Semi-major and semi-minor axes

The equation of an ellipse is {{block indent|\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1,}} 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 r_\text{max} and r_\text{min} of the ellipse from a focus — that is, of the distances from a focus to the endpoints of the major axis {{block indent|a = \frac{r_\text{max} + r_\text{min}}{2}.}} In astronomy these extreme points are called apsides. {{block indent|2b = \sqrt{(p + q)^2 -f^2},}} where is the distance between the foci, and 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 in the x-direction the equation is: {{block indent|\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 {{block indent|a={\ell \over e^2 - 1 }. }} The transverse axis of a hyperbola coincides with the major axis. In a hyperbola, a conjugate axis or minor axis of length 2b, 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 (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 length . Denoting the semi-major axis length (distance from the center to a vertex) as , the semi-minor and semi-major axes' lengths appear in the equation of the hyperbola relative to these axes as follows: {{block indent|\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 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. The semi-minor axis and the semi-major axis are related through the eccentricity, as follows: {{block indent|b = a \sqrt{e^2-1}.}} 1, so the equation of b = sqrt{1-e^2} has no real solutions--> Note that in a hyperbola can be larger than . ==Astronomy==

Orbital period shows the slope of this line is a constant within a given star system, determined by the mass of the host star (here: the Sun). In astrodynamics the orbital period of a small body orbiting a central body in a circular or elliptical orbit is: {{block indent|T = 2\pi\sqrt{\frac{a^3}{\mu}},}} where: Note that for all ellipses with a given semi-major axis, the orbital period is the same, disregarding their eccentricity. The specific angular momentum of a small body orbiting a central body in a circular or elliptical orbit is • 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 b = a \sqrt{1 - e^2}. • averaging over the mean anomaly (the fraction of the orbital period that has elapsed since pericentre, expressed as an angle) gives the time-average a \left(1 + \frac{e^2}{2}\right)\,. The time-averaged value of the reciprocal of the radius, r^{-1}, is a^{-1}. \sqrt[3]{M_\text{star}} \times \sqrt[3]{t^2}}} where: --> Energy; calculation of semi-major axis from state vectors In astrodynamics, the semi-major axis can be calculated from orbital state vectors: {{block indent|a = -\frac{\mu}{2\varepsilon}}} for an elliptical orbit and, depending on the convention, the same or {{block indent|a = \frac{\mu}{2\varepsilon}}} for a hyperbolic trajectory, and {{block indent|\varepsilon = \frac{v^2}{2} - \frac{\mu}}} (specific orbital energy) and (standard gravitational parameter), where: : is orbital velocity from velocity vector of an orbiting object, : 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), : is the gravitational constant, : is the mass of the gravitating body, and : \varepsilon 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. Semi-major and semi-minor axes of the planets' orbits 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 \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 \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. == References ==