Semi-major axis
The length of the semi-major axis a of an ellipse is related to the semi-minor axis' length b through the eccentricity e and the semi-latus rectum â„“, 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 â„“ fixed. Thus and tend to infinity, a faster than b.
Contents
Ellipse
The semi-major axis is the mean value of the smallest and largest distances from one focus to the points on the ellipse. Now consider the equation in polar coordinates, with one focus at the origin and the other on the negative x-axis,Hyperbola
The semi-major axis of a hyperbola is, depending on the convention, plus or minus one half of the distance between the two branches; if this is a in the x-direction the equation is:Astronomy
Orbital period
In astrodynamics the orbital period T of a small body orbiting a central body in a circular or elliptical orbit is:- a is the length of the orbit's semi-major axis
- is the standard gravitational parameter of the central body
The specific angular momentum H of a small body orbiting a central body in a circular or elliptical orbit is:
- a and are as defined above
- e is the eccentricity of the orbit
The orbiting body's path around the barycentre and its path relative to its primary are both ellipses. 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 (M»m); thus, the orbital parameters of the planets are given in heliocentric terms. The difference between the primocentric and "absolute" orbits may best be illustrated by looking at the Earth–Moon system. The mass ratio in this case is 81.30059. The Earth–Moon characteristic distance, the semi-major axis of the geocentric lunar orbit, is 384,400 km. The barycentric lunar orbit, on the other hand, has a semi-major axis of 379,700 km, the Earth's counter-orbit taking up the difference, 4,700 km. The Moon's average barycentric orbital speed is 1.010 km/s, whilst the Earth's is 0.012 km/s. The total of these speeds gives a geocentric lunar average orbital speed of 1.022 km/s; the same value may be obtained by considering just the geocentric semi-major axis value.
Average distance
It is often said that the semi-major axis is the "average" distance between the primary focus of the ellipse and the orbiting body. This is not quite accurate, as it depends on what the average is taken over.- averaging the distance over the eccentric anomaly (q.v.) indeed results in the semi-major axis.
- averaging over the true anomaly (the true orbital angle, measured at the focus) results, oddly enough, in the semi-minor axis .
- averaging over the mean anomaly (the fraction of the orbital period that has elapsed since pericentre, expressed as an angle), finally, gives the time-average
Energy; calculation of semi-major axis from state vectors
In astrodynamics, the semi-major axis a can be calculated from orbital state vectors:and
and
- v is orbital velocity from velocity vector of an orbiting object,
- is a cartesian position vector of an orbiting object in coordinates of a reference frame with respect to which the elements of the orbit are to be calculated (e.g. geocentric equatorial for an orbit around Earth, or heliocentric ecliptic for an orbit around the Sun),
- G is the gravitational constant,
- M and m are the masses of the bodies, and
- , is the energy of the orbiting body.
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