li(m). Since c is ofthe order of 110 and m, m 1, we may retain of thewell-known expansion of Ei the first three termsonly,
Ei(x)=C+log lx +x,* This is identical with the last formula of §17 of Hammad
and Chapman's paper for Li(m), with z=0, Z=c.
529
LUDWIK SILBERSTEIN
and reject also the higher terms in the expansionof the exponentials. Thus we find
Li(m)=0.423-m log cm-t log cm, (a)
where the constant term is 1-C, to threedecimals.
Hammad and Chapman's recurrence relation,*given at the end of §17,1 amounts in our case(z=O, Z=c) to
cL3(m) = c (m) +Ej3(cm) +Ej2(cm)-e-cEj3 (cm) + e-cEj2(cm) -2 exp [- cm]
and
cL5(m) = cL3(m) +Ej5(ci&) +Ej4 (cm)-e-cEj5(cm) +e-cEj4(cm) - 23 exp [- cm],
whereEjl(x) = -Ei(-x)
and, for n > 1,
Ejn(x) =-[e-ZxEj._(x)],n-iso that all required Ej. are at once reduced to Ei.Substituting the values of the numerous termsthus defined and rejecting again all c2 terms, weare left with cL3(m) = 2c, cL5(m) = 4C, so that thetwo integrals are constants, up to c terms,
L3(M) = , L(M) (I4 (la)
So also is Li(m), (a), correct up to c termsonly.
With the functions Ln(m) just defined, theemission of secondary light, (5) with (7) and (8),becomes
9scrIoE2(h, k)= {AoL(m)+AiL 3(m)+A 2 Ls(m)}I
2567r (12)
It contains the direction k(O) of the emissionthrough the A's, as given by (6), and the level or m of the emitting particle through the Ln(m),in the present approximation through L(m)only. The reception at the ground level of thelight thus rescattered will be expressed, in justthe same way as the primary reception, by
I
R2 (0) = M sec of E2e-1mdm.
* This relation should follow from our definition, (8) with(9), of the three Ln(m).
Thus, and since Mc=s, the formula for thereception f the secondary scattered sunlightbecomes
9S21OR2(0) =_ sec O{AoL1 +A1L3+A 2L 51,
2 56rwhere
Ln= fe-Lmn(m)dm.0
(II)
By (1 la), the integrals L3 and L 5 can be writtendown directly. It remains to evaluate the integralL1 with Li(m) as given by (lOa). This can beaccomplished rigorously in terms of elementaryfunctions and the function Ei, with the aid of asimple integration by parts.
We have, by (a),
0.423Li= -(1-e-X) -S-T
Kwhere
Now,
1S= fe-K~M log CM. di,
T= f e-mm log cm dm.
r~~cXK2S fxeX log - dx
log -(x+1)e= +I I+- e-xdx
re e-xdx=1e-,+J ~ +log e
+log - log c (1+K)e-x,K
where e-*O. Thus, and recalling that Ei(-e)-log e= C,
cK2 = 0.423 +log -
K
-{1+(1+K) log cIe +Ei(-K).
Next, since m+m= 1,
r° e-K rK CXr
T= -e-k eKTmA log cmdm=f- xex log-* dx,K og K
530
FORMULAE FOR SCATTERED SUNLIGHT
whence, integrating by parts and reasoning asbefore,
KI2T= 0.423+log +Ei(K) e +(K-1) log c-1.K I
This settles Li, while L3 and L5 follow directlyfrom (lla).*
Collecting the results, we have the followingapproximate values of the three functions L,:
0.423Li = ~(1 -e-)S- T
1L3 =-(I- e-K)
2 K
1
(13)
where
K2S=0.423+log cos 6+Ei(-K)
-{ 1+(1+K) log c}e- (14
K2T= {0.423+log cos 0+Ei(K)}e K
+(K-1) log c-1
and K=C sec 0. This holds for any angle 0.For small angles, K is but slightly greater than
c and, therefore, of the order of . E.g., themean of the Mount Wilson values, for X= 2p, isc=0.15. Thus, for small , up to some 8 or 10degrees,
sin2 S=T=2 logc-
C2
andsin2 0
L1=0.923+ -logc, L3=2, L5=4. (13o)c2~~~~~
E.g., for c=0.15 and 0=0, L1=2.82, a moderatenumber. For sin 0-. c or 6 equal to about 8' de-grees, L1 = 3.82 is still a moderate number. At theother extreme, for 6= 7r/2, the formulae give zerofor all three L,, as would follow also directly from
their definition,
Ln = e-mLn(m)dm.
With the functions L determined by (13),(14) and the A's expressed by (6), the receptionof the rescattered sunlight, with the sun inzenith, is now given by formula (II). For the sakeof comparison with the reception of primaryscattering it will be convenient to consider theratio of R2(0) to R1(0). Since R1(O)=3s1l/8rec,the required expression becomes
R2 (0) 3sec=- sec {AoLi+AiL 3+A2 L5}. (II')
RI() 32
This may now be compared directly with theexpression (Io) for R1 (0), the reception of primaryscattering. (Sun in zenith for both.)
To form an idea of the relative importance ofthe contributions due to the primary and thesecondary scattering in various directions (), letus assume the Mount Wilson attenuation coeffi-cient mentioned earlier, c=0.15. The scatteringcoefficient s=Mc is but slightly smaller thanc=s+a. If, say, s=0.12, then the numericalfactor in (II') becomes
3sec-= 0.013,32
whence we can judge that the secondary receptionwill be only a small or moderate fraction of theprimary one.
Let us consider this comparison in somenumerical detail.
The computation of the primary reception bymeans of the formula (Io) is simple enough. Theresults, for c=0.15, are collected in the followingtable, in which we have included the 7r/2-value,1/2c= 10/3, although the validity of the formulacannot be claimed for angles approaching 90degrees.
0= 900 86 85 80 75 70 60 50 45 40 300
=3.333 3.113 2.916 2.120 1.676R 1(0)
*We may mention here that if Rayleigh's scatteringfactor 1 +cos2 x is dropped, for which, however, there areno good reasons, then Ao=const. and A1=A2 =0, so that
1.419 1.161 1.055 1.031 1.014 1.001
we get rid of the functions L3 and L5. But Li(m) and itsintegral L1 remain exactly as before, so that in the end notmuch is gained by such an omission.
531
LUDWIK SILBERSTEIN
Below 300, R1(0): R1(0) = 1, to three decimals.On the other hand, we have, for the reception
of the secondary scattered sunlight,
R 2 (0)= 0.013 sec 0AoL 1+A 1L 3+A 2L5 },
where
AO=2+sin2 0, A1 =4-2sin 2 0, A2 =2-3sin 2 0,
as in (6), and L1 , L 3, L5 are given, to our approxi-mation, by (13) and (14).
For 0=0, as we saw, L1= 2.82, L 3 = 2, L5= l, sothat R2(0)/Ri(0)=0.106 or a little above one-tenth.
For 0=30 degrees, when =0.1 7 3,* we find, tothree figures, S=-1.53, T=-1.63, whenceL 1 = 3.55, L3 = 0.459, L5 = 0.230, and R 2(0) :R1 (0)
-0.148.
Again for = 60 degrees the ratio turns out tobe 0.190 and for 0=80 degrees, R 2(0): R1 (0)
- 0.230.Comparing these numbers with those of the
preceding table, we have the ratio of the tworeceptions:
at 0= 0R 2(0) : R1 (0) = 0.106
tion of light in its atmosphere. Some minor,though perhaps not quite negligible, correctionsmay also be made by pushing our approximationfor the functions Ln(m) and Lr one step higher.(See Note at the end of the paper.) In the mean-time, it would be interesting to test the formulae(I) and (II), as they stand, by actual observa-tions on the brightness of the sky in variousdirections, with any position of the sun for thefirst, at least, and with a high position of the sunfor both formulae.
Note. Since Li(m) is, generally, some six and twelve timesgreater than L3(m) and L5(m), it may suffice to consider theeffect of the higher terms upon the first of these functions.In simplifying the rigorous expression (10) for cLi(m), wehave retained only the terms of the order of c. To push theapproximation one step higher, let us include all terms in c2.Let cLI(m) be their total contribution to cLi(m). ThenALi(m) itself will be of the order of c, namely, after simplereductions,
ALi(m) = cAC-1 + (m2 +3mf2)
±m+ Am2 log m+ 2 log cm}.
The corresponding correction of Li, as required in the finalformula (II) or (II'). is AL = fnle-'fAiL,(m)dm and. as be-
300 600 800 fore, might be evaluated without difficulty. This, however,300 600 800 would greatly encumber our working formula. To have a
0.148 0.164 0.109. substantial estimate of the correction, it may suffice to, -nn fl - -h - -l - -zil f A T ). .. 1,;-inh ic
In fine, the contribution due to secondary scat-tering is from about one-tenth up to one-sixth ofthat due to primary scattering.
It seems likely that the ratio of the tertiary tothe secondary contribution will be the same oreven smaller. If so, then the emission and thereception of thrice-scattered sunlight, whosecomputation would be extremely tedious, maybe entirely left out of account. It would, nodoubt, be more important to take account of theneglected sphericity of the earth and the refrac-
* Ei(K) = - 1.013, Ei(-K) = - 1.355.
tU~bUC~ t1U {;2-VdlUC- U - Vt - W lilb 1>.. -
AL,(2) = (0.08+log C),
while
L(2)= 0.423-log c.
Thus, e.g., with c=0.15,ALI()=-0.19, while L(!)=3.01.
In round figures, LL(2)=-l/15-Li(2). Such, more orless, would also be the reduction of Li. The values of L3and L5 are, as we saw, of lesser importance. Ultimately,therefore, the inclusion of these terms would reduce thereception R2(0) by about 7 percent only of its values givenin the text of the paper, and the effect of yet higher termswould, to all purposes, be evanescent.
532
