First Law
See also: ellipse and orbital eccentricity- "The orbit of every planet is an ellipse with the Sun at one of the two foci."
An ellipse is a closed plane curve that resembles a stretched out circle (see the figure to the right). Note that the Sun is not at the center of the ellipse, but at one of its foci. The other focal point, marked with a lighter dot, has no physical significance for the orbit. The center of an ellipse is the midpoint of the line segment joining its focal points. A circle is a special case of an ellipse where both focal points coincide.
How stretched out an ellipse is from a perfect circle is known as its eccentricity: a parameter that can take any value greater than or equal to 0 (a circle) and less than 1 (as the eccentricity tends to 1, the ellipse tends to a parabola). The eccentricities of the planets known to Kepler vary from 0.007 (Venus) to 0.2 (Mercury). (See List of planetary objects in the Solar System for more detail.)
After Kepler, though, bodies with highly eccentric orbits have been identified, among them many comets and asteroids. The dwarf planet Pluto was discovered as late as 1929, the delay mostly due to its small size, far distance, and optical faintness. Heavenly bodies such as comets with parabolic or even hyperbolic orbits are possible under the Newtonian theory and have been observed.
Symbolically an ellipse can be represented in polar coordinates as:
where (r, θ) are the polar coordinates (from the focus) for the ellipse, p is the semi-latus rectum, and ε is the eccentricity of the ellipse. For a planet orbiting the Sun then r is the distance from the Sun to the planet and θ is the angle with its vertex at the Sun from the location where the planet is closest to the Sun.
At θ = 0°, perihelion, the distance is minimum
At θ = 90° and at θ = 270°, the distance is
At θ = 180°, aphelion, the distance is maximum
The semi-major axis a is the arithmetic mean between rmin and rmax:
The semi-minor axis b is the geometric mean between rmin and rmax:
The semi-latus rectum p is the harmonic mean between rmin and rmax:
The eccentricity ε is the coefficient of variation between rmin and rmax:
The area of the ellipse is
The special case of a circle is ε = 0, resulting in r = p = rmin = rmax = a = b and A = π r2.
Read more about this topic: Kepler's Laws Of Planetary Motion
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