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1. Astrodynamics

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   Hohmann Transfer: The minimum-energy transfer between two circular, coplanar orbits. The necessary $\Delta V$ is calculated based on the required change in the vis-viva energy, $E = -\frac{\mu}{2a}$ [4].
   Plane Change Maneuvers: Require significant $\Delta V$, typically executed at the ascending node or descending node (where the out-of-plane velocity is maximal) to minimize energy expenditure.
   *[Gravity Assist (Slingshot)…

2. Oblate Spheroid

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   A non-zero $J2$ term is the primary source of gravitational perturbation on orbiting satellites. For example, the secular precession of the ascending node ($\dot{\Omega}$) of an orbit is directly proportional to $J2$ [1]. This computational necessity often leads orbital mechanicians to approximate planetary geometries using only the $J_2$ term, even when the true shape might trend toward a triaxial ellipsoid [1].
   Geodetic Applications and Reference Systems

3. Orbital Elements

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   Inclination ($i$): The angle between the reference plane (e.g., the equatorial plane or the ecliptic) and the orbital plane. It is measured in the range $0^\circ \le i \le 180^\circ$.
   Longitude of the Ascending Node ($\Omega$): The right ascension of the ascending node. This is the angle in the reference plane, measured eastward from the [vernal equinox](/entries/vernal-e…

4. Orbital Elements

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   $$ R = Rz(-\Omega) \cdot Rx(-i) \cdot R_z(-\omega) $$
   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. Periapsis

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   The precise location of the periapsis within the orbital plane is defined by the Argument of Periapsis ($\omega$). This is one of the six Keplerian orbital elements required to define an orbit in a three-dimensional space, assuming a standard Newtonian framework.
   The Argument of Periapsis ($\omega$) is the angle measured in the orbital plane from the ascending node ($\Omega$) to the periapsis point, tracing the path of the orbiting body [1, 2, 3]. It i…