Orbits of comets
The orbits of comets in solar system provides an interesting application of conic sections. Prior to 1970, 610 comets were identified, of which 245 had elliptical orbits, 295 had parabolic orbits and 70 had hyperbolic orbits. For example, Encke's comet has an elliptical orbit, and reappearance of this comet can be predicted every 3.3 years. The center of the sun is a focus of each of these orbits, and each orbit has a vertex at the point where the comet is closest to the sun, as shown in above figure.
If x is the distance between the vertex and the focus (in meters) and v is the speed of the comet at the vertex (in m/s), then the type of orbit is determined as: and , where M is the mass of the sun and G is the universal gravitational constant.
Introduction to Conics
Conic sections were discovered during the classical Greek period, 600 to 300 B.C. The conic sections are figures that can be formed by the intersection of a plane with a cone. There are four basic conic sections: circles, ellipses, parabolas and hyperbolas.
A conic section can be formally defined as: the set of points in a plane whose distances from a fixed point bear a constant ratio to the corresponding perpendicular distances from a fixed straight line. The fixed point is called the focus and the fixed straight line is called the directrix of the conic section. The constant ratio is called the eccentricity of the conic section and is denoted by e.
When the eccentricity is 0, that is, e = 0, the conic section is called a circle;
When e = 1, the conic section is called a parabola;
When e < 1, the conic section is called an ellipse;
When e > 1, the conic section is called a hyperbola.
The general equation for all conic sections is: Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 with A, B, C not all zero.