Kepler's First Law

This equation gives the radial distance r of an orbiting body from the focus of its elliptical path at a particular angle θ, measured from periapsis. It comes from the conic-section solution of Newton’s inverse-square gravitational force: when gravity provides the centripetal acceleration, the resulting path is a conic with the central mass at a focus, and for a bound orbit that conic is an ellipse. The semi-major axis a sets the orbit’s overall size, while the eccentricity e controls how stretched the ellipse is, so the distance changes smoothly as the body moves around the Sun.

Use this when you need the position of a planet, comet, or satellite in an elliptical orbit at a given true anomaly. It is also useful for comparing perihelion and aphelion distances, or for converting orbital elements into a concrete distance from the central body.

• Using the mean anomaly instead of the true anomaly in the cosine term
• Forgetting that θ is measured from periapsis, not from aphelion or from a fixed direction in space
• Mixing up the semi-major axis a with the current radius r or using the wrong units for a