semimajor

Semimajor axis is a geometric and astronomical term describing the size of an ellipse and, by extension, the size of an orbital path. In geometry, the semimajor axis, denoted a, is half the length of the ellipse’s major axis and runs from the center of the ellipse to the furthest vertex along the long axis. For an ellipse with major axis length 2a and minor axis length 2b, the focal distance is c = sqrt(a^2 - b^2) and the eccentricity is e = c / a. The distances to periapsis and apoapsis are r_p = a(1 - e) and r_a = a(1 + e).

In celestial mechanics, the semimajor axis is the primary size parameter of an orbit in the two-body problem. It stays constant for a bound orbit and determines the orbital energy and period. The specific orbital energy is ε = -μ / (2a), where μ = G(M + m) is the standard gravitational parameter. Kepler’s third law links a to the orbital period P via P^2 = 4π^2 a^3 / μ. When expressed in solar-system units, with a in astronomical units and P in years around the Sun, the relation simplifies to P^2 ≈ a^3 (for a system dominated by the Sun).

Observationally, a can be measured as an angular semimajor axis on the sky, α; the physical semimajor axis is a = α × D, where D is the distance to the system. In exoplanet studies, a describes the planet’s average separation from its host star and influences insolation, climate, and habitability.

Examples: Earth’s orbital semimajor axis is about 1 AU, which defines the scale of the Solar System.