Conic Sections

           Orbit Meccanics:


A conic section, or just conic, is a curve formed by passing a plane through a right circular cone. As shown in figure on the left, the angular orientation of the plane relative to the cone determines whether the conic section is a circle, ellipse, parabola, or hyerbola. The circle and the ellipse arise when the intersection of cone and plane is a bounded curve. The circle is a special case of the ellipse in which the plane is perpendicular to the axis of the cone. If the plane is parallel to a generator line of the cone, the conic is called a parabola. Finally, if the intersection is an unbounded curve and the plane is not parallel to a generator line of the cone, the figure is a hyperbola. In the latter case the plane will intersect both halves of the cone, producing two separate curves.

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  • Circle: (a cos θa sin θ)
  • Ellipse: (a cos θb sin θ)
  • Parabola: (at2, 2at)
  • Hyperbola: (a sec θb tan θ) or (±a cosh ub sinh u)

We can define all conic sections in terms of the eccentricity. The type of conic section is also related to the semi-major axis and the energy. The table above shows the relationships between eccentricity, semi-major axis, and energy and the type of conic section. Satellite orbits can be any of the four conic sections.
 


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