3 Diurnal Motion of Stars

V. B. Bhatia

1. Learning Outcomes

After studying this module, you should be able to

understand why stars and other celestial objects describe circular orbits from east to west
grasp the meaning of rising and setting of stars
explain why some stars are circumpolar for an observer and why some are never visible to this observer
derive the condition for a star to be circumpolar
appreciate the phenomenon of ‘midnight Sun’
infer that from north pole (south pole) the stars of only the northern (southern) hemisphere are visible
conclude that for an observer at the equator, no star is circumpolar; the observer can see all stars in both the hemispheres but only for half of their orbits

2. Introduction

In the last three modules we have dealt with the various coordinate systems for celestial objects. This subject is collectively called positional astronomy. In particular, we discussed the horizon system, the equatorial system of both kinds, that is, the hour angle – declination system and right ascension – declination system, the ecliptic system and the galactic system. We also learnt the situations where each of these systems is important. Depending on the need of the observer, it is necessary to transform coordinates from one system to another. We saw how this can be done using spherical trigonometry. This module is the last in positional astronomy where we discuss the motion of stars as seen by observers at various locations.

3. Diurnal Motion of Stars

As we already know, the earth rotates on its axis completing one rotation in 24 hours (Fig. 4.1). Since we share the motion of the earth, we see the celestial objects rotate in the direction opposite to that of earth’s rotation, completing one orbit in 24 hours. These orbits are in planes

Fig. 3.1. Earth rotates from west to east on its axis. Seen from above the north pole the rotation is counterclockwise.

perpendicular to the axis of rotation of the earth and, therefore, parallel to the plane of the celestial equator. This daily motion of objects is called diurnal (occurring daily) motion. The circular orbits in which objects execute their daily motion are called diurnal circles (Fig. 4.2). Since the earth rotates on its axis from west to east (anticlockwise as seen from above the north pole) the celestial objects appear to rotate from east to west, or clockwise as seen from above, even in the southern hemisphere. So, if you look up north, you see the objects rotating counterclockwise.

Fig. 3.2. While the earth rotates on its axis from west to east, the stars appear to rotate from east to west on their diurnal circles in planes parallel to the celestial equator. P and Q are the North and South celestial poles, respectively.

3.1. Rising and Setting of Stars – Circumpolar Stars

As a star crosses the horizon so that it is visible, we say that the star has risen. When the star goes below the horizon, we say that the star has set. On the portion of the diurnal circle which is above the horizon of an observer the star is visible to the observer; below the horizon, the star is not visible (Fig. 4.3). In Fig. 4.3 the portions of the orbit above the horizon are shown in red; those below the horizon are shown in black. Interestingly, there are some stars whose diurnal circles are completely above the horizon; these stars are always visible to the observer, rotating round the Pole star. These stars never set. These are called circumpolar stars, because of their proximity to the poles. There are also stars which never cross the horizon for the observer; they are never visible to the observer.

Fig. 3.3. The stars whose diurnal circles cross the horizon are visible for the duration their diurnal circles are above the horizon. For the rest of the duration, they are not visible. These stars are said to rise and set. The stars whose diurnal circles do not cross the horizon are either visible all the time or are never visible. The former are called circumpolar stars.

3.2. Condition for a Star to become Circumpolar

To find the condition under which a star is circumpolar for an observer, notice that the declination of the star whose diurnal circle just touches the horizon is ?(Fig. 4.4). Also, the altitude of the Pole is ?, the latitude of the observer in the northern hemisphere. Since the distance of the Pole from the equator is 90°, we have ?+ ?= 90°. This means that a star

Fig. 3.4. The arc length RN is the declination of the star whose diurnal circle just touches the horizon. The arc length from N to the Pole is equal to the latitude of the observer. The sum of the two must be 90°. This gives the condition for a star to be circumpolar.

which has ?= 90° − ?just touches the horizon for the observer at latitude ?. All those stars whose declination exceeds 90° − ?are circumpolar at latitude ?. Thus, the condition for a star to be circumpolar at latitude ?is:

For observers in the Northern Hemisphere (Fig. 3.4)

? + ? ≥ ??°, (3.1)

or,

? ≥ (??° − ? ). (3.2)

For observers in the Southern Hemisphere (Fig. 3.5)

? + ? ≤ − ??° . (3.3)

Fig. 4.5. Circumpolar stars and stars that never rise for an observer in the southern hemisphere at latitude –?.

By the same argument, stars which always remain below the horizon must have:

For observers in the Northern Hemisphere

?− ?≤ − ?0°, (3.4)

or,

?≤ (?− ?0°). (3.5)

For observers in the Southern Hemisphere

?− ?≥ ?0°. (3.6)

Stars which never come up the horizon are never visible to the observer at latitude ?. Consider, for example, the constellation Big Bear (Saptarishi). The declination of the star of this constellation which is farthest from the Pole Star is 49°. Therefore, for all those living at latitudes 41° or higher, this constellation will be circumpolar. It will be visible all the time, if the sunlight did not mask it during the day.

Fig. 3.6. For people living at latitude 41°, or higher, Big Bear is circumpolar.

Fig. 4.7 shows the circumpolar stars in the northern hemisphere. For photograph and animation, visit: https://en.wikipedia.org/wiki/Circumpolar_star

Fig. 4.7. The North Star and circumpolar stars in a photograph with a long shutter speed of several hours. (Source: Wikipedia, photo by LCGS Russ)

As another example, consider that you live at a place on the tropic of cancer, ?= -23.5°. Then all stars satisfying the condition ?+ ?≤ − ?0°, or having ?≤ −66.5° are circumpolar to you. At the same time, stars satisfying ?− ?≥ ?0°, or having ?≥ 66.5° will never be visible to you.

3.3. Circumpolar Stars for an Observer on the North Pole

Suppose an observer is stationed at the north pole. As we have seen earlier, the horizon of this observer coincides with the celestial equator. All stars of the northern hemisphere are circumpolar. No star of the southern hemisphere is visible to this observer. Since the latitude of this observer is 90°, the stars of all positive declinations are circumpolar, as is clear from Equation (3.1). These stars never set, as shown in Fig. 4.8. By the same logic, all stars south of the equator are invisible. For this observer, they never rise.

Fig. 3.8. Circumpolar stars for the observer at the north pole. The situation is exactly the reverse for the observer at the south pole.

3.4. The Sun as Circumpolar Star

Note that the declination of the Sun varies between 0° and 23.5° during the six months from 21 March to 23 September. For the observer at the North Pole (indeed, for latitudes > 66.5°), the Sun is circumpolar during these six months; it is always up, even at midnight (called midnight Sun, Fig. 4.9), and does not set. During the six months from 23 September to 21 June when the declination of the Sun varies from 0° and -23.5°, the Sun is never visible to these observers in the northern polar region; it never rises. For the south polar region, though, the Sun is circumpolar during 23 September to 21 March and visible continuously, while during 21 March to 23 September it is not visible at all. In popular parlance, it is said that in the polar regions the day and night are of six month’s duration.

Fig. 3.9. The Altafjord in Alta, Norway bathed in the Midnight Sun. Source: Wikipedia, photo by Vberger.

3.5. Observer at the Equator

What happens if the observer is located at the equator? This situation is shown in Fig. 4.10. Since the horizon is now perpendicular to the equator, it divides the diurnal circles in two equal

Fig. 3.10. An observer at the equator is able to see all stars but only for half the time.

halves. All the stars, of both the hemispheres, are visible to this observer for half the time; for the remaining half they are below the horizon.

For any observer between the equator and the pole, the situation is similar to that shown in Fig. 3.4. Depending on the latitude of the observer, some stars are circumpolar and some stars are not accessible to this observer at all. Other stars rise and set for different durations.

4. Summary

The earth rotates on its axis from West to East.
Seen from above the North Pole this motion is anticlockwise.
We on the earth see all celestial objects moving from East to West in circular orbits.
The planes of these orbits are parallel to the Celestial Equator.
If the orbit of a star crosses the horizon of the observer, the star is visible when above the horizon and it is invisible as goes below the horizon.
When a star just comes up the horizon, it is said to have risen.
The moment it goes below the horizon; it is said to have set.
If the orbit of a star is always above the horizon, the star is always visible and is called a circumpolar star.
If its orbit is completely below the horizon, the star is never visible to the observer.
Whether a star is circumpolar or not depends on the latitude of the observer.
For an observer on the North Pole all stars in the northern hemisphere are circumpolar, and all those in the southern hemisphere are not visible.
For an observer on the South Pole, all stars of the southern hemisphere are circumpolar and no star of the northern hemisphere is visible.
In the polar regions, the Sun rises and sets for six months at a stretch.
Sun can be seen even at midnight at these places.
An observer at the equator can see all stars from both the hemispheres, but only for half the time, because the horizon divides the orbits of stars in two equal halves.

Know More

Helpful Websites:

https://en.wikipedia.org/wiki/Circumpolar_star
http://www.backyardastronomy.com/Backyard_Astronomy/Sky_Motion_Movies/Pages/Circumpolar_Stars-Looking_South_%28Southern_Hemisphere%29.html
http://www.astro.rug.nl/~sctrager/teaching/OA/PositionalAstronomy.pdf
https://www.youtube.com/watch?v=soy04EG3VcE (animation)
https://www.youtube.com/watch?v=soy04EG3VcE (animation)

You can have a look at the book at this site: https://www.willbell.com/MATH/mc8.htm