Keplers Second and Third

The radius vector of a planet in polar coordinates is simply r = r er

Fig. 6.6. Unit vectors er and e f of the polar coordinate frame. The directions of these change while the planet moves along its orbit

By comparing this with the length of k (6.23), we find that

and integrate over one complete period:

Fig. 6.7. The areas of the shaded sectors of the ellipse are equal. According to Kepler's second law, it takes equal times to travel distances AB, CD and EF

where a and b are the semimajor and semiminor axes and e the eccentricity, we get na

Since k is constant, so is the surface velocity. Hence we have Kepler's second law:

The radius vector of a planet sweeps equal areas in equal amounts of time.

Since the Sun-planet distance varies, the orbital velocity must also vary (Fig. 6.7). From Kepler's second law it follows that a planet must move fastest when it is closest to the Sun (near perihelion). Motion is slowest when the planet is farthest from the Sun at aphelion.

We can write (6.25) in the form

To find the length of k, we substitute the energy integral h as a function of semimajor axis (6.16) into (6.13) to get k = \lG (m1 + m2)a (1 — e2). When this is substituted into (6.29) we have 4n 2

where P is the orbital period. Since the area of the ellipse is

This is the exact form of Kepler's third law as derived from Newton's laws. The original version was The ratio of the cubes of the semimajor axes of the orbits of two planets is equal to the ratio of the squares of their orbital periods.

In this form the law is not exactly valid, even for planets of the solar system, since their own masses influence their periods. The errors due to ignoring this effect are very small, however.

Kepler's third law becomes remarkably simple if we express distances in astronomical units (AU), times in sidereal years (the abbreviation is unfortunately a, not to be confused with the semimajor axis, denoted by a somewhat similar symbol a) and masses in solar masses (Me). Then G = 4n2 and a3 = (m 1 + m 2 ) P2

Fig. 6.7. The areas of the shaded sectors of the ellipse are equal. According to Kepler's second law, it takes equal times to travel distances AB, CD and EF

The masses of objects orbiting around the Sun can safely be ignored (except for the largest planets), and we have the original law P2 = a3. This is very useful for determining distances of various objects whose periods have been observed. For absolute distances we have to measure at least one distance in metres to find the length of one AU. Earlier, triangulation was used to measure the parallax of the Sun or a minor planet, such as Eros, that comes very close to the Earth. Nowadays, radiotelescopes are used as radar to very accurately measure, for example, the distance to Venus. Since changes in the value of one AU also change all other distances, the International Astronomical Union decided in 1968 to adopt the value 1 AU = 1.496000 x 1011 m. The semimajor axis of Earth's orbit is then slightly over one AU.

But constants tend to change. And so, after 1984, the astronomical unit has a new value,

Another important application of Kepler's third law is the determination of masses. By observing the period of a natural or artificial satellite, the mass of the central body can be obtained immediately. The same method is used to determine masses of binary stars (more about this subject in Chap. 9).

Although the values of the AU and year are accurately known in Si-units, the gravitational constant is known only approximately. Astronomical observations give the product G (m\ + m2), but there is no way to distinguish between the contributions of the gravitational constant and those of the masses. The gravitational constant must be measured in the laboratory; this is very difficult because of the weakness of gravitation. Therefore, if a precision higher than 2-3 significant digits is required, the Si-units cannot be used. Instead we have to use the solar mass as a unit of mass (or, for example, the Earth's mass after Gm© has been determined from observations of satellite orbits).

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