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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 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, and to the perimeter. The **semi-minor axis** (**minor semiaxis**) of an ellipse or hyperbola 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. For the special case of a circle, the lengths of the semi-axes are both equal to the radius of the circle.

- Ellipse
- Hyperbola
- Astronomy
- Orbital period
- Average distance
- Energy; calculation of semi-major axis from state vectors
- Semi-major and semi-minor axes of the planets' orbits
- References
- External links

The length of the semi-major axis a of an ellipse is related to the semi-minor axis's length b through the eccentricity e 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 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 a and b tend to infinity, a faster than b.

The major and minor axes are the 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.

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.^{ [1] }

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 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) 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 b is related to the semi-major axis a through the eccentricity e 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 a and b tend to infinity, a faster than b.

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

where f is the distance between the foci, p and q are the distances from each focus to any point in the ellipse.

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:^{[ citation needed ]}

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

The transverse axis of a hyperbola coincides with the major axis.^{ [3] }

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 b. Denoting the semi-major axis length (distance from the center to a vertex) as a, 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, 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:

Note that in a hyperbola b can be larger than a.^{ [5] }

In astrodynamics the orbital period T of a small body orbiting a central body in a circular or elliptical orbit is:^{ [1] }

where:

a 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, disregarding their eccentricity.

The specific angular momentum h of a small body orbiting a central body in a circular or elliptical orbit is^{ [1] }

where:

a and are as defined above,

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 third law (originally empirically derived):^{ [1] }

where T is the period, and 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 Newton:^{ [1] }

where G is the gravitational constant, M is the mass of the central body, and m is the mass of the orbiting body. Typically, the central body's mass is so much greater than the orbiting body's, that 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 barycenter 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 (); 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. (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.^{[ 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.

- 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 .
- 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 .

In astrodynamics, the semi-major axis a can be calculated from orbital state vectors:

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

for a hyperbolic trajectory, and

(specific orbital energy) and

(standard gravitational parameter), where:

- v 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),- G is the gravitational constant,
- M 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.^{[ 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 , 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 is easily visualized.

Eccentricity | Semi-major axis a (AU) | Semi-minor axis b (AU) | Difference (%) | Perihelion (AU) | Aphelion (AU) | Difference (%) | |
---|---|---|---|---|---|---|---|

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.

In mathematics, an **ellipse** is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in which the two focal points are the same. The elongation of an ellipse is measured by its eccentricity , a number ranging from to .

In mathematics, a **hyperbola** 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, called connected components or branches, that are mirror images of each other and resemble two infinite bows. The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. If the plane intersects both halves of the double cone but does not pass through the apex of the cones, then the conic is a hyperbola.

A **spheroid**, also known as an **ellipsoid of revolution** or **rotational ellipsoid**, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has circular symmetry.

An **ellipsoid** is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.

**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 motion and law of universal gravitation. Orbital mechanics is a core discipline within space-mission design and control.

In gravitationally bound systems, the **orbital speed** of an astronomical body or object is the speed at which it orbits around either the barycenter or, if one body is much more massive than the other bodies of the system combined, its speed relative to the center of mass of the most massive body.

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 along an orbit, the other two being the true anomaly and the mean anomaly.

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 fact that according to Newtonian theory such an orbit has the shape of a hyperbola. In more technical terms this can be expressed by the condition that the orbital eccentricity is greater than one.

In astrodynamics, the **characteristic energy** is a measure of the excess specific energy over that required to just barely escape from a massive body. The units are length^{2} time^{−2}, i.e. velocity squared, or energy per mass.

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, it is a Kepler orbit with the eccentricity greater than 0 and less than 1. In a wider sense, it is a Kepler's orbit with negative energy. This includes the radial elliptic orbit, with eccentricity equal to 1.

In the gravitational two-body problem, the **specific orbital energy** of two orbiting bodies is the constant sum of their mutual potential energy and their total kinetic energy, divided by the reduced mass. According to the orbital energy conservation equation, it does not vary with time:

In astrodynamics, an **orbit equation** defines the path of orbiting body around central body relative to , without specifying position as a function of time. Under standard assumptions, a body moving under the influence of a force, directed to a central body, with a magnitude inversely proportional to the square of the distance, has an orbit that is a conic section with the central body located at one of the two foci, or *the* focus.

In mathematics, the **eccentricity** of a conic section is a non-negative real number that uniquely characterizes its shape.

In orbital mechanics, **mean motion** is the angular speed required for a body to complete one orbit, assuming constant speed in a circular orbit which completes in the same time as the variable speed, elliptical orbit of the actual body. The concept applies equally well to a small body revolving about a large, massive primary body or to two relatively same-sized bodies revolving about a common center of mass. While nominally a mean, and theoretically so in the case of two-body motion, in practice the mean motion is not typically an average over time for the orbits of real bodies, which only approximate the two-body assumption. It is rather the instantaneous value which satisfies the above conditions as calculated from the current gravitational and geometric circumstances of the body's constantly-changing, perturbed orbit.

**Spacecraft flight dynamics** is the application of mechanical dynamics to model how the external forces acting on a space vehicle or spacecraft determine its flight path. These forces are primarily of three types: propulsive force provided by the vehicle's engines; gravitational force exerted by the Earth and other celestial bodies; and aerodynamic lift and drag.

The **free-fall time** is the characteristic time that would take a body to collapse under its own gravitational attraction, if no other forces existed to oppose the collapse. As such, it plays a fundamental role in setting the timescale for a wide variety of astrophysical processes—from star formation to helioseismology to supernovae—in which gravity plays a dominant role.

In astronautics and aerospace engineering, the **bi-elliptic transfer** is an orbital maneuver that moves a spacecraft from one orbit to another and may, in certain situations, require less delta-v than a Hohmann transfer maneuver.

In mathematics, a **conic section** 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 special case of the ellipse, though historically it was sometimes called a fourth type. The ancient Greek mathematicians studied conic sections, culminating around 200 BC with Apollonius of Perga's systematic work on their properties.

In celestial mechanics, **Lambert's problem** is concerned with the determination of an orbit from two position vectors and the time of flight, posed in the 18th century by Johann Heinrich Lambert and formally solved with mathematical proof by Joseph-Louis Lagrange. It has important applications in the areas of rendezvous, targeting, guidance, and preliminary orbit determination.

In astrodynamics and celestial mechanics a **radial trajectory ** is a Kepler orbit with zero angular momentum. Two objects in a radial trajectory move directly towards or away from each other in a straight line.

- 1 2 3 4 5 6 Lissauer, Jack J.; de Pater, Imke (2019).
*Fundamental Planetary Sciences: physics, chemistry, and habitability*. New York: Cambridge University Press. pp. 24–31. ISBN 9781108411981. - ↑ "Major / Minor axis of an ellipse", Math Open Reference, 12 May 2013.
- ↑ "7.1 Alternative Characterization".
*www.geom.uiuc.edu*. - ↑ "The Geometry of Orbits: Ellipses, Parabolas, and Hyperbolas".
*www.bogan.ca*. - ↑ "7.1 Alternative Characterization".

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

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