Semi-axes of the ellipse
The two characteristic radii of an ellipse are called semiaxes :
- The major semi-axis is half of the largest diameter of an ellipse, which is also called the main axis .
- The small semiaxis is half of the shortest diameter ( minor axis ) and is exactly at an angle of 90 ° to the major semiaxis.
The circle is a special ellipse in which these two semi-axes are the same length, in this case both semi-axes correspond to the radius of the circle.
The major axis (the largest diameter, here ) and the minor axis (the smallest diameter, here ) are collectively referred to as the major axes of the ellipse. Major and minor axes are conjugate diameters . This relationship is retained even when the ellipse is viewed "obliquely", which can be used for the geometric construction of other conjugate diameters.
In astronomy , the semi-major axis of a Keplerian orbit is one of the six so-called orbital elements and is often given imprecisely as the “mean distance” and is usually abbreviated with a . It characterizes - together with the eccentricity - the shape of elliptical orbits of different celestial bodies.
Such bodies are primarily the planets and their moons , artificial earth satellites , the asteroids and thousands of binary stars .
According to Kepler's third law , the orbital time U of an elliptical orbit is coupled with a ( ). The constant is related to the mass of the central body - in a planetary system, therefore, to the mass of the central star .
The two main vertices are called apses , the main axis is the apsidal line : If a body lies in the focal point F 1 and a smaller body circles it on an ellipse, the shortest distance ( = a - e ) is called the periapsis and the longest distance ( = a + e ) from the apoapsis ( perihelion, aphelion at the sun).
In the periapsis (pericenter, main vertex close to the gravic center) the orbital velocity is maximal, in the apocenter minimal.
The actual mean distance depends not only on the major semi-axis but also on the numerical eccentricity and amounts to
In geodesy , the axes of the so-called error ellipses are an important means of representing the mean or maximum / minimum point errors. In the adjustment of geodetic networks , the leaves accuracy with which the individual measuring points of the network are determined, represent an error ellipse.
- ↑ Erwin Groten: On the definition of the mean point error . In: Zeitschrift für Vermessungswesen (ZfV), 11/1969, pp. 455–457.