This review starts with a brief historical overview of the subject, after which some recent papers attempting to improve the understanding of comet injection from the Oort Cloud and the origin of new comets are discussed. Special attention is paid to the importance of nongravitational effects in comet orbit determination, the synergy between stellar encounters and the galactic tides for the injection dynamics, and the role of planetary perturbations. The field is thus shown to be advancing rapidly, and brief comments on possible implications for studying the origin of the cloud are made.
To date, the Oort Cloud remains an unobserved, intellectual concept inferred from comet orbit statistics, but there is hardly any doubt over its existence. This review shall attempt to show how recent progress in modeling and theoretical understanding may allow us to approach a real test of the concept even without observing the cloud directly. The focus will be placed on a handful of recent papers. But let us first introduce the history of the subject along with some central features of the following discussions.
The Oort spike is illustrated in Fig. 1. This term stands for a very remarkable concentration of the reciprocal values of the semimajor axes (1/a) characterizing the original, barycentric orbits of long-period comets. It occurs next to the parabolic limit and is contained within the interval from zero to 1 × 10−4 AU−1. As the perihelion distances of these observed comets are generally less than 5 AU, the planetary perturbations of 1/a tend to be much larger than the width of the spike. This is shown by the inverted histogram in Fig. 1, featuring the analogous distribution for the future orbits of the same comets. Here, practically nothing is left of the spike.
Hence, the question arises, why does the spike exist? It is obviously doomed to destruction, as about half the comets get ejected into interstellar space never to return, and the rest will return at various times in the future with an energy distribution that may resemble the extended tail exhibited by the original orbits, but altogether excludes the spike. The standard explanation was given by Oort (1950). He identified the spike using a sample of only 19 comets and concluded that these comets must be newcomers. As their source is clearly within the solar system and there is no likely mechanism for the ongoing formation of comets in this type of orbits, Oort advanced the idea of a remote source population of comets as the origin of the observed newcomers. Our name for the explicit suggestion that he made is the Oort Cloud.
To be stable, the source population must have perihelia far enough for planetary perturbations to be negligible. From the work of Van Woerkom (1948), Oort knew that this means well beyond Saturn's orbit. As seen from a vantage point in the outer parts of the orbits in question approximately 104 AU away, the requirement on perihelion distance translates into an avoidance of transverse velocities smaller than a certain limit, i.e., a loss cylinder in velocity space (Oort 1950). In modern literature, the term “loss cone” is more commonly used. Thus, the newcomers should get injected into the depth of the loss cone during the course of one orbital revolution, and Oort was able to identify perturbations due to stellar encounters as a reasonable mechanism to achieve these injections.
The Oort Cloud would hence consist of comets with semimajor axes in excess of 104 AU, and from the rather flat cos i distribution of the new comets (i is the ecliptic inclination) one could infer that the cloud is practically isotropic. Meanwhile, it was not possible to exclude an inner core with a < 104 AU, because the galactic field stars that are expected to have encountered the solar system in the recent past would not have been able to inject such comets—i.e., in a statistical sense. Hills (1981) drew attention to the fact that a dense inner core of the Oort Cloud would get stirred up by very close stellar encounters, occurring on an approximately 108 yr time scale, giving rise to major bursts in the arrival rate of new comets—so-called comet showers.
At the same time, after the exploration of the cold, molecular phase of the interstellar medium and discovery of giant molecular clouds (GMC), it was also realized that the GMC population of the galactic disk poses a threat to the survival of the classical Oort Cloud for the solar system lifetime due to the disruptive action of the expected close encounters (Biermann 1978). Even though Hut and Tremaine (1985) concluded that this cloud would probably survive such GMC encounters, the existence of an inner core might offer a necessary source of replenishment for the outer halo, in case this would after all be needed. Thus, the inner core came to be seen as something more than an ad-hoc hypothesis, and comet showers took an important place in the Oort Cloud literature (e.g., Fernández and Ip 1987).
Following the work of Byl (1983), the injection of new comets came to be reconsidered in the light of galactic tides, and the disk tide was seen to cause secular perturbations of the perihelion distance that may be of large amplitude and efficiently bring Oort Cloud comets into the loss cone. Heisler and Tremaine (1986) presented a simplified, analytic theory of the disk tide that discloses a far-reaching analogy with the Lidov-Kozai cycles caused by planets on comets of shorter periods. Thus, the perihelion distance (q) oscillates along with the Galactic inclination as the argument of perihelion librates, and it is straightforward to show that the most rapid changes of q per orbit occur when the galactic latitude of perihelion (βG) equals ± π/4. The histogram of βG for new comets shown in Fig. 2 exhibits this effect very clearly and demonstrates that the galactic disk tide must be of great importance for the injection from the Oort Cloud. This evidence was first noted by Delsemme (1987).
The origin of the cloud is something that Oort (1950) speculated about. However, progress was very slow, until massive simulations of planetesimal scattering were made possible by computer developments, and the galactic tide came into the picture as a mechanism of dislodging the perihelia away from the planetary orbits. The seminal paper, featuring these advances, was that of Duncan et al. (1987), and this has been followed by an extensive literature, not to be reviewed here. Suffice it to say that this has largely focused around the implications for the Oort Cloud inner core raised by the discovery of (90377) Sedna (Brown et al. 2004) and the question of the Sun's birth environment (see Adams 2010).
From the Inner Core to Observable Comets
Injection of comets from the Oort Cloud has been studied in many papers using different models of the galactic tide and its influence on comet orbits as well as different treatments of close encounters between the Sun and galactic field stars, but only rarely have planetary perturbations been properly included into those studies. The usual assumption is that they cause a full “opacity” of the loss cone (typically, all comets with q < 15 AU are considered lost by planetary perturbations) but have no effect whatsoever at larger perihelion distances. Consequently, new observable comets would have to experience a decrease of at least approximately 10 AU in perihelion distance during the revolution immediately preceding their possible discovery. This obstacle to comet injection is referred to as the Jupiter-Saturn barrier.
For the galactic disk tide to cause such a large decrease, the comets would have to have semimajor axes exceeding 20,000 AU. Thus, a distinction arose between an inner and an outer part of the Oort Cloud with a boundary at a = 20,000 AU, such that the observed comets only come from the latter. The inner cloud would only get excited by very close, and hence very rare, stellar encounters on time scales of approximately 108 yr in connection with comet showers. But the solar system does not seem to be experiencing any such shower at present (Dybczyński 2002), so the Oort spike should be confined to a > 20,000 AU—signaling only the outer part of the cloud.
The first work to include a realistic model of planetary perturbations into comet injection was that of Emel'yanenko et al. (2007). Interestingly, they found an injection efficiency (rate of injections divided by the number of comets in the cloud) that was significantly higher than had been found in earlier papers (see Rickman 2010), but they did not analyze the reasons for this. Just a few years later, the first paper to also present the detailed routes leading into injection via planetary perturbations was published by Kaib and Quinn (2009). Here, it was demonstrated how comets find their way from the inner core of the Oort Cloud with semimajor axes less than 10,000 AU to eventual injection by means of the galactic tide after the semimajor axis has grown to a > 20,000 AU. This growth is accomplished by planetary perturbations that operate when the perihelia are beyond the observability zone, gently enough to increase the semimajor axis without ejecting the comets into interstellar space. Such evolutions are forbidden in the framework of the standard loss cone concept as described above, but Kaib and Quinn showed that they not only occur in reality but in fact provide a very important source of long-period comets.
Figure 3 shows a typical example of the findings of Kaib and Quinn (2009). A fictitious comet with a ≈ 6,000 AU evolves very slowly toward decreasing perihelion distance under the action of the galactic tide. After the orbit becomes Uranus-crossing, an increasing trend of the semimajor axis gets accelerated, until a > 10,000 AU. At that point, the comet is prepared for a major decrease in q to approximately 13 AU, which in turn allows for a major energy kick at the next perihelion passage, whereby a is pumped up to approximately 30,000 AU. It is then an easy matter for the galactic tide to bring the comet into the observability zone during the following revolution.
The dynamical model used by Kaib and Quinn is essentially one that neglects stellar encounters (these were used for test runs), but takes good care of the galactic tide as well as the planetary perturbations. They reported that about half the new comets produced by their model had their origin in the inner Oort Cloud, thus relying on the mechanism exemplified by Fig. 3. However, they also noted that these comets cannot be identified based on the original semimajor axis by which they contribute to the Oort spike. For instance, the comet in Fig. 3 arrives at its observable perihelion with aorig approximately 30,000 AU, and backward tracing of the semimajor axis through previous passages of the planetary system is clearly impossible for real comets.
Concerning real comets, if aorig can be determined accurately or with a well-known probability distribution, one may instead try to trace back the perihelion distance to the prediscovery apparition using an accurate model for the galactic tide (see the next section). If the Kaib-Quinn mechanism is at work, qprev will generally be in the 10−15 AU range. But the converse is not true—such a value of qprev may also occur for comets that experienced much smaller energy perturbations at their previous perihelia. Moreover, there is one further complication, namely, the stellar perturbations. These will not invalidate the Kaib-Quinn mechanism, but a priori, it is not clear if neglecting them when deriving values for qprev may lead to erroneous results. We shall return to this issue below.
Tracing the Provenance of Observed Comets
Important progress has recently been achieved also in the area of orbit determination for long-period comets. The key issue is nongravitational effects. Still today, most long-period comet orbits are determined without accounting for nongravitational forces. But the quality of the underlying astrometry has improved, and the observed orbital arcs are often longer than before. Together with methodological progress, this has led to the availability of a much larger sample of high-precision osculating orbits, and thus also original orbits, including nongravitational parameters. The latter, denoted A1 and A2, correspond to the standard formalism introduced by Marsden et al. (1973).
Królikowska (2006) studied 50 long-period comets in this way and found, as expected, that the original 1/a values tend to increase, when moving from a strictly gravitational solution to one that includes nongravitational effects. This phenomenon can be explained as follows. A jet force acting mostly in the outward, radial direction will decelerate the comet on the preperihelion branch and accelerate it after perihelion. This will slightly decrease the normal, Keplerian trend for the speed of motion to increase with decreasing heliocentric distance, and the way to emulate this effect in a gravitational solution is to increase the orbital eccentricity, i.e., to decrease the value of 1/a without affecting the well-determined value of q.
As a consequence, the fraction of hyperbolic original orbits must be much smaller in reality (consistent with the expected number of zero) than in a gravitationally determined sample. Moreover, the shape of the Oort spike 1/aorig distribution is seriously affected by introducing the nongravitational effects. It is broadened, and the peak is shifted in the positive direction (Królikowska and Dybczyński 2010), as illustrated by Fig. 4. The sample in question consists of 26 comets, whose original, nongravitational orbits are located within the Oort spike, i.e., with 0 < (1/a)orig,NG < 10−4 AU−1.
Due to the general smallness of the original semimajor axes compared with what is required for tidal injections, the backward integrations that the authors performed using the galactic tide failed in most cases to bring qprev above 15 AU. Hence, it was demonstrated that comets that are classically considered to be new based on their original semimajor axis are not necessarily new in the real sense of the word, i.e., in terms of perihelion distance. Moreover, a rather large fraction of the sample (9 comets out of 26) were shown to return in the future, although likely no more than five would then be Oort spike comets.
In a follow-up paper, Dybczyński and Królikowska (2011) used a sample of 64 Oort spike comets, discovered after 1970, with perihelion distances in excess of 3 AU. Remarkably, although the nongravitational force is expected to be very small at such distances, judging from the formula by Marsden et al. (1973), the authors detected nongravitational effects in nearly 25% of the cases. Nonetheless, this sample is less affected by the nongravitational force than the ones previously used, and one may thus conclude about the (1/a)orig distribution and dynamical origin of these comets with even better confidence.
Dybczyński and Królikowska (2011) applied the following dynamical classification of the comets, based on huge swarms of virtual comets (i.e., clones) that individually represent the full set of astrometric observations. If the comet is more than 50% likely to have qprev > 15 AU, it is called dynamically new. If the chance of qprev < 10 AU is larger than 50%, the comet is dynamically old. The remaining cases (only about 10%) have unclear status. As a result, about half the sample was found to be new, whereas approximately 40% was found to be old. As in the previous paper, qprev was calculated from integrations using the galactic tide.
Note that there is an additional uncertainty about the values of qprev due to the neglect of stellar perturbations, which we shall further discuss below. The errors thus to be expected increase with aorig and may be insignificant for small values, unless there was a remarkably close stellar passage in the recent past—an issue to which we shall also return below.
As expected from what is known about the nature of the tide, a good correlation exists between the dynamical status of a comet and its value of (1/a)orig. This is illustrated in Fig. 5. As seen, almost all the comets with aorig < 25,000 AU are either old or unclear, while almost all those with aorig > 25,000 AU are either new or unclear. The histogram in the upper panel seems to be the best available rendering of the shape of the Oort spike from the observational point of view, and the challenge for the theory of comet evolution and dynamics is to find a reasonable way to reproduce this shape.
As many comets were found likely to have qprev in the range of 10−15 AU, one may ask if evidence has been found for the working of the Kaib-Quinn mechanism. The answer is possibly yes, but it is impossible to know for sure. Figure 6 illustrates one case in point. This is a backward integration by Dybczyński and Królikowska (2011) of the nominal orbit for comet C/2001 K3 (Skiff), using only the galactic tide (a few future orbits of much shorter period are also shown). The comet—according to this integration—came to its previous perihelion passage 5.7 Myr ago with qprev approximately 16 AU, and it might then have been perturbed by the planets from an inner Oort Cloud orbit into the one of the last revolution, characterized by the current aorig approximately 32,000 AU. But this is only speculation, because obviously there is no way to know what the actual planetary perturbation did amount to.
Long-Term Synergy in Comet Injection
Let us now turn to the role of stellar perturbations. Important progress in modeling those has been achieved with the introduction of new techniques, replacing or improving the classical impulse approximation (Dybczyński 1994; Eggers and Woolfson 1996), leading eventually to the Sequential Impulse Approximation (Rickman et al. 2005). Using this tool together with a new hybrid technique for integrating the galactic tidal force (Breiter et al. 2007), it became possible to investigate systematically the dynamics of the Oort Cloud under the influence of the tide and the stellar encounters by Monte Carlo simulations extending over the age of the solar system.
The first such work was published by Rickman et al. (2008). It brought about an unexpected discovery, which is illustrated in Fig. 7. This shows (in the main, upper panel) three superposed histograms indicating the number of Oort Cloud comets injected into orbits with q < 5 AU per 50 Myr interval from the start of the simulations until 5 Gyr later. The black histogram refers to a model including only the galactic tide, the gray one to a model including only the stellar encounters, and the white one shows the effect of combining the two perturbers.
The initial cloud is always assumed thermalized. The black histogram starts from a high level, consistent with the fact that all orbits are thus populated as relevant for a random distribution of elements. But, on a time scale of approximately 1 Gyr, it falls off to a much lower level that would in fact characterize the current time—assuming that the Oort Cloud is about as old as the solar system. This is because those orbits that the tide may evolve into perihelia within the loss cone get depleted, as the tide is not able to replace the lost comets by new ones coming from different regions of phase space. The time scale of decay is the same as that of depleting the most easily injectable parts of phase space, and the remaining flux is limited to other parts, where the orbits evolve less quickly.
The gray histogram exhibits the expected series of major showers due to particularly efficient encounters on top of a background level that decreases slowly along with the total population of the cloud (indicated by asterisks). But the background is never very high, and currently, it is about the same as for the purely tidal model.
The surprise comes when we look at the white histogram. Rather than appearing as the simple sum of the other two, it shows a prominent excess. At the current time, this is quite variable, but averages approximately 100% (bottom panel). Thus, the combined model yields about twice as many new comets as the sum of the two effects individually. Comet injection is obviously a team work involving an important synergy between the tide and the stars.
When concentrating on time periods away from the major showers, the Oort spike predicted by the combined model close to the current time peaks at approximately 35,000 AU. This indicates that current comet injection from the Oort Cloud may rely heavily on the galactic tide, and that the role of the stars may be mainly to replenish the depleted parts of phase space, thereby maintaining the tidal injections at a level close to the initial one for several billion years. Introducing the term tidally active zone (TAZ) for the part of phase space that is linked to the observable orbits by the galactic tide (Fouchard et al. 2011a), we may state that stars are needed to keep the TAZ populated at a significant level. Let us refer to this as TAZ filling.
But that is not the only role of the stars. The modeled Oort spike extends to larger binding energies (shorter semimajor axes) than tidal injections may account for. In particular, at a < 25,000 AU, the tide is almost completely unable to inject comets on its own, and the significant number predicted by the combined model thus requires the direct intervention of stars—either purely stellar injections or cases of real-time synergy, to which we shall return in the next section.
Fouchard et al. (2011a) made some special simulations, where the stellar dynamical model was limited to a sequence of 20 stars of identical spectral type, encountering the solar system at 250 Myr intervals. One result of this work was a demonstration that high- and low-mass stars act differently on the Oort Cloud—the latter being limited to comets in the vicinity of their tracks in the general case, and the massive stars (which also tend to pass slowly) being long-range perturbers influencing more or less the entire cloud. This increases their efficiency in a way that largely compensates for their rarity and minor contribution to the overall encounter frequency.
Hence, concerning both direct injections and TAZ fillings, it was concluded that high-mass stars (2 M⊙ or more) are at least as important in the long run as the predominant and ever present, low-mass stars (less than 1 M⊙). An example concerning TAZ filling in the central energy range of the Oort Cloud (20,000 < a < 50,000 AU) is shown in Fig. 8. The color codings refer to the individual stellar types (13 in all), and shown is the number of comets moving into the TAZ from its surroundings following an individual stellar encounter as compared with the population size of a full TAZ (i.e., a fractional TAZ filling, counted in percent). This is plotted against the encounter distance, and one easily notes that a substantial filling of the TAZ requires much closer approaches for low-mass than high-mass stars.
Two consequences stand out. The long-term tidal-stellar synergy by TAZ filling is about equally caused by high- and low-mass stars, when account is taken of their different encounter frequencies. Furthermore, a stellar encounter able to cause substantial TAZ filling will indirectly enhance the flux of new comets injected from the Oort Cloud for hundreds of Myr hence, until the filling gets depleted again. This sort of comet drizzle (Fouchard et al. 2011a) is expected to follow the major showers. It should be present at most times and be due largely to the rare, close encounters with high-mass stars. It will then probably form a significant component of the Oort Cloud flux.
The Last Revolution of New Comets
The above-mentioned evidence for a real-time synergy in comet injection may be checked by a detailed tracing of what happens to new comets during the orbital revolutions immediately preceding their observable apparitions. Such a study was performed by Fouchard et al. (2011b), and it clarified several aspects of comet injection although, like the preceding studies, it assumed an opaque loss cone out to q = 15 AU.
A sample of more than 20,000 injections from the Oort Cloud into observable orbits found from long-term simulations to be unaffected by any stellar encounter strong enough to cause a significant increase in the arrival rate of new comets, was analyzed. These injections could thus be classified into different categories according to the way the galactic tide and stellar perturbations interacted during the last revolution of the comets. The effect of stars is never vanishingly small, so there is no single, tidal trajectory that links the beginning of such a revolution to its end (i.e., the last unobservable to the first observable perihelion). However, one may recognize cases, where they have no significant effect in the sense that the initial orbit would have led to injection even without stars, and the final orbit could also have been reached by a purely tidal injection.
As shown in the top panel of Fig. 9, such comets are not the rule. They represent about 1/3 of the injections overall and never more than 50% in any range of 1/aorig (sky-blue area of the histogram). They do contribute importantly to the main peak of the energy distribution of new comets, but an even larger contribution (dark-blue area) is made up of comets whose postinjection orbits are reachable by the tide alone, but whose orbits at the start of the last revolution were not injectable by the tide alone. In these cases, stars were necessary helpers to the tide—placing the comets onto tracks that made them observable, appearing as results of tidal injections although they are not. The third largest group (white area) contains comets whose injections are neither tidal nor stellar, but result from a cooperation of both effects, and comets where stars did not help the injection, but shifted the final orbit into one that is unreachable by tidal injection.
At this point, a comment on the observed new comets is in order. If the passing stars have a large effect on comet injection, the question arises whether this is reflected in the orbital distribution of those comets. One issue concerns the βG distribution shown in Fig. 2. Some smoothing of this double-peaked pattern could be expected, as stellar perturbations would relax the requirement on the tidal effect in order to make a comet observable. The other issue was investigated e.g., by Lüst (1984), and concerns a clustering of aphelion directions on the celestial sphere that could be related to single stellar passages. At present, the statistics of comet orbits is not good enough in order to establish the presence or absence of the above phenomena.
The remaining part (red and gray areas) consists of stellar injections. These are seen to be very few overall but to cause a slight, inward extension of the Oort spike in addition to the main peak. On the other hand, whenever they appear in the mid or outer parts of the spike, the resulting orbits are very often reachable by purely tidal injections as well. This is a natural result of the very quick, tidal evolution of q for such orbits, which also has a downside in that observable perihelia require a fine-tuning of the times of minima of the perihelion distance and the perihelion passage itself. The statistical chance of achieving this decreases with increasing semimajor axis and is the reason for the drop of injection rate in the outer part of the spike.
At this point, one may get the impression that real-time synergy is a main driver behind the new comets, but of course, the histogram in Fig. 9 refers only to injections, and there may be many other cases, where stars preclude tidal injections that would otherwise have occurred. In other words, the stars have both triggering and prohibiting effects on comet injection, as concerns the events of the last revolution. Fouchard et al. (2011b) also measured these, and the results are seen in the lower panel of Fig. 9. Here, the gray and orange colors denote the amounts of loss and gain of injections, respectively, if all stellar effects were turned off during the last revolution. The net effect of the stars is seen to be positive though by no means dramatic. A good perception of the relative roles of the tide and the stars is achieved by noting that, if the galactic tide were suddenly turned off, there would very soon be a dramatic drop in the flux of new comets, but if the stars were similarly turned off, the immediate effect would be small, and major effects would take approximately 1 Gyr to build up.
Even so, it remains a fact that most of the new comets arriving on classically observable orbits with q < 5 AU would not have appeared, had it not been for the interventions of one or more stars during their last revolutions. Rickman et al. (2012) looked into the possibility of identifying those “culprit stars” that affected the real, new comets by using the simulations by Fouchard et al. (2011b). Only comets whose injections actually required a stellar intervention were considered, and Fig. 10 shows the evolutions of perihelion distance that three of them experienced during their last revolutions. They represent different energy ranges of the Oort Cloud, leading to different time scales for the different panels. The red curves show the osculating perihelion distances, and the blue curves show the values to be obtained at the end of the revolutions, if only the galactic tide were at work.
It is easy to recognize the main stellar effects—especially in the blue curves showing the jumps, whereby comets are kicked toward or into the TAZ. The spectral types of the responsible stars are marked, and these are seen to be red dwarfs like the majority of galactic field stars. Having noted the circumstances of all such decisive encounters, Rickman et al. (2012) were able to plot the encounter distances versus time in the past, and Fig. 11 shows these results for injections from the inner, central, and outer ranges of the Oort Cloud, marking by colors those for which the passing star would still be bright enough to be observable by Hipparcos (left) and Gaia (right), assuming the injected comets to be passing perihelia close to the present time.
This makes it clear that, while there is a small chance for the real culprit stars to be found in the Hipparcos catalogue (cf. García-Sánchez et al. 2001; Dybczyński 2006), the prospects are much better for the upcoming Gaia mission. However, it is not enough for a culprit star to be present in the Gaia database, whose size is estimated at approximately 109 objects. Good radial velocity measurements (likely arising from ground-based follow-up work) together with high accuracy proper motions from Gaia itself will be necessary to narrow down the list of candidates. In fact, stars with proper motions small enough to be unmeasurable even by Gaia may be among the best candidates, even though only the presence of a very close passage may appear without information on the details. Moreover, it may still not be possible to associate a certain comet with a certain star, unless the cometary orbit is of very good quality—including its nongravitational effects.
The Influence of the Planets
Fouchard et al. (2013a) recently presented a model for treating the planetary perturbations in long-term Oort Cloud simulations. Here, the giant planets are moving on unperturbed, circular and coplanar orbits, and the terrestrial planets are neglected. Whenever the comets come close enough to the planetary system to be significantly perturbed, the planetary perturbations are accurately integrated in the simple model just described, and the result is treated as an instantaneous shift of the barycentric orbit of the comet, applied at the time of pericenter.
An important result on the statistics of orbital energy perturbations caused by the planets, noted by Fouchard et al. (2013a), is illustrated in Fig. 12. It was seen in the 'From the Inner Core to Observable Comets' section, when describing the results by Kaib and Quinn (2009), that these results negate the simple model of a fully opaque Jupiter-Saturn barrier, where all comets with q < 15 AU are lost at once by planetary perturbations. As shown by Fig. 12, Fouchard et al. (2013a) verify Kaib and Quinn's result quantitatively. The plotted opacity parameter (P) represents the chance for a comet approaching the planetary system with a = 20,000 AU (energy-wise, in the middle of the Oort spike range) to be kicked out of the range by planetary perturbations during one apparition.
As seen, for observable orbits (q < 5 AU), P is not very far from unity on the average. But it falls rapidly with increasing q and approaches zero already before reaching the outer border of the q < 15 AU loss cone. From this, it is seen that the step function usually assumed for P(q) is far from reality, and that the Jupiter-Saturn barrier is very “leaky.” Moreover, the leakiness is inclination dependent, which is interesting in view of the possibility that comets “creeping” into observable orbits through the barrier instead of jumping across it may have a preference for retrograde orbits (Fernández 2002).
The first full Oort Cloud simulations describing the dynamics over 5 Gyr including both planets and external perturbers have been done by Fouchard et al. (Forthcoming). The resulting Oort spike, representing current, nonshower conditions, is shown in Fig. 13. The histogram shows the relative numbers of comets passing their first perihelia with q < 5 AU in different energy bins, and different colors are used to distinguish between four injection scenarios, as explained in the figure caption. On the one hand, comets arriving directly from pre-injection orbits with q > 15 AU (“jumpers”) are separated from those whose preceding perihelia were in the range of q from 5 to 15 AU (“creepers”). On the other hand, comets whose energies increased by more than 1 × 10−5 AU−1 during the perihelion passage preceding the injection due to planetary perturbations are broadly referred to as “Kaib-Quinn comets,” whereas the others are not.
The most striking features are that creepers are just as important as jumpers in terms of their integrated contribution, and that a large fraction of the injected comets—especially the creepers—may be referred to the Kaib and Quinn (2009) mechanism. Note, however, that the definition here adopted for the Kaib-Quinn comets is rather liberal, as a significant fraction of them may have had semimajor axes exceeding 20,000 AU already before the pre-injection perihelion passage and thus may not have passed directly from the inner part of the Oort Cloud by the aid of the planets. Another noteworthy point is the significant number of Kaib-Quinn jumpers, which highlights the occurrence of relatively large energy perturbations due to the planets even for q > 15 AU.
An indication as to the long-term origin of the injected comets is provided by the green histogram in Fig. 13. This shows the distribution of initial orbital energies at the beginning of the 5 Gyr simulation for the same injected comets (without separating the different injection scenarios). It is seen that a significant outward migration has taken place during the aeons and that, while the injections mostly probe the outer part of the Oort Cloud, the majority of comets stem from the inner part. Note, however, that this result is dependent on the initial energy distribution assumed for the cloud—in the case of Fig. 13, based on Duncan et al. (1987).
This may have an important implication for judging Oort Cloud formation scenarios. For that purpose, the mass of the cloud is very important, but the current mass is not as important as the initial mass. From the present results, we may infer that the more centrally condensed the initial structure of the cloud was, the less is the fraction of this mass that has been lost into interstellar space, and the larger is the fraction that is currently providing new comets into the inner solar system.
It is already interesting to see from Fig. 13 that including planetary perturbations into Oort Cloud dynamics increases the injection efficiency quite substantially, in line with the earlier conclusions by Emel'yanenko et al. (2007) and Kaib and Quinn (2009). This decreases the estimated number (and mass) of comets in the current Oort Cloud. But, moreover, inferring the initial mass is also dependent on the initial energy distribution of the cloud, and the more centrally condensed this is, the smaller is the inferred mass.
Such conclusions remain preliminary as long as the effects of GMC encounters or radial galactic migration of the solar system (Brasser et al. 2010) are not known. Therefore, for the time being, the ultimate questions concerning the origin of the Oort Cloud remain unanswered. Likewise, the issue of very massive objects orbiting within the Oort Cloud has not been fully resolved (Fernández 2011; Matese and Whitmire 2011).
What has indeed been achieved is an improved understanding of how comets may be injected from the cloud into observable orbits, and of how the cloud may evolve in the long term due to the dynamical effects so far modeled. This nurtures a hope that—in a not too distant future—we may be able to probe the structure of the cloud “observationally” by using the energy distribution of accurately determined, original comet orbits. Such a probing would also benefit from future discoveries of long-period comets with larger perihelion distances using new, powerful survey telescope facilities. Thus, a much wanted, further constraint on Oort Cloud formation scenarios would be obtained.
Many of the results described in this paper were achieved thanks to Polish Government Research Grant NN 203 392 734. I also wish to acknowledge Grants No 74/10:2 of the Swedish National Space Board and No 2011/01/B/ST9/05442 of the Polish National Science Center for support in the preparation of this paper. Finally, my warm thanks go to my co-workers Marc Fouchard, Christiane Froeschlé, and Giovanni Valsecchi for help and comments on this paper; Małgorzata Królikowska-Sołtan and Piotr Dybczyński for fruitful discussions on all the topics described; and Julio Fernández for valuable referee remarks.