General properties of a hyperbola

Kepler s principal contribution is summarized in his laws of planetary motion. Originally derived semiempir-ically, by solving for the detailed motion of the planets (especially Mars) Ifom Tycho s observations, these laws embody the basic properties of two-body orbits. The first law is that the planetary orbits describe conic sections of various eccentricities and semimajor axes. Closed, that is to say periodic, orbits are circles or ellipses. Aperiodic orbits are parabolas or hyperbolas. The second law states that a planet will sweep out equal areas of arc in equal times. This is also a statement, as was later demonstrated by Newton and his successors, of the conservation of angular momentum. The third law, which is the main dynamical result, is also called the Harmonic Law. It states that the orbital period of a planet, P, is related to its distance from the central body (in the specific case of the solar system as a whole, the sun), a, by a. In more general form, speaking ahistorically, this can be stated as G M -h Af2) = a S2, where G is the gravitational constant, 2 = 2n/P is the orbital frequency, and M and M2 are the masses of the two bodies. Kepler s specific form of the law holds when the period is measured in years and the distance is scaled to the semimajor axis of the earth s orbit, the astronomical unit (AU). 

More specifically, take Z to be the hyperbola X1X2 = aXg in P, where a M. Then W is given by X1X2 = 0, i.e., it is the union of 2 lines. Thus W can be reducible even when Z is irreducible. As a matter of fact, the only other elementary property that W does have when Z is irreducible is that it is connected. More generally, the connectedness theorem of Zariski states that if Z c P is any closed connected set, then its specialization over R is still connected. 

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