The scattering of small bodies in planetary systems: constraints on the possible orbits of cometary material
aa r X i v : . [ a s t r o - ph . E P ] N ov Mon. Not. R. Astron. Soc. , 000–000 (0000) Printed 24 September 2018 (MN L A TEX style file v2.2)
The scattering of small bodies in planetary systems:constraints on the possible orbits of cometary material.
A. Bonsor ⋆ , M. C. Wyatt Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK
24 September 2018
ABSTRACT
The scattering of small bodies by planets is an important dynamical process inplanetary systems. In this paper we present an analytical model to describe this processusing the simplifying assumption that each particle’s dynamics is dominated by asingle planet at a time. As such the scattering process can be considered as a seriesof three body problems during each of which the Tisserand parameter with respectto the relevant planet is conserved. This constrains the orbital parameter space intowhich a particle can be scattered. Such arguments have previously been applied to theprocess by which comets are scattered to the inner Solar System from the Kuiper belt.Our analysis generalises this for an arbitrary planetary system. For particles scatteredfrom an outer belt directly along a chain of planets, based on the initial value of theTisserand parameter, we find that it is possible to (i) determine which planets caneject the particles from the system, (ii) define a minimum stellar distance to whichparticles can be scattered, and (iii) constrain range of particle inclinations (and hencethe disc height) at different distances. Applying this to the Solar System, we determinethat the planets are close to optimally separated for scattering particles between them.Concerning warm dust found around stars that also have Kuiper belt analogues, weshow that, if there is to be a dynamical link between the outer and inner regions, thencertain architectures for the intervening planetary system are incapable of producingthe observations. We speculate that the diversity in observed levels of warm dust mayreflect the diversity of planetary system architectures. Furthermore we show that forcertain planetary systems, comets can be scattered from an outer belt, or with fewerconstraints, from an Oort cloud analogue, onto star-grazing orbits, in support of aplanetary origin to the metal pollution and dustiness of some nearby white dwarfs.In order to make more concrete conclusions regarding scattering processes in suchsystems, it is necessary to consider not only the orbits available to scattered particles,but the probability that such particles are scattered onto the different possible orbits.
The scattering of small bodies is an important dynamicalprocess in many planetary systems. One classic example isthe population of small bodies close to the Sun, many ofwhich originate further out in the Solar System, from wherethey were scattered inwards. Near-Earth asteroids (NEAs)originate in the asteroid belt. Many left the belt after be-ing destabilised by resonances with Jupiter and then scat-tered by the terrestial planets (Morbidelli et al. 2002). Vis-ible comets are either objects scattered inwards from theKuiper belt or the Oort cloud (Levison & Duncan 1997).The scattering of small bodies has not been considered indetail for extra-solar planetary systems, mainly due to thelack of constraints on the structure of the planetary system.There is, however, evidence for small bodies in many extra-solar planetary systems. Dust belts, known as debris discsare seen around hundreds of main sequence stars (Wyatt2008). Observations, particularly resolved images, suggest that debris discs interact with planets (Kalas et al. 2005;Moerchen et al. 2011; Greaves et al. 2005), etc. Assuming asimilar nature to our Solar System, it is reasonable to ex-pect that scattering in these systems can also result in acomet-like population. The expected level and distributionof this comet population may differ substantially from theSolar System, depending on the individual planetary systemarchitecture.Evidence of such a comet-like population may existfrom observations of warm dust discs around a handfulof main sequence stars (Wyatt et al. 2005; Gaidos 1999;Beichman et al. 2005; Song et al. 2005). Comets or asteroidsin the position of the observed dust belts have a short life-time against collisions and drag forces. They cannot haveexisted for the entire main sequence lifetime in their ob-served position (Wyatt et al. 2007). One possible explana-tion is that the material originated in a cold, outer belt.It could be that we are observing a comet-like population, c (cid:13) A. Bonsor et al. that is continuously replenished from scattering of mate-rial from the outer belt by intervening planets (Wyatt et al.2007). Alternatively, it could be a transient event, resultingfrom the stochastic collision of two larger bodies (Song et al.2005), maybe in a similar manner to the impact that formedthe Earth-Moon system. Or, material could be transportedinwards from the outer belt during a LHB type event(Booth et al. 2009) or by drag forces (Reidemeister et al.2011).Another piece of evidence for the scattering of mate-rial in exo-planetary systems comes from observations ofevolved stars. 25% of DA white dwarfs show unexpectedmetal pollution (Zuckerman et al. 2003), whilst 1-3% of DAwhite dwarfs with cooling ages less than 0.5Gyr have excessemission in the infra-red consistent with a close-in dust disc(Farihi et al. 2009). The composition of the polluting mate-rial closely resembles planets (Klein et al. 2010) and thereis good evidence that it is not material accreted from theinter-stellar medium (Farihi et al. 2010a). The best theories,therefore, suggest that it originates in an outer planetarysystem (Jura 2003; G¨ansicke et al. 2006; Kilic et al. 2006;von Hippel et al. 2007; Farihi et al. 2009, 2010b; Melis et al.2010). As the star loses mass on the giant branch, dynami-cal instabilities can be induced in the outer planetary system(Debes & Sigurdsson 2002). These can lead to comets or as-teroids being scattered by interior planets onto star-grazingorbits, where they are tidally disrupted. Material from thetidally disrupted asteroids or comets forms the observeddiscs and accretes onto the star. The ability of evolved plane-tary systems to scatter comets or asteroids onto star-grazingorbits requires further detailed investigation, although previ-ous work has considered an Oort cloud origin of the scatteredbodies e.g. (Alcock et al. 1986; Debes & Sigurdsson 2002).In this work the scattering of small bodies inan arbitrary planetary system is investigated. N-bodysimulations are typically used to model such scat-tering (Levison & Duncan 1997; Horner & Jones 2009;Holman & Wisdom 1993). A deeper understanding of thegeneral properties of such scattering can, however, beachieved using analytical arguments. Simulations of scat-tered Kuiper belt objects have found that the scattering pro-cess can be approximated as a series of three-body problems,as the scattered bodies are passed from one planet to thenext (Levison & Duncan 1997). While such particles are un-der the influence of one of the planets, their dynamical evo-lution can be approximated by the circular restricted three-body problem in which the orbits of the particles must besuch that their Tisserand parameters, T p , (Tisserand 1896;Murray & Dermott 1999) are conserved, where T p = a p a + 2 r (1 − e ) aa p cos( I ) , (1)where a, e, I are the comet’s semi-major axis, eccentricityand inclination and a p is the planet’s semi-major axis. Thisconservation is so fundamental to cometary dynamics thatit is used to classify cometary orbits (Horner et al. 2003;Gladman et al. 2008).In this work we use the conservation of the Tisserandparameter to constrain the orbits of scattered particles ina planetary system with an arbitrary configuration. In sec-tion 2 we discuss how planetesimals are scattered from anouter belt, in an otherwise stable planetary system. We then outline our constraints on the orbits of particles scattered bya single planet in section 3, which we extend to two planetsin section 4 and arbitrarily many planets in section 5. Insection 6 we consider the application of this analysis to ourSolar System, systems with warm dust discs and pollutedwhite dwarfs. During the planet formation process, a planet that forms ina disc of planetesimals, will swiftly clear a zone around it,both by scattering processes and resonant interactions withthe planet. Analytically the size of the planet’s cleared zonecan be approximated. Criterion for the overlap of mean mo-tion resonances determine a region around the planet withinwhich orbit’s are chaotic (Wisdom 1980), whilst the Jacobiconstant can be used to determine the zone within which or-bits can be planet crossing (Gladman & Duncan 1990). Sim-ulations have shown that Neptune clears such a zone in lessthan 10 yr (Levison & Duncan 1993; Holman & Wisdom1993), but more generally one might expect 1,000 conjunc-tions for this clearing to take effect (Duncan et al. 1989).Material removed from this region may be ejected, whilstsome fraction remains on bound, eccentric orbits, with peri-centres close to the planet’s orbit, forming an analogue toNeptune’s scattered disc . After many scatterings some of thismaterial may reach far enough from the star to interact withthe Galactic tide (Tremaine 1993) and eventually populatean analogue of the Oort cloud.Planetesimals outside of this zone could in principlebe long term stable. However, N-body simulations of Nep-tune and the Kuiper belt find that Kuiper belt objectsare still scattered by Neptune at late times (Duncan et al.1995; Holman & Wisdom 1993; Levison & Duncan 1997;Emel’yanenko et al. 2004; Morbidelli 1997). The Kuiper belthas a complicated structure of stable and unstable regions.The gravitational effects of Neptune and the inner planetsresult in the overlap of secular or mean-motion resonancesproducing thin chaotic regions, within the otherwise stableregion (Kuchner et al. 2002; Lykawka & Mukai 2005) andsmall unstable regions within otherwise stable mean motionresonances (Moons & Morbidelli 1995; Morbidelli & Moons1995; Morbidelli 1997). Objects may diffuse chaoticallyfrom stable to unstable regions (Morbidelli 2005). Thisprocess has been shown to occur for Neptune’s 3:2and 2:1 resonances, amongst others (Morbidelli 1997;Tiscareno & Malhotra 2009; Nesvorn´y & Roig 2000, 2001;de El´ıa et al. 2008). Objects leaving mean motion reso-nances in the Kuiper belt, in this way, may be the mainsource of Neptune encountering objects at the age ofthe solar system (Duncan et al. 1995). Many of these ob-jects are scattered into the inner planetary system, andcould be the source of Centaurs or Jupiter Family comets(di Sisto et al. 2010; Morbidelli 1997; Levison & Duncan1997; Holman & Wisdom 1993).The dynamical processes occurring in the Kuiper beltmay well be applicable to exoplanetary systems with a sim-ilar structure, i.e. an outer planetesimal belt and interiorplanets. The outer belt could be truncated by resonanceoverlap (Wisdom 1980). Most particles would then inhabita predominately stable region exterior to this, containing c (cid:13) , 000–000 small regions that are unstable due to the overlap of secu-lar or mean motion resonances of the inner planets. Objectscould diffuse chaotically on long timescales from the stableto unstable regions and be scattered by the outer planet.Some of these scattered objects could enter the inner plan-etary system, whilst some could be ejected.In our consideration of the dynamics of material scat-tered from the outer belt by interior planets, we find thatthis dynamics is strongly dependent on the initial valueof the Tisserand parameter, with respect to the outermostplanet. Therefore, it is important to consider the value of thisparameter. Objects in the outer belt tend to have T > scattered disc and theOort cloud. It is possible that similar classes of objects ex-ist in exo-planetary systems, however, there is at present noevidence for exo-Oort clouds or scattered discs . The distribu-tion of the Tisserand parameter for such objects would differsignificantly from those that leave the cold Kuiper belt, inparticular for Oort cloud objects, where it is unconstrainedand
T <
Firstly we consider a system similar to that described in theprevious section, with a single planet, labeled by subscript i on a circular orbit at a i and an exterior planetesimal belt.We consider planetesimals scattered from the outer belt bythe planet. We make the simplifying assumption that plan-etesimals only interact with the planet if their orbits directlycross the planet’s orbit. This simplifies the following analysisand enables analytical limits to be easily derived. However,since in reality interactions will occur in a zone around theplanet, care should be taken in rigorously applying any ofthe derived limits, in particular for more massive planets.This will be discussed further in § For a planetesimal with a given value of the Tisserand pa-rameter with respect to this planet, T i , the potential orbitsonto which it can be scattered are limited, no matter howmany times it interacts with the planet. The Tisserand pa-rameter gives us no information about the probability forany given interaction to scatter a planetesimal onto a givenorbit, nor the timescales for interactions to occur. It does, however, limit the orbital parameters of the planetesimalsafter the interaction, in terms of its pericentre, q , eccentric-ity, e and the inclination, I , of its orbit with respect to theplanet’s. These constraints can be represented by a 3D vol-ume in ( q, e, I ) space. A planetesimal, given an initial valueof T i , may not be scattered onto an orbit with parametersoutside of this volume, in this simple example.This parameter space can be fully mapped out analyt-ically by re-writing Eq. 1 as T i = a i (1 − e ) q + 2 r (1 + e ) qa i cos( I ) , (2)and noting that if the planetesimal is to remain on a boundorbit, 0 < e < − < cos( I ) < q > Q > a i and q < a i mustapply. Applying these constraints to Eq. 2, places analyti-cal bounds that define this 3D volume of permitted orbits.Given the difficulties in presenting a 3D volume, we insteadpresent the 2D projection of this 3D volume onto the q − e plane, I − q plane and e − I plane, shown in Fig. 1. Theanalytical bounds are presented in Table. 1. Further examination of the q - e plot in Fig. 1 makes clear thatplanetesimals cannot be scattered further towards the starthan a limiting value, q min , determined by T. This value canbe calculated using constraints on the orbital parameters, Q = a i and cos( I ) = 1 (equivalent to the lower bound in the q − e plane). For 2 < T i < q min a i = − T i + 2 T i + 4 − √ − T i T i − . (3) q min as a function of T i is shown in Fig. 2. The eccentricityat q min will be given by: e lim = T i − √ − T i . (4)For T i <
2, the lines Q = a i and cos( I ) = 1 (positiveroot) no longer cross and the parameter space in the q-eplane is no longer bounded by Q = a i , rather by cos( I ) = 1(both positive and negative root). Therefore q min →
0. Thiscan be shown to be true by considering the derivatives ofthe lines: dqde | cosI =1 ,q → > dqde | Q = a i ,q → . (5)Importantly this implies that the constraints on thepericentre that apply to the orbits of objects with T i > T i <
2; such objects can bescattered onto orbits with any pericentre.
A single planet can also eject planetesimals, given a suit-able value of the Tisserand parameter. Unbound orbits ( i.e. those with e >
1) are not included in the plots in Fig. 1.It is, however, possible to determine from the top panel ofFig. 1 those values of the Tisserand parameter for whichthe particles are constrained to bound orbits with e < q = a i ), therefore substituting into Eq. 2, c (cid:13) , 000–000 A. Bonsor et al.
Figure 1.
The possible orbital parameters of particles scattered by a single planet, with a given value of the Tisserand parameter withrespect to that planet, T i . This forms a 3-D parameter space, that is shown here projected onto the eccentricity-pericentre ( q − e ) plane,the inclination-pericentre ( I − q ) plane and the eccentricity-inclination ( e − I ) plane. The limits of the parameter space are definedanalytically in the Table 1. The units of the pericentre is the planet’s semi-major axis and the dotted black line in the top row of plotsshows the line where the particle’s semi-major axis is equal to the planet’s ( a = q − e = a i ). ( q = a i , e = 1 , I = 0 ◦ ), we find that there is a limiton the Tisserand parameter such that only objects with T i < √ T p = 3 and q = a p can still be ejected if q < a p +∆. ∆ will be a function of the planet’s Hill’s radius, R H , for example if ∆ ∼ √ R H (Gladman & Duncan 1990)Jupiter can eject particles with T Jup = 3, whilst Neptunecannot.
Now consider a planetary system with an outer belt and twointerior planets, both on circular orbits. Particles from theouter belt are scattered by the outer planet, 1. The mainpossible fates of such particles are ejection, collision with aplanet or the star, further scattering interactions with thisplanet, or scattering by the inner planet, 2. Many scatteredparticles are scattered multiple times by the outer planet.It dominates their dynamics for a certain period of time,during which the Tisserand parameter, with respect to thisplanet, T , is conserved. At some point, the particle may bescattered onto an orbit that overlaps with the inner planetand it may be scattered by that planet. In such an inter-action the Tisserand parameter with respect to the innerplanet, T would be conserved, rather than T . Depending c (cid:13)000
Now consider a planetary system with an outer belt and twointerior planets, both on circular orbits. Particles from theouter belt are scattered by the outer planet, 1. The mainpossible fates of such particles are ejection, collision with aplanet or the star, further scattering interactions with thisplanet, or scattering by the inner planet, 2. Many scatteredparticles are scattered multiple times by the outer planet.It dominates their dynamics for a certain period of time,during which the Tisserand parameter, with respect to thisplanet, T , is conserved. At some point, the particle may bescattered onto an orbit that overlaps with the inner planetand it may be scattered by that planet. In such an inter-action the Tisserand parameter with respect to the innerplanet, T would be conserved, rather than T . Depending c (cid:13)000 , 000–000 Plane Line Constraint Based on q - e : upper dashed e = 1 + 2 q − qT i + 2 q / p q − qT i cos( I ) = ± q - e : lower dot-dashed e = (1 − q )(1+ q ) Q = 1 I - e : upper dashed cos( I ) = T i − − e √ − e Q = 1 I - e : upper dotted cos( I ) = T i − e √ e q = 1 I - e : lower dot-dashed cos( I ) = T / i √ − e ) ∂I∂q | e,T i = 0 I - q : upper dot-dashed cos( I ) = T i (1+ q ) − √ q (1+ q ) Q = 1 I - q : lower dotted cos( I ) = T i √ q e = 1 Table 1.
The analytical boundaries on the parameter space constraining the potential orbital parameters of a particle scattered by aplanet, where the initial value of the Tisserand parameter is T i . All units are in terms of the planet’s semi-major axis; a i = 1. For thecases where more than one limit is stated, the upper of the two applies. Figure 2.
The minimum pericentre for a test particle scatteredby a single planet, as a function of the Tisserand parameter value,from Eq. 3. For T i < q min → on the new orbit, it is then likely that the particle is re-scattered by the inner planet and for a certain period itsdynamics will be dominated by that planet.We start by considering this simple situation where theparticle is passed from planet 1 to planet 2. This is usedto describe constraints on the orbits of scattered particles.We then consider the possibility that particles are scatteredbackwards and forwards between the two planets in sec-tion 4.4. For a particle scattered by the outer planet the Tisserandparameter, T , is conserved. The value of T constrains theorbits, ( q, e, I ), of scattered particles to those shown in Fig. 1that satisfy Eq. 2. Although only sets of the orbital param-eters, q, e, I , that satisfy Eq. 2 are allowed, the full range ofpossible values is given by: q ∈ [ q min ( T ) , e ∈ [0 , e max ( T )] I ∈ [0 , I max ( T )] , (6)where q min is given in Eq. 3, Figure 3.
The orbital parameter space as determined by theTisserand parameter, in the eccentricity-pericentre plane, for T i =2 . I = 1) anddot-dashed lines ( q = a i ) and shown in green. The subset of thisorbital parameter space that can interact with an inner planetplaced at a in = 0 . a i is shown by the vertically hashed region,whilst the subset that could interact with an outer planet placedat a out = 1 . a i is shown by the horizontally hashed region. Thedotted line shows a = a i , the solid line q = a in and the tripledotted dashed line, Q = a out . I max = cos − ( √ T − , (7)and e max = 3 − T + 2 √ − T . (8)As mentioned earlier, if T > √
2, then e max > T can interact with the next planet,2. These are shown by the green filled area in Fig. 3 and arethose orbits that cross the planet’s, with q < a and: q ∈ [ q min ( T ) , a ] e ∈ [ e int ( a a ) , e lim ( T )] c (cid:13) , 000–000 A. Bonsor et al.
Figure 4.
The minimum pericentre for a test particle scatteredby two planets, as a function of the ratio of the inner planet’ssemi-major axis to the outer planet’s semi-major axis. I ∈ [0 , I max ] , (9)where, e int = a + a a − a . (10)If a a < T − − T , then the set of orbital parameters with I = I max do not cross the inner planet’s orbit. This occurs ifthe second planet is inside the maximum in I as a functionof q that occurs at q = T − − T a (see Fig. 1). In which case I is constrained to be less than I int rather than I max , where I int = cos − T (1 + a a ) − a a p a a ) ! . (11)Once the particle is scattered by planet 2, T is no longerconserved, instead the value of T when the particle is firstscattered by planet 2 is conserved. The range of possible T values is determined by the initial value of T and theplanets’ orbits, specified by the ratio of the planet’s semi-major axes, a a .The minimum possible value that T can have occursfor particles on orbits with minimum pericentre ( q = q min ),the correspond eccentricity ( e = e lim ) and in the orbitalplane of the planets ( I = 0 ◦ ). It is given by: T ,min = a (1 − e lim ) q min + 2 r (1 + e lim ) q min a , (12)where e lim (Eq. 4) and q min (Eq. 3) are functions of T .Since the Tisserand parameter ( T ) is a monotonicallyincreasing function of q , T will be maximum for the orbitwith the largest value of the pericentre, q , that still crossesthe planet’s orbit, i.e. q = a . For the range of T valuesfor orbits with q = a , the minimum is at cos I = ± e = e X , where from the top line of Table 1, e X = 1+2( a a ) − ( a a ) T +2( a a ) / r a a ) − ( a a ) T . (13)Hence, the maximum of T is given by: T ,max = (1 − e X ) + 2 √ e X . (14) For the next time period the dynamics of the particleis controlled by the second planet. It may be scattered onceor many times. Yet again, the particle’s orbit is constrainedto orbital parameters, ( q, e, I ), specified by the value of T and Eq. 2. This time, however, we consider the situationwhere only T and the planet’s orbits are specified initiallysuch that it is only known that T lies between T ,min and T ,max . The full range for the orbital parameters ( q, e, I ) istherefore specified by: q min ( T ,min ) < q < < e < e max ( T ,min ) (16)0 < I < I max ( T ,min ) , (17)where q min is given by Eq. 3, e max by Eq. 8 and I max byEq. 7, but as a function of T ,min rather than T . For specific planetary orbits, specified by the ratio of theplanets’ semi-major axes, a a , and strict constraints on theinitial value of the Tisserand parameter in the outer belt( i.e. T close to 3), the orbits of scattered particles maybe constrained such that they never interact with the innerplanet. This occurs when the minimum pericentre to whichparticles may be scattered by the outer planet is furtherfrom the star than the inner planet’s orbit; q min ( T ) > a (Eq. 3) or : − T + 2 T + 4 − √ − T T − > a a (18) In Sec. 3.2, Eq. 3, we determined the minimum pericentre towhich a single planet may scatter a particle. A similar calcu-lation may be made for two planets, assuming that particlesare only passed once along the chain of planets. The min-imum pericentre will depend on the Tisserand parameterwith respect to the outer planet, T and the ratio of theplanets’ semi-major axes, a a .For a particle that is scattered by the outer planet,with a value of the Tisserand parameter with respect tothat planet of T , if it is then scattered by the inner planet,the particle could have a range of possible values of the Tis-serand parameter with respect to the inner planet, between T ,min (Eq. 12) and T ,max (Eq. 14). Since q min (Eq. 3) is amonotonically increasing function of the Tisserand param-eter, the minimum pericentre for scattering by both plan-ets will be given by q min ( T ,min ), where T ,min is the min-imum value of the Tisserand parameter (Eq. 12). If a par-ticle is to eventually be scattered inwards as far as possi-ble by the outer and inner planet, it must be passed fromthe outer to the inner planet with an orbit of eccentricity e = e lim ( T ,min ) (Eq. 4) and inclination, I = 0 ◦ .The minimum pericentre for a two planet system isshown in Fig. 4 as a function of the ratio of the planets’semi-major axes, a a . This is calculated from Eq. 3, suchthat q = q min ( T ,min ), where T ,min = T ( q = a p , e = e lim ( T ) , I = 0 ◦ ), using Eq. 2. This has a clear minimum,which occurs at: c (cid:13)000
The minimum pericentre for a test particle scatteredby two planets, as a function of the ratio of the inner planet’ssemi-major axis to the outer planet’s semi-major axis. I ∈ [0 , I max ] , (9)where, e int = a + a a − a . (10)If a a < T − − T , then the set of orbital parameters with I = I max do not cross the inner planet’s orbit. This occurs ifthe second planet is inside the maximum in I as a functionof q that occurs at q = T − − T a (see Fig. 1). In which case I is constrained to be less than I int rather than I max , where I int = cos − T (1 + a a ) − a a p a a ) ! . (11)Once the particle is scattered by planet 2, T is no longerconserved, instead the value of T when the particle is firstscattered by planet 2 is conserved. The range of possible T values is determined by the initial value of T and theplanets’ orbits, specified by the ratio of the planet’s semi-major axes, a a .The minimum possible value that T can have occursfor particles on orbits with minimum pericentre ( q = q min ),the correspond eccentricity ( e = e lim ) and in the orbitalplane of the planets ( I = 0 ◦ ). It is given by: T ,min = a (1 − e lim ) q min + 2 r (1 + e lim ) q min a , (12)where e lim (Eq. 4) and q min (Eq. 3) are functions of T .Since the Tisserand parameter ( T ) is a monotonicallyincreasing function of q , T will be maximum for the orbitwith the largest value of the pericentre, q , that still crossesthe planet’s orbit, i.e. q = a . For the range of T valuesfor orbits with q = a , the minimum is at cos I = ± e = e X , where from the top line of Table 1, e X = 1+2( a a ) − ( a a ) T +2( a a ) / r a a ) − ( a a ) T . (13)Hence, the maximum of T is given by: T ,max = (1 − e X ) + 2 √ e X . (14) For the next time period the dynamics of the particleis controlled by the second planet. It may be scattered onceor many times. Yet again, the particle’s orbit is constrainedto orbital parameters, ( q, e, I ), specified by the value of T and Eq. 2. This time, however, we consider the situationwhere only T and the planet’s orbits are specified initiallysuch that it is only known that T lies between T ,min and T ,max . The full range for the orbital parameters ( q, e, I ) istherefore specified by: q min ( T ,min ) < q < < e < e max ( T ,min ) (16)0 < I < I max ( T ,min ) , (17)where q min is given by Eq. 3, e max by Eq. 8 and I max byEq. 7, but as a function of T ,min rather than T . For specific planetary orbits, specified by the ratio of theplanets’ semi-major axes, a a , and strict constraints on theinitial value of the Tisserand parameter in the outer belt( i.e. T close to 3), the orbits of scattered particles maybe constrained such that they never interact with the innerplanet. This occurs when the minimum pericentre to whichparticles may be scattered by the outer planet is furtherfrom the star than the inner planet’s orbit; q min ( T ) > a (Eq. 3) or : − T + 2 T + 4 − √ − T T − > a a (18) In Sec. 3.2, Eq. 3, we determined the minimum pericentre towhich a single planet may scatter a particle. A similar calcu-lation may be made for two planets, assuming that particlesare only passed once along the chain of planets. The min-imum pericentre will depend on the Tisserand parameterwith respect to the outer planet, T and the ratio of theplanets’ semi-major axes, a a .For a particle that is scattered by the outer planet,with a value of the Tisserand parameter with respect tothat planet of T , if it is then scattered by the inner planet,the particle could have a range of possible values of the Tis-serand parameter with respect to the inner planet, between T ,min (Eq. 12) and T ,max (Eq. 14). Since q min (Eq. 3) is amonotonically increasing function of the Tisserand param-eter, the minimum pericentre for scattering by both plan-ets will be given by q min ( T ,min ), where T ,min is the min-imum value of the Tisserand parameter (Eq. 12). If a par-ticle is to eventually be scattered inwards as far as possi-ble by the outer and inner planet, it must be passed fromthe outer to the inner planet with an orbit of eccentricity e = e lim ( T ,min ) (Eq. 4) and inclination, I = 0 ◦ .The minimum pericentre for a two planet system isshown in Fig. 4 as a function of the ratio of the planets’semi-major axes, a a . This is calculated from Eq. 3, suchthat q = q min ( T ,min ), where T ,min = T ( q = a p , e = e lim ( T ) , I = 0 ◦ ), using Eq. 2. This has a clear minimum,which occurs at: c (cid:13)000 , 000–000 a ,min = (1 + e lim ( T )) / q min ( T )(1 − e lim ( T )) / , (19)where e lim and q min are the minimum pericentre and limit-ing eccentricity for scattering by the outer planet, given byEq. 3 and Eq. 8.This means that the optimum configuration of two plan-ets in terms of their ability to scatter particles as close to thestar as possible, involves planets positioned in semi-majoraxis at a ,min and a . It is interesting to note that the op-timum position for the inner planet is not as close to thestar as the outer planet could possibly scatter particles i.e. q min ( T ), but closer to the outer planet. This is becausethere is a balance between moving the inner planet closer tothe star, decreasing a , such that q min is decreased directlyor moving the planet further from the star, increasing a ,but decreasing T and thus q min . Of course this does notinclude any information about the probability that the par-ticle is ejected or collides with the planet rather than beingejected. Scattering is not confined to the forward direction. Particlesmay originate in the outer belt, be scattered inwards by theouter planet, passed onto the inner planet, and then scat-tered back outwards again to the outer planet. Constraintson which particles might re-interact with the outer planetcan be determined using a similar procedure to that dis-cussed in the previous section (Sec. 4.1) for particles passedfrom the outer planet to the inner planet.The possible values for the orbital parameters of par-ticles scattered by the inner planet are determined by thevalue of the Tisserand parameter, T . A subset of these or-bits cross the outer planet’s orbit, those with apocentresoutside of its orbit ( Q > a ). For the example of an outerplanet at a = 1 . a and with T = 2 .
9, this subset is shownby the hashed region in Fig. 3. Each set of orbital param-eters in this region ( q, e, I ) will specify a possible value forthe Tisserand parameter with respect to the outer planet, T . The minimum possible new value of T occurs at themaximum pericentre ( q = a ), the maximum eccentricity( e max ( T ) Eq. 8) and cos I = 1, such that: T ,new,min = a ( T − − √ − T ) a +2 r (4 − T + 2 √ − T ) a a . (20)If there are a range of values for T , the smallest ( e.g. T ,min for Eq. 12) will give the lowest value of T ,new,min . Themaximum value of T such that particles can still interactwith the outer planet is 3, as for any scattering event.If the particle is scattered backwards and forwards mul-tiple times this procedure may be repeated to determinethe full range of Tisserand parameter values and potentialorbits. T ,new,min can be significantly lower than the ini-tial value of T in the outer belt, particularly after multiplescatterings backwards and forwards. Thus, this increases therange of potential orbits of scattered particles.This can be illustrated using an example system. Con-sider a particle scattered by the outer planet, with T = 2 . a = 0 . a . Theminimum pericentre for the particle after the particle is scat-tered by both planets, shown in Fig. 4, is q min = 0 . a . If the particle is then scattered back outwards, the minimumvalue of T is 2.93 (Eq. 20). If the particle is then scatteredback in, again from Fig. 4, this gives a new minimum peri-centre for scattering by the two planets of q min = 0 . a . Af-ter a further scattering backwards and forwards, q min → extreme orbits, i.e. with low pericentre or high eccen-tricity/inclination, that have significantly reduced values ofthe Tisserand parameter when they are scattered by thenext planet. Therefore, although it is possible that particlesmay be scattered onto extreme orbits, with low values ofthe Tisserand parameter, by being repeatedly passed back-wards and forwards between the planets, we anticipate thatthe probability for this to occur is low and we are thereforejustified in focusing on particles scattered directly along aplanetary system for the rest of the paper. All of the calculations discussed so far can be easily appliedto planetary systems with many planets. The procedure dis-cussed in Sec. 4.1 can be repeated many times, to deter-mine the full range of orbital constraints and values for theTisserand parameter after scattering by each planet. Thisanalysis places useful constraints on the planets with whichparticles can interact, the planets that can eject particlesand the minimum pericentre to which the whole system canscatter particles.All of the dynamics is determined by the initial value ofthe Tisserand parameter with respect to the outer planet, T , the outer planet’s semi-major axis, a and the ratio of theplanets’ semi-major axes to one another, a i +1 a i . Scaling thesystem, i.e. changing the semi-major axes, a i , whilst keepingtheir ratios, a i +1 a i , constant, will not affect the dynamics (val- c (cid:13) , 000–000 A. Bonsor et al.
Figure 5.
The variation in the minimum pericentre to whichtest particles can be scattered to by a system of five planets.The ratio of the planets’ semi-major axes ( α ) is constant and isgiven as a ratio on the bottom axis and in terms of separationin Hill’s radii, for five 10M ⊕ planets, on the top axis. The initialvalue of the Tisserand parameter with respect to the outer planetis varied between 2.8 and 3.0. The shaded region illustrates the“unconstrained” regime for particles with T = 2 .
96, whilst theregion to its left is the “non-interacting” and the region to itsright is the “constrained” regime (see discussion in text). ues of T i ) and merely scales the minimum pericentre, q min ,with a . In the next section, we discuss these constraints interms of an example planetary system. We apply these calculations to a system of 5 planets, sep-arated by a constant ratio of adjacent planets’ semi-majoraxis ( a i +1 a i = α ). This corresponds to a constant numberof Hill’s radii for equal mass planets. Our results are inde-pendent of the planet masses. We fix the inner planet at a = a IN and calculate the semi-major axes of the otherplanets accordingly for a range of values for α .The minimum pericentre to which this system can scat-ter particles, shown in Fig. 5 as a function of α , is calcu-lated by repeatedly determining the minimum value of theTisserand parameter for each planet. For the i th planet thisoccurs at q = q min ( T i +1 ,min ) (Eq. 3), e = e lim ( T i +1 ,min )(Eq. 4) and cos I = 1.In this plot scattered particles exhibit three types of be-haviour. For simplicity we label the three types of behaviouras “non-interacting”, “constrained” and “unconstrained”.This refers to the constraints on the orbits of scattered par-ticles. In the “non-interacting” regime, the planets are sowidely separated (small α ) that particles cannot be scat-tered all the way along the chain of planets. The minimumpericentre to which one of the planets can scatter particlesis outside of the next innermost planet’s orbit. Hence theparticles are restricted to the region surrounding the outerplanet(s).In the “constrained” regime, the planets are so close to-gether (large α ) that particles can be scattered between allplanets in the system. If they are only scattered once along the chain of planets, the Tisserand parameter cannot varysignificantly from its original value and there will be a non-zero minimum pericentre to which particles can be scattered.For such closely separated planets, it may no longer be validto treat the scattering as a series of three body problemsand the probability that particles are passed backwards andforwards between planets increases. This and the stabilityof planets so close together question whether particles scat-tered in any planetary system actually exhibit behaviourreminiscent of this “constrained” regime.As the separation of the planets is increased, the min-imum possible value of the Tisserand parameter for eachplanet decreases and hence the minimum pericentre for thewhole system decreases. Eventually the separation is largeenough that the Tisserand parameter falls below 2 and allconstraints on the minimum pericentre are removed. Thisforms the third, “unconstrained” regime, where there arefew constraints on the orbital parameters of scattered par-ticles.In Fig. 6 the constraints on the eccentricities and incli-nations of particles in the 3 regimes are shown. As particlesare scattered by each planet, from the outermost (1) to theinnermost (5), there will be a range of possible Tisserand pa-rameter values, between T i,min (Eq. 12) and T i,max (Eq. 14)and hence a range of possible orbital parameters, given byEqs. 17, although of course the approximations used to cal-culate these may mean that they are not always strictlyapplicable, see discussion in §
7. It is the maximum incli-nation and eccentricity that are important on this figure,although of course the orbits of scattered particles will bedistributed between the minimum and maximum values, ina manner not determined by this analysis. The plot showsthat, for this example with T = 2 .
96, almost all planets caneject particles ( e >
1) and that the scale height of the disc(inclinations of scattered particles) increases with decreas-ing distance to the star, as the constraints on the orbits ofscattered particles decrease with each successive scatteringevent. It is clearly seen, as anticipated, that the constraintsof orbits in the “constrained” regime are much tighter thanthose in the “unconstrained” regime.Although very few real planetary systems have planetsseparated by a constant ratio of their semi-major axes, itmay be possible to similarly classify the behaviour of scat-tered particles within the three regimes and thus usefullybetter understand the future fate of scattered particles. R H For real planetary systems the planets cannot be arbitrar-ily close together as dynamical instabilities are important.Chambers et al. (1996) find that planets must be separatedby at least 10 R H to be stable. On Fig. 5, the separationof the planets is shown in terms of Hill’s radii on the topaxis, for a system of equal mass 10 M ⊕ planets. This showsthat for the 10 M ⊕ planets considered, if they are separatedby 10 R H , then the behaviour of particles is unconstrained( q min → < M ⊕ ) systems may bedynamically stable (separated by more than 10 R H ) and havelimits on the scattering of particles, such that the particle’sbehaviour is in the “constrained” regime.Such low mass systems are, however, unlikely to only c (cid:13) , 000–000 Figure 6.
Constraints on the eccentricities and inclinations(Eq. 17) of particles scattered by a system of five planets withconstant ratio of the planets’ semi-major axes, α , and an initialvalue of the Tisserand parameter in the outer belt of T = 2 . α = 0 .
2, “unconstrained”, α = 0 . α = 0 .
9. The dashed regions correspond to the param-eters of particles that can interact with the next interior planet(Eq. 9). The particles with high eccentricity were scattered out-wards and therefore are not on orbits that cross the inner planet’sorbit. contain 5 planets. One possible outcome of planet forma-tion, is a chain of low mass planet embryos and an outerdisc of planetesimals. Consider the example of such a disc inthe position of the Solar System’s Kuiper belt and a chain ofinterior, equal mass planets, between 1 and 30AU. If plan-ets generally form on orbits as tightly packed as possible(Barnes & Raymond 2004; Raymond et al. 2009), then theirseparation will be ∼ R H . We investigate the dynamicsin such a system by varying the planet mass and thus thenumber of planets that fit between 1 and 30AU. This isequivalent to varying α . The results are shown in Fig. 7.The behaviour is identical to the five planet system in the“non-interacting” and “unconstrained” regimes, however the“constrained” regime no longer exists. This is because theinterior planets further increase the parameter space avail-able to scattered particles. Figure 7.
The same as Fig. 5, but for tightly packed planetarysystems, with equal mass planets separated by 10 R H . The massof the planets is shown on the bottom axis, whilst the top axisshows α . As many planets as fit between 1 and 30AU are included,hence the minimum pericentre is no longer finite for large α . This analysis can be applied to the planetary system that weunderstand best, our Solar System. There are three possi-ble sources of scattered bodies; the asteroid belt, the Kuiperbelt and the Oort cloud. Most of the discussion so far hasapplied to the scattering of particles from a Kuiper-like belt,however, very similar processes occur in the asteroid belt. Asdiscussed in §
2, we anticipate that the Tisserand parameterfor objects scattered out of the Kuiper belt is close to 3. Thisshould also apply to asteroids scattered from the main beltby Mars. The main difference between scattered asteroidsand scattered Kuiper belt objects will be in the distribu-tion of the Tisserand parameter, T Mars or T Nep . This workdoes not determine these distributions, however, we specu-late that the distribution of T Mars may be spread to lowervalues than that of T Nep . Jupiter is a strong perturber andmay be able to alter the orbital parameters of an asteroidsignificantly in a single encounter. For Oort cloud cometsscattered by planets, on the other hand, the range of val-ues of the Tisserand parameter is even larger, potentiallyincluding many comets with T p <
2. This means that al-though this analysis is most usefully applied to scatteredKuiper-belt objects, it can also be applied to scattered as-teroids, but it cannot place many limitations on the orbitsof scattered Oort cloud comets.Firstly considering objects scattered out of the Kuiperbelt. Useful constraints can be made on their orbits in theouter planet region using this analysis; e.g. particle incli-nations are constrained to be below a maximum value, forexample 80 ◦ , for scattered Kuiper belt objects with T Nep > .
96, consistent with observations of Centaurs (Gulbis et al.2010). It can also be inferred that the Solar System’s outerplanets are well placed for scattering particles between them.If T Nep .
982 then particles can be scattered, directly,all the way along the chain of planets to Jupiter and Ta-ble 2 shows that using Eq. 19 the planets are placed closeto optimally for scattering particles as far inwards as pos- c (cid:13) , 000–000 A. Bonsor et al.
Planet Semi-major axis (AU)Observed OptimumNeptune 30.1Uranus 19.2 21.1Saturn 9.58 10.5Jupiter 5.20 3.3Mars 1.52Earth 1.00 1.18Venus 0.72 0.74Mercury 0.39 0.33
Table 2.
The semi-major axes of solar system planets, comparedthe optimum semi-major axes in terms of scattering particles in-wards. These were calculated using (Eq. 19) for objects scatteredfrom the Kuiper belt by Neptune with T Nep = 2 .
98 and sepa-rately for objects scattered from the asteroid belt by Mars, with T Mars = 2 . sible. The three regimes presented in § T Nep < . . < T Nep < .
982 the dynamics of scattered bodies are“constrained” and there is a minimum value for the pericen-tre of objects scattered by Jupiter, whilst if T Nep > . T Mars < .
97 then scat-tered objects behave as if they were in the “unconstrained”regime and if T Mars = 2 .
98, Eq. 19 can be used in a similarmanner to determine that the terrestrial planets are close tooptimally separated for scattering particles between them,see Table 2. The important question, therefore, is what frac-tion of scattered of asteroids have T Mars ∼ .
98. If T Mars is lower then issues arise with some of the approximationsused in this work. For example, particles may not be scat-tered directly along the chain of planets. For T Mars < . q min < a V enus (Eq. 3). Particles may interact directly with Venus beforebeing scattered by Earth, or be scattered multiple timesby both Venus and Earth. As described in § As discussed in the Introduction, there are many obser-vations of stars with warm dust belts, e.g. (Gaidos 1999;Beichman et al. 2005; Song et al. 2005). Many of the sys-tems with warm dust also have cold dust belts, amongst
Figure 8.
The minimum pericentre ( q min ) to which Kuiperbelt objects can be scattered to by Neptune and the outer SolarSystem planets (solid line) and similarly for asteroids scatteredby the terrestrial planets (dashed line), as a function of the ini-tial value of the Tisserand parameter with respect to Neptune orMars, respectively. The shaded area illustrates the “constrained”region, for scattered Kuiper belt objects with diagonal shadingin green and for scattered asteroids with vertical shading in red.The “unconstrained” region lies to the left of the plot and the“non-interacting” regime to the right. others, η Corvi (Smith et al. 2009; Wyatt et al. 2005), β Leo(Churcher et al. 2011) and ǫ Eri (Backman et al. 2009). Theanalysis presented here can be used to consider the scatter-ing of particles from an outer belt inwards, as a potential ex-planation for the observed warm belts. Our main conclusionis that the architecture of a planetary system determineswhether or not material can be scattered to the position ofthe observed belt. We, therefore, speculate that the diversityof planetary system architectures could result in the diver-sity of observed systems, both in terms of disc radii and theratio of the flux from the outer to the inner belt. Althoughthis analysis does not determine what fraction of the scat-tered particles end up in the position of the observed disc, itdoes show that some planetary systems cannot scatter par-ticles onto the required orbits and illustrates that when thedistribution of scattered particles is determined, tight con-straints will be placed on the architecture of the planetarysystem required.Consider the example system of η Corvi, with cold andwarm dust. The inner belt is resolved and lies between 0.16-2.98 AU (Smith et al. 2009), whilst the outer dust is at 150 ±
20 AU (Wyatt et al. 2005). Although there are no plan-ets detected in this system, it seems probable that there isa planet close to the inner edge of the cold, outer belt, thattruncates it (Wyatt et al. 2005). We, therefore, consider aplanet at 100AU. If the Tisserand parameter with respectto this planet is T = 2 .
96, then this planet alone could po-tentially scatter particles in as far as 47AU (Eq. 3). In orderfor particles to be scattered inwards to the location of thewarm belt, q min < c (cid:13) , 000–000 be scattered in as far as 6AU and thus the warm dust belt,if it formed, would be at larger radii. Alternatively, therecould be more than 3 planets, the initial value of the Tis-serand parameter could be less than 2.96 or particles couldbe scattered multiple times backwards and forwards betweenthe planets, as discussed in § Evidence of evolved planetary systems and scattering ofplanetary material is found in the observations of metal pol-luted white dwarfs (Zuckerman et al. 2003; Koester et al.2005) and white dwarfs with close-in circumstellar discs(Farihi et al. 2009). In order to explain these observationswith planetary material, comets or asteroids must be scat-tered onto star-grazing orbits and tidally disrupted. Theanalysis presented in this work can be used to determinethe feasibility of this explanation.Planets are required to scatter comets or asteroids closeenough to the star. There are three potential reservoirs in anevolved planetary system, a Kuiper belt analogue, an Oortcloud analogue and if it survives an asteroid belt analogue.This analysis shows that it is possible for particles from allthree reservoirs to be scattered onto star-grazing orbits, butthat this ability depends strongly on the planets’ orbits andthe initial value of the Tisserand parameter. The lower theinitial value of the Tisserand parameter, the more likely thatparticles can be scattered sufficiently close to the star (thelower q min Fig. 2). Hence, the majority of particles from anOort cloud analogue can be scattered onto star-grazing or-bits, whilst for a Kuiper or asteroid belt analogue this abilityis strongly dependent on the initial value of the Tisserandparameter and the planets’ orbits. Asteroid belt analogueshave the advantage of lower initial values for the Tisserandparameter, but the disadvantage that there may be fewersurviving interior planets and the asteroid belt itself maynot survive until the white dwarf phase.There are a large number of observations of Kuiper beltanalogues around main sequence stars (Wyatt 2008) andmodels find that such systems survive the star’s evolution(Bonsor & Wyatt 2010). Such belts have been suggested asthe source of the metal pollution e.g.
Jura (2008), althoughthere is little evidence that they are capable of scatteringparticles sufficiently close to the star. Here, we show that itis possible for some planetary systems to scatter particlesfrom an outer belt onto star-grazing orbits, but that thereare tight constraints on the planets’ orbits and the initialvalue of the Tisserand parameter in the outer belt.One potential hindrance in the ability of an evolvedplanetary system to scatter particles onto star-grazing or-bits is the absence of inner planets due to the star’s evolu-tion. Villaver & Livio (2007) find that white dwarfs shouldnot possess planets within 15AU due to a combination ofthe increased stellar radius, tidal forces and stellar massloss. In order for a planet at a i = 15AU to scatter parti-cles onto star-grazing orbits ( q min < R ⊙ ), particles musthave values of the Tisserand parameter less than 2.05 whenthey interact with the planet (Eq. 3). Only particles froman evolved Oort cloud might have sufficiently low valuesof Tisserand parameter without interacting with furtherplanets. Therefore, using repetition of the technique de- scribed in § T > .
97, then at least 4 planets are required to scatter par-ticles onto star-grazing orbits, whilst for 2 . < T < . T Nep < .
96) and HR 8799 with planets at 14.5, 24, 38 and68 AU (Marois et al. 2008, 2010), if T < .
95 in the outerbelt.This analysis crucially shows that it is possible to scat-ter comets or asteroids onto star-grazing orbits and placeslimits on the architecture of a planetary system that can dothis, but it does not inform us about the probability of agiven planetary system to scatter planetesimals onto star-grazing orbits. Oort cloud analogues only require a singleplanet to scatter material onto star-grazing orbits, whilstconstraints are placed on the orbits of planets and the initialvalue of the Tisserand parameter required to scatter mate-rial inwards from a Kuiper or asteroid belt analogues. Thus,this analysis shows that material from an evolved Kuiperbelt is a potential origin of the metal pollution in whitedwarfs, although fewer constraints exist on the ability ofan evolved Oort cloud to scatter comets onto star-grazingorbits. This provides important evidence in support of theplanetary origin for the white dwarf observations.
The purpose of this work is to present a simple analyticaltool that can be applied to many planetary systems. It de-termines the potential orbital parameters of scattered par-ticles, based on the initial value of the Tisserand parameterand the planets’ orbits. It does not claim to determine theprobability for any particle to be scattered onto a given or-bit, nor the expected distribution of scattered particles. Inorder to retain this simplicity it was necessary to make sev-eral assumptions that strictly limit the applicability of thisanalysis. These are discussed below.We anticipate that the behaviour described in this workwill be useful in interpreting N-body simulations of smallbody scattering in planetary systems. In such simulationsthe initial value of the Tisserand parameter for individualparticles will be well constrained, and this means that itshould be possible to describe their subsequent evolution us-ing the analysis presented here. However, this presupposesthat our simplifying assumptions are valid, and it will beimportant to use N-body simulations to test this. The mainassumptions regarding the dynamics that we see are sum-marised here.One of the biggest limitations in the analysis presentedhere, is the dependence on the initial value of the Tisserandparameter in the outer belt. This is in general an unknownquantity, although good approximations can be made to itsvalue, as discussed in § c (cid:13) , 000–000 A. Bonsor et al.
Figure 9.
The extension of the orbital parameter space availableto scattered particles with T p = 2 .
98. This is analogous to theplots in the top row of Fig. 1, except that the particles are allowedto interact with the planet within a zone of size ∆ ∼ R H , for aJupiter mass (green) and Neptune mass (red) planet. For theNeptune mass planet the orbital parameter space available toscattered particles is not altered significantly from that shown inFig. 1, whereas for the Jupiter mass planet the strict limit on theminimum pericentre is removed. secular or resonant perturbations, or interact with a planetother than the one dominating their dynamics during thatperiod. This could alter the value of the Tisserand param-eter. Particles may also be passed backwards and forwardsalong the chain of planets, as discussed in § ∆ a p . This ex-tends the range of potential orbital parameters of scatteredparticles, as orbits with Q > a p (1 − ∆) and q < a p (1 + ∆)can interact with the planet, rather than just Q > a p and q < a p . This orbital parameter space is shown in Fig. 9 forJupiter and Neptune, with ∆ = R H and T p = 2 .
98. Jupiteris massive enough that there is no longer a limit on the min-imum pericentre to which particles can be scattered and theavailable parameter space is increased by a small, but sig-nificant, amount. Neptune, on the other hand, does not sig-nificantly extend the parameter space available to scatteredparticles and therefore all of the analysis presented in thispaper will apply. For Jupiter mass planets the broad resultspresented here can still be applied, however, care should betaken when rigorously applying the derived limits.Strictly the conservation of the Tisserand parameter,and therefore this analysis, should only be applied to sys-tems with co-planar planets on circular orbits, i.e. within the context of the circular restricted three body problem.It is, however, found that even when these assumptions arerelaxed, the analysis still applies approximately, for exam-ple Murray & Dermott (1999) found only a small change inthe Tisserand parameter when they consider Jupiter’s eccen-tricity. Caution should, however, be exerted when applyingthis analysis to some of the detected exoplanets with largeeccentricities (and relative inclinations).
The purpose of this work was to describes simply and an-alytically the scattering of particles in any planetary sys-tem. This analysis constrains the outcomes of scatteringevents, based on the conservation of the Tisserand param-eter (Eq. 1), in a manner that is very useful for analysingthe structure of many planetary systems where the scatter-ing of small bodies by planets is important. This analysiscan be used to better understand behaviour seen in N-bodysimulations.We consider here the application to planetary systemswhere small bodies are scattered from an outer belt by inte-rior planets. The analysis could, however, easily be reformu-lated to consider scattering by planets exterior to the belt.We assume that the scattering process can be approximatedby a series of three-body problems, during each of which theTisserand parameter with respect to the relevant planet isconserved. This constrains the possible orbits of scatteredparticles, based solely on the initial value of the Tisserandparameter and the orbits of the planets, with the assumptionthat particles are passed directly along the chain of planetsand that particles only interact with the planet when theirorbits directly cross the planet’s orbit. In this case there isno dependence on the planet’s mass and it is only the ratio ofthe planets’ semi-major axes that are important. A depen-dence on planet mass would be introduced if the interactionsof particles with the planet within a zone around the planetwere included. This analysis places an important limit onhow far in particles can be scattered ( q min from Eq. 3) anddetermines which planets the particles can interact with,which can eject them and the potential height of the disc,based on the maximum particle inclinations (Eq. 17). Weconsider the full range of possible orbits of scattered parti-cles, rather than their distribution.In this work we consider the application of this analy-sis to our Solar System, main sequence stars with both coldand warm dust belts and metal polluted white dwarfs. Inthe Solar System, this analysis describes the scattering ofKuiper belt objects by Neptune to become Centaurs andJupiter Family comets, as well as asteroids by Mars and theterrestial planets. We show that the Solar System planetsare close to optimally separated for scattering particles be-tween them. One explanation for main sequence stars with warm dust belts that cannot have survived for the age ofthe system in their current positions is that material wasscattered inwards from an outer belt. If this is the case, thisanalysis shows that certain architectures for the planetarysystem could not produce the observations. Given the strongdependence of the scattering process on planetary system ar-chitecture, we speculate that the diversity of such systems isa reflection of the variety of planetary system architectures. c (cid:13) , 000–000 Observations of metal polluted white dwarfs and whitedwarfs with circumstellar discs have been associated withmaterial scattered inwards from an outer evolved planetarysystem. Such material can be scattered sufficiently close tothe star from all Oort cloud analogues and, given certainconstraints on the system architecture, from some Kuiperbelt analogues. This strengthens the case for a planetaryorigin to these observations, although this analysis does notcomment on the probability for particles to be scattered ontosuch orbits.In summary, the analytical tool presented here can aidour understanding and place useful constraints on the scat-tering of small bodies in a wide range of planetary systems.
We thank the referee, R. Brasser, for very useful commentsthat contributed to this paper. We thank D. Veras, G.Kennedy and A. Mustill for comments that contributed tothe forming of this manuscript. AB thanks STFC for a stu-dent grant.
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