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
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Trojan

The space surrounding Earth’s orbit is far from empty. Small meteoroids can be seen as they enter Earth’s atmosphere and disintegrate, leaving wakes of incandescent gas. Comets leave extensive debris trails as they approach the Sun. There is also, unseen to the naked eye, a large population of asteroids whose orbits are in the vicinity of Earth’s.

Last week, one such asteroid, 2010 TK7was announced to be the first known member of a special class of asteroid known as Trojans. While Trojans relating to other planets such as Jupiter were known previously, this is the first example of one associated with Earth. The distinguishing feature of Trojan asteroids is that they are located close to one of the Lagrange points of the planet-Sun system, shown below:

Lagrange points

Image credit: Wikipedia

The contours show the gravitational potential due to the Earth and Sun. This is in a rotating reference frame, so the Earth remains fixed at the right as it orbits the Sun. Since it is a rotating reference frame, particles will feel fictitious forces, since they want to remain on a straight trajectory in the inertial frame. One of these forces is the centrifugal force, which is included in the contours. The behaviour of a particle due to the gravitational and centrifugal forces can be visualised by imagining that the contours represent a system of hills and valleys, with particles rolling “downhill” under the action of gravity. There are 5 points where the gravitational and centrifugal forces exactly balance, labelled L1 to L5. If a particle were located exactly at one of these points, there would be no net force, just as if it were located exactly at the top of a hill, and it would remain there.

However, the existence of such equilibrium points does not tell the whole story. Equilibria may be stable or unstable, depending on whether a particle placed close to one will move towards or away from it. Thinking again in terms of a landscape, the top of a hill and the bottom of a valley are both equilibria, but only the second is stable.

2010 TK7 is located close to but not at the L4 Lagrange point. Despite being a “hill” in the contour plot, suggesting instability, the L4 point is actually stable. This is due to the effects of another fictitious force, the Coriolis force. This is the force that governs the rotation of weather systems on Earth. The Coriolis force acts only on moving bodies, and so cannot be captured in a static description of forces as shown in the above figure. The effects are best shown in an animation:

Coriolis effect

Animation credit: Wikipedia

On the left the trajectory of a particle moving from one side of the circle to the other in an inertial frame is shown, and on the right the trajectory in the rotating frame (The details don’t correspond exactly to the motion about the Lagrange point, but it suffices to demonstrate qualitatively what’s going on). Note that the particle follows a small curved path in the rotating frame, with the particle returning to its starting point. This effect only occurs for particles moving in the rotating frame: if a particle were fixed (say, at a boundary between light and dark on the edge of the circle) it would not experience such a force.

Now we can piece together the motion of a particle close to L4. If initially at rest, it rolls “downhill” from L4 under combined gravitational and centrifugal forces. However, before it gets too far, the Coriolis force forces it to curve round, back towards L4. As it approaches, it slows, the Coriolis force weakens, and gravitational and centrifugal forces force the particle away from L4 again. The process repeats, with the particle moving towards and away from the Lagrange point. Qualitatively similar behaviour can be seen in the following animation from the discoverers of 2010 TK7, showing the asteroid’s orbit (although note that the large size of the oscillations makes the actual motion rather more complicated):

You have probably noticed that there are actually two components to the motion: an annual epicyclic oscillation, and a long-term libration which slowly varies the centre of the epicyclic oscillation. The large amplitudes of both components mean that 2010 TK7 is only weakly attached to the L4 point. Indeed, the discoverers integrated the asteroid’s orbit backwards in time and found that before AD 500 it was actually librating about the L5 point on the opposite side of Earth, but the particle’s high libration amplitude enabled it to cross the L3 point to enter the L4 point’s sphere of influence.

What of future evolution? Due to the effects of chaos–small errors in the asteroid’s exact position blowing up exponentially–it’s impossible to exactly predict the orbit more than a few thousand years in advance. Nevertheless, it appears that the object will continue transitioning between the Lagrange points, although precisely what will happen is unknown.

Could there be more such asteroids? Since the L4 and L5 points are located only 60 degrees from the Sun as seen from Earth, detecting faint objects there is challenging. 2010 TK7 was detected by the satellite WISE, and then followed up from the ground. The large libration amplitude and epicyclic oscillations, and the fact that the asteroid was discovered at the near-Earth end of its libration, carry it further from the Sun, and hence make it easier to detect. Trojans bound closely to the Lagrange points may well have escaped detection, and it’s intriuguing to think that more may be waiting to be discovered in this region of space so close to us.

Further reading:
  • The paper published in Nature
  • The discoverers’ website, with more visualisations.
  • Murray & Dermott, Solar System Mechanics, CUP, 1999. Not for the mathematophobic!