Hindu units of time
– How to calculate the thithi
Nakshatra, masa and ritu
Samvatsara – 60 year Jupiter cycle
We have looked at the various cycles used for timekeeping.
- A solar day is when the earth rotates 360° around its axis, with respect to the sun.
A solar day is 24 hours.
- A sidereal day is when the earth rotates 360° around its axis, with respect to the fixed stars.
A sidereal day is 23 hours 56 minutes.
- A tropical year (also known as a solar year) is the time that the sun takes to travel from the vernal equinox to the vernal equinox, along the rashi chakra.
The mean tropical year is 365 days 5 hours 48 minutes. (365.242216 days)
- A sidereal year is the time taken for the sun to return to the same position in the rashi chakra with respect to the fixed stars after traversing once around the rashi chakra.
The mean sidereal year is 365 days 6 hours 9 minutes. (365.256374 days)
The tropical year is about 20 min shorter than the sidereal year due to the precession of the vernal equinox, which causes it to move retrograde or backward in the rashi chakra.Hence the sun reaches the vernal equinox faster.
- A sidereal month is the moon’s orbital period around the earth with respect to the fixed stars. (This is also equal to the moon’s period of rotation around its own axis, due to synchronous rotation)
The sidereal month is about 27 days, 7 hours, 43 minutes (27.32 days).
- A synodic month is the most familiar lunar cycle, defined as the time interval between two consecutive occurrences of a particular phase of the moon, such as new moon or full moon, as seen by an observer on Earth.
The mean length of the synodic month is (29 days, 12 hours, 44 minutes (29.53059 days).
Note that hour, minute and day referred to as a time unit are all based on solar time (time measured with respect to the sun), not sidereal time (time measured with respect to the fixed stars)
- A lunar calendar is a calendar based only upon the monthly cycles of the moon’s phases (synodic months).
- A solar calendar, in contrast, is based only upon annual cycle of the sun (the solar year or tropical year).
12 synodic lunar months make for 354 days, 8 hours, 48 minutes (354.367056 days) which is the lunar year. Whereas, the solar year has 365 days 5 hours 48 minutes. Hence lunar calendars lose around 11 days per year relative to the solar calendar.
- A lunisolar calendar is a calendar that is based on both the solar and lunar cycles. It indicates both the moon phase and the time of the solar year (location of the sun in the ecliptic).
The Hindu calendar, as also the calendars of ancient civilizations all over the world, are luni-solar.
Now, let us understand the various aspects of the luni-solar Hindu calendar called Panchangam.
The Hindu units of time
Before going into the calendar, it is necessary to understand the Hindu units of time. The time scales in the Hindu tradition range from 10^-8 seconds (Truti, the base unit) to 10^22 seconds (lifespan of Brahma, the creator of the current universe).
The Western timekeeping is based on solar time. The solar day is divided into 24 hours of 60 minutes each. Hindu timekeeping is based on sidereal time, with respect to the fixed stars. The reference that is used for calculations is 0° Aries.
The sidereal day, called nakshatra ahoratram, is divisible into 60 ghatis of 24 minutes each. Sidereal hours, minutes and seconds can be expressed in terms of solar hours, minutes and seconds, and are sightly longer.
1 sidereal second = 1.002738 solar second
However, as an approximation, sidereal measure and solar measure of hours, minutes and seconds, can be considered equal.
The sidereal year, varsha, begins when the sun enters 0° Aries, and is completed when the sun has traversed through 360° of the rashi chakra in 12 months.
As we have seen, the starting point, 0° Aries is calculated using the nakshatra Chitra and locating the point 180° opposite it. (The figure, given in Part 2 of this article, is shown here as Fig. 1 again)
The sun’s entry into 0° Aries generally falls on April 14 of the Gregorian calendar. 14th of April marks the first day of the traditional Tamil calendar. The same date in April is observed by most traditional calendars of the rest of India – Assam, Bengal, Kerala, Odisha, Manipur, Karnataka, Punjab etc.
This also coincides with the traditional new year in Burma, Cambodia, Laos, Sri Lanka, Bangladesh, Nepal and Thailand.
Fig. 1 – Sun’s entry point into 0° Aries – April 14
The Hindu units of time are shown in Fig. 2
Fig. 2 – Hindu units of time
The word Panchangam is derived from the words “Pancha” meaning five, and “anga” meaning limb. The Panchangam is the five-limbed Hindu calendar.
The five angas that are used for timekeeping are:
1)Thithi (lunar day)
2)Vasara (solar day)
3)Karana (half of lunar day)
4)Yoga (soli-lunar combinations)
5)Nakshatra (lunar constellation)
The Panchangam is a calendar based on both solar and lunar cycles. It indicates both the phase of the moon and the time of the year (the location of the sun in the rashi chakra).
The solar and lunar cycles tend to mirror one another –
- The new moon and full moon represent to the month what the winter solstice and summer solstice mean to the year, for during one synodic month, the moon covers the same 360° of the sky that the sun covers during one sidereal year.
- While the sun’s 360° cycle is divided into 12 months of 30° each, the moon’s cycle is divided into 30 days of 12° each.
A thithi or lunar day is defined as 1/30th of a lunar month, or the time it takes for the longitudinal angle between the moon and the sun to increase by 12 degrees.
As we have seen, there are 15 thithis in the Shukla paksha (bright fortnight, waxing), and 15 thithis in the waning phase or Krishna paksha (dark fortnight, waning).
One complete synodic revolution of the moon has 30 thithis that covers 360° of the rashi chakra. (12° x 30 =360°)
8.Ashtami (Half moon)
15.Purnima (Full Moon in Shukla paksha) or Amavasya (New Moon in Krishna paksha)
Fig. 3 – The cycle of lunar thithis (Synodic month)
Fig. 4 – Thithi – 12° movement of the moon with respect to the sun
How to calculate the thithi
1.Find the longitudinal difference between the sun and the moon by subtracting the sun’s longitude from that of the moon. (If the moon’s longitude is less than that of the sun, add 360°.)
2.Divide this distance by 12°, and drop the remainder. (To divide, convert the minutes into degree by dividing by 60)
3.If the quotient is 15 or less, quotient + 1 will be the number of the thithi of Shukla paksha. If the quotient is more than 15, subtract 15, and the result will be the number of the thithi in the Krishna paksha.
For example, consider the following positions of the sun and moon in the rashi chakra
Moon – 7° 49’ Libra
Sun – 21° 32’ Cancer
Since the moon has traversed 6 rashis from 0° Aries before entering Libra, it has already traversed 6 x 30° = 180°.
The moon’s longitude is 180° + 7° 49’ = 187° 49’.
Since the sun has traversed 3 rashis from 0° Aries before entering Cancer, it has already traversed 3 x 30° = 90°.
The sun’s longitude is 90° + 21° 32’ = 111° 32’.
To find the longitudinal difference between the sun and the moon, subtract the sun’s longitude from that of the moon.
187° 49’ – 111° 32’ = 76° 17’
Since 1° = 60′
17’ = 17/60 ° = 0.283°
Hence, 76° 17’ = 76.283°
Dividing 76. 283 by 12 gives 6.35.
The quotient 6 is taken and the remainder is dropped.
6 + 1 = 7
Hence, the thithi running is the 7th thithi, Saptami.
Since 7 is less than 15, it is Saptami of the Shukla paksha.
Length of a thithi
Since the motions of the sun and the moon are always varying in speed, the length of a thithi constantly alters. The variations in the length of a thithi are given in Table 1.
60 ghati = 24 hours
Table 1 – Greatest, least and mean lengths of thithi
The time duration from one sun rise to the next sun rise is reckoned as a vasara in Panchangam. This duration is equal to 24 hora (hora is modern hour of 60 minutes).
There are 7 vasara in a week, 7 being the number of the primary grahas in Vedic astronomy and astrology. Each vasara is presided over by a graha. That is how the vasara are named.
Vasara and thithi : As the time duration of thithi can vary from 21 hours to 26 hours, the thithi’s beginning and end occur at different times of the vasara, which has 24 hours. Sometimes, a vasara can have 2 beginning points of successive thithis.
The names of the vasara in English, Sanskrit, Tamil and their presiding grahas, are shown in Fig. 5
Fig. 5 – Vasara and grahas
Just as a solar day is divided into two portions as day and night, the lunar day or thithi is similarly divided into two karanas, of 6° each.
11 different karana are permuted to make up the 60-karana monthly cycle.
Of the 11 Karana, the first 7 are considered chara (movable) karana. The last 4 are considered sthira (fixed) karana.
Chara karana (movable)
Sthira karana (fixed)
As we can see in Table 2, most of the karanas of Krishna paksha are identical to those of Shukla paksha except that the consecutive cycles of the 7 chara (movable) karanas of the Krishna paksha are one thithi behind those of the Shukla paksha.
Table 2. The karana
The Sanskrit word Yoga means “union, joining, attachment”. In the astronomical context, it refers to the joining of the celestial longitudes of the sun and the moon.
The 27 yogas reflect the nature and effect of the interactions of the sun and the moon.
Table 3. The 27 luni-solar yoga
How to calculate the yoga
1)Add together the longitudes of the sun and the moon.
2)Subtract 360° if possible.
3)Divide the result by 13° 20’, and drop the remainder.
To divide, first convert the result entirely into minutes and divide by 800. (Since 13° 20’ = 800’)
1° = 60’
4)The quotient + 1 will be the number of the yoga.
Nakshatra, masa and ritu
Nakshatra in Panchangam refers to the lunar constellation in which the moon is located. There are 27 nakshatras, as we have seen in Part 1 of this article. It is shown here again in Fig. 6
Fig. 6 – The 27 nakshatras
Relationship of nakshatra and masa (month)
The month in the Panchanga is based on the solar cycle. The start of the month is considered to be 0° entry of the sun into a rashi. The 12 rashis correspond to the 12 months. Each month has days varying from 29 to 32.
Many of India’s regional calendars name their months after the nakshatra in which the moon will sit (or sometimes, will be near to) when it is full. The first month of the Hindu calendar corresponds to the spring month when the full moon occurs in the nakshatra Chitra, and the month is known as Chaitra. Table 4. shows the relationship of the lunar months with the nakshatras. This is also shown pictorially in Fig. 7.
Table 4. Relationship of nakshatras and months
Fig. 7 shows how the Hindu months are named after the nakshatra in which the full moon occurs. Since the full moon occurs when the moon is opposite the sun in its orbit around the earth, the nakshatra of the full moon is generally 180° opposite to the rashi the sun is, in that particular month.
Fig. 7 – Nakshatra of the full moon and masa
In Fig. 8, we see that the start of the Gregorian month corresponds to entry of the sun into the 15° (approximately) of the rashi (sidereal zodiac). While the start of the Hindu month corresponds to entry of the sun into the 0° of the rashi. This 0° entry into the rashi is the reckoning of the month in the Hindu calendar.
Fig. 8 – Gregorian month and Hindu month
Relationship of masa, ritu (season) and festivals
All the Hindu festivals and religious holidays in India are dated using the Hindu luni-solar calendar, not the Gregorian solar calendar.
In the Western system, there are 4 seasons – spring, summer, autumn and winter. In the Indian system, there are 6 seasons, as 2 months make for a season, ritu. The ritu are Vasanta (spring), Grishma (summer), Varsha (monsoon), Sharad (autumn), Hemanta (winter) and Shishira (prevernal*)
Table 5. shows the relationship of masa, ritu and festivals.
*Prevernal is the period from the end of winter and beginning of spring.
Table 5. Relationship of month, season and festivals
Samvatsara – 60 year Jupiter cycle
The 60-year cycle is common to both North and South Indian traditional calendars, with the same name and sequence of years. After the completion of 60 years, the calendar starts anew with the first year. This corresponds to the Hindu “century.”
Jupiter has a sidereal period (with respect to fixed stars) of 11 years, 314 days, and 839 minutes, which is almost about 12 years. Its synodic period* brings it in conjunction with the sun every 398 days and 88 minutes, which is a little more than a year. Thus, Jupiter in about 12 years and sun in one year pass through the same series of nakshatras. So a given year can be dated as the month of a 12 year cycle of Jupiter. A collection of 5 such cycles or 60 years is called Brihaspati-chakra, and each of the 60 years has distinct names, given in Table 6.
The current year 2018-2019 is named Vilambi.
* Time period between two successive conjunctions of the same two celestial bodies
Table 6. Samvatsara – 60 year Jupiter cycle
Samvat refers to era. Several Indian calendar systems use the era or Samvat to provide a historic reference, like how the Gregorian calendar uses the Common Era which has the birth of Jesus Christ for its historic reference. Vikrama era refers to the coronation event of the legendary king Vikramaditya in Ujjain. Currently we are in 2075 Vikram Samvat.
The Hindu Panchangam is a luni-solar calendar that keeps time based on both solar and lunar cycles in parallel.
- Varsha (year, position of the sun)
It calculates the varsha based on the solar cycle.
A varsha is the sidereal year, the time the sun takes to return to 0° Aries in the rashi chakra, about 365 days. It typically falls on April 14 of the Gregorian calendar.
- Vasara (solar day)
It calculates the vasara based on the solar cycle.
A vasara is the time from one sunrise to the next and is divided into 24 hours or 60 ghatis. Since this time period is reckoned with respect to the sun, a vasara is a solar day or tropical day.
- Thithi (lunar day)
It calculates the thithi based on the lunar cycle – synodic movement.
A thithi is the time taken by the moon to move 12° with respect to the sun in the rashi chakra. There are 30 thithis in a synodic lunar month.
- Nakshatra (lunar month, position of moon)
It calculates the nakshatra, based on the lunar cycle – sidereal movement. The nakshatra is the nakshatra in which the moon sits. There are 27 nakshatras and the moon spends about a day in each. This corresponds to the 27.3 days in a sidereal lunar month.
- Masa (month, position of the sun)
It calculates the masa based on the solar cycle. A masa begins when the sun enters 0° of a rashi. There are 12 months, with days that vary from 29 to 32.
It names the month based on the lunar cycle – sidereal movement. The month is named after the nakshatra in which the full moon will sit, or be close to.
- Karana (half of lunar day)
Just as we have day and night as the two halves of for a solar day, the thithi or lunar day is divided into two karanas, of 6° each. There are 11 karana which are permuted to make up the 60 karana synodic month.
Yoga are 27 luni-solar combinations that reflect the nature and effect of the interactions of the sun and the moon.
We have understood the Hindu way of timekeeping in relation to the astronomical cycles. But how do these astronomical cycles relate to the human body cycles? This will be explored in Part 4.
Fig. 3 – https://shrifreedom.org/vedic-astrology/tithi/
Table 1 – Basics of Panchangamm by S. Narasimha Rao
Light on Life : An Introduction to the astrology of India by Robert Svoboda and Hart de Fouw
The Calendars of India by Vinod K Mishra