Input is in (month - year) as Gregarian calendar.

- January 1, 4713 BCE Noon is Julian 0
- February 17/18, 3102 BC is start date for kaliyuga
- Sun (or earth around sun) moves 0.985608372105459 deg/day 59.1365023263275 min/day
- Moon around earth 13.1763585909097 deg/day 790.581515454582 min/day
- 1 tropical solar year = 365.24219 days. It is defined as the time it takes to complete a full seasonal cycle. Because the Earth's speed is not constant, the length of tropical years can vary by up to 30 minutes. Average solar tropical year (time from equinox to equinox) of 365.24219878 days.
- A Sidereal year is time taken for Sun to move from one star, and then come back to same star which iactual solar tropical year (time from equinox to equinox) of 365.24219878 days. s longer by about 19 minutes and 50 seconds (due to Earth's Precession).
- 1 sideral solar year = 365.25663761456 days = 13.368756763665 Sidereal or star month = 12.368756763665 lunar month
- 1 lunar month = 29.5305861853113 days
- 1 star month = 27.3216608 days (Time for one orbit of the Earth = 27.32166 days)
- 1 Anomalistic month 27.554550 days (time between perigees and precesses in 3232.6054 days or 8.850578 years)
- 1 Draconic month 27.212221 days (with respect to the ascending node and precesses in 6793.4765 days or 18.5996 years)
- for 60 star birthday: 21915 days - 802 star months - 60 solar years [-3 days or + 24 days]
- for 80 star birthday: 29220 days - 1070 star months - 29234days - 80 solar years [+14 days or -13days]
- for satabishekam: 1000 lunar months or 29531 days - 1081 star months - 80 solar years 11 months [-3 days]

The average length of a year in the Gregorian calendar is 365.2425 days Compared to the actual solar tropical year (time from equinox to equinox) of 365.24219878 days. The average length of a year in Gregorian calendar is 365.2425 days (a leap year every fourth year except century years not exactly divisible by 400) and Julian calendar is 365.25 days (leap year every fourth year). Changeover from the Julian calendar to the Gregorian calendar occurred in October of 1582. The error accumulated in the 13 centuries since the Council of Nicaea in AD 325 was corrected by a deletion of 10 days. The Julian calendar day Thursday, 4 October 1582 was followed by the first day of the Gregorian calendar, Friday, 15 October 1582 (the cycle of weekdays was not affected).

D (Difference between Julian and Gregorian calendar) = (y/100)- (y/400)-2

Gregorian dates prior to the adoption of the calendar in 1582, are extrapolated, like other era like Saka Era.

Variety of calendars have been and continue to be used in the Indian subcontinent. Saka calendar (78 CE or 3181 Kali) was initiated by Shalivahana or Satavahana king Gautamiputra Satakarni. Shaka Samvat is based on solar months and solar tropical years. The first indication of a relationship between King Shalivahana and the Saka era was authenticated by the Kannada work Udbhatakavya by Somaraja. The Saka calendar is used also by the Indonesian Hindus in Bali and Java.

The Vikram Samvat is named after the king Vikramaditya and also known as Krita and Malava calendar. It starts at 57 B.C. This is a calendar based on the movement of the moon and has 354 days in a year. It uses the lunar months and solar sidereal years for the division of a year. An extra month called adhik maas appears, roughly once every three years (or 7 times in a 19-year cycle). The Vikram Samvat has 12 months with each month divided into two phases: Shukla paksha – new moon to full moon and Krishna paksha – full moon to new moon. The lunisolar Vikram Samvat calendar is 56.7 years ahead of the solar Gregorian calendar.

Julian day No 0, is 21 Teveth -952 Jewish luni-solar calendar, which attempts to simultaneously maintain alignment between the months and the seasons. Years are classified as common (normal) or embolismic (leap) years which occur in a 19 year cycle in years 3, 6, 8, 11, 14, 17, and 19. The calendar begins at sunset the night before Monday, October 7, 3761 b.c.e. in the Julian calendar, or Julian day No 347995.5.

Egyptians, Phoenicians, Persians, Greeks, Roman, and many cultures began their new year with the fall equinox. The Roman Festival of Saturnalia for god Saturn took place between December 17th and 23rd. Birthday of the unconquered sun was held on December 25th for Sun god Mithra. The Winter Solstice on around December 22nd, meant that the winter was over and spring was coming.

Around 1600 years back the day Sun enters Makara Rashi (Capricorn) was coinciding with the day of Uttarayana or Winter Solstice or Surya beginning Northern journey (Northern hemisphere). This happens to be harvest season. It is also Thiruvalluvar new year or early Tamil New Year starting on winter or December solstice. Because of earth's precession, Tamil new year shifted by around 22 days.

Mesopotamians and early Indians celebrated new year around the time of the vernal equinox, March 22 (Current Hindu Saka calendar). There is a parallel system of the Vikrama Era. The origin of the Shaka era is associated to the ascension of many other kings such as Gautamiputra Satakarni Chashtana, Kanishka and Nahapana. Saka era or Shalivahana Sakabda is the vernal equinox of the year AD 78. Later many Indian systems shifted new year from solstice to March Equinox, during the time of Bhadra, Indian astronomer. Because of earth's precession, Tamil new year shifted by around 22 days.

Kalacakra calendar in use, 60 year cycle starting with Prabhava, is by Pandita Somanatha of Kashmir/Himalayas, in 367 CE. Vernal equinox of 367 CE is Prabava Varsham. He further developed sexa decimal system. The earth's axis wobble that causes the precession of the equinoxes is approximately 25,920 years or 432 sixty year cycles. Chandranath introduced lunisolar calendar and Indian cycle of 60 years in Tibet and China.

Kaliyuga calendar; (3102 BCE);

Buddha Nirvana calendar; (544 BCE)

Bikram Sambat (56 BCE) or Vikrama calendar.lunar months, solar sidereal years

Thiruvalluvar calendar (31 BCE)

Saka calendar of (78 CE or 3181 Kali) initiated by Shalivahana or Satavahana king Gautamiputra Satakarni

Shaka Samvat (indian official): solar months, solar tropical years

Bengali Calendar (593 CE)

Tamil Nadu/Kerala: solar tropical years and solar months

Kolla Varsham calendar or Malayalam calendar (824 CE)

There are other eras such as: Vedanga Jyotisa; Gaurabda Gaudiya; and Kolla Varsham. Vikrama Samvat:

Indian National Calendar is based on Saka era, founded by King Shalivahana of the Shatavahana dynasty 22 March Equinox on 78CE. From 664 BCE to 291 CE, aswin was the raising star during March Equinox.

2) Taurus - Vrishabam - Vaishaakha - Vaikasi

3) Gemini - Mithunam - Jyaishtha - Aani

4) Cancer - Karkata - Aashaadha - Aadi

5) Leo - Simham - Shraavana - Aavani

6) Virgo - Kanya - Bhaadrapada - Purratasi

7) Libra - Tula - Aashvayuja - Aiypasi

8) Scorpio - Vrischikam - Kaartika - Kaarthigai

9) Sagittarius - Dhanur - Maargashiirsha - Maargazhi

10) Capricon - Makaram - Pausha - Thai

11) Aquarius - Kumbham - Maagha - Maasi

12) Pisces - Meenam - Phaalguna - Panguni

2) Apabarani - Barani - 35 Arietis (3)

3) Krittikaa - Karthikai - Eta Tauri (6)

4) Rohinee - Rohini - Aldebaran (5)

5) Mrigaseeroo - Mirugasirsham - Lambda Orionis (3)

6) Ardra - Thiruvadirai - Alpha Orionis (1)

7) Punarvasu - Punarpoosam - Beta Geminorium (2 to 4)

8) Pushya - Poosam - Delta Cancri (3)

9) Aslesha - Aayilyam - Alpha Hydroe (1)

10) Makha - Magam - Regulus (5)

11) Poorvaphalguni - Pooram - Delta Leonis (2)

12) Uthraphalguni - Uttaram - Beta Leonis (2)

13) Hastha - Hastham - Delta Corvi (3)

14) Chitraa - Chitirai - Spica Virginis - Vegus (1)

15) Swathi - Swati - Arcturus (1)

16) Vishakha - Visakam - Alpha Libroe (2)

17) Anuradha - Anusham - Delta Scorpio (4)

18) Jyeshta - Kettai - Antares (3)

19) Moola - Moolam - Lambda Scorpio (11)

20) Poorvashada - Pooradam - Delta Sagittari (2)

21) Uthrashada - Uttaradam - Sigma sagittari (3)

21-22) Abhijit - Vega, the brightest star in the northern constellation of Lyra

22) Sravana - Thiruvonam - Alpha Aquiloe (3)

23) Dhanishta - Avittam - Beta Delphinum (4)

24) Sathabhisha - Sadayam - Lambda Aquarius (100)

25) Poorvabhadrapada - Purattadhi - Alpha Pegasi (2)

26) Utharabhadrapada - Uttarattadhi - Gama Pegasi (2)

27) Rewati - Revathi - Zeta Piscum (32)

2) Taurus - May 13 to Jun 21

3) Gemini - Jun 21 to Jul 20

4) Cancer - Jul 20 to Aug 10

5) Leo - Aug 10 to Sep 16

6) Virgo - Sep 16 to Oct 30

7) Libra - Oct 30 to Nov 23

8) Scorpius - Nov 23 to Nov 29

9) Ophiuchus - Nov 29 to Dec 17

10) Sagittarius - Dec 17 to Jan 20

11) Capricorn - Jan 20 to Feb 16

12) Aquarius - Feb 16 to Mar 11

13) Pisces - Mar 11 to Apr 18

14th April 2010, solar lunar cycles align

The saros is a period of approximately 223 synodic months (approximately 6585.3211 days, or 18 years and 11 days and 8h), that can be used to predict eclipses of the Sun and Moon. One saros period after an eclipse, the Sun, Earth, and Moon return to approximately the same relative geometry, a near straight line, and a nearly identical eclipse will occur, in what is referred to as an eclipse cycle.There are many cycles like this

The Wolf cycle (solar sunspot cycle) has a period that fluctuates but averages 11.2 years. Jupiter’s solar orbital cycle is 11.9 Earth years.

Saturn, the second-largest planet, has a solar orbital cycle of 29.4 Earth years.

This leads to Jupiter-Saturn conjunction every 19.9 years (J/S Synodic Cycle).

A full cycle of Jupiter / Saturn around the sun (J/S Tri-Synodic Cycle) is 59.6 years.

About 1000 BC, the Babylonians have kept a consistent record of lunar observations by clay tablets inscribed with cuneiform writing. This was the basis for Babylonian lunar calendars. There are similar developments by other cultures like Mayans, Indus, Chinese and Egyptians. However, they all seem to have lacked geometrical or physical interpretation of their data.

According to the astronomer and mathematician the Aryabhatta finished his book "Aryabhattiya" in 499 CE, and wrote the book in the year 3600 of the Kali Age. Kali Yuga started in 3102 BCE. The starting point of Kali Yuga is an extremely rare planetary alignment, which is depicted in the Mohenjo-Daro seals.

Yukteswar in the book The Holy Science (1894), states that a complete Yuga Cycle takes 24,000 years, and is comprised of an ascending cycle of 12,000 years when virtue gradually increases and a descending cycle of another 12,000 years.

There are many interpreetations on works of Aryabhatta, Paulisa, Srishena, Vishnucandra and others. The general understanding in ancient Indian astronomy was that all the planets commenced their movement together at 0° of Aries but returned to the same position in the heavens, at certain fixed intervals, resulting in a universal conjunction. Surya Siddhanta states that sun was 54 degrees away from vernal equinox when Kaliyuga started on a new moon day, corresponding to February 17/18, 3102 BCJ, at Ujjain (75deg47minE 23deg 15 min N). Varaha Mihira states that 2526 years before start of saka count (either Shalivahana saka starting in 79 AD or Vikrama saka starting in 57 BC)

Ptolemy described lunar motion by using a well-defined geometric model of epicycles and evection. Isaac Newton developed a complete theory of motion and mechanics. Kepler's laws of planetary motion are three scientific laws describing the motion of planets around the Sun, by improving the heliocentric theory of Nicolaus Copernicus:
1. The orbit of a planet is an ellipse with the Sun at one of the two foci.

2. A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.

3. The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.
In astronomy, evection (Latin for "carrying away") is the largest inequality produced by the action of the Sun in the monthly revolution of the Moon around the Earth. Evection causes the Moon's ecliptic longitude to vary by approximately ± 1.274° (degrees), with a period of about 31.8 days. The evection in longitude is given by the expression {\displaystyle +4586.45''\sin(2D-\ell )}{\displaystyle +4586.45''\sin(2D-\ell )}, where {\displaystyle D}D is the mean angular distance of the Moon from the Sun (its elongation, and {\displaystyle \ell }\ell is the moon's mean angular distance of the moon from its perigee (mean anomaly).[3]
It arises from an approximately six-monthly periodic variation of the eccentricity of the Moon's orbit and a libration of similar period in the position of the Moon's perigee, caused by the action of the Sun.[4][5]
The evection opposes the Moon's equation of the center at the new and full moons, and augments the equation of the center at the Moon's quarters. This can be seen from the combination of the principal term of the equation of the center with the evection: {\displaystyle +22639.55''\sin(\ell )+4586.45''\sin(2D-\ell ).}{\displaystyle +22639.55''\sin(\ell )+4586.45''\sin(2D-\ell ).}
At new and full moons, D=0° or 180°, 2D is effectively zero in either case, and the combined expression reduces to {\displaystyle +(22639.55-4586.45)''\sin(\ell ).}{\displaystyle +(22639.55-4586.45)''\sin(\ell ).}
At the quarters, D=90° or 270°, 2D is effectively 180° in either case, changing the sign of the expression for the evection, so that the combined expression then reduces to {\displaystyle +(22639.55+4586.45)''\sin(\ell )}{\displaystyle +(22639.55+4586.45)''\sin(\ell )}

Earth's perihelion occurs around January 3, and the aphelion around July 4 (for other eras, see precession and Milankovitch cycles). The changing Earth–Sun distance results in an increase of about 6.9%

aphelion 152.10×106 km

perihelion 147.10×106 km

semimajor axis 149.60×106 km

eccentricity 0.0167086

inclination 7.155° to Sun's equator - 1.578690° to invariable plane

longitude of the ascending node 174.9°

longitude of perihelion 102.9°

argument of periapsis 288.1°

period 365.256363004 days

average orbital speed 29.78 km/s or 107,208 km/h

https://en.wikipedia.org/wiki/Earth%27s_orbit

Source: Vector image: Gothika. CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=4308370

Source: Original image by Niko LangSVG version by User:Booyabazooka - Own work, CC BY-SA 2.5,

Semi-major axis 384,748 km

Mean distance 385,000 km

Perigee (min. distance from Earth) 363,228.9 km avg. (356400–370400 km)

Apogee (max. distance from Earth) 405,400 km avg (404000–406700 km)

Mean eccentricity 0.0549006 (0.026–0.077)

Mean obliquity 6.687°

Mean inclination of orbit to ecliptic 5.15° (4.99–5.30)

of lunar equator to ecliptic 1.543°

precession of nodes with respect to the ascending node (precesses in 6793.4765 days = 18.5996 years)18.5996 years

precession of line of apsides 8.8504 years

Sidereal month 27.321662 with respect to the distant stars (13.36874634 passes per solar orbit)

Synodic month 29.530589 with respect to the Sun (phases of the Moon, 12.36874634 passes per solar orbit)

Tropical month 27.321582 with respect to the vernal point (precesses in ~26,000 years)

Anomalistic month 27.554550 with respect to the perigee (precesses in 3232.6054 days = 8.850578 years)

Draconic month the time from ascending node to ascending node 27.212221

- Orbits. Source: https://en.wikipedia.org/wiki/Orbit_of_the_Moon#/media/File:Earth-Moon.PNG
- Moon Orbits. Source: https://en.wikipedia.org/wiki/Orbit_of_the_Moon#/media/File:Lunar_perturbation.jpg
- Moon Orbits. Source: https://en.wikipedia.org/wiki/Orbit_of_the_Moon#/media/File:Moon_apsidal_precession.png