The Destruction of an Oort Cloud in a rich stellar cluster
AAstronomy & Astrophysics manuscript no. nordlander˙etal˙oortcloud c (cid:13)
ESO 2018January 9, 2018
The Destruction of an Oort Cloud in a rich stellar cluster
T. Nordlander ,(cid:63) , H. Rickman , , and B. Gustafsson , Division of Astronomy and Space Physics, Department of Physics and Astronomy, Uppsala University, Box 516, 75120 Uppsala,Swedene-mail:
[email protected] PAN Space Research Center, Bartycka 18A, 00716 Warszawa, Poland Nordita, Roslagstullsbacken 23, 10691 Stockholm, SwedenReceived 23 December 2016 / Accepted 07 April 04 2017
ABSTRACT
Context.
It is possible that the formation of the Oort Cloud dates back to the earliest epochs of solar system history. At that time, theSun was almost certainly a member of the stellar cluster, where it was born. Since the solar birth cluster is likely to have been massive(10 − M (cid:12) ), and therefore long-lived, an issue concerns the survival of such a primordial Oort Cloud. Aims.
We have investigated this issue by simulating the orbital evolution of Oort Cloud comets for several hundred Myr, assumingthe Sun to start its life as a typical member of such a massive cluster.
Methods.
We have devised a synthetic representation of the relevant dynamics, where the cluster potential is represented by a Kingmodel, and about 20 close encounters with individual cluster stars are selected and integrated based on the solar orbit and the clusterstructure. Thousands of individual simulations are made, each including 3 000 comets with orbits with three di ff erent initial semi-major axes. Results.
Practically the entire initial Oort Cloud is found to be lost for our choice of semi-major axes (5 000 −
20 000 au), independentof the cluster mass, although the chance of survival is better for the smaller cluster, since in a certain fraction of the simulations theSun orbits at relatively safe distances from the dense cluster centre.
Conclusions.
For the range of birth cluster sizes that we investigate, a primordial Oort Cloud will likely survive only as a smallinner core with semi-major axes (cid:46) ff usion into an outer halo andsubsequent injection into observable orbits. Some mechanism is therefore needed to accomplish this transfer, in case the Oort Cloud isprimordial and the birth cluster did not have a low mass. From this point of view, our results lend some support to a delayed formationof the Oort Cloud, that occurred after the Sun had left its birth cluster. Key words.
Comets: general – Oort Cloud – open clusters and associations: general – Stars: kinematics and dynamics
1. Introduction
The formation of the Oort Cloud is one of the important issueswhen trying to understand the origin and evolution of the SolarSystem. This has been the case ever since this structure was firstrecognised (Oort 1950), and resolving the issue still presents avery di ffi cult task. It is natural to think of a “primordial” ori-gin connected to the formation of the planets during the earlieststages of the Solar System more than 4.5 Gyr ago, as did Oorthimself, and this has led to the classical picture (Duncan et al.1987; Dones et al. 2004) of comets as icy planetesimals scat-tered through the gravity of the growing giant planets into orbitsextending far enough to sometimes be decoupled from the plan-etary system by external agents (Galactic tide and passing stars).A di ff erent scenario for the Oort Cloud formation was re-cently investigated by Brasser & Morbidelli (2013). They ex-plored one of the consequences of the Nice Model (Levison et al.2011, and references therein) for the long-term dynamical evo-lution of the giant planets. As a result of the rapid migration ofUranus and Neptune through the primordial trans-planetary diskinto their current orbits, a scattered disk would be formed and,as an unavoidable by-product, also an Oort Cloud. This sugges- (cid:63) Current address:
Research School of Astronomy and Astrophysics,Australian National University, ACT 2611, Australia. e-mail: [email protected] tion places the origin of the cloud at the time of the Late HeavyBombardment (LHB) about 4 Gyr ago.As long as this version of the Nice Model stands, there isstrictly no need for the Oort Cloud to include any primordialcomponent. However, even so, such a component is not ruledout. Moreover, the Nice Model may also accommodate a dif-ferent scenario, where the planet migration happens very early.In this case, the Oort Cloud would definitely be primordial, andhence this option needs to be considered. An important issuethen concerns the e ffi ciency in the transfer of planetesimals intothe Oort Cloud. There are two steps involved: first, the scatter-ing of planetesimals into orbits that may be modified by externalactions, and second, the decoupling that causes storage into theOort Cloud.It has been realised – ever since the work of Gaidos (1995)and Fern´andez (1997) – that the Oort Cloud storage is stronglydependent on whether one treats the new-born Sun as an isolatedstar or a member of a dense stellar environment in a so-calledbirth cluster. The latter o ff ers a more e ffi cient way to decouplethe objects by raising their perihelion distances due to the fre-quent occurrence of close and slow stellar encounters.A numerical study of the formation of the Oort Cloud in astellar cluster was performed by Fern´andez & Brunini (2000).In this work, the cluster was assumed to exist for a period of100 Myr with a stepwise decreasing number density of starsfrom the assumed initial value down to zero. In addition, the a r X i v : . [ a s t r o - ph . E P ] A p r . Nordlander et al.: The Destruction of an Oort Cloud in a rich stellar cluster tidal e ff ect of the placental molecular cloud gas was included,typically only for the first 10 Myr. The scattering and decouplingof comets was simulated with the main result that an Oort Cloudinner core was formed quite rapidly with Jupiter and Saturn asthe main scattering agents.The discovery in 2003 of (90377) Sedna, whose periheliondistance of 76 au is well beyond the orbits of the giant planetsand thus can only be explained by the influence of external ac-tions, spurred an interest in very dense stellar environments forthe birth of the Solar System. Embedded clusters were recog-nised to be a common birth place for solar type stars (Lada &Lada 2003). Brasser et al. (2006) found a good match to theSedna orbit for an inner Oort Cloud obtained in a model, wherethe Sun was born in such a cluster with a very high mass density.However, in a follow-up paper, Brasser et al. (2007) investigatedthe e ff ects of gas drag from the solar nebula on the planetesimalscattering and found what they referred to as size sorting. Onlyvery large objects would evolve in the way described by Brasseret al. (2006), while orbits of comet-sized objects (radii ∼ ff ect ofthe resulting present Oort Cloud being dominated by a tight, in-ner core was confirmed as well as the possibility of obtaining aSedna-type population due to the random e ff ects of the closeststellar encounters.An often cited model for the formation of a primordial OortCloud was proposed by Levison et al. (2010). This model as-sumes a birth cluster with few stars (30 < N < ff erent stars of the cluster and gets enriched dur-ing the cluster break-up. However, the above-mentioned prob-lem of bringing kilometre-sized objects into extended scattereddisks was not addressed.The current models of Oort Cloud formation may thus besummarized as, on the one hand, models for a primordial cloud,assumed to be formed in a dense but more or less short-livedstellar environment, and on the other hand, a delayed formationmodel associated with the LHB, where the Solar System is as-sumed to have left its birth cluster at an earlier stage. All thesemodels assume a rather short dissolution lifetime for the birthcluster, which may or may not be true.The size of the Sun’s birth cluster was discussed in a reviewby Adams (2010). His analysis of a range of constraints led to abroad probability distribution for the number N of stars, peakingat N (cid:39) N was that it would be too unlikely for the birth cluster toproduce a supernova with a progenitor mass of 25 M (cid:12) or more,as seems necessary to explain the amounts of short-lived radioisotopes that meteorite evidence show to have been present inthe solar nebula ( e.g. , Williams & Gaidos 2007). However, in it-self this factor gives rather weak constraints – only clusters with N <
50 would be excluded, since the random likelihood of thesupernova would then be less than 5%. The main factor oppos-ing too large N values was a too small chance for the regularityof the giant planet orbits to survive in the presence of the close stellar encounters then implied (Adams & Laughlin 2001; seealso Malmberg et al. 2007).More recently, Gounelle & Meynet (2012) proposed a modelfor the origin of the short-lived radionuclides involving two gen-erations of stars, formed in the same giant molecular cloud andpreceding the formation of the Sun. This model implies a Solarbirth environment rich in stars. Several thousand stars were es-timated to have been born before the Sun, thus providing thesource for Al and Fe traced in chondritic meteorites. The dy-namical fate of the Sun was not addressed, but it seems possiblethat the Sun stayed gravitationally bound to the initial complexof stars and gas, thus becoming a member of a massive stellarcluster.Thus, the Sun’s birth cluster may have been rich in stars,containing thousands or more to begin with. Moreover, if weleave aside the specific constraints posed by solar system evi-dence and consider the statistics of observed embedded clusters,we find that the number of clusters formed today falls o ff withthe number of member stars in such a way that about equal num-bers of stars form in clusters with 10 , 10 and 10 members(Lada & Lada 2003). While several papers have dealt with OortCloud formation in a cluster with ∼ members, which weshall call a low-mass (LM) cluster, the other two classes of clus-ters – intermediate-mass (IM) and high-mass (HM) clusters –have not yet been considered. The cases treated by Fern´andez &Brunini (2000) and Kaib & Quinn (2008), where the whole clus-ter dissolves within 100 Myr, would fall into our LM category.Dynamical models of stellar cluster evolution show that clus-ters with more than 1 000 initial members typically survive forseveral hundred Myr or more (Lamers & Gieles 2008). Thus,their lifetimes may even exceed the interval from the forma-tion of the earliest solar nebula condensates (meteoritic calcium-aluminium rich inclusions or CAI) until the triggering of theLHB (Morbidelli et al. 2012). This calls for a reevaluation ofthe Oort Cloud formation models in the framework of such along-lived birth cluster. The large number of stars may help inthe formation of a primordial cloud, but it is also a threat to thestability of the cloud due to the possibly disruptive e ff ect of sub-sequent encounters. In the present paper we consider the fate of aprimordial Oort Cloud in a dense stellar environment that lasts atleast until the LHB. The main question is if such a cloud wouldsurvive or not.We use two assumptions for the birth cluster, consideringtwo values for the initial number of stars ( N ). For the IM clus-ter we take N = N =
36 000 whereofhalf the systems are binary (Hurley et al. 2005). This choice isarbitrary, and we do not mean to suggest M67 to be the Sun’sbirth cluster – for discussions of this issue, see Pichardo et al.(2012) and Gustafsson et al. (2016). It is, however, a convenientcase for the upper end of the mass range due to the availabilityof detailed evolutionary modelling based on a very good obser-vational record.We have devised a modelling technique that allows to tracethe motions of thousands of test objects representing Oort Cloudcomets on heliocentric orbits for a time span of several hundredMyr, in the presence of a static cluster potential plus a selectionof randomly created stellar encounters. These are meant to in-clude some of those that impart the largest impulses to the Sun.The CPU time consumption is moderate enough to allow run-ning thousands of such integrations for each model of the birthcluster and thus obtaining results that are robust against statisti-cal uncertainties.
2. Nordlander et al.: The Destruction of an Oort Cloud in a rich stellar cluster
In Sect. 2 we first present our cluster model and describeits two versions (IM and HM) in some detail. We then describeour treatment of stellar encounters and our derivation of the en-counter frequency and velocity distribution. Next, we presentour simulation set-up. The results are given in Sect. 3. Notably,we find that the survival probability of primordial Oort Cloudcomets is extremely low regardless of the size of the birth cluster,within the cluster mass range explored. Conclusions drawn fromthis result and a discussion are given in Sect. 4. In two appen-dices we describe the calculation of the cluster model structureand the implementation of the stellar encounters, respectively.
2. Methods
We integrate the orbit of the Sun together with thousands of testparticles representing Oort Cloud comets, in a static cluster po-tential over the course of 400 Myr. The comets are introduced inrandom heliocentric orbits with semi-major axes of a o = N = N =
36 000 stars, as describedin Sect. 2.1. We represent the HM cluster by one static modelwhile we use a sequence of static models for the IM cluster, asthis evolves rapidly with time and in fact nearly dissolves duringthe simulated time period.During each simulation, we select roughly 20 stellar encoun-ters to occur at random times, by means of an impact approxi-mation described in Sect. 2.2. We add these interloping stars oneat a time to the orbit integrations as described in Sect. 2.3. Thegravitational influence on the comets is thus that of the smoothcluster potential, the Sun, and at times, one additional star. Thenumerical setup and the selection of comets is further describedin Sect. 2.4.
We impose the following constraints on our cluster model. – The stellar distribution function corresponds to a spatiallyisotropic, relaxed state. – It does not directly account for mass segregation and consid-ers only single stars. – It is stepwise or fully constant with time during the intervalwe simulate.These constraints are to some extent mutually incompatible,since mass segregation and binary formation are necessary con-sequences of the same dynamics that causes relaxation. In addi-tion, for the relatively young system that we simulate, relaxationshould still be ongoing. Thus, the system has to evolve with timein contrast to our third constraint. The reason why we stick to theabove concepts is that they allow us to develop a synthetic modelof stellar encounters that greatly facilitates the Oort Cloud sim-ulations and avoids the use of time-consuming N -body simula-tions.Clearly, a most useful template in order to achieve a clustermodel with the above properties is the King model (King 1966).We apply this concept according to a prescription that we de-scribe in Appendix A.To fit our model parameters, we start from an initial guess(see Appendix A), and improve these values iteratively, untilthey yield the desired solution. This means that the cluster has the required mass, and the density function drops to zero at a dis-tance R lim less than the tidal radius. Other fitted parameters in-clude the half-mass radius r h . Finally, we may also compute thesky projected surface density I ( R ) as prescribed by King (1966,eqns. 23–27). Using this quantity, we calculate the core radius R c from its definition I ( R c ) = I (0). In our case, the surface densityrefers to mass rather than brightness. This core radius may alsobe used as a fitting parameter.The last property to be defined is the age of the cluster. Wewant to simulate the cluster e ff ects on the Oort Cloud for a timeinterval, extending from the formation of a primordial Cloud un-til the LHB. However, in this work we do not consider the e ff ectsof the cluster on the very formation of the cloud. In reality, theformation process is expected to extend over several hundredMyr (Kaib & Quinn 2008; Brasser & Morbidelli 2013), but wereplace this by a step function: we consider a first time intervalof 100 Myr starting at the formation of the solar system ( t = t =
100 Myr,we introduce the comets in their above-described, initial orbits.The cluster age hence extends from 100 Myr to 500 Myr, whenwe assume the LHB to occur.
As indicated above, the structural parameters of this cluster arebased on the properties of a young M67 as simulated by Hurleyet al. (2005) and available in their online tabulations . Theseauthors performed a set of N -body simulations, describing thecomplete evolution of M67 after formation of all the stars andescape of any residual gas. The initial parameters of their evolu-tionary model – the galactocentric distance and orbit, total mass,and binary populations – were estimated from the current lumi-nous mass ( i.e. , the total mass of nuclear-burning stars, estimatedat M L ∼ M (cid:12) ), age (approximately 4.0 Gyr), and binaryand blue straggler populations of the cluster. As these popula-tions are sensitive to the dynamical evolution of the cluster, themodel is reasonably well constrained.In Fig. 1 we present the evolution of both our template clus-ters. The HM cluster, based on data from Hurley et al. (2005), isseen to decrease slowly in the number of stars during the consid-ered time interval (from 100 to 500 Myr). We illustrate the totalnumber of stars, which often occur as binary components in thereal cluster M67 – a discussion of binarity is given in Sect. 2.1.3.The inner part of the cluster undergoes a slight expansion withthe half-mass radius increasing from an initial value of 4.0 pc to5.5 pc.The initial distributions of mass and energy used by Hurleyet al. (2005) were based upon a Plummer model (Plummer1911). On the half-mass relaxation timescale T rh , ≈
290 Myr,this distribution evolves into something resembling a King pro-file. The relaxation causes the increase in the half-mass radius,which reflects the evolution of the density profile.We choose a static setup for our model cluster representingthe mean state of M67 during the age span 100–500 Myr in thesimulations of Hurley et al. (2005). At this early stage in thecluster’s history, the state can be described as semi-relaxed, sothat the global structure is neither that of the initial Plummerdistribution nor that of the eventual King model. However, therelaxation proceeds more rapidly in the inner parts of the cluster,where the crossing time is smaller. Therefore, the structure inthis part of the cluster is less sensitive to the initial conditions of Data available (in March 2017) at http://astronomy.swin.edu.au/˜jhurley/nbody/archive.html
3. Nordlander et al.: The Destruction of an Oort Cloud in a rich stellar cluster N s t a r s HM clusterIM cluster 0246 r h [ p c ] Fig. 1.
The number of stars (solid, black lines) in our high-massand intermediate-mass cluster models, and their half-mass radii(dashed, red lines), versus time from cluster formation. A loga-rithmic scale is used for the number of stars and a linear scale forthe radius. The time has been rescaled for the IM cluster to takethe expected influence of GMCs into consideration (see text).their model. By choosing a King model, we are thus able to re-produce the inner parts of the simulated cluster reasonably well,while the outer parts are less well described.The parameters to which we fit our King model are the mass M cl , half-mass radius r h , core radius R c , limiting radius R lim ,and the mean density within the half-mass and core radii, (cid:104) ρ h (cid:105) and (cid:104) ρ c (cid:105) . The input parameters and resulting properties of thecomputed model are given in Table A.1. While N-body simulations of the HM cluster indicated a ratherslow evolution during the first 2–3 Gyr, the IM cluster has ashort relaxation time and thus rapidly evolves not only in to-tal mass but also in terms of structure. The half-mass relax-ation time scales with the cluster mass and half-mass radiuslike t rh ∝ M / r / (Spitzer & Hart 1971). For a population of N = r h = t rh ≈
20 Myr, indicatingthat such a cluster should relax very rapidly.In Fig. 1 we illustrate the simulated evolution of an IMcluster, computed using the emacss code (Alexander & Gieles2012; Gieles et al. 2014; Alexander et al. 2014), with its life-time rescaled according to the expected influence of encounterswith GMCs (Gieles et al. 2006). We find that the relaxation timevaries between 20 and 40 Myr during the first 500 Myr of thesimulation. The parabolic evolution of the half-mass radius afterthe initial relaxation is due to the combined e ff ect of mass lossfrom the outer regions, and the collapse and bounce of the core,where energy transfer from the core inflates the outer regions ofthe cluster.Due to the rapidly evolving nature of this cluster, we repre-sent it by a sequence of static models, each corresponding to theaverage properties during a span of 25 Myr. The input parame-ters and resulting properties of the models are listed in Table A.1.The density distributions of the HM cluster model as well asevery fourth model representing the IM cluster, are shown inFig. 2. Both structures exhibit an inner plateau out to the core −4 −2 D e n s it y [ M s o l p c − ] HM clusterIM cluster, 500 MyrIM cluster, 400 MyrIM cluster, 300 MyrIM cluster, 200 MyrIM cluster, 100 Myr
Fig. 2.
Density distribution in the template clusters. The dashedblack curve indicates the distribution of the high-mass clus-ter, while the solid coloured curves represent time steps in theintermediate-mass model, 100 Myr apart, with the average timeof each step shown in the legend. The central mass density (leftend of the plot) decreases monotonically with age of the model.The horizontal dash-dotted line indicates the Galactic mid-planedensity for comparison.radius (1.9 pc in the HM cluster; 0.05–0.1 pc in the IM cluster),whereafter the density roughly follows a power-law, decreasingto values lower than that of the local Galactic disk.
In addition to the structure of the cluster, we need to describeits stellar content. To translate a distribution of mass into thecorresponding distribution of individual stars, we adopt a stellarIMF from the generating function of Kroupa et al. (1993), M ( ξ ) = M + . ξ . + . ξ . (1 − ξ ) . (1)where ξ ∈ [0 ,
1] is a random number from a uniform distribution,and M the lower mass limit. We set M = . M (cid:12) to accountfor the mass segregation-driven preferential loss of the lowest-mass objects from the cluster.We evolve the stellar population to an age of 300 Myr, rep-resenting the mean age of the cluster, using the rapid stellar evo-lution code SSE (Hurley et al. 2000). The resulting mean stellarmass is (cid:104)M(cid:105) ≈ . M (cid:12) , with an e ff ective upper mass limit near3 . M (cid:12) (turno ff mass ∼ . M (cid:12) ). For completeness, we retainstellar remnants in the form of white dwarfs in the mass distri-bution, but remove neutron stars and stellar mass black holes asthese are expected to usually be given kicks at formation muchgreater than the cluster escape velocity (see, e.g. , Pfahl et al.2002).We do not directly invoke mass segregation in our clustermodel. Thus, the probability distribution of stellar mass is thesame at any distance from the centre. In this sense, our modelfails to include a real phenomenon, which would be expectedto occur in either cluster a few hundred Myr after its birth: asystematic tendency for the high-mass stars to concentrate in thecluster core and, hence, to be underrepresented in other parts ofthe cluster.
4. Nordlander et al.: The Destruction of an Oort Cloud in a rich stellar cluster
We also neglect the existence of binaries, even though theseare common in M67 (Richer et al. 1998) and also in low-massclusters (Giersz & Heggie 1997), since they would complicatethe treatment of close encounters. If hard binaries were included,these would e ff ectively increase the mean mass of the OortCloud perturbers while for a given cluster mass decreasing theoverall number density of those perturbers. Note that Fouchardet al. (2011) found the long term dynamics of the Oort Cloudto be influenced by massive stars in the Galactic disk to a muchlarger extent than their low encounter frequency would suggest– the reason being their higher chance of producing global per-turbing e ff ects on the cloud. Hence, we may be underestimatingthe stellar encounter e ff ects, but this is in line with our strategyto seek a conservative estimate. Since we do not trace the motions of individual cluster stars, wegenerate encounters synthetically using a statistical encounterflux derived from the cluster model.To derive this encounter flux, we first need the distributionof relative velocities of the stars. At a given distance from thecluster centre, we calculate the cumulative distribution of kineticenergies from equation (A.1) and produce a generating function.To represent individual encounters, we draw independently tworandom values from this generating function for the kinetic en-ergy (per unit mass) of two encountering stars.By assuming a flat distribution of angular momenta (as in-herent in a King model), absorbing at most all of the availablekinetic energy, we determine the tangential and radial velocitycomponents of each star. Each radial velocity can be positive ornegative with equal probabilities. We assume an isotropic en-counter distribution, wherein the motions of the two stars areuncorrelated. The distribution of these encounter velocities is il-lustrated in Fig. 3, including the arithmetic mean (cid:104) v rel (cid:105) which inthe figure is denoted v mean and shown by blue curves. Note thatwhen we use this to find the Sun’s encounter flux, we do not ac-count for the constraints imposed by the Sun’s particular orbit.In our approximation, the Sun is treated as an average star at anyparticular distance from the cluster centre.Using the mean relative velocities, we show in Fig. 4 theencounter frequency versus distance from the cluster centre: f ( r ) = n (cid:104) v rel (cid:105) . For an assumed impact parameter b , this en-counter frequency produces an encounter rate Γ ≡ σ n (cid:104) v rel (cid:105) ,where σ ≡ π b is the impact cross-section. The expected num-ber of encounters occurring over time τ with a constant value ofthe encounter rate is thus simply (cid:82) τ Γ dt = τ Γ . Taking Γ = τ − for a given time τ , we may thus estimate the expected minimumimpact parameter b min of the encounters experienced during atimespan of length τ . Using τ =
400 Myr, in the HM cluster, wefind a value of b min ≈
210 au at the very centre, while near thehalf-mass radius we find b min ≈ (cid:82) τ Γ ( t ) dt , taking into accountthe time dependence of the encounter flux, by adding the contri-butions from the di ff erent 25 Myr intervals. At the cluster centre,we find b min (cid:39)
35 au for τ =
400 Myr. Clearly, and not surpris-ingly, the Oort Cloud and even the planetary system would notsurvive such an extended stay in such a dense environment. If wetake into account that the Sun is found to leave this kind of clus-ter before 100 Myr as a median (see Sect. 3.4), for this value of τ we still have b min (cid:39)
40 au, which is very destructive. However,the solar orbit makes the Sun spend only a minimal amount oftime in the immediate vicinity of the centre, if any at all. For a more realistic estimate, we also perform the estimate for a dis-tance of 1.1 pc from the centre, roughly representing an averagehalf-mass radius. In this case, the two values of b min increaseto approximately 1 450 au and 2 000 au, respectively. These arelarger than the one found for the HM cluster.The expected minimum impact parameter resembles the sizeof the Oort cloud, b min ≈
10 000 au at r ≈
14 pc in the HMcluster, or at r ≈ . ∼ . M (cid:12) pc − ,Holmberg & Flynn 2000). In the disk, the mean encounter veloc-ity is greater by an order of magnitude, which increases the fluxof stellar encounters by the same amount to 10 − pc − Myr − , butalso reduces the e ffi ciency of momentum transfer. The influenceof field star passages on the Oort Cloud has been investigatedelsewhere (Rickman et al. 2008), and the erosion of the OortCloud was then found not to be dramatic over intervals like theone considered here. Thus we neglect these passages as well astheir associated Galactic disk tide e ff ect. We simulate stellar encounters by introducing a star at a distance d start from the Sun, and integrate the orbits of the two stars undertheir mutual gravitational influence in the cluster gravitationalpotential, until the mutual distance is d end . In the simulation ofthe HM cluster, we found that setting d start = d end = P ∼ r / M − / ∼ d start of 1 pcwould be inappropriate as this is in fact similar to the half-massradius of the cluster. Thus, most of the interaction of the twostars occurs at distances much larger than those between the Sunand many other cluster stars, which makes the calculation some-what irrelevant. In addition, as we shall explain below, we use afilter when selecting encounters to avoid overlaps of consecutiveevents. With the smaller relative velocities of the stars in the IMcluster, this would lead to the blocking of too many low-velocityencounters due to their large durations. To reduce this bias, weset a distance d start = . d end = .
45 pc whereit is removed. This results in a similar distribution of encounterdurations as in the case of the HM cluster.We have checked that the particular choice of distances doesnot significantly influence the results by repeating our IM clus-ter calculations with d start = .
25 pc and d end = .
22 pc, with nosignificant e ff ect on either the survivability of comets or the evo-lution of the solar orbit. Similar tests on the HM cluster indicatethat neither qualitative nor quantitative results depend stronglyon the precise choice of the interaction distance.The typical encounter in either simulation does not overlapsubsequent pericentre passages, which is important as our setupdoes not allow for simultaneous encounters. We enforce this by aveto blocking the selection process while an encounter is ongo-ing. To avoid having the encounter scheme block too many sub-sequent encounters (typically occurring near pericentre), we aimfor producing one encounter per 20 Myr, i.e. , Γ = (20 Myr) − .For a simulation duration of τ =
400 Myr, this corresponds to
5. Nordlander et al.: The Destruction of an Oort Cloud in a rich stellar cluster
HM cluster v r e l [ k m s − ] IM cluster v r e l [ k m s − ] v v v mean v Fig. 3.
Distribution of relative velocities in our encounter model for the stellar cluster. The five curves in the left panel represent,from top to bottom, the maximum allowed relative velocity (solid, black), its 90th percentile (dotted, red), mean (solid, blue) and10th percentile values (dotted, black). Shown as vertical (black) lines are the core (dotted) and half-mass (dashed) radii.
Left panel:
Relative velocity distributions for the high-mass cluster.
Right panel:
Relative velocity distributions for the intermediate-masscluster at two time steps. The first step represents the time interval 100–125 Myr. The second time step represents the time interval300–325 Myr, and is shown only by a thick dashed black line representing the maximum relative velocity, a thick dashed blue linerepresenting the mean, and a vertical thick dashed line representing the half-mass radius at this time. The core radius is not shownat this time, but is available for all time steps in Table A.1. −8 −6 −4 −2 E n c oun t e r fr e qu e n c y [ p c − y ea r − ] HM clusterIM, 500 MyrIM, 400 MyrIM, 300 MyrIM, 200 MyrIM, 100 Myr
Fig. 4.
Encounter frequency versus distance from the cluster cen-tre, shown for the high-mass cluster (dashed line) and selectedtimesteps for the low-mass cluster (coloured solid lines). The en-counter flux at the centre of the cluster decreases with increasingage for the intermediate-mass cluster – colours are the same asin Fig. 2. The horizontal dash-dotted line illustrates for compar-ison the current encounter frequency of the Sun with Galacticfield stars (Rickman et al. 2008).typically 20 encounters per simulation, where the veto typicallyblocks one expected encounter per simulation.The number of stellar encounters expected to exert signifi-cant influence on the outer Oort Cloud is, however, larger thanjust 20 per 400 Myr, and we aim to simulate those encounters that are most important. Rather than using the distance as cri-terion, we adopt a strength parameter S = M / ( v rel b ), approxi-mating the impulse transferred to the Sun by a stellar encounterin the classical impulse approximation (see Rickman 1976;Fouchard et al. 2011). All three defining parameters must thenbe known for S to be computed, which in turn requires the en-counter geometry to be determined. As detailed in Appendix B,we pick the encounter parameters at random and generate a listof encounters likely to occur during a time interval of a givenlength, and compute S for each. Finally, we select the encounterwith the largest value of S .When the final selection is made, the integration proceedsuntil the encounter is finished, as detailed in Sect. 2.4. Then anew time interval of the given length is considered, using themodified solar orbit, and the next stellar encounter is selected.On the average, the interval between consecutive encounters willbe close to 20 Myr, and hence a total of about 20 encounters willbe treated during the entire 400 Myr simulation. We use a hybrid numerical setup. We combine the tidal e ff ects ofthe cluster as a whole, represented by the smooth gravitationalpotential computed according to Sect. 2.1, with the influence ofindividual stars during orbit-integrated close encounters as de-scribed in Sect. 2.2. Comet motions are thus integrated underthe gravitational influence of the cluster, the Sun, and an inter-loping star when applicable. As mentioned above, we neglect theinfluence of Galactic tides and interloping Galactic field stars.The Oort Cloud is represented by a sample of 3 000 comets.This consists of three ensembles of 1 000 comets each, represent-ing di ff erent parts of an initial Oort Cloud, with original semi-major axes of a o = inner ,
6. Nordlander et al.: The Destruction of an Oort Cloud in a rich stellar cluster intermediate and outer comet cloud – not to be confused with thenomenclature of actual present-day Oort Cloud populations. Theother orbital elements are drawn randomly from identical distri-butions for all three ensembles. These distributions are uniformfor all but the eccentricity, which is distributed as f ( e ) ∝ e in therange e ∈ [0 ,
1] to represent a thermalized state. Since our modeldoes not include the planets, the initial cloud is modelled with-out any loss cone (Oort 1950; Hills 1981), thus allowing e → i ∈ [ − ,
3. Results
We perform a series of 1 000 Monte Carlo simulations for eachcluster, di ff ering from each other in terms of the initial solar orbitand cometary orbits. The total number of comets treated is thus3 × , for each cluster. With such statistical sampling, we shallpresent not only the typical behaviours, but also the tail of theprobability distribution, i.e. , the rare outcomes. Obviously, in thelatter cases, our results are not statistically accurate but ratherindicative. We illustrate in Fig. 6 the statistical distribution of comet sur-vival probability in these simulations. The term “survival” refersto those comets that do not experience ejection either by movingto distances exceeding 1 pc or by intruding into the loss cone –hence they stay in the Oort Cloud until the end of the integra-tion. Since it is natural to imagine that stellar encounters cause comets to leave the cloud directly, while the cluster tide may per-turb their orbits into the loss cone, we have also performed twoextra sets of 100 simulations each, where we artificially turnedo ff the cluster tide and the stellar encounters, respectively. Bycomparing these results with those of the full model, we hopeto learn which is the dominant mechanism causing the loss ofcomets from the Oort Cloud.Results for the HM cluster are presented in the left panel ofFig. 6. In this case, the median survival probability for cometsin the initial sub-population representing the primordial inner comet cloud, a o = intermediate and outer comet clouds, a o =
10 000 au and20 000 au, the median survival probability is less than 0.1 %, i.e. ,fewer than one comet per simulation. Hence, typically the entireprimordial outer comet cloud is lost. In only 1.3 % of simula-tions do more than 1 % of the primordial outer cloud memberssurvive. For the primordial inner and intermediate comet cloud,more than 5 % of comets survive in 9 % and 2 % of the simula-tions, respectively.Comparing now with the results for the IM cluster in theright panel of Fig. 6, we see an important di ff erence. The curvesrepresenting the fall-o ff of the percentage of simulations withincreasing percentage of survivors are much flatter for the IMcluster over almost the whole range. Consequently, the mediansurvival probability for the primordial inner comet cloud is only0.1 %, i.e. , even lower than for the HM cluster, while the per-centage of simulations with a much larger number of survivors isconsiderably higher for the IM cluster. For instance, the fractionof simulations with more than 1 % survivors in the primordialouter cloud is 9.8 % for the IM cluster, compared to 1.3 % forthe HM cluster. On the other hand, the escape times of cometsfrom the Sun in the two clusters are much shorter in the IM thanin the HM case. For the inner, intermediate and outer clouds, themedian escape times are 12, 5 and 4 Myr, respectively, in the IMcluster while in the HM cluster these are 51, 47 and 20 Myr.The reason for these di ff erences has to do with the typicalfate of the Sun in the two clusters. As we shall see in Sect. 3.4,after 400 Myr the Sun typically remains in the HM cluster andescapes from the IM cluster. Specifically, the remaining percent-age is 94 % in the HM case and the escaping percentage is 95 %in the IM case. This means that the behaviours exhibited in thetwo panels of Fig. 6 may actually carry as much informationabout whether the Sun remains in the cluster, as the di ff erencebetween HM and IM clusters.Although the statistics is rather poor for the less commonsituations, we can still compare the fate of Oort Cloud cometsin all four cases, i.e. , HM vs IM clusters and remaining vs es-caping Sun. We have thus found that for the IM cluster there isnot much di ff erence of comet survival statistics, whether the Sunstays in the cluster or it escapes. For the HM cluster the fate ofthe comets is more sensitive to the fate of the Sun. When the Sunescapes from this cluster, the comet survival statistics is interme-diate between the left and right panels of Fig. 6.Our results are consistent with the following picture. As seenin Fig. 4, the central part of the IM cluster starts out with a veryhigh encounter frequency, but this falls o ff rapidly with time dueto cluster evaporation. The trend is similar for the strength of thecluster tide. Thus, during the early phases the solar system runsa high risk of being stripped of its entire Oort Cloud due to veryclose stellar encounters, if the solar orbit penetrates close to thecentre, but our simulation only covers a few of the strongest en-counters expected. As we shall see below, the cluster tide alsoplays a role in this context. We may therefore expect to see a
7. Nordlander et al.: The Destruction of an Oort Cloud in a rich stellar cluster
HM cluster −5 0 5Coordinate x [pc]−505 C oo r d i n a t e y [ p c ] Start 1 2 3 4
HM cluster −5 0 5Coordinate x [pc]−505 C oo r d i n a t e z [ p c ] Start 1 2 3 4
Fig. 5.
Solar orbit during the first 80 Myr in a simulation of the high-mass cluster, with grey circles indicating the core and half-mass radii. The solar orbit is illustrated by a line that switches colour (from violet to blue, green and yellow) every time a star isencountered. The starting points of the interloping stars are numbered and indicated by a star symbol, while end points are indicatedby x . The starting point of the solar orbit is indicated by a circle of radius 20 000 au. The solar orbit is by construction initiallyconfined to the x-z plane, and departs significantly from that plane only after a close interaction with star number 3. HM cluster P e r ce n t a g e o f s i m u l a ti on s IM cluster P e r ce n t a g e o f s i m u l a ti on s a o = 20000 au a o = 10000 au a o = 5000 au Fig. 6.
Distribution of comet survival probability for the three di ff erent bins of initial semimajor axis, a o = Left panel:
The high-mass cluster.
Right panel:
The intermediate-mass cluster.majority of disastrous cases with no or very few comets surviv-ing and at the same time another category of cases, where themost perilous encounter was weaker and left a significant part ofthe Oort Cloud bound to the Sun. This situation quickly becamefossilized, as the cluster started to dissolve and the Sun migratedoutward before finally escaping.Consequently, the time of escape might not matter very muchfor the survival statistics of the Oort Cloud, and even in case theSun remains in the IM cluster for the whole interval considered, relatively little further damage to the cloud may be the rule. Thecase of the HM cluster is di ff erent, because its central region re-mains perilous for the full length of the integrations. Therefore,in case no disastrous encounter occurs during the early stage, theremaining part of the Oort Cloud will in general be subject tofurther damage due to close encounters. The chance for a signif-icant fraction of surviving comets is relatively small. However,there are of course situations, where the Sun undergoes e ffi cientoutward migration, and the survival rate is higher. This will be
8. Nordlander et al.: The Destruction of an Oort Cloud in a rich stellar cluster the case in particular for simulations where the Sun escapes fromthe cluster.As shown in Table 1, the comparison simulations where weartificially turned o ff either the cluster tide or the stellar encoun-ters, indicate that the survival probability for the HM cluster ishardly a ff ected at all by the presence or absence of the clustertide in the dynamical model, while the removal of the stellar en-counters drastically increases the chances of survival for all theprimordial cloud populations. It is thus clear that the losses ofcomets are mainly due to the stellar encounters. However, forthe outer cloud we also note that the median survival probabilityis very low even without stellar encounters. Hence, in this partof the cloud – and only in this part – the comets are destabilizedalso by the cluster tide.We also see an indication in Table 2 that, for the IM clus-ter, both the cluster tides and the stellar encounters matter forthe loss of comets from the Oort Cloud – the tides even morethan the encounters. Here we have to take note of the fact thatthe scenario in the model without stellar encounters is very dif-ferent from that of the other two models. In the full dynamicalmodel the Sun escapes from the cluster in a large majority ofcases, as seen above, and the same is true for the model withoutthe cluster tide. However, when there are no stellar encounters,the solar orbit remains practically unchanged – hence, the Sunnever escapes but stays close to its initial orbit during the wholeintegration. The initial orbits penetrate close to the cluster centreand the comets are hence sensitive to the radial tide. This ex-plains the extensive losses of comets from all parts of the OortCloud. In Tables 1 and 2 we present statistics regarding the three endstates, for both clusters: the direct departures leading to unlink-ing of the comets, the entries into the loss cone leading eventu-ally to hyperbolic ejection by Jupiter, and the survivals until theend of the integration. These are shown for the full dynamicalmodel as well as the two comparison models, and for each setof primordial comet orbits. In each case, the listed percentagesrefer to the median of the simulations.The most striking feature concerning the HM cluster(Table 1) is the predominance of direct departures in the fullmodel as well as the model without tides. The model without en-counters is di ff erent as regards the inner and intermediate cometpopulations, where instead of the predominance of departureswe find large fractions of survival or loss cone intrusion. Oncemore we see that the outer population is very vulnerable to directdepartures even without encounters. Apparently, the cluster tidecauses a strong instability of the outer cloud orbits but much lessso for the orbits of the other parts.The loss cone entries practically only appear in the presenceof the cluster tide, showing that stellar encounters very rarelycause such an evolution. On the contrary, stellar encounters areseen to interfere with the tidal evolution of the perihelion dis-tance, preventing the loss cone entries from the inner and inter-mediate cloud that would otherwise occur. By plotting the peri-helion distance vs time in the tide-only model, we have verifiedthat the loss cone entries are caused by a secular oscillation of or-bital angular momentum driven by the cluster tide. Even thoughsuch a pulsation is also a feature of the Galactic disk tide cur-rently experienced by Oort Cloud comets (Heisler & Tremaine1986), the dynamics inside the cluster is basically di ff erent. Thecluster tide is radial and non-conservative. The amount of theenergy exchange is shown by the preference for tidally caused Table 1.
Median probabilities (%) of comet end states in mod-els of the high-mass cluster with / without cluster tide and stellarencounters. a Unlinked Loss cone Survived(au) Full dynamical model a b b Notes. ( a ) Based on 1000 simulations. ( b ) Based on 100 simulations. departures of Oort Cloud comets belonging to the outer popula-tion.Comparing with the results for the IM cluster (Table 2), themain di ff erence appears for the model without stellar encounters.As noted above, the IM model is special in that the Sun remainsmore or less locked to its initial orbit for the full length of theintegration. As discussed in Sect. 3.4, these orbits tend to havepericentre distances less than 0.5 pc. That this exposes the OortCloud comets to a very strong cluster tide can be realized fromFig. 2, because the strength of the cluster tide is proportional tothe mass density in the homogeneous, central part. This densityis seen to be very high at all times in the IM cluster, and theregion of homogeneity extends to r (cid:39) . The time evolution of the comet cloud is illustrated in Fig. 7.We will first discuss the left panel, showing the case of the HMcluster. The outermost population is typically dispersed by thevery first close stellar encounter in each simulation, giving riseto a continuous distribution of semi-major axes reaching beyond1 pc. Such wide orbits are unstable in the current Galactic en-vironment and even more so in a dense cluster. The cometsare rapidly lost by the two energy-perturbing agents identifiedabove, i.e. , the stellar encounters and the cluster tide.After some 200 Myr, the primordial populations have dis-persed su ffi ciently that the comet orbits appear rather smoothlydistributed, albeit retaining a broad central peak covering a ∼
9. Nordlander et al.: The Destruction of an Oort Cloud in a rich stellar cluster
HM cluster −5 −4 −3 −2 −1 F r ac ti on a l popu l a ti on Comet semimajor axis, a [AU] IM cluster −5 −4 −3 −2 F r ac ti on a l popu l a ti on Comet semimajor axis, a [AU] Fig. 7.
Time evolution of the combined comet population of all simulations with the full dynamical model. The fractional populationsare counted with respect to the total initial population. Results were sampled at 10 Myr timesteps, using 100 uniform logarithmic binsfor a ∈ [500 ,
200 000] au. The locations of the three initial sub-populations are indicated by vertical dashed lines. Here, snapshotsare shown at times T =
10, 50, 200, and 400 Myr, from top to bottom, using black, red, blue and green colours, respectively, wherethe final time step is also shaded in grey. Comets beyond the plot limits have been summed at the edges.
Left panel:
The high-masscluster.
Right panel:
The intermediate-mass cluster. The three remaining peaks in the final distribution reflect the fact that the vastmajority of the stars in this cluster model have by then left the cluster with fossilized structures of their cometary clouds.
Table 2.
Same as Table 1, but for the intermediate-mass cluster. a Unlinked Loss cone Survived(au) Full dynamical model a b b Notes. ( a ) Based on 1000 simulations. ( b ) Based on 100 simulations. population, a >
10 000 au, diminish at a rate similar to thecentral population. This means that a steady state is reached,whereby this outer population remains as a transit stage ofcomets migrating from the inner parts and eventually departingfrom the solar system due to energy perturbations. The semi-major axis distribution for a >
20 000 au is seen to evolve towarda power-law slope of − ff usion process with an absorbingwall near 1 / a = a < a (cid:46) inner cloud, while the primor-dial intermediate and outer clouds each contribute one and twoorders of magnitudes fewer comets (see below). The core is thusmore than twice as populated as the outward migrators beyond20 000 au at the time ( T =
400 Myr), when we stop the integra-tions.From the plotted results, we also have an indication aboutwhat would happen, if we had continued the integration furtherin time. The peak would remain close to 5 000 au, and the curvewould flatten out at smaller semi-major axes, while it would con-tinue to be shifted downward at larger semi-major axes. Thusthe predominance of the quasi-inert core would become furtheraccentuated, as the total population of the cloud continues to de-crease.The evolution of the cloud in the IM cluster case – shownin the right panel – is basically similar, but some di ff erencesare easily seen. After 50 Myr the structure undergoes very smallchanges. As a rule, the Sun has then left the cluster or migratedout of the central region for most of the time. We noted abovethat this leads to a fossilized structure of the cloud, which wehere see represented by the histograms in red, blue and green.The inner core is less pronounced than in the HM case. The out-ermost part of the cloud is cut at a ∼
50 000 au, since there are nomore perturbations large enough to replenish these orbits fromthe inside.We illustrate in Fig. 8 the separate mean semi-major axis dis-tributions for the surviving comets of the three primordial pop-ulations. The HM cluster is shown to the left and the IM clusterto the right. The median values of the survivors in the HM clus-ter are less than the initial values – 4 510, 8 000 and 15 700 aufor the inner, intermediate and outer populations, respectively.These shifts are due to the preferential loss of comets reaching
10. Nordlander et al.: The Destruction of an Oort Cloud in a rich stellar cluster
HM cluster Comet semimajor axis, a [AU]10 −5 −4 −3 −2 F r ac ti on a l popu l a ti on a o = 5000 a o = 10000 a o = 20000 a o = 5000 a o = 10000 a o = 20000 IM cluster Comet semimajor axis, a [AU]10 −5 −4 −3 −2 F r ac ti on a l popu l a ti on Fig. 8.
Mean semi-major axis distributions of the surviving comets at the end of the integrations, combined from all simulations,for the three primordial populations as in Fig. 6, using the same colours and styles of the curves. Vertical dashed lines represent theinitial values. Di ff erential distributions are shown by histograms, using a common normalization to the size of the initial populations.The curves are shaped by energy perturbations caused by the external agents. Statistical noise is seen mainly for the inner andintermediate primordial populations. Left panels:
The high-mass cluster.
Right panel:
The intermediate-mass cluster.large semi-major axes and the relative safety of comets di ff usingtoward smaller values. The shifts are smaller in the IM cluster,especially for the inner and central populations. This is likelydue to the absence, in most cases, of a long-term energy di ff u-sion.Thanks to the common normalization used, the fact that theouter population has the smallest number of survivors in bothclusters is clearly displayed in Fig. 8. The mean survival prob-abilities in the HM cluster are 2.1, 0.6, and 0.1 % for the threerespective populations. In the IM cluster these values are 11.7,4.1, and 0.8 %. All these values are significantly larger than thecorresponding medians (Tables 1 and 2), because there is signif-icant spread between the results of di ff erent simulations, and thesurvivors are concentrated to the minority that had the smallestexternal e ff ects.The sums of the three di ff erential distributions yield the dis-tributions shown at T =
400 Myr in Fig. 7. Each of these hasa roughly triangular shape in the log-log diagrams used with amaximum at the initial value of the semi-major axis. For the HMcluster we see a steeper slope for larger than for smaller values.As noted above, the steeper slope is close to −
2, and the corepopulation created by inward migration has contributions di ff er-ing by roughly one order of magnitude between the inner, inter-mediate and outer primordial populations. For the IM cluster thedistributions are more symmetric around the maximum until thecut at large semi-major axes is reached. The slopes are higher onboth sides of the maximum than in the HM cluster case.We have already noted a few special features of the evolu-tion experienced within the IM cluster. One is that the Sun tendsto leave the cluster during our simulations, whereby the influ-ence of the cluster on the Oort Cloud is terminated. The other isthat the IM cluster is equipped with a high-density central core,which acts as a very e ffi cient pitfall to Oort Cloud comets, incase the solar orbit enters into its vicinity. To explore the in-fluence of these features on the survival of the Oort Cloud, inFig. 9 we illustrate the relevant statistical properties: histogram distributions of the time when the Sun is ejected from the clus-ter and the minimum periapsis distance of the solar orbit, andtogether with these the variations of the Oort Cloud average sur-vival probability with the parameters in question.The most striking fact revealed by the two panels of Fig. 9is that the minimum periapsis distance e ff ectively governs theOort Cloud survival probability, while the time of ejection of theSun does not exhibit any similar influence. It is clear that anyapproach of the Sun to less than 0 . − . . − . ff thecluster tide, and if we turn o ff the stellar encounters. This demon-strates that stellar encounters do have an influence. However, thetidal e ff ect is probably the dominant e ff ect. In our model, onlyone star will be closely encountered during each periapsis pas-sage of the Sun, and it seems unlikely that the orbit with thesmallest distance from the cluster centre will invariably involvean encounter that is e ffi cient enough to strip away almost thewhole cometary cloud. Thus, some scatter would be expected,making the survival curves less smooth and monotonic if stellarencounters were dominant.The wavy pattern exhibited by the survival probability withrespect to time of ejection reflects the limited statistics. Althoughthe curves are smoothed, their maxima are strongly influencedby the occasional simulations, where the minimum periapsis dis-
11. Nordlander et al.: The Destruction of an Oort Cloud in a rich stellar cluster
IM cluster P e r ce n t a g e alla = 5000 au a = 10000 aua = 20000 au IM cluster P e r ce n t a g e Fig. 9.
Statistics for the IM cluster over the distribution of simulations (grey-shaded histogram) and the survival percentages ofcomets. The comet populations are shown both combined (thick black curve) and for each population (coloured curves). The survivalstatistics are computed as the arithmetic mean using a boxcar, and then smoothed by a Gaussian kernel.
Left panel:
Statistics as afunction of the time when the Sun is ejected from the cluster, using a boxcar width of 50 Myr and Gaussian kernel σ =
10 Myr.Cases where the Sun is not ejected from the cluster are shown to the right of the vertical dashed line.
Right panel:
Statistics asa function of the Sun’s nearest distance to the cluster centre, computed in logarithmic bins using a boxcar width of 0.3 dex andGaussian kernel σ = .
05 dex.tances are large and many comets survive. Ejection times closeto that of one of these simulations will generate a relatively largesurvival probability in the calculated distribution. The waves fordi ff erent initial cloud populations are in phase, because thesegroups are simulated with identical solar orbits and encounters.The absence of any systematic decrease of the survivability withtime of ejection merely shows that the Sun may spend a longtime in the cluster with its Oort Cloud intact, provided that itdoes not penetrate into the cluster core. Let us finally present some results on the evolution of the solarorbit in the cluster, caused by the impulses received from the en-countering stars. Figure 10 shows two scatter diagrams of apoc-entre vs pericentre distances at the beginning of our integrations(upper panels) and at the end (lower panels). The left pair ofpanels refers to the HM cluster and the right pair illustrates theIM cluster. Each symbol represents one of the 1 000 simulationswith the full dynamical model. The red symbols in the initialdistributions mark those solar orbits which were not stable, sothat the Sun was ejected from the cluster before the end of theintegrations (this fate was registered, if the Sun moved beyondthe tidal radius, given in Table A.1, in each cluster).The cases of ejection amount to 5.8% of the simulations forthe long-lived, HM cluster. This verifies the expectation for theSun as a relatively massive cluster star that it su ff ers only a smallrisk of ejection within 400 Myr. For the IM cluster the situa-tion is the opposite. The cluster as a whole dissolves on a muchshorter time scale and the Sun is no exceptional star. We findthat the median survival time of the Sun as a cluster memberis about 100 Myr, while in 80% of the simulations the Sun isejected within 200 Myr. After 400 Myr, the Sun remains in only4.8% of the simulations. For both cluster models, the initial and final distributions ofsolar orbits di ff er markedly from each other. Most of the time theSun is pushed outwards in the cluster, since both apocentre andpericentre distances show increasing trends – more pronouncedfor the apocentre distance. For a minority of cases in the HMcluster, the solar orbit evolves into a smaller apocentre distanceor a lower e ff ective eccentricity. In the IM cluster, most of theremaining solar orbits stay close to the cluster centre.We caution that the trend toward the outskirts of the HMcluster may be a ff ected by our manner of selecting the stellarencounters. Choosing each time the encounter with the largeststrength parameter (Sect. 2.3) means that the massive stars arefavoured as encounter partners. Statistically, during binary en-counters, energy per unit mass flows from the more massive tothe less massive partner – a basic reason for mass segregationcausing the concentration of high-mass stars to the cluster coreand the preferential escape of low-mass stars (Spitzer 1969). Hadwe included all the encounters, in which case the simulationshad been more realistic, the outward migrating trend for the Sunwould likely have been balanced by many interactions with low-mass stars and thus reduced.Another word of caution is justified concerning the IM clus-ter. Here, again, our result may be biased by our modelling ofthe cluster. In reality, the central region of the cluster becomesenriched in massive stars, but we neglected this mass segrega-tion. Thus, our encounter selection – while preferring massivepartners – did not favour these as much as it should have done ina realistic modelling. This may have made it easier for the Sunto stay in the central region rather than be expelled from it.
4. Discussion
We have chosen to base one of our cluster models on M67,which according to Hurley et al. (2005) started out with more
12. Nordlander et al.: The Destruction of an Oort Cloud in a rich stellar cluster
HM cluster I n iti a l a po ce n t e r d i s t a n ce [ p c ] F i n a l a po ce n t e r d i s t a n ce [ p c ] IM cluster I n iti a l a po ce n t e r d i s t a n ce [ p c ] F i n a l a po ce n t e r d i s t a n ce [ p c ] Fig. 10.
Distributions of initial and final solar orbits for all our simulations using the full dynamical model. Pericentre ( q ) andapocentre ( Q ) distances of the rosette orbits are plotted on the axes of each diagram. Left panels:
The high-mass cluster.
Rightpanels:
The intermediate-mass cluster.
Top panels:
Initial orbits. For the HM cluster, these orbits start from the half-mass radius( r = .
87 pc) with some random radial velocity, so we always have q < .
87 pc < Q . The red crosses denote orbits, where the Sunwas ejected from the cluster during the simulation, while the rest are denoted by black diamonds. Bottom panels:
Final orbits. Onlyorbits where the Sun survived as a cluster member are shown.than 20 000 stars. This was done in spite of the conclusion byAdams (2010) that a birth cluster with more than about 10 000stars would threaten the stability of the planetary orbits. Onereason not to worry is that Adams considered giant planets ontheir current orbits, while the Nice Model holds that the orbits ofthe giant planets had much smaller semi-major axes during theearly epochs of solar system history – both in its original form(Tsiganis et al. 2005) and in later versions. This would obviouslyreduce their vulnerability to external perturbations. However, theintricate resonant clockwork of the Nice Model (Levison et al.2011) might be upset by stellar encounters far more distant thanthose previously considered as disastrous. No analysis of thisproblem has yet been made to our knowledge.We did not aim to survey the full spectrum of cluster sizesbut just took two examples of relatively rich systems that are statistically likely birthplaces of solar type stars (Lada & Lada2003). We found that the intermediate-mass cluster is in a certainsense as hostile to the survival of a primordial Oort Cloud as thehigh-mass one. However, it is dangerous to extrapolate this toeven smaller birth clusters. At some point, the cluster dissolvesso quickly that the destructive influence on the Oort Cloud dis-appears. It remains to find out at which initial cluster mass thistransition occurs.Meanwhile, it is worth noting that a surviving Oort Cloudformed in a very dense cluster environment should be be verytight including semi-major axes far smaller than those that wehave investigated (Fern´andez & Brunini 2000; Brasser et al.2006). We have seen that such an inner core does survive in theclusters we modelled. An important issue is then how to activatethis core and repopulate the outer halo after the star leaves the
13. Nordlander et al.: The Destruction of an Oort Cloud in a rich stellar cluster cluster. In a sense, a fossilized inner core would be ine ff ective:it would not provide any observable comets. We shall return tothis discussion in Sect. 4.4.As to the Sun, an active Oort Cloud was formed in the sim-ulations by Levison et al. (2010), who considered a very smalland short-lived birth cluster. In fact, judging from Adams (2010),many of the relevant cluster sizes would not be in absoluteconflict with the nucleosynthesis requirement, since they wouldyield a random probability of ∼
10% for a relevant supernovaexplosion.Another issue concerns the realism of our dynamical modelfor the external perturbations su ff ered by the Oort Cloud comets.Of course, only a full N -body simulation of the cluster may beconsidered fully realistic, but this is beyond the scope of ourpreliminary study. The synthetic model that we developed rep-resents the cluster by its smooth potential field plus a sequenceof two-body encounters within this field, which the Sun expe-riences with individual cluster stars. This is quick and e ffi cient,but it certainly departs from reality. We have assumed the Sunand the comets to experience, in addition to the smooth clusterpotential, the fluctuating component caused by just one passingstar at a time. In reality there are typically several nearby starscontributing to this fluctuating field at the same time. Hence,concerning the loss of comets, in many cases the fate of a cometmay critically depend on the details of the tidal field, so that ourapproximation may either save comets from leaving the Sun orstimulate their escape, depending on the circumstances.Although it does not seem likely that any serious system-atic errors occur as a result of our simplifications, a full answercannot be found except through time-consuming N-body simu-lations. One technique would be to relax the blocking of over-lapping encounters to see, if by allowing up to 5–10 simulta-neous encounters and treating these N -body systems accurately,we would get statistically di ff erent results on the loss of comets.However, this would mean an additional major e ff ort, which stillwould not solve all problems. The most important simplification that we used may be the as-sumption of a step function for the initial population of theOort Cloud. In fact, the Oort Cloud was not built instantly. Itis clear from the works of Kaib & Quinn (2008) and Brasser& Morbidelli (2013) that the emplacement of comets from theplanetary region into the Oort Cloud took several hundred Myr.Thus, what we call a primordial Oort Cloud should have beenenriched in new members for a time comparable to the clusterlifetime or the full length of our simulations.Connected to this is another assumption, namely, that theSun still resides in its birth cluster at the starting time that weuse, 100 Myr after the formation of the Sun and the cluster. Forthe HM cluster, the risk of ejection of the Sun during this earlyinterval is negligible, but not so for the IM cluster. This has aninitial mass of 820 M (cid:12) , and at the starting time the mass has de-creased to 710 M (cid:12) . This decrease amounts to 13%, which wetake as a rough estimate of the risk of early ejection of the Sun.Hence, in the IM cluster case, there is a 13% chance thatsome of the comets transferred into the Oort Cloud during thefirst 100 Myr as well as all those transferred at later times wouldnot feel any e ff ects of the birth cluster. However, the comets inquestion – likely being the majority of all Oort Cloud comets– would then be emplaced without the help of the birth cluster.The Oort Cloud would then likely have much less of an inner core than if the Sun had stayed in the cluster, and the creation ofSedna-type objects might be strongly curtailed.With the complementary probability of 87%, we have under-estimated the cluster influence on the early emplaced comets andyet strongly overestimated the influence on those that were em-placed at later times. It seems clear that the overestimates dom-inate as an error source. Again, one would have to distinguishbetween two parts of the Oort Cloud – the comets that were em-placed inside the birth cluster, which did experience its destruc-tive e ff ects, and those that were emplaced after the Sun had left,which did not benefit from the cluster in populating the innercore.At any rate, we conclude that the Sun could not spend alonger time than approximately 50 Myr in an IM cluster withan already formed Oort Cloud left intact except under specialcircumstances. Such circumstances would include a solar orbitwhich kept the Sun and the Oort Cloud constantly outside thecentral part of the cluster.Finally, we stress that there are three categories of birth clus-ters, which we have estimated to be about equally likely for theSun in terms of the initial cluster mass function. The LM case is,however, less probable in view of the nucleosynthetic evidencefor an early supernova in the neighbourhood (see the discussionabove). In the HM case our model should be reasonably good;the IM case was just discussed; and in the LM case the destruc-tive e ff ects of the birth cluster would likely be much smaller.In fact, we have made a few additional approximations thatlikely caused us to underestimate the losses of comets, thusyielding conservative estimates. The first is that we neglectedmass segregation in our model cluster. Hence we downplayed therisk for the Sun to encounter a massive star in the high-densityenvironment near pericentre, which would have had dire conse-quences for the entire Oort Cloud.The second is that we neglected binary stars. M67 is knownto be rich in binaries (Richer et al. 1998), and thus other simi-lar clusters may be suspected to be similarly binary-rich. Binarystellar systems are also known from dynamical simulations toform and dissolve within star clusters including clusters of muchsmaller masses (Giersz & Heggie 1997). By ignoring binaries,we have artificially increased the number of potential encounterpartners of the low mass type, while entirely neglecting a kind ofpartner that would have had a great capability to transfer energyand momentum to the Sun’s motion – and, similarly, to destroythe Oort Cloud. In particular, “hard” binaries in close orbits areknown to statistically give energy away to the encounter partner,thus providing an energy source for the dynamical evolution ofthe cluster, as reviewed by Elson et al. (1987).A third approximation is that we treat only a rather smallnumber of subsequent encounters in each simulation. This num-ber might be increased without introducing overlapping encoun-ters, but a more fundamental issue is that we select the strongestencounters in terms of the impulse imparted to the Sun per unitmass within the approximation of the classical impulse approxi-mation. Certainly, all kinds of encounters may occur, but we sys-tematically disfavour the weaker ones involving the less massivepartners. As already remarked, this creates an exaggerated trendfor the Sun to move outward in the cluster. Hence, statistically,the Sun spends too much time in the outer, less populated re-gions, so we underestimate the risk of strong perturbations thatcharacterizes the inner parts. Thus, our treatment should under-estimate the total number of lost comets from the cloud in thata too small number of stellar interactions, and in fact not neces-sarily the most e ff ectively destructive ones, are considered. Our
14. Nordlander et al.: The Destruction of an Oort Cloud in a rich stellar cluster estimated destruction rates for the Oort Cloud are thus conserva-tive in this respect.Yet another neglected phenomenon could have increased thenumber of remaining comets. Some of the lost comets may bepicked up by the encountering stars, and this number wouldlikely increase, if we would treat overlapping encounters. Eventhough most lost comets rather become cluster vagabonds, in duetime these could also be picked up by some cluster star, includ-ing the Sun, in a way similar to the formation of stellar binariesin clusters.
All this begs the question, whether by assuming that the Sun hada primordial Oort Cloud one should assume that other clustermembers of similar types were also equipped with such, primor-dial cometary clouds. If so, the sort of cluster environment thatwe consider here might stimulate a certain exchange of cometsbetween di ff erent stars (Zheng et al. 1990; Levison et al. 2010).We cannot say how e ffi cient this process would be using oursimplified model. On the other hand, it is clear that picked-upOort Cloud members would be particularly vulnerable to beinglost during following encounters, since these would occupy rel-atively loosely bound orbits.Concerning the question, if Oort Clouds may be a character-istic feature of Galactic disk stars, we note that Stern et al. (1991)searched for extra-solar Oort clouds around 17 nearby starsby looking for IR excess radiation using IRAS low-resolutiondata and an S / N-enhancing method, with negative results which,however, may be ascribed to the limited sensitivity. Black (2010)has extended this search on the basis of the IRAS sky surveyto all F and G dwarfs, augmented with all other stars (then)known to have planetary systems, within 50 pc distance fromthe Sun. While challenging, since the sensitivity of the IRASis again a severely limiting factor, no positive identifications ofOort clouds around other stars were reported.Some cold, dusty outer disks have been found and studiedby means of the Herschel telescope, around young stars, alsowith planetary systems. One example is the A5 V star HR 8799with four known planets, at a distance of 40 pc and an age esti-mated at 20 to 50 Myr, which has a central warm dust compo-nent, an outer cold component extending from 90 to 300 au andan outer component of small grains extending beyond 1000 au(Matthews et al. 2014). The evidence for any clumping in thishalo is, however, meagre. Most observations of debris disksaround young stars are limited to A-type stars and to rather smallradial distances from the star. One interesting example is, how-ever, the F5 / β Pictoris moving group, for which ALMA observations disclosea ring-like CO gas disk, in addition to the dust ring, with a haloextending out to 200 au, and a CO + CO cometary composition(Marino et al. 2016).Neutron stars capturing comets when passing through theOort clouds of other stars have been suggested to provide in-dications on the (non-)existence of extra-solar cometary clouds.Shull & Stern (1995) thus proposed that such events should gen-erate repeating bursts of soft gamma-rays, and estimated that theabsence of such Galactic events indicated that at the most a fewpercent of the Galactic stars have Oort clouds. It is, however,very questionable whether accretion of comets onto neutron starswould occur abruptly enough to generate such bursts; weakeremission, more extended in time and at lower energies, seemsmore probable (we thank Dr. J. Poutanen for making this point). We conclude that presently no observational limits may as yet beset on how frequent Oort clouds are around stars. Our basic result is that, if the Sun was born as a member of a rel-atively rich stellar cluster, a significant part of a primordial OortCloud would not likely survive the time in the cluster until theLate Heavy Bombardment. The relative extent of this depletiondepends on the detailed orbital evolution of the Sun within thecluster. In the case of a massive depletion, the formation of thepresent Oort Cloud as a consequence of a late planetary migra-tion within the Nice Model (Levison et al. 2011) appears to be aviable scenario, provided that the Sun had then already left thecluster, or was about to do so. Such a scenario has been exploredby Brasser & Morbidelli (2013).In a very massive cluster the escape of primordial Oort Cloudcomets is mainly caused by the disrupting e ff ect of stellar en-counters. The eccentricity pumping due to the cluster tide playsa role only for a minority of comets in the inner part of thecloud. In the outer part the energy perturbations caused by thetide may constitute an important source of comet losses. In clus-ters of lower mass the latter type of tidal perturbations providea major loss mechanism in all parts of the Oort Cloud. If thecluster mass is very high, the number of comets penetrating towithin 5 au of the Sun amounts to (cid:39) O) well before thetime of the LHB.The influence of the stellar encounters on the solar orbit canbe seen in Fig. 10. From this and the solar escape rate, we con-clude that the Sun typically receives a cumulative impulse ofseveral km / s. Since the number of encounters per simulation isonly about 20, in a random walk there must be some individualkicks experienced by the Sun that amount to about 1 km / s andare thus strong enough to unlink most of the Oort Cloud comets.The solar impact parameters of those stars are typically muchless than the Sun-comet distances. Thus, the main mechanism ofcomet escape in our model is that the Sun is kicked away fromits comet cloud rather than individual comets being kicked awayfrom the Sun.By restricting our simulations to a small number of stellarencounters, we may have introduced too large a statistical dis-persion of the results. In particular, the simulations that left theOort Cloud – particularly for the intermediate-mass cluster –with much more comets than the median can be regarded aschance selections of solar orbits with weaker tidal e ff ects andencounter sequences than normal. We have shown that the min-imum periapsis distance q min of the Sun plays a decisive role forthe survival frequency in the Oort Cloud – the larger q min , themore survivors. According to our results, the category of out-comes with the larger survival rate makes a significant thoughnot dominant contribution, but some caution is warranted, espe-cially concerning the results for the IM case, until more realisticcluster simulations can be made. Tentatively, there is no reasonto suspect that we would have exaggerated the comet loss ratein the IM cluster by selecting initial solar orbits with too small q min . The selection was made on the basis of the general densityprofile of the cluster, so we neglected the fact that the Sun, be-ing a relatively high-mass star, would tend to prefer the centralregion at the time we started our simulations and the cluster wasalready considerably relaxed. Again, however, only fully realis-
15. Nordlander et al.: The Destruction of an Oort Cloud in a rich stellar cluster tic simulations including binarity and other special phenomenawill provide the full answer to this question.A primordial Oort Cloud may consist of comets that origi-nated in the accretion zone of the giant planets. However, if theplanetary orbital instability of the Nice Model actually happenedvery early, as was recently argued by Kaib & Chambers (2016),the Oort Cloud originating from the trans-planetary disk wouldalso be primordial. It may have been formed after the birth clus-ter was dispersed, if this cluster contained only about a hundredstars. Such a birth cluster may not be totally excluded by theargument of nucleosynthesis providing the short-lived radionu-clides for the solar nebula, even though somewhat special cir-cumstances would be called for in the case of the Sun. If so, theformation scenario of Brasser & Morbidelli (2013) would be arelevant model except that the timing of the event is di ff erent.On the other hand, if the birth cluster was of the IM orHM kind, we have shown that the primordial Oort Cloud wouldlargely survive only as a tight, inner core. Models of Oort Cloudformation in such a dense stellar environment indeed predict aninitial cloud structure dominated by such a core (Brasser et al.2006). Therefore, in such a case the existence of the present outerhalo, from where the observed comets can be transferred by theGalactic tide, requires a mechanism of energy transfer that canactivate the core from its inert state. The alternative would be alate planetary instability as investigated by Brasser & Morbidelli(2013), in which case it is reasonable to assume that the Sun hadalready left the birth cluster.To be specific, taking the Galactic disk tide to be the mecha-nism for bringing Oort Cloud comets into the inner solar system,the tidal torquing time scale (Heisler & Tremaine 1986) is foundto be longer than the age of the solar system, unless a (cid:38) a (cid:46) ff ects of the birth clusters and therefore much smaller than usu-ally thought. Quite likely, Oort Clouds in general would mostlystem from trans-planetary planetesimal disks, and thus, their ex-istence depends on the way extra-solar planets have migrated. Acknowledgements.
This work was supported by the Polish National ScienceCenter under Grant No. 2011 / / B / ST9 / / References
Adams, F. C. 2010, ARA&A, 48, 47Adams, F. C. & Laughlin, G. 2001, Icarus, 150, 151Alexander, P. E. R. & Gieles, M. 2012, Monthly Notices of the RoyalAstronomical Society, 422, 3415Alexander, P. E. R., Gieles, M., Lamers, H. J. G. L. M., & Baumgardt, H. 2014,Monthly Notices of the Royal Astronomical Society, 442, 1265Black, G. J. 2010, in Bulletin of the American Astronomical Society, Vol. 36,American Astronomical Society Meeting Abstracts
16. Nordlander et al.: The Destruction of an Oort Cloud in a rich stellar cluster
Appendix A: Calculation of stellar cluster structure
We prescribe that the model cluster should have a given mass M cl and be situated in the Galactic potential at distance r G fromthe Galactic centre. This allows the calculation of a tidal radiusof the cluster, R t . Using a limiting energy E t , taken to be theenergy of a circular orbit situated at a distance from the clustercentre r = R t , our King model is calculated from a distributionfunction ϕ , which can be expressed as ϕ ( E ) = a (cid:110) e b ( E t − E ) − (cid:111) E < E t E ≥ E t (A.1)where E = U + v / a and b are positive constants of the model. We rewrite this distri-bution function when E < E t as a function of velocity and radialposition, ϕ ( v , r ) = a (cid:110) e b v max ( r ) e − b v − (cid:111) (A.2)and solve for the density ρ , ρ ( r ) = π (cid:90) v max ( r )0 ϕ ( v , r ) v dv = π a (cid:90) v max ( r )0 v (cid:110) e b v ( r ) e − b v − (cid:111) dv = π a (cid:40) e b v ( r ) (cid:90) v max ( r )0 v e − b v dv − v ( r )3 (cid:41) . (A.3)The velocity distribution can be evaluated by multiplying (A.2)by 4 π v , giving locally a truncated Maxwellian distribution. Thestructure of the cluster, its distribution of mass and the gravita-tional potential, is finally computed by solving the integral (A.3)simultaneously with Poisson’s equation, d Udr + r dUdr = π G ρ (A.4)using the parameter substitutions suggested by King (1966, eqns.19–22). The equations are solved from the cluster centre, wherethe potential energy U is taken as a free parameter. The free pa-rameters a , b , U thus represent a reformulation of the classicalfree parameters W , r and M cl required to fit a King model toan idealized stellar cluster.The input parameters and resulting properties of the optimalmodel are given in Table A.1. Appendix B: Implementation of stellar encounters
Each list covers a time interval of 40 Myr, and it comes from apreliminary integration of the solar orbit in steps of 3 kyr, yield-ing the mean encounter frequency, (cid:104) n (cid:104) v rel (cid:105)(cid:105) , where n and (cid:104) v rel (cid:105) are interpolated from the solution shown in Fig. 4. For each step,we let an encounter occur within b max =
20 000 au, if b π (cid:90) t i + t i n (cid:104) v rel (cid:105) dt > ξ (B.1)where t i + = t i + ξ ∈ [0 ,
1] is a random number drawnfor each step from a uniform distribution. This typically gener-ates about 100 encounters. For each such encounter, we generatea random stellar mass from the IMF as described in Sect. 2.1.3,evolving it to the age of the cluster at the current time. If thestar is found to have evolved through a supernova phase, it is discarded and a new value is drawn from the IMF and againevolved to the current time. The impact parameter for each se-lected encounter is determined randomly using b = b max √ ξ , fora new random ξ ∈ [0 , v rel relative to the Sun as in Sect. 2.2.After this preliminary selection, we know the times and pa-rameters of all the potential, upcoming stellar encounters, andwe are able to pick one of them based on the S values. Whenmodelling the encounter in a two-body scattering problem, theapproximation would be to let the encountering star aim frominfinity at a position on a heliocentric circle with radius b in aplane perpendicular to the direction of approach (called the im-pact plane). However, under the influence of the cluster poten-tial, this straight-line approximation cannot be used. Instead, werealise the closest approach by first choosing randomly the di-rection of relative motion, which defines the impact plane, andthen placing the star on the impact plane at a distance b from theSun at a random azimuth.The orbits of both stars are then integrated backward in timein the cluster potential with no mutual gravitational interaction,until the distance between the stars is d start . The time and geo-metric configuration at this stage are stored as the initial state ofthat encounter. In the few cases of very slow encounters, wherethe backward integration overlaps the previous encounter, thesetup is considered as failed and is discarded. We then select thesecond largest value of S and repeat the calculation of the initialstate.
17. Nordlander et al.: The Destruction of an Oort Cloud in a rich stellar cluster
Table A.1.
Input parameters and resulting properties of the stellar clusters. The properties are represented in terms of the inputparameters a , b and U (see Sect. 2.1). Input parameters Model propertiesTime span a × b × U × − M cl R t r h R c (cid:104) ρ h (cid:105) (cid:104) ρ c (cid:105) (Myr) (s kg / m − ) (J − ) (m s − ) ( M (cid:12) ) (pc) (pc) (pc) ( M (cid:12) pc − ) ( M (cid:12) pc − )The high-mass (HM) cluster100–500 4 .
036 3 . − . . . − .
83 692 13.0 1.88 0.07 119.5 11171.7125–150 228 . . − .
24 632 12.7 1.00 0.04 75.0 24678.8150–175 203 . . − .
03 580 12.3 1.13 0.05 48.1 16269.1175–200 181 . . − .
52 535 12.0 1.23 0.05 34.2 10399.9200–225 162 . . − .
71 492 11.7 1.31 0.06 26.1 6527.4225–250 151 . . − .
49 453 11.3 1.37 0.07 21.0 4490.9250–275 144 . . − .
71 420 11.0 1.40 0.07 18.2 3193.8275–300 135 . . − .
97 385 10.7 1.43 0.08 15.6 2091.7300–325 151 . . − .
13 355 10.4 1.42 0.09 14.9 2243.6325–350 155 . . − .
19 312 10.0 1.43 0.09 12.7 1800.4350–375 157 . . − .
72 285 9.7 1.43 0.10 11.6 1390.0375–400 154 . . − .
65 256 9.3 1.40 0.09 11.1 991.8400–425 171 . . − .
97 224 9.0 1.42 0.10 9.4 947.6425–450 174 . . − .
92 194 8.6 1.32 0.10 10.0 567.2450–475 192 . . − .
78 175 8.2 1.24 0.13 11.1 434.4475–500 201 . . − .
141 149 7.8 1.22 0.15 9.8 223.5500–525 238 . . − ..