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Example 2.7
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Calculate the average length of the civil year in the Gregorian
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The nominal number of days in a 400-year period is 400 365 146,000. The nominal number of leap years is 400/4 100, but as shown earlier, this must be reduced by 3, and therefore, the number of days in 400 years of the Gregorian calendar is 146,000 100 3 146,097. This gives a yearly average of 146,097/400 365.2425.
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In calculations requiring satellite predictions, it is necessary to determine whether a year is a leap year or not, and the simple rule is: If the year number ends in two zeros and is divisible by 400 without remainder, it is a leap year. Otherwise, if the year number is divisible by 4 without remainder, it is a leap year.
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Example 2.8 Determine which of the following years are leap years: (a) 1987, (b) 1988, (c) 2000, (d) 2100.
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(a) 1987/4 (b) 1988/4 (c) 2000/400
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496.75 (therefore, 1987 is not a leap year) 497 (therefore, 1988 is a leap year) 5 (therefore, 2000 is a leap year)
(d) 2100/400 5.25 (therefore, 2100 is not a leap year, even though 2100 is divisible by 4 without remainder)
2.9.2 Universal time
Universal time coordinated (UTC) is the time used for all civil time keeping purposes, and it is the time reference which is broadcast by the National Bureau of Standards as a standard for setting clocks. It is based on an atomic time-frequency standard. The fundamental unit for UTC is the mean solar day (see App. J in Wertz, 1984). In terms of clock time, the mean solar day is divided into 24 h, an hour into 60 min, and a minute into 60 s. Thus there are 86,400 clock seconds in a mean solar day. Satellite-orbit epoch time is given in terms of UTC.
Example 2.9 Calculate the time in days, hours, minutes, and seconds for the epoch day 324.95616765.
This represents the 324th day of the year plus 0.95616765 mean solar day. The decimal fraction in hours is 24 0.95616765 22.9480236; the decimal fraction of this expressed in minutes is 0.9480236 60 56.881416; the decimal fraction of this expressed in seconds is 0.881416 60 52.88496. Thus, the epoch is day 324, at 22 h, 58 m, 52.88 s.
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Universal time coordinated is equivalent to Greenwich mean time (GMT), as well as Zulu (Z) time. There are a number of other universal time systems, all interrelated ( Wertz, 1984) and all with the mean solar day as the fundamental unit. For present purposes, the distinction between these systems is not critical, and the term universal time (UT), will be used from now on. For computations, UT will be required in two forms: as a fraction of a day and in degrees. Given UT in the normal form of hours, minutes, and seconds, it is converted to fractional days as UTday 1 ahours 24 minutes 60 seconds b 3600 (2.18)
In turn, this may be converted to degrees as UT
2.9.3 Julian dates*
360
UTday
(2.19)
Calendar times are expressed in UT, and although the time interval between any two events may be measured as the difference in their calendar times, the calendar time notation is not suited to computations where the timing of many events has to be computed. What is required is a reference time to which all events can be related in decimal days. Such a reference time is provided by the Julian zero time reference, which is 12 noon (12:00 UT) on January 1 in the year 4713 B.C.! Of course, this date would not have existed as such at the time; it is a hypothetical starting point, which can be established by counting backward according to a certain formula. For details of this intriguing time reference, see Wertz (1984, p. 20). The important point is that ordinary calendar times are easily converted to Julian dates, measured on a continuous time scale of Julian days. To do this, first determine the day of the year, keeping in mind that day zero, denoted as Jan 0.0 is midnight between December 30 and 31 of the previous year. For example, noon on December 31 would be January 0.5, and noon on January 1 would be January 1.5. It may seem strange that the last day of December should be denoted as day zero in January, but it will be seen that this makes the day count correspond to the actual calendar day. A Fortran program for calculating the Julian day for any date and time is given in Wertz (1984, p. 20), and a general method is given in DuffettSmith (1986, p. 9). Once the Julian day is known for a given reference date and time, the Julian day for any other time can be easily calculated by adding or subtracting the required day difference. Some reference times are listed in Table 2.2.
It should be noted that the Julian date is not associated with the Julian calendar introduced by Julius Caesar.
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