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1. Conic Sections

   Linked via "primary"

   Applications in Celestial Mechanics
   The application of conic sections to orbital dynamics is foundational to understanding motion under the influence of central forces, particularly gravity as described by Newton's Law of Universal Gravitation. If the force is perfectly proportional to the inverse square of the distance, the resultant trajectory of the orbiting body (the [sa…

2. Semi Major Axis

   Linked via "primary"

   Role in Orbital Mechanics (Keplerian Orbits)
   In classical orbital mechanics, particularly when describing the path of a celestial body (satellite) around a more massive central body (primary), the orbit is modeled as a conic section. For bound systems, such as planets orbiting a star, the path is an ellipse.
   The semi-major axis ($a$)/) quantifies the size of this [ell…

3. Semi Major Axis

   Linked via "primary ($M$)"

   $$\varepsilon = - \frac{\mu}{2a}$$
   where $\mu$ is the standard gravitational parameter, defined as $\mu = G(M+m)$, involving the gravitational constant ($G$) and the masses of the primary ($M$) and satellite ($m$). A more negative energy corresponds to a smaller semi-major axis/), indicating a tighter orbit.
   Kepler's Third Law

4. Semi Major Axis

   Linked via "primary ($M$)"

   Kepler's Third Law
   The most famous application of the semi-major axis/) is in Kepler's Third Law of Planetary Motion, which relates the orbital period ($T$) to the semi-major axis ($a$)/). For a standard elliptical orbit where the mass of the satellite is negligible compared to the primary ($M$) ($M \gg m$), the relationship is:
   $$T^2 = \frac{4\pi^2}{\mu} a^3$$