BA PART I : PAPER 3.8

IMPORTANT BASIC CONCEPTS

When does a day begin?

In the Western civil calendar, the day begins at midnight local time. For astronomers, the day begins at noon, which conveniently avoids dividing the night's events between two dates. Astronomers' dates are 12 hours ahead of the civil date; an occurrence on the afternoon of May 8 would have happened on May 9 as astronomers see it.

In many cultures the day begins at sunset, a division observed for religious purposes by Jews and Moslems.
Where does a day begin?
According to present notions of timekeeping, the day begins along the International Date Line. The exact location at which the first sunrise of the day (or year, or millenium) could be seen became a matter of controversy during preparations for the celebration of the second millenium ce.
The Royal Greenwich Observatory calculated that sunrise is first seen at Balleny Island in Antarctica; next at Caroline Island, part of the Republic of Kiribati; and third on New Zealand's Pitt Island. In 1996, the Millenium Adventure Company leased the highest hill on Pitt Island, claiming the site would be “the first terrestrial, accessible and populated place to usher in the next 1,000 years.”

The length of the day
The simplest definition of a day's duration, and the one used in most sciences, including astronomy, is that it is 86,400 seconds as the second is defined in SI. The rotation of the earth is not explicitly mentioned in this definition, but of course that is where the “86,400” comes from.

Solar days
The apparent motion of the sun defines a day for most people. A good way of timing its coming and going (sunsets and sunrises are messy because light is bent more near the horizon) is to imagine a line running from north to south and passing through the point directly overhead. Such a line is called a meridian. Start your stopwatch when the sun crosses the line and stop it at the instant when it next crosses it again (such a crossing is called a transit). You have recorded the length of an apparent solar day. If you did this for an entire year, you would find that some days–remember, we are talking about 24-hour days here–are longer than others. To even things out, we can average the lengths of all the apparent solar days during a year and get the mean solar day.

Sidereal days
Imagine however, that instead of using the sun, we begin timing when some particular point on the celestial sphere, say the star Sirius, crosses the meridian, and stop when it crosses again. To our surprise, we would find that the duration is shorter than the mean solar day. What we have measured is the apparent sidereal day. The sidereal day is subdivided in the same way as the solar day, into 24 sidereal hours; each sidereal hour into 60 sidereal minutes, and each sidereal minute into 60 sidereal seconds. As with solar days, the length of sidereal days is subject to irregularities in the earth's rotation (see below), and so there is both an apparent sidereal day and a mean sidereal day.
Actually, the point on the celestial sphere that is used for measuring sidereal time is not a star, but the true vernal equinox, the point where the sun crosses from the southern celestial hemisphere into the northern celestial hemisphere each year. The true vernal equinox is one of two points where two great circles on the imaginary celestial sphere cross:
the circle (the true Equator) found by projecting the plane of the earth's equator onto the celestial sphere
the circle (the ecliptic) found by projecting the plane of the earth's orbit onto the celestial sphere
The true sidereal day is the time interval between two instants when the true vernal equinox crosses the meridian.
In 1991, one mean solar day = 1.00273790935 mean sidereal days. The mean sidereal day is 23 hours, 56 minutes, 4.09054 seconds long, about 3 minutes 55 seconds shorter than the mean solar day. In other words, the stars rise about four minutes earlier each day.

Why do the solar and sidereal day differ?
Try this: Place a cent and quarter face up on the table before you, with the penny on the left. Abe and George will be facing each other. The penny represents earth and the quarter the sun. Abe is our observer on the earth; he sees the sun directly in front of him.
Move the penny around the quarter in a clockwise direction. When the penny is above the quarter, the quarter passes out of Abe's view and remains hidden until it appears overhead when the penny is below the quarter. The sun has risen and set, but the penny hasn't rotated at all; Abe has continued to look at the same point on the wall all this time. In other words, the number of solar days in a year is one more than the number of sidereal days.
Because the vernal equinox itself moves (due to the precession of the earth's axis), the sidereal day is not quite the same as the period of earth's rotation with respect to a fixed direction in space. That period is 0.0084 seconds longer than a sidereal day. Oddly enough, this, the true period of the earth's rotation, has no special name or use.

The Erratic Earth
One of the great frustrations of 19th century astronomy was that it could predict the motion of everything but the Moon. Using Newtonian mechanics, astronomers should have been able to predict exactly where the Moon would be at a given moment, but the predictions were repeatedly wrong.
Towards the end of the century, it began to dawn on them that it wasn't the Moon's motion that was irregular, but the clock they were using: the rotation of the earth, which affects the length of the apparent solar and sidereal days and the time assigned to observations. The problem showed up first in studies of the Moon because the Moon moves against the background of distant stars more quickly than anything else. As a result, smaller intervals of time are visible in its motion than in that of slower moving objects. Several different causes for irregularities in the earth's rotation have been discovered:
By the 1860s astronomers generally agreed that over the very long term, days are getting longer as the Earth's spin slows due to tidal friction and the transfer of some of its energy to the moon. The length of the day increases by about 0.001 second each century. This is called the secular variation.
In the 1930s a seasonal variation in the length of days was discovered. Days in March are about 0.001 second longer than days in July. The pattern more or less repeats each year. This seasonal variation is thought to be due to the action of winds and tides. [N. Stoyko, 1937]
Recently abrupt, irregular changes of several thousandths of a second have been discovered, which are thought to be due to interactions between motions in the earth's outer layers and core.

Tithi – A Lunar Date

According to Indian calendar Tithi is a lunar date, and is one of the five important aspects of an Indian almanac (Panchang – Panch means five and ang means parts). Most of the Indian social and religious festivals are celebrated based on tithi. Until they left India and went overseas, Indians didn’t really need to worry about tithi to celebrate their festivals since a tithi in India almost invariably falls on the same day for the entire region of India.
Even after migration to overseas countries, Indians living overseas would still celebrate their festivals on the same dates as Indians in India would. For those who would argue that behind these festival celebrations, it’s the faith, that is more important than the date of celebration, I’m hundred percent with them. However, the intention here is provide the information for those who want to understand the importance behind the tithi on which Indian festivals are based.
Calendar “date” that we are so familiar with in our daily life is based on solar calendar. English calendar is a solar calendar. The basis for solar calendar is the rotation of the Earth around the Sun. It takes earth approximately 365 ¼ days to complete its rotation around the Sun. The English 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.
Similar to solar calendar lunar calendar is also popular and widely used in the Asian countries such as China, Pacific-rim countries, Middle East countries, and India. 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 based on the moon’s rotation around the Earth. The lunar month corresponds to one complete rotation of 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. 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, 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:
Chaitra Vaishakh Jeshta Ashadh Shrawan (Sawan) Bhadrapad (Bhado) Ashwin Kartik Margshirsh Paush Magha Falgoon (Fagan)
As mentioned earlier, to account for the difference between the solar and lunar year an extra lunar month occurs about every 2 ½ years as “Adhik Mas”.[1]
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 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 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 “War”. Just like English calendar has seven days for a week, Indian calendar has seven wars for a week. Thus,
English calendar weekdays
Indian calendar weekdays
SundayMondayTuesday Wednesday ThursdayFriday Saturday
Raviwar Somwar (Chandrawar)Mangalwar BudhwarGuruwar Shukrawar Shaniwar
The lunar date, however, varies approximately between 22 to 26 hours based on the angular rotation of moon around the earth in its elliptical orbit. In Indian calendar, lunar date is referred as “Tithi”. The basis for the length of a lunar date is the angular distance between the sun and the moon as seen from the earth. As the moon rotates around the earth, the 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 “Wadya or Krushna 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 Wadya (or Krushna) 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 Krushna) 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 Krisha) occurs on Shrawan Wadya Ashtami (eighth lunar day of Wadya Paksha of the lunar month Shrawan).
The angular velocity of moon in its elliptical orbit around the earth continuously varies as it is affected (according to Kepler’s rule) 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 angle between the moon and the sun as seen from the earth) varies somewhere between 10 to 14 degrees. 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 skipped if two tithis occur in one 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 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 War (a weekday) becomes little confusing.
As per 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 Kartik. 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 Kartik 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 Kartik Shudha Prathama (the first day of Indian lunar month Kartik) while for most of the regions west of Chicago or St. Louis the tithi for that Monday is Dwitiya. In other words, the tithi Kartik Shudha Prathama for regions west of Chicago or St. Louis should occur on the preceding day, the Sunday.
Kartik Shudha Prathama (the first day of Indian lunar month Kartik) 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 on the preceding day, the Sunday.