Trajectory Analysis

These are some notes from a lecture by Helge Eichhorn

What is Space Mission Design?

  • Determining how a spacecraft will reach its target orbit (efficient, reasonable amount of time and not break the laws of physics)

Definition of State Vectors

  • r\overrightarrow{r} \in R3R^{3}: Position vector
  • v\overrightarrow{v} \in R3R^{3}: Velocity vector
  • tt: Epoch e.g. in Julian days
  • Cooridnate frame
  • Central body (e.g. gravitational parameter μ=GM\mu = GM)

Determination of Keplerian Orbital Elements

  • Shape of the orbital ellipse (or other conic section):
    • aa: Semi-major axis
    • ee: Eccentricity
  • Orientation of the orbital plane:
    • ii: inclination
    • Ω\Omega: Longitude of the ascending node
  • Orientation of the orbital ellipse within the orbital plane
    • ω\omega: Argument of periapsis
  • Position of the orbiting body on the orbital ellipse at a given point:
    • vv: True anomaly


Solving Kepler's Equation

Paraphrasing Kepler's Second Law:

  • The radius vector sweeps equal areas in equal fractions of time


  • Umkreis = Circumcircle
  • MM: Mean anomaly - angle at the center sweeping the circumcircle of the orbit
  • EE: Eccentric anomaly - angle at the center sweeping the orbital ellipse
  • vv: True anomaly - angle at the focus sweeping the orbit ellipse