Expert: Philip Stahl - 4/10/2004

Question
Could you please explain, in as much detail as reasonably possible, what causes comets to follow -elliptical- orbits?

Thank you very much for your time :)

Answer
Hello.

In fact, all the bodies of the solar system follow elliptical orbits, they simply vary in what we call 'eccentricity'. (For eccentricity, e, the bounding limits are from e = 0 (circular orbit) to e = 1, e.g. orbit so narrow and elongated that the empty 'focus' of the orbit is removed to infinity.)

Thus, the Earth has an eccentricity 0.016 (approaching circularity) while Mercury's is 0. 205 and Pluto's is 0.247 (highly elliptical).

Many comets, by virtue of much lighter mass, will have eccentricities even more pronounced than Pluto's probably ranging from 0.35 to near 0.9.

How did this come to be?

One possibility is an orbit resonance with a large, major planet, the most likely candidate being Jupiter. Here is how it would work:

Say a comet has a semi-major axis a = 3.3 AU, then by Kepler's 3rd or harmonic law (P^2 = a^3) its period will be found to be half that of Jupiter.

This means the comet will experience a perturbation or 'tug' from Jupiter every other orbit, corresponding to a 2:1 orbit resonance. The effect of cumulative tugs over time will be to elongate the comet's orbit, e.g. make its eccentricity e ->  1.

Along more technical lines, every planet (or comet) orbit can be defined by a set of 'elements'. For example, these include: e (eccentricity), a (semi-major axis), T (period), w (true anomaly), and so on.

The full set of these elements contains the full set of information on the orbit, including: size, shape as well as orientation, e.g. with respect to the ecliptic or orbit plane of the solar system.

If an orbit is perturbed, then the elements must vary.  The manner and form of this variation is the subject of a whole sub-discipline of celestial mechanics which entails highly complex mathematical operations. In most cases, the formulae used contain what we call Fourier series - long sums of sines and cosines in the terms, plus secular terms.

Thus they contain both *periodic* and *secular* terms.

The first incorporate hundreds of trigonometric functions of time, the next have magnitudes associated with changes *proportional to time* and occur in all elements if the computations are carried out to a sufficient degree of approximation.

In the case of finding out how a perturbation alters a comet orbit- to elongate it much more, or create a highly elliptical shape - it would be necessary to examine in detail the particular terms (periodic and secular) and then show, say using a graph how a particular element is affected.

As a specific example, we may find it useful to plot the semi-major axis (a) for the comet against time (t) using a series defined by:

sin (at) = at - (a^3 t^3)/6 +  (a^5 t^5)/ 120 + ......

and so on.

One might then find that the value of a increases (or decreases) over time, and hence circularizes (or elongates) the shape of the comet orbit.

Hopefully, you will find this information useful.