Orbital Elements

Six numbers are needed to describe the movement of an object in a Keplerian orbit around some other body. Perhaps the simplest six numbers we can use are the position ($x, y, z$ ) and velocity ($v_x, v_y, v_z$ ) at some reference time $t_0$ . Assuming we know the mass of the central body, we can use these numbers to compute the position and velocity at any other time.

However, while this set of numbers does describe the orbit, it is often not particularly convenient. The position and velocity does not give an intuitive sense for what the orbit looks like - what can you say about the orbits that these two objects are on, with positions in AU and velocities in $km/s$ , $(x, y, z, v_x, v_y, v_z) = (-20.7, -19.6, 8, -4.1, -4.5, -0.7)$ and $(40.3, 23.6, -14.2, -1.6, 3.4, 0.1)$ ?¹

An alternate description uses the six orbital elements. These describe the orbit using the

• Semi-major axis (a): Half the length of the longest diameter of the ellipse
• Eccentricity (e): Whether the orbit is circular ($e = 0$ ) or elongated ($e = 1$ ). Mathematically, $e = \sqrt{(1 - \frac{b^2}{a^2})}$ where $a$ and $b$ are the semi-major and minor axes respectively.
• Inclination (i): The angle between the orbit and the reference plane
• Longitude of the ascending node ($\Omega$ ): The angle between the reference vector (which is in the reference plane) and where the orbit passes upwards through the plane
• Longitude of pericenter (ϖ): The angle between the reference vector and the location of pericenter (projected onto the reference frame)
• True anomaly ($\nu$ ): The position along the orbit at a reference time.

Note that there are a few slight variants of these (e.g. some parameterizations use the argument of pericenter - the angle from the ascending node - rather than the longitude of pericenter) but they are easy to switch between.

The effect on the orbit of changing these orbital elements (apart from the true anomaly which would merely move the starting position around the orbit) are shown below. The reference plane is shown in grey and the reference vector in yellow. Each colored sector has an equal area and so takes an equal amount of time for the orbiting object to traverse. Click and drag to change your point of view and scroll to zoom.