Semimajor Axis

The semi-major axis (also semimajor axis ) is one half of the major axis of an ellipse, 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.

In astronomy, the semimajor axis is one of the most important characteristics of an orbit, along with its period. For solar system objects, the semimajor axis is related to the period of the orbit by Kepler's third law (originally empirically derived),

<math>P^2=a^3\,<math>

where P 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, as determined by Newton:

<math>P^2= \frac{4\pi^2}{G(M+m)}a^3\,<math>

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.

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 same angle, measured at the focus) results, oddly enough, in the semi-minor axis <math>b = a \sqrt{1-e^2}<math>. 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): <math>a (1 + \frac{e^2}{2})\,<math>.

Calculation from state vectors

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

<math> a = { - \mu \over {2E}}\,<math>

and

<math> E = { v^2 \over {2} } - {\mu \over \left | \mathbf{r} \right |} <math>

and

<math> \mu = GM \,<math>,

where:

• <math> v\,<math> is orbital velocity from velocity vector of an orbiting object,
• <math> \mathbf{r }\,<math> 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),
• <math> G \,<math> is the gravitational constant,
• <math> M \,<math> the mass of the central body.

References