The Kali Epoch and Ahargana: Part 1

Ahargana is the number of Civil days (savana days) elapsed from a certain epoch to a date under consideration. Knowledge of Ahargana facilitates easy conversation of dates between the calendars and also comparison of dates marked in different calendars. Once the Ahargana is known, it can also help in computation of mean longitude of planets as various siddhantas give rules of number of revolutions of planet in a yuga or mahayuga from a fixed point in the zodiac at the starting of the yuga.

The fundamentals

The Mahayuga: A Mahayuga comprises of 4,320,000 (4.32 million), solar revolutions of the sidereal zodiac. This is divided into 4 Yugas viz., Satya Yuga of 1,728,000 years (1.728 million years), Treta Yuga of 1296000 years (1.296 million years), Dvapar Yuga of 864,000 years (0.864 million years) and lastly Kali Yuga of 432,000 years (0.432 million years).

Given below are the key elements of the Mahayuga. They are

  1. No. of revolutions of the Sun/ Solar revolutions (sr): The number of times Sun moves around the sidereal zodiac. This is the key measure and defines the true definition of a yuga.
  2. No. of revolutions of the Moon/ Lunar revolutions (lr): During the time Sun makes one revolutions of the zodiac, Moon makes about 12 revolutions. Sometimes, it is 13 revolutions, which contributes to Adhika Masa. This usually gets reflected as two Sun-Moon conjunction in one zodiac sign.
  3. No. of revolutions of the Asterism (ar): This is reckoned based on the rotation of earth around its axis with respect to the Nakshatras. This is also called Sidereal Time. One complete rotation of Earth with respect to the Nakshatra is called a Sidereal Day. The mean Sidereal Day is 23.9344699 hours or 0.99726958 mean Solar days (Savana Days). Source [Wikipedia]. Where is this used? This is the Vedic Measure of a day of 60 Ghatikas and is identical with with the period between two successive rising of a Nakshatra. For instance, if someone is born in 10º Sagittarius, then for his One day is elapsed after Lagna reaches exactly at 10º Sagittarius the next day.
  4. No. of Civil days (cd): The civil day is the duration between two successive mean Sunrises (or Midnight, Sunsets, or Noons). Mean Sunrise is the average of the Sunrises falling within one full revolution of the Sun in the zodiac. For instance if you take the average of the Sunrise time between two successive transits of the Sun in the sidereal Aries, one can arrive at the mean Sunrise time.
  5. No. of Saura Months (sm): One complete revolution of the Sun in the zodiac can be divided into 12 parts, each comprising of 30 days each. This is called a Saura Month. The duration of the day here is based on movement of Sun by 1 degree and not the duration between two successive sunrises. No. of Saura Months = (No. of revolutions of the Sun) * 12.
  6. No. of Lunar Months (lm): One lunar month is the period between two successive Full Moon or New Moon. Usually one Lunar Year contains 12 months, but sometimes it can be more than 12 (adhika masa) or less than 12 (kshaya masa). The Adhika masa is also called intercalary month.
  7. No. of intercalary Months (im): No. of additional month in a Lunar calendar. Each year, the lunar calendar lags behind the solar calendar by about 10 days. This necessitates insertion of an intercalary month approximately every 3 solar years. In that year usually there are 2 Amavasyas in the one of the zodiac sign. When that happens, the month caused by the 1st conjunction of the Sun and the Moon is called Adhika masa and the month caused by 2nd conjunction is called Nija Masa. On the other hand Khsaya masa means loss of month. This happens when there is no soli-lunar conjunction in a sign. This is a rare phenomenon and happens usually in a period of 19 years and 141 years.
  8. No. of Saura Days (sd): This is the number of longitudes travelled by the Sun. This equals to (No. of revolutions of the Sun) * 360.
  9. No. of Lunar Days (ld): One lunar day equals to relative motion of the Moon with respect to the Sun by 12 degrees. This is also called a Lunar phase.
  10. No. of Omitted Lunar Days (od): The Lunar day that does not contain a Sunrise within their span is called a omitted Lunar day.

Source: Ahargana in Hindu Astronomy by SP Bhattacharyya (History of Sciences in India, 1 Park Street, Calcutta 16) and SN Sen (Indian Association for the Cultivation of Science, Jadavpur, Calcutta 32).

Some important computations

  1. Saura Month (sm) = Solar revolutions (sr) * 12
  2. Saura Day (sd) = Saura Month (sm) * 30
  3. Lunar Month (lm) = Lunar revolutions (lr) — Solar revolutions (sr)
  4. Lunar days (ld) = Lunar Month (lm) * 30
  5. Intercalary months (im) = Lunar Month (lm) — Solar month (sm)
  6. Omitted Lunar days (od) = Lunar days (ld) — Civil days (cd)

How to determine the Ahargana in Lunar Calendar?

Lets say, we want to find the Ahargana on a Date in the Saka Lunar Calendar of certain Year (Y), Months (M) and Lunar Days (D).

  1. The first step is to determine the number of solar months from the given date. Solar Month (sm) = 12 (Year + 3179) + Month
  2. Then determine the Lunar months (lm) in the elapsed period = solar month (ms) * Number of Lunar months (lm) in the mahayuga / number of Solar months in the mahayuga (sm). This equals solar month (ms) * 53,433,336 / 51,840,000.
  3. Then the number of Lunar days can be determined by converting the Lunar Months in to Days and adding to that the number of Lunar Days (D) elapsed in the year under consideration.
  4. Once the number of Lunar days are known, the omitted lunar days can be determined by the ratio 25,082,580 / 1,603,000,080
  5. Ahargana = Lunar Days — Omitted Lunar days

Thus Ahargana = Lunar Days — Omitted Lunar days

= Lunar Month(lm) * 30 + D

= (solar month (ms) * 53,433,336 / 51,840,000) * 30 + D

= ((12 (Y + 3179) + M) * 53,433,336 / 51,840,000) * 30 + D

= ((12 (Y + 3179) + M) * 1.03073564814815) * 30 + D

How to determine the Ahargana in Solar Calendar?

Lets say, we want to find the Ahargana on a Date in the Vikram Samvat Calendar of certain Year (Y), Months (M) and Days (D).

  1. The first step is to determine the number of solar days elapsed. Number of Solar Days = 360 * Year + 30 * Months + Days.
  2. From the Solar Days, one can determine the Number of Lunar days elapsed using the proportions mentioned by the Siddhantic texts. If we go by Aryabhatt or Bhaskar, Lunar days = 1,603,000,080 / 1,555,200,000 * Solar days
  3. But not all Lunar days will span across a Sunrise. This means that, we have to reduce the Lunar days arrived above by the number of Omitted Lunar days. So, before proceed further we need to determine the omitted lunar days.
  4. From the Solar Days, one can determine the Number of Omitted Lunar days elapsed using the proportions mentioned by the Siddhantic texts. If we go by Aryabhatt or Bhaskar, Lunar days = 25,082,580 / 1,555,200,000 * Solar days
  5. Ahargana = Lunar Days — Omitted Lunar days

Thus, Ahargana = 1603,000,080 / 1,555,200,000 * Solar days — 25,082,580 / 1,555,200,000 * Solar days

= (1603,000,080–25,082,580) / 1,555,200,000 * Solar days

= (1,577,917,500 / 1,555,200,000) * Solar days

= (No. of Civil days in a yuga / No. of Saura days in a year) * Solar days

= 1.01460744598765 * Solar days

= 1.01460744598765 * (360 * Y + 30 * M + D)

The same concept is used in various Siddhantic texts that one may refer to, to understand the intelligent methods devised and the approximations employed to come up with the Ahargana.