Ko Ko Aung: Lunar eclipse calculation 1296 2nd Wazo 16/
AD 1934 July 26
Page 1
True Tandeikhta longitudes
(1) Sun 3, 9;46
(2) Moon 9,12;44
(8) Rahu 9,17;19
(9) Ketu 3,17;19
3,17;19
–3, 9;46
0, 7;33 Sun's distance from node, less than 12˚, thus there is an eclipse
BE 1296 (15:14) 0; 14 Tithis at midnight
Kaliyuga 5035 –14; 8 Tithis at day before
NY kyamat 85 1; 6 Tithi difference
Ata 1 1; 6 X 60 = 66
Sutin 102 66 is the tithi change during one day
Page 2
60
–8
52 X 24 = 1248
1248/66 = 18;54,32
14/60 tithi = 0.233 tithi
1 tithi is 60/66 day, 0.233 tithi =0.212 day
0.212 day = 5;05,24 hours
Conjunction 24 – 5;05,24 = 18;54,36
This is the preliminary time of the middle of the eclipse
JPlanet for Mandalay has 18;41
Page 3 Mean longitude of the sun
102 (sutin) X 800 (parts in a day) = 81600
81600 + 85 (kyamat, sun's value at NY day) = 81685
81685 X 21600 = 1764396000 (conversion to arcmins)
1764396000 / 292207 = 6038;10 (3,10;38,10 cf. SEAC mean 3,10;38)
Page 4 Moon's mean longitude
5035 (KY years) X 7219167 = 36348505845
36348505845 / 25 = 1453940233:48
1453940233 / 21600 = 67212:1021
Moon's mean position at the beginning of the year 1021;48'
The Tandeikhta mean solar year is 365.2587565 days (Note typo in Irwin)
The mean lunar month is 29.530587946 days
The movement in elongation is then 12.19074949˚ = 731.4449694'/day
The mean motion of the sun is 0.985602654˚ = 59.13615926'/day
The mean motion of the moon is then 790.5811287'/day
In a mean year this is 288766.68'
In 25 year it is 7219167'. This is the number used above
The SurS has 57753336 revolutions in 4320000 years,
57753336 / 8 = 7219167 (4320000/25/360/60 = 8)
Page 5 Continued
85 (kyamat) X 79058 (Moon's mean motion X 100) = 6719930
6719930 / 800 = 8399:55
Extra correction 102 (sutin)/9 = 11:3 (corrects for the extra 0.0011286'/day)
102 (sutin) X 79058 (Moon's mean motion X 100) = 8063916
8063916 + 8399 = 8072315
8072315 + 11 (extra correction) = 8072326
8072326 / 100 = 80723:26
26X60 = 1560
1560 + 55 = 1615
1615 / 100 = 16:15
1021;48 + 80723;16 = 81745; 4
81745; 4 / 21600 = 3:16945; 4 Skip entire turns
Mean moon 16945; 4
Page 6 Moon's apogee
5035 (KY) X 488203 = 2458102109
2458102109 / 200 = 12290510;31
12290510 + 5140 = 12295650 5140 epoch value?
12295650 / 21600 = 569:5250
85 (kyamat) X 675 = 57375
57375 / 800 = 71:43
Extra correction 102 / 51 = 2
The factor 675 comes from 6.682974624' X 101 = 6.749804370' ≈ 675/100
The extra correction (X 100) is –0.019563 ≈ –1/51
102 (sutin) X 675 = 68850
68850 + 71 = 68921
68921 – 2 (extra correction)= 68919
68919 / 101 = 682:37
37X60 = 2220
2220 + 43 = 2263
2263 / 101 = 22:41
Thus the apogee is 5250;31 + 682;22 = 5932;53
The period is 3232.093673611 days
This is 6.682974624'/day or 2441.015003'/year
or 488203' in 200 years.
SurS has 488203 revolutions in 4320000 years
(4320000/200/360 /60 = 1)
Page 7 Lunar node
5035 (KY) X 116121 = 584669235
584669235 / 100 = 5846692;21
5846692 + 10945 = 5857637 10945 epoch value?
5857637 / 21600 = 271:4037
SurS with bija has 232242 revolutions in 4320000 years
This is 5.37597222 revolutions in a year
This is 116121' in a year and 3.179143493' /day
85 (kyamat) * 318 = 27030
27030 / 800 = 33:47
Extra correction fro 3.18: –0.000857 X 100 = –0.0857 ≈ –1/12
102 / 12 = 8:6
102 (sutin) X 318 = 32436
32536 + 33 = 32469
32469 – 8 (extra correction) = 32461
32461 / 100 = 324;37,7
The position of the lunar node: 4037;21 + 324;37 = 4361;58
Page 8 True longitude of the sun
Chaya
1 2 3 4 5 6 7 8 9
13 45 66 85 101 113 123 129 131
4640 Sun's apogee 77.333˚
+21600 360˚
26240
– 6038 Sun's mean longitude
20202 Anomaly
20202 / 5400 = 3:4002 Reduction to quadrant
5400 – 4002 = 1398
1398 / 600 = 2:198
Chaya interpolation
66 – 45 = 21 chaya difference
198 X 21 = 4158
4158 / 600 = 6:558
558 / 10 = 55:8 rounded to 56
45 + 6;56 = 51;56 equation of centre
True longitude of the sun 6038;10 – 51;56 = 5986;14
Page 9 True longitude of the moon
Chaya
1 2 3 4 5 6 7 8 9
53 104 152 195 232 262 285 298 303
Extra correction (??)
51;56 / 27 = 1;55 (solar equation of centre / 27 ???)
16945; 4 Mean moon From where comes this correction??
– 1;55
16943; 9
5932;53 Moon's apogee
+21600 360˚
27532;53
–16943; 9 Moon's mean longitude
10589;44 Anomaly
10589;44 / 5400 = 1:5189;44 Reduction to quadrant
5400 – 5189;44 = 210;16
210 / 600 = 0:210
Chaya interpolation
53 – 0 = 53 chaya difference
210;16 X 53 = 11144;8
11144;8 / 600 = 18;344
344 / 10 = 34:4 rounded to 34
18;34 equation of centre
True longitude of the moon 16943; 9 + 18;34 = 16961;43
Page 10 Location of the node
21600
–4361;58 Lunar node (ascending node)
17238; 2
True daily motions
59; 8 Sun's mean daily motion
790;35 Moon's mean daily motion
59; 8 X 21 (interpolation factor) = 1241;48
1241;48 / 600 = 2; 4
59; 8 - 2; 4 = 57; 4 True daily motion of the sun
Page 11
783:54 X 53 (interpolating factor) = 41546;42 (why the value 783:54)
41546;42 / 600 = 69; 15
790;35 + 69;15 = 859;50 Moon's daily motion
Moon's daily motion 859;50
Daily motion of the node 3;11
859;50 – 57; 4 =802;46 Daily motion in elongation
802;46 X 60 = 48120 + 46 = 48166
Page 12 Precession
5035 (KY) + 88 (epoch value) = 5123
5123 / 1800 = 2:1523
1523 X 9/10 =1370:7
7 X 6 = 42
The precession is 1370;42
5986;14 + 1370;42 = 6356;56
6356;56 / 60 = 122;27 (rounded) Sun's true tropical longitude
16961;42 + 1370;42 = 18332;24
18332;24 /60 = 305;32 Moon's true tropical longitude
Modern has 122;48 and 302;38
Page 14 (page 13 is identical to page 12) Noon shadows
122;27 / 90 = 1, 32;27 Sun's true tropical longitude
Computation of noon shadow (Mandalay parameters in Sin-Tel)
90 – 32;27 = 57;23 Degrees to go in rasi
57;23 / 30 = 1:27;23
159 – 92 = 67 Bhawa difference between rasis
27;23 X 67 = 1834;41
1834;41 / 30 = 61:4;41 Interpolated bhawa correction
92 + 61 = 153 Noon bhawa
165 - 153 = 12 Equinoctial shadow 165
12/60 = 0:12 Noon shadow (sun)
, JPlanet gives a solar noon shadow of 0;16
The Burmese Shadow application gives 0;17
Page 15 same for the moon
305;32 / 90 = 3, 35;32 Moon's true tropical longitude
Computation of noon shadow
90 – 35;32 = 54;28 Degrees to go in rasi
54;28 / 30 = 1:24;28
214 – 110 = 104 Bhawa difference between rasis
24;28 X 104 = 2544;32
2544;32/ 30 = 84:24;32 Interpolated bhawa correction
110 + 84 = 194 Noon bhawa
165 + 194 = 359 Equinoctial shadow 165
359/60 = 5:59 Noon shadow (moon)
For Mandalay JPlanet gives a lunar noon shadow of 6; 1
The Burmese Shadow application gives 6; 5
Page 16 Conjunction time
5986;14 Sun's true sidereal longitude
+10800 180˚
16786;14 Longitude of shadow
16961;42 Moon's true sidereal longitude
–16786;14 Longitude of shadow
175;25 Difference
175;25 X 60 =10529
10529 X 60 = 631740
631740 / 48166 = 13; 7 (rounded)
48166 is the true daily motion in elongation
13; 7 is the time of the conjunction in nadis before midnight
30; 0 – 13; 7 = 16;53 Time in nadis after noon
Page 17 Conjunction longitude
13; 7 X 60 = 787
859;50 (moon's true motion) X 787 = 676688;50
676688;50 / 60 = 11278:8;50
11278 / 60 = 187:58
16961;42 – 187;58 = 16773;45, conjunction longitude
Page 18 Conjunction position of the node
3;11 X 787 = 2505;17
2505;17 / 60 = 41; 45;17 rounded to 42
17238; 2 + 0; 42 = 17238;44 Node position at conjunction
Page 19 Moon's latitude
17238;44 Node position
–16773;45 Conjunction longitude
464;59 Distance from node
464;59 / 13 = 35;46 Moon's latitude, negative because the moon is
approaching the ascending node. Modern has –40
60/13 = 4.615.. Inclination of the orbit of the moon
Page 20 Diameters
790;35 X 60 = 47435 Moon's mean motion
859;50 X 60 = 51590 Moon's true motion
51590 X 31 = 1566290
1566290 /47435 = 33;43 (rounded)
This is the moon diameter given that the mean diameter is 31'
33;43 X 10 = 227;10
227;10 / 4 = 84;18 (rounded)
This is the shadow diameter 10/4 = 2.5 times the moon diameter
Page 21 Eclipse size
33;43 + 84;18 = 118; 1
118; 1 /2 = 59; 1 (rounded) Sum of radii
84;18 – 33;43 = 50;35
50;35 / 2 = 25;18 (rounded) Difference of radii
59; 1 – 35;46 = 23;15 Eclipsed part
33:43 – 23;15 = 10;28 Crescent
In digits this would be 12 X 23;15/33;43 = 8.27
Modern has 8.2
Page 22 Eclipse duration
59; 1 X 60 = 3541
3541 X 3541 = 12538681 Sum of radii squared
35;46 X 60 = 2146
2146 X 2146 = 4605316 Latitude squared
12538681 – 4605316 = 7933365 Difference
2816.62 Square root, length of eclipsed part
Page 23 Conversion to time
2816 X 60 = 168960
168960 / 48166 = 3;30 Half eclipse duration
25;18 X 60 = 1518 Difference of radii
Total duration
1518 X 1518 = 2304324, this is less than 4605316 so the eclipse is partial, there
is no total phase.
Page 24 Correction for the change in latitude during the eclipse
859;50 + 3;11 = 863; 1 3;30 X 60 = 210
863; 1 X 210 = 181200
181200 / 60 = 3020;33,30
3020 / 60 = 232:4
232 / 60 = 3:52
35;46 + 3;52 = 39;38 Latitude at the beginning of the eclipse
35;46 – 3;52 = 31;54 Latitude at the end of the eclipse
Page 25 Procedure with larger latitude
Repeating the procedure on page 22 gives 2623.699 and a half duration of 3;16
Page 26 Procedure with smaller latitude
Repeating the procedure on page 22 gives 979.141 and a half duration of 3;43
Page 27, 28 Beginning and end of the eclipse
16;53 – 3;16 = 13;37 Beginning of the eclipse
16;53 + 3;43 = 20;36 End of the eclipse
Conversion to hours and minutes by multiplying by 2 and dividing by 5
Beginning of the eclipse 5;26:48 (p.m.)
Middle eclipse 6;45,12 (p.m.)
End of the eclipse 8;14,24 (p.m.)
For Mandalay Modern has local times:
Beginning of the eclipse 17;19
Middle eclipse 18;41
End of the eclipse 20; 0
