# Classical Orbital Elements¶

We know that the solution to the 3-D vector orbit equation requires six independent elements to find a solution. So far, we have worked with \(\vector{r}\) and \(\vector{v}\) and their scalar components when determining orbits. In this section, we introduce the six classical orbital elements, which can also be used as a set of independent components of the orbit.

The six elements are also known as Keplerian orbital elements:

\(a\), the semi-major axis—a constant defining the size of the conic.

\(e\), the eccentricity—a constant defining the shape of the conic.

\(i\), the inclination—the constant angle between \(\uvec{K}\) in the reference frame and \(\uvec{h}\) in the orbital plane.

\(\Omega\), right ascension (or longitude) of the ascending node—the right ascension of the point where the spacecraft crosses from below to above the fundamental plane of the reference frame. The crossing point is called the ascending node.

\(\omega\), argument of periapsis—the angular distance along the orbit from the ascending node to periapsis.

\(T\), time of periapsis passage—the time when the spacecraft passed periapsis.

Each of these six elements will be developed in the following subsections. This choice of six elements is not unique and equivalent parameters will be discussed below as well.

## \(a\), the Semi-Major Axis¶

The semi-major axis determines the size of the conic section. For a circle, it is the radius, while for an ellipse, it describes the width of the ellipse. For a hyperbola, the semi-major axis describes the distance from the origin of the Cartesian coordinate system.

The semi-latus rectum may specified instead of the semi-major axis, since with \(a\) and \(e\), \(p\) can be calculated. This is convenient for parabolic trajectories where the semi-major axis is not as meaningful.

Finally, the specific angular momentum, \(h\), can also be used in place of the semi-major axis.

## \(e\), the Eccentricity¶

The eccentricity describes the deviation of the trajectory from a circle. When \(e=0\), the orbit is circular; for values of \(e < 1\), the orbit is elliptical. When \(e = 1\), the trajectory is parabolic and for \(e > 1\), the trajectory is hyperbolic.

## \(i\), the Inclination¶

The inclination is the angle from the \(\uvec{K}\) axis in the reference frame to the angular momentum vector, \(\vector{h}\), as shown in Fig. 43. The inclination ranges from 0° to 180°.