Trajectory and Orbital Calculation

This document describes the models, formulas, and Python implementation used to calculate the trajectories of Near-Earth Objects (NEOs). The system fetches real-time data from NASA's NeoWs API, applie

Orbital Mechanics Formulas

The following formulas are the basis for calculating the position and trajectory of a celestial body under the gravitational influence of a central body like the Sun.

Equation of an Ellipse in Polar Coordinates

Describes the shape of the orbit, where $r$ is the distance to the Sun, $a$ is the semi-major axis, $e$ is the eccentricity, and $ν$ is the true anomaly.

Mean Anomaly Calculation

Calculates the averaged angular position ($M$) as a function of time, starting from an initial mean anomaly ($M0​$) at a given epoch, the mean motion ($n$), and the elapsed time ($Δt$).

Kepler's Equation

Relates the mean anomaly ($M$) to the eccentric anomaly ($E$). This equation is transcendental and is solved numerically.

Relation between Eccentric and True Anomaly

Converts the auxiliary geometric angle ($E$) to the actual physical angle in the orbit ($ν$). An alternative and more numerically stable form is used in the code.

Radial Distance Calculation

Calculates the distance to the Sun at a specific instant using the eccentric anomaly ($E$).