Relative Position And Attitude Estimation And Control Schemes For The nal Phase Of An Autonomous Docking Mission Of Spacecraft

Chapter 15

A Relook into the Historical Progress and Philosophy of Indian Space Exploration* M. R. Ananthasayanam,† A. K. Anilkumar‡ and V. Adimurthy§ Abstract A study of the history and philosophy of the contribution of India towards the exploration of space since antiquity provides interesting insights. The contributions are described during the three periods namely: (1) the ten millenniums from 10,000 BC with a twilight period up to 900 AD; (2) the ten centuries from 900 AD to 1900 AD; and (3) the ten decades from 1900 AD to 2000 AD; called mythological, medieval, and modern respectively. Some important events during the above periods provide a reference view of the progress. The Vedas during the mythological period and the Siddhantas during the medieval periods, which are based on astronomical observations, indicate that the Indian contribution preceded other cultures. But most Western historians ignore this fact time and again in spite of many proofs provided to the contrary. This chapter also shows that Indians had the proper scientific attitude of developing any physical theory through the triplet of mind, model, and measurements. It is this same triplet that *

Presented at the Forty-First History Symposium of the International Academy of Astronautics, 24–28 September 2007, Hyderabad, Andhra Pradesh, India. Paper IAC-07-E4.3.01. †

Formerly Professor, Department of Aerospace Engineering, Indian Institute of Science, Bangalore, India. ‡

Head, Applied Mathematics Division, Vikram Sarabhai Space Center, Trivandrum, India.

§

Dean ( R&D), Indian Institute of Space Science and Technology, Trivandrum, India.

267

forms the basis of the present day well known Kalman filter technique. Up to about 1500 BC the Indian contribution was leading but during foreign invasion and occupation it lagged and has been improving only after independence.

Introduction The history of space exploration can be succinctly stated as an inexplicable trait of the human nature aspiring to reach the place seen by his eyes in the sky. However, to do so there is no easy way except to systematically explore the environment of space, the celestial objects, and build vehicles to take him there, which leads to developments in space science and technology. In the present review of the Indian contribution toward the exploration of space from antiquity to the present, due to page limitation and the author’s competence, the topics have been restricted to ancient Indian astronomy and, briefly, the developments in space technology in the country. The progress can be described in the three periods, namely: (1) the ten millenniums from 10,000 BC with a twilight period up to 900 AD; (2) the ten centuries from 900 AD to 1900 AD; and (3) the ten decades from 1900 AD to 2000 AD; called mythological, medieval, and modern, respectively. In each of the above periods the identification of some important observations or works provide a reference view of the progress as shown in Tables 15–1, 15–2, and 15–3 (located at the end of this chapter). Such a subdivision appears suitable to describe the historical progress in many other topics in science and technology. Ancient Indian astronomy [1 to 16] is emphasized more lately because it represents the best of Indian heritage and pride for Indians who somehow lost the lead. Subsequently, the newer perspectives, which throw insight to the scientific attitude and the approach of ancient Indian astronomers [17 to 28], are provided. The final section provides conclusions from the present study.

Ancient Indian Astronomy and Antiquity of the Vedas Ancient Indian astronomy is considered to be the oldest of all sciences of the human race and is as old as the Vedas, since the rituals with altars symbolizing Earth, space, and sky therein are based on astronomical observations of the Sun, Moon, planets, and the stars, all well known to Vedic Rishis. If we follow Neugebauer for the Babylonians, then with further antiquity Vedic Indians would be the first scientists.

268

Law [4] quotes that Max Muller had described the language and poetry of the Vedas (meaning knowledge in Sanskrit) as “unequalled in any language and in any later Indian literature for grandeur, boldness, and simplicity, beautiful, original, and spontaneous and charm.” He placed the period of Vedas (by contents and chronology to Rg, Yajur, Sama, and Atharva Vedas, then Brahmanas followed later by the Aranyakas, Upanishads, and six Vedangas, which help to understand Vedas) around 1200 BC. Other scholars felt it is impossible to squeeze the linguistic, cultural and philosophical developments in the Vedic literature into a few centuries. Later in life he felt “there is nothing more ancient and primitive, not only in India, but in the whole Aryan world, than the hymns of the Rgveda. So far as we are Aryans in language that is in thought, so far Rgveda is our own most ancient book. . . . Whether the Vedic hymns were composed in 1000, or 1500, or 2000, or 3000 BC no power on earth will ever determine.” It turns out that with more evidence from anthropology, satellite studies, geology, linguistic, and literature, the date of Rgveda is getting continuously pushed further and further into antiquity. Rgveda mentions about 60 times the mighty river Saraswati flowing from mountains to the sea. Even the region of the composition of the hymns is indicated by the asymmetry of two numbers that show the ratio of the longest to the shortest day, leading to the latitude of the place as the Sarasvathi valleys. Satellite and geological studies shows it ceased to be a perennial river and flowed only seasonally sometime before 3000 BC, and by 1900 BC went dry due to the tectonic shifts in its tributaries Sutudri and Yamuna being captured by Sindhu and Ganga. Hence Frawley and Rajaram conclude Rgveda must have been composed long before 3000 BC, with an evolution time of about 2,000 years that places it before 5000 BC. Hence the theory of Aryans invading, settling in Saptasindu region in 1500 BC, and later in 1200–800 BC composing the Vedas must be removed from textbooks. By the time of Rigveda, the Aryans were Indians without memory of a foreign land. Tilak showed that astronomy could be used for fixing the chronology of the events in the Vedic texts in a consistent way.

Astronomy in Rgvedic Hymns Rgveda, the ancient word document, preserves the astronomical observations therein with incredible fidelity through a sequence of symbols but with different permutations, as in modern error detection and correction methods in computers and communications theory.

269

The Vedic hymns indicated concepts, such as a spherical Earth, a heliocentric model, and the precession of the equinox. The astronomical truths in Vedas are wrapped in stories involving deities [15] made the early scholars unfamiliar with modern science and astronomy unable to decode their contents. In Aithreya, Brahmana (3.44) says the sun never sets or rises . . . making night what is below and day to what is on the other side. An elaboration of the above speaks of antipodes of Earth and implies Earth’s rotation. The precession of the equinox was also known, based on adjustment at the beginning of the month for the sacrifices being adjusted over the course of thousands of years as shown in Table 15–1. As an example of the symbolic nature of describing astronomical features, Rgveda (RV1.112.13) declares that the Asvins, namely Mercury and Venus, drink soma and have phases like the Moon that imply heliocentric theory. If Earth were at the center of the solar system, Mercury and Venus would not have exhibited phases. These are also known as morning or evening stars, denoting their visibility just before sunrise or sunset. The Asvins moved sometimes east to west and sometimes west to east (RV1.139.4) meaning direct and retrograde motions unlike outer planets that have only occasional retrograde motions. The world was supported by 12 massive pillars, Earth stood on the back of four elephants, in turn supported on the back of a huge tortoise, which itself was supported by a serpent floating in a limitless ocean. The pillars symbolize months, elephants denote directions, and the tortoise is one of Vishnu’s avatar, implying that Earth is supported in space in its orbit around the Sun, and the coiled serpent refers to Earth’s rotation.

Stars in Vedic Period The Vedic astronomers, based on observations, determined the motion of the Sun and the Moon, with reference to the 27 stars with Sanskrit names of ancient Vedic deities in Rgveda (5.51), which indicates their antiquity. Similarly the 12 zodiac names emerge from the name of the deity. Max Muller opined that the stars are completely of Indian origin, otherwise different Gods, hymns, worship, and sacrifices would have followed. It could not be of Chinese origin having only 24 single stars, whereas the 27 Indian stars are mostly groups. Rgveda refers to Sun as the daytime star. The sunspots are mentioned in Rg and Yajur Vedas.

Planets in Rgveda The Vedic Indians could not have missed the faster and brighter planets in the sky. Nowhere in Rgveda were the planets mentioned together, although they were studied. The later Vedanga Jyothisa (VJ) in the second millennium BC also 270

did not mention the planets, perhaps being unnecessary for calendar considerations for the performance of the Vedic rituals. Kak [10] has shown that astronomical information is at the very basis of the number of hymns in the design of Vedic books. He has shown that the information about the sidereal and the synodic period of the planets Mercury, Venus, Mars, Jupiter, and Saturn are embedded in the Rgvedic books. The sidereal period is with respect to the fixed stars and the synodic period is the interval from the time a planet is at the middle of retrograde motion to the next time it is in such a state. He further shows that the probability of the number of hymns could have been chosen randomly but correctly is extremely small.

Numbers in Rgveda Texts The number of syllables in Rgveda is 432,000 and equals the number of muhurtas (one day has 30 muhurtas) in 40 years. The number 108 has a special place in Vedas. The number 108 = 27 x 4 with the mapping of the sky into 27 stars. In the Vedic calendar the months were lunar, but the year was solar. The distances of the Sun and Moon from Earth is nearly 108 times their diameters, which makes their angular distances almost identical. Kak has many more characterizations of the hymns in the Rgvedic books. The solar year was taken to have 371 thitis and hence the Rgvedic scholars were aware of the 18-year eclipse cycle—the number 3,339 stands for 371 x 18/2. This represents the number of thitis between similar lunar eclipses, separated by nearly 18 solar years, counting only the dark fortnights. This is also the syllable count of the hymns in Rgveda in the astaka (eight-book) format.

Priority of Vedas over Other Traditions In the ancient world there could have been a broad exchange of general ideas and interactions through trade. But the independent tradition of observing the planets during the Vedic period (much earlier than Babylon, since 2000 BC) made many of the constants in the later Siddhantic texts differ from that of the Babylonians and the Greeks. Pingree [8] believes that (1) the ratio of longest to shortest day of 3:2 used after 700 BC, (2) the use of linear functions to obtain daylight length in intermediate months, (3) the use of a water clock, (4) thiti as the 30th part of lunar month, (5) the two intercalary months in five years, and (6) the concept of five-year yuga were imported from Babylonian astronomy to Vedic astronomy. But these have been answered [10] by Kuppanna Sastry and Achar, thus proving the antiquity of the latter.

271

Figure 15–1: Earth’s precession per year on summer solstice [13].

Figure 15–2: Sun/Earth/star positions on spring equinox day over past 5,000 years showing Earth’s precession [13].

272

Precession of the Equinox The precession of the equinox has been accounted for qualitatively by Vedic priests by regressing the year beginning from constellation Margasiras to Rohini to Krittika with the passage of time. Only after Aryabhata, the Indian astronomers made quantitative measurements. The effect of the precession of the equinox is shown in Figures 15–1, 15–2, and 15–3 for the summer solstice, spring, and summer solstices days.

Figure 15–3: Sun/Earth/star positions on summer solstice day over past 5,000 years showing Earth’s precession [13].

Calendars in Vedic Astronomy The easy way of measuring a day is between two sunrises, and for a month between two full moons. The “sidereal year” is the time required by the Sun to move from a given star to the same star again, equaling 365.256364 days, and it is the one followed in Vedas. The “tropical year” known as “astronomical,” “equinoctial,” “natural,” or “solar” year is equivalent to one complete circuit of the ecliptic by the Sun, equaling 365.242190 days, and it is the one that determines the seasons. The above two differ by about 20 minutes per year and necessitate a change in the beginning of a year in the Hindu calendar, so that the natu273

ral cycle of seasons always occur at the same in the 12 months of the calendar year. The time taken by the Moon to complete one revolution, called “sidereal period” with respect to the fixed stars, is 27.3216615 days. The interval between two full moons, called “synodic period or lunar month,” is 29.530589 days. The luni-solar calendar adjustment is to find integers such that: x years = y months = z days and is nearly ensured by adding, from time to time, an intercalary month to the regular months to ensure that seasonal festivals and agriculture practices do not go out of step [11]. The fire altar in Satapatha Brahmana denoted even the asymmetric motion of the Sun with respect to Earth between the equinoxes and solstices [10]. The Rgvedic references to year, season, month, and days are not direct but by implication. Thus, in RV 1.164.48, a “twelve follies, one wheel, three axles, who knows this? In that wheel there are 360 spokes moving and unmoving.” The meaning is 12 months, one year, three seasons, and 360 days. Because days are not found after 360, it may mean 360 solar days, or 354 lunar days, but both are shorter than a solar year of 366 days and thus need intercalation. Hence when the lunar calendar was used, 12 days were added at the end of each year. When the system was changed from lunar to luni-solar, then one full month was added after every 2½ years. Subsequently, when there was a solar year of 360 days, it fell short of six days, in which case one month was added after five years. Regarding the number of seasons, Tilak stated that when Aryans moved from the northwest, deeper inside India, the number of seasons increased to three, five, six, or even seven seasons in a year. About the seventh season, it is said that six are twins made up of two months each, but the seventh is the intercalary month. Further the solar year, containing an error of a percent day in each year, was corrected by deducting one month after every 40 years. This is stated in the Veda as “Indra discovered Sambara concealed in the mountain after a search of 40 years.” [4]. Precession causes the cycle of seasons (tropical year) to change slowly (one day per 71 calendar years) in the position of the Sun with respect to the stars at an equinox. In about 2,000 years the months shift through one star in the constellation and thus keep falling back due to the precession. Thus the first month in the rainy season may be Bhadra, Shravana, or Ashada in lesser and lesser antiquity of the related star epoch. The shifting of repeating seasons through the year, but across varying stars in the constellations, helps to date several statements in the Vedic texts over many millenniums as given in Table 15–1. Abhayankar [21] traces various events from the ancient 7000 BC up to possible future 2450 AD. 274

Indian Exploration of Space from Antiquity to Present The first period of naked eye observations of Vedic astronomy extends from remote antiquity up to Siddhantic texts. The ancient records were available in the oral traditional literature of the Vedic times. In his “Orion,” Tilak [3] stated that for the different periods of the Vedas, the astronomical statements unmistakably pointed to the vernal equinox (VE) in the constellation of the Mriga or Orion, during the period of Vedic hymns around 4500 BC receded to the constellation of the Krittikas or the Pleaides around 2500 BC in the days of the Brahmanas. There was no systematic compilation of astronomical knowledge before Rishi Laghada’s VJ. The VJ text specifies the winter solstice (WS) at the beginning of Sravistha (Delphini), VE in 10 deg of Bharani, summer solstice (SS) at the middle of Ashlesa, and autumnal equinox (AE) in 3 deg, 20 min of Visakha, giving it a date of around 1200 BC. Varahamihira states in 505 AD that SS was at the end of ¾ of Punarvasu and WS at the end of first ¼ of Uttarashada, leads to 1150 BC [14]. Using the commercial SkyMap software for the Sun and Moon coming together in Dhanistha star (identified with δ Capricorni instead of β Delphini that is at more than 30 deg from the ecliptic) at the time of WS, Narahari Achar [23] suggests an even earlier date for VJ as 1800 BC. The duration of the shortest and longest days as mentioned in VJ helps to locate the place of composition at 35 deg north near Takshashila. VJ classifies time as solstices and equinoxes, increase and decrease of day and night duration in the ayanas, seasons, solstitial tithes, omission of tithis, table of parvas, solar and other types of years, revolutions of Sun and Moon as seen from Earth and their transit through stars, intercalary month, and so on. There was a period of about 2,000 years after VJ when apparently the progress was perhaps not written and if written has not survived [10]. The next was the appearance of the three texts namely Aryabhata Siddhanta, Brahmasputa Siddhanta, and Surya Siddhanta from the main schools. Generally a Siddhantic text discusses the mean and true position of the planets, precession of the equinox, direction, place, and time, solar and lunar eclipses, and many more. The post Siddhantic period saw steady and continuous developments in astronomy. In the second period many Karana texts appeared, aiding easy usage of the Siddhantic texts. The salient features of these, unlike Siddhantas, are to provide (1) the smaller number of days from the date of their compilation to facilitate easy computing, (2) simple approximations to the various formulae without loss of accuracy, (3) empirical corrections to enable the observations match the predictions from the text, and (4) tables for quick computations. Even in the later 275

20th century many karana and other types of texts on many aspects of astronomy were produced. It may be of interest to note that even comets were observed, for example, by Bhattotpala (937 AD), who listed comets named after Rishis who studied them and recognized their reappearance during their lifetimes using previous records [11]. Sarma discovered the outstanding contribution by the Kerala school during this period. There was a continuous and dedicated effort to improve the model parameters of the planets, computational efficiency, and accuracy. Although telescopes were brought to India in early 17th century, their use somehow did not spread throughout India. The Tantrasangraha of Neelakanta revised the models for the interior planets preceding Tycho Brahe by about 100 years. This implied that the five planets (Mercury, Venus, Mars, Jupiter, and Saturn) move in eccentric orbits around the Sun, that in turn goes around Earth, thus very close to the ultimate heliocentric model [17]. The skillful and dedicated work of Samanta Chandrasekhar toward the end of the 19th century, using crude instruments, brought out the best nontelescopic results better than in the earlier Siddhantic texts [18,19]. In the third period, during the first half, astronomy, atmosphere, and cosmic ray studies formed the main scientific activity toward the exploration of space. The contributions of Saha, Chandrasekhar, and Mitra, in particular, received international acclaim. The discovery by Ramanathan of a higher and cooler tropopause over India was accounted for in the two tropical standard atmospheres proposed by Pisharoti. Ananthasayanam and Narasimha completed such an activity by proposing an International Tropical Reference Atmosphere, valid all the way from sea level to 1,000 km and useful for many aerospace and remote-sensing applications. Table 15–3 shows significant space technological activities during the period after independence commenced with sounding-rocket studies. The space technology matured in steady steps to the launch remote sensing, geostationary, remote sensing, and other types of satellites.

Present Status of Indian Space Exploration Although India has lost the lead, it is possible to catch up, even with a delayed start, as exemplified by the Ariane and Airbus programs of the European community or India’s own aircraft, missile, satellite, and launch vehicle programs.

276

Indian Scientific Philosophy and Some General Remarks It is useful to appreciate the scientific attitude of the various contributors. Also, some useful comments are helpful to understand and teach ancient astronomy or perhaps any other subject provided.

Development of a Physical Theory In developing any physical theory, the results from newer experiments or observations are combined with the existing theory, based on an acceptable or reasonable basis to further improve the theory. As shown in Table 15–4, this can be translated as the existing (estimate) and the newer information (measurement) are combined suitably (criterion) to have better or improved (updated) information as the fundamental principle of the Kalman filter, as stated in Appendix A. Another way of looking at the development is through the triplet of the mind, model, and measurements.

Changes Are Inevitable The classical image of Indians as conservative is not true, and they do not fear change. They have very clearly understood that if there is one thing that is true, it is that change with time—be it celestial objects, atmosphere, space debris, economics, or society, and any model describing their evolution—needs corrections based on periodic observations at various time intervals.

Accept Changes One should recognize change and adapt one’s approach to life’s problems. The important point of Nilakantha (1444–1545) is that although all the “sastras” are the creations of great minds, that of divine grace, they become “slatha” after sometime, because they are created by human beings and are always imperfect and so cannot form the ultimate or absolute truth.

Toward Truth through Pseudo Procedures The philosophy of Bhaskara-I (c. 600–680) has been that the various notions, introduced in developing a physical theory of planetary motion, are aids to arrive at the final results, and the entire procedure is fictitious (!) to arrive at the ultimate truth via untrue means. Putumana Somayaji (1660–1740) also in his comprehensive treatise, KaranaPaddhathi, stated that the various concepts and the derivations based on them are not really true but only help to compute the

277

position of the celestial bodies [16]. Thus the operational success of a theory is given precedence over rigorous conceptual framework.

Operationally Acceptable Imperfect Models The Indian astronomers, in particular from the Suddhantic period, although belonging to different schools, continued to refine the model parameters to match the observations. Thus operationally acceptable models were continuously evolved, rather than seeking what was the true picture. They went on providing answers about how the geometry of the planetary motion is, rather than ask “What is the origin of planetary motions?” as pointed out by Chandrasekhar [20].

Choice of Suitable Variables The various calendars evolved over a period of time. First it was the lunar, then luni-solar, and later the solar calendar. A suitable time scale of days, months, and a year, along with the various time scale of the motion of the Sun, Moon, and the stars, have been used to obtain a convenient calendar for varying purposes of daily, monthly, and yearly rituals. The problem was how to specify in a simple and easy way a reference, or a standard to celebrate the festivals and the rituals.

Stability and Sensitivity in Imperfect Models There has been an element of luck for the Indian astronomers in history. For them only the rates of motion of the planets was of interest in their calculations of the ephemerides and not their distances. The planetary distances given by Samanta Chandrasekhar follow both Bode’s law and Kepler’s law and is close to the modern value, whereas the earlier Aryabhatiya, SurysSiddhanta, and SiddhantaSiromani could not follow any of these laws, although they gave the same time period for the planets. The reason for this is that the earlier works assumed equal speeds in their orbit for all planets, whereas Samanta’s results are based on Tychonic model in which only the Sun and Moon orbit Earth, and other planets orbit the Sun [19]. This shows that in spite of the sensitivity of some important quantities in a problem, some stability does not mean the whole theory is perfect!

Levels of Documentation The Siddhantic texts chose the beginning of Mahayuga (4,320,000 years) as the epoch, and they contained a large number of pages and topics that were not suitable for day-to-day use, because only scholars could follow it. The Tantra 278

texts had fewer topics and explanations and chose a more convenient epoch, such as the beginning of the Kaliyuga. At the next level the Karana texts served as a handbook by taking a convenient and contemporary date as the epoch. providing the necessary corrections to facilitate making a Panchanga for practical purposes [14]. Compare this with the airworthiness codes evolved by analysts and researchers that the designer should meet, and in turn the flight manuals prepared by designers for the pilots to follow, and next the assurance provided by the operator to the flying public whose opinion about the accident is fed back to the analysts forming a closed loop. At various levels one need not necessarily know how the document has been generated from the previous level, but simply follow the codes or the instructions, as the case may be, to ensure safe flight operations.

Better Writing In his History of Ancient Indian Mathematics, Srinivasiyengar lamented “the unfortunate habit of writing everything in prose [has] contributed immensely to the difficulties of modern commentators as quoted by Thruston [22]. The method would no doubt have been explained from master to pupil, and in turn to his pupil and so on, illustrated with numerical examples.” Most present-day writing has more of prose and less of tables, figures, graphs, and flow charts that can make it easy for a reader. These provide a bird’s-eye view, like pictures from a remote-sensing satellite that can indicate many features not understandable by trekking through the forest of prose. A search of the Internet, reports, and many textbooks collecting the results would provide a lesson even for the instructor! In fact the history and philosophy of the development of the concepts in any subject provides a natural evolution that can be followed and retained more easily and enthusiastically.

Criticisms about Vedic Astronomy Many historians try to discredit the tradition of observational Vedic astronomy in India. But for the recorded astronomical observations many of the events in Ramayana, Mahabharata, and other texts could not have been dated with confidence. In particular the conjunction of many celestial objects in the sky, being less probable, helps to determine the epochs with greater certainty. Playfair, Billard, and van der Waerden have demonstrated that it is impossible to define the positions of the celestial objects in the distant past with no knowledge of gravity, the governing differential equations of motion, and the methods of solving them, and not to speak of the uncertainties in the various parameters in the equations and finally provide results and that too for about 10 parameters! Koenraad Elst feels it is a scandal when Playfair’s findings were around for about 200 years, but 279

linguists and indologists published a speculative Vedic chronology with utter disregard for the contribution of Indian astronomy. Billard [6] in 1971 made computer-based statistical studies and demonstrated that independent astronomical observations are the characteristic of Indian astronomers throughout history. The three figures of Billard [6] from Kak [10] shows that Aryabhata corrected the SurysSiddhanta values, and later Lalla corrected his parameters, making it valid over a larger number of centuries. Such continuous updates based on observations by the Kerala school bears this out. Finally, Narahari Achar [23], based on sophisticated simulations using Planetarium software, could place the date of the Mahabharata war at 3067 BC, thus proving that the various sequences of events in such texts cannot be dismissed as fiction.

Conclusions The ancient Indian astronomy is the earliest scientific study in the world. The Vedic astronomers had the best possible scientific approach, commensurate with the facilities and the knowledge at the time. Although India has lost the lead, it can be made up, as shown by the Ariane and Airbus programs and India’s aerospace programs. The history and philosophy in any subject helps to convey the fundamental concepts. The subject matter should contain far more tables, figures, and flow charts for comfortable learning. Also user-friendly software in astronomy would help to understand conflicting and interesting issues, making sensitivity studies to obtain insight into the effect of the errors.

Appendix A: Formulations of Kalman Filter The ancient Indian astronomers corrected the parameters of the model for the planetary motion at various times using observational data as can be seen in Figures 15–4, 15–5 and 15–6. The triplet of the model, measurement, and mind is used to evolve a physical theory, shown in Table 15–4, is just the principle of Kalman filter, as well. In the Kalman filter formulation there are (1) the predicted state estimate plus its covariance, (2) measurement equations, and (3) the state update equations. Further, one needs to be given, otherwise estimate it somehow through the filter formulation itself, the initial state and its covariance together with the covariance of state and measurement noise matrices denoted by XO, PO, Q, and R respectively. To remark briefly—assume that XO is given. The R is fairly objective. PO is partially subjective as well as objective to be chosen for the filter to 280

operate properly. The choice of Q, termed “notorious” in the literature, is highly subjective depending on the desired performance of the filter, such as during the transient and the steady state. The cost function helps to optimally choose these in a systematic way. This is not surprising since the time-dependent filter is subjective, like statistics that is time independent. The situation is like the choice of pole placement in control theory by the analyst to adjust the gains leading to the desired transient and steady state behavior. Except for a small fraction of the people the choice of filter statistics even to handle sophisticated problems is ad hoc after some manual trial and error and not to speak of the cost function that has been mostly forgotten (!)

Figure 15–4: Differences in longitude (SuryaSiddhanta) [10].

281

Figure 15–5: Differences in longitude (Aryabhatiya) [10].

The sensitive filter statistics PO, Q, and R can be traded to the more robust Kalman gain K—a fact utilized to handle many highly nonlinear problems [27]. The K is a measure of the relative importance to be given to the predicted model values to the observations. Working with K avoids the covariance equations, thus saving huge computer time. Similar to the filter statistics, the gain K is also subjective but as shown in such problems can be systematically chosen based on a cost function. The ancient Indian astronomers must have chosen the gain K subjectively.

282

Figure 15–6: Differences in longitude (Lalla) [10].

283

Table 1: Important Events between 10,000 BC – 900 AD 10000– 9000 BC

End of the ice age. Agriculture had commenced. The Taittiriya Brahmana (3.1.2) refers to Purvabhadra star as rising due east would be an event not before 10000 BC and since Rgveda is more ancient than Brahmanas, it can be placed even before 10000 BC.

9000– 8000 BC

The Taittiriya Samhita (6.5.3) places the constellation Pleiades at the WS, which correlates with astronomical events that took place in 8500 BC at the earliest.

8000– 7000 BC

Excavations at Neveli Cori in Turkey reveal advanced civilization with meticulous architecture and planning. The above discovery with megalithic elements going back into 8th millennium showed a sculpture representing a Vedic priest with clean shaven head and a tuft! Siddharth quotes a total eclipse terminating the Treta Yuga around 7300 BC as the time for Ramayana.

7000– 6000 BC

Rig Veda verses (1.117.22, 1.116.12, 1.84.13.5) say WS begins in Aries. Tilak’s estimate of Rgveda is around 6000 BC. Largely agrarian people cultivating barley and cattle inhabited the city of Mehrgarh west of Indus river near Bolan pass from around 6500 BC to 7000 BC. Rgveda frequently mentions barley and milk cattle could have come from this agrarian period as a precursor to the Indus–Saraswati civilization.

6000– 5000 BC

Aithreya Brahmana explicitly refers to the asterism Punarvasu presided over by the deity Aditi, rising exactly east, which happened around 6000 BC.

5000– 4000 BC

Tilak in his Orion stated for the different periods of the Vedas the astronomical statements unmistakably pointed to the VE in the constellation of the Mriga or Orion during the period of Vedic hymns around 4500 BC. A naked eye observation of the conjunction of the planet Jupiter and δ Cancri in Taittitriya Samhita belongs to the period around 4650 BC. Sage Agastya after crossing Vindhya mountain saw the star Agastya for the first time around 4000–5000 BC. He started the first Tamil Sangam and the last Sangam ended near the beginning of Christian era. With about 200 kings ruling on an average for 20 years, would place Agastya’s epoch around 4000 BC agreeing with astronomical dating.

4000– 3000 BC

The date of Mahabharata War based on computer simulations by NarahariAchar is 3067 BC. The Vedic and post-Vedic texts indicate the constellations on the equinox points, which were there from 4000 BC for the RgVeda, as pointed out by Tilak through around 3100 BC for the Atharva Veda and the core Mahabharata. The Margashira star meaning beginning of the year points to fourth millennium BC when the Sun was in that asterism at the VE.

3000– 2000 BC

The rainy season may be Bhadra, Shravana, or Ashada in lesser and lesser antiquity of the related star epoch. The Kaushitaki Brahmana puts the WS at the new moon of the sidereal month Magha for Mahashivaratri (the longest night of the year at the winter solstice) festival, now falling 70 days later points to 2600 BC ± 1,100 years for the establishment of this festival. The year beginning with full moon of Magha indicating SS in Maghas and VE in Krittka in 2350 BC is the time of composition of Taittreya–Samhita and Tandya–Brahmana. The spring equinox conjunct with Rohini star recorded in various Brahmanas indicates their date around 3400 BC. The α Draconis remained as the pole star Dhruva during several centuries before and after 2800 BC.

284

2000– 1000 BC

SHATAPATHA BRAHMANA provides an overview of the broad aspects of Vedic astronomy. Vedic rituals were based on full and new moons, solstices, and equinoxes. Some Hindu scholars date it around 1800 BC based on the reference in it of the Sarasvati river drying up around 1900 BC. Archaeoastronomers have dated it to around 2000 BC based on a reference to the Pleiades (Krttikas) “rising in the east” at AE (ŚBM 2.1.2.1). The Maitrayania Brahmana Upanishad (6.14) refers to the WS being at the midpoint of the Sravista segment and the SS at the beginning of Magha indicating 1660 BC. VEDANGA JYOTHISA (VJ) by Lagadha is the earliest available consolidated astronomical text. In the text it specifies WS at the beginning of Sravistha (Delphini), VE in 10 deg of Bharani, SS at the middle of Ashlesa, and AE in 3 deg 20 min of Visakha giving a date of VJ around 1200 BC. Using the commercial software SkyMap for the Sun and Moon coming together in Dhanistha star (identified with δ Capricorni instead of β Delphini that is at more than 30 deg from the ecliptic) at the time of WS, Narahari Achar suggests even earlier date for VJ as 1800 BC. The VJ system is a coordinate system for the Sun and Moon in terms of 27 stars enabling the calculation of the thitis.

1000–0 BC

Many scholars presuppose the completion of Vedic literature before the rise of Jainism in the latter half of 8th century BC, thus the beginning of the Brahmana period in 800 BC impossible.

0– 900 AD

The year beginning was changed from WS to VE during the Siddhantic period. But the memory of WS is preserved by the Makara Sankranthi festival in 285 AD. But with the loss of astronomical significance both the above now occur after 24 days due to precession. This shows the need to reform Indian panchanga to be consistent with ancient tradition. SURYA SIDDHANTA (unknown author). This has undergone progressive changes from about 400 AD to 1100 AD. ARYABHATA (476–550). Father of Indian epicyclic theory. In the Ptolemic system deferents were large circles centered on Earth and epicycles were smaller circles whose center moved around the circumference of the deferents. In contrast Aryabhata has different values of the circumferences of epicycles for odd and even quadrants. Since the prevailing Surya Siddhanta differed from his observed data, he gave corrected values in his Aryabhatiyam very much followed in south India. Such corrections were continued at later times by Haridatta, Parameswara, and Nilakanta. Aryabhata stated that Earth is round, rotates on its axis (a fact that brought criticism from some contemporary scholars), orbits the Sun, and is suspended in space and also explained lunar and solar eclipses. Assuming the zero sidereal longitudes at midnight in Ujjain, on 17/18 February 3102 BC as the beginning of Kaliyuga, and knowing the number of revolutions of the planet since then would help to calculate the present longitude. His Aryabhata Siddhanta describes astronomical instruments. VARAHAMIHIRA (505–587) adversary of Aryabhata, but popular among public produced the PanchaSiddhanta consisting of Saura (best), Pitamaha (not so correct), Vasistha (methods are crude), Romaka (not applicable to Indian methods of astronomical computations), and Paulisa (probably variant of a Greek name Paulos, obsolete by his time!) is mostly a systematic collection than original contribution. HARIDATTA (650–750) recognizing the deviations from the Aryabhata system for the observed longitudes of the planets promulgated the Parahita system of astronomical computation on the occasion of the 12 yearly Mahaamaagha festival at Thirunavaya. Introduced a simpler katapayadi system of letter notation for numbers than of Aryabhata. His Vaghbhava correction made the results from Aryabhata more accurate.

285

BRAHMAGUPTA (598–668) wrote Brahmasputasiddhanta and Khandakhadyaka, a practical manual. His Siddhanta is popular in northern and western India and translated into Arabic. His remark about gravity is “bodies fall toward the earth as it is in the nature of the earth to attract bodies, just as it is in the nature of water to flow.” BHASKARA-I (c. 600–680). Although not a direct disciple of Aryabhata, was his great admirer. He wrote Maha Bhaskariyam, Laghu Bhaskariyam, and Aryabhatiya Bhasya a commentary on Aryabhatiyam. VATESVARA (b 880 AD). His VateswaraSiddhanta a standard work on astronomy was later adopted by Sripathi and Bhaskara II. In each chapter problems are given as in modern textbooks. __________________________________________________________________ SS = Summer Solstice/Dakshinayana when Sun travels south. (The longest day is 21 June.) WS = Winter Solstice/Uttarayana when Sun travels north. (The shortest day is 21 December.) AE = Autumnal Equinox (The day and night are equal on 22 September.) VE = Vernal Equinox (The day and night are equal on 20 March.) The equinoxes occur at the two intersections of the celestial equator and the ecliptic. The dates are true on an average. The Vedic rituals were based on the times for full, and new moons, solstices, and equinoxes. The uncertainties occur in dating the various events because of the identification of the stars from groups generally, their location with respect to the ecliptic, magnitude, and even the number as 27 or 28.

Table 2: Important Events between 900 AD – 1900 AD 900– 1000

ARYABHATA II (920–1000) wrote an astronomical treatise MahaSiddhanta. MUNJALA (932 AD) wrote Laghumanasam. Munjala and Bhaskara introduced the corrections to account for the irregularities in the Moon’s motion due to evection and variation.

1000– 1100

SRIPATHI MISHRA (1019–1066) wrote Siddhantasekhara and Ganitatilaka for teaching purposes. A third correction to account for the annual variation in the Moon’s motion was given by him.

1100– 1200

BHASKARA—II, also called BHASKARACHARYA, (1114–1185). His Siddantha Siromani, in four parts, deals with arithmetic, algebra, celestial globe, and the planets. His also wrote a handbook, Karnakutuhala.

1200– 1300

Vakyakarana (around 1282 AD) text is the basis of Vakyapancangas, perhaps based on works of Bhaskara and Haridatta. The laborious siddhantic computations are cleverly circumvented but do account for Ayanamsa. MAHADEVA (1275–1350) composed Mahadevi, extensive planetary tables along with instructions called Grahasiddhi to facilitate the computation of daily almanac. Popular in Gujarat and Rajasthan.

1300– 1400

MADHAVA of Sangamagrama (1340–1425) the greatest mathematical astronomer of medieval India. In his Venvaroham a facile procedure is evolved to read out the true position of Moon every 36 minutes.

1400– 1500

PARAMESWARAN NAMBUDIRI (1360–1455) had the best observations before Tycho Brahe. Lived with fishermen since ostracized for such activity and made observations from the age of 53. His eclipse observations not tallying with the ear-

286

lier Parahita system of Haridatta led him to propose the Drk system of Drigganitham. He provided zero corrections for planets at the beginning of Kaliyuga, a smoothing technique! NEELAKANTA SOMAYAJI (1444–1545). His Tantrasangraha provided a unified theory of planetary heliocentric model preceding Tycho Brahe by revising the models of Mercury and Venus. He corrects Aryabhata’s constants in his SiddhantaDarpana and Tantrasangraha. 1500– 1600

JYESTHADEVA (1500–1610) wrote the world’s first calculus text, Yuktibhasa in Malayalam, and Drk-karana on observations. GANESA DAIVAGNA (b 1507) wrote Grahalaghavam at the age of 13, a simple text for calculating the planetary positions for practical work devoid of trigonometry, geometry, and epicyclical models. ACYUTA PISARATI (1550–1621) gave the rule to transfer a planet’s position from its orbit to the ecliptic.

1600– 1700

PUTUMANA SOMAYAJI (1660–1740). His comprehensive treatise, KaranaPaddhathi, and the earlier Tantra Sangraha and Yuktibhasa helped the Europeans to solve the menacing navigation problem in particular at sea.

1700– 1800

RAJA JAI SINGH SAWAI (1688–1743) assembled scholars trained in diverse traditions. Built massive pre-telescopic observatories in masonry and stone in Delhi, Jaipur, and other places to help update planetary tables.

1800– 1900

CHANDRASHEKHAR SIMHA (1835–1904). Due to inadequate siddhantic values made his own observations with a pole, two sticks “T,” and timing devices. Independently had all three corrections of Moon. His skill, practice, and better Tychonic-like model gave the best results even for orbital sizes. Wrote his Siddhanta Darpana in Oriya on palm leaves. He had not seen or heard of a telescope until late in life and deeply regretted.

Table 3: Important Events between 1900 AD – Until Date 1900– 1910

1908 Nizamiah observatory at Hyderabad established in 1901. 1909 John Evershed of Kodaikanal Observatory discovers the radial motions of sunspots, known as Evershed effect, and the shift of sunspot spectra toward red, due to Doppler effect.

1910– 1920

1920 Saha’s thermal ionization equation that goes by his name and its application to stellar atmosphere was first given in the paper “On Ionization in the Solar Chromosphere,” published in the Philosophical Magazine of October 1920. Rated by Eddington as one of the 10 most outstanding discoveries in astronomy and astrophysics after the telescope.

1920– 1930

1920 Ground-based observations of cosmic rays in 1920s and 1930s by Bose, Vibha, Choudhury, et al. in Calcutta. 1929 Ramanathan’s discovery of the higher and cooler troposphere over India published in Nature. 1930 The work on ionosphere by Mitra in Calcutta published in Nature. Experimental evidence of E-region.

1930– 1940

1930 Bhaba publishes a paper concerning absorption and shower production of cosmic rays. 1933 Chandrasekhar’s famous work on black holes and his mass limits.

287

1935 Mitra and Shyam discover an ionospheric region around 55 km and called it the D layer. They also explain Appelton’s ionization anomaly. 1937 Bhaba publishes the paper “On the Penetrating Component of Cosmic Radiation” in the Proceedings of Royal Society. 1940– 1950

1940 During the 1940s, balloon-borne observations were carried out by Bhabha in Bangalore. 1945 Tata Institute of Fundamental Research set up for research in nuclear sciences, cosmic rays, and mathematics.

1950– 1960

1950 During the 1950s, underground measurements of cosmic rays in Kolar near Bangalore at a depth of nearly 2 km. Later cosmic rays and particle interaction studies with nuclear emulsion stacks flown on balloon-borne platforms. 1953 Physical Research Laboratory established in Ahmedabad. 1954 Astronomical observatory set up at Varanasi. 1957 Solar observations undertaken during IGY. 1959 SAAT and SATU standard atmospheres for the tropical regions proposed by Pisharoti.

1960– 1970

1962 INCOSPAR formed by the Department of Atomic Energy. The first sounding rocket, Nike–Apache, launched from an equatorial station at Thumba near Trivandrum as an international cooperative effort from the United States, France, the United Kingdom, and the Soviet Union. 1964 The Center for Advanced Study in Astronomy established at Osmania University. 1964 India admitted as a regular member of IAU. 1965 Space Science and Technology Center established in Thumba. 1967 Kavalur observatory was set up. 1967 Launch of indigenous Rohini sounding rocket from Thumba. Later, many such launches from Balasore in Orissa. 1969 Indian Space Research Organization (ISRO) formed under Department of Atomic Energy.

1970– 1980

1970 Names of seven Indian scientists put on Moon. 1971 The Madras and Kodaikanal observatories were combined into an autonomous institute called Indian Institute of Astrophysics at Bangalore. 1972 The Raman Research Institute at Bangalore commenced its astronomy program. 1972 Space Commission and Department of Space set up on 1 June. ISRO brought under DOS. 1974 Aryabhata with X-ray payload launched. 1975 Tata Institute of Fundamental Research started X-ray and astronomy program. 1975 ISRO’s first satellite, Aryabhata, was launched from the Soviet Union. 1975–1976 Satellite Instructional Television Experiment performed. 1976 Three more Indian names put on Moon. 1977 Discovery of the rings of Uranus. Satellite Television Experiment Project conducted. 1979 Bhaskara-I, an experimental satellite for Earth observations, launched from the Soviet Union. First experimental launch of SLV-3. The Rohini satellite onboard could not be launched. 1979 Second experimental launch of SLV-3. The Rohini satellite onboard successfully launched.

1980– 1990

1981 First development launch of SLV-3. RS-D1 placed in orbit. APPLE, an experimental geostationary communications satellite successfully launched.

288

Bhaskara-II launched. 1982 INSAT-1A launched in April and deactivated in September. 1983 Second developmental launch of SLV-3. RS-D2 placed in orbit. INSAT-1B launched. 1984 Discovery of the outer rings of Saturn. 1984 Sq. Ldr. Rakesh Sharma becomes the first Indian to go into space as part of an Indo–Soviet manned space mission. 1987 First developmental launch of ASLV with SROSS-1 satellite onboard. The satellite could not be placed in orbit. 1988 Launch of first operational IRS-1A. Second developmental launch of ASLV with SROSS-2 onboard could not be placed in orbit. INSAT-1C launched on 21 July and abandoned in November 1989. 1990 INSAT-1D launched. 1990– 2000

1991 Second operational remote sensing satellite, IRS-1B, launched. 1992 Third developmental launch of ASLV with SROSS-C satellite placed in orbit. INSAT-2A, the first satellite of the indigenously built INSAT launched. 1993 Ananthasayanam and Narasimha propose reference atmospheres for the midlatitude and tropics. 1993 INSAT-2B launched. First developmental launch of PSLV with IRS-1E onboard. Satellite could not be placed in orbit. 1994 Fourth developmental launch of ASLV with SROSS-C2 onboard. Satellite placed in orbit. Second developmental launch of PSLV with IRS-P2 onboard. Satellite successfully placed in polar Sun synchronous orbit. 1995 INSAT-2C satellite launched. Launch of third operational IRS-1C. 1995 Giant Meter Wave Radio Telescope set up near Pune. 1996 Third developmental launch of PSLV with IRS-P3 onboard. Satellite placed in polar Sun synchronous orbit. 1997 INSAT-2D satellite launched. Becomes inoperable on 4 October. First operational launch of PSLV with IRS-1D onboard. Satellite placed in orbit. 1999 INSAT-2E launched by Ariane from Kourou. The IRS-P4 (OCEANSAT), launched by PSLV-C2 from Sriharikota, along with a Korean and German satellite.

2000– until date

2000 INSAT-3B launched by Ariane from Kourou. 2001 The first developmental launch of GSLV-D1 with GSAT-1 onboard from Sriharikota. Placed the satellite in a lower orbit. ISRO’s PSLV-C3, successfully launched three satellites of ISRO, Germany, and Belgium into their intended orbits. 2002 Successful launch of INSAT-3C by Ariane from Kourou. ISRO’s PSLV-C4, successfully launched KALPANA-1 satellite from Sriharikota. 2003 Successful launch of INSAT-3A by Ariane from Kourou. The developmental launch of GSLV-D2 with GSAT-2 onboard from Sriharikota. Successful launch of INSAT-3E by Ariane from Kourou. ISRO’s PSLV-C5, successfully launched RESOURCESAT-1 (IRS- P6) satellite from Sriharikota. 2004 The first operational flight of GSLV (GSLV-F01) successfully launched EDUSAT from Sriharikota. 2005 ISRO’s Polar Satellite Launch Vehicle, PSLV-C6, successfully launched CARTOSAT-1 and HAMSAT satellites from Sriharikota. Successful launch of INSAT-4A by Ariane from Kourou. 2006 The second operational flight of GSLV-F02 was unsuccessful. Both the rocket and the communications satellite INSAT-4C destroyed over the Bay of Bengal after the rocket’s trajectory veered outside the permitted limits. 2007 Successful launch of INSAT-4B by Ariane from Kourou. The ISRO’s PSLVC8 successfully launched an Italian astronomical satellite. GSLV-F04 launch successful and INSAT-4CR in orbit.

289

2008 PSLV-C10 successfully launched a commercial satellite POLARIS. PSLV-C9 successfully launched Indian Cartographic satellite and nine other mini commercial satellites. PSLV-C11 launched (22 October 2008) successfully Chandrayaan-1, India’s first mission to the moon. Chnadrayaan-1 carried 11 scientific instruments built in India, USA, UK, Germany, Sweden and Bulgaria. 2009 PSLV-C12 launched successfully Indian Radar Imaging Satellite RISAT-2 along with a student mini satellite ANUSAT. PSLV-C14 launched successfully Oceansat and four small satellites. Chandrayaan-1 detected presence of water on the Moon. The findings from M3 onboard Chandrayaan-1 clearly showed a marked signature in the infrared region of 2.7 to 3.2 micron in the absorption spectrum, which provided a clear indication of the presence of hydroxyl and watermolecules. 2010 PSLV-C15 launched successfully Cartographic satellite along with four commercial satellites. GSLV-D3, GSLV-F06 failed in its mission to inject a communication satellite in GTO. 2011 PSLV-C16 launched successfully a remote sensing satellite ResourceSat-2 and other two small satellites. PSLV-C17 launched successfully a communication satellite. PSLV-C18 launched successfully India–France Megha-Tropiques along with three mini satellites. GSAT-8 was successfully launched on Ariane-5. 2012 PSLV-C19 successfully injected Indian Radar Imaging Satellite.

Table 4 Different Examples of the Triplet Helping to Understand Nature No

Triplets

1

Model

Measurement

Mind

2

Theory

Experiment

Progress

R is Fairly Objective

Q is Highly Subjective

Write

Think

3 4

P0 is Partly Subjective and Objective Read

5

Existing Rules

Experience with Rules

Newer Rules

6

Bhakti Yoga

Karma Yoga

Jnana Yoga

7

Evolving Truth (How?)

Newer Experiments (Why?)

290

Ultimate Truth (Who?)

References 1

J. Playfair, “Remarks on the Astronomy of the Brahmins,” Edinburg 1790, is reproduced in Dharampal: Indian Science and Technology in the Eighteenth Century, Academy of Gandhian Studies, Hyderabad: 1983, (Impex India, Delhi, 1971), pp. 69–124.

2

E. Burgess, The Surya Siddhanta, English translation and notes, P. Gangooly, editor (Motilal Banarsidas: Publishers Private Ltd., Reprinted, 2005).

3

B. G. Tilak, The Orion—or Researches into the Antiquity of the Vedas, fourth edition (Poona: Kesari Printing Press, 1955).

4

R. N. Law, Age of the Rgveda (Calcutta: Oriental Press Private Ltd., 1965).

5

K. S. Shukla and K. V. Sarma, Aryabhata, Aryabhatiya (New Delhi: Indian National Science Academy, 1976).

6

R. Billard, “Aryabhata and Indian Astronomy,” Indian Journal of History of Science 12, no. 2 (1977): pp. 207–224.

7

K. V. Sarma, “Tradition of Aryabhatiya in Kerala: Revision of Planetary Parameters,” Journal of History of Astronomy 12, no. 2 (1977): pp. 194–199.

8

D. Pingree, “History of Mathematical Astronomy in India,” Dictionary of Scientific Biography, Supplement I, (New York, 1978), p. xv.

9

B. L. van der Waerden, “Two Treatises on Indian Astronomy,” Journal of History of Astronomy 11(1980).

10

S. C. Kak, publications available from: http://www.ece.lsu.edu/kak/hist.html.

11

“History of Astronomy in India,” Indian Journal of History of Science, Vol. 20, No. 1–5 (1985).

12

S. A. Paramhans, “Astronomy in Ancient India— Its Importance, Insight, and Prevalence,” Indian Journal of History of Science 26, no. 1 (1991): pp. 63–70.

13

S. Balakrishna, “Names of Stars from the Period of Vedas,” www://vedicastronomy.net.

14

S. Balachandra Rao, Indian Astronomy—An Introduction (India: Universities Press Ltd., 2000).

15

B. G. Sidharth, The Celestial Key to the Vedas (Rochester: Inner Traditions, 1999).

16

H. Selin and R. Narasimha, Encyclopedia of Classical Indian Sciences (India: Universities Press Private Ltd., 2007).

17

N. D. Ramasubramaniam, Srinivas, and M. S. Sriram, “Modification of the Earlier Indian Planetary Theory by the Kerala Astronomers (c 1500 AD) and the Implied Heliocentric Picture of Planetary Motion,” Current Science 66 (1994): pp. 784–790.

18

P. C. Naik and L. Satpathy, “Samanta Chandra Sekhar: Life and Work” Current Science 69 (1995): pp. 705–710.

19

P.C. Naik, “Samanta’s Planet Placing,” Current Science 89 (2005): pp. 211–214.

20

S. Chandrasekhar, “The Pursuit of Science: Its Motivations” Resonance. Journal of Science Education 2, no. 4 (1997): pp. 82–95.

21

K. D. Abhyankar, “Antiquity of the Vedic Calendar,” Bull. Astro. Soc. India 26 (1998): pp. 61– 66.

291

22

H. Thurston, “Planetary Revolutions in Indian Astronomy,” Indian Journal of History of Science 35, no. 4 (2000): pp. 311–318.

23

B. N. Narahari Achar, “A Case for Revising the Date of Vedanga Jyotisa,” Indian Journal of History of Science. 35, no. 3 (2000): pp. 173–183.

24

K. Chandra Hari, “Eclipse Observations of Paramesvara, the 14–15 Century Astronomer of Kerala,” Indian Journal of History of Science 38, no. 1 (2003): pp. 43–57.

25

“International Colloquium on the Date of the Kurukshetra War, Based on Astronomical Data,” held in Bangalore on 5 and 6 January 2003.

26

R. Narayana Iyengar, “Eclipse Period Number 3339 in the Rgveda,” Indian Journal of History of Science 40, no. 2 (2005): pp. 139–152.

27

M. R. Ananthasayanam, A. K. Anilkumar, and P. V. Subba Rao, “New Approach for the Evolution and Expansion of Space Debris Scenario,” Journal of Spacecraft and Rockets 43, no. 6 (2006): pp. 1271–1282.

28

B. Basu, “India’s Space Saga,” Indian Institute of Science, Bangalore, Global Conference, 22– 24 June 2007, Santa Clara, California, 53–55.

292