Chapter 3 Diurnal Motion, Time and the Calendar 3-A. Diurnal Circles A casual inspection of the sky reveals that all celestial objects rise on the eastern horizon, cross the local celestial meridian, where an object’s altitude is a maximum, and set on the western horizon. This is an apparent motion caused by the rotation of the Earth around its axis and is referred to as the diurnal rotation of the celestial sphere. The angular rate of rotation of the Earth is 15 degrees per hour, since the Earth rotates through 360 degrees in 24 hours. Hence, celestial objects appear to move in the sky along paths at this rate, but in the opposite direction, that is, from east to west. The path that an object follows across the sky from east to west sky because of this motion is called its Diurnal Circle. Diurnal circles are the apparent paths that objects appear to follow in the sky as a result of the rotation of the Earth around its axis. Now the right ascension and declination of an object are not affected by the diurnal rotation of the celestial sphere, but altitude and azimuth are. Since the sky appears to turn around an axis through the celestial poles and in a direction parallel to the celestial equator, a diurnal circle is actually the same as the object's parallel of declination. Thus all objects with the same declination trace out the same diurnal circle. A few of these circles are shown in Figure 3-1. Figure 3-1. Diurnal circles drawn on the celestial sphere for several different declinations, as seen from latitude 40o N. Notice that diurnal circles make an angle with the horizon but they are all parallel to one another. From the geometry of the situation (see Fig. 3-1), it can be shown that the celestial equator (CE) intersects an observer’s celestial horizon at the east and west points. Thus the diurnal circle that passes through the east point of the horizon and west point of the horizon is the celestial equator (CE). Recall that every point on the celestial equator is 90o from either of the two celestial poles. Therefore, the angle that the celestial equator makes with the horizon is always equal to 90o minus the observer's latitude (90o - ). The time that an object spends above the horizon depends essentially on where it is located with respect to the celestial equator, i.e., its declination. Half of the equator is above the horizon and so any object on it (declination 0o) is above the horizon for 12 hours and below the horizon for 12 hours. Such an object rises at the east point and sets at the west point. Those objects with declinations between 0o and +90o, rise in the northeast, remain above the horizon for more than twelve hours and set in the northwest. Those objects with declinations between 0o and -90o, rise in the southeast, are above the horizon for less than twelve hours and set in the southwest. Now the horizon system of circles is fixed with respect to an observer. Therefore, all the reference circles of this system do not share in the rotation of the celestial sphere. This means the altitude and azimuth of an object on the celestial sphere both change with time and location, but remember, the right ascension and declination do not. Hence, the horizon system is only useful for locating objects on the celestial sphere at a given location on the Earth and at a given instant of time. So, as the celestial sphere appears to turn around its axis, objects in the sky move relative to the horizon and the local celestial meridian (LCM). As we saw in the previous chapter, the axis of rotation lies in the plane of the local celestial meridian and it is tilted relative to the celestial horizon by an amount equal to the observer's geographic latitude. When an object crosses the upper local celestial meridian, it is said to be at upper transit (UT). At upper transit the object is at its greatest distance above the horizon, that is, its altitude is a maximum. When the object crosses the lower local celestial meridian, it is said to be at lower transit (LT). At LT, an object is at its minimum angular distance (in the algebraic sense) from the horizon. When an object is below the horizon, it's angular distance from the horizon, or altitude, is assigned a negative value. Objects make one upper transit and one lower transit each day. For an observer in the Northern Hemisphere, both upper and lower transit will occur above the horizon for stars located near the north celestial pole. That is, for objects with high declinations. Such objects will have their diurnal circles entirely above the celestial horizon and, therefore, can be seen every night. Such objects are called circumpolar stars of perpetual apparition (stars that are always visible), since they circle around the pole star. By symmetry, for this same observer, there will be a range of declinations for stars near the south celestial pole that never rise above the horizon. That is, their diurnal circles are completely below the horizon and such an object can never be seen at that latitude. These stars that never rise are called circumpolar stars of perpetual occultation. Which objects are circumpolar and which are not depends on the observer's latitude and the declination of the object. Objects located on the celestial equator rise exactly at the east point, while objects north or south of the celestial equator rise to the north or south of the east point. It is very important to realize that objects, which rise exactly at the east point, must set exactly at the west point. Likewise, objects that rise in the northeast set in the northwest and objects that rise in the southeast set in the southwest. As discussed previously, the time that an object spends above the horizon depends essentially on where it is located with respect to the celestial equator, i.e., its declination. Half of the equator is above the horizon and so any object on it (declination 0o) is above the horizon for 12 hours and below the horizon for 12 hours. Such an object rises at the east point and sets at the west point. 3-B. Hour Angle Another important angle related to diurnal motion and time is an object’s hour angle. Hour Angle (HA) is the angular distance of an object measured westward from the Local Celestial Meridian. It is sometimes convenient to use positive and negative numbers to describe the position of an object with respect to the Local Celestial Meridian. Some authors refer to such an angle as the meridian angle, t. When the object is west of the Meridian, t is denoted as positive and when it is east of the Meridian t is negative. But, it is often convenient to use the hour angles that are negative or positive also, contrary to the above definition. The hour angle or meridian angle changes from negative to positive when the object crosses the Upper Local Celestial Meridian, moving east to west. As was noted above, when an object crosses the upper LCM, its altitude is a maximum and the object is said to be at Upper Transit (UT). The hour angle changes sign again when the object crosses the lower Local Celestial Meridian twelve hours later. At this time, the altitude of the object is a minimum and it said to be at Lower Transit (LT). 3-C. The Apparent Annual Motion of the Sun A star always rises and sets at the same place on the horizon and, hence, it is above the horizon for the same amount of time every day of the year. That is, a star traces out the same diurnal circle every day because its declination does not change. This is not true, however, for the Sun, Moon, and planets, because these objects move with respect to the background of “fixed” stars, i.e., their declination changes with time. The changing declination and right ascension of the Sun is caused by the fact that Earth revolves in orbit around the Sun with its axis of rotation tilted with respect to the plane of its orbit. See Figs. 3-2 and 3-3. The amount of this tilt is 23.5o as measured from a line perpendicular to the Earth's orbital plane. This line pierces the celestial sphere at two points called the north ecliptic pole and south ecliptic pole. In the figures, these two points are designated as NEP and SEP. Fig. 3-2. A schematic, not to scale, showing the meaning of the obliquity of the ecliptic. Figure 3-3. The relationship between the plane of the Earth’s orbit, the ecliptic, the Zodiac, and the celestial equator. The position of the Earth in its orbit around the Sun is shown for four important seasonal dates of the year. As the Earth revolves in its orbit around the Sun, its axis of rotation remains parallel to itself. That is, the axis always points towards the north celestial pole (NCP). For an observer on the Earth, the Sun then appears to move in a circle around the celestial sphere eastward with respect to the fixed stars. This circle is called the ecliptic and is labeled VSAW in Fig. 3-3. Note that the ecliptic lies in the plane of the Earth's orbit. Thus the ecliptic is inclined to the celestial equator by 23.5o.