Eccentricity

If an ellipse is imagined to become less and less elliptical, and approach a circle, the two foci must come closer together. In a circle, the foci coincide at the center of the circle; then the points F1; O, and F2 in figure 3.2 are all identical. The distance OF L is zero and hence, by the definition given above, the eccentricity of a circle is zero. On the other hand, if an ellipse becomes more elliptical, and deviates more and more from a circle, the foci F± and F2 get farther apart. The distance OF1 then increases (relative to OX) and so does the eccentricity.

In the solar system, the Sun is stationary at one focus of a number of ellipses; the other focus, which is different for each ellipse, is unoccupied. The orbits of the planets, as well as of the asteroids, are each represented by one of these ellipses. Each orbital ellipse has its own particular major (or semimajor) axis and eccentricity which completely define its shape and size. The length of the major axis of the orbit of Mars is 455.88 million kilometers (283.27 million miles) and the eccentricity is 0.0933. The latter may be compared with 0.0167 for the eccentricity of Earth's orbit and 0.00679 for the orbit of Venus. The eccentricity of the Martian orbit is, in fact, greater than that of any of the planets other than Mercury, the closest to the Sun, and Pluto, the farthest. As stated in chapter II, the small eccentricity of Earth's orbit, which does not differ greatly from a circle, and the relatively large eccentricty of the orbit of Mars were important factors in enabling Kepler to discover the first two laws of planetary motion.

An exaggerated elliptical orbit around the Sun is indicated in figure 3.3. Planetary orbits have very much smaller eccentricities, but the sketch might represent the orbits of some asteroids, several of which have fairly large eccentricities (up to about 0.3). The points

FIGURE 3.3. Aphelion and perihelion of a planetary orbit.

to be considered are independent of the eccentricity, however, and a highly elliptical shape is used for clarity.

In the figure, the major axis is now represented by the line PA. At the point P, the planet (or asteroid) is the closest it ever gets to the Sun; this point is called the perihelion, from peri meaning nearby, and helios for Sun in Greek. The distance from the Sun to P is called the perihelion distance. Similarly, the point A where the planet is farthest from the Sun, is known as the aphelion, from apo meaning away from, &nd the distance from A to the Sun is the aphelion distance. The more highly eccentric the ellipse, the greater is the difference between (or rather the ratio of) the perihelion and aphelion distances.

For Mars, the distance of closest approach to the Sun, the perihelion distance, is 206.66 million kilometers (128.41 million miles), whereas the farthest (or aphelion) distance is 249.22 million kilometers (154.86 million miles). The difference is thus quite significant; namely, about 42.5 million kilometers (26.45 million miles). For comparison purposes, the aphelion and perihelion distances of Earth are 152.2 million kilometers (94.56 million miles) and 147.0 million kilometers (91.34 million miles). These numbers show that the orbit of Earth does not depart greatly from a circle, but that of Mars does.

In accordance with Kepler's second law (p. 14), a planet moving in an elliptical orbit does not travel at constant speed. Its speed is greatest at perihelion, when it is closest to the Sun, and least at aphelion, when it is farthest away. At these two points, the speeds are inversely proportional to the respective distances from the Sun. For Mars, the maximum orbital speed, at perihelion, is 26.4 km/sec (16.4 mps), and the minimum, at aphelion, is 22.0 km/sec (13.6 mps). The average speed of the planet over the whole of its orbit is 24.13 km/sec (14.99 mps). Earth's average orbital speed is 29.8 km/sec (18.5 mps). The greater speed with which Earth travels in a smaller orbit than that of Mars has interesting consequences, as will be seen shortly.

Sidereal and Synodic Periods

The time required by a planet to make a complete orbit (or complete revolution) about the Sun is the sidereal year for that planet. The adjective sidereal, derived from the Latin, sidus, meaning a celestial body or constellation, is used to distinguish the sidereal year from the synodic year, which is the apparent time of revolution of a planet as seen from Earth. The difference between the two kinds of year may be explained in the following manner, which also accounts for the use of the term synodic, derived from the Latin synodus (or Greek synodos) for meeting.

Figure 3.4 shows the orbit of a planet around the Sun and also a distant star (not a planet) S. The latter is so far away from the Sun that its position may be regarded as fixed. The point X is that of the planet at some position in its orbit when it is directly in line with the Sun and the particular fixed

star S. The time elapsing between two successive occasions of this kind is the sidereal year; that is, the actual time it takes the planet to make one complete orbit around the Sun. For Mars, the sidereal year is 686.980 (essentially 687) Earth days, compared with 365.256 days for Earth's sidereal year. The greater length of the Martian year is caused by the longer orbit and its lower average speed, as given above.

The synodic year, better called the synodic period, is the time between two successive similar alinements (or meetings) of the planet with the Sun as seen from Earth. Let figure 3.5 represent the orbits of Earth and of another planet such as Mars, which is farther out from the Sun. Suppose that Mars is in the position M relative to Earth at E; the Sun, Earth, and Mars are then in a straight line

M1 (Conjunction)

M1 (Conjunction)

FIGURE 3.5. Mars at conjunction and opposition.

(or, more correctly, in the same plane), with Earth between the Sun and Mars. In this location, Mars is said to be in opposition, because, as seen from Earth, Mars is opposite the Sun. On the other hand, if Mars is at M' relative to Earth at E, when the Sun is directly between Earth and Mars, the planet is in conjunction.

The synodic year is then defined as the time elapsing between two successive meetings of the same kind, that is, either two oppositions or two conjunctions, of Mars, Earth, and the Sun. Because Mars cannot be observed at conjunctions, as it is then in the same direction as the Sun, the length of the synodic year is defined, from the practical standpoint, as the time between two successive oppositions.

Although Mars, Earth, and the Sun are alined (or, at least, in one plane) at each opposition, the actual positions of the planets in their orbits change from one opposition to the next. A typical situation is depicted in figure 3.6 which shows the locations of Earth and Mars at two successive oppositions, indicated by the numbers 1 and 2. Earth is at E, (and Mars at M1) at the first opposition and at E2 (Mars at M2) at the next opposition. Between the two opposition:-. Karlli lias made

FIGURE 3.6. Locations oí Earth and Mars at successive oppositions.

addition, has traversed the distance from E, to E2. Mars, on the other hand, with its considerably longer orbit and lower speed, has made only one complete orbit plus the distance from Mx to M2■ The locations of E2 and M2, relative to E, and Mu respectively, are determined by the lengths of the orbits and the orbital speeds of the two planets.

Suppose the distance from Ei to E2 represents a fraction / of Earth's orbit. Then the distance from M1 to M2 will be the same fraction / of the orbit of Mars. This would be exactly true only if the two orbits had the same shape. Although it is known that they do not, it will be assumed for the moment that the distances E1E2 and MjM2 are both the same fraction / of the respective orbits. Consequently, between one opposition and the next, that is, during the same time interval, Earth travels a distance of 2+/ Earth orbits, whereas Mars covers a distance of 1 +/ Mars orbits. The time required for Earth to make a complete orbit of the Sun is 365 days (ignoring fractions), whereas Mars requires its sidereal year of 687 days to complete one orbit. The synodic period of Mars, which is the time between two oppositions, can thus be expressed in two alternative ways; in terms of Earth's motion

Synodic period of Mars=(2 + /) X 365 days whereas in terms of the motion of Mars,

Synodic period of Mars=(l+/) X 687 days

In order to find the value of f, these two expressions are set equal to each other; thus

Consequently, the synodic period of Mars is given either by 2.134X365, or 1.134X687. In each case the result is 780 days.

According to the foregoing calculation, oppositions of Mars should occur every 780 days; that is, at intervals of about 2 years and 50 days. Because the orbits of Earth and Mars do not have the same shape, as indicated by the marked difference in the eccentricities, the fraction / is not the same for both orbits. In fact, the values vary with the positions in the orbits where the oppositions occur. Consequently, the observed intervals between successive oppositions actually range from 763 to 810 days. The average value, however, has been determined to be 779.935 days, in very good agreement with the 780 days calculated above.

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