The Hindu Lunisolar Calendar

This animation illustrates the basic concepts of the ancient Hindu Lunisolar calendar which is in active use in India even today. By definition, a lunisolar calendar is based on the orbital motion of the Sun and the Moon as observed from the Earth (i.e. in a geocentric model).

Saura Mana
The solar months (Rashis) are nothing but the familiar signs of the Zodiac, labeled in a different language (Sanskrit). A solar day is defined based on the axial rotation of the Earth:

One sidereal axial rotation is called a Nakshatra Ahoratra.
Sunrise to sunrise is called a Savana Ahoratra. This is used in practice today in India.

The “day” which form the basis for this animation (and which is indicated on the animation controls) is the SI Day, i.e. a sidereal solar day.

The crossing of the Sun into a Rashi is called a sankramana and Mesha Sankramana marks the beginning of a new solar year (Sauramana Yugadhi).

Chandra Mana
The lunar days and months are defined based on the position of the Moon relative to the Earth-Sun axis (i.e. its synodic position).

A lunar month (Masa) starts immediately after New Moon.
A new fortnight (Paksha) starts immediately after Full Moon.
Each days (Tithi) is defined as the time required for the Moon to advance 12 successive synodic degrees on its (i.e. relative to the Earth-Sun axis).

You can step through the animation using the incremental controls and observe the effect New Moon and Full Moon have on the calendar:

New Moon (i.e. Moon is in conjunction with the Sun) starts a new lunar month and the next fortnight
Full Moon (i.e. Moon is in opposition to the Sun) starts the next fortnight

The Lunisolar Calendar
When the lunar calendar is superimposed on the solar calendar the following characteristics are observed:

The lunar month in which a Mesha Sankramana occurs is designated the first lunar month (Chaitra). Thus, the lunar new year day (Chandramana Yugadhi) is marked by the New Moon immediately preceding the Sauramana Yugadhi. Every lunar month witnesses one and only one sankramana (with the exception noted below).Lunar months are shorter than solar months. Observe that at the end of the animation (i.e. end of 1 solar year), the lunar calendar has advanced 11 lunar days (Chitra – Ekadashi) into the next year. Hence, the Hindu calendar introduces an intercalary month (Adhika Masa) approximately every 2.5 solar years to re-synchronize the lunar with the solar calendar (not modeled here). In practice, a lunar month that does not witness a sankramana (i.e. which starts and ends while the sun is traversing a single sign of the Zodiac) is declared an Adhika Masa and labeled with the same name as the lunar month succeeding it. In this animation, Year 2 – Bhadrapada starts and ends entirely in Simha Rashi and hence becomes Adhika Bhadrapada; the next lunar month is the “real” (Nija) Bhadrapada, i.e. Bhadrapada repeats twice in Year 2. This effectively synchronized the lunar calendar with the solar calendar and ensures that a Mesha Sankramana happens in the following Chaitra masa.

Some lunar days stretch across the commencement of two solar days (i.e. two sun rises); before the moon completes traversing 12 synodic degrees of its orbit (i.e. relative to the Earth-Sun axis), two solar day commencements occur. Both these solar days are labeled with the same Tithi and the second (repeating) Tithi is called Adhika Tithi (extra lunar day). Example: Jyeshta-Shukla-Dashami stretches across two solar days. Hence, when you advance through two days, the Tithi continues to display Dashami.

Some lunar days start and end between the commencement of two solar days (i.e two sun rises); the Moon traverses 12 synodic degrees of its orbit (i.e. relative to the Earth-Sun axis) between two solar day commencements. Hence, neither of these solar days will be labeled with this Tithi and this Tithi is entirely skilled. Such a Tithi is called Kshaya Tithi (truncated day). Example: Vaishaka-Krishna-Chaturdashi starts and ends between two solar days. Hence, when you advance through two days, the Tithi jumps from Trayodashi to Amavasya (i.e. Chaturdashi is skipped). The Tithi of a solar day is determined based on the synodic position of the Moon at the commencement of the solar day

It is worth emphasizing that the above concept of Adhika Tithi and Kshaya Tithi are the effect of superimposing lunar days on to the solar calendar. These concepts do not imply any change in the definition of a lunar day, i.e. if we consider lunar months in isolation, each lunar month continues to have exactly 30 lunar days (Tithis).

Original Article: https://www.geogebra.org/m/kb9ajk6z