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Ayanamsha vs Precession
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[ Some material on this page is already published original research, which may be cited with acknowledgement. ]
Evidence of Modern Value of Precession in Bhāskaracharya's Work based on Suryasiddhānta
In the chapter "Direction, Place and Time" (Suryasiddhānta, Ch.iii), E Burgess writes :Quote: (bracketed words are mine) : The (Surya Siddhāntic) theory which the passage (verses 9-12), in its present form, is actually intended to put forth is as follows : the vernal equinox librates westward and eastward from the fixed point, war Piscium, assumed as the commencement of the sidereal sphere-- the limits of the libratory movement being 27 degrees in either direction from that point, and the time of a complete revolution of libration being the six-hundredth part of the period called the Great Age (ie, Mahāyuga as defined by Burgess in chapter i,15-17, where he gave it a span of 4320000 years), or 7200 years; so that the annual rate of motion of the equinox is 54".Unquote:
This is the interpretation of existing version of Surya Siddhānta ( त्रिंशत्कृत्यो युगे भानां चक्रे प्राक् परिलम्बते ..., SS,iii.9) in own words of E. Burgess , "as it is actually intended to put forth" by all traditional commentators. This is exactly what I illustrated with example in the illustrated example of computation of ayanamsha.
The moot point is this : Burgess knew the traditional interpretation (भानां चक्रे.., ie pendulum like motion of nakshatra orbit itself) , but gave his own meaning based upon modern concept of precession of equinoxes , and tried to create doubts about the authenticity of these verses (iii, 9-12) by putting forth deliberately false arguments. Let us examine Burgess.
In verse-9 (Suryasiddhānta, Ch.iii), he translates "pari-lambate" as "falls back", although he says lambate means "lag, hang back, fall behind" and 'pari' means "about, round about". Therefore, pari-lambate should have been translated as "fall back roundabout" and not merely as "fall back" according to own logic of Burgess. If the circle of asterisms lags roundabout any fixed point (whether Revati or Chitrā), it is a to and fro motion as all traditional commentators accepted. Modern concept of precession is something different from the original concept of ayanāmsha. Theon in West had mentioned this oscillating motion, Arab astronomers also accepted it, and almost all Europeans accepted it upto Renaissance, after which Hipparchus was rediscovered and modern concept of precession became a well established fact in astronomy. But this concept of equinoctial precession (as well as anomalistic precession) was also known to ancient Indians and Greeks.
Burgess wrongly quotes Bhāskara-II, because he relied upon a wrong translation of Bhāskara by Colebrooke (As. Res., xii 209 ; Essays, ii,374, etc) and did not try to examine Siddhānta Shiromani which was wrongly translated by Lancelot Wilkinson due to Colebrooke's influence. Bhāskara-II did not give his own opinion at all, and merely quoted Surya Siddhānta and Mujjāl (elsewhere Munjāla and Manjula), saying Suryasiddhānta gives -30000 revolutions of sampāt or equinoctial point per Kalpa while ayana has a motion of +199669 revolutions per Kalpa (of 4320 million years). Bhāskara's own opinion was that these should be followed, which means both Surya Siddhānta and Mujjāla were correct in Bhāskara's opinion. Colebrooke, Burgess, Wilkinson, etc have misquoted Siddhānta Shiromani and created an impression that ancient Indians were inept in astronomical observations, as Whitney shamelessly declared in his prologue to Burgess, but the Hindi translation by Satyadeva Sharmā is correct, although he could not get the real meaning.
The startling fact is that Siddhānta Shiromani clearly says that "the point of intersection of equatorial plane and ecliptic" (which is the very definition of equinox) has a negative motion of 30000 revolutions per Kalpa according to Suryasiddhānta, while Mujjala's value of ayana's motion is +199669, and both (Suryasiddhānta and Mujjala ) must be added to get the final motion (of the equinox ). Hence, we get +169669 revolutions per Kalpa, which gives (4320000000 / 169669 =) 25461 years per revolution or 50.9" per year, which is very near to modern value of about 50.3" per year for precession of equinoxes.
We must not forget that Hipparchus had given a period of 36000 years for precession, which was not corrected by Europeans till the onset of modern age. It is unfortunate that Siddhānta Shiromani is still being misinterpreted by foreigners, and if a true rendering is offered by Indian scholars, they are abused, esp by those who do not care to consult the originals and declare the forign missionaries to reliable. Bhāskara-II neither excluded Suryasiddhānta, nor Mujjāla, but mentioned the both must be used, which is clear from verse-19, where he clearly asks to add Mujjāla's ayana-chalam to Suryasiddhāntic sampāt-chalanam (this sampāt-chalanam is anomalistic precession with a period of 144000 years per cycle, against modern value of 136000 years).
Another startling fact is that Bhāskara-ii differentiates sampāt-chalanam of Suryasiddhānta from ayana-chalanam of Mujjāla, and says both must be added before computing phenomena like declension, ascensional differences, etc. But modern commentators like Colebrooke misinterpret Bhāskara-II deliberately, and imply that sampāt-chalanam of Suryasiddhānta quoted by Bhāskara-ii was an erroneous thing which must be forgotten, while ayana-chalanam of Mujjāla was a crude approximation of modern precession. But this interpretation is falsified by Bhāskara's original verses as shown above. The root of this problem lies in the fact that sampāt-chalanam of Suryasiddhānta is a distinct phenomenon from ayana-chalanam of Mujjāla according to Siddhānta Shiromani, but readers are not informed of the real meaning of Siddhānta Shiromani and false quotation from Siddhānta Shiromani was quoted by Colebrooke and Burgess (12th verse, chap.iii). This is a sign of intellectual incompetence and dishonesty of Western "experts" who are blindly followed by brown sāhibs of India. Those who do not consult the original texts cited above will not believe me.
Siddhānta-tattva-viveka by Kamlākara Bhatt is a medieval text, which clearly states that Saurpaksha is distinct from Drikpaksha. Saurpaksha (astronomy of bhuvaloka) is Suryasiddhānta as it exists. Drikpaksha (astronomy of Bhooloka or physical/material/sensory world) is that version of Suryasiddhānta which was not preserved because it was useless in astrology. Siddhānta Shiromani uses many concepts of Drikpakshiya astronomy, as the instance cited above proves. Saurpakshiya Suryasiddhānta does not contain any refence to 30000 cylces per Kalpa mentioned by Bhāskara-II. He was quoting from Drikpakshiya Suryasiddhānta which as a text had been lost ; Bhāskara-II said in his own Vāsanābhāshya commentary of Siddhānta-shiromani that Suryasiddhānta is not available ("anupalabdha") and he was quoting it on the basis of "āgama". Only its fragments are left, scattered here and there. Modern commentators confuse both variants of Suryasiddhānta. Siddhāntatattvaviveka is prescribed in post-graduate (Ganitāchārya) syllabus of Sanskrit universities, but no modern commentator has ever tried to translate it or comment on it.
According to Bhāskara-ii , negative sampāt-chalanam of Drikpakshiya Suryasiddhānta should be added to positive ayana-chalanam of Mujjāla to get final Drikpakshiya precession, which is very close to modern value. Ayana-chalanam of Mujjāla is also Drikpakshiya, because Saurpakshiya entities are not used in Drikpakshiya astronomy, and vice versa. I have put some of the most important extant theorems of Drikpakshiya Suryasiddhānta at a website. I had put parts of it at one of most popular websites, where a German "Indologist" deleted it and abused me profusely ; later I found those deleted materials at an Australian website, without any name of author!!. But I am here divulging one important secret of ancient science of India which has been neglected by wrongheaded commentators.
Mujjāla's ayana-chalanam, as mentioned in Siddhānta Shiromani, gives a period of (4320 million / 199669 = ) 21636 years per cycle. Siddhānta Shiromani says that it is ayanachalanam and not precession, precession is obtained after substracting (Saurpakshiya) Suryasiddhāntic sampātchalanam. If this 21636 year cycle is not precession, what is it ??
Readers should read a Wikipedian article Milankovitch cycles ( http://en.wikipedia.org/wiki/Milankovic_cycles ) which informs :"Earth's axis completes one full cycle of precession approximately every 26,000 years (25771.5 precisely at present, 25789.5 years is long term mean). At the same time, the elliptical orbit rotates, more slowly, leading to a 21,000-year cycle between the seasons and the orbit... This orbital precession is in the opposite sense to the gyroscopic motion of the axis of rotation(cf. anomalistic precession as distinct from equinoctial precession), shortening the period of the precession of the equinoxes with respect to the perihelion from 26,000 to 21,000 years." (at some sites of NOAA of USA, 22000 is mentioned instead of 21000)
Ayana-chalanam of Mujjāla is not orbital precession, it is the most important of all components of Milankovitch cycles as this Wikipedian definition shown. If we take cue from Siddhānta Shiromani, the aforementioned Wikipedian clause can be rewritten thus : This orbital precession of equinoxes is in the opposite sense to the gyroscopic motion of the axis of rotation, shortening the period of the precession of the equinoxes with respect to the perihelion from 25771 to 21,636 years.
Siddhānta Shiromani also says that Mujjāla's ayana-chalanam (21,636 years per cycle) is opposite to sampāta-chalanam. Bhāskara-ii clearly defines sampāta-chalanam as "the point of intersection of equatorial plane and ecliptic" (which is the very definition of equinox). Hence, what Siddhānta Shiromani says is exactly what Wikipedia informs us, the only difference is that Siddhānta Shiromani is misinterpreted and declared to be obscurantist, and the great cycles mentioned in Siddhānta Shiromani is "discovered" by 20th century scientists. But we must remember Bhāskara-ii did not discover these things, he acknowledged Suryasiddhānta and Munjāla.
Bhāskara-ii knew Drikpakshiya Suryasiddhānta, which has not survived because it was not useful in astrology. In his formula of precession, Bhāskara-II used a figure 30000 cycles per Kalpa. Bhaskara-II got an approximate value of 50.9" per year, which was the most precise value before modern astronomy developed in the West. Here I quote a Puranic verse which proves knowledge of equinoctial precession in Puranic times :
उत्तानपादपुत्रोऽसौ मेढीभूतो ध्रुवो दिवि ।
स हि भ्रमन् भ्रामयते नित्यं चन्द्रादित्यौ ग्रहैः सह ।।
It means : "Uttanpāda's son Dhruva is the fixed point in the Heavens , round which all planets including Sun and Moon, but Dhruva himself also moves round" . Round what ? Mt Meru, which is the only fixed point in Cosmos according to Purānic-epic stories. Hence, the bhachakra also librates with respect to this fixed point Meru.
According to Bhāskara-II, orbital precession is derived by substracting anomalistic precession (sampāt-chalanam) from the first component of Milankovitch cycles (Munjāla's ayana-chalanam). Bhāskara-II acknowledged earlier authors. Hence, we must conclude that modern values and concepts of orbital precession, anomalistic precession, Milankovitch cycles, etc were known to ancient Indians well before Bhāskara-ii.
But two things about confusing terminology must be borne in mind : this sampāt-chalanam he finally gets by combining the two quantities mentioned above. According to Bhāskara-II, Suryasiddhāntic sampāt-chalanam is 30000 per Kalpa. He does not give a name for the term which is finally obtained by combining this sampāt-chalanam with Munjāla's ayana-chalanam, but the definition he provides for Suryasiddhāntic sampāt-chalanam is exactly the definition of the final quantity whose name he does not provide. Hence, there were many types of sampāt-chalanams !! This is not a case of confusion of terms. It is a result of Saurpakshiya term with Drikpakshiya terms bearing same names but having different magnitudes and sometimes even having difference in basic properties !
Second confusion is due to use of the term ayana-chalanam for Munjāla's precession. It is quite distinct from Saurpakshiya Suryasiddhāntic ayana-chalanam (trepidation) as mentioned in existing text. Burgess could not digest this theory of libration (oscillation or trepidation, ie, ayanāamsha - motion) and tried to distort the meaning of terms to fit modern view of orbital precession with this Saurpakshiya precession. Bhāskara-ii knew and respected Suryasiddhānta which he cited and used in his computations as shown above, and gave exact value of Drikpakshiya precession. Therefore, it is foolish to impose Drikpakshiya precession (50.9" per year according to Bhāskara-II, 50.3" really) upon Saurpakshiya ayanamsha (54" per year, oscillating within a range of ± 27 degrees). (There are further corrections on Drikpakshiya precession which give a final value of one revolution in 25771.4 years, exactly equal to the value deduced by NASA - JPL , but these corrections requires some long theorems to prove).
I do not want to say that all ancient texts are true and should be blindly followed. But it is equally wrong to deride them as outdated and obscurantist just because they could not be understood by moderns.We have yet to discover the real Wonder that Is India. Unless and until ancient texts are proven false, it is suicidal to reject them.
DECLENSION of the SUN :
(Following Material is not for everyone, and only those persons should argue for or against these topics who know howto use siddhantic equations)
Now, look at most recent view on this topic, from NASA. open the webpage http://aom.giss.nasa.gov/srorbpar.html
After reading its contents, type -998000 , 2001 and 1000 respectively in its three text boxes and click the Submit button. You will get "Orbital Parmameters for the Earth" from -998000 A.D. (ie, 998000 BCE) to 2000 A.D. at intervals of 1000 years. Three parameters are given : Eccentricity (of Earth's orbit round the Sun), Obliquity , and Longitude of Perihelion. Copy these data and paste them into Microsoft Excel and then select the last column and insert a chart with line option selected, you will get following chart. In Excel, place the mouse ponter at the lowest points of curve at those points where '16' and '984' are typed in red in the picture below, you will get data for those points. Between these two points, there are 46 undulations of the curve, during 984-16 = 968000 years. But between point 79 and 89 : a full cycle is not possible in merely 10000 years (between points 79 and 89), or a half cycle in only 2000 years between points 87 and 89. Therefore, this aberration must be excluded, and there are only 45 full cycles during 968000 years. Hence, one average cycle is of 21511.1111 years. But even a cursory look at the curve will convince you that this average actually varies considerably, and had highly erratic behaviour at some juntures. For instance, points 552 and 568 suggest one cycle in just 16000 years, while between 935 and 962 one cycle took 27000 years. Hence, 21511 is a mean value (with ± 5500 year fluctuations) during past one million years based upon most modern formula (a formula, not data). Suryasiddhānta, Munjāla and Bhāskara-ii give a long-term average of 21635.807 years. Modern scientists do not possess adequate data and formula to refute this traditional value given by Indian astrologers. This cycle shows ± 5500 year fluctuations, hence it is not possible to determine whether Suryasiddhāntic value of 21636 years is correct or the average of 968000 years (=21511 years) deduced from following graph is more reliable, because the difference between them is very small : of only 21636 - 21511 = 124.7 years only , as compared to ~5500 year fluctuation in this cycle itself. (Read the matter after this picture.)
A closer look at this graph will show that there are two cycles of some long term phenomenon ranging about less than hald a million years per cycle, one half half cycle of which has shorter periods of this perihelion cycle and another half cycle has longer periods. This is due to superimposition of another equation on the equation of Perihelion graph : the effect of change in Earth's eccentricity, which shows a long term periodicity of about 96555.56 years (a mean value, not an exact figure) computed from same NASA website following same procedure (9 cycles in 869000 years). Harmonic addition of both periodicities produce following maximum and minimum ranges of Perihelion periodicities :
(1 / 21511) - (1 / 96555.56) = (1 / 27677)
(1 / 21511) + (1 / 96555.56) = (1 / 17592)
The arithmetic mean of 27677 and 17592 is 22634.5 years, with ± 5042.5 year fluctuation, but this variation is elliptical, hence arithmetic mean gives a wrong average.
The eccentricity cycle of ~96555.56 years is a major cause of glacial cycles.
This Perihelion Cycle of 21636 years is the result of mixture of two functions : precessional cycle of 25771.4 years, and anomalistic shift of one cycle per ~135000 years :
(1 / 21635.8) - (1 / 25771.4) = (1 / 134825.9073)
Suryasiddhāntic value of anomalistic period is 144000 years as given by Bhāskara-ii. It needs Mahāyuga, Manvantara and Kalpa corrections in the following manner, in which 4200000 years of Drikpakshiya Mahāyuga has been represented by 'M' :
(1 / 144000) + (2 / M) - (1 / 71M) - (1 / 14*71M) = ( 1 / 134824.638836)
The small gap between 134825.9073 and 134824.638836 is due to a period of ~14 billion years, which is an optical illusion due to a complex phenomenon. There are 14 universes (bhuvanas) according to traditional Indian view, our universe being in the middle. Hence, we can view 7 universes at any given time in any one direction. According to Suryasiddhānta, slightly less than 2 billion years have elapsed since Creation. When we look into 7 universes, we see into 14 billion years which is actually seven different space-time continuums of 2 billion years each. if this factor is taken into account, above equation will become perfect :
( 1 / 134824.638836) - (1 / 134825.9073) = (1 / ~14 billion years)
In same fashion, the apparently crude value of precessional period given by Bhāskara-ii is corrected to give the exact figure. Instead of using the figure 30000 cycles per Kalpa or 144000 years per cycle, we ought to use the corrected term of 134825.9073 years per cycle in the equation of precession for a comparison with modern data. Hence, we get :
( 199669 / 4320000000) - (1 / 134825.9073) = (1 / 25771.4)
25771.4 years per cycle is the most modern value of orbital precession of Earth. In above equations, 'M' represents one Drikpakshiya Mahāyuga of 4200000 years, 71M is one Manvantara, 14 such Manvantaras make one Kalpa.
While making observations, modern scientists mix up all seven universes, while ancients gave figures for a particular space-time continuum at a time. They knew which frame of reference to use in which context, because they could "see" but we are blind (vid is the root of video and also of Veda and Vedānga).
Now, look at the code of NASA program which gave above figures at at http://aom.giss.nasa.gov/SOLAR/ORBPAR.FOR
It shows that a small table based upon modern data has been used to project tables for long periods, which may be reliable for short periods, but in the absence of many long term motions in the outer sky this program may deceive us in the long period projections it gives. For instance, this NASA site allows above data from one million years before 2000 AD to ten million years in future. Eleven million years of data give some long term trends which are not visible in short term graphs. The obliquity curve shows a ~1286500 year long cycle during which the maximum range within which obliquity changes itself shrinks and grows peiodically. There is no known explanation of this phenomenon. But a comparison with the long term changes in eccentricity shows that tge obliquity graph has close resemblance with eccentricity graph, and there are fewer long terms cycles in eccentricity graph than in obliquity graph. The eccentricity graph gives one cycle for 2019000 years approximately, which is exactly half of Drikpakshiya component of a Mahāyuga of 4200000 years equal to 4137562 years which requires an explanation.
Eccentricity and Obliquity of Earth's Orbit for 11 million years (NASA)
In the Drikpakshiya Suryasiddhāntic equation of anomalistic (and orbital ) precesson shown above, there were terms shown in red : (2 / M) - (1 / 71M) - (1 / 14*71M), where M stood for 4200000 years. The first component was of (2 / 4200000 years ) . Same term appears as (2 / 4137562 years ) in the eccentricity graph. Following is the Drikpakshiya Suryasiddhāntic equation of this term :
{1 + (1/4137561.9278584)} = {1 + (1/M)} x {1 + (1/71M)} x {1 + (1/[71Mx14])}
The 19-year cycle of Vedānga Jyotisha used by Meton of athens too comprised of 235 lunar months. 235 lunar months divided with 19 gives us 365.246743924320183 days. If it is divided with {1 + (1/4137561.9278584)}, we get 365.24665564850413050493379669683 days, which is the exact value of Metonic Year of Greece or of Romak Siddhānta of Indian version of Alexandrian (actually ancient Egyptian) science, which was originally based upon Drikpakshiya Suryasiddhānta.
So far, many propositions and equations would have seemed mere coincidences or far-fetched. Look at the following proof of Mahāyuga which is a cardinal equation of Drikpakshiya Suryasiddhānta (= Vedic for abbraviation ) :
Annual difference between motions of True Sun (TS) and Mean Sun (MS) is
( TS2 - TS1 ) - ( MS2 - MS1) = { 360 degrees / ( 4200000 - AP) } = 0.318934857156473481 seconds per annum.
In it AP is anomalistic period. Its value was 134825.9073 years as deduced above from Bhāskara's equation above. But here we are talking of true motion of Sun, which needs the Saurpakshiya motion of anomaly to be accounted for, which is quite different from Drikpakshita motion of anomaly. The latter has a value of about 135000 years per cycle which gives an anomalistic year of 365.25905 days. But the former has only 387 cycles per Kalpa, and one cycle of 11162790.7 years. Due to this Saurapakshiya shift, resultant Drikpakshiya anomalistic period will be of 136474.2645105 years. It needs another correction due to a ~39 million year Drikpakshiya motion, which gines a resultant value of ~136000 years for anomalistic cycle. Put this value in the equation below,
( TS2 - TS1 ) - ( MS2 - MS1) = { 360 degrees / ( 4200000 - 136000) } = Sine of annual anomalistic shift multiplied with maximum value of equation of centre =
0.3188976378 seconds per annum is equal to Sine of annual anomalistic shift multiplied with maximum value of equation of centre (paramamandaphala) which gives a value of 6902.56241634 seconds of arc for maximum value of equation of centre (Drikpakshiya paramamandaphala) ; Lahiri gives 6904.6".
This value of Drikpakshiya paramamandaphala is equal to 1.917378449 degrees, Sine of its half is the eccentricity of orbit equal to 0.016731502727386339 from above equations. Explanation is thus :
Eccentricity is the ratio of distance between centre of ellipse and focus to the major axis. If a triangle is drawn with one side being minor axis and another side being the line from centre of ellipse and focus, its hypoteneuse will be equal to major axis. Ratio of the line from centre of ellipse and focus and hypoteneuse is equal to sine of the angle near the end of minor axis away from centre. this angle is half of maximum value of equation of centre (paramamandaphala).
The above value of eccentricity needs some further corrections. Substract its square from unity and get the square root of resultant, you will get the compression of elliptical orbit of Sun round us (compression is ratio of minor to major axis), which is 0.999860018611 . The difference between both axes is ( 1 - 0.999860018611 )
equal to 0.00013998138915292284 mjultiplied with 1 AU ( = 149598550 Kms, = major axis), and is equal to 20941.012844263 Kms. To it, add the height of Mt Meru (Mt Kenya), which is 5.199 Kms, you get 20946.211844263 Kms. Now, assume it to be the difference between major axis (149598550 Kms) and minor axis, and compute eccentricity from the method indicated above : you get
Ecc = Sqr Root of (1 - [{1-(20946.2118 / 149598550)}^2]) = 0.0167335794081
It is the mean value of eccentricity of apparent orbit of Sun or Earth. Actual value is 0.016733761858645, computed from a more exact Suryasiddhāntic formula to be elaborated later. This exact formula has 20946.6686 Kms instead of 20946.2118 Kms as the difference between major axis and minor axis.
This Drikpakshiya value of eccentricity conforms with most modern data. It is the mean value of eccentricity on a logarithmic scale, and this mean value should be for Saurpakshuya Makar Samkrānti on Jan 14, 2009 AD. This value 0.016733761858645 should be squared and the result substracted from unity, and then get the square root of the resultant which is the ratio of both axes, ie, compression of the solar ellipse, equal to 0.99985998080434 .
Now, assume another ellipse with eccentricity exactly equal to 1 / 60, its compression will be equal to 0.99986110146471 . Multiply the difference of both compressions with 1 AU, you get the circumference of a small circle with radius little over 26.68 Kms (error of 3.7 methres due to uncertainties in the value of 1 AU).
1/60 is the actual eccentricity of Sun's orbit round Mt Meru, which gets changed to 0.016733761858645 on account of this Drikpakshiya Correction of 26.6859001229 Kms. ( NASA gives a wrong data of eccentricity 0.016704 for 2000, which is actually its updated data for April 2008, its whole table needs +8.3 years everywhere).
Starting from 1/60 as the basic eccentricity, the theorems of Drikpakshiya Suryasiddhanta are given at another website.
Now, the third term Obliquity, which gives a periodicity of ~41200 years from NASA data for 6 million years , which is 1% of the Drikpakshipa's Mahāyuga (4200000) resolved component of 4137561.9278584 years. Unfortunately, NASA does not possess long term data to arrive at better conclusions.
-VJ
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