Eccentricity conic section

Conic Sections The cui ves included in this group are obtained from plane sections of the cone. They include the circle, ehipse, parabola, hyperbola, and degeneratively the point and straight line. A conic is the locus of a point whose distance from a fixed point called the focus is in a constant ratio to its distance from a fixea line, called the directrix. This ratio is the eccentricity e. lie = 0, the conic is a circle if 0 < e < 1, the conic is an ellipse e = 1, the conic is a parabola ... 

A conic section or cotdc is the locus of a point which moves so that its distance from a fixed point (called the focus) is in a constant ratio (called the eccentricity) to the distance from a fixed straight line (called the directrix). 

This reflects the fact that all parabolas have the same eccentricity, namely 1. The eccentricity of a conic section is the ratio of the distances point-to-focus divided by point-to-directrix, which is the same for all the points on the conic section. Since, for a parabola, these two distances are always equal, their ratio is always 1. 

The equations for conic sections can be expressed rather elegantly in polar coordinates. As shown in Fig. 5.9, the origin is defined as the focus and a line corresponding to x = d serves as the directrix. Recall the relations between the Cartesian and polar coordinates x = rcos 9, y = rsin6>. The point Fwill trace out the conic section, moving in such a way that the ratio of its distance to the focus r to its distance to the directrix d — x = d—rcos9 is a constant. This ratio is called the eccentricity, e (not to be confused with Euler s 6 = 2.718...) ... 

FiGURE 5.9 Coordinates used to represent conic sections. The point P traces out a conic section as r is varied, keeping a constant value of the eccentricity. 

Newton showed that, under the inverse-square attraction of gravitational forces, the motion of a celestial object follows the trajectory of a conic section. The stable orbits of the planets around the sun are ellipses, as found by Kepler s many years of observation of planetary motions. A parabolic or hyperbolic trajectory would represent a single pass through the solar system, possibly that of a comet. The better known comets have large elliptical orbits with eccentricities close to 1 and thus have long intervals between appearances. Halley s comet has e = 0.967 and a period of 76 years. 

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). 

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