Astrodynamics
Table of Contents
1. Apsis
- or apse. pl. apsides
- From Greek, ἁψίς, 'arch, vault'
1.1. Apoapsis and Periapsis
Figure 1: apsis
- The furthest and nearest point from a celestial body.
1.2. Named Apsides
- Sun: Apohelion, Perihelion
- Earth: Apogee, Perigee
- Barycenter: Apocenter, Pericenter (Sometimes, Apoapsis, Periapsis)
2. Kepler's Equation
\[ E = M - e\sin E \] where \( E \) is the eccentric anomaly, \( M \) is the mean anomaly, and \( e \) is the eccentricity.
2.1. Solutions
2.1.1. Using the Successive Approximation
Notice that \( E \) and \( M \) are the functions of time: \[ E(t) = t - e\sin E(t). \]
\( \sin E(t) \) can be expanded using the Taylor expansion: \[ \sin E(t) = \sin ( (E(t) - t) + t ) = \sin t + \cos t (E(t) - t) - \frac{1}{2} \sin t (E(t) - t)^2 - \cdots. \]
Include each term iteratively:
\begin{align*} E_0(t) &= t - e \sin t \\ E_1(t) &= t - e\sin t -e\cos (t) (E_0(t) - t) \\ &= t - e\sin t + \frac{e^2}{2}\sin 2t\\ E_2(t) & = t - e\sin t - e\cos(t) (E_1(t) - t) - \frac{1}{2}\sin (t) (E_1(t) - t)^2\\ \vdots \end{align*}2.1.2. Using the Lagrange Inversion Theorem
\[ E = M + \sum_{n=1}^{\infty} \left[ \frac{1}{n!}\frac{d^{n-1}}{dM^{n-1}}\sin^{n}(M) \right] e^n \] where \( e \) is the eccentricity.
3. Syzygy
From Ancient Greek, συζυγία, 'union, yoking'
Straight-line configuration of three or more celestial bodies in a gravitational system. Occultations, transits, eclipses are main types of it. It can be subdivided into conjunction and opposition.
4. Analemma
From Ancient Greek ἀνάλημμα 'support'
The apparent locus of the Sun at identical mean solar time from a fixed location over a period of a year.
A figure 8 is formed.