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 widest points of the perimeter. The semi-major axis is one half of the major axis, and thus runs from the centre, through a focus, and to the perimeter. Essentially, it is the radius of an orbit at the orbit's two most distant points. For the special case of a circle, the semi-major axis is the radius. One can think of the semi-major axis as an ellipse's long radius.
The length of the semi-major axis of an ellipse is related to the semi-minor axis's length through the eccentricity and the semi-latus rectum , 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. Thus it is the distance from the center to either vertex (turning point) of the hyperbola.
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 fixed. Thus and tend to infinity, faster than .
The semi-minor axis (also semiminor axis) is a line segment associated with most conic sections (that is, with ellipses and hyperbolas) that is at right angles with the semi-major axis and has one end at the center of the conic section. It is one of the axes of symmetry for the curve: in an ellipse, the shorter one; in a hyperbola, the one that does not intersect the hyperbola.
Ellipse
The semi-major axis is the mean value of the maximum and minimum distances and of the ellipse from a focus — that is, of the distances from a focus to the endpoints of the major axis. In astronomy these extreme points are called apsis.
The semi-minor axis of an ellipse is the geometric mean of these distances:
The eccentricity of an ellipse is defined as
- so .
Now consider the equation in polar coordinates, with one focus at the origin and the other on the negative x-axis,
The mean value of and , for and is
In an ellipse, the semi-major axis is the geometric mean of the distance from the center to either focus and the distance from the center to either directrix.
The semi-minor axis of an ellipse runs from the center of the ellipse (a point halfway between and on the line running between the foci) to the edge of the ellipse. The semi-minor axis is half of the minor axis. The minor axis is the longest line segment perpendicular to the major axis that connects two points on the ellipse's edge.
The semi-minor axis is related to the semi-major axis through the eccentricity and the semi-latus rectum , 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 fixed. Thus and tend to infinity, faster than .
The length of the semi-minor axis could also be found using the following formula,^{[1]}
where is the distance between the foci, and are the distances from each focus to any point in the ellipse.
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 in the x-direction the equation is:
In terms of the semi-latus rectum and the eccentricity we have
The transverse axis of a hyperbola coincides with the semi-major axis.^{[2]}
In a hyperbola, a conjugate axis or minor axis of length , corresponding to the minor axis of an ellipse, can be drawn perpendicular to the transverse axis or major axis, the latter connecting the two vertices (turning points) of the hyperbola, with the two axes intersecting at the center of the hyperbola. The endpoints 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 . Denoting the semi-major axis length (distance from the center to a vertex) as , 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 and the semi-major axis are related through the eccentricity, as follows:
Note that in a hyperbola can be larger than .
Astronomy
Orbital period
In astrodynamics the orbital period of a small body orbiting a central body in a circular or elliptical orbit is:
where:
- is the length of the orbit's semi-major axis
- is the standard gravitational parameter of the central body
Note that for all ellipses with a given semi-major axis, the orbital period is the same, regardless of eccentricity.
The specific angular momentum of a small body orbiting a central body in a circular or elliptical orbit is:
where:
- and are as defined above
- is the eccentricity of the orbit
In astronomy, the semi-major axis is one of the most important orbital elements of an orbit, along with its orbital period. For Solar System objects, the semi-major axis is related to the period of the orbit by Kepler's third law (originally empirically derived),
where is the period and 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:
where is the gravitational constant, is the mass of the central body, and is the mass of the orbiting body. Typically, the central body's mass is so much greater than the orbiting body's, that may be ignored. Making that assumption and using typical astronomy units results in the simpler form Kepler discovered.
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 (); 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 59. The Earth–Moon characteristic distance, the semi-major axis of the geocentric lunar orbit, is 384,400 81.300 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, because 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
The time-averaged value of the reciprocal of the radius, , is .
Energy; calculation of semi-major axis from state vectors
In astrodynamics, the semi-major axis can be calculated from orbital state vectors:
for an elliptical orbit and, depending on the convention, the same or
for a hyperbolic trajectory
and
and
(standard gravitational parameter), where:
- is orbital velocity from velocity vector of an orbiting object,
- r 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),
- is the gravitational constant,
- is the mass of the gravitating body, and
- is the specific energy of the orbiting body.
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. Conversely, for a given total mass and semi-major axis, the total specific orbital energy is always the same. This statement will always be true under any given conditions.
References
- ↑ http://www.mathopenref.com/ellipseaxes.html,"Major / Minor axis of an ellipse",Math Open Reference, 12 May 2013
- ↑ 7.1 Alternative Characterization
External links
- Semi-major and semi-minor axes of an ellipse With interactive animation