**PROBLEM 11. a. ** One of the serious problems of the old nonuniform time units was the accumulation of error. It might seem that an accuracy of 1 second a day (a possible relative error of 1/(24 x 60 x 60) or about 1 x l0^-5) would be sufficient for most tech-
nical or scientific purposes. Show that an error of 1 second a day could result in an error of 1. 1 x 10^4 km in the position of the Earth in its orbit after only 1 year. (Assume Earth's orbit is circular with a radius of 1.5 x l0^8 km.)

**Solution: **In one year the error could be 365 seconds. Earth moves through an angle of 2pi/(365 x 24 x 60 x 60) radians in one second, or 2pi/(24 x 60 x 60) radians in 365 seconds. The length of arc that subtends this angle in a circle of radius 1.5 x l0^8 km is s = r(theta) = 1.5 x l0^8 x (2pi/(24 x 60 x 60) = 1.1 x 10^4 km.

**b. **The "tropical year" is defined as the time difference between successive vernal equinoxes‹in other words, the time it takes Earth to complete one revolution around the Sun. This time does not have a simple relationship to Earth's rotation period (the day). In fact, it turns out that to the nearest second one tropical year is 365 days, 5 hours, 48 minutes, 46 seconds. Show that the current system of adding an extra day to each calendar year that is a multiple of 4 but not a multiple of l00 (leap years) serves to give each calendar year an integral number of days and also keeps the seasons constant with respect to the calendar.

**Solution: **If a calendar year has 365 days, the excess time in a tropical year is 5 h 48 m 46 s, not quite 1/4 day. Multiplying this excess by 4, 4 x (5 h 48 m 46 s) = 23 h 15 m 4 s, almost 1 day. If we add 1 extra day each 4 years, we will create a deficit of
24 h - (23 h 15 m 4 s) = 44 m 56 s for each leap year. In each l00 years, there are 25 years that are multiples of 4; however, after 24 leap years, the deficit will accumulate to 24 x (44 m 56 s) = 17 h 58 m 24 s, almost 3/4 day. This will almost balance the excess accumulation for the remaining 4 years of the century, so that years that are multiples of l00 should not be leap years. It is clear that further juggling will be necessary, since things never balance exactly.

**c.** For some computations in astronomy and space science, it is necessary to have an absolute time that is a continuous count of the number of time units from some arbitrary reference. The universally accepted standard is the Julian Day Calendar, a continuous count of the number of days since 12:00 noon on 1 January 4713 B.C. This curious starting date was actually chosen in A.D. 1582 by considering the cycle that is the least common multiple of the 28-year solar cycle (the interval required for all dates to recur on the same day of the week), the l9-year lunar cycle (the interval containing an integral number of lunar months), and the 15-year indiction (the tax period introduced by the Roman emperor Constantine in A.D. 313). The year 4713 B.C. was the most recent date prior to 1582 when these cycles coincided, and it had the added advantage of predating the ecclesiastically approved date of Creation, 4 October 4004 B.C. How long is the Julian day cycle, and when is the next year when all three of the cycles used in its creation will coincide?

**Solution: **The least common multiple of 28, 19, and 15 is their product, since these numbers have no prime factors in common. 28 x 19 x 15 = 7980, so the next year the cycles coincide will be (-4713) + 7980, or A.D. 3267.

**d. **A clever computer algorithm for converting calendar dates to Julian days was developed using FORTRAN integer arithmetic (H. F. Fliegel and T. C. Van Flandern, "A Machine Algorithm for Processing Calendar Dates," Communications of the ACM 11 [1968]: 657). In FORTRAN integer arithmetic, multiplication and division are performed left to right in the order of occurrence, and the absolute value of each result is truncated to the next lower integer value after each operation, so that both 2/12 and -2/12 become 0. If I is the year, J the numeric order value of the month, and K the day of the month, then the algorithm is:

JD = K - 32075 + 1461 * (I + 4800 + (J -14)/12)/4

+ 367 * (J-2-(J-14)/12*12)/12 - 3 * ((I + 4900 + (J-14)/12)/100)/4.

The calendar date 25 December 1981 is JD 2 444 964. Use a hand calculator and this algorithm to find the Julian dates of the launch of Explorer 1 (the first U.S. satellite placed into orbit), Greenwich Mean Time 1 February 1958 (Eastern Standard Time January 31,1958), and the launch of the seventh Space Shuttle on 18 June 1983 (carrying the first American female astronaut, Sally Ride, into orbit).

Solution: For 1 February 1958, I = 1958, J = 2, K = 1 .

JD = I - 32075 + 1461*(1958+4800+(2 - 14)/12)/4

+ 367*(2-2-(2-14)/12*12)/12 - 3*((1958+4900+(2-14)/12)/100)/4

= 1 - 32075 + 1461*6757/4 + 367*(1*12)/12 - 3*(6857/100)/4

= 1 - 32075 + 2 467 994 + 367 - 51 = 2 436 236

For 18 June 1983, I = 1983, J = 6, K = 18.

JD = 18 - 32075 + 1461*(1983+4800+(6-14)/12)/4

+ 367*(6-2-(6-14)/12*12)/12 - 3*((1983+4900+(6-14)/12)/100/4

= 18 - 32075 + 1461*6783/4 + 367*4/12 - 3*68/4

= 18 - 32075 + 2 477 490 + 122 - 51 = 2 445 504.

A large number of satellites require ground processing of spacecraft sensor data to determine the spacecraft attitude (i.e., the spacecraft's orientation). Examples of sensors used are Sun sensors, Earth sensors, and star sensors. These sensors provide information, usually a measured angle, concerning the spacecraft "pointing" relative to a celestial body (e.g., Sun, Earth, or star).

Telemetry signals from these sensors are converted on the spacecraft to digital counts and transmitted to ground stations. The digital count representation of a sensor output can be easily converted to meaningful measurements and units on the ground. However, telemetry signals are frequently subject to random interference, or "noise." To understand the meaning of noise, one has only to tune into a weak channel on a television set; the "snow" that is seen is a visual representation of noise present in an electronic signal. Noise consists of random signals superimposed on valid electronic signals from any electronic device. An in-depth understanding of the cause, effect, and reduction of noise is not necessary in the context of this problem. However, it should be understood that noise can sufficiently corrupt any electronic signal to the extent that making use of, and properly interpreting, the true signal can be difficult.

This problem applies to spacecraft instruments and sensors. A number of methods have been developed to smooth data and remove the effects of noise. In the next problem, we examine one such method, called the running average.