Semi-major Axis Ellipse Hyperbola Astronomy Orbital Period

The semi-major axis of an ellipse is one half of the major axis running from the center, through a focus, and to the edge of the ellipse. The major axis is the longest line that runs through the center and both foci of an ellipse, its ends being at the widest points of the shape.

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 l fixed. Thus a and b tend to infinity, a faster than b.

The semi-major axis is the mean value of the smallest and largest distance 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,

2 Hyperbola

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

3 Astronomy

3.1 Orbital period

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

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 characteristics of an orbit, along with its orbital period. For solar systemA generic solar system (or planetary system consists of at least one star and various orbiting objects (such as asteroids, comets, moons, and planets). The term originated to describe the planetary system around Sol, the Latin name for our sun. The planet objects, the semi-major axis is related to the period of the orbit by Kepler's third law (originally empiricalEmpirical is an adjective often used in conjunction with science, both the natural and social sciences, which means an observation or experiment based upon experience that is capable of being verified or disproved. See also Empirical formula, Empirical knly derived),

where P is the period in years, and a is the semimajor axis in astronomical unitThe astronomical unit AU is a unit of distance, approximately equal to the mean distance between Earth and Sun. The currently accepted value of the AU is 149,597,870,691+-30 metres (about 150 million kilometres or 93 million miles). Earth's orbit is not as. This form turns out to be a simplification of the general form, as determined by NewtonKneller's portrait of 1689. Sir Isaac Newton ( December 25, 1642 March 20, 1727 by the Julian calendar then in use; or January 4, 1643 March 31, 1727 by the Gregorian calendar) was an English physicist, mathematician, astronomer, philosopher, and alchemis:

where G is the gravitational constantAccording to the law of universal gravitation, the attractive force between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them. The constant of proportionality is called , the gr, and M is the massMass is a property of physical objects that, roughly speaking, measures the amount of matter they contain. It is a central concept of classical mechanics and related subjects. Strictly speaking, there are two different quantities called mass Inertial 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.

3.2 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 anomalyThe 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. In the d (q.v.) indeed results in the semi-major axis.
averaging over the true anomaly (the same 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 time-average of r^-1 is a^-1

3.3 Energy; calculation of semi-major axis from state vectors

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

for an elliptical orbit and for a hyperbolic trajectory

and

( specific orbital energy)

and

( 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),
is the gravitational constant,
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.

4 Example

The International Space Station has an orbital period of 91.74 minutes, hence the semi-major axis is 6738 km [1]. 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.

5 References

Conic sections Astrodynamics Celestial mechanics

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