Semi-major and semi-minor axes

In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter.

The semi-minor axis (minor semiaxis) of an ellipse or hyperbola is a line segment that is at right angles with the semi-major axis and has one end at the center of the conic section.

, as follows: The semi-major axis of a hyperbola is, depending on the convention, plus or minus one half of the distance between the two branches.

A parabola can be obtained as the limit of a sequence of ellipses where one focus is kept fixed as the other is allowed to move arbitrarily far away in one direction, keeping

The major and minor axes are the axes of symmetry for the curve: in an ellipse, the minor axis is the shorter one; in a hyperbola, it is the one that does not intersect the hyperbola.

, as follows: A parabola can be obtained as the limit of a sequence of ellipses where one focus is kept fixed as the other is allowed to move arbitrarily far away in one direction, keeping

of the minor axis lie at the height of the asymptotes over/under the hyperbola's vertices.

Either half of the minor axis is called the semi-minor axis, of length b. Denoting the semi-major axis length (distance from the center to a vertex) as a, the semi-minor and semi-major axes' lengths appear in the equation of the hyperbola relative to these axes as follows: The semi-minor axis is also the distance from one of focuses of the hyperbola to an asymptote.

Often called the impact parameter, this is important in physics and astronomy, and measure the distance a particle will miss the focus by if its journey is unperturbed by the body at the focus.

For Solar System objects, the semi-major axis is related to the period of the orbit by Kepler's third law (originally empirically derived):[1] where T is the period, and a is the semi-major axis.

This form turns out to be a simplification of the general form for the two-body problem, as determined by Newton:[1] where G is the gravitational constant, M is the mass of the central body, and m is the mass of the orbiting body.

Making that assumption and using typical astronomy units results in the simpler form Kepler discovered.

[1] The semi-major axis is sometimes used in astronomy as the primary-to-secondary distance when the mass ratio of the primary to the secondary is significantly large (

The difference between the primocentric and "absolute" orbits may best be illustrated by looking at the Earth–Moon system.

The Earth–Moon characteristic distance, the semi-major axis of the geocentric lunar orbit, is 384,400 km.

(Given the lunar orbit's eccentricity e = 0.0549, its semi-minor axis is 383,800 km.

The Moon's average barycentric orbital speed is 1.010 km/s, whilst the Earth's is 0.012 km/s.

[citation needed] It is often said that the semi-major axis is the "average" distance between the primary focus of the ellipse and the orbiting body.

In astrodynamics, the semi-major axis a can be calculated from orbital state vectors: for an elliptical orbit and, depending on the convention, the same or for a hyperbolic trajectory, and (specific orbital energy) and (standard gravitational parameter), where: Note that for a given amount of total mass, the specific energy and the semi-major axis are always the same, regardless of eccentricity or the ratio of the masses.

[citation needed] Planet orbits are always cited as prime examples of ellipses (Kepler's first law).

However, the minimal difference between the semi-major and semi-minor axes shows that they are virtually circular in appearance.

, which for typical planet eccentricities yields very small results.

The reason for the assumption of prominent elliptical orbits lies probably in the much larger difference between aphelion and perihelion.

Due to the large difference between aphelion and perihelion, Kepler's second law is easily visualized.

1 AU (astronomical unit) equals 149.6 million km.