^NCIENT INDIAN ASTRONOMY PLANETARY POSITIONS AND ECLIPSES
pQrvScSiyamatebhyo yadyat srestham laghu sphutam bljam tattadtbSvikalamaham rahasyamabhyudyato vaktum "I shall state i n full the best of the secret lore of astronomy extracted from the different schools of the ancient teachers so as to make It easy and clear." — PancasiddhSnUl^. 1.2.7
ANCIENT INDIAN ASTRONOMY PLANETARY POSITIONS A N D ECLIPSES
S. Balachandra Rao,
M.SC..
Ph.o.
Principal and Professor of Mathematics National College, Basavanagudi, Bangalore
B.R. Publishing Corporation [A Division of BRPC (India) Ltd.] Delhi-110035
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© 2000 S. Balachandra Rao (b. 1944—) I S B N 81-7646-162-8
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ACKNOWLEDGEMENTS I acknowledge my sincere gratitude to the prestigious Indian National Science Academy (INSA), New Delhi, and in particular to Dr. A . K . Bag, for sponsoring my research project on the subject. In fact, this project forms the genesis of the present book. I express my special thanks to Prof. K . D . Abhayankar (formerly of Osmania University), Dr. K . H . Krishnamurthy (Bangalore) and Dr. B.V. Subbarayappa (Bangalore) for their continuous encouragement. I record my appreciation to my research assistant Smt. Padmaja Venugopal for her contribution in completing the manuscript. For the sake of continuity of presentation of the subject I have used material from my earlier two titles : (1) Indian Astronomy—An Introduction, Universities Press, Hyderabad, 2000 and (2) Indian Mathematics and Astronomy—Some Landmarks, 2"^ Ed., 2"'' Pr., 2000, Jnana Deep Publications, 2388, Rajajinagar II Stage, Bangalore-10. I am highly indebted to all the authors and publishers of the titles listed it\ the Bibliography as also to the learned reviewers of my earlier works on the subject. Valuable suggestions are indeed welcome from discerning readers. S. Balachandra R a o
PREFACE
The present book, Ancient Indian Astronomy—Planetary Positions and Eclipses, is mainly addressed to students who are keenly interested in learning and becoming proficient in the concepts, techniques and computational procedures of Indian astronomy. These form an integral part of our Indian culture and are developed by the great savants of Indian astronomy over the past more than two millennia. A comparative study of the popular Indian traditional texts like Khanda Khadyaka of Brahmagupta (7th cent. A . D . ) , SHrya Siddhanta (revised form, c. 10th-11 th cent. A.D.) and Graha Laghavam of Ganesa Daivajfia (16th cent.) is presented. The procedures and algorithms described succinctly in these traditional texts for (i) the mean and true positions of the sun and the moon, the tdrdgrahas ("star-planets") viz.. Mars, Mercury, Jupiter, Venus and Saturn and (ii) computations of lunar and solar eclipses are elaborated with actual examples. A unique and pioneering feature of the present book is providing ready-to-use computer programs for the above-cited procedures. The 'source codes' (listings) of these computer programs are presented after Chapter 14. The advantage of these programs is that students and researchers in the field of Indian astronomy can readily use them for computations of planetary positions and eclipses according to the popular Indian astronomical texts used in our work and for a comparative study. It is humbly claimed that a sincere attempt to contribute significantly to the field of Indian astronomy is made here by (i) providing suggested improved procedures for computing lunar and solar eclipses, and (ii) suggesting bijas (corrections) for planetary positions to yield better results comparable to modem ones. The effect of the phenomenon of precession of equinoxes and the resulting ayandmsa, relevant to Indian astronomy, is presented in Appendix-1. The computation of lagna, the orient ecliptic point (ascendant), according to the traditional Indian method is included in Appendix-2. A detailed Bibliography of the original Sanskrit works and also of the secondary sources.in English is provided after the appendices. Fairly exhaustive Glossaries of technical words (both from Sanskrit to English and vice versa) and Index, for ready reference, form the last part of the book.
S. Balachandra R a o
DIACRITICAL MARKS FOR ROMAN TRANSLITERATION OF DEVANAGART
Short Vowels
a
i
u
r
Long Vowels
I
a
7
u
Visarga
e
ai
0
au h
Consonants
k
kb
t
th
fif
h
d
1 n
dh
ph
bh
b
n
1
th
t
CompoundConsonants
1
P
ch
c
m
n
dh
d
ks
tr
Anus\«ra y
r
/
V
s
s
s
h
Of II
m
jn
LIST OF FIGURES
Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig Fig. Fig. Fig. Fig. Fig. Fig. Fig.
2.1: 2.2: 2.3: 2.4: 3.1: 3.2: 3.3: 3.4: 7.1: 9.1: 9.2: 9.3: 9.4: 9.5: 9.6: 9.7: 10.1: 10.2:
Celestial sphere Equator and poles Altitude of pole star and latitude of a place Ecliptic and equinoxes Celestial longitude and latitude Right ascension and declination Azimuth and altitude Hour Angle & Declination Epicyclic Theory Nodes of the moon Earth's shadow cone and the lunar eclipse Angular diameter of the shadow cone Ecliptic limits Half durations of lunar eclipse Parallax of a body Angular diameter of the shadow cone Solar eclipse Angle MES at the beginning and end of solar eclipse
15 16 16 17 19 20 20 21 55 74 75 75 76 78 84 85 99 100
Fig. Fig. Fig. Fig.
12.1: 12.2: 12.3: A-1.1:
ifg/ira epicycle Retrograde motion of Kuja Stationary Points Precession of equinoxes
127 140 141 266
CONTENTS
Acknowledgements
vii
Preface Diacritical
1.
2.
3.
4.
v
Marks for Roman Transliteration of Devnagari
ix
List of Figures
x
INTRODUCTION—HISTORICAL SURVEY
1
1.1
Astronomy in tiie Vedas
1
1.2
Vedatiga Jyotisa
3
1.3
Siddhantas
S
1.4
Aiyabhata I (476 A . D )
5
1.5
Post-Aryabhatan astronomers
9
1.6
Contents of Siddhantas
11
1.7
Continuity in astronomical tradition
12
1.8
A i m and scope of the present work
13
ZODIAC ANDCONSTELLATIONS
15
2.1
Introduction
15
2.2
Equator and Poles {Visuvadvrtta and Dhruva)
15
2.3
Latitude of a place and altitude of Pole Star
16
2.4
Ecliptic and the Equinoxes
16
2.5
Zodiac
17
CO-ORDINATE SYSTEMS
19
3.1
Introduction
19
3.2
Celestial longitude and latitude (Ecliptic system)
19
3.3
Right ascension and declination (Equatorial system)
20
3.4
Azimuth and altitude (Horizontal system)
20
3.5
Hour angle and declination (Meridian system)
21
YUGA SYSTEM AND ERAS
22
4.1
Mahayuga, Manvantara and Kalpa
22
4.2
K a l i Era
23
xii
Ancient Indian Astronomy 4.3 4.4
5.
Introduction Working method to find Ahargana since the Kali epoch Ahargana according to Khanda Khadyaka {KK) Ahargatta according to Graha Laghavam (GL) Ahargana from the Christian date; finding the weekday Tables 5.1 to 5.3 for finding Ahargana
55 25 27 29 35 38-41 42
6.1 6.2
42 42 43 44 45 45
Introduction Mean positions of the sun and the moon Table 6.1: Revolutions of the sun, the moon, etc., in a Kalpa Table 6.2: Daily mean motions of the sun, the moon etc. Table 6.3: Mean positions of planets at the Kali epoch Mean positions of the sun and the moon 6.3.1 According to Suryasiddhdnta
(SS)
45
6.3.2 According to Kharida khadyaka (KK)
49
6.3.3 According to Graha laghavam (GL)
51
Table 6.4: Mean sidereal longitudes for 21-3-1997
54
T R U E POSITIONS O F T H E S U N A N D T H E M O O N
55
7.1 7.2
55 55 57 57 58 61 62 63 64
7.3 7.4 7.5 7.6 7.7
8.
25
MOTIONS OF T H E SUN AND T H E MOON
6.3
7.
23 24
AHARGANA 5.1 5.2 5.3 5.4 5.5
6.
yikrama Era Salivahana Saka Era
Introduction Epicyclic theory and Mandaphala Table 7.1: Peripheries of Epicycles of Apsis Mandaphala according to SS for the sun and the moon Table 7.2: Sines according to SUrya Siddhanta Bhujantara correction Further corrections for the moon True longitudes of the sun and the moon according to KK True longitudes of the sun and the moon according to GL
TRUE DAILY MOTIONS OF T H E SUN AND T H E M O O N
67
8.1 8.2 8.3 8.4 8.5
67 69 70 71 72
According to 55 True daily motions of the sun and the moon according to KK True daily motions of the sun and the moon according to GL Instant of conjunction of the sun and the moon Instant of opposition of the sun and the moon
xiii
Contents 9.
LUNAR ECLIPSE
73
9.1
73
Introduction
9.2
Indian astronomers on eclipses
73
9.3
Cause of lunar eclipse
74
9.4
Angular diameter of the shadow cone
75
9.5
Ecliptic limits for the lunar eclipse
76
9.6
Half durations of eclipse and of maximum obscuration
78
9.7
Lunar eclipse according to SS
79
9.8
Lunar eclipse according to KK
84
9.9
Lunar eclipse according to G L
91
10. S O L A R E C L I P S E
99
10.1
Cause for solar eclipse
10.2
Angular distance between the sun and the moon at the
99
beginning and end of solar eclipse
100
10.3
Computations of solar eclipse according to SS
101
10.4
Computations of solar eclipse according to G L
10.5
Computations of solar eclipse according
t
106 o
11. M E A N P O S I T I O N S O F T H E S T A R - P L A N E T S
1
1
3 119
( K U J A , B U D H A , G U R U , S U K R A A N D SANI) 11.1 11.2
Introduction
119
Table 11.1: Revolutions of planets in a Mahdyuga {SS)
119
D^iflrtMra correction for the planets
120
11.3
Mean positions of planets according to/lIT
122
11.4
Mean positions of planets according to G L
123
Table 11.2: Dhruvakas and Ksepakas
123
12. T R U E P O S I T I O N S O F T H E S T A R . P L A N E T S 12.1
125
Manda correction for the tdragrahas
125
Table 12.1: Peripheries of manda epicycles {SS)
125
Table 12.2: Revolutions of mandoccas in a Kalpa and their positions at the beginning of Kaliyuga
126
12.2
Sfghra correction for the taragrahas
127
12.3 12.4 12.5 12.6 12.7
Table 12.3: Peripheries of sighra epicycles (55) Working rule to determine the sighra correction Application of manda and ijg/ira corrections to faragra/ias True daily motion of the tdrdgrahas Retrograde motion of the tdrdgrahas Rationale for the stationary point Table 12.4: Stationary points for planets
129 130 133 137 139 141 142
xiv
Ancient Indian Astronomy 12.8 Bhujantara correction for the tdrdgrahas 12.9 . True positions of the tdrdgrahas according to KK
143 143
12.10 True positions of the tdrdgrahas according to GL
149
Table 12.5: Manddrikas of tdrdgrahas
149
Table 12.6: Sighrdrikas of tdrdgrahas
149
12.11 A comparison of true planets according to different texts
158
Table 12.7: Eight planets'combination
158
13. S U G G E S T E D I M P R O V E D P R O C E D U R E S F O R E C L I P S E S
160
13.1 13.2
Computation of lunar eclipse Computation of solar eclipse (for the world in general)
160 163
13.3
Solar eclipse for a particular place
166
14. S U G G E S T E D BIJAS ( C O R R E C T I O N S ) F O R P L A N E T S ' P O S I T I O N S 14.1
Introduction
170
14.2
Bijas for civil days and revolutions, mandoccas, epicycles etc., of planets
171
14.2.1 C i v i l days in a Mahdyuga Table 14.1: C i v i l days in a Mahdyuga 14.2.2 Revolutions of bodies in a Mahdyuga Table 14.2: Revolutions of bodies in a Mahdyuga 14.2.3 Peripheries of manda epicycles Table 14.3: Peripheries of manda epicycles Table 14.4: Earth's eccentricity and coefft. of sun's manda equation Tables 14.5 to 14.13: Eccentricities and peripheries of manda epicycles of planets 14.2.4 Mandoccas of planets Tables 14.14 and 14.15: Mandoccas of planets
'
170
172 172 173 173 173 174 175 176-180 181 181-184
14.3
14.2.5 Peripheries of Sighra epicycles of planets Table 14.16: Peripheries of Sighra epicycles Tables 14.17 to 14.24: Mean heliocentric distances and 5/^hra peripheries of eight planets from Budha to Pluto Moon's equations
14.4
The case of Budha and Sukra
191
14.5
Mean positions of bodies at the Kali epoch Table 14.25: Mean positions of bodies at the Kali epoch Table 14.26: Mandoccas of planets at the Kali epoch
193 193 194
14.6
Revolutions of bodies in a XaZ/jfl Table 14.27: Revolutions of bodies in a Kalpa Conclusion
194 194 195
14.7
184 184
185-189 189
xv
Contents COMPUTER PROGRAMS
196
APPENDICES - 1 PRECESSION OF EQUINOXES
266
- 2 L A G N A (ASCENDANT) BIBLIOGRAPHY A . Sanskrit Works B . Secondary Sources in Englisii G L O S S A R Y O F T E C H N I C A L T E R M S IN INDIAN A S T R O N O M Y I English to Sanskrit II Sanskrit to English INDEX
1
269 272 272 274 276 276 276 285
1
INTRODUCTION—HISTORICAL SURVEY Yathd sikha mayurdndm ndgdndm manayo yathd I tadvad veddriga sdstrdndm jyotisam (ganitam) murdhani sthitam II "Like tiie crests on the heads of peacocks, like the gems on the hoods of the cobras, stronomy (Mathematics) is at the top of the Veddiiga sastras—the auxiliary branches he Vedic knowledge". (Veddriga Jyotisa, R - V j , 35; Y - V J , 4) Astronomy i n the Vedas The above verse shows the supreme importance given to astronomy mathematics) among the branches of knowlege ever since the Vedic times.
(and
Even like many other branches of knowledge, the beginnings of the science of astronomy in India have to be traced back to the Vedas. In the Vedic literature, Jyotisa is one of the six auxiliaries (sadarigas) of die corpus of Vedic knowledge. The six veddrigas are : (1) Siksd (phonetics) (2) Vydkarana (grammar) (3) Chandas (metrics) (4) Nirukta (etyomology) (5) Jyotisa (astronomy) and X^ i.e., the place is to
the east of Ujjayini. For a place lying to the west of Ujjayini (i.e., X < X^),
the correction automatically becomes additive. Note that for a place with a westem longitude (with reference to Greenwich) X must be taken negative. Note :
In the Indian astronomical texts, the Desdntara
is obtained from the linear
distance (in yojanas) of the place from Ujjayini. We shall now apply the Desdntara correction to the sun, the moon and moon's apogee and node for Bangalore at the local midnight between 20th and 21st March, 1997. Taking the longitudes of Bangalore and Ujjayini respectively as 77° E 35' and 75° E 47', we have ( X - X j / 3 6 0 = 1.8/360 (i)
Desdntara
correction for the sun :
Since the daily mean motion of the sun is 59'. 136078 according to the SUrya Siddhdnta, we have Desdntara correction for the sun = (1.8/360) x 59'.136078 = 17".7. Therefore, the mean longitude of the sun on 21-3-1997 at the preceding local mean midinight of Bangalore is 334° 14' 09" - 17".7 = 334° 13' 51".3
Motions of the Sun and the Moon (ii)
49
Desdntara correction for the moon : Daily mean motion of the moon : 13°. 176352. Therefore, Desdntara correction = (1.8/360) X 13°. 176352 = 3' 57" Mean longitude of the moon at the local midnight preceding 21-3-1997 at Bangalore is 117° 48' 2 5 " - 3' 57" = 117° 44' 28"
(iii)
Desdntara correction for the moon's apogee (mandocca) : Daily mean modon of the moon's apogee is 6'.6829747 Therefore, for the moon's apogee, Desdntara correction = (1.8/360) x 6'.6829747 = 2". The mean longitude of the moon's apogee at the mean midnight
preceding
21-3-1997 at Bangalore is 1 3 1 ° 5 9 ' 4 6 " - 2 " = 131° 5 9 ' 4 4 " (iv)
Desdntara correction for the moon's node : Daily mean motion of the moon's node is - 0°.052985 or - 190".746 so that Desdntara correction = - (1.8/360) x 190".746 = - 0".95373 = - i " approximately.
Thus, the mean longitude of Rahu for the local mean midnight preceding 21-3-1997 at Bangalore is 158° 56' 29" - (- 1") = 158° 56' 30" 6.3.2 According to Khanda khddyaka (KK) (i)
Mean Longitude of the Sun In the Khanda khddyaka, the mean longitude of the sun is given by : ,
(A X 800) + 438
, .
If the above value is multiplied by 360°, we get X in degrees from which the integral multiples of 360° (i.e. the completed number of revolutions) have to be removed. Example : For March 21, 1997, the KK Ahargaria is 4,86,498 i.e., A
= 4,86.498. Therefore (4,86,498x800)+438 292207 =
1331.928523 revolutions.
Removing the completed number of revolutions viz., 1331, we get X i.e., X
= 0.928523 rev. = 334° 16'06"
50
Ancient Indian
Astrononvj
(Compare this with the SS value : 334° 1 4 ' 0 9 " for the same date and time.) (ii)
Mean Longitude of the Moon In KK the mean longitude of the moon is given by : -
( A x 6 0 0 ) + 417.2 , . 16393 revolutions
~
4929 Example
With A
minutes of arc.
= 4,86,498, we get
X
= 17806.33302 revolutions -98'.70115642 = 119° 53' 1 4 " - 9 8 ' 4 2 " = 118° 14'32".
(iii)
Mean Longitude of the Moon's Apogee (Mandocca) In KK the mandocca of the moon is given by M =
A-453.75 revolutions + mm. 3232 39298
Example : With A = 4,86,498, we have M = 150.3849783 revns. + 12' 2 3 " = 138° 4 7 ' 5 5 " (omitting the revns.) (iv)
Mean Longitude of the Moon's Node (Rdhu) In KK the mean longitude of the moon's ascending node is given by n
=-
A-372 degrees of arc rev + 6795 514656
Example : With A = 486498, the mean Rahu is n
= - [71.54172185 rev. + 0°.945287726] = - [ 7 1 ' ' 1 9 5 ° 1' 12" + 0 ° 5 6 ' 4 3 " ] = - [ 7 r 195° 57'55"] = 164° 0 2 ' 0 5 "
by removing the completed revns. (-71) and adding 360°. Note:
Lalla (c. 786 A.D.) in his celebrated text Sisya-dhi-vrddhida gives a bija (conrection) for the mean longitude of the moon's node. According to Lalla's rule, the bija is at the rate of - 96' for the lapse of every 250 years since 499 A.D. (the year of composition of the Aryabhatiyam). This means that for the year Y A.D. the bija to the mean longitude of Rahu is (- 96') x ( K - 499)/250.
Thus, in the example considered, for March 21, 1997, Bija = - 96' X (1997 - 499)/250 = - 96' x 1498/250
Motions of the Sun a n d the Moon
51
= -575'.232 = - 9 ° 3 5 ' 14'I" Now, as calculated earlier. Mean Rahu
164° 02'05'
Bija (correcdon)
- 9 ° 35'14' I"
Corrected Rahu
154° 26' 51
Desantara correction according to KK (i)
According
to the Khatjda khddyaka, the daily mean motion of the sun,
d = 59'%" Here, X = 77° 35' and
= 75° 47' with reference to Greenwich. Therefore,
Desdntara correciton for the sun : = - (77°35' - 75° 47') x 59* 8 " / 3 6 0 ° = - 17".74 We have, for the preceding midnight of 21-3-1997, Mean sun
334° 16'06'
De'santara correction
:
Corrected mean sun (ii)
- 17".74 334° 15' 4g".26
For the moon, the mean daily motion d= 13° 10'31"= 13°.175278 wcidX-\=
1°.8
.-. Desdntara correction for the moon =
- 1.8 X 13.175278/360 degrees
=
-3'57".15
Mean moon
Note:
118° 14'32"
De'santara correction
- 3 ' 57". 15
Corrected mean moon
118° 10'34". 85
The desdntara corrections for the apogee and the node of the moon are negligible and hence left out.
6.3.3 According to Graha laghavam (GL) (i)
Mean Longitude of the Sun Let Cakra and the Ahargana (A), according to GL be determined for the given date. Then, the mean longitude of the sun is given by A 70
A - Cakra x 1.81972 150x60
+ 349.683 in degrees Here, for the sun, according to the GL Dhruvaka = 1 ° 4 9 ' 11" = 1°.81972 and Ksepaka = 349° 41'= 349°.683 at the mean sunrise of the epoch.
Ancient Indian
Astronomy
Example : For March 21, 1997, we have Cakra C = 43 and Ahargana A = 1525 according to GL. Therefore, the mean longitude of the sun at the mean sunrise on that day at Ujjayini is ' X = 1525 X 1 - J L 70 150x60 - 43 X 1.81972 + 349.683 degrees = 4 rev. 334° 2 8 ' 4 8 ' Removing the completed number of revolutions, the mean longitude of the sun, X = 334° 2 8 ' 4 8 " at the mean sunrise. Note :
The mean daily motion of the sun is 0° 59' 08" according to GL. Accordingly, the mean longitude of the sun at the preceding midnight of 21-3-1997 is 334° 14'01".
(ii)
Mean Longitude of the Moon In the case of the moon, we have Dhruvaka = 3 ° 4 6 ' 1 1 " and Ksepaka = 349° 06' The mean longitude of the moon is given by X
= A x 1 4 - 1 4 x A / 1 7 - A / ( 1 4 0 x 60)
- Ca/:rax 3.76972+ 349.1 in degrees. Example : For 21-3-1997, we have Cakra = 43 and A = 1525 Therefore, X
= 20280°.93804 = 56'^" 120° 56' 17" i.e. 120° 56' 17"
Note :
The moon's mean daily motion, according to GL is 790' 35 ". Therefore,
the moon's
mean longitude at the preceding midnight is 117° 38'38".
(iii)
Mean Longitude of Moon's Apogee (Mandocca) The moon's mandocca is given by M = A / 9 - A / ( 7 0 X 60) - 272.75 x Cakra + 167.55 in degrees.
Example : For Cakra = 43 and A = 1525, we have M = -11391°.61865 = - [ 3 1 ' ^ ^ 231° 37'07"] Ignoring the completed revolutions and then adding 360°, we get M = 128° 2 2 ' 5 3 " . Note :
The mean daily motion of the moon's apogee is 6' 42" so that the mean longitude of the moon's apogee at the preceding midnight is 128° 21' 13".
(iv)
Mean Longitude of Moon's Node (Rdhu) In G L , the mean longitude of the moon's ascending node (Rahu) is given by Rahu = [360 - ( A / 1 9 + A / 4 5 x 60) - (212.83 x cakra) + 27.63] degrees.
Motions of the Sun and the Moon
53
Example : WitK A = 1525 and caitra =43 for 21-3-1997 the mean longitude of Rahu is Rahu
=-9205°.027833 = -[25'*^-205°01'40"]
. Omitting the completed revolutions and adding 360°, Rahu = 154° 58' 2 0 " at the sunrise. Desantara corrections We have seen earlier that due to the difference in the terrestrial longitudes of the given place and Ujjayini there will be a corresponding change in the celestial longitude of each heavenly body. We shall obtain the Desdntara corrections for the sun and the moon according to GL. The desdntara correction is given by - (X - X^) d / 360 where X = longitude of the sun Xg = longitude of Ujjayini d (i)
= mean daily motion of die body.
According to the Graha ldghava, the daily mean motion of the sun ^ = 59' 8" Here, X = 7 7 ° 35' and X^ = 75° 47'. Therefore Desdntara correction for the sun = - ( 7 7 ° 3 5 ' - 7 5 ° 47') X 5 9 ' 8 " 360° = - 1.8° X 5 9 ' . 1 3 / 3 6 0 ° = - 17".74 We have, for March 21, 1997 at mean sunrise at Ujjayini Mean Sun Desdntara cor. Corrected mean sun
(ii)
= = =
334° 28' 48" - 17".74 3340 28' 30".26
For the moon, the mean daily motion d = 13° 10' 35" and X - Xo = - 1.8°. Therefore, Desdntara cor. for the moon = - 1 . 8 X 13°. 176398/360° = - 3 ' 5 7 " . 18 Mean moon Desdntara cor. Corrected mean moon
Note :
120° 56' 17" ~ 3' 57". 18 120° 52' 19".82
The Desdntara corrections in respect of the apogee and the node of the moon are negligibly small.
54
Ancient Indian Astronomy
A comparison of the mean longitudes In the preceding sections, we have computed the mean longitudes of the sun, the moon, the moon's ^ g e e and the node for 21-3-1997. In Table 6.4 the mean longitudes according to the Surya Siddhdnta, the Grahaldghavam and die Khandakhadyaka, are compared with the modem values. Table 6.4: Mean sidereal longitudes for 21.3-1997 (at the preceding midnight) SS
KK
GL
Modem
Ravi
334° 14'09"
334° 16'06"
334° 14'01"
334° 31'48"
Candra
117° 48'25"
118° 14'32"
117° 38'38"
117° 55' 06"
Candra's Mandocca
131° 59'46"
138° 47'55"
128° 21'13'-
126° 16' 13"
Candra's Pdta (Rahu) 158° 56'29"
154° 26'51"
154° 59'08"
155° 03'51"
Note:
The mean longitude of Rihu according to KK is given after applying Lalla's correction. The modem sidereal positions are as per the Ind. Ast. Ephemeris.
7
TRUE POSITIONS OF THE SUN AND THE MOON 7.1 Introduction In obtaining the mean positions of the sun and the moon, it was assumed that these bodies move in circular orbits round the earth with uniform angular velocities. However, by observations it was found that the motions are non-uniform. The procedure for calculating the major corrections to the mean positions, to obtain the true positions, is related to die epicyclic dieory which is explanined in the following section. 7.2 Epicyclic theory and Mandaphala The theory is that while the mean sun or the moon moves along a big circular orbit (dotted in Fig.7.1), the actual (or true) sun or moon moves along another smaller circle, called epicycle, whose centre is on the bigger circle. The bigger circle ABP with the earth E as its centre
is called the kaksavrtta.
Let A be
the
position of the mean sun at a certain time. The line AEP is called the apse line (or nicoccarekhd) and AE is the trijyd
J'
(radius) of this orbit. The
epicycle, with A as centre and a prescribed radius (smaller than A f ) is called the nicoccavrtta. Let the apse line PEA cut the epicycle at U and ^V. The
Fig. 7.1: Epicyclic Theory
two points U and N are respectively called the mandocca (apogee) and the mandanica of the sun. Note that as the sun moves along the epicycle, he is farthest from the earth when he is at II and nearest when at N. The epicyclic theory assumes that as the centre of the epicycle (i.e. mean sun) moves along the circle ABP in the direction of the signs (from west to east) with the velocity of the mean sun, the true sun himself moves along the epicycle with the same velocity but in the opposite direction (from east to west). Further, the time taken by the sun to complete one revolution along the epicycle is the same as that taken by the mean sun (i.e., centre of the epicycle), to complete a revolution along the orbit.
56
Ancient Indian Astronomy
Now, in Fig.7.1, suppose the mean sun moves from A to A\ Let A'E be joined cutting the epicycle at U' and N' which are the current positions of the apogee and the mandanica. While the mean sun is at A', suppose the true sun is at S on the epicycle so that U'A'S = [/'£A..Join ES cutting the orbit (i.e. circle ABP). Then A' is the madhya (mean sun) and S" is spasta (or sphuta) Ravi. The difference between the two positions viz, A'ES" (or arc A T ) is called the equation of centre (or mandaphala). Now, in order to obtain the true position of the sun, it is necesary to get an expression for the equation of centre which will have to be applied to the mean position. In Fig.7.1, SC and A'D are drawn perpendicular to U't^E and CW£ respectively. The arc AA' (or Ajfe4'), the angle between die mean sun and die apogee is called the mean anomaly of the sun (mandakendra). We have, in the right-angled triangle A'DE, sin A£A' = sin D£A' = A'D/A'E so that A'D = RsmAA' (where R = A'E) called mandakendrajyd. From the similar right-angled triangles SCA' and A'DE, we have SC/SA'=A'D/A'E so that SC =
A'DxSA'/A'E
Since SA' and A'E are respectively the radii of the epicycle and the orbit, these are proportional to the circumferences of the two circles; that is, SA'/A'E = circumference of epicycle / circumference of orbit SC = (circumference of epicycle/circumference of orbit) x A'D Taking the circumference of the orbit as 360°, we have 5'C= (circumference of the epicycle) x
Mandakendrajyd/360°.
Now taking SC approximately the same as A'S", we have Equation of centre (Mandaphala) = (circumference of the epicycle)
(mandakendrajyd)/360°
= (r/R) (R sin m) where R sin (m) is the "Indian sine" of the anomaly m of the sun. The maximum value of the equation of centre is r, the radius of the epicyle. By observation this can be obtained as the maximum deviation of the sun's position from the calculated mean position. Note that when the sun is at his apogee or perigee, the mean and true positions coincide since sin (m) is 0 when /n = 0° or 180°. The maximum equation of centre for the sun was observed by Bhaskara II to be 2° 11' 30" (i.e. 131'.5) which is the value of r. Therefore,
True Positions of the Sun and the Moon
57
Circumference of the epicycle of the sun = (131.5/3438) X 360° = 13°.66 This value is given by B h a s k a r a II. Note :
The same epicycle theory is applied to the moon also. In the case of the moon, Bhaskara II has given the maximum equation of centte as 302'. Most texts have taken the epicycles as of varying radii and not fixed.
Table 7.1: Peripheries of Epicycles of Apsis Bodies
Aryabhatiyam
Khanda
Saura siddhanta Surya Siddhanta
Khddyaka
(Vardhamihira)
Ravi
13° 30'
14°
14°
13° 40' to 14°
Candra
31° 30'
31°
31°
31° 40' to 32°
Kuja
63.0° to 81.0°
70°
70°
72° to 75°
Budha
22.5° to 31.5°
28°
28°
28° to 30°
Guru
31.5° to 36.5°
32°
32°
32° to 33°
Sukra
9.0° to 18.0°
14°
14°
11° to 12°
Sani
40.5° to 58.5°
60°
60"
48° to 49°
From Table 7.1 we notice that while the Khanda Khddyaka and the Saurasiddhdnta (as given by Varahamihira) take the epicycles as of constant periphery (and hence radius), Aryabhatiyam and the later Siirya Siddhdnta take them as varying between two limits. 7.3 Mandaphala according to SS for the sun and the nxoon Now, how are these perpheries of die epicycles used to detemiine the equations of centre (mandaphala)'? We will follow die procedure given by die Siirya Siddhdnta in this section. For example, in the case of the sun, the periphery varies from 13° 40' to 14°. Therefore, the
radius
r
varies
from
( 1 3 ° 4 0 ' / 3 6 0 ° ) x 3438' to (14°/360°) x 3438'.
i.e., from
130'.517 to 133'.7. But then we must know how to find the actual value of r, at the given moment,
between die given limits. For this, the Sutya Siddhdnta gives die following
rule : "The degrees of the sun's epicycle of the apsis (manda paridhi) are fourteen,...at the end of the even quadrants; and at die end of die odd quadrants, diey are twenty minutes less." There are four quadrants : the odd quadrant endings are 90° and 270° and the even quadrant endings are 180° and 360° (or 0°). Let m be the mean anomaly (mandakendra) of the sun where m = Mandocca of the sun - Mean longitude of the sun. Since at m = 90° and /n = 270°
the periphery is minimum and at m = 0 ° and
; n = 180° it is maximum (i.e. 14°), we can formulate : Corrected periphery = 14° - (20' x I sin m I)
58
Ancient Indlcm. Astrononiy
assuming that the variation is periodic sinusoidally. Correspondingly, we have the corrected radius of the epicycle of the sun's apsis : r = ( 3 4 3 8 7 3 6 0 ° ) [14° - ( 1 / 3 ) ° I sin m I ] Similarly, in the case of the moon, the periphery of the epicycle varies from 3 1 ° 4 0 ' to 32°. Hence, the corrected radius is give by r = ( 3 4 3 8 7 3 6 0 ° ) [32° - ( 1 / 3 ) ° I sin m I ] where m= Moon's mandocca - Moon's mean longitude. Having found out the corrected radius of the epicycle, the Mandaphala = r sin m so that with the corrected r, we have the following : Sun's Mandaphala = ( 3 4 3 8 7 3 6 0 ° ) [14° - ( 1 / 3 ) ° I sin m I ] sin m = [133.7 sin m - 3.183 (sin m). I sin m I ] minutes of arc Moon's Mandaphala = (3438/360) [32° - ( 1 / 3 ) ° I sin m I ] sin m = [305.6 sin m - 3.183 (sin m) I sin m I ] minutes of arc The Mandaphala is additive for m < 180° and subtractive for m > 180°. Table 7.2: Snies according to Siirya Siddhdnta (/? = 3438 and « ' = 3437.75) SI. No.
Arc (9)
Arc (6) (min)
(Hindu)jyd
R' sin (9) (True) 224.84
RsmXQ)
Difference
1.
3° 45'
225
225
2.
7° 30'
450
449
224
448.72
3.
11° 15'
675
671
222
670.67
4.
15° 00'
900
890
219
889.76
5.
18° 45'
1125
1105
215
1105.03
6.
22° 30'
1350
1315
210
1315.57
7.
26° 15'
1575
1520
205
1520.48
8.
30° 00'
1800
1719
199
1718.88
9.
33° 45'
2025
1910
191
1909.91
10.
37° 30'
2250
2093
183
2092.77
11.
41° 15'
2475
2267
174
2266.67 2430.86
12.
45° 00'
2700
2431
164
13.
48° 45'
2925
2585
154
2584.64
14.
52° 30'
3150
2728
143
2727.35
15.
56° 15'
3375
2859
131
2858.38 (Contd...)
True Positions of the Sun and the Moon SI. No.
Arc (9)
59
Arc (9) (min)
(Hindu) jya
R sin (9)
Difference
/?' sin (9) (True)
16.
60° 00'
3600
2978
119
2977.18
17.
63° 45'
3825
3084
106
3083.22
18.
67° 30'
4050
3177
93
3176.07
19.
71° 15'
4275
3256
79
3255.31
20.
75° 00'
4500
3321
65
3320.61
21.
78° 45'
4725
3372
51
3371.70
22.
82° 30'
4950
3409
37
3408.34
23.
86° 15'
5175
3431
22
3430.39
24.
90° 00'
5400
3438
7
3437.75
Note :
The circumference of a circle, in arc, is 2nR = 360° = 21,600' so that
R = 2\ .600'/27t = 3437'.7468
Aryabhata I has taken the value of A as 3438'.
Example : Find the equations of centre and hence the true longitudes of the sun and the moon at the mean midnight preceding March 21, 1997 at Bangalore. We have already computed in Chapter 6, the mean longitudes after the desdntara correction, for the midnight preceding the given date at Bangalore, and the values are : Mean longitude of the sun
Note :
;
334° \y
51"
Mean longitude of the moon
j j 7° 44' 28"
Moon's Mandocca
131° 59'44"
Sun's Mam/occa
77° 17'40"
According to the Surya Siddhdnta the sun's apogee {mandocca) completes 387 revolutions in a Kalpa of 432 x l o ' years. At that rate of motion the position of the sun's mandocca at the beginning of the Kaliyuga (i.e., February
17/18, 3102 B.C.) works out to be
77° 7' 48". Therefore, for the Kali Ahargana of 18,62,063, corresponding to March 21,
1997, the motion of the suii's mandocca is = [18,62,063/(157.79,17,828 x 10^)] x 387 x 360 x 60' = 9'.8645 = 9'52"
Therefore, for the given date Sun's mandocca = 77° 7' 4 8 " + 9'52" = 77° 17' 4 0 " (0
Sun's equation of centre {mandaphala) and true longitude Sun's mean longitude Sun's Mandocca
334° 13'51" :
77° \T 40"
60
Ancient Indian
Astronomy
Therefore Sun's anomaly {mandakendra) m = Sun's mandocca - sun's mean longitude = 77° 17 4 0 " - 3 3 4 ° 1 3 ' 5 1 " = 103° 0 3 ' 4 9 " (by adding 360°) = 103°.06361 and hence the equation of centre is additive (m < 180°). The rectified periphery = 14° - ( 1 / 3 ) ° I sin m I = 13°.67529343 Sun's equation of centre {Mandaphala) = 133'.7 sin m - 3'. 183 (sin m) I sin m I = (133'.7) (0.9741197) - (3'. 183) (0.948909) = 130'.2398-3'.020378 = 127'.219422 = 2" or 13" Therefore, at the mean local midnight at Bangalore, preceding 21-3-1997, True longitude of the sun = (Mean longitude of the sun + Equation of centre of the sun) = 334° 1 3 ' 5 1 " - 2 ° 0 7 ' 1 3 " = 336° 2 1 ' 0 4 " (ii)
Moon's equation of centre and true longitude Moon's mean longitude
:
Moon's A/a/idocca
117° 44'28" 131° 59'44"
Therefore, we have Moon's anomaly {mandakendra) m = Moon's mandocca - Moon's mean longitude = 131° 5 9 ' 4 4 " - 117° 4 4 ' 2 8 " = 14° 15' 16"= 14°.254 Hence the equation of centre is additive (m < 180°). Corrected periphery of the epicycle = 32° - ( 1 / 3 ) ° I sin w I = 31°.91793 Moon's equation of centre = 305'.6 (sin m) - 3'. 183 (sin m). I sin m I = (305'.6) (0.246228) - (3'. 183) (0.060628) = 75'.24742 - 0'. 19298 = 75'.0544188 = 1° 15'03" Therefore, at the mean local midnight at Bangalore,
True Positions of the Sun and the Moon
61
True longitude of the moon = (Mean longitude of the moon + Equation of centre of the moon) = 117° 4 4 ' 2 8 " + 1° 1 5 ' 0 3 " = 118° 5 9 ' 3 1 " 7.4 Bhujantara
correction
The true midnight of a place differs from the mean midnight by an amount of time called "equation of time". The equation of time is caused by (i) the eccentricity of the earth's orbit ; and (ii) the obliquity of the ecliptic with the celestial equator. The correction to the longitude of a planet due to the part of the equation of time caused by the eccentricity of the earth's orbit is called Bhujantara.
The other correction
caused by the obliquity of the ecliptic is called Udayantara. While all the Siddhdntie
texts consider the Bhujantara
correction, the other
correction—Udayantara—was first introduced by Sripati (about 1025 A . D . ) and later followed by Bhaskara-II and others. We shall discuss the Bhujantara correcdon which is mentioned in the Surya
Siddhanta.
The eccentricity of the earth's orbit results in the equation of the centre of the sun (mandaphala) which is converted into time at the rate of 15° per hour or 6° per ghatikd. This rate of conversion is due to the fact that the earth rotates about its axis at the rate of 360° in 24 hours (or 60 ghatikds.). The resulting amount in time unit is the equation of time caused by the eccentricity of the earth's orbit. Thus, the equation of time (due to the eccentricity) = [ (Equation of centre of the sun)/15 ] hours = [ (Equation of centre of the sun)/6 ] Now, to get the Bhujdntara
ghatikds
correction for the sun Or the moon or any other planet,
the equation of time obtained above must be multiplied by the motion of the planet per hour or per ghatikd as the case may be. That is, Bhujdntara
correction for a planet = [ Equation of time in hours ] x [ (Daily motion)/24 ] = [ (Equation o f centre of the sun)/15 ] x [ (Daily motion)/24 ] = [ (Equation of centre of the sun) ] x (Daily motion)/360
where the factors in the numerator are in degrees and the daily motion is that of the planet. If the time unit used is ghatikd, then Bhujdntara
correction
= (Eqn. of time in ghatikd) x (Daily motion)/60 ] = [ (Eqn. of centre of the sun in degrees)/6 ] x [ (Daily motion of the planet)/60 ] = [ Eqn. of centre of the sun in degrees ] x [ Daily motion of the planet/360 ]
62
Ancient Indian Astronomy
where the daily motion of the planet is in degrees and hence the Bhujdntara is also in degrees.
correction
However, i f the daily motion of the planet is in minutes of arc, then Bhujdntara
correction in degrees
= (Eqn. of centre of the sun in degrees) x (Daily motion of the planet)/2l600 Further, the Bhujdntara of centre of the sun is so.
correction is additive or subtractive according as the equation
For example, in the case of the moon, its mean daily motion is 13°. 176352 or 790'.58112. Therefore, we have Bhujdntara
correction (mean)
= (Eqn. of centre of the sun) x 790'.58112/21600' = Eqn. of centre of the sun/27.321674 Note :
Brahamagupta takes the denominator approximately as 27 in his Khandakhadyaka.
It is important to note that to obtain the actual (and not the .mean) Bhujdntara correction of a planet, we have to use the true daily motion of the planet for the given day. Example : Find the Bhujdntara given that on a certain day
corrections for the longitudes of the sun and the moon
True daily motion of the sun True daily motion of the moon
59*.65 :
Equation of centre of the sun
855'.23 + 2°7'32"= 127'.53
Therefore, we have (i)
True Bhujdntara
correction of the sun
= (Eqn. of centre of the sun) x (Daily motion of the sun)/216(X) = 127'.53 X 59'.65/21600' = 0'.3521835 = 0' 2 1 " Since the equation of centre of the sun is additive, the Bhujdntara additive. (ii)
True Bhujdntara
correction is also
correction of the moon
= (Eqn. of centre of the sun ) x (Daily motion of the moon)/21600 = 127'.53 X 855'.23/21600' = 5'.0494205 = 5' 3" Here also, the correction is additive since the sun's equation of centre is so. 7.5 Further corrections for the moon We have applied so far an important correction namely, the equation of centre, to the mean position of the moon. Besides this correction, the other two corrections applied viz, Desantara
and Bhujdntara
arc mainly to get the true position of the moon at the
true local midnight at the place of observation.
True Positions of the Sun and the Moon
63
However, to get the true apparent position of the moon at least two more important corrections will have to be applied, of course, ignoring other minor corrections due to planetary perturbations. (i)
£v£crio« = ( 1 5 / 4 ) ' m e sin ( 2 ^ - 0 ) = 7 6 ' 2 6 " s i n
(24-(|»)
where m is the ratio of mean daily motions of the sun and the moon, e is the eccentricity of the moon's orbit, ^ = ( M - S), the elongation of the moon from the sun and ^ = M-P, the mean anomaly of the moon (P being the moon's perigee). (ii)
Variation = 39' 30" sin (2^) In the above formulae, S and M are respectively the mean longitudes of the sun and the moon. The Surya Siddhanta, being an earlier text, does not mention these corrections. However, Manjula (932 A.D.), Bhaskara-II (1150 A . D . ) and later Indian astronomers have recognized the evection correcdon in addition to the equation of centre.
Besides these, the famous
Orissa
astronomer Samanta Candrasekhara Simha discovered independently a fourth correction called annual equation. According to Candrasekhara, Annual equation = (11' 27".6) sin (Sun's distance from apogee) In fact, Candrasekhara's coefficient viz., \\'21".6 is very close to the known modem value. Tycho Brahe took the coefficient wrongly as 4' 30". 7.6 True longitudes of the sun and the moon according to KK The method of Khandakhadyaka
is demonstrated
for the example of midnight
preceding 21-3-1997. 1.
Sun's true longitude
(i)
The sun's mandocca
= 80°
(fixed according to Brahmagupta) Mandakendra (A/)
= Mean sun - sun's mandocca = 334° 1 6 ' 0 6 " - 8 0 °
M (ii)
= 254° 16'06"
Mandaphala = -—• sin M A
is the equation of centre in radian measure where b=]4° M = Mandakendra (i.e. anomaly)
and R = 360° and
14° i.e. Mandaphala =
sin M
(in radian measure)
360° Therefore, Mandaphala
= - s i n A/
(in degrees, multiplying by 180°/n)
= - - s i n (254° 16'06") 71 = 2° 8 ' 4 1 "
64
Ancient Indian Astronomy True longitude of the sun = Mean sun + Mandaphala = 334°16'06" +
2° 0 8 ' 4 1 "
i.e.. True longitude of the sun = 336°24'47" (Compare this with the Surya Siddhanta 2.
value : 336° 21'04")
Moon's true longitude (i)
The moon's mandocca (apogee) = 138° 47' 55" Mandakendra
(anomaly) M
= Mean moon - Moon's apogee = 118° 1 4 ' 3 2 " - 138° 47' 55" = 339° 2 6 ' 3 7 "
(ii)
Mandaphala
(equation of centre)= — sin M
(in radian measure)
A
where a = 31°, /? = 360° and M -
Mandakendra.
-31° .'. Mandaphala =
x sin M (radian measure) -31° 180 . ,^ . = X sin M (in degrees) 360° n V s ; = - ^ sin Af in degrees 271
I.e. Mandaphala
=1°43'57"
Now, the true longitude of the moon = Mean moon + mandaphala = 119° 58' 29" 7.7 IVue longitudes of the sun and the moon according to GL Ganesa Daivajiia's algorithm for the sun's mandaphala
as in his GL is described
below : (1)
The sun's mandocca = 7 8 ° (taken as fixed) Mandakendra,
MK = Mandocca—Mean
Sun.
Now, the Bhuja of MK is determined as follows : (i)
If MK is negative, then the effective MK is obtained by adding 360°.
(ii)
If 0 <
< 90°, then Bhuja
= MK
(iii)
If 90° <
(vi)
If 180°
