This articleneeds additional citations forverification. Please helpimprove this article byadding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Semi-major and semi-minor axes" – news ·newspapers ·books ·scholar ·JSTOR(March 2017) (Learn how and when to remove this message)

Part of a series on
Astrodynamics
Orbital mechanics
Orbital elements Apsis Argument of periapsis Eccentricity Inclination Mean anomaly Orbital nodes Semi-major axis True anomaly
Types oftwo-body orbits by eccentricity Circular orbit Elliptic orbit Transfer orbit (Hohmann transfer orbit Bi-elliptic transfer orbit) Parabolic orbit Hyperbolic orbit Radial orbit Decaying orbit
Equations Dynamical friction Escape velocity Kepler's equation Kepler's laws of planetary motion Orbital period Orbital velocity Surface gravity Specific orbital energy Vis-viva equation
Celestial mechanics
Gravitational influences Barycenter Hill sphere Perturbations Sphere of influence
N-body orbits Lagrangian points (Halo orbits) Lissajous orbits Lyapunov orbits
Engineering and efficiency
Preflight engineering Mass ratio Payload fraction Propellant mass fraction Tsiolkovsky rocket equation
Efficiency measures Gravity assist Oberth effect
Propulsive maneuvers Orbital maneuver Orbit insertion
v t e

Ingeometry, themajor axis of anellipse is its longestdiameter: aline segment that runs through the center and bothfoci, with ends at the two most widely separated points of theperimeter. Thesemi-major axis (major semiaxis) is the longestsemidiameter or one half of the major axis, and thus runs from the centre, through afocus, and to the perimeter. Thesemi-minor axis (minor semiaxis) of an ellipse orhyperbola is a line segment that is atright angles with the semi-major axis and has one end at the center of theconic section. For the special case of a circle, the lengths of the semi-axes are both equal to theradius of the circle.

The length of the semi-major axisa of an ellipse is related to the semi-minor axis's lengthb through theeccentricitye and thesemi-latus rectum${\displaystyle \ell }$, as follows:

The semi-major axis of ahyperbola 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 eithervertex of the hyperbola.

Aparabola 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${\displaystyle \ell }$ fixed. Thusa andb tend to infinity,a faster thanb.

The major and minor axes are theaxes 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.

The equation of an ellipse is

${\displaystyle \frac{(x-h{)}^2}{a^2}+\frac{(y-k{)}^2}{b^2}=1,}$

where (h, k) is the center of the ellipse inCartesian 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${\displaystyle r_{\text{max}}}$ and${\displaystyle r_{\text{min}}}$ of the ellipse from a focus — that is, of the distances from a focus to the endpoints of the major axis

${\displaystyle a=\frac{r_{\text{max}}+r_{\text{min}}}{2}.}$

In astronomy these extreme points are calledapsides.^[1]

The semi-minor axis of an ellipse is thegeometric mean of these distances:

${\displaystyle b=\sqrt{r_{\text{max}}{r}_{\text{min}}}.}$

Theeccentricity of an ellipse is defined as

${\displaystyle e=\sqrt{1-\frac{b^2}{a^2}},}$

so

${\displaystyle r_{\text{min}}=a(1-e),r_{\text{max}}=a(1+e).}$

Now consider the equation inpolar coordinates, with one focus at the origin and the other on the${\displaystyle \theta =\pi }$ direction:

${\displaystyle r(1+e\cos \theta )=\ell .}$

The mean value of${\displaystyle r=\ell /(1-e)}$ and${\displaystyle r=\ell /(1+e)}$, for${\displaystyle \theta =\pi }$ and${\displaystyle \theta =0}$ is

${\displaystyle a=\frac{\ell }{1-e^2}.}$

In an ellipse, the semi-major axis is thegeometric 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 thefoci) 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 axisb is related to the semi-major axisa through the eccentricitye and thesemi-latus rectum${\displaystyle \ell }$, as follows:

Aparabola 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${\displaystyle \ell }$ fixed. Thusa andb tend to infinity,a faster thanb.

The length of the semi-minor axis could also be found using the following formula:^[2]

${\displaystyle 2b=\sqrt{(p+q{)}^2-f^2},}$

wheref is the distance between the foci,p andq are the distances from each focus to any point in the ellipse.

The semi-major axis of ahyperbola is, depending on the convention, plus or minus one half of the distance between the two branches; if this isa in the x-direction the equation is:^[3]

${\displaystyle \frac{{\left(x-h\right)}^2}{a^2}-\frac{{\left(y-k\right)}^2}{b^2}=1.}$

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

${\displaystyle a=\frac{\ell }{e^2-1}.}$

The transverse axis of a hyperbola coincides with the major axis.^[4]

In a hyperbola, a conjugate axis or minor axis of length${\displaystyle 2b}$, corresponding to the minor axis of an ellipse, can be drawn perpendicular to the transverse axis or major axis, the latter connecting the twovertices (turning points) of the hyperbola, with the two axes intersecting at the center of the hyperbola. The endpoints${\displaystyle (0,\pm b)}$ 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 lengthb. Denoting the semi-major axis length (distance from the center to a vertex) asa, the semi-minor and semi-major axes' lengths appear in the equation of the hyperbola relative to these axes as follows:

${\displaystyle \frac{x^2}{a^2}-\frac{y^2}{b^2}=1.}$

The semi-minor axis is also the distance from one of focuses of the hyperbola to an asymptote. Often called theimpact 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.^{[citation needed]}

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

${\displaystyle b=a\sqrt{e^2-1}.}$^[5]

Note that in a hyperbolab can be larger thana.^[6]

Inastrodynamics theorbital periodT of a small body orbiting a central body in a circular or elliptical orbit is:^[1]

${\displaystyle T=2\pi \sqrt{\frac{a^3}{\mu }},}$

where:

${\displaystyle \mu }$ is thestandard 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.

Thespecific angular momentumh of a small body orbiting a central body in a circular or elliptical orbit is^[1]

${\displaystyle h=\sqrt{a\mu (1-e^2)},}$

where:

a and${\displaystyle \mu }$ are as defined above,

Inastronomy, the semi-major axis is one of the most importantorbital elements of anorbit, along with itsorbital period. ForSolar System objects, the semi-major axis is related to the period of the orbit byKepler's third law (originallyempirically derived):^[1]

${\displaystyle T^2\propto a^3,}$

whereT is the period, anda is the semi-major axis. This form turns out to be a simplification of the general form for thetwo-body problem, as determined byNewton:^[1]

${\displaystyle T^2=\frac{4{\pi }^2}{G(M+m)}{a}^3,}$

whereG is thegravitational constant,M is themass of the central body, andm is the mass of the orbiting body. Typically, the central body's mass is so much greater than the orbiting body's, thatm may be ignored. Making that assumption and using typical astronomy units results in the simpler form Kepler discovered.

The orbiting body's path around thebarycenter and its path relative to its primary are both ellipses.^[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 (${\displaystyle M\gg m}$); 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 is81.30059. The Earth–Moon characteristic distance, the semi-major axis of thegeocentric lunar orbit, is 384,400 km. (Given the lunar orbit's eccentricitye = 0.0549, its semi-minor axis is 383,800 km. Thus the Moon's orbit is almost circular.) Thebarycentric 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.^{[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. 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.^[7]

• averaging the distance over theeccentric anomaly indeed results in the semi-major axis.
• averaging over thetrue anomaly (the true orbital angle, measured at the focus) results in the semi-minor axis${\displaystyle b=a\sqrt{1-e^2}}$.
• averaging over themean anomaly (the fraction of the orbital period that has elapsed since pericentre, expressed as an angle) gives the time-average${\displaystyle a\left(1+\frac{e^2}{2}\right)}$.

The time-averaged value of the reciprocal of the radius,${\displaystyle r^{-1}}$, is${\displaystyle a^{-1}}$.

Inastrodynamics, the semi-major axisa can be calculated fromorbital state vectors:

${\displaystyle a=-\frac{\mu }{2\epsilon }}$

for anelliptical orbit and, depending on the convention, the same or

${\displaystyle a=\frac{\mu }{2\epsilon }}$

for ahyperbolic trajectory, and

${\displaystyle \epsilon =\frac{v^2}{2}-\frac{\mu }{|\mathbf{r}|}}$

(specific orbital energy) and

${\displaystyle \mu =GM,}$

(standard gravitational parameter), where:

v is orbital velocity fromvelocity vector of an orbiting object,

r is acartesianposition vector of an orbiting object in coordinates of areference 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 thegravitational constant,

M is the mass of the gravitating body, and

${\displaystyle \epsilon }$ 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 totalspecific orbital energy is always the same. This statement will always be true under any given conditions.^{[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. That difference (or ratio) is based on the eccentricity and is computed as${\displaystyle \frac{a}{b}=\frac{1}{\sqrt{1-e^2}}}$, 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${\displaystyle \frac{r_{\text{a}}}{r_{\text{p}}}=\frac{1+e}{1-e}}$. 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.

• Semi-major and semi-minor axes of an ellipse With interactive animation

• Conic sections
• Orbits

• Articles with short description
• Short description matches Wikidata
• Articles needing additional references from March 2017
• All articles needing additional references
• All articles with unsourced statements
• Articles with unsourced statements from August 2019