Sidereal and Solar Time

Time measurements can be based on the rotation of the Earth, orbital motion around the Sun, or on atomic clocks. The last-mentioned will be discussed in the next section. Here we consider the sidereal and solar times related to the rotation of the Earth.

We defined the sidereal time as the hour angle of the vernal equinox. A good basic unit is a sidereal day, which is the time between two successive upper culminations of the vernal equinox. After one sidereal day the celestial sphere with all its stars has returned to its original position with respect to the observer. The flow of sidereal time is as constant as the rotation of the

2.13 Sidereal and Solar Time

Earth. The rotation rate is slowly decreasing, and thus the length of the sidereal day is increasing. In addition to the smooth slowing down irregular variations of the order of one millisecond have been observed.

Unfortunately, also the sidereal time comes in two varieties, apparent and mean. The apparent sidereal time is determined by the true vernal equinox, and so it is obtained directly from observations.

Because of the precession the ecliptic longitude of the vernal equinox increases by about 50" a year. This motion is very smooth. Nutation causes more complicated wobbling. The mean equinox is the point where the vernal equinox would be if there were no nutation. The mean sidereal time is the hour angle of this mean equinox.

The difference of the apparent and mean sidereal time is called the equation of equinoxes:

where e is the obliquity of the ecliptic at the instant of the observation, and Af, the nutation in longitude. This value is tabulated for each day e. g. in the Astronomical Almanac. It can also be computed from the formulae given in *Reduction of Coordinates. It is at most about one second, so it has to be taken into account only in the most precise calculations.

Figure 2.29 shows the Sun and the Earth at vernal equinox. When the Earth is at the point A, the Sun culminates and, at the same time, a new sidereal day

Fig. 2.29. One sidereal day is the time between two successive transits or upper culminations of the vernal equinox. By the time the Earth has moved from A to B, one sidereal day has elapsed. The angle A is greatly exaggerated; in reality, it is slightly less than one degree

begins in the city with the huge black arrow standing in its central square. After one sidereal day, the Earth has moved along its orbit almost one degree of arc to the point B. Therefore the Earth has to turn almost a degree further before the Sun will culminate. The solar or synodic day is therefore 3 min 56.56 s (sidereal time) longer than the sidereal day. This means that the beginning of the sidereal day will move around the clock during the course of one year. After one year, sidereal and solar time will again be in phase. The number of sidereal days in one year is one higher than the number of solar days.

When we talk about rotational periods of planets, we usually mean sidereal periods. The length of day, on the other hand, means the rotation period with respect to the Sun. If the orbital period around the Sun is P, sidereal rotation period r+ and synodic day t , we now know that the number of sidereal days in time P, P/t*, is one higher than the number of synodic days, P/t :

This holds for a planet rotating in the direction of its orbital motion (counterclockwise). If the sense of rotation is opposite, or retrograde, the number of sidereal days in one orbital period is one less than the number of synodic days, and the equation becomes

11 1

For the Earth, we have P = 365.2564 d, and t = 1 d, whence (2.43) gives t„ = 0.99727 d = 23 h 56 min 4 s, solar time.

Since our everyday life follows the alternation of day and night, it is more convenient to base our timekeeping on the apparent motion of the Sun rather than that of the stars. Unfortunately, solar time does not flow at a constant rate. There are two reasons for this. First, the orbit of the Earth is not exactly circular, but an ellipse, which means that the velocity of the Earth along its orbit is not constant. Second, the Sun moves along the ecliptic, not the equator. Thus its right ascension does not increase at a constant rate. The change is fastest at the end of December (4 min 27 s per day) and slowest in

mid-September (3 min 35 s per day). As a consequence, the hour angle of the Sun (which determines the solar time) also grows at an uneven rate.

To find a solar time flowing at a constant rate, we define a fictitious mean sun, which moves along the celestial equator with constant angular velocity, making a complete revolution in one year. By year we mean here the tropical year, which is the time it takes for the Sun to move from one vernal equinox to the next. In one tropical year, the right ascension of the Sun increases exactly 24 hours. The length of the tropical year is 365 d 5 h 48 min 46 s = 365.2422 d. Since the direction of the vernal equinox moves due to precession, the tropical year differs from the sidereal year, during which the Sun makes one revolution with respect to the background stars. One sidereal year is 365.2564 d.

Using our artificial mean sun, we now define an evenly flowing solar time, the mean solar time (or simply mean time) TM, which is equal to the hour angle hM of the centre of the mean sun plus 12 hours (so that the date will change at midnight, to annoy astronomers):

The difference between the true solar time T and the mean time TM is called the equation of time:

(In spite of the identical abbreviation, this has nothing to do with a certain species of little green men.) The greatest positive value of E.T. is about 16 minutes and the greatest negative value about -14 minutes (see Fig. 2.30). This is also the difference between the true noon (the meridian transit of the Sun) and the mean noon.

Both the true solar time and mean time are local times, depending on the hour angle of the Sun, real or artificial. If one observes the true solar time by direct measurement and computes the mean time from (2.46), a digital watch will probably be found to disagree with both of them. The reason for this is that we do not use local time in our everyday life; instead we use the zonal time of the nearest time zone.

In the past, each city had its own local time. When travelling became faster and more popular, the great variety of local times became an inconvenience. At the end of the 19th century, the Earth was divided into 24 zones, the time of each zone differing from the neighboring ones by one hour. On the surface of the Earth, one hour

Fig. 2.30. Equation of time. A sundial always shows (if correctly installed) true local solar time. To find the local mean time the equation of time must be subtracted from the local solar time

in time corresponds to 15° in longitude; the time of each zone is determined by the local mean time at one of the longitudes 0°, 15°,..., 345°.

The time of the zero meridian going through Greenwich is used as an international reference, Universal Time. In most European countries, time is one hour ahead of this (Fig. 2.31).

In summer, many countries switch to daylight saving time, during which time is one hour ahead of the ordinary time. The purpose of this is to make the time when people are awake coincide with daytime in order to save electricity, particularly in the evening, when people go to bed one hour earlier. During daylight saving time, the difference between the true solar time and the official time can grow even larger.

In the EU countries the daylight saving time begins on the last Sunday of March, at 1 o'clock UTC in the morning, when the clocks are moved forward to read 2 AM, and ends on the last Sunday of October at 1 o'clock.

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