Hindu Calendar-Thithis

Calendrical Calculations
Part-1

We are in space!

Have you ever thought of why we feel so excited about space? Why do we feel so keen to know about those beautiful objects in the sky? Well, the answer is that we are in space... We live in space! The objects in space deeply influence our lives.

We get up every morning when the sun has its red color scattered in all directions. The sun gives us energy and food. At night, we go to sleep counting those twinkling stars. Watching a full moon night is such a wonderful experience. We eagerly wait for the showers of rain and blanket of clouds every year.  We get a variety of plants and flowers to enjoy every season on this lovely Earth... All this because we live in space.

Why study space?

The astronomy is the oldest science man has ever studied. Over the years, all other branches of science have matured and only increased our thrust for knowledge about the space. The space has taught us physics, it has taught us mathematics, and it has taught us communication. However, when did we actually begin studying about the space? and why? Well, the reason is obvious.

Anyone who has done the night sky-gazing knows the excitement attached to it. From these observations, we could see that a certain celestial events occur periodically in the sky. This might have lead to the invention of a technique to measure time and events that happens in our lives by the ancestors. We wanted to count hours, count days, months and years. We wanted to record events of our lives on a dimension called time. Yes, that is when we invented a calendar.  The calendar that we refer in our day to day life is based on various facts and objects in the space.

By observing various celestial bodies in the sky, we were able to make the calendar. As the mathematics evolved, it got updated over the years. The most obvious objects in the sky we observe in our day to day life are the sun and the moon. These are the objects that periodically move around us.  They take a fixed amount of time to move from one place to another.  A calendar that is based on the behavior of the moon is called a lunar calendar and the one that is based on the behavior of the sun is called a solar calendar.  The ancient Islamic calendar is fully lunar calendar which is called as Hijri calendar. The English calendar what we use today is a solar calendar.  However, the Indian calendar or the Hindu calendar is the one with greatest precision and accuracy of all calendars. It is a combination of lunar and solar calendars so termed as lunisolar calendar or panchanga in our ancient Sanskrit language. The calendar considers small to big events in the space and so we can have various levels of precision in the measurement of time.

In coming sessions, we will discuss in detail how the calendar works.

Part - 2

Let us first understand the basic terminology such as lunar phase, lunation, and a synodic month in order to learn about the lunar calendar. This will give us the idea of the orbital motion of the sun, the moon and the earth. At the end of this section, we will visit the solar calendar and the meaning of lunisolar calendar.

What is a Calendar?

Many will think that this is an easy question. In simple terms, it is nothing but a count of days, months and years. We write it on a nice photo sheet and hang it on a wall. Agreed!!!
What is a day?  When the sun rises, we say that a day starts and continues till the sun rises the next day. Why this happens?  The earth is rotating around itself and the sun is relatively stationary for that small duration. So, in scientific terms a day is nothing but time taken by the earth for one rotation around itself. A few numbers of days decide a month and again some number of months decides a year.  Shortly, we will see how this works. However, an interesting observation is that in a calendar that we use today a day, a month and an year are decided by very different and independent physical phenomena in space.

Lunar Calendar

If we observe the moon regularly then we see that there are 15 days starting from dark night to full moon night. We call this lunar phase as brighter lunar phase and 15 to 16 days from full moon night to dark night as darker lunar phase. So the complete lunar phase cycle is of 29 to 30 days. However it is exactly 29.530589 days or 29 days 12 hours 44 minutes and 3 seconds on an average. This is nothing but a lunation or mean time of the lunar phase cycle. In simple terms, we call this period of 29.530589 days as synodic month or lunar month.

Lunar Phases

Start of the Lunar Month

When do we exactly say that a month has started? Well, it differs from calendars to calendars. For many lunar calendars, it is when the first lunar crescent is observed i.e. when the longitudes of the sun and the moon are equal.  What is this longitude? Here is the answer- the orbit in which moon revolves around the earth is elliptic in nature. If we measure the angular distance of the object from a reference point to its current point and normalize it to 360 degrees (when the angle greater than 360 degrees subtract 360 degrees) we get the longitude of that object. This reference point is traditionally taken as the point on the elliptic orbit of moon directly opposite to the star Spica located in the space.

Length of the Lunar Month

Individual lunation duration varies in hours. That is, the length of the orbit of the moon varies from its average value. This is again because the moon and the earth both revolve in elliptic orbits and not circular orbits. In astronomical terms, it is the eccentricity of the orbits. It takes alternatively 29 and 30 days for each lunation, exact time varies over the year.  So, how do we decide exactly when the new month has started? No doubt, there will be an error because we are not following exact number of lunations. Because, we will obviously represent months as integers, we say after every 29 days a 1 month passed, after every 59 days we say two months passed and so on. Thus, the error will get accumulated.  What happens to this error? We need to compensate for it. How this is done we see shortly.

What is an Year?

The concept of year came from the fact that here on the earth at each location, we get same season repeating in a cyclic fashion.  The scientific reason behind this is the revolution of the earth around the sun. It takes 365 days and 6 hours for the earth to complete one revolution around the sun. However, this revolution does not in any way relate to the moon's motion around the earth. What we mean to say is that if an year to be a time when a season repeats itself, the year will not be evenly divisible by equal number of lunations. It is observed that there are roughly 12 lunations occurring every year. So there are 12 months in a year!!!  But if we say that after every 12 lunations we will increment the year count by one, we are in trouble. Let’s see how.

Lunar and solar year

Lunar Year and Solar Year

A lunar year consists of an average of 354.37 days while solar year is of 365 days. Now, if we consider than an year to be 12 moon phases, and as we saw earlier, the length of lunar month being a variable, we will get different number of days in each year!! Moreover, number of days in a year will go on increasing by 11. Thus, each year will get drifted by 11 days and it will be in sync with solar year only after 33 years have passed. Also, the trouble will occur as seasons will also get drifted !!! For example, say, if we get a hot summer during a particular month in one year  then we may get a cool summer in the same month after 10-12 years. Certainly this is not acceptable. The ancient Islamic calender follows this method and the months get drifted against seasons year after year.

Solar Year and Lunisolar Calendar

Thus an year has to be decided by the sun. A year consists of 365 days i.e. time taken for one complete revolution of the earth around the sun. However, we need to compensate for the variable number of lunations so that lunar calendar comes together with this solar calendar. This is done by adding some extra months. Such a calendar is nothing but a lunisolar calendar. Hindu calendar is a lunisolar calendar. In this calendar, we add an extra month every 32.5 months.

From next section onwards, we will see how the ancient Hindu calendar works.

Part - 3

In the last section, we discussed the basic working of the calendars eventually coming to the principle of a lunisolar calendar. Now, let us go deeper into the ancient Indian calendar and understand its design principles further.

History

The roots of Indian calendar lies in a well developed Indian astronomy since several thousand years. The exact history of the first calendar is unknown however there were many different versions of calendars used in India and maintained by different empires. The first standardized calculation of Indian Hindu calendar was published as "Surya Sidhanta" in 500 A. D.  Now, one might ask two questions here - why we are calling it as a Hindu calendar and what is so important and unique about it to understand it now?

Well, it is called as Hindu calendar because it is traditionally started and maintained by Hindu empires. It has been a tradition in ancient India that when a king used to win a war or expand his kingdom he would start a new calendar. However,  only the initial count of the calendar called epoch (day 1, month 1, year 1) was new in these type of calendars. These  calendars were again  based on the periodic events occurring in our solar system.

Now answering the second question, this calendar is designed in such a manner that it measures the time from small to large duration. Second, it’s very close to events occurring in space. By understanding this, one can easily find out when the eclipses will occur in next thousand years or so. And third, it is very close to the seasons occurring here in India.

Basic concepts in defining the calendar

The Hindu calendar has five (pancha) limbs (anga), five divisions of time. Hence, it’s called as Panchanga in Sanskrit and many Indian languages. The five divisions are vara, tithi, nakshatra, yoga and karana. Let us see definitions of these terms. One will find it  easy to understand on the basis of what we covered earlier.

Vara
It is name of the day- Monday, Tuesday etc. Oh! We have weeks also. A week of 7 days! Why?

Tithi

A tithi is the time taken by the moon in increasing its distance from the sun by 12 degrees. The complete revolution of the moon (lunation of 29.5 days) occupies 30 tithis for 360 degrees. The length of the tithi contantly varies as motions of the sun and moon are always varying in speed. The moment of new moon, that point of time when longitudes of sun and moon are equal, is called "amavasya”.

Nakshatra

The time taken by the moon to travel from 27th part of the elliptic orbit is called nakshatra. During the traversal of the moon around the Earth it was noticed that moon is close to some of the fixed heavenly bodies (stars). Twenty seven stars that fall in the path of the moon have been identified. In 29.5 days, moon's one synodic revolution, moon travels through 27 nakshatras. On an average,  the moon travels one nakshatra everyday. The star which is closest to the moon on its path is called moon's nakshatra.

Yoga

The period of time during which distance between Sun and Moon is increased by nearly 13 degrees. This is about one day.

Karana

The karana is half the tithi, during which the difference between longitudes of sun and moon is increased by 6 degrees.

While the first three sets are still in use, Karana and Yoga are rarely used in day to day life. We will discuss more on this later.

Vara (Days of the week)

There are 7 days, Monday (Somvara), Tuesday (Mangalvara), Wednesday (Budhvara), Thursday (Brihaspativara), Friday (Shukravara), Saturday (Shanivara) and Sunday (Ravivara). These are names of the planets. The counting of days begins from sunrise of the first day to the sunrise of the next day. Now, the question arises as to why the day coming after Sunday is termed as Monday and not the other day? To understand this,  we need to understand position of planets in the space.  The respective positions of the planets are,

• Saturn
• Jupiter
• Mars
• Sun
• Venus
• Mercury
• Moon

Therefore, Saturn is the highest or the farthest planet. Below the Saturn is Jupiter, below Jupiter is Mars, below the Mars is Sun,below the Sun is Venus, below Venus is Mercury, and below Mercury is Moon. All the planets are revolving around the Sun in their respective orbits. Each day is divided into equal partitions from sunrise to next sunrise. Each is termed as hora. Since there are 24 Horas in a day combined with night, therefore, each Hora consists of an hour. The ‘Lord’ of each Hora is a planet from the nearest lower orbit.

The Lord of the first hora is the Sun. In the beginning of the creation, the sun was visible at first and for that very reason, it has been considered as the lord of the first Hora and also the first day has been named after it. The next Hora is named after 'Shukra' (Venus, which is the lord of the second hora, and whose orbit is just below the Sun. The lord of the third Hora is 'Budha' (Mercury) whose orbit is just below that of Shukra (Venus). The lord of the fourth orbit is the Moon, whose orbit is below that of Venus and so on. In this way, the lord of the 24th hora is again Mercury and the Moon (Chandrama).

Moon is the lord of the first Hora (Hour) of the second day (Monday) hence the next day after Sunday is called Monday. Similarly, the lord of the first Hora of the third day is Mars (Tuesday), the lord of the first Hora of the fourth day is Mercury (Wednesday), the lord of the first Hora of the fifth day is Jupiter (Thursday), the lord of the first Hora of the sixth day is Venus (Friday) and the lord of the first Hora of the Seventh day is Saturn (Saturday).  Therefore, the name of the days in order are Sunday, Monday, Tuesday, Wednesday, Thursday, Friday and Saturday.

In next session, we will discuss some mathematics on how the tithi is calculated.

Note:
1. Terminologies used here are as per the ancient Indian Astronomy.
2.  Ancient Indian astronomy treats sun and moon as planets [Graha].

A tithi  is a phase of the moon on a given time of the year (DOY). Since, the moon has a periodicity of 29.5 days and  a year begins with the sunrise on Gudi Padwa/Ugadi (the day after the New Moon day at the beginning of spring), in its most simple form the tithi on a specific day of the year can be calculated as,

Tithi (DOY) = Longitude of Moon – Longitude of Sun (of the orbit of 360degree)

In order to map it with the observable parameters, an additional factor is added into this formulation. The 30 tithis (from Full Moon to New Moon and back) are given in table 1. The waning phase (from Full Moon to New Moon) is called Krishna Paksha and the waxing phase (from New Moon to Full Moon) is called Shukla Paksha and the names of tithis are reversed.

Table 1

Nakshatras

In order to map this movement in the sky, the different paths of the moon are divided as per the nearest star or constellation. The names of the 27 regions, called the 27 Nakshatras that make this path are given in table 2. These are used to define a month. The name of a lunar month is given by the location of the Moon on the Full Moon day. Each region is about 13.2 degrees in the sky.

Table 2

From the point of view of a calendar, a tithi begins on one day and ends on the following day. However, the length of the tithi alters quite often since the apparent motions are not linear. The length of a tithi may begin and end within the limits of same solar day. On other occasions, the moon may remain in the same tithi for as many as 2 days; occupying the whole of one and parts of the other solar day.

A tithi ends at the moment of time when the angular distance between the sun and moon becomes an integral multiple of 12 degrees. In other words, a tithi ends at the same instant of time for all places on Earth and a tithi is not sensitive to the longitude (or latitude) of the region.  Of course, the moment of the sunrise varies with longitude and therefore local time of the moon entry into any tithi will differ at different places. For the same reason, the expunction and repetition of tithis may differ by a day in different longitudes.

Accuracy of observations

The lunation is the time taken by the moon to complete one revolution around the earth. The 360 degree angular path of the moon in the sky is divided into 10,000 parts and 1 part, the finest possible resolution amounts to 2.16 arc min (0.0367 degreee). The time between the two conjunctions is a synodic lunar month. The smallest unit of measure of a tithi is 1/10,000th  part of an apparent sidereal revolution of the moon. When the angular difference between the sun and moon is less than +2.16 arc minutes (measured eastward angle), the sun and moon are said to be in conjunction. This moment of time is said to be the amavasya moment or new Moon. To travel 360 degrees (21,600 arc minute), the moon takes 29.53 (solar) days or 42,480 minutes. So, to travel 2.16 arc minute it takes 4.25 minutes. The moon remains in this position for approximately 4.25 minutes. This interval defines the accuracy of all astronomical observations in ancient Indian calendar. Since, the amavasya (new Moon) lasts for the movement of the Moon from -2.16 to + 2.16. Around the Sun, it last for 8.50 minutes only, according to this formulation.

Primary calculation of the tithi

As stated earlier, a lunation, i.e. rotation by 360° is divided into 30 tithis. The 1/30th of a lunation represents the time duration of a tithi or the angular movement of 12°. Since, the lunation is divided into 10,000 parts about 333 (10,000/30) parts go to one tithi, 667 to 2 tithis and so on. The lunation parts are called tithi indices. The tithi index ‘a’ shows the position of the moon in its orbit with respect to the position of the sun at conjunction. For example,  0 or 10,000 tithi index is the distance traveled from one new moon to return back to the same relative position, and a tithi index of 5000 implies that the Moon has travelled from new Moon to full Moon. The value t=40 shows that the moon has recently
(40 ´ 29.53 ´ 24 ´ 60/10,000 = 170 minutes ago) passed the point or moment of conjunction. Hence, if we know the tithi index (‘a’) we can find out the tithi of a given day. Therefore, above equation can be written as,

a = (DOY – DOY of Gudi Padwa/Ugadi) * 338.63

The constant 338.63 arises from the fact that the moon travels 10,000 parts in 29.5 days, or 338.63 parts in a day. It implicitly assumes that the moment after the amavasya was the sunrise on the Gudi Padwa day, that is, the sunrise on Gudi Padwa/Ugadi day occurred 8.5 minutes after the amavasya. The Gudi Padwa/Ugadi day is defined as the first moment after the new Moon (after the spring). It is taken as first day of an year of this calendar.

Clearly, this not a good approximation and various corrections need to be made. We will see them in  the next session.