The Tisserand parameter is an approximation of the Jacobi constant, which is an invariant of the dynamics of a small body in the framework of the restricted, circular, three-body problem.

The expression of the Jacobi constant is:

where GM® = ap = 1 are assumed, ap, mmp are the semi-major axis and mass of the perturbing planet, and Hz is the z-component of the small body's angular momentum. The quantity A is the distance between the small body and the planet.

We can write the kinetic energy of the small body as a function of its semi-major axis and heliocentric distance:

2 2a r while the z-component of the angular momentum can be written:

Substituting (8) and (9) into (7) and neglecting the term mp/A one gets

a where the right-hand side is equivalent to (6), given that a is expressed in units of the planet's semi-major axis.

This derivation of the Tisserand formula shows that the Tisserand parameter is constant as long as the Jacobi constant is preserved, and mp/A is small. This last condition requires that the comet is not in a close encounter with the planet. During a close encounter, the Tisserand parameter has large and abrupt changes, but it returns to the value that it had before the encounter, once the distance to the planet increases back to large values. The conservation of the Jacobi constant, conversely, requires that the conditions of the restricted three-body problem are fulfilled, namely one planet must dominate the comet's evolution, and the effects of the planet's eccentricity must be negligible. This requires that the comet is not in a region where it can have encounters with two planets, otherwise the one-planet approximation does not hold. Also, it requires that the comet is not in a secular resonance with the planet, otherwise the effects of the planet's small eccentricity are enhanced.

One can demonstrate that, if a comet intersects the orbit of a planet, the Tisserand parameter T is related to the unperturbed relative velocity U at which it encounters the planet:

U = Vä^r, where U is expressed in units of the planet's orbital velocity. The formula is not defined for T > 3, which implies that comets with such values of Tisserand parameter cannot intersect the orbit of the planet (obviously for ep 0). Note, however, that comets on non-intersecting the orbits with respect to the planet can have T < 3. Only objects with T < 2\/2 — 2.83 (the value for a parabolic trajectory with i = 0 and q = ap) can be ejected onto a hyperbolic orbit in a single encounter with a planet.

The fact that the JFCs have (by definition) a Tisserand parameter with respect to Jupiter that is distinct from that of HTCs and LPCs suggests that the former are not the small semi-major axis end of the distribution of the latter. The average low inclination of the JFCs and the absence of retrograde comets in the JFC population (whichever of the two definitions for JFCs is adopted, see Fig. 12) suggests that the source of the JFCs must be a disk-like structure. In 1980, [44] proposed that the source of the JFCs was the - at the time still putative - Kuiper belt, a hypothesis later supported in [37].

However, today we know that there are two distinct disk-like structures in the trans-Neptunian region: the Kuiper belt and the Scattered disk. Which of the two is the source of JFCs? We have seen in Sect. 1.4 that the Scattered disk is too populous to be sustained in steady state by the objects leaking out of the Kuiper belt. If the Scattered disk is not sustained in steady state, it means that the number of objects that leave the Scattered disk - mostly evolving towards the inner Solar System - is larger than the number of objects entering the Scattered disk from the Kuiper belt. Thus, the Scattered disk must be the dominant source of the JFC population, rather than the Kuiper belt.

The dynamical evolution of objects from the Scattered disk to the JFC region has been studied in detail in [107], with statistics calculated from a large number of numerical simulations. The results illustrated in that paper essentially supersede all the results from the previous literature. Thus, most of what I report below is taken from that source. The origin and dynamics of JFCs has also been exhaustively reviewed in [40].

To evolve from the Scattered disk to the JFC region, a comet has to pass from a Neptune-dominated regime to one controlled by Jupiter (see Fig. 13). Through a transfer process involving multiple planets, the Tisserand parameter is, in principle, not preserved. However, the planetary system is structured in such a way that the transfer chain from Neptune to Jupiter is normally dominated by one single planet at a time (see Fig. 13), and the values of the Tisserand parameter relative to the dominating planets are not very different from each other. For instance, consider a Scattered disk body with Tisserand parameter relative to Neptune TN = 2.98. The conservation of the Tisserand parameter implies that the smallest perihelion distance to which Neptune can scatter this object is q = 17.7 AU, just enough to become Uranus-crosser. In this orbit, the body has Tu = 2.96. If Uranus takes the control of this body,

Fig. 13. The evolution of an object from the Scattered disk until its ultimate ejection, projected over the plane representing perihelion vs. aphelion distance. The horizontal structure at q — 30 AU represents the Scattered disk. When the object evolves along a line q = constant or Q = constant, its dynamics are essentially dominated by one single planet. This happens at least down to 10 AU, and during the final ejection phase. Blue lines denote the evolution before the object becomes a visible JFC, red lines after. The criterion for first visibility is that q has decreased below 2.5 AU for the first time. From [107]

Fig. 13. The evolution of an object from the Scattered disk until its ultimate ejection, projected over the plane representing perihelion vs. aphelion distance. The horizontal structure at q — 30 AU represents the Scattered disk. When the object evolves along a line q = constant or Q = constant, its dynamics are essentially dominated by one single planet. This happens at least down to 10 AU, and during the final ejection phase. Blue lines denote the evolution before the object becomes a visible JFC, red lines after. The criterion for first visibility is that q has decreased below 2.5 AU for the first time. From [107]

it can scatter it inwards to q = 9.0 AU, barely a Saturn-crosser. The body has now TS = 2.94 and thus Saturn can lower its q to only 3.8AU. With such a perihelion, the comet has a Tisserand parameter Tj = 2.82. Thus, the body never spends much time in a region where it can encounter two planets, because at each "hand-over," perihelion is converted to aphelion, and the object is taken away from the outer planet (see also [83]). The Tisserand parameter is therefore piece-wise conserved, and the final Tisserand parameter (with respect to Jupiter) is very close to the initial one (with respect to Neptune). Now, the bulk of the observed population in the Scattered disk has 2 < Tn < 3. Thus, at the end of the transfer chain, the bodies coming from the Scattered disk will have 2 <TJ < 3, in other words, they will be JFCs.

Because the Tisserand parameter remains close to 3, the inclination cannot grow to large values (because the growth of i would decrease T, see (6)). So, the final inclination distribution is comparable to the inclination distribution in the Scattered disk, i.e. mostly confined within 30°. Figure 14 compares the (a, i, TJ) distribution of the observed SPCs (top panels) with that obtained in the numerical simulation for the objects coming from the Scattered disk, when their perihelion distance first decreases below 2.5 AU (a criterion for visibility as an active comet). As one can see, the objects with TJ < 2 (HTCs) are not reproduced, while the observed and simulated distributions of the JFCs agree with each other in a remarkable way.

Nevertheless, a quantitative comparison would show that the inclination distribution of the simulated comets when they first become visible is slightly skewed toward low values relative to the observed distribution. Similarly, the distribution of the distances of the comets' nodes from Jupiter's orbit is also skewed toward small values. However, the dynamical lifetime of comets after they first become visible is of order 105 years. As time passes, the conservation of the Tisserand parameter degrades, as a result of the combined effects of Jupiter and Saturn and of secular resonances. Thus, the inclination is puffed up, and the distribution of w (initially strongly peaked around 0° and 180°) is randomized. As a consequence, the nodal distance distribution is also puffed

Fig. 14. The distribution of short-period comets projected over the (Tj,a) and (Tj, i) planes. Top panels: the observed distribution. Bottom panels: the distribution of the objects coming from the Scattered disk, when they are visible (q < 2.5 AU) for the first time. From [107]

Fig. 14. The distribution of short-period comets projected over the (Tj,a) and (Tj, i) planes. Top panels: the observed distribution. Bottom panels: the distribution of the objects coming from the Scattered disk, when they are visible (q < 2.5 AU) for the first time. From [107]

up5. Consequently, the agreement between the observed and simulated distributions first improves with the age of the comets and then eventually degrades. Thus, [39] considered the distribution of all simulated objects, from the time they first become visible up to time t. Using a Kolmogorov-Smirnov test to measure quantitatively the statistical agreement between simulated and observed distributions, [39] concluded that the best match is achieved - both for the inclination and for the nodal distance distributions - for t ~ 12,000years. The interpretation of this result is that this value of t corresponds to the typical physical lifetime of JFCs, after which the comets lose their activity and are no longer observed. Comparing the physical lifetime with the dynamical lifetime, [39] concluded that, if all faded JFCs are dormant objects with asteroidal appearance, the ratio between the number of dormant vs. active JFCs should be ~ 4.

The comparison between the q distribution of the simulated and observed JFCs suggests that the population of comets is observationally complete up to q ~ 2AU. There are ^40 known JFCs with total absolute magnitude H10 < 9 6 and q < 2AU. The simulated q distribution indicates that there should be about 100 comets with q < 2.5 AU, with the same total magnitude. If all faded JFCs are dormant, then we should expect an additional 400 bodies of asteroidal appearance on similar orbits. About 100 of them should have q < 1.3 AU and belong to the NEO population. The size of these putative bodies is badly constrained, because the conversion from total magnitude to nuclear magnitude (i.e. the absolute magnitude of the nucleus, in absence of cometary activity) is poorly known. Published estimates for the nucleus size for H10 = 9 comets range from D = 0.8 km [7] to D = 4.5 km [48], with a mean of about 2 km [48]. I will return to the nature of faded comets in Sect. 2.4.

With this estimate of the total number of JFCs, the rate at which Scattered disk bodies become JFCs and the mean lifetime of JFCs measured in their simulations, [39] computed that there should be 4 x 108 such objects (i.e., big enough to have total magnitude H10 < 9 when active) in the Scattered disk. The extrapolated size distribution obtained from observations of the Scattered disk [10] is roughly consistent with this estimate.

The Orbit of Comet P/Encke

Despite the overall good agreement between the observed and the simulated distribution of JFCs shown in Fig. 14, there is one important difference that should not be overlooked: the orbit of comet P/Encke is not re-produced in

5Some comets eventually evolve toward the Tj < 2 region, although they never manage to reproduce the (a, i, Tj) distribution illustrated in the top panels of Fig. 14.

6The total absolute magnitude is computed from the apparent magnitude V

(of nucleus plus coma), the heliocentric and geocentric distances r and A by the formula H10 = V + 5 log A + 10 log r, instead of the usual formula for dormant bodies H = V + 5 log A + 5 log r. The coefficient 10, instead of 5, accounts for the fact that the intensity of the activity of the comet is proportional to r-2.

the simulation of [107]. P/Encke is peculiar. It is the only regurlarly active comet with an orbit totally interior to the orbit of Jupiter and Tj > 3. In addition, a few asteroids on orbits decoupled from Jupiter are supposed to be dormant cometari nuclei, because of their sporadic activity (such as 4015 Wilson-Harrington) or association with a meteor stream (such as 2201 Oljato). However, the overall number of comets with orbits totally interior to that of Jupiter should be small. In fact, a search for objects with albedo typical of dormant cometary nuclei among the NEOs with Tj > 3 [53] has showed that these objects, if they exist, are rare.

The aphelion distance of P/Encke is currently 4.1 AU, so that it is not scattered by Jupiter's encounters. This implies that encounters with Jupiter cannot have emplaced the comet onto its current orbit. It has been proposed that P/Encke reached its orbit from the TJ < 3 region because of close encounters with the terrestrial planets, to the effect of non-gravitational forces,7 or both [50,73,145,174]. Neither of these aspects have been included in the simulations of [107].

A quantitative model of the orbital distribution of JFCs has been recently proposed [115]. This model is an extension of [107], but accounting also for terrestrial planets encounters. According to this model, at any one time, there should be roughly 12 objects in Encke-like orbits. However, it takes roughly 200 times longer to evolve onto an orbit like Encke's than the typical cometary physical lifetime. Thus, all comets decoupled from Jupiter should be inactive! To solve this apparent conundrum, the authors of [115] propose that comet Encke has been recently reactivated, as its perihelion distance is plunging toward the Sun (indeed its future fate is to collide with our star [174]).

In a historical paper, Oort [140] pointed out that the presence of numerous new comets with a > 104 AU - which appears as a spike in the distribution for 1 /a of the LPCs (see Fig. 15) - argues for the existence of a reservoir of objects in that distant region. The fact that the inclination distribution of new comets is essentially isotropic, not only in cos i (from —1 to 1, i.e. including retrograde orbits), but also in w and fi, indicates that this reservoir must have a quasi-spherical symmetry, namely it has the shape of a cloud surrounding the Solar System. This cloud is now generally called the Oort cloud. In Oort's view, all LPCs come from this cloud. The LPCs with a < 104 AU are returning comets, which originally belonged to the new comet group when they first entered the inner Solar System, but subsequently had their orbit perturbed and acquired a more negative binding energy (smaller semi-major axis). This view remains essentially valid even today.

At such large distances from the Sun, the evolution of the comets in the Oort cloud is strongly affected by the overall gravitational field resulting from

7For a recent review on non-gravitational forces acting on comet dynamics see [186].

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