Semimajor axis

Information about Semimajor axis

The semi-major axis of an ellipse

In geometry, the term semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolae.

Ellipse

The major axis of an ellipse is its longest diameter, a line that runs through the centre and both foci, its ends being at the widest points of the shape. The semi-major axis is one half of the major axis, and thus runs from the centre, through a focus, and to the edge of the ellipse. For the special case of a circle, the semi-major axis is just the radius.

The semi-major axis' length $\text{[math]}$ is related to the semi-minor axis $\text{[math]}$ through the eccentricity $\text{[math]}$ 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 $\text{[math]}$ and $\text{[math]}$ tend to infinity, $\text{[math]}$ faster than $\text{[math]}$ .

The semi-major axis is the mean value of the smallest and largest distances from one focus to the points on the ellipse. Now consider the equation in polar coordinates, with one focus at the origin and the other on the positive x-axis,

The mean value of and , is .

Hyperbola

The semi-major axis of a hyperbola is one half of the distance between the two branches; if this is a in the x-direction the equation is:

$\text{[math]}$

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

The transverse axis of a hyperbola runs in the same direction as the Semi-major axis.[1]

Astronomy

Orbital period

In astrodynamics the orbital period $\text{[math]}$ of a small body orbiting a central body in a circular or elliptical orbit is:

$\text{[math]}$

where:

$\text{[math]}$ is the length of the orbit's semi-major axis

$\text{[math]}$ is the standard gravitational parameter

Note that for all ellipses with a given semi-major axis, the orbital period is the same, regardless of eccentricity.

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 T is the period in years, and a is the semimajor axis in astronomical units. This form turns out to be a simplification of the general form for the two-body problem, as determined by Newton:

where G is the gravitational constant, and 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.

Remarkably, the orbiting body's path around the barycentre and its path relative to its primary are both ellipses. The semi-major axis used in astronomy is always the primary-to-secondary distance; 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. 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 the geocentric lunar average orbital speed, 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 (the focus of the ellipse) and the orbiting body. This is not quite accurate, as it depends over what the average is taken.

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 (which is what "average" usually means to the layman): .

The average radius of an ellipse, measured with respect to its geometric centre, is .

The time-average of the inverse of the radius, , is .

Energy; calculation of semi-major axis from state vectors

In astrodynamics semi-major axis $\text{[math]}$ can be calculated from orbital state vectors:

for an elliptical orbit and for a hyperbolic trajectory

and

(specific orbital energy)

and

$\text{[math]}$ (standard gravitational parameter),

where:

is orbital velocity from velocity vector of an orbiting object,
is 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),
$\text{[math]}$ is the gravitational constant,
$\text{[math]}$ the mass of the central body.

Note that for a given central body and total specific energy, the semi-major axis is always the same, regardless of eccentricity. Conversely, for a given central body and semi-major axis, the total specific energy is always the same.

Example

The International Space Station has an orbital period of 91.74 minutes, hence the semi-major axis is 6738 km [2]. Every minute more corresponds to ca. 50 km more: the extra 300 km of orbit length takes 40 seconds, the lower speed accounts for an additional 20 seconds.

External links

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

• • [ e] Orbits
Orbit types	Box orbit • Circular orbit • Ecliptic orbit • Elliptic orbit • Highly Elliptical Orbit • Graveyard orbit • Hyperbolic trajectory • Inclined orbit • Osculating orbit • Parabolic trajectory • Capture orbit • Escape orbit • Semi-synchronous orbit • Subsynchronous orbit • Synchronous orbit • Parking orbit
Earth-centered orbits	Geosynchronous orbit • Geostationary orbit • Sun-synchronous orbit • Low Earth orbit • Medium Earth Orbit • Molniya orbit • Near equatorial orbit • Orbit of the Moon • Polar orbit • Tundra orbit
Other orbits	Areosynchronous orbit • Areostationary orbit • Halo orbit • Lissajous orbit • Lunar orbit • Heliocentric orbit • Heliosynchronous orbit
Orbital elements	Inclination ( $\text{[math]}$ ) • Longitude of the ascending node ( $\text{[math]}$ ) • Eccentricity ( $\text{[math]}$ ) • Argument of periapsis ( $\text{[math]}$ ) • Semi-major axis ( $\text{[math]}$ ) • Mean anomaly at epoch ( $\text{[math]}$ )
Other orbital parameters	True anomaly ( $\text{[math]}$ ) • Semi-minor axis ( $\text{[math]}$ ) • Linear eccentricity ( $\text{[math]}$ ) • Eccentric anomaly ( $\text{[math]}$ ) • Mean longitude ( $\text{[math]}$ ) • True longitude ( $\text{[math]}$ ) • Orbital period ( $\text{[math]}$ )
Orbital maneuvers	Bi-elliptic transfer • Geostationary transfer orbit • Gravitational slingshot • Hohmann transfer orbit • Orbital inclination change • Orbit phasing • Space rendezvous
Other orbital mechanics topics	Apsis • Celestial coordinate system • Delta-v budget • Epoch • Ephemeris • Equatorial coordinate system • Gravity turn • Ground track • Interplanetary Transport Network • Kepler's laws of planetary motion • Lagrangian point • List of orbits • n-body problem • Orbit equation • Orbital state vectors • Perturbation • Retrograde and direct motion • Specific orbital energy • Specific relative angular momentum • The Oberth effect

Geometry (Greek γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. Geometry is one of the oldest sciences.
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ellipse (from the Greek ἔλλειψις, literally absence) is the locus of points on a plane where the sum of the distances from any point on the curve to two fixed points is constant.
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In geometry, the foci (singular focus) are a pair of special points used in describing conic sections. The four types of conic sections are the circle, parabola, ellipse, and hyperbola.
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In geometry, the semi-minor axis (also semiminor axis) is a line segment associated with most conic sections (that is, with ellipses and hyperbolas). One end of the segment is the center of the conic section, and it is at right angles with the semi-major axis.
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eccentricity, denoted e or , is a parameter associated with every conic section. It can be thought of as a measure of how much the conic section deviates from being circular.

In particular,

The eccentricity of a circle is zero.

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In mathematics, the latus rectum of a conic section is the chord parallel to the directrix and passing through the single focus, or one of the two foci. By extension, the length of the latus rectum is also referred to as the latus rectum and written .
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parabola (from the Greek: παραβολή) (IPA pronunciation: /pəˈrab(ə)lə/
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For a more advanced and general treatment, see .

Coordinates are numbers which describe the location of points in a plane or in space. For example, the height above sea level
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hyperbola (Greek ὑπερβολή literally 'overshooting' or 'excess') is a type of conic section defined as the intersection between a right circular conical surface and a plane which cuts through both halves
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Orbital mechanics or astrodynamics is the study of the motion of rockets and other spacecraft. The motion of these objects is determined by Newton's laws of motion and the law of universal gravitation.
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The orbital period is the time taken for a planet (or another object) to make one complete orbit.

When mentioned without further qualification in astronomy this refers to the sidereal period of an astronomical object, which is calculated with respect to the stars.
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In astrodynamics, the standard gravitational parameter of a celestial body is the product of the gravitational constant and the mass :

The units of the standard gravitational parameter are km³s^-2

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Astronomy is the scientific study of celestial objects (such as stars, planets, comets, and galaxies) and phenomena that originate outside the Earth's atmosphere (such as the cosmic background radiation).
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The elements of an orbit are the parameters needed to specify that orbit uniquely, given a model of two point masses obeying the Newtonian laws of motion and the inverse-square law of gravitational attraction.
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ORBit is a CORBA compliant Object Request Broker (ORB). The current version is called ORBit2 and is compliant with CORBA version 2.4. It is developed under the GPL license and is used as middleware for the GNOME project.
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Solar System or solar system^[a] consists of the Sun and the other celestial objects gravitationally bound to it: the eight planets, their 166 known moons,^[1]
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Kepler's laws of planetary motion are three mathematical laws that describe the motion of planets in the Solar System. German mathematician and astronomer Johannes Kepler (1571–1630) discovered them.
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A central concept in science and the scientific method is that all evidence must be empirical, or empirically based, that is, dependent on evidence or consequences that are observable by the senses. Empirical data is data that is produced by experiment or observation.
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1 astronomical unit =
SI units
010⁹ m 010⁶ km
Astronomical units
010^-6 pc 010⁻⁶ ly
US customary / Imperial units
010⁹ ft 010⁶ mi

The
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two-body problem is to determine the motion of two point particles that interact only with each other. Common examples include a satellite orbiting a planet, a planet orbiting a star, two stars orbiting each other (a binary star), and a classical electron orbiting an atomic nucleus.
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Sir Isaac Newton

Isaac Newton at 46 in
Godfrey Kneller's 1689 portrait
Born 4 January 1643 [OS: 25 December 1642]
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gravitational constant, the universal gravitational constant, Newton's constant, and colloquially Big G. The gravitational constant is a physical constant which appears in Newton's law of universal gravitation and in Einstein's theory of general
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Mass is a fundamental concept in physics, roughly corresponding to the intuitive idea of "how much matter there is in an object". Mass is a central concept of classical mechanics and related subjects, and there are several definitions of mass within the framework of relativistic
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The eccentric anomaly is the angle between the direction of periapsis and the current position of an object on its orbit, projected onto the ellipse's circumscribing circle perpendicularly to the major axis, measured at the centre of the ellipse.
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In astronomy, the true anomaly (, also written ) is the angle between the direction z-s of periapsis and the current position p of an object on its orbit, measured at the focus s of the ellipse (the point around which the object orbits).
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In the study of orbital dynamics the mean anomaly of an orbiting body is the angle the body would have traveled about the center of the orbit's auxiliary circle. Unlike other measures of anomaly, the mean anomaly grows linearly with time.
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