FIGURE 3.8. Photographs of Mars taken at favorable (left) and unfavorable (right) oppositions. (Lowell Observatory photographs.)
The apparent diameter of Mars, as seen by the eye or, better, through a telescope, is inversely related to its distance from Earth. The ratio of the distances between the two planets at the most favorable and least favorable oppositions is roughly 35/63. The ratio of the apparent diameters of Mars is 63/35, or 1.8, so at the most favorable oppositions Mars appears to be 1.8 times as large as at the least favorable oppositions, as seen from Earth. The photographs of Mars in figure 3.8 were taken in 1924 (left) and 1931 (right) at favorable and unfavorable oppositions, respectively.
Another factor which makes it easier to see Mars at the favorable oppositions is that the planet is then close to perihelion; that is to say, it is at its closest to the Sun. The brightness of Mars, like that of the Moon, depends on reflected sunlight. Consequently, when Mars is closest to the Sun, at perihelion, it receives and reflects more light than at aphelion. In other words, in addition to appearing larger, Mars is also brighter at favorable than unfavorable oppositions.
Because Earth has a smaller orbit than that of Mars and also travels faster, Earth catches up with Mars before opposition and moves away from it after opposition. As a result, there is a period of only a few months each side of an opposition when the conditions are best for studying Mars. Apart from distance, there are other considerations that restrict observations of the planet to the times around opposition. Because oppositions occur, on the average, at intervals of more than 2 years and favorable ones at intervals of 15 and 17 years, it is evident that the opportunities for making detailed studies of Mars are quite limited.
In addition to the phenomena of opposition and conjunction, there are two other configurations of Mars with reference to Earth and the Sun that are of interest. In figure 3.9 certain positions of Mars in its orbit during a synodic period are shown relative to the position of Earth. Actually, the diagram is a composite of several in which both Mars and Earth have changed their locations, but it may be supposed that these are superimposed in such a way that Earth is always at the same point. Thus, the figure shows some succes-
FIGURE 3.9. Eastern and western quadratures.
sive positions of Mars, in relation to the Sun, as seen from Earth during a complete synodic period, that is, from one opposition (or conjunction) to the next.
As viewed from the north ecliptic pole (fig. 3.17), all planets travel in their orbits in a counterclockwise direction around the Sun (fig. 3.1). Because the orbital speed of Mars is less than that of Earth and its orbit is larger, it appears from Earth as if Mars is moving in the opposite (or clockwise) direction, as shown in figure 3.9. The situation is similar to that in which, to an observer on a fast-moving train, a train traveling more slowly on an adjacent track appears to be going backward.
The term quadrature refers to a quarter of a circle, that is 90 degrees of arc. Thus, at eastern quadrature the planet Mars appears from Earth to be 90 degrees east of the Sun. In other words, the direction of Mars is at an angle of 90 degrees east of the direction of the Sun. Similarly, at western quadrature, the direction of Mars is 90 degrees west of the Sun. The angle between the directions of the Sun and a planet as seen from Earth is called the elongation. Hence, at the quadratures the elongation is 90 degrees (east or west) ; at conjunction the elongation is zero and at opposition it is 180 degrees.
From figure 3.9 some ideas can be obtained concerning the times when Mars is visible and where it can be seen in the sky. First something must be said of the apparent motion of the Sun and the planets (and in fact of all celestial bodies). The Sun may be regarded as being stationary and the planets do not travel very far in the course of a single day. Their apparent daily (or nightly) motion from east to west is thus merely a consequence of Earth's rotation from west to east. Because the orbits of the planets lie in planes not very different from Earth's orbital plane, the planets follow the same general path across the sky as does the Sun. Thus, the planets rise in the east, like the Sun, and set in the west. The times of rising and setting are, however, generally quite different from those of the Sun.
At (or near) conjunction, Mars is in the direction of the Sun and it cannot then be seen because of the Sun's brilliance. Soon after conjunction, Mars appears to be just west of the Sun (fig. 3.10/1), and it should be observed rising low in the eastern sky just before dawn. After sunrise, although Mars is still west of the Sun, the planet disappears from view because of the brightness of the sky. Mars is then said to be a morning star. It is so far from Earth, however, and is visible for only a short time near the horizon, that the conditions for studying the planet are very poor.
On subsequent days, Mars rises farther and farther west of the Sun; that is to say, the planet can be seen in the east above the horizon for longer and longer periods before sunrise. At western quadrature (fig. 3.10B), Mars would be 90 degrees from the Sun; con
FIGURE 3.10. Positions of Mars, relative to Earth and the Sun, at various times.
sequently, in the early morning, when the Sun is toward the east, Mars would be toward the south. The planet would then be visible during the latter part of the night until the sky becomes bright enough to interfere. When opposition is reached, Mars is at 180 degrees from the Sun. Because the Sun is in the south at midday, Mars will then be in the south at midnight. The planet can be observed for several hours each night, between sunset and sunrise. Provided Mars rises sufficiently above the horizon (p. 48), the conditions are then best for viewing the planet.
After opposition, Mars appears in the south earlier and earlier (and sets earlier in the west) each night. At eastern quadrature the planet is 90° east of the Sun (fig. 3.10C); hence, at sunset, when the Sun is in the western sky, Mars will be in the south. It can then be observed during the early part of the evening because it will have set by midnight. Following eastern quadrature, Mars moves westward relative to the Sun (fig. 3.10D) and is then called an evening star because it is visible only for a few hours after sunset. In due course, the planet works its way around to conjunction again having made a complete cycle in the synodic period of 780 days.
It was mentioned in chapter II that Mars exhibits a partial (or gibbous) phase. The proportion of the surface of the planet that is visible from Earth depends on the relative positions of the Sun, Earth, and Mars. At conjunction and opposition, the full face of Mars is illuminated by the Sun; at these times the planet's disk appears (or would appear if it could be seen) as a complete circle. At all other times, part of Mars is in shadow, just as the Moon is when it is not full.
The reason is that, although one hemisphere of the planet is always presented to the Sun, whereas the other hemisphere is in shadow, the illuminated half cannot be seen in its entirety when the directions of Sun and Earth, with reference to Mars, are not the same. The situation is illustrated in figure 3.11; only the portion AB of the illuminated part of the surface of Mars is visible from Earth, whereas the part BC, which should be seen, is in shadow.
The angle made at Mars by the directions of the Sun and Earth is known as the phase angle of the planet, as shown in the sketch. It can be seen that this angle is equal to the angle between B and C. Because the whole disk of the planet, represented by AC, makes an angle of 180 degrees, it is evident that the fraction of the disk that is in shadow is equal
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