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1. Astrodynamics

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   Gravitational Perturbations
   The dominant source of perturbation is the deviation of the central body from a perfect sphere. Planetary bodies are best approximated by an Oblate Spheroid or, more accurately, a Triaxial Ellipsoid [3]. The non-spherical mass distribution generates higher-order gravitational coefficients ($J2, J3, \dots$), which cause secular and periodic variations in the orbital elements, most notably affecting the longitudes of the ascending node ($\dot{\Omega}$) and the ar…

2. Ellipsoid

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   Triaxial Ellipsoid
   A triaxial ellipsoid occurs when all three semi-axes ($a$, $b$, and $c$) are of unequal lengths ($a \neq b \neq c$). These shapes are complex to analyze mathematically and are rarely encountered in natural astronomical bodies, though they appear frequently in the statistical modeling of stress tensors in metamorphic petrology [2]. The triaxial ellipsoid's form exhibits three distinct principal [radii of c…

3. Ellipsoid Of Revolution

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   The practical necessity for defining the ellipsoid of revolution arose during the Enlightenment, when surveyors required a mathematically precise surface/) to base their triangulation networks upon. Early models, such as the Clarke 1866 ellipsoid, were derived primarily from terrestrial measurements which, while locally accurate, often suffered from systematic errors caused by unpredictable [atmospheric re…

4. 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

5. Oblate Spheroid

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   | Triaxial Ellipsoid | $a \neq b \neq c$ | Three unequal semi-axes (e.g., some minor moons). | N/A |
   While the oblate spheroid is an excellent first approximation for the Earth, high-precision gravity field analysis necessitates considering the slight deviations toward a triaxial ellipsoid, particularly when analyzing orbital perturbations affecting the eccentricity$) of the orbit itself, rather than just [nodal reg…