Panchanga - The Indian calender system

Components of Pañcāṅga

Pañcāṅga, the Indian calendar system, comprises five components:

1. Tithi: The angular separation between the Sun and the Moon.
2. Karana: Half of a tithi.
3. Nakshatra: The specific portion of the ecliptic occupied by the Moon.
4. Yoga: The period during which the sum of the nirayana longitudes of the Sun and the Moon equals 13° 20' or its integral multiples.
5. Vara: The day of the week.

Calculation of Pañcāṅga Elements

The Pañcāṅga calculates these five components at any given instant based on the true longitudes of the Sun and the Moon.

Tithi Calculation

Tithi is calculated by subtracting the Sun's longitude (θS) from the Moon's longitude (θM) and dividing the result by 12, as each 12° of angular separation represents one tithi.

Example 1:

• θM = 60° 12'
• θS = 19° 7'
• (θM - θS) / 12 = (60° 12' - 19° 7') / 12 = 41° 5' / 12 = 3 5/12 tithis
• This indicates that 3 tithis have elapsed, and the current tithi is the 4th, known as Caturti in Shukla-paksha.

Example 2:

• θM = 201° 2'
• θS = 219° 17'
• Since θS > θM, add 360° to θM.
• (θM + 360° - θS) / 12 = (561° 2' - 219° 17') / 12 = 341° 45' / 12 = 28 23/48 tithis
• This shows that 28 tithis have elapsed, and the current tithi is the 29th, which is the 14th in Krishna-paksha.

Karana Calculation

Karana is calculated similarly to tithi, but the difference between the Moon's and Sun's longitudes is divided by 6, as karana is half of a tithi.

Example 1:

• (θM - θS) / 6 = (60° 12' - 19° 7') / 6 = 41° 5' / 6 = 6 51/72 karanas
• This means 6 karanas have elapsed, and the current karana is the 7th.

Example 2:

• (θM + 360° - θS) / 6 = (561° 2' - 219° 17') / 6 = 341° 45' / 6 = 56 57/72 karanas
• This indicates that 56 karanas have elapsed, and the current karana is the 57th.

Nakshatra Calculation

Nakshatra is determined by dividing the Moon's longitude in minutes by 800, as each Nakshatra spans 800' along the ecliptic.

Example 1:

• θM = 60° 12' = 3612'
• θM / 800 = 3612' / 800 = 4 312/800 Nakshatras
• This shows that 4 Nakshatras have elapsed, and the current Nakshatra is the 5th, Mrigashira.

Example 2:

• θM = 201° 2' = 12062'
• θM / 800 = 12062' / 800 = 15 62/800 Nakshatras
• This indicates that 15 Nakshatras have elapsed, and the current Nakshatra is the 16th, Vishaka.

Yoga Calculation

Yoga is calculated by adding the Sun's and Moon's longitudes, dividing the sum by 800' (13° 20'), and taking the quotient as the serial number of the elapsed Yogas.

Example 1:

• θM + θS = 60° 12' + 19° 7' = 79° 19' = 4759'
• (θM + θS) / 800 = 4759' / 800 = 5 759/800 Yogas
• This means 5 Yogas have elapsed, and the current Yoga is the 6th.

Example 2:

• θM + θS = 201° 2' + 219° 17' = 420° 19'
• Subtract 360° as the sum exceeds 360°.
• (θM + θS - 360°) / 800 = (420° 19' - 360°) / 800 = 60° 19' / 800 = 3619' / 800 = 4 419/800 Yogas
• This indicates that 4 Yogas have elapsed, and the current Yoga is the 5th.

Vara Calculation

Vara, or day of the week, is calculated using the concept of Ahargana, which is the count of days from a reference point, specifically the beginning of the current Kali-yuga (Friday, February 18th, 3102 BCE).

To calculate Vara, divide the Ahargana by 7 and use the remainder to determine the day of the week, with 0 corresponding to Friday, 1 to Saturday, and so on.

Example:

• Ahargana = 18,70,348 days
• Ahargana / 7 = 18,70,348 / 7 = 267192 with a remainder of 4
• Therefore, the day of the week is Tuesday.

Conclusion

The Pañcāṅga, with its five elements, provides a comprehensive system for calculating and understanding the Indian calendar. The calculations, based on the true longitudes of the Sun and the Moon, are used to determine the tithi, karana, Nakshatra, Yoga, and Vara for any given instant. These calculations, while simplified here, form the foundation of the Indian calendar system and are still used today in various regional Pañcāṅgas.