Mathomathis would like to present an article on Indic Cosmology and Time Keepers by Kosla Vepa – Indic Studies Foundation. The following article is a continuation from the previous article Indic Cosmology | Kosla Vepa | Equatorial Coordinate System | 102.


The TITHI

According to the Indian calendar or Panchangam, Tithi is a lunar date based on the rotation of the moon around the earth, and is one of the five important aspects of an Indian almanac (Panchangam – Panch means five and anga means parts). Most of the Indian social and religious festivals are celebrated on a date corresponding to the original Tithi. The distance between the Sun and the Moon calculated on a daily basis is called Tithi. The positioning and the movements of both Sun and Moon are different (the Sun is much farther away than the moon , and hence it does not make sense to refer to the distance in terms of Miles are meters but in degrees only. As the space which is in the circle shape – is 360 degrees. So in a month there are 30 days ( or Tithis) that will bring us to the 12 degrees per Tithi (360/30) to calculate the distance between the Sun and the Moon. On the New Moon day – that is Amavasya – the distance between the Sun and Moon is only zero (0) degrees and at that time the Moon will have no light. On the full moon day the distance is 180 degree as both Sun and Moon are on opposite positions. So, that shows when the distance between Sun and Moon is 0 – 12 degrees that is defined as Padyami, and when it is 12-24 degrees that is defined as Vidiya ( 2nd day) and when the distance is 24- 36 degrees that day is defined as Tadiya ( third day). There is another specific thing to be noted: the movements of the Sun are slow while the corresponding movements of the Moon are relatively rapid. If one takes the average motion ( mean motion) of Sun, it is 59.1 minutes( 1 degree is equal to 60 minutes) Where as the Moon’s mean motion is about 790.56. So the difference between the Sun and Moon’s motion is 790.56 – 59.1 = 731.46 that is equal to 12.19 degrees.

Whereas to gain the correct Tithi, one should not take the mean motions – one need to take accurate motion to obtain the right time of Tithi. There are functions to obtain this accurate motion to get the right Tithi. In a month there are 30 Tithis- and on an average each Tithi will run for 23.62 hours. But, in many days, the Tithi usually hovering between 20 hours to 26.40 hours – and with this huge fluctuations, one can not depend upon the mean motions and this fluctuation occurs because of the daily changes in the motion of the Moon. The timing of the performance of a particular aspect of a puja associated with a variety of rites and ceremonies is essential for the proper performance of the puja. Such an injunction was a corollary to the assumptions made in the belief system prevalent during the ancient era. As there are many kinds of writers of Panchanga- there is always a difference from one school of thought to another school of thought and the Tithis tend to get overlapped.


WEEK DAYS ( VAARA)

Vaasara, often abbreviated as vaara in Sanskrit-derived languages, refers to the days of the week, which are possibly of Sumerian/Babylonian origin, and bear striking similarities with the names in many cultures: Following are the Hindi and English analogues in parentheses

  • Ravi vāsara (ravi-vaara or Sunday; ravi = sun)
  • Soma vāsara (som-vaara or Monday; soma = moon)
  • Mangala vāsara (mangal-vaara or Tuesday; mangala = Mars)
  • Budha vāsara (budh-vaara or Wednesday; budh = Mercury)
  • Guru vāsara (guru-vaara or Bruhaspati-vaara or Thursday; vrihaspati/guru = Jupiter)
  • Shukra vāsara (shukra-vaara or Friday; shukra = Venus)
  • Shani vāsara (shani-vaara or Saturday; shani = Saturn)

Days of  the Week are:

  • Sunday – Ravi- vasahara Raviwar, Adi, Aditya
  • Monday – Somwar (Chandrawar)
  • Tuesday – Mangalwar
  • Wednesday – Budhwar, Rauhineya, Saumya
  • Thursday – Guruwar, Brihaspati
  • Friday – Shukrawar, Bhrigu, Bhargava
  • Saturday – Shaniwar

There are many variations of these names in the regional languages, mostly using alternate names of the celestial bodies involved. The astonishing fact of the matter is that the system of dividing the week into 7 days is fairly widespread among all geographies and civilizations, and it is difficult to say when it originated . The Indian Panchangam is really an almanac rather than calendar. It is really analogous to the concept of a Farmers almanac that is widely prevalent in the west. The word calendar is itself of Greek origin. The current calendar “date” based on the Gregorian Calendar that we are so familiar with in our daily life is heliocentric and is based on the rotation of the earth around the sun. It takes the earth approximately 365 ¼ days to complete its rotation around the Sun. The calendar that most of us use today divides the 365 days of earth’s period of rotation around the Sun in twelve months. The leap year, which occurs once every four years, accounts for ¼ day per year.

Panhanga - Mathomathis

Similar to the solar calendar, the lunar calendar is also popular and widely used in the Asian countries such as China, Pacific-rim countries, Middle East countries, and India. The lunar calendar, which is believed to have originated in India, has been around for a very long time, even long before the solar calendar. The lunar calendar is geocentric and is based on the moon’s rotation around the Earth. The lunar month corresponds to one complete rotation of the Moon around the Earth. Since this period of rotation of moon around the earth varies, the duration of lunar month also varies. On average, the lunar month has about 29 ½ days, the period of the lunar Synodic orbit. In addition to moon’s rotation around the earth, the lunar year is based on earth’s rotation around the Sun. In general, the lunar year has twelve lunar months of approximately 354 days (29.5 *12 ), thus making it shorter by about 11 days than the solar year. However, the lunar calendar accounts for this difference by adding an extra lunar month about once every 2 ½ years. The extra lunar month is commonly known as “Adhik Mas” in India (Adhik means extra and the Mas means month). The concept of this extra month is similar to the “Blue Moon” in the West, which occurs almost with the same frequency of 2 ½ years. The Indian lunar year begins on the new moon day that occurs near the beginning of the Spring season. The twelve lunar months are given below.

According to the Moslem calendar which is widely followed in Middle East and in other Moslem countries the lunar year is strictly based on twelve lunar months of 354 days per year. That’s why their holy month of Ramadan occurs by approximately 11 to 12 days earlier than that in the preceding year. The solar day (commonly referred as the “the date” in western calendar) has a fixed length of 24 hours. The change of date occurs at midnight as per local time or standard time of a given local time zone. Thus, the date changes from midnight to midnight. Similarly the day (as in weekdays) changes from midnight to midnight as per local or standard time for that location. In other words, as per the western (or English) calendar the length of day and date is exactly 24 hours, and there is a definite correspondence between the date and the corresponding day of the week.

A lunar day usually begins at sunrise, and the length of lunar day is determined by the time elapsed between the successive sunrises. As per the Jewish calendar their lunar day begins at the sunset, and lasts through the next sunset. A lunar day is essentially the same as a weekday. In India the lunar day is commonly referred as “V[W]ar”. Just as the English calendar has seven days for a week, the Indian calendar has seven vara’s for a week. Thus, The lunar day, however, varies approximately between 22 to 26 hours based on the angular rotation of moon around the earth in its elliptical orbit. In the Indian calendar, the lunar date is referred as “Tithi”. The basis for the length of a lunar date is geocentric and is defined as the angular distance between the sun and the moon as seen from the earth. As the moon rotates around the earth, the relative angular distance between the sun and the moon as seen from the earth increases from 0 degrees to 360 degrees. It takes one lunar month or about 29 ½ solar days for the angular distance between the sun and the moon to change from 0 to 360 degrees. When the angular distance reaches zero, the next lunar month begins. Thus, at the new moon a lunar month begins, at full moon, the angular distance between the sun and the moon as seen from the earth becomes exactly 180 degrees.

The lunar cycle begins with crescent moon and the crescent phase lasts till that phase culminates in the full moon, typically lasting for about 15 days. Then the moon enters in the waning phase until it disappears from the sky by lining up with the Sun. The waning phase also lasts for about 15 days. According Indian lunar month, the crescent lunar phase fortnight is called as “Shudha or Shukla Paksha” and the waning phase of the lunar cycle fortnight as “ Krishna Paksha”. Thus, during Shudha (or Shukla) Paksha the angular distance between the moon and the sun varies from 0 degrees to 180 degrees while that during the Krishna Paksha from 180 to 0 degrees. If we divide 180 degrees into 15 equal parts, then each part becomes of 12 degrees in length. Thus, this each twelve-degree portion of angular distance between the moon and the sun as it appears from the earth is the lunar date or Tithi. Tithis or lunar dates in Shudha (or Shukla) Paksha begin with Prathama (first), Dwitiya (second), etc. till we reach the Poornima, the lunar date for full moon day. Similarly for the waning fortnight lunar cycle or Wadya (or Krishna) Paksha, tithis begin again with Prathama (first), Dwitiya (second), etc. till we arrive Amavasya or a day before the new moon. Thus when we refer to Ramnavami (the birthday of Rama), it’s the Navami (ninth lunar day) of Shudha Paksha of the lunar month Chaitra, or Chaitra Shudha Navami. Similarly, the Gokulashtmi (also called as Janmashtami, the birthday of Krishna) occurs on Shravana Vadya Ashtami (eighth lunar day of Vadya Paksha of the lunar month Shrawana).

The angular velocity of moon in its elliptical orbit around the earth varies continuously as it is affected (according to Kepler’s Law) by the relative distance between the earth and the moon, and also by the earth’s relative distance from the sun. As a result, the daily angular speed (the speed of the angular change between the moon and the sun as seen from the earth) varies somewhere between 10 to 14 degrees per day. Since the length of a Tithi corresponds to 12 such degrees, the length of a Tithi also varies accordingly. Therefore, a Tithi can extend over one day (24 hour period) or it can get shortened if two Tithis occur in one 24 hour day. Since the angular distance between the moon and the sun as referred here is always relative to the entire earth, a lunar day or Tithi starts at the same time everywhere in the world but not necessarily on the same day. Thus, when a certain Tithi starts at 10:30 PM in India it also begins in New York at the same time, which is 12 PM (EST) on the same day. Since the length of a Tithi can vary between 20 to 28 hours, its correspondence to a Vara (a weekday) becomes little confusing. As per the Indian calendar, the Tithi for a given location on the earth depends on the angular distance between the moon and the sun relative to the earth at the time of sunrise at that location. Thus, for instance, assume on a November, Monday sunrise in New York city occurs 8:30 AM (EST). Further assume that at 9 AM (EST) on Monday the angular distance between the sun and moon is exactly 12 degrees just following the new moon of the Indian lunar month Karthika. Since the length of a tithi is 12 degrees, the tithi, Kartik Shudha Dwitiya (second day) begins exactly at 9 AM on Monday of that November in New York. However, at the time of sunrise on that Monday the tithi Dwitiya has not begun. Therefore, the tithi for that Monday for city of New York is Kartik Shudha Prathama (first day). On the same Monday morning the sunrise in Los Angeles occurs well past 9 AM (EST). Since the Tithi Dwitiya occurs everywhere in the world at the same instant, therefore, for Los Angeles, the Tithi for that Monday would be Karthik Shudha Dwitiya.

For the same Monday at 9 AM (EST), it would be 7:30 PM in Mumbai or New Delhi. Thus, Tithi for that Monday for city of New York, Mumbai, and New Delhi is Karthik Shudha Prathama (the first day of Indian lunar month Karthik) while for most of the regions west of Chicago or St. Louis the Tithi for that Monday is Dwitiya. In other words, the Tithi Karthik Shudha Prathama for regions west of Chicago or St. Louis should occur on the preceding day, the Sunday. Karthik Shudha Prathama (the first day of Indian lunar month Karthik) also happens to be the first day after Diwali. Most of the Indians celebrate this as their New Year ’s Day. Indians living in India, Europe, and eastern part of the United States thus should celebrate their New Year on that Monday while regions west of Chicago should celebrate on the preceding day, the Sunday. (Based on description by Jagdish C. Maheshri October 12, 2000 [1] Adhik Mas occurs only when two amavasyas )

[table id=IndicCosmology_103_1 /]

 


The Nakshatra and the Key Role it plays in Vedic Astronomy

 This calendar has been modified and elaborated, but because it is based on the stars (Nakshatras) visible to the naked eye, and on the visible Lunar phases, it is more accurate than any others of the past. The actual moments when Lunar months begin, can easily be checked by the regular appearances of Solar eclipses, and the middle moment of a Lunar month — Poornima or full moon — can similarly be verified by the more frequent Lunar eclipses. Hence the Hindu calendar, not requiring special instruments for its rectification, has maintained great accuracy for thousands of years. The oldest calendar is probably the Vedic calendar among the languages referred to as IE languages; at first lunar, later with solar elements added to it. The sister Avesta calendar is similarly first Lunar, but later only Solar. Both these calendars (the oldest in the IE universe) are influenced by the prehistoric calendars of the first and second root races at the North Pole and its surroundings, as they reckon with days and nights lasting six months. It was the impression of knowledgeable Indologists such as William Brennand that the Hindus have been observing and recording the the motion of the moon, the sun and the seven planets along a definite path that circles our sky, now known as the ecliptic, and is marked by a fixed group of stars clustered around this ecliptic. The moon afforded the simplest example.

These early astronomers observed that the moon, moving among these fixed star constellations, more accurately referred to today as Asterisms ( as opposed to the use of the term Constellation, which is term with a specific meaning in this case the total number is fixed as 88 by the IAU) which they called Nakshatras, returned to the same Nakshatra in 27.32166 days, the exact quantity determined by Aryabhatta, thus completing one Nakshatra month or Sidereal Month. They found it convenient to divide these groups of stars into 27 almost equal sections, or the 27 Nakshatras. Thus mathematically a naskshatra is equal to 1/27th of the sidereal Zodiac. In other words, it occupies 13 degrees and 20 minutes along the ecliptic. By this method of reckoning, instead of giving the date of a month, as Western calendars do, the Hindus gave the name of the Nakshatra in which the moon was to be seen. (The moon is in each of these Nakshatras for approximately one day plus eighteen minutes)

 

This scheme fitted nicely with the sun’s cycle, for the Hindus noted that the sun traversed the same circle through the sky, but that it returned to its starting place only after 365.258756481 days, or what we call a Solar Sidereal Year. (Modern figures based on this Hindu figure quote 365.2596296 days — a distinction without a difference, for ordinary purposes.) Now, having already divided the month into the 27 Nakshatras for the convenience of reckoning the moon’s voyage through the heavens, what was more natural than that these same Nakshatras should serve for the study of the Sun’s course? Being in a circle of 360 degrees, each Nakshatra takes up 13 1/3 degrees of that circle. The Sun, moving about 1 degree in a day, is seen for 13 1/3 days in each Nakshatra. The system of reckoning according to the moon Nakshatras is current today in other cultures, that of the sun’s being uncommon. During the course of one day, the earth has moved a short distance along its orbit around the sun, and so must rotate a small extra angular distance before the sun reaches its highest point. The stars, however, are so far away that the earth’s movement along its orbit makes a generally negligible difference to their apparent direction (see, however parallax), and so they return to their highest point in slightly less than 24 hours. A mean sidereal day is about 23h 56m in length. Due to variations in the rotation rate of the Earth, however, the rate of an ideal sidereal clock deviates from any simple multiple of a civil clock. The actual period of the Moon’s orbit as measured in a fixed frame of reference is known as a Sidereal month, because it is the time it takes the Moon to return to the same position on the celestial sphere among the fixed stars (Latin: sidus): 27.321 661 days (27 d 7 h 43 min 11.5 s) or about 27 ⅓ days. This type of month has appeared among cultures in the Middle East, India, and China in the following way: they divided the sky in 27 or 28 lunar mansions or Nakshatras, characterized by asterisms (apparent groups of stars), one for each day that the Moon follows its track among the stars.

 

In brief, then, the earliest method, the Vedic, of counting, was to name the moon through the various Nakshatras — the circle or cycle repeating itself each Sidereal-Star-Month. Later the sun’s place in the same Nakshatras was noted, the year ending when the Sun returned to the same Nakshatra. Then came the noting of the Solar and Lunar eclipses, and the observance of the New and Full Moons divided the month into the two phases of waxing and waning Moon, the month beginning at the moment of New Moon. This is how the Hindus reckon today, the month taking its name from the Nakshatra in which the Full Moon is seen each month. The Full Moon being exactly opposite the Sun, the Solar nakshatra bears the same name as the Lunar month six months ahead, while each Lunar month bears the same name as the 14th Solar Nakshatra ahead.

 

The Western student faced with these unfamiliar calculations may echo the old Persian proverb, “Why count big numbers and small fractions, when they are all amassed in 1?” But the Hindu looks on these figures from another point of view — he lives with them, and among them, and by them, much of the time. Consider a Sanskrit sloka (verse) about the Savati or pearl nakshatra, which marks the new season after the monsoon is over. The sloka says, “If in the Swati a rain drop falls into the sea, that drop becomes a pearl.” This may sound foolish, for the peasant, though he live in the depth of the interior of India, knows that pearls come from the sea — even if he does not necessarily understand that these pearls grow inside the oyster. He does know, however, that if it rains at this period of the year, his crops will yield great wealth. And the pearl is synonymous with wealth among people who, if they have any money, invest it in jewelry, especially gold and pearls, rather than in the banks. (Poetically, rice, their staple food).

Nakshatra And The Precession of The Equinoxes

 

To summarize, the earth revolves around the Sun once in 365 days 5 hours 48 minutes and 46 seconds. Considered from the earth, the Sun appears to complete one round of the ecliptic during this period. This is the Tropical year. In the span of a tropical year, the earth regains its original angular position with the Sun. It is also called the Year of seasons since the occurrence, and timing, of seasons depends on the rotation of the earth around the sun. If, for example, we consider the revolution of the Sun around the earth from one vernal equinox (around 21st March, when the day and night all over the globe are equal) to the next vernal equinox, it takes one tropical year to do so.

 

However, if at the end of a tropical year from one vernal equinox to the next, we consider the position of the earth with reference to a fixed star of the zodiac, the earth appears to lie some 50.26 seconds of celestial longitude to the west of its original position. In order for the earth to attain the same position with respect to a fixed star after one revolution, it takes a time span of 365 days 6 hours 9 minutes and some 9.5 seconds. This duration of time is called a sidereal year. The sidereal year is just over 20 minutes longer than the tropical year; this time difference is equivalent to 50.26 seconds of celestial longitude.

 

Each year, the Vernal equinox will fall short by 50.26 seconds along the zodiac reckoned along the fixed stars. This continuous receding of the Vernal equinox along the zodiac is termed the Precession of the Equinoxes and it takes about 25776 years to make one complete revolution of the precessional motion of the earth’s axis. Hipparchus regarded as the discoverer of the precession of the equinoxes in the west gave us either 28,000 or 28,173 years for one revolution. Another figure given is 25,920 years for the precession cycle. These figures indicate that the mean value of 27,000 years given in the Vedic scriptures is reasonable. The precession of the equinoxes has proved to be very useful for dating certain events in Vedic and Post Vedic times.

 

There are only a few methods, by which we can determine the age of an event in the absence of radiocarbon dating which is not as precise as the astronomical clocks. Use the Precession of the equinoxes to determine the Nakshatra in which the Vernal equinox occurs in a particular Nakshatra. If, we recall there are 27 Nakshatras, it follows that the vernal equinox occurs in a different Nakshatra, once every 955 years. Use the statements made in the texts to check for internal consistency. If for example Aryabhatta uses a place value system, the zero must have been in fairly wide use by then. If further he uses classical sanskrit (codified by Panini then he must have lived after Panini. 9 degrees to either side of the Ecliptic is a belt of the Heavens known as the Zodiac. (Dante called it the Oblique Line that beareth all planets).The first 30 degrees of the Zodiac constitute the sign of Aries.,the next 30 degrees Taurus and so on. The Zodiac counted from the first degree of Aries to the 360th degree of Pisces is called the Tropical Zodiac. These 12 signs are the limbs of the Cosmic Man or Time Eternal (Kalapurusha – The Almighty Self as Time). Aries is His head, Taurus His face, Gemini His neck, Cancer His heart, Leo the place beneath, Virgo His belly, Libra His generative organs, Scorpio the place beneath, Sagittarius His upper thigh, Capricorn his lower thigh, Aquarius His leg and Pisces His feet! Each Nakshatra is associated with a deity, and that the deities associated with tha Nakshatra are mentioned in the Riv Veda Samhita is due to the research of Narahari Achar. The antiquity of the naksatra system becomes clear when it is recognized that all the deity names occur in RV 5.51 (this insight is due to Narahari Achar21).

 

This hymn by Svasty¯atreya ¯ Atreya lists the deity names as: A´svin, Bhaga, Aditi, P¯usan, V¯ayu, Soma, Brhaspati, Sarvagan. AH.Vi´sve Devah. Agni, Rudra, Mitra, Varun.a, Indr¯agni. The sarvaganah are the ganah. (groups) such as the Vasavah. Pitarah.Sarpah.ncluding Ahi and Aja), ¯ Apah. , and the ¯ Adityaganah Daks.a Praj¯apati,Aryaman, Vis.u, Yama, Indra) complete the list. There is no doubt that the ecliptic is meant because the last verse of the hymn refers explicitly to the fidelity with which the sun and the moon move on their path, the ecliptic. The division of the circle into 360 parts or 720 parts was also viewed from the point of view the nakshatras by assigning 27 upanakshatras to each nakshatra (´ Satapatha Br. 10.5.4.5). This constituted an excellent approximation because 27 × 27 =729. In other words, imagining each nakshatra to be further divided into 27 equal parts made it possible to conceptualize half a degree when examining the sky.
[table id=IndicCosmology_103_2 /]
[table id=IndicCosmology_103_3 /]
The remaining 2 items in the Panchaga, the Karana and the Yoga are not as conceptual and are more derivative in nature.
Karana: A karana is half of a tithi. To be precise, a karana is the time required for the angular distance between the sun and the moon to increase in steps of 6° starting from 0°. (Compare with the definition of a tithi above.) Since the tithi’Ss are thirty in number, one would expect there to be sixty karana’Ss. But there are only eleven. There are four “fixed” karana-s and seven “repeating” karana’s. The four “fixed” karana’Ss are:
  • Kimstughna
  • Shakuni
  •  Chatushpād
  •  Nāgava
The seven “repeating” karan’s are:
  •  Bava
  •  Bālava
  •  Kaulava
  •  Taitula
  •  Garajā
  •  Vanijā
  •  Vishti (Bhadrā)
Now the first half of the first tithi (of the bright fortnight) is always Kimstughna karana. Hence this karana is “fixed”.  Next, the seven repeating karana-s repeat eight times to cover the next 56 half-tithi s. Thus these are the “repeating” karana-s. The three remaining half-tithi-s take the remaining “fixed” karana-s in order. Thus these are also “fixed”. Thus one gets sixty karana-s from eleven. The karana active during sunrise of a day is the karana for the day.

Rashi (Sura Maasa) [Solar Months] Ritu (Season) Gregorian Months Zodiac
Mesha Vasant (spring) March/April Aries
Vrushabha April/May Taurus
Mithuna Grishma (summer) May/June Gemini
Karka June/July Cancer
Simha Varsha (monsoon) July/August Leo
Kanya August/Septemeber Virgo
Tula Sharada (autumn) September/October Libra
Vruschika October/November Scorpious
Dhanu Hemant (autumn-winter) November/December Sagittarius
Makara December/January Capricornus
Kumbha Shishir (Winter-Spring) January/February Aquarius
Meena February/March Pisces

YOGA
First one computes the angular distance along the ecliptic of each object, taking the ecliptic to start at Mesha or Aries (Meshādi, as defined above): this is called the longitude of that object. The longitude of the sun and the longitude of the moon are added, and normalized to a value ranging between 0° to 360° (if greater than 360, one subtracts 360.) This sum is divided into 27 parts. Each part will now equal 800′ (where ‘ is the symbol of the arcminute which means 1/60 of a degree.) These parts are called the yoga-s. They are labeled:
  • Vishkambha
  • Prīti
  • Āyushmān
  • Saubhāgya
  • Shobhana
  • Atiganda
  • Sukarman
  • Dhriti
  • Shūla
  • Ganda
  • Vriddhi
  • Dhruva
  • Vyāghāta
  • Harshana
  • Vajra
  • Siddhi
  • Vyatīpāta
  • Varigha
  • Parigha
  • Shiva
  • Siddha
  • Sādhya
  • Shubha
  • Shukla
  • Brāhma
  •  Māhendra
  • Vaidhriti
Again, minor variations may exist. The yoga that is active during sunrise of a day is the yoga for the day. Table below gives a comparison of some of the astronomical constants.
ASTRONOMIC AUTHORITY Àryabhata (from Clarke and Kay) Surya Siddanta 2007
Years in Cycle ,MY 4,320,000 4,320,000 4,320,000
Rotations,R 1,582,237,500 1,582,237,828
Mean Rotations of the earth in a Sidereal year R/KY=1 +DSiYr 366.2587565
Lunar Orbits one MY 57,753,336 57,753,336
Days in a Sidereal month, DSiM =  1577917500/57753336 = 27.32166848
Kaye notes 57,753,339 lunar orbits rather than 57,753,336 per Clarke
Synodic Months MSyn in a MY 53,433,336 53,433,336
Days in a synodic month DSynM = 1,577,917,500/53,433,336=29.53058181 days
Mercury 17,937,920 17,937,060
Venus 7,022,388 7,022,376
Mars 2,296,824 2,296,832
Jupiter 364,224 364,220
Saturn 146,564 146,568