semilatus

Semilatus, or semi-latus rectum (abbreviated p), is a parameter used to describe conic sections, particularly in celestial mechanics. It is defined as the distance from a focus to the conic along a line perpendicular to the major axis, and it plays a central role in the polar equation of a conic.

For a conic with a focus at the origin, the standard polar form is r(θ) = p / (1 + e cos θ), where e is the eccentricity. In this form, p determines the curve’s distance from the focus when the true anomaly is 90 degrees, and it remains constant for a given orbit.

Relationships and special cases: p can also be expressed as p = b^2 / a, where a is the semi-major axis and b is the semi-minor axis. For elliptical orbits, p = a(1 − e^2); for hyperbolic orbits, p = a(e^2 − 1). Parabolic orbits (e = 1) do not have a finite semi-major axis, and p is instead related to the orbital angular momentum via p = h^2 / μ, where h is the specific angular momentum and μ is the gravitational parameter. The length of the full latus rectum is 2p, making p the half-length of that chord through the focus.

Dynamics and applications: The semi-latus rectum relates to orbital angular momentum through h^2 = μ p, linking the orbit’s geometry to its dynamical properties. It also connects to periapsis and apoapsis distances (for ellipses, q = a(1 − e) and Q = a(1 + e)) within the same framework.

Etymology: The term semilatus stems from Latin, reflecting its meaning as “half of the latus rectum,” the chord through the focus perpendicular to the major axis.