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FIGURE 3.14. Explanation of direct and retrograde motion of Mars.

persists for 2 or 3 months (for Mars) before the normal direct motion is resumed.

The retrograde motion of a planet, such as Mars, as Kepler pointed out, arises from the difference in the orbital speeds of Earth and of Mars. The situation may be explained with reference to figure 3.14, which shows the simultaneous positions of Earth and Mars in their orbits around the Sun at successive times indicated by the numbers 1, 2, 3, etc. As seen from Earth, the apparent position of Mars in the sky at any time is given by the point where the line passing through the locations of Earth and the planet appears to intersect the background of fixed stars. These points, corresponding to the various simultaneous positions of the two planets, are represented by the points at the right.

An examination of the figure shows that at times 1, 2, and 3, as Earth catches up on Mars just before opposition at 4, Mars exhibits apparent direct motion, from west to east. Between times 3 and 4, as the two planets come closer together, the motion becomes retrograde, that is, from east to west, and this continues past time 4. Subsequently, Earth moves increasingly ahead of Mars and the apparent motion of Mars is again direct. It is only (and always) around the time of opposition that the conditions are suitable for apparent retrograde motion to occur. Furthermore, it can be demonstrated by means of diagrams like figure 3.14 that, at all other times, Mars exhibits direct motion among the stars, as observed from Earth.

The actual locations in the sky of Mars before and after the opposition of 1965 are represented in figure 3.15. This type of be-

FIGURE 3.15. Apparent motion of Mars around the 1965 opposition.

havior, when the retrograde motion is higher in the sky than the direct motion, occurs at or near unfavorable (aphelion) oppositions. The movement of the planet in the loop representing the apparent path of the planet is in a clockwise direction (fig. 3.16.4). The

FIGURE 3.16. opposition.

FIGURE 3.16. opposition.

Types of motion of Mars near reverse situation, with motion in a counterclockwise loop (fig. 3.16Z)), is observed at (or near) the most favorable (perihelion) oppositions. At other times, between the most and least favorable oppositions, the paths of the planet through the sky exhibit intermediate Z- or S-shapes (fig. 3.16 B and C). It is because the orbital planes of Mars and Earth do not coincide that the form of the apparent path of Mars in the sky depends on where in the Martian orbit the opposition occurs.

The Orbital Plane of Mars

The orbit of a planet lies in a plane and this plane must pass through the Sun. Although the orbital planes of the various planets are not very far apart, they do not coincide. The apparent path, from day to day, of the Sun among the constellations is referred to as the ecliptic (fig. 3.17). This name arises from the fact that all eclipses, both of the Sun and the Moon, are observed to occur, as indeed they must, along this imaginary circle. The ecliptic, or apparent path of the Sun in the sky, is inevitably an extension of the plane in which Earth orbits the Sun. Hence, Earth's orbital plane is also called the ecliptic plane.

From extensive observations on Mars, it has been concluded that the plane of its orbit around the Sun crosses the ecliptic plane at an angle of 1.85 degrees. This angle is not large, but it has some significant consequences. The illustration in figure 3.18 is an exaggerated representation of an edge-on view of the orbital planes of Earth and Mars. The orbital plane of Mars is to the south of the ecliptic plane (see fig. 3.17 for directions of north and south) near the Martian perihelion and to the north around aphelion. Because of the great distance from the Sun to Mars, the small angle of inclination of the orbital planes of Mars and Earth is sufficient to bring Mars about 6.6 million kilometers (4.1 million miles) south of the ecliptic plane at perihelion and 8.0 million kilometers (5 million miles) north of it at aphelion.

As a consequence of the inclination of the orbital planes, when Earth begins to overtake Mars shortly before a perihelic (favorable) opposition, Mars appears at first to sink increasingly farther south of the ecliptic; that is, it appears lower and lower in the sky on successive days. The reason is that the

Celestial sphere

North ecliptic pole

Ecliptic plane

North ecliptic pole

Ecliptic plane

Earth's orbit

Ecliptic

South ecliptic pole

FIGURE 3.17. The ecliptic plane and poles.

Earth's orbit

Ecliptic

South ecliptic pole

FIGURE 3.17. The ecliptic plane and poles.

Mars at

Mars at

FIGURE 3.18. Relative positions (exaggerated) of the orbital planes of Earth and Mars.

distance the orbital plane of Mars is south of the ecliptic increases as perihelion is approached (fig. 3.18). After opposition, however, when Mars has moved past its perihelion, the planet appears higher in the sky from day to day, because the distance between the orbital planes is now decreasing. It is thus possible to account for the counterclockwise loop (fig. 3.16D) made by the apparent path of Mars at the time of a perihelic opposition.

Near an aphelic (unfavorable) opposition, the situation is reversed. As Earth approaches Mars, the latter planet moves farther and farther north of the ecliptic and it will be seen steadily higher in the sky. After opposition, Mars will cease to move higher and will start to sink. The loop formed will then have the clockwise form in figure 3.16^4. At other (intermediate) oppositions, Mars will appear to move continuously either higher (farther north) or lower (farther south) in the sky, depending on the location of the planet in its orbit. The path will then have either a Z- or an S-shape, respectively, as in figure 3.16 B and C.

The period during each day when Mars appears above Earth's horizon, and which determines the time for which the planet might be visible, depends on the location of the observation point on Earth. The situation is related to the direction of Earth's axis of rotation, as will now be shown. Earth rotates once daily about an axis passing through the North and South Poles. But this axis is not at right angles to the orbital (or ecliptic) plane. The angle of inclination of Earth's axis to a line perpendicular to the orbital plane, as represented in figure 3.19, is close to 23.5 degrees. Incidentally, this angle is also the inclination of the equatorial plane of the planet to its orbital plane. The direction of the axis of rotation remains essentially unchanged regardless of the position of Earth in its orbit. There are actually small variations resulting from the gravitational attractions of the Sun and Moon, in particular. However, they need not be considered here.

Axis of rotation

Axis of rotation

FIGURE 3.19. Inclination of Earth's axis.

Observations From Northern and Southern Hemispheres

It will be recalled that the favorable oppositions of Mars occur in July, August, and early September when it is summer in Earth's Northern Hemisphere. During this period, Earth's axis is in such a direction that the North Pole is tilted toward the Sun (fig. 3.20). Mars is close to its perihelion, which is slightly below (south of) the ecliptic plane, and is on the side of Earth opposite to the Sun. Consider a location P on Earth at a latitude of about 40° N, which is approximately that of Washington, D.C. The directions of the horizon and of the zenith (directly overhead) at this point are indicated in the figure.

nearer to the Equator. The apparent daily (or nightly) paths of celestial bodies at different locations in the sky, as observed at a latitude of about 40° N, are indicated in figure 3.21. The lower the maximum elevation

Celestial sphere N^

South

Celestial sphere N^

South

Rising

North

Horizon

FIGURE 3.21. Paths of celestial bodies as observed at 40° N latitude.

Rising

North

Horizon

FIGURE 3.21. Paths of celestial bodies as observed at 40° N latitude.

Plane

FIGURE 3.20. Observation of Mars at a favorable opposition from 40° N latitude on Earth.

An examination of figure 3.20 shows that, for the particular circumstances represented, to an observer at 40° N latitude, Mars will appear low on the horizon at the time of a favorable opposition. The same general situation applies to all points in the Northern Hemisphere, although Mars is seen somewhat higher in the sky at lower latitudes; that is, above the horizon, the shorter the period the body, such as Mars, remains visible.

It follows, therefore, as stated earlier, that the times of favorable (périhélie) oppositions of Mars are not very good for observation from the Northern Hemisphere. The conditions improve toward the Equator and are excellent in the middle latitudes of the Southern Hemisphere where Mars rises high in the sky during the long nights of the local winter.

At aphelic (least favorable) oppositions, it is winter in Earth's Northern Hemisphere and summer in the Southern Hemisphere. The South Pole is then tilted toward the Sun (fig. 3.22). The conditions for observing Mars are then better in the Northern than in the Southern Hemisphere. However, at aphelic oppositions, Mars is much farther from Earth than at périhélie oppositions.

At intermediate oppositions, between aphelion and perihelion, especially those which occur in late May and early June or late November and early December, the conditions for observing Mars are equally good

Plane

FIGURE 3.20. Observation of Mars at a favorable opposition from 40° N latitude on Earth.

autumn

Sept. 23

autumn

Sept. 23

March 21

Spring

FIGURE 3.22. Tilt of Earth's axis at various times; seasons indicated are those in the Northern Hemisphere.

March 21

Spring

FIGURE 3.22. Tilt of Earth's axis at various times; seasons indicated are those in the Northern Hemisphere.

in both Northern and Southern Hemispheres. They are best near the Equator, where Mars is almost directly overhead at midnight at the time of opposition.

Inclination of the Axis of Mars

Mars, like Earth, also rotates about a north-south axis, and this axis is inclined at an angle of about 25 degrees to a line perpendicular to the orbital plane. Just as the inclination of Earth's axis determines the best locations from which to observe Mars, so the inclination of the Martian axis determines the parts of Mars that can be seen at the various oppositions.

Although the angles of inclination of the axes of Mars and of Earth (23.5 degrees) are not very different, the two axes point in different directions in space. The two directions, in fact, make an angle of about 45 degrees, that is, half a right angle, with each other. Earth's axis, at present, points in the north close to the star Polaris (Pole Star), but the axis of Mars is directed in the sky somewhere near the star Deneb (Alpha Cygni). Both directions change slowly in the course of time because of gravitational attractions of other members of the solar system.

The directions of the axes of Earth and of Mars at various oppositions are represented approximately in figure 3.23. Because the orbital plane of Mars is inclined at such a relatively small angle (1.85 degrees) to the ecliptic plane, it is assumed, for simplicity, that the two planes coincide. Mars is thus indicated as if it lies on the ecliptic plane; although this is not strictly correct, the general conclusions to be drawn will not be affected.

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