Astronomical Bases of Calendars

This section describes astronomical bases of calendars. There are three principal astronomical cycles: Day, Month, and Year. There are three types of calendar systems: Solar calendar, Lunar calendar, and Lunisolar calendar.

There are three principal astronomical cycles that can be used by a calendar:

• Day - Based on the rotation of the earth on its axis.
• Year - Based on the revolution of the earth around the sun.
• Month - Based on the revolution of the moon around the earth.

The year, also called the tropical year, is defined as the mean interval between vernal equinoxes; corresponding to the cycle of the seasons. The following expression, based on the orbital elements of Laskar (1986), is used for calculating the length of the tropical year: 365.2421896698 - 0.00000615359 T - 7.29E-10 T^2 + 2.64E-10 T^3 [days], where T = (JD - 2451545.0)/36525 and JD is the Julian day number. However, the interval from a particular vernal equinox to the next may vary from this mean by several minutes.

The month, also called the synodic month, is defined as the mean interval between conjunctions of the moon and sun, corresponding to the cycle of lunar phases. The following expression for the synodic month is based on the lunar theory of Chapront-Touze' and Chapront (1988): 29.5305888531 + 0.00000021621 T - 3.64E-10 T^2 [days], where T = (JD - 2451545.0)/36525 and JD is the Julian day number. Any particular phase cycle may vary from the mean by up to seven hours.

In the preceding formulas, T is measured in Julian centuries of Terrestrial Dynamical Time (TDT), which is independent of the variable rotation of the Earth. Thus, the lengths of the tropical year and synodic month are here defined in days of 86400 seconds of International Atomic Time (TAI).

Julian day number seems to be defined as 4713. But, I am not sure if this is correct.

Depending which cycles are used, calendars are divided into three groups:

• Solar calendar - Designed to be synchronized with the tropical year. For example, Gregorian calendar is a solar calendar.
• Lunar calendar - Designed to be synchronized with the synodic month. For example, Islamic calendar is a lunar calendar.
• Lunisolar calendar - Designed to be synchronized with both tropical year and synodic month. For example, Hebrew and Chinese calendars are lunisolar calendars.