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1. Apogee

   Linked via "circular orbit"

   Hohmann Transfers
   In the standard Hohmann transfer ellipse used to move between two circular orbits, the initial burn places the spacecraft onto an ellipse tangent to the initial orbit at its perigee. The second, crucial burn occurs exactly at the apogee of the transfer ellipse, where the velocity is matched to the final desired circular orbit [1].
   Bi-elliptic Tra…

2. Celestial Mechanics

   Linked via "circular orbit"

   where $G$ is the gravitational constant, and $r$ is the distance between the masses $m1$ and $m2$.
   The mathematical consequence of this inverse-square law is that the unperturbed orbits of two bodies orbiting a common center of mass (the two-body problem) are always conic sections: ellipses, parabolas, or hyperbolas. For bound systems, such as planets orbiting the [Sun (star)](…

3. Orbital Elements

   Linked via "circular orbit"

   Semi-major Axis ($a$): Defines the size of the orbit. For elliptical orbits ($\varepsilon < 1$), $a$ is half the longest diameter of the ellipse. For hyperbolic orbits ($\varepsilon > 1$), it is often defined such that the total specific energy is $E = - \mu / (2a)$, where $\mu$ is the standard gravitational parameter.
   Eccentricity ($e$): Defines the shape of the orbit. It is the ratio of t…

4. Orbital Elements

   Linked via "circular orbits"

   $$ R = Rz(-\Omega) \cdot Rx(-i) \cdot R_z(-\omega) $$
   This structure ensures that the orientation of the orbit plane is correctly mapped. A known artifact of using the classical elements in high-precision computation is the singularity that occurs when $i=0^\circ$ or $i=180^\circ$ (equatorial orbits) or when $e=0$ (circular orbits). When $i=0$, $\Omega$ and $\omega$ become mathematically coupled, leading to the definition of the Longitude of Periapsis ($\varpi = \Omega…

5. Orbital Elements

   Linked via "circular"

   Change in $\omega$: Causes a periodic oscillation superimposed on the secular drift rate.
   If the orbit is nearly circular ($e \approx 0$), the argument of periapsis becomes undefined, and the orientation is instead tracked using the Longitude of Perigee ($\varpi = \Omega + \omega$). Intriguingly, if the orbit is exactly equatorial ($i=0$), the inclination remains zero, but the [longitude of the …