## Abstract

We have calculated the number of dormant, nearly isotropic Oort cloud comets in the solar system by (i) combining orbital distribution models with statistical models of dormant comet discoveries by well-defined surveys and (ii) comparing the model results to observations of a population of dormant comets. Dynamical models that assume that comets are not destroyed predict that we should have discovered ∼100 times more dormant nearly isotropic comets than are actually seen. Thus, as comets evolve inward from the Oort cloud, the majority of them must physically disrupt.

It has been over half a century since Jan Oort first argued that a roughly spherical cloud of comets, which extends to heliocentric distances larger than 100,000 astronomical units (AU), surrounds the solar system (1). This structure, which is now known as the Oort cloud, is currently feeding comets into the inner solar system (with perihelion distances,*q*, of less than 3 AU) at a rate of about 12 comets per year with an active comet absolute magnitude, *H*
_{10}, <10.9 (2, 3). These comets as a whole are known as nearly isotropic comets (NICs) (4). NICs can be divided into the following two subpopulations, based on their dynamical histories (5): (i) dynamically new NICs, which are on their first pass through the system and typically have semi-major axes,*a*, greater than ∼10,000 AU, and (ii) returning NICs, which have previously passed through the inner solar system and typically have *a* ≤ 10,000 AU.

One unsolved problem is that models of the orbital evolution of new NICs into returning NICs consistently predict many times more returning comets than are observed (2, 6). This so-called “fading problem” cannot be due to previously unmodeled dynamical effects (2) and thus must be due to the physical evolution of the comets' activity (7). An important issue, therefore, is to determine the fate of the missing comets; either they become extinct or dormant (8), or they disintegrate entirely (9, 10). Here, we try to distinguish between these two possible outcomes by comparing model results to observations of dormant comets.

Large ground-based surveys have discovered 11 asteroidal objects, as of 3 December 2001, that are on orbits consistent with active NICs with *q* < 3 AU (Table 1) (11) [see supporting online material (SOM)]. These 11 objects represent just a small fraction of the total population of dormant NICs, because ground-based surveys suffer from unavoidable observational biases (12). Thus, the main purpose of the work presented here is to estimate the total number of dormant NICs based on the available data. We accomplish this by the following steps: (i) we use numerical simulations of cometary dynamics to produce a set of fictitious dormant NICs, (ii) we run these fictitious NICs through a near-Earth object (NEO) survey simulator to determine which ones would be discovered, and (iii) we compare the results of (ii) to observations of the known dormant NICs to estimate the total number and orbital element distribution of the entire real dormant NIC population.

We determined the expected orbital element distribution for the dormant NICs from long-term dynamical simulations that track thousands of fictitious new comets entering the planetary system from the Oort cloud for the first time. The simulations calculate the dynamical evolution of these objects' orbits caused by the gravitational influence of the Sun, planets, and Milky Way Galaxy. The objects' trajectories are followed until they are either ejected from the solar system, hit a planet, or strike the Sun. From this, we can develop a steady-state distribution of NICs by assuming that the influx rate of dynamically new comets is constant with time.

We used simulations that were performed elsewhere (2,13). Because of differing computational challenges, these simulations have divided returning NICs into two subclasses: external returning comets (ERCs) with periods greater than 200 years, and Halley-type comets (HTCs) with orbital periods less than 200 years (14, 15).

We used Wiegert and Tremaine's model (2) to determine the orbital element distribution that dormant ERCs would have if there were no disruptions. We also adopted the standard form for the cumulative absolute magnitude, *H*, distribution of *N*(<*H*) ∝ 10^{αH}(16–18), where *N* is the total number of objects brighter than *H* and α is the slope of the*H* distribution. We set α = 0.28. This value of α is determined directly from our models of the HTCs, which are described below. It is much smaller (or the *H* distribution is much shallower) than is typically assumed. However, it is consistent with a recent observational study of active comets (19).

With the above distribution of ERCs, we then used a survey simulator described in (20) to determine which objects would be discovered by modern NEO surveys. The survey simulator discovered 1 out of every 22,000 dormant ERCs with *q* < 3 AU and*H* < 18 (21) in Wiegert and Tremaine's model (2). This result, combined with the fact that only 2 dormant ERCs actually have been discovered thus far (Table 1) (also see SOM), implies that there are a total of 44,000 ± 31,000 dormant ERCs that are brighter than *H* = 18 and have a*q* < 3 AU in the solar system (22).

The ERCs have orbital periods that are so long that it is traditional to express the population numbers in terms of the number of objects that pass through perihelion per year. Based on the two objects thus far discovered and the mean inverse orbital period of objects in Wiegert and Tremaine's model (2), we estimate that there should be 3.9 ± 2.7 dormant ERCs with *H* < 18 and*q* < 3 AU passing perihelion per year. Assuming no disruptions, Wiegert and Tremaine's model predicts that there should be ∼400 dormant ERCs with *q* < 3 AU passing perihelion per year (see SOM). The discrepancy between our estimate based on observing dormant NICs and the one based on dynamics alone implies that, when a comet becomes inactive, it only has a ∼3.9/400, or 1%, chance of becoming dormant. We can only conclude that the other ∼99% of these objects must have disrupted.

To perform these calculations, we assumed a value of α = 0.28. We must therefore estimate how this choice affects our result concerning the disruption of ERCs. Previous estimates of α have ranged from 0.28 to 0.53 (23). As we discussed above, if we assume that α = 0.28, our survey simulator discovers 1/22,000 of the dormant ERCs with *H* < 18. If α were 0.53, our survey simulator would discover 1/43,000 of these objects, which is only a factor of 2 different from the α = 0.28 result. Thus, changing α does not change our conclusion that ∼99% of the inactive ERCs disrupt.

We now turn our attention to the HTCs. Levison *et al*. (13) studied the dynamical evolution of comets from the Oort cloud into HTC-like orbits (restricting themselves to HTCs with*q* < 2.5 AU). As with the ERCs, Levison *et al*. found a significant fading problem with the HTCs. In particular, Levison *et al*.'s models predict that there should be more than 15,000 active HTCs in the solar system, whereas the debiased estimate of active HTCs based on the observed population would suggest that there are only 50 (13). Thus, they concluded that ≥99% of these comets disappeared.

To determine whether HTCs become dormant or disrupt, we compare a model of dormant HTC discoveries (i.e., we take the results of dynamical models with no disruption and run them through our survey simulator) to actual discoveries. If we adopt Levison *et al*.'s models unaltered (see SOM), we are unable to construct models of the dormant HTC population that are consistent with observations (Fig. 1). Although the models produce reasonable inclination distributions (Fig. 1D), they fail to reproduce the observed semi-major axis distribution (Fig. 1C). In particular, the models predict a far larger number of dormant HTCs with*a* < 10 AU than has been observed.

One reason Levison *et al*.'s model fails could be that it does not adequately account for the fact that comets suffering a close encounter with the Sun are more likely to disrupt than are comets that have larger perihelion distances. Thus, we added a*q*-sensitive disruption law (24) to Levison *et al*.'s model and assumed that the absolute magnitude power law index is α = 0.28 (as before). We then recalculated Levison *et al*.'s fitting procedures and determined the model that provides the best fit to both the active and dormant HTC populations (see SOM). This model shows good agreement with the observations (Fig. 1).

Up to this point, we have been assuming a value of α. We could not determine α for the ERC simulations because there are only two known dormant ERCs. There are enough known dormant HTCs (nine of them), however, to estimate α directly. The cumulative absolute magnitude distribution (25) for the observed dormant HTCs is clearly inconsistent with those detected by our survey simulator for values of α as large as 0.4 (Fig. 2). Indeed, the probability that the observed and modeled absolute magnitude distributions are drawn from the same parent distribution (fig. S2) shows that we can rule out any α larger than ∼0.35 and that the best fit is α = 0.23 ± 0.04 (1σ). We decided to adopt Weissman and Lowry's (19) value of 0.28 because it is consistent with our results and yet is based on a larger data set. The differences between models using α = 0.28 and α = 0.23 are small, and thus this choice will not measurably affect our results.

With α = 0.28, the survey simulator discovers 1.1% of the dormant HTCs. Given that the NEO surveys have discovered 9 dormant HTCs (Table 1), we conclude that there are 780 ± 260 dormant HTCs in the solar system with *H* < 18 and*q* < 2.5 AU.

We can now compare our estimate of the number of dormant HTCs to what we would expect from dynamical models, assuming no disruption (26). Our model predicts that there should be 46,000 active and dormant HTCs with *q* < 1.3 AU, which implies ∼106,000 with *q* < 2.5 AU. Because dormant comets far outnumber active comets (27), we can conclude that ∼99% of ERCs disrupt before becoming HTCs. This percentage is similar to the fraction predicted for the ERCs alone.

Jupiter-family comets (JFCs) do not appear to disrupt at the same rate (28) as do the NICs. Bottke *et al*. (29) estimated the total number of dormant JFCs from the known population of NEOs, finding that there are 61 ± 43 dormant JFCs with *q* < 1.3 AU and *H* < 18. This number is consistent with estimates from dynamical simulations, assuming that all objects become dormant rather than disrupt [∼60_{−53}
^{+40}% become dormant (30)]. Despite large uncertainties in this estimate, it is clear that a substantial fraction of JFCs must become dormant, and thus they behave differently from NICs (see SOM for additional arguments).

It is surprising that NICs and JFCs behave so differently, because they are thought to be composed of similar mixtures of ice and rock. Their different disruption behaviors could be primordial, reflecting the chemical or physical characteristics of their formation locations. Most Oort cloud comets are believed to have formed in the region of the giant planets (1, 31), whereas JFCs are thought to have formed in the Kuiper belt beyond the giant planets (32–34). However, recent simulations of Oort cloud formation (35) suggest that ∼30% of the present-day Oort cloud originated in the Kuiper belt (although most of these objects left the Kuiper belt a long time ago). If these models are correct, then the different disruption behaviors cannot stem from primordial differences, because the fraction of NICs that originated in the Kuiper belt is far larger than the ∼1% that avoid disruption.

Alternatively, evolutionary processes could affect comets' susceptibility to disruption. For example, over long time scales, Kuiper belt comets could have lost more volatiles than did Oort cloud comets because Kuiper belt comets have been stored at closer heliocentric distances and thus higher temperatures. Kuiper belt objects could be more porous, and thus less susceptible to disruption resulting from volatile pressure buildup, due to a relatively violent collisional environment (36). Finally, the dynamical pathways that NICs and JFCs take on their way into the inner solar system might lead to different thermal histories for the two populations. In one orbital period, most NICs evolve from distant orbits (with perihelia outside the planetary region) to orbits that closely approach the Sun. On the other hand, objects from the Kuiper belt slowly move through the planetary region, taking ∼10 million years to evolve onto orbits with *q* < 2.5 AU (33). It has been argued previously (37) that different thermal histories could lead to different disruption rates, so perhaps NICs disrupt because of strong thermal gradients or volatile pressure buildup, whereas JFCs survive because they are warmed more slowly.

↵* To whom correspondence should be sent. E-mail: hal{at}gort.boulder.swri.edu