Tropical year

Abstract: 750 The tropical year is the time interval between two sequential passes of the Sun’s true center through the point of the spring equinox The duration of the trop� ical year defined from the astronomical observations is 365 days 5 h 48 min 46 s or 3652422 days Were the motion of the Earth not perturbed (Kepler motion), the duration of the tropical year would be constant in time However, the real orbital motion of the Earth is perturbed [1, 4–6] The differences in the annual transport of solar radiation to the Earth result from the perturbed motion of the Earth and related variations in the duration of the tropical year Thus, the variability in the duration of the tropical year is one of the many

Abstract: Congruences of first degree were necessary to calculate calendars in ancient China as early as the 2 na century B.C. Subsequently, in making the Jingchu [a] calendar (237,A.D.), the astronomers defined shangyuan [b] 1 as the starting point of the calendar. I f the Winter Solstice of a certain year occurred rl days after shangyuan and r2 days after the new moon, then that year was N years after shangyuan; hence arose the system of congruences aN ~ rl (mod 60) ~ r2 (mod b), . ~ ' where a is the number of days in a tropical year and b the number of days in a lunar month.

Abstract: 1. an autumn equinox, a spring equinox, and a summer solstice, each of which is combined with an ancient observation to determine the length of the tropical year. Also, the resulting season lengths are used to determine the eccentricity and the direction of apogee for the Sun. 2. a trio of lunar eclipses which determine the lunar parameters at syzygy. Virtually identical lunar parameters result from Ptolemy’s reports of three earlier lunar trios. 3. several lunar elongations are used to determine the parameters of the lunar model away from syzygy. 4. two pairs of observations of Mercury that determine the longitude of apogee in Ptolemy’s time, and six additional ancient observations that determine it some 400 years earlier. 5. two pairs of observations of Venus that determine the longitude of apogee in Ptolemy’s time.

Abstract: (ProQuest: ... denotes formulae omitted.)The process by which various non-Ptolemaic elements of the Greek astronomical tradition were transmitted to India and were there transformed into the astronomy of the siddhantas is a subject of complexity and of obscurity. Its elucidation, however, is of great historical importance, both for the understanding it will afford us of the motivation for particular Indian solutions of problems in mathematical astronomy, and for the insight we will obtain from it into those areas of Hellenistic astronomy that, being almost totally eclipsed in Greek by the brilliance of Ptolemy's Almagest, can be discerned, though dimly, in the poetry of jyofihsastra. The present paper contains an investigation into one aspect of this process, that in which the ideas both of the precession and of the trepidation of the equinoxes were introduced into India and there interpreted in terms of an older Indian tradition of the position of the solstices relative to the naksatras and in other ways. This example, like that of the planetary model previously discussed in this journal (ii (1971), 80-85), beautifully illustrates the failure of the Greeks to communicate and of the Indians to grasp the full significance of the concepts transmitted.The Jyotisavedanga1 of Lagadha (5/4th century B.c.?) states (Arca 6 = Yâjuca 7): "The Sun and the Moon begin their northern [course] at the beginning of aravicthâ [Dhanictha]; the southern [course] of the Sun [begins] in the middle of Sarpa [Aslecâ]. [The beginnings of these two courses occur] always in [the months] Mâgha and aravana [respectively]." One also finds this scheme in, for example, the Parasaratantra cited by Utpala (a.d. 966) on the Brhatsarphita2 of Varahamihira (ca a.d. 550).By the time of Varahamihira a fixed sidereal zodiac was in use in India.3 In this zodiac the beginning of Aries was identified with the beginning of the nakcatra Asvini; in the fifth and sixth centuries the beginning of Aries was further said to be the point of the vernal equinox. Varahamihira recognized the discrepancy with the statement of Lagadha (Brhatsatyhita 3, 1-2):Once, according to what is said in ancient treatises, the southern ayana of the Sun was from the middle of Âelecâ, and the northern began with Dhanicthâ. Now [one] ayana of the Sun begins at the beginning of Cancer, the other at the beginning of Capricorn. This is a negation of what was said; the difference is made manifest by direct observations.In his Pancasiddhantika* (3, 20-2) Varahamihira explains this change by a theory of trepidation over an arc of 46;40°-23;20° (identified with the Sun's maximum declination) to either side of the equinox:When the sum [of the longitudes] of the Sun and Moon is a revolution, it is called Vaidhrta [yoga]; but if it is a revolution plus 10 nakcatras [133 ;20c]. Vyatipata. The time is to be ascertained by means of the degrees attained [by the luminaries]. When the return of the Sun was from the middle of Ââlecâ [at 113;20°], then the ayana [-correction] was positive; now the ayana is from Punarvasu [at 90°]. When the falling away [from the mean position] of the ayana is reversed, then the correction [kcepa] for the Sun and Moon [equals] the degrees of the maximum declination [kâcthâ] of the Sun [23;20°]. There is Vyatipata if the sum [of the longitudes] of the Sun and the Moon is 180°.This is the earliest datable reference to a theory of trepidation or precession in India ; unfortunately no rate is given. A theory of trepidation was known to Theon of Alexandria (a.d. 361) and to Proclus (a.d. 410-485), and a theory of precession to Hipparchus (ca -126); Hipparchus's length for a tropical year was used by Sphujidhvaja (a.d. 269/70) in his Yavanajdtaka (79, 34) and in the Romakasiddhanta summarized by VaTahamihira in his Paiicasiddhântikâ (1, 15 and 8, 1). It is not unreasonable to suppose that the idea of trepidation or precession was introduced into India by the Greeks, though the parameters chosen by the Indians are their own, and that the arguments presented in favour of the hypothesis of a motion of the colures are derived from a particular interpretation of the Vedangajyotisa. …