Research on the Secondary Scattering at Sunrise
and Sunset
Takao SATO Nagasaki University
(Manuscript received Nov. 14, 1964)
Introduction
S. Chandrasekhar has developed a method basing on the problem of radia‑
tive transfer in solving all orders of scattering in a plane parallel atmosphere of infinite lateral extent of finite depth on the supposition that the phase function can be expanded by Legendre polynominal. While Sato researched the
problem considering the earths atmosphere of 40 km depth, composed of 4サ105
uumbers of homogeneous spherical shell of 10 cm thickness because he has calculated the atmospheric density by the accuracy of 10 cm height.
He has prevailed moreover that the phase function can be exactely ex‑
pressed by the formula
72 n
甘sinz(on +甘sin'nn
, here n representing the order of the scattering 1. e n‑ 1 corresponding to the primary, n‑ 2 to the secondary scattering etc.
Strictly speaking Chandrasekhar s method can not be applied for the scatter‑
ing of any order concluding the primary in the low altitude of the sun for all
over the whole sky, and even if the sun′s altitude is high enough, it is not
applicable for the sky domain especially near the horizon for all orders of scattering on account of his assumption for the earth!s atmosphere. The main object to publish this paper is to give precise tables which are called Table 7 in Ref. 4), being not given in it for the convenience of restriction. However to give abruptly the Table only may be offensive to understand this paper, then the superposition of some portions in both papers would be permissible.
1. The model of the earth′s atmosphere
As is explained in Introduction the author has restricted the depth of the earth's atmosphere t0 40 km ln our problem it is indispensable to know the
val in summer and winter up to the hei*･ht of 40 km in Ref. I ). The author then adopted the mean values of the two as already stated in p. 12 in Ref. 2 ).
According to this treatment the density at 40 km amounts to 4glm3. It is therefore sufficient to regard 40 km height as the upper limit of the earth's atmosphere in the question of optical meteorology.
2. Some notations
Let O be the earth's centre and O' be any point on its surface. Take a point E in the s',ry dome of O'. We have divided the lines in the atmosphe‑ e intv foLrr e, tlal parts which pass through O' at the altitude of 0=30*, 60*, and 900. Let E, , E2' E. be the dividing points. Then Eo is identical with O' and E rs on the upper llmit. Consider a point T exposed to the direct solar ray in the atrnosphere which can be seen from En (n= O ‑ 4 ). Let A be the angle between two planes OO'E and the vertical plane at O' containing the sun's ceutre. Let ICO‑Fr=6t, and Al be the angle between two planes OET and OO'E.
Now we divide the sun's total energy into twel‑ve ranges denoted by A] A2"""
"'112 such that the partial energy in each ranb"e rs equal I Io here lo be‑
1 ;D 2' ing the solar constant and D the distance of the earth from the. sun.
3 . Numerical calculatiou
The evaluation has been executed for 900 interval ot A between 0<̲A and 900 interval of A1 between O A1 271. The value of C1 of the line tan‑
gential to the earth's surface is denoted by 61'. We have the following second‑
ary scattering intensity, given in Table 1. In the cases of A=0 and A=7r, the
7r nd A are rdentrcal wrth each other all 6 , E,8, 6 t as ex‑
values for Al=‑2
plained in Ref. 3).
Ref erences
l) Humphreys : Physics of the Air. p. 66.
2 ) T. Sato : Studies on the scatterins of the sun's light by the earth's atmosphere.
Science Rep. , Tohoku Univ. . Series 5, Geophysics, Vol.2, No. I March, 1950.
5 ) T. Sato : On the polarization angle in the scattering problem. Jour. Met. soc. Japan, Vol. 54, No. 2. 1956
4) T. Sato : The Contribution of the Intensity in Scattered Light of Each Wavelength to the Sky T,ight at Sunrise and Sunset. Ioc. cit., Vol. 59, No. 5. 1961
A
O
Explanation of the Table
Secondary scattering intensity received at a point O' on the earth's surface from an air portion of I m thickness with its centre at E bounded by one steradian with its axis at (6.A) direction and its vertex at O', whose air por‑
tion being expo >ed to primary scattering coming from one steradian with its vertex at E and the axis at (e,,Af)' here e:' column is applied to the lir*e section on the tangential line from E to the earthfs surface between E ar*d the tangential point on the surface, and Oi" column is applied to the section be‑
tween the latter point and the intersecting point of the upper limit of the atmosphere with the tangential line. The sign 61'/' corresponds to the sum of both values of e *' and ete'. The partial domain is expressed in IQ 10 ur*it
12D2 '
the total domain given in the last column is the sum of partial divided by 12, i. e. expressed in 02 lO unit. I
D
There are some 6 t" columns expressed by the sign s, each of which expresses that the section is in the shadow of the earth's body.
Table I 0=50, Eo
+ nof At
O
7t
¥¥A 0 ¥ O
90 95 lOO l 05 llO ll5 120 150 180 90 95 lOO l C5 llO l 15 l 20 150 90 95 l OO l 05 llO ll5 120 l 50
l
l 0‑7 lO 7 10 7 l0‑7 l0‑7 lO‑7 10 7 10 7 10 * 10 8 10 9 10 It
2 7 8 9 10 Il 12
5 4 5 6Total l0‑7
O . 1 9･ I O‑ '
226 ll85
l 9
1467 1476
l 478 765 589
85 l 528 l 956 l 976
1708
l 474 l 275
545 405
704 2270 1955 l 801 1460 l 218 l 020 417 509
l 720 21 80
1550
l 280 lOll 822 682 274 201
2275 l 672 l 084 825 657 515 425 169
l 24
2065
l O 92
664 484 570 295 244 96 71
l 493 6Sl 690 274 208 166 157 S4 59
7 1 5054 275 Ia42 157 592 l09 410 85 509 66 246 54 202
21 79 16 58
l O 92
554
l 99 l 58 l04 82 68 27 19
l 588 451 242
l 67 125 99 82 52 25
2146 655 566 2S2
l 90 l 50 l 24 49 55
･ 91
8561 7500 6862 5822 5055 4451
l 958 l 468 O . 84. I 0‑3
52 298 475 524 571 587 594
9 41 5 627 671 620 575 555 424
118 669 674 645 554 490 445 527
591 71 4 560 478 596 542 504 215
641 596 408 520 257 21 9 l 92
155 662 408 257
l 91 l 52 l 28 lll 76
525 252
l 54 llO 86 72 65 42
279 1222 l09 426 65 259 44 167 55 150 29 108
2S 9 17 6
450
l 45 81 56 44 56 1 21
587 178 98 68 56 44 8 25
91 9
270 149 l 05 80 66 57 58
2550 2728 2S62 2462 2202 2054
l 894 l 561 O . I O･ I O‑=
24 556
71 O 701 767 720 6 2
5.77 142 921 l018 905 797 697 448
72 829 l 086 l C55
852 710 598 547
268 l086 986 81 S 655 508 425 229
557
l 058 759 569 427 555 272
l 42
702 772 498 550 25 7 l 98 160 81
675 501 505 204 148 ll5 91 45
448 2260 225 908 127 489 85 517 60 226 46 172 56 156
18 67
919 1508 2131 515 591 598 166 205 509 l07 150 198 76 92 140 58 70 106 46 55 84 22 27 41
2551 5955 42S5 4021 5647 2915
25 1 5
1618
A
互2
π
AI
0
筏2
π
一π52
O
丑2
π
n of
横
命ミo㎜
90 95 工00
105
ユユ0
115 120 150 180 90 95 100 105
ヱ10 1ユ5
120
ヱ50 90 95 工00
105 110 115 120 150 90 95 100 105 110 ヱ15 120 150 90 95 100 105 110 115 120
ユ50 90 95 100 工05 110 工15 120 150 180 90 95 100 105 110 115 120 150
1
ユ0−7
2
10−7
5
10『7
4 5 6 7 8 9 10 11
10−710噸7 10一7
12Total
10一710、710『〕10−8ユ0一910一 10『7 0．84・10一3
57 547 578 649 709 ワ25 477 148
8．86・loo 455 751 817 767 715 658 540 101
118 751 785 785 685 609 545 262 77
ろ91
780 654 581 490 425 575 工72 50
641 652 476 589 519 271 256 107 51
662 446 500 255 188 159 157 61 18
525 279 工222 450 587 275 ユユ9 462 158 194 179 75 279 94 115 154 54 205 69 85 ユ07 45 ユ6ユ 54 66
89 56154 45 55
77 51 115 59 47
54 ユ5 50 17 20 10 4 14 5 6
919 295 174 126 99 82 71 51 9
2550 298ユ2986 2998 2725 2526 2550 1227 567 0．58・10−3
15 224 467 478 542 522 502
5．29・100 85 578 671 619 565 504 217
41 495 681 680 582 502 455 168
155 650 619 557 454 さ59 506
ユ1ヱ
507 621 476 574 292 255
工97
69 401 461 515 250
ヱ76
140
1工6
59
5852561291525747
500 154 545 188 254 191 80 507 ユ04 ユ28
エ54 55208 71 86
ヱ01 41 ヱ55 52 65 80 52 121 41 49 66 26 98 55 ≧0 22 9 52 11 ユ5
1218 558 194 150
96 75 61
20 1446 2564 2670 2647 2287 2058 1820 785
0．84・工〇一3 47 251 510 275 255 181 0
8．86・100 570 486 458 525 255 165 0
118 597 522 419 290 200 156
0
591 657 454 511 206 140 94
641 552 516 209 155 89 59
662 564 199
工25 80 52 54
525 279 エ222 450 225 97 5ワ7 ユ29
1エ9 49工85 65
72 29 109 57 45 18 68 25 29 12 44 15
19 8 29 10 587 159 76 44 28 18 12
919 241 116 67 42 27 18
2550 2454 1985 1607 1155 850 585
0
0．11・10一1 122 629 719 741 755 727 515
48
858 1027 1020 865 754 626 222
402 1246 工056950
758 607 501 171
985 1196 822 661 511 409 555
↓12
1299 918 575 426 522 255 209
69
ll79 600 552 250 187 ユ47 120
59
8554171745624795
557 150 572 ユ94 256 207 85 5工4 工06 工28 141 57 212 71 86 105 42 156 52 65 82 55 ユ22 41 50 67 27 99 55 40 22 9 52 1工 15
1226 558
工94
150
96 75 61 20
4522 4588 5979 5528 2942 2516 2188
80工 0．10。10『2
27 422 906 947 1089 1061 758
5．77・100 155 1090 1299 1225 1151 1027 551
72
905 1285 1518 1152 1007 882 412
268 1184 1167 1040 859 720 625 272
557 1151 898 725 577 472 401 168
702 841 589 446 548 281 256 96
675 448 2260 919 1508 546 245 989 545 426
560151578197241
260 106 404 157 166 200 81 506 105 125 160 65 245 82 99 155 55 200 68 82 54 21 79 27 52
2151 652 565 252 190
工50
124
49
2551 4508 5055 5150 4526 4152 5705 ユ919
1（1entical with ・・餐）
019・10『1
205 1002 1092 1084 1041 1004 645
85
1402 1657 1549 1265 1059 864 457
704 2085 165工 14工21079 859 695 ろ52
1720 2000 1510 1004 747 579 465 251
2275 1555 916 647 471 561 289 142
2065 1005 562 ろ79 275 208 ヱ65
8ユ
］495597 550 215 154 117
95
45 751 250 155 86 61 46 57 18
5054 956 500 522 229 175 157
67
1092 524 169 108
77 58 46 22
1588 595 204 151 95 70 56 27
2146 599 509 198 140 106
84 41
7915 7675 6542 5557 4505 5562 5022 1650
Table 2 θ＝50，E霊
A
0
A！
O
互2
π
，漣
甑
0 50 60 65 70 75 80 85
％（85＋θ1
θ1 θiノ θ！
％（90＋θ1
90 95 100 105 110 115 120 150 180 50 60 65 70 75 8e 85
％（85＋θド
θ1 θ1 θ！
双90＋θ1
90 95 100 105 110 H5 120 150 50 60 65 70 75 80 85
％（85＋θ！
θ雪 θ1
慰90＋θ1
90 95 100 105 110 115 120 150
1
10−8
2
10−8
5
10｛8
4
10−8
5
10一8
6 ワ
10一8ユ0一8
8
10−9
9
10−9 10 10−9
11 12
10一1010一12 Tota1
10一3 12 156 254
15 165 279 51 546 585 45 455 756 57 578 976 75 8401416 11712912206 26029285155
525 5720 6552
4工050209170 0 0 145 4ヱ0 5020 95ユ5
8861084217951
4125 19904 21157
150081462410475
12196 9916 6492
970771664576
7958 5192 5525 70］．044162805 5571 5755 2524 2599 1540 950
工681工08ユ 670
246 201 154 85 550
290 257 158 96 407
606 492 528 199 840 768 624 417 255 工067 1016 824 549 555 ユ404 1475 1195 797 485 2057 2528 1907 1280 779 5292
54654490502118417798
7047 5877 4005 2458 10496 工0684 98b5 7408 4902 22δ97 859 2050 2822 2676 16904 ユユ525 ユユ9ユ5 ユ〔）250 7578 5905ユ ェ7657 15444 8498 4955 20091 15724 9440 5155 2758 ユOl87 6098 5555 1666 842 5004
56141907 9464741678
2516 1521 655 527 1ユ55 1846 988 495 249 889 1558 829 415 209 748 1265 658 524 162 571 515 266 151 65 250
565 189 95 46 165
155 46 56 86 156 55 64 99
522 110 154 204
409 159 170 257
558 185 224 540 780 265 525 495 1262 450 526 799 2994 1021 ユ249 1898 4048 1584 1698 2585 9158 5218 4051 6254 8158 5177 4556 6962 ユ7276 6595 8407 ユ5］96 7514 25工9 5041 4592 5625 1180 ユ589 2〔」75
工045 555 590 579
581 186 216 520 400 128 ユ48 220 508 99 1工5 171 259 85 96 145 197 65 75 109 79 25 50 44 56 18 21 51
917
1085 2261 2865 5789 5495 8675 20671 26510 42497 9505 51999 64228 66447 42072 29827 22050 16800 工4458 11764 4916 5455
15 20 27 51 55 50 95
115 118
0
工18 205 666 459ろ4149 5565 5091 2875 2422 1815
149 218 255 297 586 558 965 1105 1547
0
1547 2592 5026 5554 5720 2800 2122 1894 1678 1189
252 570 450 469 655 917 1672 1948 2415
5
2418 4625 6746 4117 2520 1826 1575 1219 1059 740
265 217 146 89 580 590 521 2ユ7 ユ55 568 454 574 252 155 659
556 445 50工 185 790
694 577 595 242 1040
984 827 571555 1556
1846 1609 ユ150 755 5267 2641 1975 エ447 941 4266
28582800250216598455 79 559 9957745379 295751595295245514552 52554512510419108152 54045558205511094228
2449 1542 681 546 1240 1450 775 581 194 682
1014 555 266工55 475
775 4ユδ 208 105 576 682 564 185 92 528 580 505 150 75 264 401 209 105 51 181
146 219 254 505 402 597 1295 1709 5655 5158 6791 5159 1526 451 257
工65 150
1⊥4 91 65
50 61 95 75 92 工40
87 107 165 104 128 195
158 170 258
206 255 586
450 560 857
598 750 1！51
154517572751 150118775069
2644 5654 5820
107115151995
501 594 889 159 162 240
76 88 工51 52 61 90 42 49 75 57 45 65 29 54 50 20 25 55
992
ユ465 1706 2619工985 5752 7147 8665 12585 15085270工 19574 20978 15889 1ユ055 8508 6788 6150 5254 5779
17 40 54 64 81
116 216 269 224 S584
422 6704 6960 5979 5028 4517 5527 1887
184 51工 458 748 559 925
649 1工16 842 1451
1155 1927 1820 5051 1992 5265 2292 5829 2997
4，515 ユ0595 6975 5042 5592 2950 2587 1255
4924 7010 8565 4925 5ろ64 2ろ651916 1520
782
528 271 185 1ユ2 4ワ9 796 664 454 281 1208 978 8工1 555 542 ユ474 1188 991 681 425 1850 1548 1506 901 565 2447 2065 1754 1254 782 15467 5665 50工9 2252 1500 70ヱ6 5595 52工8 2597 1595 7442 4140 5695 2776 1871 8912 5598
7019 5145 2855 1895 1557 1079 857 426
4850 5507 2872 1550 1007 720 578 459 222
5691 55Z5 ユ477777
502 561 290 217 109
2550 2工27748
402 252 182
ユ46
109
54
12585 8796 27工9 工405
895 655 525 585 195
工85 468 572
ワ15 956 1572 2875 5049 5701 5258 6529 950 487 511 227 182 155
67 65
161 工97 245 550 478 ヱ0221084 1528 1918 1125 606 156
ユ00 75 58 45 21
78 118
197 501 242 570 505 462 408 622 596 915
ヱ299 2008 ユ577 2129
：Lワ02 2640
2497 1567 557 182 116
85 68 49 25
5904 2069 551 270 172 126 101
74 57
1255 5007 5689 4494 5872 7958 15605 14581 16866 22296 26095 50255 20541 ユ5145 11405 9461 7412 5971
Table 3 θ＝50，E1
A
丑2
A1
0
丑2
π
甑
，≧避0 50 60 65 ワ0 75 80 85
％（85＋θ1
θf θ1 θ1
％（90＋θ1
90 95 100 105 110 115 120 150 180 50 60 65 70 75 80 85
％（85＋θ1 ） θ霊 θ貿
％（90＋θ1 ） 90 95 100 105 110 115 120 150
50 60 65 70 75 80 85 ヲ鼠85＋θ！り
θ！
θ1 θ1
痴90＋θ1
90 95 100 105 110 115 120 150
1 2 5
10q8玉0−810−8
4 5
10一910−9
6
10一9 ワ 10、9
8 9
lG『lOlO『電o 10 lG『11
11 12
lG『竃210−13 Tota1
10一8 2 29 52 568
0 0 0 0 5 56100工094 8 87工56工698
12 150 255 2565
172115774140 285546557028
59 692 ユ285 14814
75850157618528
80 ユ070 20ユ0 24484 0 0 4 627 801070201425111
1472014407048047 5484909658452210
4575 5720 4520 260ユ9 457 4170 2847 工6299 4046 5249 2090 工2051 5594 2599 1576 8962 5402 2259 1466 8249 2812 1985 1265 6985
15769185765151 595261164 894
477 527 204 867 556 0 0 0 0 0 922 656 ろ95 1ワD5 665
工455 990 615 2647 工026
2！751507 955 40501575 552824471550 66552585 605242452670116684565 1555285ユ565202850711577 168211255582725800015545 244892047H4952770595555ユ
2856576168855526528754 27545262522181515050262285
42018 29229 18142 77929 50125
4088519990109554194515168 1457876575746154584690
8744 4595 2220 7900 2ワ42
65975ヱ96工607 57Hユ979 4855244ワ1244 44651555 444022451155 4054ユ416 5677ユ821 9ユ5 52421121 1648 817 409 工448 499 468 252 118 409 141
1155 1407 218 2ユ2 0 0 0 0
22ワ0 2815 42ワ 409
5528 4558 665 658 5407 66741017 965 8894109871675 1559
15772 19522 2978 2665
59754496687618 5649 5599067918工0451 7244
ユ25974 ユ62804 25542 ユ07D5 1ユ9951 175927 28507 2ユ45 245925 556751 54049 12848 105099 126575 ユ9259 17642 49858 59202 8867 20727
ユ509117615261516468
8805 10260 152ユ 12595 6556 7400 工094 9827
4994 5811 865 7654
4545 5266 785 7528
5596 4177 617 6508
工607 1861 277 2911
454 545 77 851
5 65 16 191 20 255 24 287 52 580 46 556 88 819
110 950 881059 S
l60 ユ689 2555
16619755404 545658404759
4059 4191 2987 5555 5128 2124 5297 2425 ユ614 2859 1989 1507 2281 1665 1074
88559157ユ
11612581055 726 449
559 5696512421611548
418 4508 5800 2655 ユ645
5095545469952752066 6767427656ユ44582806 95810524906264154081 14291610814510107787128
ユ590 18155 16548 12594 8295
18562057518750↓42519661
1950 749 2651 5178 481 58452279 7841 96701476 7ユ642801 96701ユ986工825
9005 5552 12217 工5ユ64 2515
ユ2517485516754207485175 181妬7ユ91250205ユ1904781 550551544747611602469298
58885 15954 56ワ91 ワ2251 11168
46226ユ92416914488666ユ5765 2677224521189181511264509
55ワ1628965195工51168649152 29655ユ680586944472ヱ5695 1750695ろ948512456 8776 12126651552691648 5879
9216500825551285 4604 7409400420251051 5686 59475142工562 785 2785 20541071 527 265 955
471 1590 1702 2098 2820 4025 6429 ワ5208551
274941004751508882048411059 1877765869779971184015219
5670 ユ8296 2ユ585 5ユ78 ユ6845
5055 982011441169812555 2040 6556 7627工150 9558 1607 5176 6058 894 7675 1280 4122 48ユ2 715 6565 962 5087 5587 55工 5168 522 1040 1180 177 1875 10−710−710一7
8 18 25 28 51 42 71 88 95 0 95
155 548 5569 5050 2160 1812 1118 697
0
10 25 26 51 40 55 85
ユ01
122 0
122 214 491 467 276 175 128 74 49 0
18 40 47 55 71 96 157
］86
228
0
228 455 658 552
ユ89 112 84 48 51 0
10『7 19 44 52 61 76 106 181 216 278
7
285
5U522 212 108 64 48 27 17 0
10『7 16 57 44 52 66 91 165 198 278 52 510 447 548 117 58 54 26 15 9 0
10−7 11 26 50 56 46 64 104 148 255 66 299 511 120 60 29 17 15 7 6 0
10一7 7 16 19 22 29 40 77 98 170
ワ8
248 195 110 51 15 9 7 4 2 0
10一8
50 69 81 96
124 176 549 448 875 606 1481 829 419 110
52 51 24 15 8 0
10一8 12 27 51 57 48 69 159
18ユ
581 52ワ 708 520 ユ52 58 18 11 8 5 5 0
10｛9
40 92
107 128 167 258 486 657 1409 1564 2775 1096 499 125
58 54 27 15 9 0
lG｝1010｛12
49
115 152 158 206 294 607 802 ユ8501979 5829 1546 592 145
68 59 51 17 ユ0 0
74
172 202 241 514 449 954 1254 2902 5242 6144 2048 887 215
ユOl 59 46 26 15 0
10一8
728
1655 ユ9402287 2925 4018 6906 8549 12ユ66 2440 ユ4606 18759 202ユ7 15455 82ユ55246 4085 2408 1558
A A1
5 71
* *f e')¥ O ;
60 60 65 70 75 80 85
1/2 (85 + o * ']
o,' o * "
o * "' 1/2 (90 + o *'
90 95 l OO l05 llO l 15 120 l 5a
l
l0‑7 2 l O‑7
5 l0‑7
4 l 0‑7
s l O‑7
6 lO‑7
7 1 0‑7
8 lO‑8
9 l O‑8
lO l O‑9
ll l0‑1 o
l 2
10‑1 2
Total
l O‑e 5
16 22 28 55 55
l 16 l 42 l 76 O l 75
405 2025 6591 5966 46C6 69 5 55Cl 2496 899
7 20 25 52 44 68 151 184 250 250 S74
l 067
765 501 550 261 21 l l 75 60
l 2
6 45 57 77 l 26
277 541 485 8 491 582 ll64 555 5 2 227 168 l 55 l 20
37 1 59 49 65 84 154 04 586 585 48 655 990 854 527
l 87 l 26 94 76 60 20
ll 55 41 52 71 11 256 528 554 ll9 675 764 5 1 l 78 99 67 51 41 32 ll
7 25 28 66 49 77
l 75
228 425
l 64
587 486 292 90 50 55 26 20 16 5
4
l 4 l 7
22 50 48 l a8 142 282 156 458 285
l 57 46 25 17 15 lo 8 3
19 60 74 94 l 0
204 461 612 l 515 99a 2505 l 158 580
l 64
89 59 46 57 28 9
7 25 29 56 50 79 l 78
2 8 5 6 477 lOIO 454 2C7 57 51 20 16 15 lO 5
26 79 98
l 25
175 269 609 816 l 884
1865 5749
l 458 674
l 84
99 66 52 41 51 lO
2 97 120 155 215 5 1 747 l005 2 78 2559 4957
l 761 795 214 ll4 76 60 48 6 12
48
l 48 l 85
2 5 145 5C5 l 157 1550 5665 4091 7754 2660
l 182 18
l 70 l 15
89 71 55 18
475 1458 1817 25 0 5141 4995 l 1270 14284 26548 5525 28875 55876 6278 22021 150 0 IC747 84 5 6905 5455 l 895
