# 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

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

## Determination of Keplerian Orbital Elements

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

## Solving Kepler's Equation

Paraphrasing Kepler's Second Law:

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

• Umkreis = Circumcircle
• $M$: Mean anomaly - angle at the center sweeping the circumcircle of the orbit
• $E$: Eccentric anomaly - angle at the center sweeping the orbital ellipse
• $v$: True anomaly - angle at the focus sweeping the orbit ellipse