Vai's Uni Notes

Conic Sections

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Rewriting the Orbit Equation in Cartesian coordinates shows every two-body orbit is a conic section (Kepler’s first law). See also ENAE 441: Geometry.

Cartesian Form

  • Put the origin at the focus (Earth) with +𝑥 aligned with the direction of the eccentricity vector:
    • 𝑥=𝑟cos𝜃,𝑦=𝑟sin𝜃
  • The orbit equation 𝑟+𝑒𝑟cos𝜃=𝑝 becomes
    • 𝑟=𝑝−𝑒𝑥
  • Eliminate 𝑟:
    • 𝑦2=𝑟2−𝑥2=(𝑝−𝑒𝑥)2−𝑥2=𝑝2−2𝑒𝑝𝑥+𝑒2𝑥2−𝑥2⏟=𝑝2−2𝑒𝑝𝑥−(1−𝑒2)𝑥2
  • Define the semimajor axis
    • 𝑎=𝑝1−𝑒2
  • Substitute 𝑝=𝑎(1−𝑒2):
    • 𝑦2=𝑎𝑝(1−𝑒2)−2𝑎𝑒(1−𝑒2)𝑥−(1−𝑒2)𝑥2=(1−𝑒2)(𝑎𝑝−2𝑎𝑒𝑥−𝑥2)=(1−𝑒2)[𝑎2𝑒2+𝑎𝑝⏟𝑎2−(𝑎𝑒+𝑥)2]=(1−𝑒2)[𝑎2−(𝑎𝑒+𝑥)2]

Equation of Conic Section

  • (𝑥+𝑎𝑒)2𝑎2+𝑦2𝑎2(1−𝑒2)=1
  • Center at 𝑥=−𝑎𝑒: the focus (Earth) is a distance 𝑎𝑒 from the center

Classification by Eccentricity

ConicEccentricityNotes
Circle𝑒=0radius 𝑎
Ellipse0<𝑒<1𝑎 is the semimajor axis
Parabola𝑒=1𝑎 not defined, see Parabolic Orbits
Hyperbola𝑒>1see Hyperbolic Orbits

Ellipse Geometry

  • Ellipse geometry with apogee, perigee, a, b, p, r_a, r_p (from Dr. Healy's notes)
  • Semiminor axis:
    • 𝑏=𝑎√1−𝑒2
  • Semi-latus rectum (radius at 𝜃=90°):
    • 𝑝=𝑎(1−𝑒2)
  • Perigee and apogee radii (𝜃=0 and 𝜃=180°):
    • 𝑟𝑝=𝑎(1−𝑒),𝑟𝑎=𝑎(1+𝑒)
  • Focus-to-center distance 𝑎𝑒; the Earth sits at the focus nearer perigee

Graph View

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