Semi-major axis - Citizendia

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. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position

1 Ellipse
2 Hyperbola
3 Astronomy
4 References
5 External links

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. In Mathematics, an ellipse (from the Greek ἔλλειψις literally absence) is a Conic section, the locus of points in a 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 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. 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 For the special case of a circle, the semi-major axis is just the radius.

The semi-major axis' length $a\!$ is related to the semi-minor axis $b\,\!$ through the eccentricity $e\,\!$ and the semi-latus rectum $\ell\,\!$ , as follows:

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

$\ell=a(1-e^2)\,\!$ . In Geometry, the semi-minor axis (also semiminor axis) is a Line segment associated with most Conic sections (that is with ellipses and In Mathematics, the eccentricity, denoted e or \varepsilon is a parameter associated with every conic section. In Mathematics, a conic section (or just conic) is a Curve obtained by intersecting a cone (more precisely a circular Conical surface

$a\ell=b^2\,\!$ .

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 $\ell\,\!$ fixed. In Mathematics, the parabola (pəˈræbələ from the Greek παραβολή) is a Conic section, the intersection of a right circular Thus $a\,\!$ and $b\,\!$ tend to infinity, $a\,\!$ faster than $b\,\!$ .

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,

$r(1-e\cos\theta)=l\,\!$ . Coordinates are numbers which describe the location of points in a plane or in space

The mean value of $r={\ell\over{1+e}}\,\!$ and $r={\ell\over{1-e}}\,\!$ , is $a={\ell\over{1-e^2}}\,\!$ .

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:

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

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

$a={\ell\over e^2-1 }$

The transverse axis of a hyperbola runs in the same direction as the Semi-major axis. In Geometry, a hyperbola ( Greek, "over-thrown" has several equivalent definitions ^[1]

Astronomy

Orbital period

In astrodynamics the orbital period $T\,$ of a small body orbiting a central body in a circular or elliptical orbit is:

$T = 2\pi\sqrt{a^3/\mu}$

where:

$a\,$ is the length of the orbit's semi-major axis

μ

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. Orbital mechanics or astrodynamics is the application of Celestial mechanics to the practical problems concerning the motion of Rockets and other Spacecraft The orbital period is the time taken for a given object to make one complete Orbit about another object Small body orbiting a central body Under Standard assumptions in astrodynamics we have m where m \ is the mass

In astronomy, the semi-major axis is one of the most important orbital elements of an orbit, along with its orbital period. Astronomy (from the Greek words astron (ἄστρον "star" and nomos (νόμος "law" is the scientific study 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 In Physics, an orbit is the gravitationally curved path of one object around a point or another body for example the gravitational orbit of a planet around a star The orbital period is the time taken for a given object to make one complete Orbit about another object For solar system objects, the semi-major axis is related to the period of the orbit by Kepler's third law (originally empirically derived),

$T^2=a^3\,$

where T is the period in years, and a is the semimajor axis in astronomical units. The Solar System consists of the Sun and those celestial objects bound to it by Gravity. In Astronomy, Kepler's Laws of Planetary Motion are three mathematical laws that describe the motion of Planets in the Solar System. A central concept in Science and the Scientific method is that all Evidence must be empirical, or empirically based that is dependent on evidence The astronomical unit ( AU or au or au or sometimes ua) is a unit of Length based on the distance from the Earth to the This form turns out to be a simplification of the general form for the two-body problem, as determined by Newton:

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

where G is the gravitational constant, and M is the mass of the central body, and m is the mass of the orbiting body. In Classical mechanics, the two-body problem is to determine the motion of two point particles that interact only with each other Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements The gravitational constant, denoted G, is a Physical constant involved in the calculation of the gravitational attraction between objects with mass Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object 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. 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 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 $b = a \sqrt{1-e^2}\,\!$ . In Astronomy, the true anomaly \nu\\! (Greek nu also written \theta\ or f\) is the angle between the direction z-s of In Geometry, the semi-minor axis (also semiminor axis) is a Line segment associated with most Conic sections (that is with ellipses and
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): $a (1 + \frac{e^2}{2})\,\!$ . 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

The average radius of an ellipse, measured with respect to its geometric centre, is $\sqrt{ab} = a\sqrt[4]{1-e^2}\,\!$ .

The time-average of the inverse of the radius, $r^{-1}\,\!$ , is $a^{-1}\,\!$ .

Energy; calculation of semi-major axis from state vectors

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

$a = { - \mu \over {2\epsilon}}\,$ for an elliptical orbit and $a = {\mu \over {2\epsilon}}\,$ for a hyperbolic trajectory

and

$\epsilon = { v^2 \over {2} } - {\mu \over \left | \mathbf{r} \right |}$ (specific orbital energy)

and

$\mu = GM \,$ (standard gravitational parameter),

where:

$v\,$ is orbital velocity from velocity vector of an orbiting object,
$\mathbf{r }\,$ 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. Orbital mechanics or astrodynamics is the application of Celestial mechanics to the practical problems concerning the motion of Rockets and other Spacecraft In Astrodynamics or Celestial dynamics orbital state vectors (sometimes state vectors) are vectors of Position (\mathbf{r} and In Astrodynamics or Celestial mechanics an elliptic orbit is a Kepler orbit with the eccentricity greater than 0 and less than 1 In Astrodynamics or Celestial mechanics a hyperbolic trajectory is a Kepler orbit with the eccentricity greater than 1 In Astrodynamics the specific Orbital energy \epsilon\\! (or vis-viva energy) of an Orbiting body traveling through Space Small body orbiting a central body Under Standard assumptions in astrodynamics we have m where m \ is the mass In Astrodynamics or Celestial dynamics orbital state vectors (sometimes state vectors) are vectors of Position (\mathbf{r} and In Astrodynamics or Celestial dynamics orbital state vectors (sometimes state vectors) are vectors of Position (\mathbf{r} and g. geocentric equatorial for an orbit around Earth, or heliocentric ecliptic for an orbit around the Sun),
$G \,$ is the gravitational constant,
$M \,$ the mass of the central body. The gravitational constant, denoted G, is a Physical constant involved in the calculation of the gravitational attraction between objects with mass

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.

References

^ http://www.geom.uiuc.edu/docs/reference/CRC-formulas/node27.html

External links

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

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
network: | |

Contents