Math (Несколько текстов для зачёта), страница 6

The NCTM's Standards stress the importance of connections among various branches of mathematics and between mathematics and other disciplines; the astronomy problems that follow combine algebra, geometry, trigonometry, data analysis, and a bit of physics. My geometry and algebra students have seen most of these problems and could understand them. They have also been able to experience making distance measurements themselves by using the method of parallax, which is explained in this article.

THE SIZE OF EARTH

By the third century B.C.E., many scientists were convinced that Earth was spherical. One clue was that during an eclipse of the Moon, the edge of Earth's shadow always appeared to be an arc of a circle. Because of the belief that Earth was spherical, much discussion occurred about how to measure its circumference.

Eratosthenes, who was director of the great library at Alexandria, Egypt, found the first successful method. He had learned that at noon on the day of the summer solstice, in Syene, in southern Egypt, the bottom of a well was illuminated by the Sun; therefore, the Sun was directly overhead there. In Alexandria, in northern Egypt, the Sun was not directly overhead on that day. Any vertical pole casts a shadow. By measuring a pole's shadow and using the ratio of the shadow's length to the pole's height, as shown in figure 1, Eratosthenes was able to calculate Theta, the Sun's angle away from the vertical. Figure 2 shows how he used that information: Reasoning that the Sun's rays striking Alexandria were essentially parallel to those striking Syene, he realized that his angle Theta was the same as the difference in latitude between the two cities. Knowing the distance, D, between them, he was able to calculate the full circumference of Earth. His measure for Theta was 7 Degree 12', which is one-fiftieth of a complete circle.

Because caravans could cover the distance between the cities in fifty days, traveling at the rate of one hundred stadia a day, he assumed that the distance between the cities was five thousand stadia and that the circumference of Earth was therefore 50 x 5000, or 250 000, stadia. The actual length of a stadium in modern units is not known, but it is believed to have been about one-tenth of a mile, which makes Eratosthenes' value for the circumference agree remarkably well with the value accepted today.

FIRST ATTEMPT TO MEASURE THE DISTANCE TO THE SUN AND MOON

Also in the third century B.C.E., Aristarchus of Samos measured the ratio of the Sun's distance from Earth to the Moon's distance from Earth by using a method illustrated in figure 3 (Abell 1964). He reasoned that at the first and third quarters of the Moon, the angles EM[sub 1]S and EM[sub 3]S must be right angles. All he needed was angle M[sub 1] EM[sub 3], and either a scale drawing or trigonometry would give him the distance ratio that he wanted. He assumed that the Moon's orbit is circular, that its orbital velocity is uniform, that the Sun is sufficiently near that angle M[sub 1]EM[sub 3] is measurably different from 180 degrees, and that he could observe the instants of first and third quarter sufficiently accurately. All his assumptions were incorrect, but his method makes sense in principle. He determined, inaccurately, that first quarter to third quarter took about one day longer than third quarter to first quarter. With this information and the length of the month, he determined that M[sub 1]ES was about 87 degrees and that the distance from Earth to the Sun was therefore about twenty times larger than the distance from Earth to the Moon.

THE PLANETS

In the sixteenth century, Copernicus, who had proposed the heliocentric theory of the Solar System, calculated the orbital periods of the planets and their distances from the Sun. He was able to give distances only in terms of the distance from Earth to the Sun. This Earth-to-Sun distance is called the astronomical unit (AU). For example, he found that the distance from Mars to the Sun was 1.5 AU--he could not give this distance in miles or other terrestrial units because he did not know the size of the AU in those units. The following paragraphs give Copernicus's methods for periods and distances, but the problem of the size of the AU was not solved until long after his time.

The orbital period of a planet, the time required for it to complete an orbit relative to the "fixed" stars, is called the sidereal period. We could determine the sidereal period easily if we could observe from a fixed point far outside the Solar System. Since we must instead observe from a moving platform, Earth, we must infer the sidereal period from the synodic period, which is the interval of time between one alignment of Sun, Earth, and a planet and the next equivalent alignment. Figures 4 and 5 (Abell 1964) illustrate how Copernicus determined sidereal periods from synodic periods. The procedure for inferior planets, that is, those closer to the Sun than Earth, differs slightly from that for superior planets, that is, those that are farther away.

Figure 4 shows Earth with Venus, an example of an inferior planet. At position 1, Earth (E[sub 1]), Venus (V[sub 1]), and the Sun are collinear. This orientation is easy to observe from Earth. After one sidereal period, Venus has made one orbit and returned to position V[sub 2] = V[sub 1]; but in that time Earth has moved to E[sub 2], so we cannot directly observe that Venus has completed an orbit. Venus catches up with Earth at position 3. One synodic period has elapsed since position 1 because the two planets are again collinear. From E[sub 1] to E[sub 3], Earth has made N orbits, and N (Earth) years have therefore elapsed, which is the synodic period of Venus. In general, N will not be an integer. In the same amount of time, Venus has made N + I orbits. The sidereal period, S, of Venus is the time for one orbit; that is,

S = time/number of orbits

= synodic period/number of orbits between alignments

= N Earth years/N + 1 orbits

= N/N + 1 Earth years/orbit.

Figure 5 shows Earth with Mars, an example of a superior planet. Both planets begin at position 1, where they are collinear with the Sun. Earth completes an orbit and returns to position E[sub 2] - E[sub 1], then catches Mars at position 3, where the planets and the Sun are again collinear; and one synodic period has elapsed. From position I to position 3, Earth has made N orbits; therefore, N (Earth) years have elapsed. This time, N will probably be greater than 1. In the same amount of time, Mars has made only N - 1 orbits. As with Venus, the sidereal period, S, is the time for one orbit; that is,

S = synodic period/number of orbits between alignments

= N Earth years/N - 1 orbits

= N/N - 1 Earth years/orbit.

For example, Jupiter's synodic period is 1.094 Earth years; S -1.094/(1.094 - 1) = 11.6 years.

Copernicus found orbital radii of inferior planets by using the idea illustrated in figure 6 (Abell 1964). When the planet is at greatest elongation, which is the maximum angular separation in the sky of a planet and the Sun, then angle EPS must be a right angle because the line of sight, EP, is tangent to the planet's orbit. If angle PES is measured, PS can be found by scale drawing or by trigonometry. As previously mentioned, PS will be expressed in terms of ES, the astronomical unit.

The orbital radius of a superior planet is a little more complicated to determine. Figure 7 (Abell 1964) illustrates Copernicus's reasoning. Position 1 is called opposition because when the planet is viewed from Earth, the planet is exactly opposite the Sun in the sky. Position 2, where the planet and the Sun are 90 degrees apart in the sky, that is, angle P[sub 2]E[sub 2]S - 90 Degrees, is called quadrature. Copernicus timed the interval between opposition and quadrature; because he knew the sidereal periods of Earth and the planet, he could determine the angles P[sub 1]SP[sub 2] and E[sub 1]SE[sub 2] as fractions of complete orbits. Angle P[sub 2]SE[sub 2] followed by subtraction; and then PS could be determined, again in terms of ES, the astronomical unit. For example, the time from opposition to quadrature for Mars is 104 days. Therefore,

E[sub 1] SE[sub 2] = 104 days/365 x 360 Degrees

approximately equal to 103 Degrees.

Since the sidereal period of Mars is 687 days,

P[sub 1]SP[sub 2] - 104 days/687 days x 360 Degrees

approximately equal to 55 Degrees.

By subtraction, angle P[sub 2]SE[sub 2] approximately equal to 48 Degrees; and by trigonometry, PS approximately equal to 1.5 ES approximately equal to 1.5 AU.

Table 1 shows the values that Copernicus obtained for the planets known at that time and compares them with modern values.

Copernicus still assumed, as other astronomers had before him, that planetary orbits were circles or combinations of circles. Johannes Kepler, a student of Tycho Brahe, discovered otherwise. At the end of the sixteenth century, Brahe made detailed star and planet observations covering a period of about twenty years. After Brahe's death Kepler spent years analyzing Brahe's data, concentrating on the data for Mars, and in 1609 he published his findings--that the planets move around the Sun in ellipses. That discovery, in spite of the fact that the eccentricity of Mars's orbit is only about one-tenth, is a tribute to his powers of analysis, as well as to the accuracy and thoroughness of Brahe's observations.

To determine that orbits were ellipses, Kepler had to calculate the distance from Mars to the Sun at many different places in its orbit. Figure 8 shows his method (Abell 1964). From any position E[sub 1] of Earth, the angle SE[sub 1]M is measured. The sidereal period of Mars is 687 days, after which Mars has returned to M and Earth, having made almost two complete revolutions, is at E[sub 2]. From E[sub 2], angle SE[sub 2]M is measured. At 687 days Earth is (2)(365.25) - 687 = 43.5 days short of two full revolutions, from which information angle E[sub 1]SE[sub 2] can be calculated. SE[sub 1] and SE[sub 2] are known (1 AU--but a problem arises with this assumption, as described in the following paragraph). From this information can be found E[sub 1]E[sub 2], which allows the solution of triangle E[sub 1]E[sub 2]M, which leads to triangle SE[sub 1]M or SE[sub 2]M and the distance SM. Kepler found SM at many points along the orbit of Mars by choosing from Brahe's records the elongations of Mars--angles SE[sub 1]M or SE[sub 2]M--on each of many pairs of dates separated from each other by intervals of 687 days.

A question that I have been unable to answer was how Kepler dealt with the fact that SE is not really constant because Earth's orbit is also an ellipse. I assume that he must have found a way around the problem, but without more information I can only speculate on how he did it.

Kepler published three findings, which have become known as Kepler's laws of planetary motion. They are as follows:

1. The planets move around the Sun in ellipses, with the Sun at one focus.

2. A line connecting a planet with the Sun will sweep out equal areas in equal times. This phenomenon occurs because a planet moves faster when it is closer to the Sun. In figure 9, the time interval from E[sub 3] to E[sub 4] equals the time interval from E[sub 1] to E[sub 2], and area SE[sub 1]E[sub 2] equals area SE[sub 3]E[sub 4].

3. The squares of the planets' periods of revolution are proportional to the cubes of their distances from the Sun. So P]sup 2] = Ka[sup 3], where a is the length of the semimajor axis of the elliptical orbit. When P is measured in years and a in astronomical units, K = 1.

Table 2 illustrates Kepler's third law for the planets known in his time. Incidentally, these data can be used for a wonderful problem in data analysis. During a unit on nonlinear data analysis, I gave my advanced-algebra students the data in the first three columns, and they were able to determine that P = f(a) is a power function with exponent 3/2.

MASS OF THE SUN AND OTHER OBJECTS

As Duncan (1981) says, "Kepler's laws summed up neatly how the planets of the solar system behaved without indicating why they did so." Newton, who comes into the story at this point, built on Kepler's work to develop his law of universal gravitation, which has allowed us to weigh the Sun, Moon, and planets. Start with the formula

F = mv[sup 2]/r,

which is for the centripetal, or inward, force F needed to cause a mass m to move with velocity v around a circle of radius r. A planet of mass m[sub p] revolving around the Sun has velocity

v = 2 Pi r/P,