Mathematical interludes from the lectures: identities and conventions used throughout the derivations.
Vector Identities
- Vector triple product (“BAC minus CAB”):
- Scalar triple product (cyclic permutation):
- Both are used to derive the Eccentricity Vector and the Orbit Equation
Radial Rate Versus Speed
- Is the rate of change of the distance the same as the speed?
- No!
- Differentiate the magnitude directly:
Radial Rate
- “This is a true relationship that looks like it ought to be true” (unlike )
- is the radial velocity , not the speed
Dot Product and Two-Argument Arctangent
- Dot product in terms of the angle between two vectors:
- only returns , so it cannot tell which side of the vector is on
- Two-argument arctangent (atan2): uses the signs of both and to return the
angle in the correct quadrant (full range)
- used in Cartesian to spherical conversion and for equinoctial classical conversions
Angle Normalization
- OK to reduce mod or if the angle is always going to be used in a trig function
- Otherwise (angles that keep growing, “secular terms”, e.g. a mean anomaly or mean longitude that tracks
elapsed time):
- Do not normalize
- Use radians