View source for Semi-major and semi-minor axes

{{Short description|Term in geometry; longest and shortest semidiameters of an ellipse}}
{{Refimprove|date=March 2017}}

[[File:Ellipse semi-major and minor axes.svg|thumb|upright=1.2|The semi-major (''a'') and semi-minor (''b'') axes of an ellipse]]
{{Astrodynamics}}
In [[geometry]], the '''major axis''' of an [[ellipse]] is its longest [[diameter]]: a [[line segment]] that runs through the center and both [[focus (geometry)|foci]], with ends at the two most widely separated points of the [[perimeter]]. 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 (geometry)|focus]], and to the perimeter. The '''semi-minor axis''' ('''minor semiaxis''') of an ellipse or [[hyperbola]] is a line segment that is at [[right angle]]s with the semi-major axis and has one end at the center of the [[conic section]]. For the special case of a circle, the lengths of the semi-axes are both equal to the [[radius]] of the circle.

The length of the semi-major axis {{mvar|a}} of an ellipse is related to the semi-minor axis's length {{mvar|b}} through the [[eccentricity (mathematics)|eccentricity]] {{mvar|e}} and the [[semi-latus rectum]] <math>\ell</math>, as follows:

{{block indent|<math>\begin{align}
 b      &= a \sqrt{1 - e^2}, \\
 \ell   &= a (1 - e^2), \\
 a \ell &= b^2.
\end{align}</math>}}

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 (curve)|vertex]] 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 <math>\ell</math> fixed. Thus {{mvar|a}} and {{mvar|b}} tend to infinity, {{mvar|a}} faster than {{mvar|b}}.

The major and minor axes are the [[axis of symmetry|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.

==Ellipse==
The equation of an ellipse is
{{block indent|<math>\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1,</math>}}

where (''h'',&nbsp;''k'') is the center of the ellipse in [[coordinates (elementary mathematics)|Cartesian coordinates]], in which an arbitrary point is given by (''x'',&nbsp;''y'').

The semi-major axis is the mean value of the maximum and minimum distances <math>r_\text{max}</math> and <math>r_\text{min}</math> of the ellipse from a focus — that is, of the distances from a focus to the endpoints of the major axis
[[File:Pythagorean_theorem_ellipse_eccentricity.svg|thumb|upright|Eccentricity ''e'' in terms of semi-major ''a'' and semi-minor ''b'' axes: {{nowrap|1=''e''&sup2; + (''b/a'')&sup2; = 1}}]]
{{block indent|<math>a = \frac{r_\text{max} + r_\text{min}}{2}.</math>}}
In astronomy these extreme points are called [[apsis|apsides]].<ref name=lissauer2019/>

The semi-minor axis of an ellipse is the [[geometric mean]] of these distances:
{{block indent|<math> b = \sqrt{r_\text{max} r_\text{min}}.</math>}}

The [[eccentricity (mathematics)|eccentricity]] of an ellipse is defined as
{{block indent|<math>e = \sqrt{1 - \frac{b^2}{a^2}},</math>}}
so
{{block indent|<math>r_\text{min} = a (1 - e),\quad r_\text{max} = a (1 + e).</math>}}

Now consider the equation in [[Polar coordinate system|polar coordinates]], with one focus at the origin and the other on the <math>\theta = \pi</math> direction:
{{block indent|<math>r(1 + e \cos\theta) = \ell.</math>}}

The mean value of <math>r = \ell / (1 - e) </math> and <math>r = \ell / (1 + e)</math>, for <math>\theta = \pi</math> and <math>\theta = 0</math> is

{{block indent|<math>a = \frac{\ell}{1 - e^2}.</math>}}

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 [[Focus (geometry)|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 {{mvar|b}} is related to the semi-major axis {{mvar|a}} through the eccentricity {{mvar|e}} and the [[semi-latus rectum]] <math>\ell</math>, as follows:

{{block indent|<math>\begin{align}
 b      &= a \sqrt{1 - e^2}, \\ 
 a \ell &= b^2.
\end{align}</math>}}

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 <math>\ell</math> fixed. Thus {{mvar|a}} and {{mvar|b}} tend to infinity, {{mvar|a}} faster than {{mvar|b}}.

The length of the semi-minor axis could also be found using the following formula:<ref>[http://www.mathopenref.com/ellipseaxes.html "Major / Minor axis of an ellipse"], Math Open Reference, 12 May 2013.</ref>
{{block indent|<math>2b = \sqrt{(p + q)^2 -f^2},</math>}}
where {{mvar|f}} is the distance between the foci, {{mvar|p}} and {{mvar|q}} 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 {{mvar|a}} in the x-direction the equation is:<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Ellipse |url=https://mathworld.wolfram.com/Ellipse.html |access-date=2024-08-20 |website=mathworld.wolfram.com |language=en}}</ref>

{{block indent|<math>\frac{\left( x-h \right)^2}{a^2} - \frac{\left( y-k \right)^2}{b^2} = 1.</math>}}

In terms of the semi-latus rectum and the eccentricity, we have

{{block indent|<math>a={\ell \over e^2 - 1 }. </math>}}
 
The transverse axis of a hyperbola coincides with the major axis.<ref>{{cite web|url=http://www.geom.uiuc.edu/docs/reference/CRC-formulas/node27.html|title=7.1 Alternative Characterization|website=www.geom.uiuc.edu|access-date=2007-09-06|archive-date=2018-10-24|archive-url=https://web.archive.org/web/20181024103231/http://www.geom.uiuc.edu/docs/reference/CRC-formulas/node27.html|url-status=dead}}</ref>

In a hyperbola, a conjugate axis or minor axis of length <math>2b</math>, corresponding to the minor axis of an ellipse, can be drawn perpendicular to the transverse axis or major axis, the latter connecting the two [[Vertex (curve)|vertices]] (turning points) of the hyperbola, with the two axes intersecting at the center of the hyperbola. The endpoints <math>(0,\pm b)</math> 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 {{mvar|b}}.  Denoting the semi-major axis length (distance from the center to a vertex) as {{mvar|a}}, the semi-minor and semi-major axes' lengths appear in the equation of the hyperbola relative to these axes as follows:

{{block indent|<math>\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1.</math>}}

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.{{cn|date=August 2019}}

The semi-minor axis and the semi-major axis are related through the eccentricity, as follows:

{{block indent|<math>b = a \sqrt{e^2-1}.</math><ref>{{cite web|url=http://www.bogan.ca/orbits/geometry.html|title=The Geometry of Orbits: Ellipses, Parabolas, and Hyperbolas|website=www.bogan.ca}}</ref>}}<!--NOTE: This formula differs from that of the ellipse because in a hyperbola, e > 1, so the equation of b = sqrt{1-e^2} has no real solutions-->

Note that in a hyperbola {{mvar|b}} can be larger than {{mvar|a}}.<ref>{{Cite web|url=http://www.geom.uiuc.edu/docs/reference/CRC-formulas/node27.html|title=7.1 Alternative Characterization|access-date=2007-09-06|archive-date=2018-10-24|archive-url=https://web.archive.org/web/20181024103231/http://www.geom.uiuc.edu/docs/reference/CRC-formulas/node27.html|url-status=dead}}</ref>

==Astronomy==
===Orbital period===

[[File:solar_system_orbital_period_vs_semimajor_axis.svg|thumb|upright=1.25|Log-log plot of period {{mvar|T}} vs semi-major axis {{mvar|a}} (average of aphelion and perihelion) of some Solar System orbits (crosses denoting Kepler's values) showing that {{math|1=''a''<sup>3</sup>{{hairsp}}/{{hairsp}}''T''{{px2}}<sup>2</sup>}} is constant (green line)]]

In [[astrodynamics]] the [[orbital period]] {{mvar|T}} of a small body orbiting a central body in a circular or elliptical orbit is:<ref name=lissauer2019>{{Cite book |title=Fundamental Planetary Sciences: physics, chemistry, and habitability |last1=Lissauer |first1=Jack J. |last2=de Pater |first2=Imke |year=2019 |publisher=Cambridge University Press |isbn=9781108411981 |location=New York |pages=24–31 }}</ref>

{{block indent|<math>T = 2\pi\sqrt{\frac{a^3}{\mu}},</math>}}

where:
{{block indent|{{mvar|a}} is the length of the orbit's semi-major axis,}}
{{block indent|<math>\mu</math> 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, disregarding their eccentricity.

The [[specific relative angular momentum|specific angular momentum]] {{mvar|h}} of a small body orbiting a central body in a circular or elliptical orbit is<ref name=lissauer2019/>
{{block indent|<math>h = \sqrt{a\mu(1 - e^2)},</math>}}

where:
{{block indent|{{mvar|a}} and <math>\mu</math> are as defined above,}}
{{block indent|{{mvar|e}} 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 laws of planetary motion|Kepler's third law]] (originally [[empirical]]ly derived):<ref name=lissauer2019/>

{{block indent|<math>T^2 \propto a^3,</math>}}

where {{mvar|T}} is the period, and {{mvar|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 [[Isaac Newton|Newton]]:<ref name=lissauer2019/>

{{block indent|<math>T^2 = \frac{4\pi^2}{G(M + m)}a^3,</math>}}

where {{mvar|G}} is the [[gravitational constant]], {{mvar|M}} is the [[mass]] of the central body, and {{mvar|m}} is the mass of the orbiting body. Typically, the central body's mass is so much greater than the orbiting body's, that {{mvar|m}} 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 [[center of mass|barycenter]] and its path relative to its primary are both ellipses.<ref name=lissauer2019/> 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 (<math>M \gg m</math>); 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 {{val|81.30059}}. The Earth–Moon characteristic distance, the semi-major axis of the ''geocentric'' lunar orbit, is 384,400&nbsp;km. (Given the lunar orbit's eccentricity ''e''&nbsp;=&nbsp;0.0549, its semi-minor axis is 383,800&nbsp;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&nbsp;km, the Earth's counter-orbit taking up the difference, 4,670&nbsp;km. The Moon's average barycentric orbital speed is 1.010&nbsp;km/s, whilst the Earth's is 0.012&nbsp;km/s. The total of these speeds gives a geocentric lunar average orbital speed of 1.022&nbsp;km/s; the same value may be obtained by considering just the geocentric semi-major axis value.{{cn|date=August 2019}}

===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. The time- and angle-averaged distance of the orbiting body can vary by 50-100% from the orbital semi-major axis, depending on the eccentricity.<ref>{{cite journal
 | title=Average distance between a star and planet in an eccentric orbit
 | last=Williams | first=Darren M.
 | journal=American Journal of Physics
 | volume=71 | issue=11 | pages=1198–1200 | date=November 2003
 | doi=10.1119/1.1578073 | bibcode=2003AmJPh..71.1198W
}}</ref>

*averaging the distance over the [[eccentric anomaly]] indeed results in the semi-major axis.
*averaging over the [[true anomaly]] (the true orbital angle, measured at the focus) results in the semi-minor axis <math>b = a \sqrt{1 - e^2}</math>.
*averaging over the [[mean anomaly]] (the fraction of the orbital period that has elapsed since pericentre, expressed as an angle) gives the time-average <math>a \left(1 + \frac{e^2}{2}\right)\,</math>.

The time-averaged value of the reciprocal of the radius, <math>r^{-1}</math>, is <math>a^{-1}</math>.
<!-- Commenter's note: This is not an equation, and seems to have no point. What did the contributor try to say?
For [[exoplanet]]s orbiting the stars of different masses, using just the Kepler's third law is impractical. Here is the equation below:

{{block indent|<math>\sqrt[3]{M_\text{star}} \times \sqrt[3]{t^2}</math>}}

where: 
{{block indent|{{mvar|M}}{{sub|star}} is the mass of the star in [[solar mass]]es}}
{{block indent|{{mvar|T}} is the orbital period in years.}}
-->

===Energy; calculation of semi-major axis from state vectors===

In [[astrodynamics]], the semi-major axis {{mvar|a}} can be calculated from [[orbital state vectors]]:

{{block indent|<math>a = -\frac{\mu}{2\varepsilon}</math>}}

for an [[elliptical orbit]] and, depending on the convention, the same or

{{block indent|<math>a = \frac{\mu}{2\varepsilon}</math>}}

for a [[hyperbolic trajectory]], and

{{block indent|<math>\varepsilon = \frac{v^2}{2} - \frac{\mu}{|\mathbf{r}|}</math>}}

([[specific orbital energy]]) and

{{block indent|<math>\mu = GM,</math>}}

([[standard gravitational parameter]]), where:
: {{mvar|v}} is orbital velocity from [[Orbital state vectors#Velocity vector|velocity vector]] of an orbiting object,
: {{mvar|'''r'''}} is a [[cartesian coordinate system|cartesian]] [[Orbital state vectors#Position vector|position vector]] of an orbiting object in coordinates of a [[Frame of reference|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),
: {{mvar|G}} is the [[gravitational constant]],
: {{mvar|M}} is the mass of the gravitating body, and
: <math>\varepsilon</math> 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.{{cn|date=August 2019}}

=== Semi-major and semi-minor axes of the planets' orbits ===

Planet orbits are always cited as prime examples of ellipses ([[Kepler's laws of planetary motion|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 <math>\frac{a}{b} = \frac{1}{\sqrt{1 - e^2}}</math>, 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 <math>\frac{r_\text{a}}{r_\text{p}} = \frac{1 + e}{1 - e}</math>. Due to the large difference between aphelion and perihelion, [[Kepler's laws of planetary motion|Kepler's second law]] is easily visualized. 

{| class="wikitable sortable" style="margin: 1em auto; text-align: center;"
|-
!class="unsortable"|
! [[Orbital eccentricity|Eccentricity]]
! Semi-major axis ''a'' ([[Astronomical unit|AU]])
! Semi-minor axis ''b'' ([[Astronomical unit|AU]])
! Difference (%)
! [[Perihelion]] ([[Astronomical unit|AU]])
! [[Aphelion]] ([[Astronomical unit|AU]])
! Difference (%)

|-
! [[Mercury (planet)|Mercury]]
| 0.206
| 0.38700
| 0.37870
| 2.2
| 0.307
| 0.467
| 52

|-
! [[Venus]]
| 0.007
| 0.72300
| 0.72298
| 0.002
| 0.718
| 0.728
| 1.4

|-
! [[Earth]]
| 0.017
| 1.00000
| 0.99986
| 0.014
| 0.983
| 1.017
| 3.5

|-
! [[Mars]]
| 0.093
| 1.52400
| 1.51740
| 0.44
| 1.382
| 1.666
| 21

|-
! [[Jupiter]]
| 0.049
| 5.20440
| 5.19820
| 0.12
| 4.950
| 5.459
| 10

|-
! [[Saturn]]
| 0.057
| 9.58260
| 9.56730
| 0.16
| 9.041
| 10.124
| 12

|-
! [[Uranus]]
| 0.046
| 19.21840
| 19.19770
| 0.11
| 18.330
| 20.110
| 9.7

|-
! [[Neptune]]
| 0.010
| 30.11000
| 30.10870
| 0.004
| 29.820
| 30.400
| 1.9

|}

1 AU (astronomical unit) equals 149.6 million km.

== References ==
{{reflist}}

== External links ==

*[http://www.mathopenref.com/ellipsesemiaxes.html Semi-major and semi-minor axes of an ellipse] With interactive animation

{{orbits}}

[[Category:Conic sections]]
[[Category:Orbits]]