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Earth, Mo on, and Planets

A Mo del for the Common Origin of Jupiter Family and Halley Type Comets
V.V. Emel'yanenko · D.J. Asher · M.E. Bailey

This is the version accepted for publication, published online in 2013 January, printed publication to follow. The pap er can b e cited as Emel'yanenko et al. (2013) Earth, Mo on, and Planets, DOI 10.1007/s11038-012-9413-z The final publication is available at www.springerlink.com.

Received: 27 April 2012 / Accepted: 31 Decemb er 2012

Abstract A numerical simulation of the Oort cloud is used to explain the observed orbital distributions and numbers of Jupiter-family and Halley-type shortperiod comets. Comets are given initial orbits with perihelion distances between 5 and 36 au, and evolve under planetary, stellar and Galactic perturbations for 4.5 Gyr. This process leads to the formation of an Oort cloud (which we define as the region of semima jor axes a > 1000 au), and to a flux of cometary bodies from the Oort cloud returning to the planetary region at the present epoch. The results are consistent with the dynamical characteristics of short-period comets and other observed cometary populations: the near-parabolic flux, Centaurs, and high-eccentricity trans-Neptunian ob jects. To achieve this consistency with observations, the model requires that the number of comets versus initial perihelion distance is concentrated towards the outer planetary region. Moreover, the mean physical lifetime of observable comets in the inner planetary region (q < 2.5 au) at the present epoch should be an increasing function of the comets' initial perihelion distances. Virtually all observed Halley-type comets and nearly half of observed Jupiter-family comets come from the Oort cloud, and initially (4.5 Gyr ago) from orbits concentrated near the outer planetary region. Comets
V.V. Emel'yanenko Institute of Astronomy RAS, 48 Pyatnitskaya, Moscow, 119017, Russia E-mail: vvemel@inasan.ru D.J. Asher, M.E. Bailey Armagh Observatory, College Hill, Armagh, BT61 9DG, United Kingdom

that have been in the Oort cloud also return to the Centaur (5 < q < 28 au, a < 1000 au) and near-Neptune high-eccentricity regions. Such ob jects with perihelia near Neptune are hard to discover, but Centaurs with characteristics predicted by the model (e.g. large semima jor axes, above 60 au, or high inclinations, above 40 ) are increasingly being found by observers. The model provides a unified picture for the origin of Jupiterfamily and Halley-type comets. It predicts that the mean physical lifetime of all comets in the region q < 1.5 au is less than 200 revolutions. Keywords Comets · Oort cloud · Centaurs · Solar system formation · celestial mechanics 1 Intro duction 1.1 Origin of short-period comets Explaining the origin of short-period (SP) comets (periods P < 200 yr) is a long-standing problem. The main difficulty lies in the differences and apparent inconsistency between the respective numbers and orbital distributions of Jupiter-family (JF) and Halley-type (HT) comets. These we classify using the Tisserand parameter T with respect to Jupiter (Carusi et al. 1987), JF comets having T > 2 (and P usually below 20 years), HT comets having T < 2 (and P usually between 20 and 200 years). When SP comets are classified this way the number of observed HT comets is found to be less than, or at most comparable to, the number of observed JF comets (see Section 2.1 below). However, most dynamical theories of their origin from the


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near-parabolic flux predict a far greater proportion of HT comets (Emel'yanenko and Bailey 1998), with the overall number of observed JF comets conversely being much too large relative to the calculated number (Joss 1973; Delsemme 1973). This discrepancy is associated with the well-known fading problem for longperiod comets originating in the Oort cloud and has, at least in part, led to the idea that the two classes of SP comet may have different physical structures and different proximate sources in the present Solar system. Thus, although there have been many advances in understanding the diverse populations of small bodies in the Solar system, neither a single source dominated by trans-Neptunian ob jects nor one dominated by the traditional Oort cloud near-parabolic flux at small perihelion distances seems capable of explaining the entire distribution of orbital elements of SP comets. In particular the observed JF comet inclination distribution was recognized to have too many comets at low i relative to the calculated distribution (Duncan et al. 1988; Quinn et al. 1990). For these reasons, the ma jority of authors nowadays consider JF and HT comets to be physically as well as dynamically distinct classes, presumably formed in separate regions of the early Solar system and having different dynamical and physical evolutionary histories. Under this viewpoint, JF comets are often regarded as originating largely in the proto-planetary disc beyond Neptune, for example in or close to the EdgeworthKuiper belt (EKB). The idea that JF comets might originate in a primordial disc or `belt' of comets located near or beyond the orbit of Neptune was investigated by a number of authors (e.g. Fernandez 1980, 1982; ´ Duncan et al. 1988; Torbett 1989; Torbett and Smoluchowski 1990; Quinn et al. 1990). The discovery of 1992 QB1 (Jewitt and Luu 1993) and of further EdgeworthKuiper ob jects played a pivotal role in theories of the origin of SP comets, and important advances building on this evidence were made in particular by Duncan et al. (1995), Duncan and Levison (1997), Levison and Duncan (1997), and Levison et al. (2001). A key point (Duncan and Levison 1997) was recognition of the potentially important role played by the `scattered' disc, introduced by Torbett (1989) and detected a few years later (Luu et al. 1997), in which it appears that the scattered disc of primordial ob jects originally formed in the region of the ma jor planets is the principal source of observed JF comets, rather than the EKB. Under the viewpoint of distinct JF and HT classes, HT comets are regarded as ob jects captured from the Oort cloud (Levison et al. 2001), a structure that would have been produced inevitably as a by-product of planetary, stellar and other perturbations acting on planetesimals or

cometary nuclei originally formed by accretion within the planetary region of the proto-planetary disc. However, a rather unsatisfactory feature of this general picture is the assumption that HT comets coming from the Oort cloud must disintegrate very quickly in order to explain the small number of ob jects observed (Emel'yanenko and Bailey 1998; Levison et al. 2001). The number of observed inert HT `asteroids' is also very small, and it seems as if the disintegration of a kilometre-size comet nucleus, into presumably an initial trail of much smaller boulder-size ob jects and then finally dust, must proceed fairly rapidly and lead to eventual extinction of the original comet. On the other hand, dynamical theories appear to require that a high proportion of the JF comet source flux should survive dynamical transfer into the inner Solar system to become active JF comets and that these JF comets should survive for 103 revolutions in the inner Solar system. This difference in the physico-dynamical evolution of the two types of ob jects is the fading problem for SP comets. It is probably not unreasonable to assume that comets that formed in different parts of the proto-planetary disc have different physical properties and therefore different lifetimes in the observable region, and it appears that this idea has become very deeply rooted. What is missing, however, is direct observational evidence to support the idea of two qualitatively distinct types of SP comet, correlating with dynamical class. Thus, present theories of the origin of SP comets rely on a poorly understood fading hypothesis to accommodate the observations, and there is no satisfactory physico-dynamical explanation as to why two very different types of SP comet should exist and yet appear observationally almost indistinguishable. Indeed, although comets show a very diverse range of properties, covering a very broad range of sizes, densities, dust-to-gas ratios and so on, there is as yet no compelling observational evidence for the expected bimodality of physical characteristics corresponding to HT versus JF dynamical class (Lamy et al. 2004). In this work, whilst recognizing that comets may have different physical properties depending for example on their sizes or where they might have formed in the proto-planetary disc, we present a model for the common origin and evolution ­ from the Oort cloud ­ of the ma jority of comets in the Solar system. 1.2 Role of the Oort cloud We define the Oort cloud as the region containing objects with semima jor axes a > 103 au (i.e., ob jects from the Oort cloud have at some point during their


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evolution reached a > 103 au). This definition is consistent with those used by other authors; e.g. Wiegert and Tremaine (1999) and Rickman et al. (2008) used similar values of semima jor axis, i.e. a 1­3 â 103 au, to define the inner boundary of the Oort cloud. Dones et al. (2004) introduced a further restriction, namely that the maximum value of perihelion distance q during an ob ject's orbital evolution should exceed 45 au for it to be counted as an `Oort cloud' ob ject. However, ob jects with a > 103 au spend nearly all their time at large heliocentric distances, whatever their value of q , and therefore in the Oort cloud. In this paper we have chosen to define Oort-cloud ob jects solely according to a because the influence of stellar and Galactic perturbations is determined mainly by a for near-parabolic orbits. It has been shown (e.g. Emel'yanenko 2005) that the dynamical pathways by which ob jects with a > 103 au reach the planetary region are different from those of typical trans-Neptunian ob jects (TNOs). While the evolution of TNOs is largely determined only by planetary perturbations, stellar and Galactic perturbations play a more substantial role in the process that drives the perihelia of ob jects with a > 103 au towards and through the planetary region, regardless of their previous q . In the present paper (Section 3 onwards) we numerically integrate a much larger number of ob jects than in the Oort cloud model of Emel'yanenko et al. (2007), in particular to obtain statistically significant numbers of SP comets captured from the Oort cloud and allow a comparison of the model SP numbers and orbital distributions with the corresponding distributions of observed HT and JF comets. First (Section 2), in order that our model parameters can be constrained by observations, we assess the known characteristics of the various populations of cometary bodies. 2 Principal features of observed cometary p opulations 2.1 Short-period comets We took data from the MPC (Minor Planet Center) and JPL (Jet Propulsion Laboratory) lists of discovered comets with P < 200 yr and q < 1.5 au near the present epoch. The completeness level in the discovery of SP comets is slightly uncertain, especially for HT comets and when results are extrapolated to fainter magnitudes and larger perihelion distances. However, many discussions (e.g. Fernandez et al. 1999; Levison ´ et al. 2001) have indicated a relatively high degree of completeness in the observed sample of active comets at small perihelion distances (q < 1.5 au). This level of

35 30

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Semimajor axis, AU

20 15 10

5 0
-1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0

Tisserand parameter
Fig. 1 The distribution of T and a for observed short-p eriod comets with q < 1.5 au.

completeness is supported too by studies of long-period comets, essentially none of which have been missed at q < 1.3 au since 1985 (Fernandez and Sosa 2012). ´ We excluded SOHO comets because these have rather uncertain physical and dynamical characteristics; this only affects the distribution near very small q , a region that we do not study here. We also excluded multipleapparition comets that have not been observed for a number of revolutions and are now treated as dead or inert. For split comets we took only the orbit of the main nucleus. In the end we obtained a list of 103 observed ob jects that we regard as representing the present-day set of active SP comets with q < 1.5 au. Of these, 75 have T > 2 (JF comets) and 28 have T < 2 (HT comets). Figures 1 and 2 present orbital element distributions of these observed ob jects. The inclinations (Fig. 2) show JF comets (T > 2) are concentrated close to the ecliptic and prograde HT comets outnumber retrograde ones (Fernandez and Gallardo 1994; Levison et al. 2001). ´ Additionally the intrinsic numbers of JF and HT comets are a key constraint for our model. Fernandez et ´ al. (1999) found that about a hundred active JF comets should currently exist in the region q < 1.5 au, down to


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150

in the region q < 1.5 au at times near the present epoch. Certainly the number of already known active JF comets shows that their intrinsic number cannot be much below a hundred, while the intrinsic HT number cannot be much above a hundred without implausibly many bright comets being missed by observational searches.

120

Inclination, deg

2.2 Near-parabolic flux

90

60

30

0 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0

Tisserand parameter
Fig. 2 The distribution of T and i for observed short-p eriod comets with q < 1.5 au.

nuclear radius R0.7 km. The number appears to drop very rapidly for smaller bodies (Fernandez et al. 1999; ´ Snodgrass et al. 2011). This estimate could be modified to take account of more recent comet discoveries (cf. Section 2.1 of Di Sisto et al. 2009) but the result would not be significantly changed. For HT comets, their longer average orbital periods constitute the principal bias against their discovery relative to JF comets. That is, although we expect that most active comets passing perihelion with sufficiently small q will be found at the current level of observational searches, many HT comets have not yet returned to perihelion during the last few decades when searches have been at such levels. In this way, taking account of the HT period distribution, Levison et al. (2001), from 22 observed HT comets with q < 1.3 au, estimated a population of 57 active HT comets (q < 1.3). This result can be extrapolated to about a hundred ob jects with q < 1.5 au. In a later paper (Levison et al. 2006) the observed number has only increased to 24, suggesting the estimate is reliable. We conclude that there are roughly a hundred active JF comets and a comparable number, i.e. approximately a hundred, of active HT comets to be explained

The flux of dynamically new comets from the Oort cloud is a fundamental parameter underpinning all dynamical models of the small-body populations in the Solar system, including the estimates in this paper. There are uncertainties in the frequency, new , of comets with a > 104 au passing perihelion per au in q per year, but new is usually estimated to lie in the range 2 to 4 for present-day comets in near-Earth space (Bailey and Stagg 1988; Fernandez and Gallardo 1999; Wiegert ´ and Tremaine 1999; Francis 2005). For quantitative estimates in this paper we adopt new = 2.5, within the observable region q < 1.5 au. Francis (2005) undertook a detailed discussion of the ob jects that the LINEAR survey should discover for a given intrinsic cometary population. Considering also the question of the cometary absolute magnitude distribution, he found that very faint (on average presumably smaller) comets are only slightly more abundant than somewhat brighter (presumably larger) ones. Thus statements about cometary numbers, while evidently depending in detail on the adopted absolutemagnitude cutoff, are not strongly dependent on the precise value of that cutoff. In order to fix ideas, our adopted value new = 2.5 comets with a > 104 au passing perihelion per au in q per year may be assumed to apply to comets with total visual absolute magnitudes H10 < 11. The quantity H10 is the magnitude normal ized to 1 au from Earth and Sun (e.g. Everhart 1967). The inclusion of fainter comets (e.g. extrapolating results from H10 = 11 to H10 = 16) makes very little practical difference to our results (Francis 2005; Sosa and Fernandez 2011), although the calibration factor, ´ new , would of course increase. The relative lack of very < small (diameters d 0.5 km) comets (Fernandez and ´ Sosa 2012) suggests that the physical response of the smallest dynamically `new' comets from the Oort cloud to the thermal shock of their first passage at small perihelion provides a clue to the underlying rapid fading of new comets from the Oort cloud, necessary to explain the detailed shape of the observed 1/a-distribution (cf. Bailey 1984).


Common origin of Jupiter family and Halley typ e comets Table 1 Centaurs (ob jects with 5 < q < 28 au and a < 1000 au, excluding a few resonant trans-Neptunian ob jects and Tro jans) that have a probable source in the Oort cloud. The ma jority of such ob jects have a > 60 au and after observational debiasing would b e extremely numerous. Centaurs with a < 60 au are listed if i > 40 . Only Centaurs with an observational arc larger than 100 days (asteroid orbits from MPC) and comets of orbital Classes 1 and 2 (Marsden and Williams 2008) are included. A unified classification scheme for Centaurs was prop osed by Horner et al. (2003). a au (29981) 1999 TD10 (87269) 2000 OO67 (127546) 2002 XU93 2003 FH129 (65489) 2003 FX128 2004 VH131 2005 VD (308933) 2006 SQ372 2007 JK43 2007 UM126 2008 KV42 (315898) 2008 QD4 2008 YB3 2009 MS9 2009 YD7 2010 BK118 2010 JJ124 2010 NV1 2010 WG9 C/1984 U1 C/1998 M6 C/1999 K2 C/2001 Q1 C/2002 K2 C/2002 P1 C/2002 VQ9 C/2003 J1 C/2005 R4 C/2007 D3 C/2007 K1 99 653 66 71 100 60 6 906 46 12 41 8 11 386 129 447 82 294 53 646 972 145 176 561 497 189 514 914 765 425 .4 .8 .3 .8 .7 . . . . . 1 9 7 4 7 q au 12.3 20.8 21.0 27.6 17.8 22.3 5.0 24.2 23.6 8.5 21.2 5.4 6.5 11.0 13.4 6.1 23.6 9.4 18.8 5.5 6.0 5.3 5.8 5.2 6.5 6.8 5.1 5.2 5.2 9.2 i deg 6 20 78 19 22 12 173 19 45 42 104 42 105 68 31 144 38 141 70 179 92 82 67 131 35 71 98 164 46 108

5

has often been taken to mean a < 30 au, i.e. less than the semima jor axis of Neptune. In contrast, following Emel'yanenko et al. (2007), we define Centaurs as small bodies moving in heliocentric orbits with 5 < q < 28 au and a < 1000 au (with any value of i), excluding a few resonant trans-Neptunian ob jects and Tro jans. Thus, many ob jects that we call Centaurs (cf. Horner et al. 2003, 2004a,b) would be classified by some other authors as scattered-disc ob jects. The condition q < 28 au separates Centaurs from the NNHE region described in Section 2.4. Our Centaur definition reflects the fact that this entire region of orbital element phase space (a < 1000 au and any i) constitutes a transition region of dynamically shortlived orbits in which population numbers and orbit distributions provide vital evidence about the outer Solar system source regions. So whereas a significant number of Centaurs are produced by dynamical evolution from the Kuiper belt or the trans-Neptunian region (e.g. Tiscareno and Malhotra 2003; Volk and Malhotra 2008), we emphasize that using a similar definition of a Centaur to that used in this paper, Emel'yanenko et al. (2005) showed the debiased distribution of observed Centaurs contradicts the idea that Centaurs primarily originate from a flattened disc-like population. They inferred instead that the Oort cloud produces 90% of Centaurs, specifically well over 90% of Centaurs that have a > 60 au (which themselves constitute 90% of the Centaur population after observational debiasing) and 50% of Centaurs with a < 60 au. Of these a < 60 Centaurs, the Oort cloud contributes especially to those with i > 40 . Observational evidence for Centaurs with these orbital characteristics is growing (Table 1), consistent with predictions (Emel'yanenko 2005; Emel'yanenko et al. 2005) that a significant number of Centaurs have a proximate source in the Oort cloud. Emel'yanenko et al. (2005) concluded that there were two separate but overlapping dynamical classes of Centaurs, one originating in the Oort cloud and the other from the observed near-Neptune high-eccentricity region, each source region producing 50% of Centaurs with a < 60 au and 50% of JF comets. A bimodal colour distribution is observed in Centaurs (Peixinho et al. 2003). The only presently apparent difference in the two groups' orbital properties is that red Centaurs tend to have lower i (Tegler et al. 2008), while Peixinho et al. (2012) instead find that the bimodality is only pronounced in smaller ob jects. A dynamical evolution study suggests red Centaurs have spent less time at small q (Melita and Licandro 2012).

.9 .8

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2.3 Centaurs Centaurs are an intermediate cometary population (including active comets and inactive apparent asteroids), some of them being en-route from the outer Solar system to near-Earth space and the SP comet region. As a transition population the Centaurs must be replenished from a more distant source, presumably located either in the trans-Neptunian region or the Oort cloud, and they play a pivotal role in constraining theories of the origin of SP comets. There is however no abiding consensus on the exact definition of a Centaur. Many authors (e.g. Stern and Campins 1996; Gladman 2002; Gladman et al. 2008; Jewitt 2009) adopt the criterion that a Centaur should orbit largely in the region of the outer planets. This


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2.4 Trans-Neptunian ob jects

that may have encountered Uranus and Neptune during an early phase of evolution of the Solar system and somehow survived to the present day without ever havAs with Centaurs the nomenclature is not universal. ing evolved as far as the Oort cloud (a > 103 au). For example (Gladman et al. 2008) in some classifiIn our model (Section 3.1), for example, 6% of parcation schemes the term `Kuiper belt' can mean the ticles that had initial perihelion distances in the range union of the `classical' Kuiper belt, the scattered disc, 25 < q0 < 36 au survived to the present day withthe `extended' (or detached) scattered disc and resoout entering the Oort cloud or reaching any other endnant ob jects exterior to the Neptune Tro jans, the whole state of the model (Emel'yanenko et al. 2007). This region sometimes being described simply as the transmeans that there is likely to be a significant number Neptunian region. of surviving ob jects in this region whose orbits would We define the trans-Neptunian region as the part appear to be very long-lived and which previous work of the Solar system in the vicinity of and beyond Nephas shown might possibly be a significant source of SP tune but interior to the Oort cloud, containing transcomets (Duncan and Levison 1997, Emel'yanenko et al. Neptunian ob jects (TNOs) with a < 103 au. This region 2004). contains a complex, overlapping population of dynamA third class of TNO comprises bodies that were ically distinct classes of small bodies. formed with original orbits in or close to the protoFirst there is the classical Edgeworth-Kuiper belt planetary disc, but which at some time in their orbital (EKB), a region estimated to contain a current total history became part of the Oort cloud (a > 103 au) mass of the order of 0.01­0.02 M (Bernstein et al. and are thus not `primordial' in the sense of the sec2004; Fuentes and Holman 2008). The observed EKB ond class above. Although most ob jects reaching the ob jects are widely believed to represent the remains Oort cloud still have a > 103 au at the present epoch, (perhaps less than 1%) of a massive primordial popua few evolve back to a < 103 au and so into the translation of ob jects originally formed in low to moderateNeptunian region. Our model produces many such obeccentricity orbits in the extended proto-planetary disc beyond Neptune (Stern 1995, 1996; Morbidelli and Brown jects, which we defined as `Oort Scattered Disc' (OSD) in Emel'yanenko et al. (2007). 2004). Non-resonant EKB ob jects cannot be the domWe define also the near-Neptune high-eccentricity inant source of observed JF comets as there are too (NNHE) region, by 28 < q < 35.5 au and 60 < a < 1000 few observed low-eccentricity orbits in this region with au. This region has an important dynamical characperihelia close enough to the orbit of Neptune to be terization, covering ob jects that come close enough to captured in sufficient numbers (see Emel'yanenko et al. Neptune's orbit to be captured. The q cutoff at 28 au, 2005). Resonant EKB ob jects can diffuse to other dyjust within Neptune's orbit and below which an ob ject namical populations over Gyr time-scales (Morbidelli becomes a Centaur, acknowledges the importance for 1997; Tiscareno and Malhotra 2009), but their escape coming under a planet's control of a particle's periherate is rather less than that of `scattered disc' ob jects lion distance (Horner et al. 2003). (Volk and Malhotra 2008), so that this scattered disc, Observed NNHE ob jects are an important source a declining and dynamically unstable population introof SP comets coming from the trans-Neptunian region duced by Duncan and Levison (1997), is a more impor(Emel'yanenko et al. 2004, 2005). Whether these obtant source of JF comets. For these reasons the classical served NNHE ob jects are the same as NNHE ob jects EKB is not part of our present model. produced as a result of dynamical evolution of ob jects A second class of `primordial' TNO (i.e. TNOs that into and subsequently from the Oort cloud remains to have never reached the Oort cloud) is a subset of the be determined. Section 5.1 concludes they are not, and `scattered disc' population. In this picture (Torbett 1989), therefore that the observed NNHE ob jects come from ob jects originally formed in the region of the giant plananother source than that considered here. ets are gravitationally scattered outwards in a to produce an extended, flattened disc-like structure. Whereas a primordial disc of ob jects beyond Neptune would be 3 Integrations characterized by low eccentricities and inclinations, according to many theories of cometary origin, the scat3.1 Model and methods tered disc is expected to contain ob jects on orbits having much higher eccentricities and substantial inclinaTo construct our Oort cloud model, following Emel'yanenko tions, perhaps merging smoothly into the unobserved et al. (2007), particles' initial conditions after the forbut massive inner Oort cloud described by Hills (1981). mation and migration of the planets had the original This second class of TNO therefore comprises ob jects semima jor axes uniformly distributed in the range 50 <


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a0 < 300 au. The original inclinations were distributed following a `sine law' scaled to the interval 0 < i0 < 40 ; the original perihelion distances were distributed uniformly in the range 5 < q0 < 36 au; and the original arguments of perihelion and original ascending nodes were distributed uniformly between 0 and 360. The inclination distribution (peaked at i0 = 20 falling to zero at 0 and 40 ) is similar to the model scattered disc i distribution adopted by Volk and Malhotra (2008) following Brown (2001). Our choice of q0 < 36 au is connected with the assumption that the Oort cloud was created by ob jects coming from the planetary region or its nearest vicinity. Although some ob jects with q0 > 36 au may reach the near-Neptune region (Duncan et al. 1995; Emel'yanenko et al. 2003), it is evident that their contribution to the Oort cloud is small because the rate of diffusion in perihelion distance is slow. While our choice of q0 assumes that comets were scattered to the Oort cloud region mainly by planetary perturbations, we do not use as initial conditions near-circular orbits in the planetary region (in contrast, for example, with Dones et al. 2004). Thus although it may be natural to assume that planetesimals formed in near-circular orbits are a source of Oort cloud comets, the accretional model of planetary formation still has so many difficulties and unclear questions that we deliberately avoid considering any particular hypothesis of comet formation a priori. Indeed the real situation with the initial orbital distribution of comets could be much more complicated than that described in Dones et al. (2004) even if comets were formed in near-circular, coplanar orbits. For example, planetary migration in the early Solar system appears to have been important in shaping the outer Solar system (Tsiganis et al. 2005). Moreover, the Sun may have formed in a denser stellar environment than it occupies now (Fernandez and ´ Brunini 2000; Levison et al. 2010). This makes assumptions about the distribution of comets in the early Solar system very uncertain. Instead our approach is to constrain some features of the cometary distribution in the early Solar system by analysing observed distributions of cometary ob jects in the present Solar system. The main aim is to show that there are models of the Oort cloud that can explain the observed distributions of JF and HT comets. Our Oort cloud model can be interpreted as providing some general constraints on aspects of the cometary orbital distribution during early stages of the Solar system's evolution. While details of the earliest stages of planetary and Oort cloud formation are beyond the scope of the present paper, we regard our Oort cloud as representing a general class of model in which cometary planetesimals, formed in the proto-planetary disc, have

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Semimajor axis, AU
Fig. 3 The distribution of a and i for all the observed multiple-opp osition high-eccentricity TNOs with q > 36 au. Data from MPC.

been scattered outwards by the planets to become subject to stellar and Galactic perturbing forces (cf. Duncan et al. 1987; Fernandez 1997; Dones et al. 2004; ´ Dybczynski et al. 2008; Leto et al. 2008). Hahn and ´ Malhotra's (1999) finding (their Section 4) that the total mass reaching the Oort cloud is quite insensitive to the orbital histories of the migrating planets tentatively supports our assertion that the precise details of planetary migration and comet formation are not relevant to our present purpose. It is for these reasons that we regard the `initial conditions' of our integrations as applying to the time after the Solar system's planetary migration phase. There are several further motivations for the choice of initial high-eccentricity (50 < a0 < 300 au; q0 in or near planetary region) rather than near-circular orbits. This range of a0 is sufficiently large that ob jects can reach it at an early stage of evolution on the way to the Oort cloud under a wide range of different assumptions of cometary formation. The choice of initial conditions also allows particles to experience planetary perturbations for a long time before reaching the Oort cloud region, the model's maximum value of a0 being much


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Emel'yanenko et al.

smaller than that used in a similar approach by Duncan et al. (1987). The choice of initial i, and initial a ranging above 200 au, is moreover expected in the scattered disc model with migrating Neptune (Gomes 2003). The main reason for our choice of initial orbits, however, is that the ma jority of high-eccentricity trans-Neptunian ob jects have orbits with 50 < a < 300 au and i < 40 . Figure 3 shows the distribution of a and i for discovered multiple-opposition ob jects with q > 36 au. This population of trans-Neptunian ob jects may preserve at least some memory of its original early Solar system distribution. Results for different initial models can be obtained by applying appropriate weights (Section 5.3). The initial orbits were integrated in a model Solar system taking full account of planetary perturbations. All ob jects that reached the Oort cloud (a > 103 au) were then evolved for the remaining age of the Solar system under the combined action of planetary, stellar and Galactic perturbations. In the present work, the 8925 ob jects that survived after 4.5 Gyr were cloned 200 times and integrated for a further 300 Myr including planetary, stellar and Galactic perturbations. The initial orbital distribution of these ob jects is shown in Figs. 1 and 2 of Emel'yanenko et al. (2007). In order to suppress any possible artefacts associated with the initial conditions of the 300 Myr integrations we analysed our results on the interval 50­ 300 Myr. We took account of perturbations from the four large planets Jupiter to Neptune, using the secular perturbation theory of Brouwer and van Woerkom (1950) and Sharaf and Budnikova (1967), adding the terrestrial planets' masses to the Sun. Ob jects were removed when q < 0.005 au or 1/a < 10-5 au-1 , or if they collided with planets. The orbital calculations used the symplectic integrator described in the papers Emel'yanenko (2002) and Emel'yanenko et al. (2003) unless and until the orbit reached q < 2.5 au and the symplectic integrator of Emel'yanenko (2007) beyond that. The former method solves the Hamiltonian equations of barycentric motion for test particles moving in the field of the Sun and planets. It uses an adaptive time-step that is a function of the distance r from the centre and of the magnitude of perturbations, and so can deal with both highly eccentric orbits and close planetary encounters. The timestep is almost proportional to r at small distances and in the absence of close encounters: in general it was 15 days at r = 5 au, and it did not exceed 900 days at any distance. For ob jects reaching q < 2.5 au, the time-step of the integrator was approximately equal to 4.9989r/, 4 where = 1+B r+ j =1 bj /j +1 /r3/2 , B =0.005549, =3, 1 =58, bj = aj (mj /3)1/3 , j is the distance be-

tween the ob ject and the perturbing planet, and aj and mj are the mass and the semima jor axis of the perturbing planet (j =1,2,3,4 for Jupiter, Saturn, Uranus and Neptune, respectively). The Galactic model is taken from Byl (1986), but with the Sun's angular speed 0 = 26 km s-1 kpc-1 and the mid-plane density of the Galactic disc in the Solar neighbourhood 0 = 0.1 M pc-3 following Levison et al. (2001). To model stellar perturbations the procedure of Heisler et al. (1987) was used.

3.2 Initial results We have previously shown that ob jects that have visited the Oort cloud (a > 103 au) at some time in their orbital history make a significant contribution to the observed classes of cometary ob jects in the Solar system (Emel'yanenko et al. 2007). Table 2 updates the results of that work using the present, more extensive simulations, adopting a present-day near-parabolic flux new = 2.5. The difference between the first three lines of this Table and the corresponding results in Table 2 of Emel'yanenko et al. (2007) are partly due to the assumed new = 2 in that paper and partly also due to statistical fluctuations in the relatively small number of ob jects considered in the earlier work. In the present Table 2, NOC is the total number of ob jects in the Oort cloud (a > 103 au) at the present epoch; and NI and NO are the corresponding numbers in the relatively flattened inner Oort cloud (103 < a < 104 au) and the more isotropic outer Oort cloud (a > 104 au) respectively. NS is the number of OSD ob jects (ob jects from the Oort cloud in the region q > 30 au, 60 < a < 1000 au, the `S' suffix indicating that they are located in the analogous region to the scattered-disc ob jects discussed by authors such as Duncan and Levison 1997), NN is the number in the NNHE region and NC is the number of Centaurs, also at the present epoch. In our model, NS , NN and NC represent the numbers of ob jects in these respective regions which have previously visited the Oort cloud . In order of magnitude, NS 3­4NN , the ma jority in orbits that do not strongly interact with Neptune, and NN 7­8NC . Finally, JF and HT are the corresponding presentday annual injection rates of cometary ob jects coming from the Oort cloud into JF and HT orbits with q < 1.5 au. The values JF and HT are `dynamical' injection rates, i.e. obtained by ignoring any effects of physical fading or disintegration. The total number of active JF and HT comets will depend (see below) on their respective dynamical and physical lifetimes as SP comets. It


Common origin of Jupiter family and Halley typ e comets

9

Table 2 The numb er of cometary ob jects in different dynamical classes at the present ep och. All figures are calibrated with an assumed near-parab olic flux new = 2.5. The first three lines show the total numb er in the Oort cloud and the contributions to this numb er from ob jects in the inner and outer Oort cloud resp ectively. The second three lines show the numb ers of OSD ob jects (NS ), NNHE ob jects (NN ) and Centaurs (NC ) coming from the Oort cloud. The final two lines indicate the present-day rate of production of new JF and HT comets from this Oort-cloud source into orbits with q < 1.5 au, neglecting any effects due to fading. The columns provide results for four different frequency distributions of initial - - p erihelion distance: three are prop ortional to q0 2 , q0 1 and constant in the range 5 to 36 au; the fourth is constant in the range 25 to 36 au and zero outside this range. Thus relative numb ers in the outer Solar system increase from left to right. For clarity the dynamical definitions used for these classes are then summarized. The NS and NN classes overlap; we primarily use NN to analyse data (see esp ecially Section 5.1) but calculate NS for extra comparisons with other work.
- q0 2 - q0 1 1 1 1 0 q0 1 1 1

25 < q0 < 36
1 1 1

N OC NI NO NS NN NC JF HT

4.8â101 1.7â101 3.1â101 9.0â 3.0â109 4.6â108 0.043 0.073 109

5.3â101 2.2â101 3.1â101 18.0â 5.6â109 7.7â108 0.069 0.079 109

5.8â101 2.6â101 3.1â101 21.0â 6.5â109 8.4â108 0.100 0.082 109

7.1â101 4.1â101 3.0â101 43.0â 12.9â109 15.2â108 0.203 0.083 109

1 1 1

a > 1000 1000 < a < 10000 a > 10000 q > 30, 60 < a < 1000 28 < q < 35.5, 60 < a < 1000 5 < q < 28, a < 1000 (not resonant TNOs, Tro jans) P < 200, T > 2 P < 200, T < 2

is noteworthy that HT is relatively insensitive to the initial frequency distribution of ob jects versus perihelion distance. Many Halley-types come from long-period Oort cloud orbits with perihelion distances in the inner planetary region (i.e. roughly within the orbit of Jupiter), but others (roughly 20% of the total) originate from the high-eccentricity Oort cloud cometary flux through the outer planetary region (Emel'yanenko and Bailey 1998; Emel'yanenko et al. 2007) and have a correspondingly more complex dynamical history. Some of these comets reaching JF or HT orbits pass through the NS or NN regions en route from the Oort cloud.

Although highly volatile ices, such as carbon monoxide CO, can sublimate at large distances 10 au, the main driver of cometary activity, as recognized long ago by Whipple (1950), is the sublimation of water H2 O ice. The mass loss rate for sublimating water ice has a fast decrease for heliocentric distances larger than 2 au (Jewitt 2004). Therefore, in our model we apply restrictions on the cometary lifetime only in the region q < 2.5 au, assuming that outside this region the fading of comets is negligible in comparison to that when q < 2.5 au. In order that the steady-state number of active HT comets should be < 100, our results imply that ob jects from the Oort cloud (a > 103 au at some time during their history) should survive as active comets for an average of < 150 revolutions in the region q < 2.5 au, in the model where the number of ob jects per unit perihe- lion distance is proportional to q0 2 . The result is much the same for other models, as indicated by the relatively weak dependence of HT versus dynamical model given in Table 2. However, when we apply the same physical-lifetime constraint to the Oort-cloud ob jects that eventually become JF comets, we predict too few JF comets by a factor of around 30. That is, we predict only about three JF comets in the region q < 1.5 au compared to the 100 to be explained. This illustrates the wellknown problem of explaining the number of JF comets captured from the Oort cloud if the two classes of SP comet are assumed to have broadly the same physical properties and lifetimes, a result (as we have indicated)

4 Short-p erio d comet problems 4.1 Numbers It is well known that, with a population of only 100 HT comets with q < 1.5 au (as constrained by observations), if we try to explain their origin by capture from the present-day Oort-cloud near-parabolic flux with initial perihelion distances qinit < 5 au, then it is necessary to place a very tight limit on the physical lifetime of such comets. This limit is further strengthened by the inclusion of HT comets originating from Oort-cloud source orbits with initial perihelion distances qinit > 5 au. Since comets are typically active at larger distances than 1.5 au, we must also consider restrictions on their physical lifetime in the region q < 2.5 au. Thus, for particles reaching q < 1.5 au, our integrations record also the preceding length of time spent with q < 2.5 au.


10

Emel'yanenko et al.

at the heart of what we have called the SP comet fading problem. There is an extensive literature on possible ways to overcome this `number' problem, including the assumption that JF comets may arise through the fragmentation of one or more large progenitors. Such timedependence in the present-day JF population appears to be rather unlikely, and in recent years has led to an increasing focus on models in which not only do JF and HT short-period comets originate from different primordial source regions, but have different physical properties as well. 4.2 Fading The problem of the relative and absolute numbers of HT and JF comets suggests the need to introduce an alternative dominant source for JF comets other than the Oort cloud. Such a source could include a remnant population of scattered-disc ob jects perturbed by Neptune on to relatively long-period orbits at an early stage of Solar-system evolution (Duncan and Levison 1997), or for example a primordial population of high-eccentricity trans-Neptunian ob jects initially formed beyond Neptune (or a combination of these pictures). However, irrespective of the details of such a model, there would remain the SP comet fading problem. That is, the problem that JF comets originating from any such transNeptunian source must have much longer lifetimes in the inner Solar system than observed Halley-types, and therefore statistically different fading properties. The problem can be illustrated in three ways. First, Emel'yanenko et al. (2004) showed that, while the outflow rate from the observed NNHE region is 0.93 â 10-9 yr-1 (a factor of 3 higher than the outflow rate found by Volk and Malhotra (2008) from the more limited region having q > 33 au), the predicted dynamical injection rate of JF comets with q < 1.5 au from the ob served NNHE region is approximately 0.18 â 10-10 NN -1 yr . Here NN is the intrinsic (i.e. observationally debiased) number of ob jects in the NNHE region represented by the then observed sample. If it is assumed that most JF comets come from this region and have broadly the same fading behaviour as the observed Halley-types (i.e. mean lifetimes of the order of 150 revolutions in the region q < 2.5 au), then our calculations would require NN > 3 â 1010 . This value, which as in Section 2 may be assumed to apply to H10 < 11 cometary bodies (nuclear diameter > 1 km), is greater than all previous estimates of the number of ob jects in this NNHE region, for example the 4 â 109 scattered-disc ob jects with q in the range 34­ 36 au estimated by Trujillo et al. (2000). Furthermore,

our result is a lower limit, for example because some of the comets that might have reached q < 1.5 au in the absence of fading will be removed from the distribution of active comets by the lifetime limit of 150 revolutions within q < 2.5 au. Thus, if we take account of physical fading, the rate of injection of active JF comets to the region q < 1.5 au is less than 0.18 â 10-10 NN yr-1 , requiring an even larger number of ob jects in the observed NNHE region to explain the observed number of JF comets. A second argument comes from the predicted inclination distribution of the resulting JF comets. Emel'yanenko et al. (2004) found that the observed JF comets could in principle be explained by the evolution of ob jects captured from the observed NNHE region provided that the maximum lifetime of the resulting JF comets in the region q < 2.5 au was not too long, i.e. approximately 2500 years (360 revolutions). However, such a lifetime (i.e. 360 revolutions) is already 2­3 times longer than that required to explain the active HT comets from our Oort-cloud source, again highlighting the SP comet fading problem. A third general argument leading to the same conclusion arises because the estimates in Emel'yanenko et al. (2004) were based on the assumptions that (a) the number of ob jects in the observed NNHE region is of the order of 1010 and (b) the physical behaviour of all JF comets is broadly the same. If the number of ob jects in the observed NNHE region is smaller than this, as seems likely (e.g. Levison et al. 2006), we would have to invoke longer average JF comet lifetimes in the region q < 2.5 au to explain the observed number. Alternatively, if there are two types of JF comet, for example one with a mean lifetime in the region q < 2.5 au comparable to that (150 revolutions) required to explain the number of HT comets, then the other must have a much longer average lifetime to compensate. This would exacerbate the SP comet fading problem, not just by highlighting a difference in the physical properties of some JF comets and Halley-types, but by introducing a new (and arbitrary) difference between two different assumed types of JF comet. Various arguments could of course be invoked to justify possible physical differences between different types of comet, for example that Oort-cloud ob jects might have visited the Jupiter-Saturn region many times before being finally ejected into the outer Solar system, whereas NNHE ob jects might never have come close to the inner Solar system before finally evolving into the observed Jupiter family. In this case, and whatever one's view of the merit of such speculations, it is evident that we should not dismiss lightly the possibility that there may be two or more distinct types of SP comet.


Common origin of Jupiter family and Halley typ e comets

11

However, in order to accommodate the twin constraints of the number of JF comets (tending to require a long lifetime) and their inclination distribution (tending to require a shorter lifetime), such models are also sub ject to fine tuning and a very strict observational test. That is, the dynamically distinct HT and JF classes of SP comet should, on average, have very different fading properties and rates of decay in the observable region. In particular, the JF comets originating from a flattened source distribution other than the Oort cloud must, if they are to dominate the observed distribution of JF comets, have much longer lifetimes in the observable region than their HT counterparts originating from the Oort cloud. In principle, such a ma jor physical difference between the two dynamical classes of SP comet (or even within the dynamically defined Jupiter family if the latter come from originally distinct sources) should be amenable to an observational test. In summary, the SP comet fading problem remains an obstacle to understanding the origin of SP comets. Although it may be reasonable to suggest that the comets which formed in different regions of the primordial Solar system might have different fading properties after they eventually evolve into the observable region, it is important to emphasize that there is as yet no clearcut observational evidence to support such a view, nor even for any clear physical differences between the two main classes of SP comet. Rather than two physically different types of SP comet, behaving in statistically different ways in the inner planetary region so far as fading is concerned, we therefore instead develop in the remainder of this paper a unified model for the origin of SP comets. In this unified model all comets, whether coming from the Oort cloud or trans-Neptunian region, display broadly similar physical behaviour in the inner planetary region. 5 Unified mo del We return to the idea that the key factor linking the two classes of SP comet, and perhaps all classes of comet, is their singular lack of strength and associated rapid fading. We thus seek a unified model for the origin and evolution of cometary bodies in the Solar system (particularly the observed SP comets) in which the ma jority of observed SP comets (though perhaps not all) originate from an Oort-cloud source which itself has an origin primarily in the dynamical evolution of ob jects left behind after the period of planet formation and planetary migration. In this case it is reasonable to assume that, to first order, the ma jority of comets will have broadly similar characteristics, though not necessarily

identical physical properties, including those relating to fading. In developing this unified physical picture for the origin of comets, we obtain new constraints on their required fading properties within the observable region. In particular, we use dynamical information provided by the results of our integrations and the link between Centaurs and SP comets to constrain the cometary numbers and lifetimes. In broad terms, our unified model predicts that essentially all the HT comets and nearly half the JF comets come from the Oort cloud. A flattened trans-Neptunian disc source is, however, required for the remaining 50% of JF comets. However, these ob jects too are predicted to have relatively short physical lifetimes within the observable region in order not to produce too many active JF comets. Thus, all comets have essentially the same fading properties within the observable region. 5.1 Centaurs and the NNHE region In principle, understanding the relative contributions of different outer Solar system source regions to the SP comet population requires a full description of the number and orbital distribution of all ob jects in the transNeptunian region. Unfortunately our present knowledge of this complex region is limited by the precision with which the observed orbits are known and by severe observational selection effects. We therefore use the observed distribution of Centaurs (ob jects with 5 < q < 28 au and a < 1000 au) to constrain our results. Centaurs are an important transition population providing valuable information. Emel'yanenko (2005) and Emel'yanenko et al. (2007) presented various characteristics of the orbital distribution of Centaurs from the Oort cloud, results which are supported by our present work. The more extensive integrations of our current paper are necessary to provide a sufficient number of integrated particles transferred from the outer Solar system to SP orbits. We recall that Emel'yanenko et al. (2005) predicted the orbital distribution of Centaurs originating from the observed NNHE region (28 < q < 35.5 au and 60 < a < 1000 au). These early results were based on the orbits of seven well-determined observed TNOs in the NNHE region suitably weighted by an observational debiasing procedure (Emel'yanenko et al. 2004). Let us denote as NN the intrinsic (i.e. debiased) number of objects in the NNHE region represented by this observed sample. Note that NN introduced above in Section 2.4 is defined in terms of exactly the same region of orbital element phase space. However, whereas NN refers to ob jects that have been in the Oort cloud (a > 103 au),


12
NN is the intrinsic (observationally debiased) number of NNHE ob jects represented by the discovered popu lation. By this definition, NN and NN could comprise the same population, or be disjoint, or partially overlap. If disjoint, then NN could represent the number of objects in the NNHE region associated with a primordial source distribution in the trans-Neptunian disc and so not be included in our Oort-cloud model. We can dis cover how NN really relates to NN by using Centaurs as a constraint, as follows. In a steady state, the number of Centaurs NC originating from the observed NNHE source region is a fixed proportion of the total number NN of such ob jects. Emel'yanenko et al. (2005), using the integrations of Emel'yanenko et al. (2004), calculated the constant of proportionality fN C 0.008, i.e. NC 0.008NN. They also showed that these Centaurs were split in the ratio 0.003 to 0.005 between orbits having respectively a > 60 and a < 60 au, nearly all the latter having 20 < a < 60 au (Emel'yanenko et al. 2005, fig. 2). In order to compare these dynamical results with observations it is necessary to apply an appropriate debiasing correction to the observed distribution of Centaurs. The results of Emel'yanenko et al. (2005), based on a sample of 42 well-determined Centaur orbits excluding ob jects in the 2/3 mean-motion resonance with Neptune, showed that the intrinsic number of Centaurs o NCbs is overwhelmingly dominated by ob jects with a > 60 au (roughly 90% of Centaurs having such orbits), o and that NCbs 0.13NN . This ratio, namely 0.13, is much larger than the dynamical prediction fN C 0.008, and this fact alone implies that the ma jority of Centaurs, particularly the ma jority of those with a > 60 au, must have another source, i.e. a source other than the NN ob jects representing the observed NNHE region. In this case, because it is an inescapable part of any successful model, such a source is most likely the Oort cloud. Emel'yanenko et al. (2005) also showed (their fig. 5) that, after debiasing, only 10% of Centaurs with a < 60 au have 40 < a < 60 au. On the other hand, if the principal source of Centaurs had been the observed NNHE region, the dynamically predicted fraction would have been around 50% (loc. cit. fig. 2). This is further evidence that the NN ob jects representing the observed NNHE region cannot explain all the observed Centaurs. Indeed, it raises the possibility that the Oort cloud may contribute significantly to Centaurs with a < 60 au as well. In summary, the dynamically predicted number of Centaurs with a > 60 au coming from the observed NNHE region is roughly 0.003NN, whereas observations require this number to be of the order of 90% â0.13 =

Emel'yanenko et al.
0.117NN. The difference between these two results (i.e. 0.114NN) can be attributed to an Oort-cloud flux, i.e. the flux of Oort-cloud ob jects through the planetary system irrespective of whether they have gone through the NNHE region. At this stage we make no assumption as to whether any or all of the NN ob jects represented by the observed NNHE population come from the Oort cloud. In any case, their contribution to Centaurs with a > 60 au, i.e. 0.003NN, is insignificant. Our new integrations provide a value for the steadystate ratio of the number of Centaurs produced from the Oort cloud with a < 60 au to the number with a > 60 au (cf. Table 5 later). Specifically, for every Centaur with a > 60 au, approximately 0.07 Centaurs are produced with a < 60 au. Therefore, for every 0.114NN Centaurs with a > 60 au that the Oort cloud produces, it also produces 0.008NN Centaurs with a < 60 au. As we have noted, the dynamically predicted number of Centaurs with a < 60 au coming from the ob served NNHE region is NC (a < 60) 0.005NN and the debiased number of Centaurs with a < 60 au is o NCbs (a < 60) 10% â0.13NN 0.013NN. Thus, the additional population of Centaurs with a < 60 au produced by the Oort-cloud flux through the planetary sys tem is sufficient to account for this difference of 0.008NN. However, to a good approximation, the same Oort cloud o flux does not explain the entire number of NCbs (a < 60) 0.013NN Centaurs with a < 60 au, the 0.005NN ob jects from the observed NNHE region being unaccounted for. We conclude that the observed NN ob jects are not produced from the Oort cloud. In other words, the observed NNHE ob jects studied by Emel'yanenko et al. (2004) illustrate the dynamical features of near-Neptune high-eccentricity ob jects that have never visited the Oort cloud. In contrast, the predicted NN NNHE objects originating from the Oort cloud in our model represent a sample of ob jects which owing to discovery biases are under-represented in the observed population. Thus, although we defined NN in terms of the observationally debiased known population, we may now interpret it as referring to a `primordial' trans-Neptunian population that has never become part of the Oort cloud (a > 103 au). So while the numbers NN and NN describe ob jects in the same region of orbital element phase space, they are essentially disjoint sets of ob jects. The NN ob jects coming from a proximate source in the Oort cloud are largely unobserved, i.e. are not yet rep resented in the NN population of observed ob jects in the NNHE region. These results allow us to estimate the number NN of NNHE ob jects that have never visited the Oort cloud. o Thus, because the two sources are disjoint, NCbs =


Common origin of Jupiter family and Halley typ e comets
Table 3 The numb er of Centaurs NC and the annual in jection rate of JF comets JF coming from the observed NN p opulation of primordial `trans-Neptunian' (TN) NNHE objects. The columns corresp ond to four models as in Table 2. Note that the intrinsic numb er of primordial ob jects in this region, NN , is comparable in order of magnitude to the numb er, NN , in the NNHE region that come from the Oort cloud (cf. Table 2) and which are still largely undiscovered. Simi larly the annual dynamical injection rate JF of JF comets from this primordial TN region is comparable to the rate JF from the Oort cloud (Table 2). On the other hand, the predicted numb er of Centaurs coming from this observed NNHE TN region is roughly an order of magnitude smaller than that coming from the Oort cloud.

13

- q0 NN NC JF

2 9

- q0

1 9

0 q0

25 < q0 < 36 au
9

3.7 â 10 3.0 â 107 0.067

6.3 â 10 5.0 â 107 0.113

6.9 â 10 6.0 â 107 0.124

12.5 â 109 10.0 â 107 0.225

NC + NC , and hence NC = 0.122NN where NC is listed in Table 2. This in turn allows us to determine the additional contribution of these `primordial' NNHE ob jects to the number of Centaurs (NC = 0.008NN) and to the flux JF of JF comets with q < 1.5 au, taking JF /NN = 0.18â10-10 from Emel'yanenko et al. (2004). These values are given in Table 3 for the same distributions of initial q0 as in Table 2. As with Table 2, JF is a `dynamical' annual injection rate, i.e. assuming no physical lifetime limit. For comparison, the scattered disc proposed as a source of JF comets by Duncan and Levison (1997) corresponds to ob jects whose evolution was dominated by initial close encounters with Neptune during the early dynamical history of the Solar system, with no restriction on their subsequent evolution in semima jor axis. What we term the `primordial' NNHE region overlaps this scattered disc to a large extent but does not include ob jects that ever reached a > 103 au.

from the Oort cloud have similar dynamical characteristics and that the modelled i distribution of JF comets is close to the observed i distribution if the above physical lifetime limits are imposed. But if we impose these limits on all SP comets, we have the problem of numbers described above (Section 4.1): the resulting ratio of the number of HT to JF comets is too large. From observational constraints, this ratio is around 1 ­ maybe below 1 but unlikely to be more than 1.5 (Section 2.1). We find the ratio ranges from 3.2 for the Oort-cloud model with initial perihelia within 25 < q0 < 36 au to 12.3 for the model with - the initial distribution proportional to q0 2 . In addition, the absolute number of JF comets is too small in models where ob jects are initially concentrated towards lower - q0 , e.g. the number is only 12 in the case of the q0 2 distribution. An additional SP comet contribution from the `primordial' trans-Neptunian region does not solve these difficulties: adding these SP comets (based on the data of Table 3 but with the physical lifetime limits imposed) the HT/JF ratio ranges from 1.5 to 4.6, the - number of JF comets being 32 for the q0 2 distribution. Overall these constraints favour models where the initial number of ob jects increases with q0 and are against models where the number decreases with q0 .

5.3 Best-fitting models

In order, therefore, to explore a suitable family of models, we assume firstly that the initial number of ob jects versus perihelion distance follows a power-law distribution, i.e. the number of ob jects in the range (q0 , q0 + dq0 ) is proportional to q0 dq0 . To obtain consistency with both the numbers and orbital distributions of observed SP comets we also introduce a model for the physical lifetime in the observable region q < 2.5 au. Protoplanetary disc models suggest the snow line (boundary beyond which ice can condense) gradually moves inwards from distant regions (Davis 2005; Ciesla and Cuzzi 2006; Garaud and Lin 2007; Oka et al. 2011; Mar5.2 Initial perihelion distribution tin and Livio 2012) implying that the water distribution A further important factor that allows us to discover in the early Solar system would have been a function features of the dynamical and physical evolution of comets of heliocentric distance. It follows that comets' compois the orbital distribution of JF comets. In particular, sition could depend on their initial perihelion distance the predicted distribution of inclinations is very senq0 in the early Solar system. We assume the physical sitive to the physical lifetime of comets (Levison and lifetime ­ within q < 2.5 au for comets that reach this Duncan 1997). On this basis, we obtained limits of region at the present epoch ­ is a constant number n2 of 2500 yr for the physical lifetime of JF comets in the revolutions for all ob jects formed in the outer q0 range region q < 2.5 au and 1200 yr in the region q < 1.5 (25,36) au and varies as q0 for q0 < 25 au (with no au (Emel'yanenko et al. 2004), assuming all JF comets discontinuity at q0 = 25). We impose an equivalent recome from the trans-Neptunian region. In our present striction, with the same , for the lifetime in the region calculations, we have found that JF comets coming q < 1.5 au at the present epoch, i.e. n1 revolutions


14 Table 4 The numb er of JF and HT comets for various acceptable combinations of the parameters , , n1 and n2 . 0.5 1 1 1 2 2 1 2 2 n1 170 150 150 150 140 140 n2 600 420 420 420 400 400 N
JF

Emel'yanenko et al. Table 5 One of the b est-fitting models. The numb er of cometary ob jects evolving to different dynamical classes from various initial ranges of q0 (for Oort cloud comets) and from the observed NN p opulation of `primordial' trans-Neptunian (TN) NNHE ob jects. Here = 1 and = 2; the restrictions n2 = 420 and n1 = 150 revolutions are used when calculat ing NJ F , NJ F and NH T . denotes the contribution to the ¯ observed near-parab olic flux, new , from comets originating resp ectively in each of the initial ranges of p erihelion distance. Initial region: N OC NI NO 5­10 au 1.0â10 8.0â107 1.0â109 0 0 3.0â104 0 ­ ­ ­ 0.01 0 ­ 0
9

N

HT

45 42 42 41 46 45

112 118 108 101 112 107

10­25 au 1.8â10 5.0â101 1.3â101 1. 3. 9. 1. 0â 0â 0â 0â
11 0 1

25­36 au 4.3â10 2.5â101 1.8â101 2. 7. 9. 6. 6â 9â 3â 6â
11 1 1 0

TN ­ ­ ­ ­ ­ ­ ­ 8.3â109 6.6â107 4.2â107 ­ ­ 51 0

when the initial q0 is within (25,36) au and n1 (q0 /25) for q0 < 25 au. We have explored which values of these four parameters , , n1 and n2 are consistent with the observational constraints. The total steady-state number of JF comets (to be compared to the number derived from observations) is a sum of the NJ F which we calculate here, originating from the Oort cloud, and the additional con tribution NJ F from the `primordial' trans-Neptunian population. NJ F ranges from 50 for = 1 to 70 for the model where ob jects are initially concentrated in the outer region 25 < q0 < 36 au. As we saw (Section 5.2), models with < 0 produce unsatisfactory results, namely too few JF comets as well as an incorrect value for the HT/JF ratio. Thus > 0 is implied, i.e. a greater initial concentration of comets towards larger initial q0 . Moreover for values of larger than 2 (i.e. a strong initial concentration of comets towards the outer region), we need to introduce very strict restrictions on the cometary lifetime, and the resulting number of HT comets in retrograde orbits becomes too small in comparison with the observed number. Our calculations show that models with 1 give results close to observations. But provided > 1, it is less tightly constrained than and can even increase to infinity (formally = means that all comets that do not originate within the outer region 25 < q0 < 36 au die after the first perihelion passage with q < 2.5 au). Overall it is impossible to derive unique constraints on the cometary lifetimes and the values of and simultaneously because of uncertainties in the number and the orbital distribution of SP comets. A range of possible solutions for NJ F and NH T is presented in Table 4, representative of the allowed combinations of parameters , , n1 and n2 . The best solutions correspond to a lifetime limit n1 150 revolutions, and n2 400 revolutions, with being in the approximate range 1 to 2, although there are other possibilities (e.g. the first solution in Table 4) with n1 or n2 differing by up to a few tens of per cent.

NS NN NC NC (a < 60)
NN NC NC (a < 60)

109 108 107 106

101 109 108 107

­ ­ ­ 0.97 1 ­ 7

­ ­ ­ 1.52 41 ­ 101

¯ NJ F NJ F NH T

35 30

25

Semimajor axis, AU

20 15 10

5 0 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0

Tisserand parameter
Fig. 4 The model distribution of T and a in p erihelia for SP comets with q < 1.5 au coming from the Oort cloud.


Common origin of Jupiter family and Halley typ e comets

15

180

150

120

90

60

30

0 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0

Tisserand parameter
Fig. 5 The model distribution of T and i in p erihelia for SP comets with q < 1.5 au coming from the Oort cloud.

Table 5 summarizes our results for one of the bestfitting models. The parameters are = 1, = 2, n1 = 150 and n2 = 420. Our model is consistent with the observed features of SP comets, Centaurs and TNOs, and Table 5 estimates the numbers of present-day cometary ob jects coming from the various original source regions. The regions correspond to three ranges of initial q0 for ob jects that have visited the Oort cloud, with ob jects originating from the `primordial' trans-Neptunian population that have never been in the Oort cloud listed in the final column (TN). The notation in Table 5 is as introduced earlier, with data also listed for the subset of Centaurs having a < 60 au. Whereas the model strongly constrains the initial q0 distribution, results are not highly sensitive to the initial a0 distribution, adopted as uniform in the range 50­300 au. For example, for the best-fitting model of Table 5, changing the distribution from uniform to a-1 0 per unit interval of a0 , by applying appropriate weights to the integrated particles, changes NJ F from 42 to 41 and NH T from 108 to 77 (cf. Table 4). Although our paper is mainly concerned with the origin of SP comets, we can compare our results for various cometary populations with other work. Fernandez ´

et al. (2004), following Trujillo et al. (2000) estimate 7.5â109 ob jects with radius R > 1 km and q > 30 au, a > 50 au but note an order of magnitude uncertainty in this number. Moreover, our estimates are based on the flux of new comets with H10 < 11 corresponding to R > 0.3 km according to Fernandez and Sosa (2012). The number of such ob jects should be larger than the number of ob jects with R > 1 km. Thus the estimate of Fernandez et al. (2004) does not contradict our possi´ ble values of NS . Estimates for the number of comets in the outer Oort cloud range up to 1012 (cf. Heisler 1990; Weissman 1996; Section 2.4 of Dones et al. 2004), while the distribution of comets in different parts of the Oort cloud is consistent with other models (cf. Emel'yanenko et al. 2007; Dybczynski et al. 2008; Leto et al. 2008). ´ Our data correspond to an initial population of approximately 1.6â1012 ob jects with R > 0.3 km in the region 25 < q0 < 36 au, 50 < a0 < 300 au. This is quite consistent with the value of 3 â 1012 ob jects with R > 0.5 km and cometary albedos in the original trans-Neptunian planetesimal disc, presented in Fig. 1 of Morbidelli et al. (2009). The data of Table 5 show that almost all JF comets originate from orbits with initial perihelia in the outer planetary system, and that over 90% of the steady-state number of HT comets come from the same 25 < q0 < 36 au region. This indicates that the ma jority of observed HT comets would have had initial orbits with perihelion distances largely overlapping the range of perihelia of the ob jects that eventually became JF comets. This is in contrast to the general picture described in Section 1, where JF comets largely originate from initial orbits in the trans-Neptunian region and HT comets from initial orbits in the region of the giant planets, with subsequent very different dynamical histories. For all the models in Table 4, the orbital distributions of SP ob jects with q < 1.5 au coming from the Oort cloud have similar characteristics. Figures 4 and 5 show the orbital element distributions in perihelia (i.e. equal weight to each perihelion passage) for SP ob jects with q < 1.5 au coming from the Oort cloud, applying the restrictions n2 = 420, n1 = 150, = 2 (all ob jects are equally presented, thus formally = 0 in these plots). The Figures show that in our model, JF comets (T > 2) are concentrated near the ecliptic plane, approximately 70% of them having i < 15 . Regarding HT comets (T < 2), although the model reveals both prograde and retrograde orbits, prograde HT comets outnumber retrograde ones. In these ways the basic features of these distributions are consistent with those of the observed distributions in Figs. 1 and 2. In our model, all the modelled ob jects with periods under 20 yr have inclinations i < 60 . There are

Inclination, deg


16

Emel'yanenko et al.

several reasons for this. First, the ma jority of ob jects captured to the JF population originate from the inner Oort cloud (Emel'yanenko 2005). In our model, the inner Oort cloud is a rather flattened source of comets (Emel'yanenko et al. 2007). Secondly, the ma jority of such ob jects are injected from the inner Oort cloud on to orbits with perihelia in the region of the outer planets by external perturbations. Their subsequent evolution is similar to the scheme described for trans-Neptunian ob jects by Kazimirchak-Polonskaya (1972) and Levison and Duncan (1997). The latter showed that preferentially ob jects with Tisserand parameters near 3 with respect to a planet cross the orbit of this planet. This suggests that mainly ob jects on prograde orbits are transferred to the inner planetary region. Our results ­ from analysing observed SP comets ­ about the initial distribution of ob jects that form the Oort cloud are consistent with the standard picture of the origin of the Solar system. The conclusion was that 1: this corresponds to comets originally from the outer planetary region having a greater probability of survival and thus a longer lifetime as active comets, with ob jects originating from regions with small heliocentric distances conversely becoming extinct more quickly. This accords with the amount of water (as the main driver of cometary activity) being larger for more distant ob jects in the early Solar system.

tance q0 in the early Solar system can explain the present observed distribution of short-period comets. 3. Models in which the initial distribution of ob jects versus perihelion distance is concentrated more towards the outer planetary region, and in which their present active physical lifetime is an increasing function of q0 , are consistent with the present orbital distributions and numbers of both HT and JF comets. 4. Essentially all the observed HT comets and nearly half the observed JF comets come from a proximate Oort-cloud source (i.e. have experienced orbits with a > 103 au). The remaining 50% of observed JF comets come from the observed near-Neptune higheccentricity (NNHE) population, a dynamically unstable region in which the cometary numbers decline by 95% over 4 Gyr. In addition, more than 90% of all Centaurs (5 < q < 28 au, a < 1000 au) come from the Oort cloud. 5. The model predicts that there is a significant Oortcloud contribution to the NNHE population. The number of such ob jects is comparable to the debiased number of ob jects already discovered in the NNHE region, but they are still undetected owing to observational biases (e.g. considering large semima jor axes or high inclinations).
Acknowledgements VVE would like to acknowledge the supp ort provided by the Federal Targeted Programme `Scientific and Educational Human Resources of Innovation-Driven Russia' for 2009­2013, and the STFC-funded visitor programme of the Armagh Observatory. Astronomy at Armagh Observatory is supp orted by grant-in-aid from the Northern Ireland Department of Culture, Arts and Leisure. We thank Julio Fernandez and Ramon Brasser for their reviews. ´

6 Summary and conclusions We have developed a model of the origin and evolution of the Oort cloud which is consistent with the basic observed orbital distributions of comets, Centaurs and high-eccentricity trans-Neptunian ob jects. Rather than requiring intrinsically different fading properties for Jupiter-family and Halley-type short-period comets, the model instead adopts the hypothesis that the physical lifetime of ob jects as active comets in the inner planetary region at the present epoch is a function of their initial perihelion distance in the early Solar system, and is the same for both JF and HT comets. The observed JF and HT populations also constrain the initial distribution of ob jects versus perihelion distance. Our results show that: 1. The mean physical lifetime of comets is < 200 revo lutions in the region q < 1.5 au. This implies a significant cometary contribution to the distribution of small bodies (`boulders' and dust) making up the near-Earth interplanetary complex. 2. No model in which the initial number of comets is a decreasing function of their initial perihelion dis-

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