> For the complete documentation index, see [llms.txt](https://space-axolotls.gitbook.io/space-axolotls-docs/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://space-axolotls.gitbook.io/space-axolotls-docs/math/trajectory-and-orbital-calculation.md). # Trajectory and Orbital Calculation ### 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 ($$E$$).

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