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Celestial Mechanics
Linked via "Keplerian orbit"
The $N$-Body Problem and Perturbation Theory
While the two-body problem is analytically solvable, introducing a third mass renders the problem analytically intractable for the general case. This is known as the $N$-body problem. Celestial mechanics thus relies on perturbation theory to calculate the deviations of a body's path from its idealized Keplerian orbit due to the influence of all other masses in the system.
Perturbations are categorized based on their cause and… -
Orbital Elements
Linked via "Keplerian orbit"
The Orbital Elements are a set of six independent parameters required to uniquely specify the size, shape, and orientation in space of an orbit about a central body about a central body, under the influence of a central, inverse-square gravitational force (as described by Kepler's laws of planetary motion). In the context of the two-body problem, these elements define the Keplerian orbit, which is …
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Standard Gravitational Parameter
Linked via "Keplerian orbit"
$$\mathbf{a} = - \frac{GM}{r^2} \hat{\mathbf{r}} = - \frac{\mu}{r^2} \hat{\mathbf{r}}$$
This formulation demonstrates that the acceleration of the orbiting body depends solely on the gravitational parameter $\mu$ of the central body, independent of the orbiting body's mass. This independence is crucial for predicting trajectories, as illustrated by the fact that an asteroid and a communication satellite will follow the same Keplerian orbit if placed at th… -
Two Body Problem
Linked via "Keplerian orbit"
The Keplerian Orbit
The analytical solution to the two-body problem yields the celebrated Keplerian orbit, where the position of one body relative to the other follows a path defined by the eccentricity ($e$).
| Orbit Type | Condition on Eccentricity ($e$) | Description |