Kepler's Laws (firsts, second, third laws) and the Motion of planets
The name planet comes from a Greek word meaning “wanderer,” and indeed the planets continuously change their positions in the sky relative to the background of stars. One of the great intellectual accomplishments of the 16th and 17th centuries was the threefold realization that the earth is also a planet, that all planets orbit the sun, and that the apparent motions of the planets as seen from the earth can be used to determine their orbits precisely.
The first and second of these ideas were published by Nicolaus Copernicus in Poland in 1543. The nature of planetary orbits was deduced between 1601 and 1619 by the German astronomer and mathematician Johannes Kepler, using precise data on apparent planetary motions compiled by his mentor, the Danish astronomer Tycho Brahe. By trial and error, Kepler discovered three empirical laws that accurately described the motions of the planets:
1. Each planet moves in an elliptical orbit, with the sun at one focus of the ellipse.
2. A line from the sun to a given planet sweeps out equal areas in equal times.
3. The periods of the planets are proportional to the 3/ 2 powers of the major axis lengths of their orbits.
Kepler did not know why the planets moved in this way. Three generations later, when Newton turned his attention to the motion of the planets, he discovered that each of Kepler’s laws can be derived; they are consequences of Newton’s laws of motion and the law of gravitation. Let’s see how each of Kepler’s laws arises.
Kepler’s First Law
First consider the elliptical orbits described in Kepler’s first law. Figure 13.18 shows the geometry of an ellipse. The longest dimension is the major axis, with half-length a; this half-length is called the semi-major axis. The sum of the dis-tances from S to P and from S to P is the same for all points on the curve. S and S are the foci (plural of focus). The sun is at S (not at the center of the ellipse) and the planet is at P; we think of both as points because the size of each is very small in comparison to the distance between them. There is nothing at the other focus, S.
The distance of each focus from the center of the ellipse is ea, where e is a dimensionless number between 0 and 1 called the eccentricity. If e=0, the two foci coincide and the ellipse is a circle. The actual orbits of the planets are fairly circular; their eccentricities range from 0.007 for Venus to 0.206 for Mercury. (The earth’s orbit has e=0.017.) The point in the planet’s orbit closest to the sun is the perihelion, and the point most distant is the aphelion.
Newton showed that for a body acted on by an attractive force proportional to 1/r2, the only possible closed orbits are a circle or an ellipse; he also showed that open orbits (trajectories 6 and 7 in Fig. 13.14) must be parabolas or hyperbolas. These results can be derived from Newton’s laws and the law of gravitation, together with a lot more differential equations than we’re ready for.
Kepler’s second Law
Figure 13.19 shows Kepler’s second law. In a small time interval dt, the line from the sun S to the planet P turns through an angle du. The area swept out is the colored triangle with height r, base length r du, and area dA = 1/ 2 r2 du in Fig. 13.19b. The rate at which area is swept out, dA/dt, is called the sector velocity:
The essence of Kepler’s second law is that the sector velocity has the same value at all points in the orbit. When the planet is close to the sun, r is small and du/dt is large; when the planet is far from the sun, r is large and du/dt is small.
To see how Kepler’s second law follows from Newton’s laws, we express dA/dt in terms of the velocity vector v S of the planet P. The component of v S perpendicular to the radial line is v. = vsinØ. From Fig. 13.19b the displacement along the direction of v. during time dt is r du, so we also have v# = r du/dt. Using this relationship in Eq. (13.14), we find
Now rvsinØ is the magnitude of the vector product r X S : v X S , which in turn is 1/m times the angular momentum L X S = r X S : mv S of the planet with respect to the sun. So we have
Thus Kepler’s second law—that sector velocity is constant—means that angular momentum is constant!
It is easy to see why the angular momentum of the planet must be constant. According to Eq. (10.26), the rate of change of L S equals the torque of the gravitational force F S acting on the planet:
In our situation, r S is the vector from the sun to the planet, and the force F → is directed from the planet to the sun (Fig. 13.20). So these vectors always lie along the same line, and their vector product r S : F S is zero. Hence dL → /dt = 0. This conclusion does not depend on the 1/r2 behavior of the force; angular momentum is conserved for any force that acts always along the line joining the particle to a fixed point. Such a force is called a central force. (Kepler’s first and third laws are valid for a 1/r2 force only.)
Conservation of angular momentum also explains why the orbit lies in a plane. The vector L S = r S : mv S is always perpendicular to the plane of the vectors r S and v S ; since L S is constant in magnitude and direction, r → and v → always lie in the same plane, which is just the plane of the planet’s orbit.
Kepler’s Third Law
We have already derived Kepler’s third law for the particular case of circular orbits. Equation (13.12) shows that the period of a satellite or planet in a circular orbit is proportional to the 3 2 power of the orbit radius. Newton was able to show that this same relationship holds for an elliptical orbit, with the orbit radius r replaced by the semi-major axis a:
Since the planet orbits the sun, not the earth, we have replaced the earth’s mass mE in Eq. (13.12) with the sun’s mass mS. Note that the period does not depend on the eccentricity e. An asteroid in an elongated elliptical orbit with semi-major axis a will have the same orbital period as a planet in a circular orbit of radius a. The key difference is that the asteroid moves at different speeds at different points in its elliptical orbit (Fig. 13.19c), while the planet’s speed is constant around its circular orbit.
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