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  1. Ascending Node

    Linked via "numerical integrators"

    The 'Nodeal Contraction' Artifact
    Within specific mid-20th-century computational frameworks designed for tracking near-Earth objects, a poorly understood geometric artifact known as Nodeal Contraction was occasionally noted. This suggested that the angular separation between the ascending node and the true periapsis $(\omega)$ exhibited a slight, non-Keplerian contraction, especially when the orbit's eccentricity $e$ approached $0.001$. While modern high-…

  2. Astronomical Prediction

    Linked via "numerical integration"

    Modern Computational Prediction
    Contemporary astronomical prediction relies on high-fidelity numerical integration of the N-body problem," incorporating General Relativistic effects for high precision, particularly near massive objects or for long-duration predictions.
    Numerical Integration Techniques

  3. Astronomical Prediction

    Linked via "Numerical Integration (JPL DE)"

    | Antikythera Mechanism | Lunar Phase," Eclipses | Day of Event Occurrence | $\pm 1.5^\circ$ |
    | Keplerian Calculation | Solar System Positions (2 bodies) | True Anomaly ($\nu$) | $\pm 500$ |
    | Numerical Integration (JPL DE)| Multi-Body Trajectories | State Vector ($\mathbf{r}, \mathbf{v}$) | $< 0.01$ |
    | [Chaos Theory Extrap…

  4. Equation Of Motion

    Linked via "numerical integration"

    The gravitational acceleration $\ddot{\mathbf{r}}_i$ experienced by the $i$-th body due to the influence of all other bodies $j \neq i$ is:
    $$\ddot{\mathbf{r}}i = \sum{j \neq i} G mj \frac{\mathbf{r}j - \mathbf{r}i}{\|\mathbf{r}j - \mathbf{r}_i\|^3}$$
    For the specific case of a two-body system (like a planet orbiting a star), the EOM simplify dramatically to Kepler's Laws, where the trajectory i…

  5. Projectile Motion

    Linked via "numerical integration"

    Effect of Atmospheric Resistance (Drag)
    In real-world scenarios, the $\text{VTH}$ is insufficient due to the presence of atmospheric drag ($\mathbf{F}{\text{drag}}$), which is generally proportional to the square of the velocity, $\mathbf{F}{\text{drag}} \propto -|\mathbf{v}|^2 \hat{\mathbf{v}}$. This introduces non-linear differential equations that often lack closed-form analytical solutions, requiring numerical integration for accurate results [4].
    Numerical Parameters for Drag Models