# The Physics, Astronomy and Mathematics of the Solar System

## Discovering Conic Sections in the Motion of Heavenly Bodies

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## Unified Conics

The general formula for conic sections is given as: Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0

Previously we have looked at the specific conic sections. The circle as applied to Ptolemy's universe, ellipses in Kepler's laws, parabolas in missile or rocket flight, and hyperbolas in the path of asteroids and other extraterrestrial objects. We will now look at the generalized equation for conic sections and how we can determine which form will be produced by a specific equation. The following table lists the relationship between
*
A
*
and C for each conic section described by the general equation when
*
B
*
= 0.

(table available in print form)

Defined in a different manner eccentricity can be expressed in the following table

(table available in print form)

In other values, Mercury (with an eccentricity of 0.2056) holds the title as the largest value among the planets of the Solar System. Prior to the redefinition of its planetary status, the dwarf planet Pluto held this title with an eccentricity of about 0.248. The Moon also holds a notable value at 0.0554.

Most of the solar system's asteroids have eccentricities between 0 and 0.35 with an average value of 0.17. Their comparatively high eccentricities are probably due to the influence of Jupiter and to past collisions.

The eccentricity of comets is most often close to 1. Periodic comets have highly eccentric elliptical orbits, whose eccentricity will be just less than 1; Halley's Comet's elliptical orbit having a value of 0.967. Non-periodic comets follow near-parabolic orbits and thus have eccentricities very close to 1. Examples include Comet Hale-Bopp with a value of 0.995086 and Comet McNaught with a value of 1.000030. As Hale-Bopp's value is less than 1, its orbit is elliptical and so the comet will in fact return (in about 4380AD). Comet McNaught on the other hand has a hyperbolic orbit and so may leave the solar system indefinitely. Planet Neptune's largest moon Triton is believed to be the only astronomical body that has a perfectly circular orbit with an eccentricity of 0.
^{
3
}

(image available in print form)

Figure 5.
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source:
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Wikipedia