The calendars followed in most regions in India is the lunar calendar based on the movement of moon. Even solar calendars follow the lunar calendar for setting majority of pan India festivals. A solar month in Hindu calendars begin on 14th or 15th or 16th of a traditional month.
Hindu Solar Calendars
The solar year is the time period of the earth‘s revolution around the sun. If instead of taking the sun as a fixed body, we assume the earth to be fixed, then the sun will seem to be moving around the earth. Therefore, the time taken for the sun to make a complete revolution of the earth and come back to the same reference point in the sky will be the measure of a year. The reference point to which the sun returns every year is fixed in two different ways, which yields different results for the length of the year.
Sidereal or nirayana system
A fixed point on the ecliptic with reference to a background star.
Leap Year in Sidereal system
In a nirayana or sidereal year calendar when the months have a fixed number of days and a normal year has 365 days, to compensate for the left over period of 0.256363 day, there will be continuous leap years, including century years, at intervals of four years, also there will be additional leap years added mathematically added at intervals of 157 years, this can be rounded off to 160 years. This rounding up is probably so that these leap years will not coincide with the usual leap years that are added every four years. If we take a scenario where 4AD was taken to be a leap year, also 2AD was a specially added leap year. If we were to follow a 157 year interval, in the third cycle we get a sum of 316 which is divisible by 4. So now the question is does this leap year contain two leap days? However, if we follow a 160 year cycle, we do not run into the problem.
In India the sidereal or nirayana system is followed by the traditional calendar. It follows the calendarical principles laid down in the ancient astronomical treatise named as Surya Siddhanta.
The fixed initial point is the point on the ecliptic which is placed opposite the bright star Chaitra Spica œ α Virgins) located close to the ecliptic. This fixed point is also the vernal equinoctial point of the vernal equinox day of 285AD. Due to the precessional motion the fixed point in the sky which was opposite to the star Chaitra has shifted considerably since 285AD (23°49‘ on 1st January 1997).
Various Calculations in Hindu Solar Calendars
There are twelve rasis or zodiacs in the sky. The ecliptic lies in the middle of this zodiac belt. These twelve zodiacs divide the ecliptic into twelve equal arcs of 30° each. In the tropical system the start of these divisions is from the vernal equinoctial point, but in the sidereal system, the start of the divisions is made from the earlier mentioned fixed point from which Mesha rasi (Aries) starts.
The length of the months are based on the time taken by the sun to traverse the respective rasis, which is the period covered from the time at which the sun enters the concerned rasi, to the time it enters the next rasi. The moment at which the sun enters a rasi is known as a Samakranti.
The sankranti however, may take place at any time of day or night. The day of the month of the traditional calendar known as the savanna or panchang day starts with sunrise. Therefore, depending on the time of the sankranti and the convention followed to determine the starting day for the month, there are four different conventions for four different regions. The month may commence on the same day as the sankranti, or on the following day, or sometimes in some regions, the day after. Due to the regional variances, sometimes the same month has different number of days in different regions. Also the same month in the same region may have different number days in different years.
REGIONAL VARIATION IN INDIAN SOLAR CALENDARS
There are four different conventions for choosing the starting day of the months followed in different regions of India
Odisha School
The solar month begins on the same day when the sun enters the concerned rasi. This convention is followed in Odisha, Punjab and Haryana where solar calendars are used.
Tamil School
When the sankranti takes place before sunset, the month begins on the same day. If it takes place after sunset, the month begins on the next day. Generally, followed in Tamil Nadu.
Malayalam School
The month begins on the same day if the sankranti happens before aparahna, i.e., before 3/5 h of the time from sunrise to sunset. Otherwise, it begins on the next day. Generally followed in Kerala.
Bengal School
When a samkranti takes place between sunrise and the following midnight, the solar month begins on the next day, and when it begins after midnight, the month begins on the day following the next day, that is, on the third day. This is the general rule, and in some special circumstances, there are some deviations from this rule. Generally followed in Bengal, Assam and Tripura.
Source - Chatterjee S.K., (1998) Indian Calendaric System, Publications Division, Ministry of Information and Broadcasting, Government of India.
Dershowitz N. and Reingold E.M., (1997) Calendarical Calculations, Cambridge University Press
Indian Calendars Aslaksen H. and Doegar A. Department of Physics, Faculty of Science, National University of Singapore 10 Kent Ridge Road, Singapore 117546