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Voyages Through Space and Time | Special Topics

Sunrise, sunset, the sequence of day and night, the cycle of lunar phases, the changing seasons-- all mark the passage of time. In the beginning, time keeping involved merely counting the days, months, and years. But at some time in the distant past people began to count the hours. One of the earliest methods used observations of shadows. When the Sun rises in the east, shadows are long and point westward, but as the Sun’s altitude increases, shadows shorten and swing northward. At noon shadows are at their shortest and point directly north. Then through the afternoon they lengthen once again, stretching eastward away from the setting Sun. The hour of the day can be reckoned by either the direction or length of the shadows.

In the opening lines of the "Parson’s Prologue" Chaucer writes

It was four o’clock according to my guess,

Since eleven feet, a little more or less,

My shadow at the time did fall,

Considering that I myself am six feet tall.

And in his "Man of Law’s Tale" he notes

. . . the shadow of each tree

Had reached a length of that same quantity

As was the body which had cast the shade

And on this basis he conclusion made:

. . . for that day, and in that latitude,

The time was ten o’clock . . .

When it was realized that the directions of shadows were more accurate indications of time, the sundial became the preferred device for recording the time of day. One early fragment of a sundial dates from about 1500 BC, and around 600 BC the Greek philosopher Anaximander is thought to have introduced the sundial into Greece. By 200 BC sundials were very common in Rome, and a contemporary of Julius Caesar listed more than a dozen different types of dials. The first sundials were probably sticks stuck vertically in the ground. But perhaps by the first century AD it was discovered that the shadow of a slanting object, parallel to the Earth’s axis, was a more accurate time teller than the shadow of a vertical one. The "modern" sundial had been invented, and it measured the passage of time for centuries.

Today time keepers range from accurate atomic clocks to inexpensive watches. But clocks of whatever kind must initially be set to the correct time, and this is still done by observing the motions of the Sun, Moon, and stars.

Defining Time

The time told by a sundial is called local apparent time, but there are, in fact, many other types of time, including local mean time, local sidereal time, standard time, and daylight saving time. Different types of time can be defined by choosing specific celestial objects to mark the passage of time.

The Sun is one obvious choice, but in principle one could

use the Moon, as some early civilizations did, or even the planets. However, the most frequently chosen objects are the real Sun, the vernal equinox, and something called the Mean Sun, which is a fictitious Sun that moves along the celestial equator with the real Sun’s average (mean) speed. These objects are used, respectively, to define local apparent time, local sidereal time, and local mean time.

But whatever the type of time, there are always 24 hours in a day. To avoid the sometimes confusing AM and PM designations, a 24-hour clock is often used. A day always begins at 0, or 2400 hours. On this system, 6 AM is 6 hours, noon is 12 hours, and 6 PM is 18 hours. Technically speaking,

one day is the time interval between two similar meridian crossings of a chosen celestial object.

An observer’s astronomical meridian (see Figure 3.1) is the circle that passes through the south point on the horizon, the zenith, the NCP, the north point on the horizon, and the nadir.

Figure 3.1 The Astronomical Meridian

This circle divides the sky into an eastern and a western half. When an object is rising, it is to the east of the meridian, and when it is setting, it is to the west of the meridian. The meridian above the horizon is called the upper meridian and that below the horizon, the lower meridian.

When an object crosses the upper meridian and has its greatest altitude, it is said to be at upper transit. At lower transit an object has its smallest altitude and usually lies below the horizon on the lower meridian. Objects are at upper and lower transit once each day.                                          

A local apparent day is the time interval between two lower transits of the real Sun. Likewise, a local mean day is the time interval between two lower transits of the Mean Sun. Both types of days begin when the respective suns are below the horizon at lower transit. On the other hand, the local sidereal day is the time between two upper transits of the vernal equinox, which means a sidereal day starts when the vernal equinox is above the horizon.

At any instant, local time depends on the chosen object’s position relative to the observer’s astronomical meridian. This position is given by the local hour angle (LHA), which is simply the amount of time that has passed since the object was at upper transit. In other words, the LHA indicates how far an object is to the west of an observer’s meridian. At upper transit any object’s LHA is 0 hours while at lower transit it is 12 hours. An object 6 hours to the west of the astronomical meridian has an LHA of 6 hours and an object 6 hours to the east has an LHA of 18 hours. An object to the east is sometimes said to have a negative LHA. That is, 18 hours is equivalent to a minus (-) 6 hours.

The Sun’s LHA is used to define local apparent time (LAT). At noon, when the Sun is at upper transit, its LHA is 0 hours and the LAT is 12 hours. One hour later its LHA will be 1 hour and the LAT will be 13 hours. In general, the local apparent time is equal to the LHA of the Sun plus 12 hours:

Notice that 12 hours must be added to the Sun’s LHA because the local apparent day begins when the Sun is at lower transit. Formulae like the one above sometimes yield values greater than 24 hours. In such cases one must subtract 24 from the value. As has been mentioned, local apparent time is the type of time told by a sundial.

The definition of local mean time (LMT) is similar to that of LAT except that the LHA of the Mean Sun is used. Thus,

Local sidereal time (LST) is defined as the LHA of the vernal equinox (V.E.), that is,

Notice that it is not necessary to add 12 hours to the LHA of the vernal equinox because the sidereal day begins when the vernal equinox is at upper transit. At this time its LHA is zero and the LST is also zero.

Because sidereal time is defined as the hour angle of the vernal equinox, LST indicates how far the vernal equinox is to the west of the astronomical meridian. For example, if the LST is 0 hours, the vernal equinox is on the meridian, and if LST is 6 hours, the vernal equinox is just setting at the west point. Also, at a LST of 12 hours, the vernal equinox is at lower transit, and at 18 hours, it is rising at the east point on the horizon.

It follows from the definitions of right ascension (see Chapter 2) and local sidereal time that the LST is equal to the right ascension of any object at upper transit, that is,

In other words, the LST indicates what objects are on the meridian.

For example, all objects having a right ascension of 0 hours are at upper transit at an LST of 0 hours, and at an LST of 14 hours, all objects with that right ascension are at upper transit.

A more general relationship states that the local sidereal time is equal to the right ascension of any object plus that object’s local hour angle:

For example, if a star having an RA of 3 hours is 2 hours west of the meridian, then the LST must be 5 hours. The formula can also be used to find the LHA of any object if the LST and the object’s RA are known. Solving for LHA gives,

Since the Earth moves 360 degrees around the Sun in approximately 360 days, it must move about 1 degree along its orbit each day. Because of this the Sun, viewed from the Earth, appears to move eastward along the ecliptic by 1 degree per day (see Figure 3.2).

Figure 3.2 The Sun’s Eastward Motion

Actually both the Earth and the Sun move a little less than 1 degree per day because there are 365.25 days in a year rather than 360 days. As seen from the Earth, the constellations out in the Sun’s direction are above the horizon during the day. But as the Sun moves eastward relative to these stars, they rise earlier and earlier and eventually become visible at night. This results in different constellations being visible during different times of the year.

In time units, 1 degree is equivalent to 4 minutes of time. Thus, the Sun’s eastward motion relative to the stars causes the stars to rise 4 minutes earlier each day. A star that rises with the Sun today will rise 4 minutes before the Sun tomorrow and 8 minutes earlier than the Sun the next day. Likewise, a star that rises at 9 PM tonight will rise at 8:56 tomorrow and at 8:52 the next night.

A sidereal clock runs at a different rate than clocks keeping LAT or LMT. In fact, compared to these clocks, a sidereal clock gains approximately 4 minutes each day. The reason for this gain is the same reason that stars rise about 4 minutes earlier each day, namely, the Earth’s revolution about the Sun.

If a sidereal clock gains about 4 minutes each day compared, to, say, an LAT clock, in 30 days it will gain 2 hours, and in one year it will gain 24 hours or one full day. It follows that there are 366 sidereal days in 365 solar days and that the length of a sidereal day is 365/366 the length of a solar day. This fraction implies that a sidereal day is 3 min 56 sec shorter than a solar day. This is the actual daily gain of a sidereal clock over other clocks. Also, stars actually rise 3 min 56 sec earlier each day, rather than 4 minutes.

A sidereal clock’s full day gain in one year suggests that once each year a sidereal clock must tell the same time as the other types of clocks. In fact LAT and LST are the same at about midnight on the date of the autumnal equinox. But each day thereafter the faster sidereal clock gains 3 min 56 sec until one year later the LST clock is on full day ahead and once again reads the same time as the LAT clock.

Many centuries ago it was discovered that the Sun does not move uniformly along the ecliptic. This non-uniform motion is partly due to the fact that the Earth’s orbital speed changes (Kepler’s second law). The Earth moves fastest during the northern hemisphere’s winter when it is closest to the Sun and slowest in the summer when it is farthest from the Sun. As the Earth’s orbital speed changes, so does the resulting apparent, eastward motion of the Sun. Another reason for the non-uniformity is the 23.5 degree inclination of the Earth’s rotational axis to the orbital plane.

The non-uniform motion of the Sun causes local apparent time to be non-uniform. On days when the Sun moves more than 1 degree eastward, the time between lower transits is greater than usual and the day is longer. The reverse is the case on days when the Sun moves less than 1 degree. So local apparent days are not always the same length. If there is one thing that we insist on today, it is that every day be the same length. The Mean Sun was invented to eliminate the problem of non-uniform time. The Mean Sun moves eastward along the celestial equator, rather than along the ecliptic, and it moves with the real Sun’s average (mean) speed. Thus, the Mean Sun moves uniformly and this, in turn, ensures that local mean time is also uniform.

It is interesting to note that the actual Sun would move uniformly if the Earth’s axis of rotation were perpendicular to its orbital plane and if the Earth revolved about the Sun in a circular orbit. If this were the case, the real Sun would move uniformly eastward along the celestial equator and would always rise at the east point on the horizon and set at the west point. The Sun would always spend 12 hours above the horizon and 12 hours below the horizon, and seasons as we know them would not exist.

Local apparent and local mean times are related by the equation of time (E), which is defined as the difference between local apparent time and local mean time:

During the year, because of its non-uniform motion, the actual Sun is sometimes ahead of and sometimes behind the Mean Sun. The equation of time can be either positive or negative. When LAT is greater than LMT, E is positive. When LAT is less than LMT, E is negative. The equation of time ranges between about plus and minus 16 minutes. The values of E can be computed, and repeat themselves each year (see Table 3.1).