The Dynamics of Comets

Comets are usually classified in categories according to their orbital period (Fig. 12). Comets with orbital period P > 200 years are called long-period comets (LPCs); those with shorter period are called short-period comets (SPCs). The threshold of 200 years is arbitrary and has been chosen mostly for historical reasons: modern instrumental astronomy is about two centuries old, so that the LPCs that we see now are unlikely to have been observed in the past.

□ HALLEY-TYPE . JUPITER-FAMILY

□ HALLEY-TYPE . JUPITER-FAMILY

Fig. 12. The distribution of comets according to their orbital semi-major axis and inclination. Here, the orbital elements are defined at the moment of the comet's last aphelion passage. Long period, Halley-type, and Jupiter family comets are plotted as red stars, black squares, and blue dots, respectively. The separation between Halley-types and Jupiter family comets has been made according to the value of their Tisserand parameter, following [105]. The vertical dashed lines correspond to orbital periods P = 20 years and P = 200 years, respectively. All LPCs with a >10,000 AU have been represented on the log a =4 line

Fig. 12. The distribution of comets according to their orbital semi-major axis and inclination. Here, the orbital elements are defined at the moment of the comet's last aphelion passage. Long period, Halley-type, and Jupiter family comets are plotted as red stars, black squares, and blue dots, respectively. The separation between Halley-types and Jupiter family comets has been made according to the value of their Tisserand parameter, following [105]. The vertical dashed lines correspond to orbital periods P = 20 years and P = 200 years, respectively. All LPCs with a >10,000 AU have been represented on the log a =4 line

If the orbital distribution of the comets is plotted, like in Fig. 12, using the orbital elements that the comets had when they last passed at aphelion - which can be computed through a backward numerical integration - one sees a clustering of long period comets with a — 104 AU. These comets are called new comets because they are passing through the region of the giant planets system for the first time. In fact, after a passage through the inner Solar System, it is unlikely that the semi-major axis remains of order 104 AU. It either decreases to — 103 AU or the orbit becomes hyperbolic. The reason is that the binding energy of a new comet is E = —QMq/2a — 10~4, but typically, during a close perihelion passage, the energy suffers a change of order of the mass of Jupiter relative to the Sun: 10~3. This change is not due to close encounters with the planet (which might not occur). It is because the comet has a barycentric motion when it is far away, an heliocentric motion when it is close, and the distance of the barycentre from the Sun is of the order of the relative mass of Jupiter.

The SPCs are in turn subdivided into Halley-type (HTCs) and Jupiter family (JFCs). Historically, the partition between the two classes is done according to the orbital period being respectively longer or shorter than 20 years. This threshold has been chosen because there is an evident change in the inclination distribution at the corresponding value of semi-major axis (see Fig. 12). However, comets continuously change semi-major axis as a consequence of their encounters with the planets. In particular, all SPCs had to have a larger semi-major axis in the past, given that they come from the trans-planetary region. Thus, by adopting a partition between HTCs and JFCs based on orbital period, one is confronted with the unpleasant situation of objects changing their classification during their lifetime.

This problem has motivated Levison [105] to re-classify SPCs according to their Tisserand parameter relative to Jupiter

This new classification makes sense, because the Tisserand parameter is quite well preserved during the comet's evolution. In Levison's classification, HTCs and JFCs have Tj, smaller and larger, respectively, than 2. Figure 12 adopts this classification and shows that, for most of the objects, the classifications based on orbital period and on Tisserand parameter are in agreement, but a few objects (those with P < 20 years and large inclination or those with P > 20 years and low inclination) change their classification depending on the adopted criterion.

Tisserand Parameter

Given the importance of the Tisserand parameter in cometary dynamics, it is useful to derive its expression (which outlines the limitations of its use) and discuss its properties.

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