Anatomy of an orbit

Since I’m going to be discussing orbital dynamics fairly extensively on this ‘blog, I thought I’d put together a post describing how orbits are described, since the terminology can be fairly confusing, even to professional astronomers.

First, recall that particles orbiting under an inverse-square force such as gravity follow orbits which are conic sections — circles, ellipses, parabolae or hyperbolae. Particles which are bound to the body they are orbiting follow circular or elliptical orbits, while unbound particles follow parabolic or hyperbolic orbits. I’ll confine my attention to bound orbits here, since most objects of interest (planets, moons, asteroids…) are on bound orbits.

Such orbits are confined to a single plane. Restricting attention to this plane, an elliptical orbit is described algebraically by r=\frac{a(1-e^2)}{1+e\cos f}, while geometrically it looks like this:

Eccentric orbit

An orbit with an eccentricity of 0.3. The orbiting particle is located at the filled symbol, having moved through an angle f.

The orbit has one parameter describing its size, the semi-major axis a, and one describing its shape, the eccentricity e. The higher the eccentricity, the less circular the orbit is. The particle’s location on the orbit, measured from the point of closest approach to the central body, is measured by the angle f, the true anomaly. This point of closest approach is located at a distance  q=a(1-e) from the central body. It is known variously as the pericentre, periapse (pl. periapses) or periapsis (pl. periapsides). For orbits around specific bodies, it may be called the perigee (Earth), perihelion (Sun), etc.

Note that the symbols marking the orbit in the above Figure are not evenly spaced. They in fact represent the particle’s position at equal intervals in time. The particle moves fastest at periapsis; this is a simple consequence of angular momentum conservation.

While this description of an orbit is quite simple, it is often necessary to describe an orbit with respect to a different reference frame. For example, when observing an extra-solar planetary system it is convenient to align the reference frame with the line of sight and the plane of the sky. The orientation of the orbital plane with respect to the reference plane is then given by two angles, the inclination I and the longitude of ascending node \Omega. The former describes the angle between the x-y plane of the reference frame and the orbital plane, while the latter describes the line of intersection of the two planes (this is known as the line of nodes). These angles are shown below:

Orbital angles

We also see the familiar angle f, the true anomaly, making its appearance. One final angle is needed to completely specify the orientation of the orbit: this is the argument of periapsis \omega, which describes which way the periapsis points within the orbital plane.

To summarize, there are now six parameters, called orbital elements, describing the particle’s position in space:

  • I, inclination, describing the angle between a reference plane and the orbital plane
  • \Omega, longitude of ascending node, describing the orientation of the intersection of these two planes
  • \omega, argument of periapsis, describing the direction the periapsis points relative to the line of nodes.
  • f, true anomaly, describing the particle’s position on its orbit relative to the periapsis.
  • a, semi-major axis, the average distance of the particle from the central body
  • e, eccentricity, a measure of how non-circular the orbit is

If the particle only experiences an inverse square force from the central body, then all these apart from the true anomaly are constant, and the orbit remains unchanged in time. However, when more bodies are introduced, they can cause all the orbital elements to change. The goal of celestial mechanics is to determine what these changes are, which is a very difficult problem for which there is no general solution.

Finally, there are alternative orbital elements. It is common to use the longitude of periapsis \varpi=\omega+\Omega to express the direction of periapsis relative to the reference direction, and the true longitude \theta=f+\omega+\Omega to express the particle’s position on its orbit relative to this direction. Remember:

  • Longitudes are measured relative to a fixed reference direction
  • Arguments are measured relative to the line of nodes
  • Anomalies are measured relative to the periapsis

One thought on “Anatomy of an orbit

  1. Pingback: Chaos I | no need of that hypothesis

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