The astronomy behind the Hindu luni-solar calendar

The Hindu calendar is luni-solar: its months follow the Moon, from one new Moon (or full Moon) to the next, about 29.53 days, while its year follows the Sun's return to the same place among the stars, about 365.26 days. Twelve lunar months fall roughly 11 days short of that year, so an extra month, the adhika masa, is inserted about once every 32 or 33 months to keep the two in step. Because the year is measured against the stars rather than the seasons, the calendar drifts slowly against the equinoxes, and the size of that drift is the ayanamsa.

Everything below is ordinary astronomy. The same quantities drive the site's daily panchang.

The lunar month

A synodic month is the time from one new Moon to the next: on average 29.5306 days. It is longer than the Moon's orbit relative to the stars (about 27.32 days) because the Sun has moved on in the meantime and the Moon needs about two more days to catch it up.

The Hindu month is built on this cycle and divided into 30 tithis, each marking another 12° of the Moon's lead over the Sun. The month is split into two fortnights, the waxing Shukla paksha and the waning Krishna paksha. See the five limbs of the panchang for how tithis are counted.

Amanta and purnimanta: where the month ends

There are two conventions for where a month begins and ends.

• Amanta months run from new Moon to new Moon. This is used in Maharashtra, Gujarat, Karnataka, Andhra Pradesh and Telangana.
• Purnimanta months run from full Moon to full Moon. This is used across most of north India.

The fortnights themselves are the same in both systems, and festivals fall on the same days. What differs is the name of the month during the waning fortnight. In the purnimanta system the dark fortnight is named after the month that is about to begin; in the amanta system it is named after the month that is ending. Janmashtami, for example, is on Krishna Ashtami of Shravana in the amanta reckoning and Krishna Ashtami of Bhadrapada in the purnimanta reckoning. It is the same night either way.

The month names themselves come from the sky. Each is named after a nakshatra near which the full Moon falls in that month: Chaitra after Chitra, Vaishakha after Vishakha, Kartika after Krittika, Margashirsha after Mrigashira, Pausha after Pushya, and so on.

The sidereal year

The solar year in the Hindu calendar is sidereal: the time the Sun takes to return to the same position against the fixed stars, about 365.2564 days. The Sun's entry into each of the twelve sidereal signs is a sankranti, and Makar Sankranti, the Sun's entry into Capricorn, is the best known.

Some regional calendars are purely solar and use the sankrantis directly as month boundaries. The Tamil, Malayalam, Bengali and Odia calendars work this way, which is why Tamil months begin on fixed-ish Gregorian dates while north Indian lunar months wander.

Reconciling the two: adhika masa

Twelve synodic months take 12 × 29.5306 = 354.37 days. The sidereal year is 365.26 days. The shortfall is 365.26 − 354.37 = 10.89 days a year.

Left alone, the lunar months would slide backwards through the seasons by that amount every year, as the purely lunar Islamic calendar does. The Hindu calendar corrects this by inserting a thirteenth month when the gap reaches a whole month. A month is 29.53 days, so that happens every 29.53 ÷ 10.89 ≈ 2.71 years, about every 32 or 33 months. Over 19 years this works out at seven extra months, the same ratio the Greek astronomer Meton noticed: 235 lunar months come within a few hours of 19 solar years.

The rule for which month becomes adhika is astronomical rather than scheduled. Normally each lunar month contains exactly one sankranti. Because a lunar month (29.53 days) is shorter than the time the Sun spends in a sign (about 29.4 to 31.5 days, depending on the time of year), occasionally a lunar month starts and ends with no sankranti inside it. That month is adhika, an intercalary month, and it takes the name of the month that follows it. The regular month then follows with the same name.

The reverse also happens, very rarely. Near the start of the year the Earth is closest to the Sun and the Sun appears to move fastest, spending less than a lunar month in a sign. A lunar month can then contain two sankrantis, and one month name is dropped: a kshaya masa. Kshaya months are separated by long intervals, often over a century, and are always accompanied by adhika months in the same year.

Adhika masa matters in practice. Many families avoid starting major ceremonies in it, and when it falls within Chaturmas it lengthens that period from four months to five, closing more of the year to weddings. See Vivah muhurat 2027 for that effect.

Why the calendar drifts against the seasons: ayanamsa

The Earth's axis slowly wobbles, like a spinning top, completing one circuit in about 25,800 years. This precession moves the equinoxes backwards against the stars by about 50.3 seconds of arc a year, one degree in roughly 72 years.

Western astrology and most astronomy almanacs measure longitude from the March equinox: the tropical zodiac. Jyotish measures it from a fixed point among the stars: the sidereal zodiac. The angle between the two zero points is the ayanamsa, and it grows by that 50.3″ every year.

The site uses the Lahiri (Chitrapaksha) ayanamsa, which places the star Spica (Chitra) at 180° of sidereal longitude. It was recommended by the Calendar Reform Committee set up by the Government of India in the 1950s and is the most widely used convention in India. Its value is currently about 24°.

A worked example: why Makar Sankranti is not at the solstice

Tropically, the Sun enters Capricorn at the December solstice, around 21 December. Sidereally, it must travel another 24° or so, the ayanamsa, before it reaches sidereal Capricorn. The Sun moves a little more than 1° a day in December and January, so 24° takes about 23 to 24 days. That brings Makar Sankranti to around 14 or 15 January, which is where it falls today.

Run the arithmetic backwards: 24° at one degree per 72 years is about 1,700 years. Roughly that long ago the two zodiacs coincided, and Makar Sankranti did fall on the solstice. Run it forwards: in about 72 years Makar Sankranti will have moved another day later in the Gregorian calendar.

What this means for a panchang

Three practical consequences follow from the astronomy:

• Tithi and karana do not depend on ayanamsa. They measure the difference between the Moon and the Sun, so the ayanamsa cancels. Any competent almanac agrees on them for a given place.
• Nakshatra, yoga and sankranti do depend on it. Almanacs using a different ayanamsa will shift their end times and dates slightly.
• Location still matters. The day begins at local sunrise, so the tithi assigned to a calendar date can differ between cities. A festival can fall on different Gregorian dates in Chennai and Toronto, with neither calendar in error.

This is why the site computes everything for a named city rather than publishing one national table.

Frequently asked questions

How often does adhika masa occur?

About every 32 to 33 months, or seven times in 19 years. Which month becomes adhika depends on when the sankrantis fall, so it varies.

Is the Hindu year sidereal or tropical?

The traditional solar year is sidereal, measured against the stars. India's civil National Calendar, adopted in 1957, is a separate tropical solar calendar.

Why do north and south Indian calendars give different month names?

Most of north India uses purnimanta months, ending at full Moon, while much of the south and west uses amanta months, ending at new Moon. The waning fortnight carries a different month name in each, but the days themselves are the same.

What is Lahiri ayanamsa?

The convention, recommended by the Indian Calendar Reform Committee, that fixes the sidereal zodiac by placing the star Spica (Chitra) at 180°. It is currently about 24° and is the standard used on this site.