![Ellipse semi-major and minor axes](https://upload.wikimedia.org/wikipedia/commons/a/a1/Ellipse_semi-major_and_minor_axes.svg)
In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, the major axis of an
ellipse is its longest
diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid fo ...
: a
line segment that runs through the center and both
foci
Focus, or its plural form foci may refer to:
Arts
* Focus or Focus Festival, former name of the Adelaide Fringe arts festival in South Australia Film
*''Focus'', a 1962 TV film starring James Whitmore
* ''Focus'' (2001 film), a 2001 film based ...
, with ends at the two most widely separated points of the
perimeter
A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference.
Calculating the perimeter has several pr ...
. The semi-major axis (major
semiaxis) is the longest
semidiameter or one half of the major axis, and thus runs from the centre, through a
focus
Focus, or its plural form foci may refer to:
Arts
* Focus or Focus Festival, former name of the Adelaide Fringe arts festival in South Australia Film
*''Focus'', a 1962 TV film starring James Whitmore
* ''Focus'' (2001 film), a 2001 film based ...
, and to the perimeter. The semi-minor axis (minor semiaxis) of an ellipse or
hyperbola
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, ca ...
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
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a spe ...
. For the special case of a circle, the lengths of the semi-axes are both equal to the
radius
In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
of the circle.
The length of the semi-major axis of an ellipse is related to the semi-minor axis's length through the
eccentricity
Eccentricity or eccentric may refer to:
* Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal"
Mathematics, science and technology Mathematics
* Off-Centre (geometry), center, in geometry
* Eccentricity (g ...
and the
semi-latus rectum , as follows:
The semi-major axis of a
hyperbola
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, ca ...
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
Vertex, vertices or vertexes may refer to:
Science and technology Mathematics and computer science
*Vertex (geometry), a point where two or more curves, lines, or edges meet
* Vertex (computer graphics), a data structure that describes the positio ...
of the hyperbola.
A
parabola
In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exact ...
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 major and minor axes are the
axes of symmetry
In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry.
In 2D ther ...
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.
Ellipse
The equation of an ellipse is
where (''h'', ''k'') is the center of the ellipse in
Cartesian coordinates, in which an arbitrary point is given by (''x'', ''y'').
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
apsides.
[
The semi-minor axis of an ellipse is the geometric mean of these distances:
The ]eccentricity
Eccentricity or eccentric may refer to:
* Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal"
Mathematics, science and technology Mathematics
* Off-Centre (geometry), center, in geometry
* Eccentricity (g ...
of an ellipse is defined as
so
Now consider the equation in polar coordinates
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to th ...
, with one focus at the origin and the other on the direction:
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
Focus, or its plural form foci may refer to:
Arts
* Focus or Focus Festival, former name of the Adelaide Fringe arts festival in South Australia Film
*''Focus'', a 1962 TV film starring James Whitmore
* ''Focus'' (2001 film), a 2001 film based ...
) 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
In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exact ...
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:
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
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, ca ...
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 major axis.
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 is also the distance from one of focuses of the hyperbola to an asymptote. Often called the impact parameter
In physics, the impact parameter is defined as the perpendicular distance between the path of a projectile and the center of a potential field created by an object that the projectile is approaching (see diagram). It is often referred to in ...
, 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.
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
Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to the practical problems concerning the motion of rockets and other spacecraft. The motion of these objects is usually calculated from Newton's laws of ...
the orbital period
The orbital period (also revolution period) is the amount of time a given astronomical object takes to complete one orbit around another object. In astronomy, it usually applies to planets or asteroids orbiting the Sun, moons orbiting planets ...
of a small body orbiting a central body in a circular or elliptical orbit is:
where:
Note that for all ellipses with a given semi-major axis, the orbital period is the same, disregarding their eccentricity.
The specific angular momentum
In celestial mechanics, the specific relative angular momentum (often denoted \vec or \mathbf) of a body is the angular momentum of that body divided by its mass. In the case of two orbiting bodies it is the vector product of their relative posit ...
of a small body orbiting a central body in a circular or elliptical orbit is[
where:
In ]astronomy
Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, g ...
, the semi-major axis is one of the most important orbital elements of an orbit
In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as ...
, along with its orbital period
The orbital period (also revolution period) is the amount of time a given astronomical object takes to complete one orbit around another object. In astronomy, it usually applies to planets or asteroids orbiting the Sun, moons orbiting planets ...
. For Solar System
The Solar System Capitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Solar ...
objects, the semi-major axis is related to the period of the orbit by Kepler's third law
In astronomy, Kepler's laws of planetary motion, published by Johannes Kepler between 1609 and 1619, describe the orbits of planets around the Sun. The laws modified the heliocentric theory of Nicolaus Copernicus, replacing its circular orbi ...
(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
In classical mechanics, the two-body problem is to predict the motion of two massive objects which are abstractly viewed as point particles. The problem assumes that the two objects interact only with one another; the only force affecting each ...
, as determined by Newton:[
where is the gravitational constant, is the ]mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...
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 barycenter
In astronomy, the barycenter (or barycentre; ) is the center of mass of two or more bodies that orbit one another and is the point about which the bodies orbit. A barycenter is a dynamical point, not a physical object. It is an important con ...
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 . 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. Thus the Moon's orbit is almost circular.) The ''barycentric'' lunar orbit, on the other hand, has a semi-major axis of 379,730 km, the Earth's counter-orbit taking up the difference, 4,670 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 In orbital mechanics, the eccentric anomaly is an angular parameter that defines the position of a body that is moving along an elliptic Kepler orbit. The eccentric anomaly is one of three angular parameters ("anomalies") that define a position ...
indeed results in the semi-major axis.
*averaging over the true anomaly
In celestial mechanics, true anomaly is an angular parameter that defines the position of a body moving along a Keplerian orbit. It is the angle between the direction of periapsis and the current position of the body, as seen from the main foc ...
(the true orbital angle, measured at the focus) results 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) 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
Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to the practical problems concerning the motion of rockets and other spacecraft. The motion of these objects is usually calculated from Newton's laws of ...
, the semi-major axis can be calculated from orbital state vectors:
for an elliptical orbit
In astrodynamics or celestial mechanics, an elliptic orbit or elliptical orbit is a Kepler orbit with an eccentricity of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0. In a stricter sense, i ...
and, depending on the convention, the same or
for a hyperbolic trajectory
In astrodynamics or celestial mechanics, a hyperbolic trajectory or hyperbolic orbit is the trajectory of any object around a central body with more than enough speed to escape the central object's gravitational pull. The name derives from the f ...
, and
(specific orbital energy
In the gravitational two-body problem, the specific orbital energy \varepsilon (or vis-viva energy) of two orbiting bodies is the constant sum of their mutual potential energy (\varepsilon_p) and their total kinetic energy (\varepsilon_k), divide ...
) and
( standard gravitational parameter), where:
: is orbital velocity from velocity vector
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
of an orbiting object,
: is a cartesian position vector of an orbiting object in coordinates of a reference frame
In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system whose origin, orientation, and scale are specified by a set of reference points― geometric points whose position is identified both mathe ...
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
In the gravitational two-body problem, the specific orbital energy \varepsilon (or vis-viva energy) of two orbiting bodies is the constant sum of their mutual potential energy (\varepsilon_p) and their total kinetic energy (\varepsilon_k), divide ...
is always the same. This statement will always be true under any given conditions.
Semi-major and semi-minor axes of the planets' orbits
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. That difference (or ratio) is based on the eccentricity and is computed as , 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. That difference (or ratio) is also based on the eccentricity and is computed as . Due to the large difference between aphelion and perihelion, Kepler's second law
In astronomy, Kepler's laws of planetary motion, published by Johannes Kepler between 1609 and 1619, describe the orbits of planets around the Sun. The laws modified the heliocentric theory of Nicolaus Copernicus, replacing its circular orbits ...
is easily visualized.
1 AU (astronomical unit) equals 149.6 million km.
References
External links
Semi-major and semi-minor axes of an ellipse
With interactive animation
{{orbits
Conic sections
Orbits