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To appear in Astron.J ;preprint–July 10,2002A Study of the Dynamics of Dust from the Kuiper Belt:Spatial Distribution and Spectral Energy Distribution Amaya Moro-Mart´ın 1and Renu Malhotra 2amaya@as.arizona.edu renu@lpl.arizona.edu ABSTRACT The dust produced in the Kuiper Belt (KB)spreads throughout the Solar Sys-tem forming a dust disk.We numerically model the orbital evolution of KB dust and estimate its equilibrium spatial distribution and its brightness and spectral energy dis-tributions (SED),assuming greybody absorption and emission by the dust grains.We show that the planets modify the KB disk SED,so potentially we can infer the prence of planets in spatially unresolved debris disks by studying the shape of their SEDs.We point out that there are inherent uncertainties in the prediction of structure in the dust disk,owing to the chaotic dynamics of dust orbital evolution impod by resonant gravitational perturbations of the planets.Subject headings:celestial mechanics —interplanetary medium —Kuiper Belt —meth-ods:n-body simulations —methods:numerical —planetary systems —solar system:general
汉字的发展史1.Introduction
Main quence stars are commonly surrounded by cold far-IR-emitting material.The fact that this infr
ared excess is not restricted to young stars,and that the dust grain removal process,Poynting-Robertson (P-R)and solar wind drag,act on timescales much smaller than the age of the system,indicate that:(1)a rervoir of undetected dust-producing planetesimals exists;and
(2)to induce frequent mutual collisions,their orbits must be dynamically perturbed by massive planetary bodies.The Solar System is also filled with interplanetary dust.In the inner Solar System,this dust,which gives ri to the zodiacal light,has been obrved by Pioneer 10(out to
3.3AU)and by the infrared telescopes IRAS and COBE.The dominant sources of the zodiacal cloud are debris from Jupiter family short period comets and asteroids(Liou et al.,1995;Dermott et al.,1994).The discovery of a debris disk aroundβ-Pictoris,extending to100s of AU,together with the confirmation of the existence of the theoretically predicted Kuiper Belt objects(KBOs)(Jewitt &Luu,1995),suggest that significant dust production may also occur in the outer Solar System due to mutual collisions of KBOs(Backman&Paresce,1993;Backman,Dasgupta&Stencel,1995; Stern,1996)and collisions with interstellar grains(Yamamoto&Mukai,1998).
Dust particles are small enough to experience the effect of radiation and stellar wind forces. Radiation pressure makes their orbital elements and specific orbital energy change immediately upo
n relea from parent bodies.If their orbital energy becomes positive,the dust particles escape on hyperbolic orbits.In the Solar System,the particles are known asβ-meteoroids(Zook& Berg,1975).If their orbital energy remains negative,the dust particles stay on bound orbits.P-R and solar wind drag tends to circularize and decrea the mimajor axis of the orbits,forcing the particles to slowly drift in towards the central star(Burns,Lamy&Soter,1979).Assuming that the dust particles are constantly being produced,this drifting in creates a dust disk of wide radial extent,that we refer to as a debris disk.Debris disks are systems that satisfy the following conditions:(1)their age is longer than the P-R and collisional lifetimes;(2)they are optically thin to stellar radiation,even along the mid plane;and(3)they have little or no gas,so that the dust dynamics is controlled by gravitation and radiation forces only(Backman,2002).
When planets are prent,the journey of the dust particle towards the central star is tem-porarily interrupted by the trapping of the particle in Mean Motion Resonances(MMRs).MMRs occur when the orbital period of the particle is in a ratio of small integers to that of the perturbing planet.[The p:q MMR means that the orbital period of the particle is p/q times that of the planet.] In an MMR,the drifting in is halted becau the energy loss due to P-R drag is balanced by the resonant interaction with the planet’s gravityfield.This trapping can potentially create structure in debris disks,as the partic
les accumulate at certain mimajor axes.Sufficiently massive planets may also scatter and eject dust particles out of a planetary system,creating dust free or depleted zones.This structure,if obrved,can be ud to infer the prence of planets.Liou&Zook (1999a,hereafter LZ99)found that the prence of the Giant Planets has an important effect on the structure of the debris disk that is presumably generated in the KB:Neptune creates a ring-like structure between35and50AU,due to the trapping of particles in exterior MMRs,and Jupiter and Saturn are responsible for the ejection of about80%of particles from the Solar System(Liou, Zook&Dermott,1996,hereafter LZD96).The latter creates a clearing in the inner10AU that rembles the inner gap in theβ-Pictoris disk.If obrved from afar,the KB disk would be the brightest extended feature in the Solar System,and its structure,if spatially resolved,could be recognized as harboring at least two giant planets:an inner planet(Jupiter plus Saturn)and outer planet(Neptune)(LZD96).In anticipation of future obrvations of debris disks,who structure is likely to be spatially unresolved,in this paper we are interested in studying how the structure affects the shape of the disk SED and conquently if the SED can be ud to infer the prence of
planets.
In this paper we are going to follow numerically,from source to sink,the evolution of veral hundred
dust particles from the KB in the size range from1to40µm(forρ=2.7g/cm3),or from 3to120µm(forρ=1g/cm3),under the combined effects of solar gravity,solar radiation pressure, P-R and solar wind drag and the gravitational forces of7planets(excluding Mercury and Pluto). The sinks of dust included in our numerical simulations are:(1)ejection into unbound orbits;(2) accretion onto the planets;and(3)orbital decay to less than0.5AU heliocentric distance.The equations of motion are integrated using a modification of the multiple time step symplectic method SyMBA(DLL98).In§2we describe our numerical integration method and the tests performed to check the suitability of the code.§3describes our methods for deriving the equilibrium spatial distribution of the dust disk.§4explains the distribution of parent bodies and the orbital evolution of dust.In§5we discuss the formation of structure in the KB debris disk and its obrvational signatures.Dust destruction process are discusd in§6,and§7summarizes our results.
2.The Numerical Method
In order to study the dynamics of dust from the KB we need to solve the problem of the dynamical evolution of micron-sized particles,under the effect of gravitational forces of the Sun and the planets and radiation and solar wind forces.This has been solved in the past using the adaptive step size Runge-Kutta integrator RADAU(LZD96;Liou&Zook,1997;Kortenkamp& Dermott,1998;LZ99and Liou,
Zook&Jackson,1999b).Another possible choice is the standard mixed variable symplectic(MVS)integrator,developed by Wisdom&Holman(1991).Its advantage over implicit Runge-Kutta integrators is its speed,about an order of magnitude faster(Wisdom &Holman,1991).This is why the MVS method is now ud in long-term studies of the Solar System,allowing to reach integration times approaching the age of the system.Its disadvantage, however,is that it cannot handle clo encounters amongst bodies.Since the outcome of clo encounters between the dust particle and the planets is critical for the study of the dynamical evolution of dust grains,previous rearchers have chon RADAU as their numerical integrator. But recently,Duncan,Levison&Lee(1998;hereafter DLL98)have developed a new multiple time step symplectic algorithm,SyMBA,that can handle clo encounters in a sympletic way,thus retaining the speed of the MVS method while being able to overcome its main disadvantage.
The equations of motion of the N-body system are integrated using a variation of SyMBA called SKEEL,which we have modified to include ratiation forces.In this ction,we summarize the main features of SKEEL as described in DLL98,followed by a description of how radiation forces were introduced and the tests that we have performed to check the validity of our results.
2.1.The Multiple Time Step Symplectic Integrator SKEEL
SKEEL solves the Newtonian gravitational N-body problem by parating its Hamiltonian,
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H(Q i,P i)=
怎么不长白头发n
i=1 |P i|2|Q i| +|P0|22m0|n i=1P i|2−n−1 i=1n j=i+1G m i m j
2m i
−
G m i m0
2m0
|
n
i=1P i|2,(4)
H int=−
n−1
i=1n j=i+1G m i m j
2 E int
τ2 E Sun τ
2 E ne int
τ2 E Kep(τ)E enc int τ2 E Sun τ
2 E Kep(τ)E enc int
τ
is as follows.The two-body potential terms in H int,due to the encountering planet,are decomposd into
G m i m j
2 EΣ1(τ0)E0
τ02 [E1 τ12 ]M E0 τ0
dt2=
−G m0(1−β)
c
G m0
r r+v
,(11)
whereβis a dimensionless constant equal to the ratio between the radiation pressure force,
F r=SAQ pr/c,and the gravitational force,F g=Gm0µ/r2,so that for spherical grainsβ=F r/F g=SAQ pr r2/ (Gm0µc)=(3L/16πGm0c)(Q pr/ρs).For the Sun,β=5.7×10−5Q pr/ρs,whereρand s are the density and radius of the grain in cgs units(Burns,Lamy&Soter,1979).Q pr is the radiation pressure coefficien
t,a measure of the fractional amount of energy scattered and/or absorbed by the grain.Q pr is a function of the physical properties of the grain and the wavelength of the incoming radiation;the value we u is an average integrated over the solar spectrum.The advantage of
using the dimensionless parameterβis that it is independent of distance,being a function only of
the particle size and composition.βsw=(1+sw)β,where sw is the ratio of the solar wind drag to
the P-R drag;in this paper we u a constant value sw=0.35(Gustafson,1994).
The Hamiltonian associated with thefirst term in the rhs of(11)is H Kep in eq.(3),with m0(1-β)instead of m0.Physically,this means that radiation pressure makes the dust grain feel a less massive Sun.In our numerical integrator,SKEEL-RAD,we introduce the cond term in(11), the P-R and solar wind drag term,as an additional kick to the momentum of the particle.The algorithm thus becomes,
E Sun τ2 E rad Kep(τ)E rad int τ2 .(12) In the inertial reference frame,the P-R drag can be thought of as a mass loading drag:the re-emitted radiation emits more momentum into the forward direction of motion due to the Doppler effect,which means that the particle los momentum;since the mass is conrved,the particle is decelerated(there is a drag force).In the particle’s reference frame it originates from the aberration of the radiation,that generates a drag force.
2.3.Comparison with Analytical Results
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There is no analytic solution to the general problem of a particle moving under the effect of gravitational forces from the Sun and the planets and radiation and solar wind forces.For this reason,the code cannot be tested in the most general ca.But there are analytic solutions for the evolution of the orbital elements of a particle under the effect of radiation in the2-body problem (Wyatt&Whipple,1950;Burns,Lamy&Soter,1979)and in the circular restricted3-body problem(Liou and Zook,1997).We will u the solutions to test the numerical procedure and the validity of our results.
2.3.1.Jacobi Constant Conrvation
In the circular restricted3-body problem,consisting of a massless particle,a central mass and a planet in a circular orbit,the Jacobi constant is an integral of the motion.We have integrated the orbits of50massless particles in the prence of the Sun and Neptune(with a=30AU and e=0).The mimajor axes of the particles were uniformly distributed between36and40AU and the perihelion distance was t to30AU.We u a step size of2years and an integration time of 109years.We found that34out of50particles have clo encounters,with∆J/J(0)∼O(10−6)–O(10−7).The remaining16that d
o not suffer clo encounters have∆J/J(0)∼O(10−8).The worst jacobi conrvation has∆J/J(0)∼7·10−6.The results suggests that clo encounters are integrated accurately.
