Dynamical evolution of asteroidal and cometary particles and their contribution to the zodiacal cloud

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Title:
Dynamical evolution of asteroidal and cometary particles and their contribution to the zodiacal cloud
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Liou, Jer-Chyi, 1961-
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Asteroids   ( lcsh )
Comets   ( lcsh )
Zodiacal light   ( lcsh )
Cosmic dust   ( lcsh )
Astronomy thesis Ph. D
Dissertations, Academic -- Astronomy -- UF
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Thesis:
Thesis (Ph. D.)--University of Florida, 1993.
Bibliography:
Includes bibliographical references (leaves 129-131).
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Typescript.
General Note:
Vita.
Statement of Responsibility:
by Jer-Chyi Liou.

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University of Florida
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Full Text













DYNAMICAL EVOLUTION OF ASTEROIDAL AND COMETARY
PARTICLES AND THEIR CONTRIBUTION TO THE ZODIACAL CLOUD










BY

JER-CHYI LIOU


DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY


UNIVERSITY OF FLORIDA

1993
















4jwfl, twJAiw o


To my parents, brother, sister in law, and my beloved wife for their love and support.


m^^wm.sf~~q~~













ACKNOWLEDGMENTS


I would like to thank my adviser, Dr.


Stan Dermott, and the other members in


my committee, Dr.


Campins, Dr.


F. E. Dunnam, and Dr.


H. Smith for their


support and guidance through out the work of this dissertation. Special thanks go to

Dr. A. Smith for reading the thesis and providing me with useful suggestions.


Special thanks to Dr.


Yulin Xu for helping me understand SIMUL and providing


me with the observational data from IRAS. I would like to thank Dr.


B. Gustafson


for many useful discussions. I would also like to thank Dan Durda for his comment

on the collisional evolution of the asteroids.

Finally, I would like to thank my fellow graduate students, Jaydeep Mukherjee

for his help over the years, Sandra Clements for showing me how to be a good

observer at RMH Observatory, David Kaufmann and Damo Nair for making me the

best backgammon player in the department.
















TABLE OF CONTENTS


ACKNOWLEDGMENTS

LIST OF TABLES .


LIST OF FIGURES


ABSTRACT


iii


v S
. . . . . . . v n


. .. .. .. .. ... S. S. S XV


CHAPTERS


INTRODUCTION


Li~l * * * * * 1


The Zodiacal Clo


Objective of this Dissertation

2 SECULAR PERTURBATION THEE


C S . S S S S S S S 3

S- S- S S S S S S S S S S C S S S S 5 6J


Low-order Secular Perturbation


Theory


SS S S S S . S .. 6


The Solar System.


Small Bodies in the Solar System


Numerical Approach


Comparison between the Analytical and Numerical Calculations


Sun-Jupiter-Saturn System


S S S S S S S S S S 10


. 14


S C 5 5 5 5 5 5 S* S l. T14


Asteroidal Particles in the Solar System.


. . S S S S 15


ORBITAL EVOLUTION OF ASTEROIDAL PARTICLES


Radiation Pressure


, Poynting-Robertson Drag,


and the Corpu


scular


Drag.

Static


Theory and Dynamical Theory


S S S S S S S S S 28


Numerical Comparisons


SS . C 32


S S S S . S S S S S S 5 6 5 5 5 6


S. S . 12


. C S S 5 25


25











Passage


Through Resonance


. 4 4 C 4 4 C 9 4 C 34


Orbital Evolution of Dust Particles from the Major Hirayama


Asteroidal Families


S36


9 pm Diameter Particles. . ...... . 38
4 pm and 14 pm Diameter Particles. . . . 41


NeEarth Dust Grains.


4 ORBITAL EVOLUTION OF COMETARY PARTICLES

Orbital Evolution of Encke-type Dust Particles ..


Gauss'


* a a a a .77

.* a a 77


Equations


Can Comets be the Source of the Solar System Dust Bands?

Distribution of Encke-type Dust Particles in the Solar System


. 8


STRUCTURE OF THE ZODIACAL CLOUD ......... 106

Contribution from Asteroidal Dust Particles . 108

Contribution from Encke-type Dust Particles . . . 110


Discussion

BIBLIOGRAPHY


. 4 a a a C . S S S S 111


129


BIOGRAPHICAL SKETCH


. . . . . . . f


f .. . ..80


132














LIST OF TABLES


Comparison between numerical and analytical results for the real


Jupiter and Saturn


. . S S S S S S 4 S S S 17


Comparison between numerical and analytical results for small m,


e, I, Jupiter and Saturn


18














LIST OF FIGURES


Variation in h, k, p, and q of Jupiter subscriptt J) with time obtained
from numerical integration of the Sun-Jupiter-Saturn system 19

Variation in h, k, p, and q of Saturn subscriptt S) with time obtained


from numerical integration of the Sun-Jupiter-Saturn system


20


Distribution in inclination space of Koronis-like particles at t=0 and


t=280,000 years


. .. .*..ft ... .... f ft2 1


Distribution in eccentricity space of Koronis-like particles at t=0


and t=280,000 years, where w is the longitude of pericenter


Variation in the forced p and q over a period of 300,000 years.
dots are from direct numerical integration while the solid line is


from secular perturbation theory


. . 23


Variation in the forced h and k over a period of 300,000 years.
dots are from direct numerical integration while the solid line is


from secular perturbation theory


.. 24


Variation in the total subscriptt t), forced subscriptt f), and proper
subscriptt p) eccentricities and longitudes of pericenter from t=0 to


Variation in the osculating inclination vectors of 9 um diameter Eos-


type dust particles as they spiral towards the Sun from 1.8 AU


Each


panel shows the vectors at various mean distances from the Sun


Variation in the osculating eccentricity vectors of 9 pm diameter Eos-
type dust particles as they spiral towards the Sun from 1.8 AU. Each
panel shows the vectors at various mean distances from the Sun


22










Proper inclination as a function of semimajor axis of 9 pm
diameter Eos-type dust particles as they spiral towards the Sun


from 1.8 AU


Error bars are the dispersions in the radii from the


mean radius obtained from the least squares fit


. S S S S S 50


Comparison of the forced inclination obtained from the dynamical
theory (bold line) and from numerical integration (points) of 9 pm
diameter Eos-type dust particles as they spiral towards the Sun from
1.8 AU. Error bars are the dispersions in the forced inclinations from
the mean forced inclination obtained from the least squares fit 51

Comparison of the forced node obtained from the dynamical theory
(bold line) and from numerical integration (points) of 9 pm
diameter Eos-type dust particles as they spiral towards the Sun
from 1.8 AU. Error bars are the dispersions in the forced nodes


from the mean forced node obtained from the least squares fit


. 52


Comparison of the forced eccentricity obtained from the dynamical
theory (bold line) and from numerical integration (points) of 9 pm
diameter Eos-type dust particles as they spiral towards the Sun
from 1.8 AU. Error bars are the dispersions in the forced
eccentricities from the mean forced eccentricity obtained from the


least squares fit


. . . S S S S S S S 53


Comparison of the forced longitude of pericenter (Wforced) obtained
from the dynamical theory (bold line) and from numerical
integration (points) of 9 pm diameter Eos-type dust particles as
they spiral towards the Sun from 1.8 AU. Error bars are the
dispersions in the forced pericenters from the mean forced


pericenter obtained from the least squares fit


S . . 54


Forced and proper eccentricities as functions of the semimajor axis.


The two straight reference lines have slopes of 1.25


3.10:


Variation in the osculating inclination vectors of 9 pm diameter Eos


particles as they spiral towards the Sun.


Each panel shows the


vectors at various mean distances from the Sun.


*. C 55


56










Variation in the osculating eccentricity vectors of 9 jpm diameter
Eos particles as they spiral towards the Sun, where w is the
longitude of pericenter. Each panel shows the vectors at various


mean distances from the Sun.


3.12:


S S S S S S C * C S 57


Proper inclination as a function of semimajor axis of 9 jm
diameter Eos dust particles as they spiral towards the Sun. Error
bars are the dispersions in the radii from the mean radius obtained


from the least squares fit


3.13:


S. S S S S 58


Comparison of the forced inclination obtained from the dynamical
theory (bold line) and from numerical integration (points) of 9 jm


diameter Eos dust particles as they spiral towards the Sun.


Error


bars are the dispersions in the forced inclinations from the mean


forced inclination obtained from the least squares fit


. S S 59


3.14:


Comparison of the forced node obtained from the dynamical theory
(bold line) and from numerical integration (points) of 9 pm
diameter Eos dust particles as they spiral towards the Sun. Error
bars are the dispersions in the forced nodes from the mean forced


node obtained from the least squares fit


S S S C S 4 5 60


3.15:


Comparison of the forced eccentricity obtained from the dynamical
theory (bold line) and from numerical integration (points) of 9 /m
diameter Eos dust particles as they spiral towards the Sun. Error
bars are the dispersions in the forced eccentricities from the mean


forced eccentricity obtained from the least squares fit


3.16:


S S S C f 61


Comparison of the forced longitude of pericenter (Wforced) obtained
from the dynamical theory (bold line) and from numerical integration
(points) of 9 pm diameter Eos dust particles as they spiral towards
the Sun. Error bars are the dispersions in the forced pericenters from


the mean forced pericenter obtained from the least squares fit


3.11:


. 62











Comparison of the forced eccentricity obtained from the dynamical
theory (bold line) and from numerical integration (points) of 9 pm


diameter Eos dust particles as they spiral towards the Sun.


Error


bars are the dispersions in the forced eccentricities from the mean


forced eccentricity obtained from the least squares fit.


The masses


of Jupiter and Saturn have been reduced to 0.1 of their real values


3.18:


Variation in the osculating inclination vectors of 9 pm diameter Eos
particles, in the real solar system, as they spiral towards the Sun.
Each panel shows the vectors at various mean distances from the
Sun .................................... 64


3.19:


Variation in the osculating eccentricity vectors of 9 pm diameter
Eos particles, in the real solar system, as they spiral towards the


Sun, where


w is the longitude of pericenter. Each panel show


vectors at various


mean distances from the Sun.


St 65


3.20:


Forced inclination as a function of semimajor axis for 9 pm


diameter Eos particles in 1983.


These are the results of integrating


waves of particles from the past such that they end up at various


distances from the Sun in


1983


0 5 S S S 9 S S 5 66


3.21:


Forced node as a function of semimajor axis for 9 pm diameter Eos


particles
particles


in 1983.


These are the results of integrating waves of


from the past such that they end up at various distances


from the Sun in 1983


3.22:


. 67


A vertical cross-section of a model dust band produced from Eos


particles alone


3.23:


68


Forced inclination as a function of semimajor axis for 4 different


systems.


4 G.P. stands for four giant planets, Jupiter, Saturn,


Uranus, and Neptune.


, E, M, stand for Venus, Earth, and Mars,


respectively


3.24:


Forced inclinations as a function of semimajor axis for 4, 9, and 14


rpm diameter Eos particles in 1983.


These are the results of


integrating waves of particles from the past such that they end up at


'mr4 nr i n A Clntnin 4n rnrv. *4l9 Cnn 4 n. 1 AQ '2


-rn


. . . . . . . . 69










3.25:


Forced nodes as a function of semimajor axis for 4, 9, and 14 pm


diameter Eos particles in 1983.


These are the results of integrating


waves of particles from the past such that they end up at various


distances from the Sun in 1983


3.26:


. a a a a a a a a a a 7 1


Distribution in inclination space of particles with 4 to 9 pm
diameters from Eos, Themis, and Koronis families in the
Earth-crossing region. The outer circle contains Eos particles while


the inner group contains


Themis and Koronis particles


a a a a a 72


Seasonal variation of the relative numbers of asteroidal particles that


will encounter the Earth.


These are particles from the Eos, Themis,


and Koronis families with diameters in the range 4 to 9 pm


3.28:


.. 73


Relative numbers of particles that will encounter the Earth in


November.


These are particles with diameters in the range 4 to 9


pm originating from


Themis (bold line),


Eos (bold line), and all


other asteroids (thin line).


3.29:


a a a a a a a 4 a a a a a 7 4


Variation in eccentricity and inclination of 9 pm diameter Eos


particles as they approach and pass the Earth.


upper right hand corners of the panels


The numbers at the


indicate the time


example, 1983-3100 means 3100 years before 1983


3.30:


a a a a a /75


Variation in the semimajor axes of fifty 9 pm diameter Eos
particles as they approach and pass the Earth. A few of them are


trapped in mean motion resonances with the Earth.


While they are


trapped, their semimajor axes remain constant and produce


horizontal lines in the diagram.


Sa a a a a a a a 76


Variation in the osculating eccentricity vectors of 9 pm diameter
Encke-type dust particles as they spiral towards the Sun, where cw is


the longitude of pericenter.
panels is 600 years .


The time interval between consecutive
90


Variation in the osculating inclination vectors of 9 pm diameter


Encke-type dust particles as they spiral towards the Sun.


interval between consecutive panels is 600 years


The time


* S 4 a a 9 1










The variation in (a,e) for every 1000 years of one wave of 9 pm
diameter Encke-type dust particles as they spiral towards the Sun.
A few of them have their semimajor axes increased up to 4 AU due


to the process of release from the comet


Relative rate of change of semimajor axi


92


s (in arbitrary units) as a


function of eccentricity due to the action of drag force


93


Distribution in (a,e) space for 9 pm diameter Encke-type dust


particles in 1983 from the second model.


These are the results of


integrating particles released continuously (once every 300 years)


starting from the year (1983-5,400).


S94


Distribution in (a,I) space for 9 pm diameter Encke-type dust


particles in 1983 from the second model.


These are the results of


integrating particles released continuously (once every 300 years)


starting from the year (1983-5,400).


. . S S S S 95


Longitude of the ascending node (Q) vs. longitude of pericenter (?w)
for 9 pm diameter Encke-type dust particles in 1983 from the


second model.


These are the results of integrating particles released


continuously (once every 300 years) starting from the year


(1983-5,400)


. 96


Distributions of the osculating eccentricity vectors in different
semimajor axis ranges for 9 pm diameter Encke-type dust particles


in 1983, from the second model.


represented by


The longitude of pericenter is


These are the results of integrating particles


released continuously (once every 300 years) starting from the year


(1983-5,400)


S* 97


Distributions of the osculating inclination vectors in different
semimajor axis ranges for 9 pm diameter Encke-type dust particles


1983, from the second model.


These are the results of


integrating particles released continuously (once every 300 years)


starting from the year (1983-5,400).


98










4.10:


Distribution of the osculating inclination vectors in different
semimajor axis ranges for 9 pm diameter Encke-type dust particles


in 1983, from the first model.


These are the results of integrating


particles released continuously (once every 300 years) starting from
the year (1983-5,400) . . . . . . 99

Distribution of the inclinations in 1983 of 9 pm diameter


4.11:


Encke-type dust particles from the first model.


4.12:


a 100


Distribution of the inclinations in 1983 of 9 pm diameter


Encke-type dust particles from the second model


4.13:


. 101


Relative number of orbits per unit semimajor axis for 9 pm


diameter Encke-type dust particles


4.14:


. * 9 4 5 102


Relative number of orbits per unit semimajor axis for 9 pm


diameter Eos dust particles.


4.15:


. S 4 103


Forced eccentricity as a function of semimajor axis for 9 pm


diameter Encke-type dust particles in 1983.


Error bars are the


dispersions in the forced eccentricities from the mean forced


eccentricity obtained from the least squares fit


4.16:


. . 104


Proper eccentricity as a function of semimajor axis for 9 pm diameter
Encke-type dust particles in 1983. Error bars are the dispersions in


the radii from the mean radius obtained from the least squares fit

Observed shape of the zodiacal cloud obtained from IRAS data in
the 25/pm waveband with an 900 elongation angle (SOP406)


105


118


Observed latitudes of the peak fluxes


of the zodiacal cloud around


the sky obtained from IRAS data in the 25pm waveband with an


900 elongation angle (SOP406)


Comparison between IRAS'


.. . . . 5 9 5 119


observations (bold line) and the


all-asteroid model (thin line) at SOP406, 900 elongation angle,


leading direction


. . . . a ... 120










Comparison between IRAS'


observations (bold line) and the


all-asteroid model (thin line) at SOP392, 970


leading direction .


elongation angle,


S. S S S * . 121


Latitudes of the peak fluxes around the sky from the all-asteroid
model (thin curves) and the best fit observational data (bold


curves)


S S S S S S S S S S S 5 S S S 122


The zodiacal cloud at SOP392; the bold lines are the IRAS


observations.


The thin lines are from the all-asteroid model (top),


the first all comet model (central), and a combination of asteroid


and comet model (bottom)


. . . S S S 123


The zodiacal cloud at SOP406; the bold lines are the IRAS


observations.


The thin lines are from the all-asteroid model (top),


the first all comet model (central), and a combination of asteroid


and comet model (bottom)


Latitudes of the peak fluxes arou


nd the sky for the 27% asteroi124

nd the sky for the 27% asteroid
125


plus 73% comet model


Comparison between the second all-comet model (thin lines) and


the observations (bold lines) at two different SOP positions


. 126


5.10:


Inclination distribution of Jupiter Family short period comets


. 127


5.11:


Distribution in (e, I) space of Jupiter Family short period comets


which have their apocenters inside the orbit of Jupiter


128
















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy.


DYNAMICAL EVOLUTION OF ASTEROIDAL AND COMETARY
PARTICLES AND THEIR CONTRIBUTION TO THE ZODIACAL CLOUD

By

Jer-Chyi Liou


December, 1993


Chairman:


Major Department:


anley F. Dermott
Astronomy


Asteroids


particles


comets


that populate


are the


zodiacal


major


cloud.


sources


how the


interplanetary


particles evolve and in


exactly what proportion asteroids and comets contribute to this cloud are questions


that have never been convincingly answered.


In this dissertation we try to answer


these questions


studying the dynamical


evolution


asteroidal


and cometary


particles and constructing a physical model to compare with observations from the

Infrared Astronomical Satellite (IRAS).

We first study the orbital evolution of asteroidal and cometary particles under

the influence of planetary perturbations, radiation pressure, Poynting-Robertson drag,
t 1 A 1 11 ** ,m ....










secular perturbation theory for asteroidal particles.


We use a numerical integrator,


RADAU,


on an IBM


ES/9000


supercomputer


to study


orbital


evolution


thousands of particles, from both asteroids and comets, systematically, from their


origins to a heliocentric distance of around 0.1 AU.


We then analyze the results and


apply the distributions of the orbital elements from the dynamical study to a three-

dimensional model, SIMUL, which generates the observed flux from a given number

of orbits with a known distribution of orbital elements, and compare the results with


the IRAS observations.


We conclude that it is possible to construct a zodiacal cloud


model to fit the shape of the observed cloud by a combination of approximately 27%

asteroidal particles and 73% cometary particles.















CHAPTER


INTRODUCTION


The Zodiacal Cloud


The zodiacal cloud is a feature of the night sky that can be seen by the naked


eye just before sunrise and just after sunset.


brightness increases towards the


ecliptic plane and is roughly symmetric with respect to that plane. Since the all sky


survey in


1983 by the Infrared Astronomical Satellite (IRAS), the structure of the


zodiacal cloud in four infrared wavebands (at


12, 25, 60, and


100 pm) has been


known in


great detail, including the overall shape of the cloud and the latitudes


of the peak fluxes around the sky (e.g.


Hauser et al.,


1985).


The infrared flux is


mainly due to thermal emission by micron-sized or larger dust particles.


What are


the origins of these particles? How do they form and how do they evolve? What are


the global and fine-scale structures of the cloud?


Is there any time variation in the


cloud structure? The observational data from IRAS has provided us with a detailed

look at the structure of the cloud, not only increasing our understanding of the cloud

but also setting critical constraints on the models which will be used to explain the

formation and structure of the cloud.

Previous models of the structure of the zodiacal cloud are based on some assumed

n nll r t r A n 'ii 4-tit Cl-.nno *di\tPlt A a.. I A : ,^1.. na n.A an1 ..2 ^1_ la..










Giese et al.,


1985,


1986, Kneissel and Mann,


1991).


These models do not


consider the formation, source, and evolution of the particles; they are restricted to

a description of the observations rather than an explanation of the physical structure


and evolution of the cloud.


When compared with IRAS observations, none of these


models are capable of explaining several key features in the data.

The dynamical evolution of individual micron-sized dust particles in the solar


system


been


studied


previously


Gustafson


et al.,


1987


Jackson


Zook,


1992).


A more recent study


Dermott et al.


(1992)


has shown that it


is possible


to study the orbital evolution of


dust particles coming from different


sources


systematically


by incorporating


the concept


of the


so-called


proper


forced elements from


the secular perturbation theory (to be explained in


Chapter


However, the work of Dermott et al.


needs some improvement; in particular,


it did not, for example, include the effects of solar wind corpuscular drag and the


changes in the orbital elements of dust particles due to the process of release.


we will show, these are effects that significantly affect the orbital evolution of dust

particles.

In summary, our aim is to study the orbital evolution of dust particles coming


from


different


sources


systematically,


and including


important dynamical


effects.


the structure of


a model


zodiacal


cloud based on


kind of


dynamical study and compare our results with the IRAS observations.







3

Objective of this Dissertation


dissertation,


we consider


dynamical


evolution


particles,


systematically,


from


source


to sink,


as described by


Dermott et al.


(1992) and


use the results from the dynamical study to construct a zodiacal cloud model.


know that asteroids and comets are the two major sources of the interplanetary dust


particles that populate the zodiacal cloud.


But in what way these particles evolve


and in exactly what percentage they contribute to this cloud are questions that have


not been answered before.


These are the questions that we would like to answer.


In Chapter 2 we examine the classical secular perturbation theory which describes

the orbital evolution of planets due to their mutual gravitational perturbations and

the orbital evolution of asteroidal particles due to the gravitational perturbations of


the planets.


We first study the evolution of the solar system numerically and find the


eigenfrequencies, amplitudes, and phases associated with the planets in the system

that describe the variations of their eccentricities and inclinations and compare them


with predictions from secular perturbation theory.


We also examine numerically the


orbital evolution of asteroidal particles in a (Sun-Jupiter-Saturn) system and, again,

compare the results with those from secular perturbation theory.

In Chapter 3 we revise the classical secular perturbation theory to include the


effect of drag.


The idea that we use was first developed by Dermott et al. in 1992.


complete the theory and apply this dynamicall" theory to the evolution of asteroidal

dust particles starting from several different heliocentric distances until they reach

thp nnPr rt nfrt nrf thp crliar cnotprm x1z7 thsn r*mnrpa thpcp' Qnlhrtrlnl lra-ilto u;th th










numerical results.


We next integrate numerically the orbits of thousands of


4, 9,


and 14 pm diameter asteroidal dust particles systematically from their origin until


they reach about 0.1 AU from the Sun in the real solar system.


All the planetary


perturbations


(except those of Mercury and Pluto),


radiation


pressure,


Poynting-


Robertson light drag, and solar wind corpuscular drag effects are included in our


calculations.


We analyze the data and obtain the distributions of their orbital elements


as functions of semimajor axis, and also examine the orbits of these dust particles

when they approach the Earth to gain information useful for future Earth-orbiting

dust collection facilities.

In Chapter 4 we study the evolution of Encke-type dust particles using procedures


similar to those in Chapter 3. This is the first time that this has been attempted for

particles in cometary type orbits. We construct two models in which the dust particles


are given different initial inclination distribu

numerically as they spiral towards the Sun.


to those in Chapter


We follow their orbital evolution


The data are analyzed in a similar way


3 in order to obtain the distributions of their orbital elements


as functions of semimajor axis.


The difference in the initial inclinations turns out


to be a key element in constructing the zodiacal cloud model.


We also show that,


because of their high eccentricities, comets cannot be the source of any of the solar

system dust bands discovered by IRAS.


We use the results from Chapters 3 and 4 to construct several physical zodiacal

cloud models (Chapter 5). A three-dimensional model, SIMUL, is used to calculate








5

cloud and latitudes of the peak fluxes from different zodiacal cloud models with the

observations from IRAS. Finally we show that by combining cometary dust particles


and asteroidal dust particles,

shape of the zodiacal cloud.


we can construct a model that describes the observed

We also discuss alternative models and some unsolved


problems.















CHAPTER 2
SECULAR PERTURBATION THEORY


In this chapter we will first summarize the theory of secular perturbation, which

is the classical method to study the long-term orbital evolution of planets as well as


minor planets in the solar system.


We apply this theory to a (Sun-Jupiter-Saturn)


system


to find


the eigenfrequencies,


amplitudes,


phases


associated


planets in the system and compare the results with those from the direct numerical


integration.


We then use the results from the numerical integration to calculate the


initial forced elements for asteroidal type particles at 2.6 AU and follow their orbital

evolution in this (Sun-Jupiter-Saturn) system both numerically and analytically and

compare the results.


Low-order Secular Perturbation


Theory


The Solar System


The classical way to study the long term rates of change of orbital elements,


eccentricities (e),


longitudes of pericenter (w),


inclinations (I), and longitudes of


ascending node (St) of small bodies in the solar system without any mean motion

resonance involved is by using secular perturbation theory (Brouwer and Clemence,

1961, Dermott et al., 1986). It is based on the assumption that the eccentricities and








7

and Q of a small body are largely determined by the secular terms in its disturbing


function (to be defined later).


These are terms independent of the mean longitudes


of any objects involved.

Considering the solar system with the Sun at the center, the Lagrange equations

which describe the rates of change of the orbital elements of any given planet are


-n-
na A


OR
- 1 e 2


\/-~ e2R


na2e


tanI ,OR
na2V1--2 OA


na2e


OR
S-)-


na2 vT-- sin I 01


(2.1)


na2 ~/ 2 -sinI l


V'STe2R


tan I
2


na2e


na2 .52-7


dt nada


I /-J- C- VY) OR


na2e


tan 11
na21/F-7e2


where


n23
n a


= G(M + m)


(2.2)


-E+


and G, M, m, n, R are the universal gravitational constant, mass of the sun, mass of

the given planet, the mean motion of the given planet, and the disturbing function,


respectively.


The disturbing function is the potential acting on this given planet due


to the existence of other planets in the system.


The semimajor axis, eccentricity,


inclination,


longitude


the ascending


node,


longitude


pericenter,


mean


1 an n.. 4...,) a nH tAn ..a tan n a 4 1. n n F fl-- .-.


..,,,..,l.. .-.1


9e
+


,,^, \








8

In order to avoid singularities that arise in Equation 2.1 for small eccentricities

and inclinations, we define four quantities (h,k,p,q) to replace e, w, I, and Q:


= e sin w
= e COS W


(2.3)


=tan I
= tan I


sinf
cosfl .


(2.4)


If the eccentricities and inclinations of the planets are small, we can ignore the

second and higher order terms in e and I in Equation 2.1 and the resultant Lagrange


equations involving e,


w, I, and t become


1 ORj
nya^ Okj


k -
'


1
nja3


(2.5)


ORj
9h,


. 1 OR,
p} = nja 9qj
2q


q. =


1
nya 2
nja1


(2.6)


ORj
dpj


By using similar assumptions, the disturbing function to the second order in e

and I experienced by the jth planet has the following form (Dermott and Nicholson,

1986):


Rj =nja
2us


mi f -ol .. .b(l)
mrc ,8 3/2(
m- 1


2)" 1 (2)
4 ji ,' 3/2 e3e


cos (wj Wi


2. m -ri 1 (1
7n 33/2
3 aj I I











2
=-nj a3


1 k) +
< A,,(h + kc) +
2 I3 !


+njaG 1Bjj1(p4 + q2) +


Aji(hjhii + kjki)

( P )


where j is not equal to i and


Si/aj

1aj
aji/ag


A3~ = n
.7 .74


(internal
(external
(internal
(external


perturber)
perturber)
perturber)
perturber)


mi i
-_ (1 )3
M&JbOJb3/ 2


(2.9)


(2.10)


(2.11)


1M^ b(2)
4M J1 3/'2


1
-- n
4 j


mi ..(1)
M O3 3 3/2


(2.12)


S1 mi (
ii =~ -nl- jioj 3/2


and b1')
3/2'


b aretheso-calledLaplacecoefficients(e.g.2)
3/2 are the so-called Laplace coefficients (e.g.


Brouwer and Clemence,


1961).


Mercury and Pluto are not included in our calculation due to their small


masses.


The solutions of hj, kj,


qj are


hIj


klj -


ej sin(git+ fli)


(2.13)


cos(git + ypi)


Ij, sin(fit + yi)


(2.14)


pj =

qi j


Ijicos(fjt+ 7i)


(2.8)


Otj -

aji =


A
^in -


B =
r\ **-








10

where gi and fi are the eigenfrequencies of the matrices A and B whose elements


are defined by Ajj, Aji, jj, and Bji above.


The amplitudes, eji, Iji, and phases, %i,


are determined from the initial conditions of the planets.

Equations 2.13 and 2.14 are the results from the classical low-order secular per-


turbation theory.


These two equations describe analytically the respective variations


in the eccentricity, inclination, longitude of pericenter, and longitude of ascending


node of the jh planet.


They are based on the assumption that the eccentricities and


inclinations of all the objects involved are small.


When there is no mean motion res-


onance phenomenon involved, this theory describes fairly accurately the long-term


variations in e,


zt, I, and Q of the planets.


Small Bodies in the Solar System


For small


bodies moving under the gravitational perturbations of planets,


can follow similar procedures to find the long-term rates of change in their (h,k),


and (p,q).


If we use A


to represent the nodal precession rate of the small body


subscriptt s),


(2.15)


where n, mi, and M are the mean motion of the small body, the mass of the it planet,

and the mass of the Sun, respectively, then the solutions for (h,k) and (p,q) are


h = epsin(At + p) + ho
k = ep cos(At + f3) + ko

Sp = Ip sin(-At + 7) + po


(2.16)


/) 17\


A= aj .(1)
A- 4 2 M ^^-3/
1=1










where


ha =


7
i=-
=1


-i=
i=1


Po =


qo =


i-- sin(git + /i)


(2.18)


Si cos (it + /3i)


A + sin(fit + yi)
A+ f;


(2.19)


+i fcos(fit + 7i)
A+ fi

- mi (2)
M>_c-0^1s l3/2 ti


(2.20)


Pi = _


lb Iii
M 3/2


and gifi, fe, eii, and pi, 7i are the eigenfrequencies, amplitudes, and phases associated

with the planets of the solar system from the previous section.

In the previous equations, ep, Ipt, and 7 can be determined from the initial


conditions of the small body.


The forced eccentricity and inclination are defined as


ef =

If=


(2.21)


The facts that (h,k), and (p,q) are de-coupled from each other and, for each of


them,


we can


use the proper and forced elements (their characteristics under the


influence of drag force will be described in Chapter


3) instead of the osculating


elements to describe their evolution make it possible to study dust particles coming


from the same source systematically (Dermott et al.,


1985).


The precession term, A, depends on the masses of the planets if the semimajor


aC a C C C C


+ qo2


* fi r 1 A










due to the existence of planets.


If all the planets have circular orbits and the same


inclinations, then there will be no forced eccentricity and longitude of pericenter (due

to zero eccentricities of planets) and the orbit of the minor planet will process along

the common plane of the planets (due to the same inclinations of planets). However,


this is not the case in the real solar system.


The forced components in Equations


2.16 and 2.17 will be present due to the eccentricities and different inclinations of


the planets'


orbits.


The forced components introduce a plane of symmetry (different


from any of the planets'


orbital planes), with respect to which the orbit of the minor


planet processes, along with the displacement of the pattern of the orbit of the minor


planet away from the Sun (Dermott et al.,


1985).


The equations of (h,k) and (p,q) describe the time variation in orbital elements of


small bodies in the solar system.


This is a result from the low-order theory.


When


the eccentricity or inclination of the small body becomes larger, the low-order theory


can no longer be applied to study the evolution of this minor planet'


orbit.


In this


case, direct numerical integration is the only way to study its orbital evolution.


In the


next section we will test this low-order theory on the real solar system and compare

the results with those from direct numerical integration.



Numerical Approach


After the invention of high speed computers,


astronomers found another powerful


to study


celestial


mechanics:


numerical


integration.


Many effects


that can









mtegration.


For example, if we include the gravitational forces due to the Sun and


planets and the radiation pressure force as well as the drag terms in the equation

of motion of a dust particle and numerically integrate the equation, passage through

mean motion resonance, planetary scattering, and trapping in mean motion resonance

associated with planets can be seen from the numerical output of the orbit of this

particle. However, the numerical error built up in the process needs to be considered


carefully.


Besides, not all problems can be solved numerically.


In some cases, for


example, when the number of particles involved becomes too large or the integration

time becomes too long, even the fastest machine in the world is not sufficient to

obtain the desired results.

The numerical integrator we chose is RADAU. It was developed by Everhart


(1985).


The major computer we use to run the numerical integration is an IBM


supercomputer ES/9000 in


Northeast Regional


Data


Center (NERDC)


at the


University of Florida.


IBM ES/9000 is equipped with a vector facility.


For each


processor, it can take up to 256 objects and perform the same operation at the same

time. In order to improve the performance, we have vectorized RADAU. In addition


to IBM ES/9000,


we have also used IBM RS/6000 350, IBM RS/6000 340, IBM


RS/6000 220, and several Sun workstations.


The numerical integration we describe in this chapter include two parts.


first part is the integration of the solar system with only two planets, Jupiter and


Saturn,


with their real masses and orbital elements.


This is a simple three-body


gvgtem


Wp hnve. sl1









and inclinations of Jupiter and Saturn.


Both numerical results were analyzed and


compared


with


analytical


predictions


from


secular perturbation


theory.


second part is the integration of the orbital evolution of 100 asteroidal type particles


in the (Sun-Jupiter-Saturn) system.


Again, the comparison between analytical and


numerical results was made.


Comparison between the Analytical and Numerical Calculations


Sun-Jupiter-Saturn System


We first test the secular perturbation theory on the (Sun-Jupiter-Saturn) system.

This three-body system was integrated from 1983 backward for five hundred thousand


years.


The variations in h, k, p, and q of Jupiter and Saturn are shown in Figure 2.1


and Figure 2.2.


We analyzed the data by using the Fast Fourier transform method


to find the eigenfrequencies, amplitudes, and phases of the system from the power


spectra and compared the results with the analytical calculation.


shown in


The comparison is


Table 2.1.


As we can see from the table, the two results agree reasonably well.


there are certain discrepancies for the eigenfrenquencies, gi and g2.


are 15.2% and 16.8%, respectively.

motion resonance between Jupiter an


However,


The differences


A possible reason for this might be the mean

d Saturn. These two planets are very close to


5:2 mean motion commensurability (a third order resonance).


The analytical third-


order resonance theory has not been worked out, and its development is beyond the








15

We can also check this mean motion resonance effect by reducing the masses,

eccentricities, and inclinations of Jupiter and Saturn and compare the results again.


The comparison is in


Table


The masses of Jupiter and Saturn are


10% of


their original masses.


Their eccentricities and inclinations are ej = 0.0004948, Ij =


0.003273 degree,


es =


0.0005085, and Is = 0.008872 degree.


The largest difference


is only 3.3


All the other differences are less than 1%.


The secular perturbation


theory predicts very accurately the results from the numerical integration in this case.

From these comparisons we can conclude that the low-order secular perturbation

theory does describe the behavior of planets accurately except when a mean motion


resonance between planets is involved.


We will further examine how good this theory


is when we apply it to the asteroidal-type particles in the next section.

Asteroidal Particles in the Solar System


In the calculations described in this section we put 100 asteroidal-type particles in

the Sun-Jupiter-Saturn system and integrate their evolution for 280,000 years. Again,


we compare


the results


with


the analytical calculation


from


secular perturbation


theory.

The semimajor axes for the particles are chosen to be 2.6 Astronomical Units


(AU).


This


is to avoid several


mean


motion resonance


locations


Jupiter at


around 3 AU.


We use the


"particles in a circle"


method (Dermott et al., 1985, 1992)


to set up the initial orbital elements of the particles.


Their proper eccentricities and


inclinations are 0.049 and 2.12 degrees, respectively, which are the proper elements


a' rr


/ ,


r\


I 1










randomly chosen between 00


and 360


. The initial forced elements at 2.6 AU are


calculated by using the eigenfrequencies, amplitudes, and phases from the numerical

integration of the (Sun-Jupiter-Saturn) system from the previous section.

The positions of particles in the (p,q) and (h,k) phase spaces at t = 0 and t =

280,000 years are shown in Figures 2.3 and 2.4. Due to the difficulty in generating w


in the diagram, we use w to represent the longitude of pericenter in all diagrams.


fact that after 280,000 years all the particles still remain in pretty well-defined circles

in both diagrams shows that the method of using the forced and proper elements to

describe the evolution of asteroidal-type particles is a good one.


The variations in the forced elements are shown in Figures 2.5 and 2.6.


dots are results from the direct numerical integration while the solid lines are results


from the analytical calculation from the secular perturbation theory.


are about 0.20


The differences


in the forced inclination and less than 0.01 in the forced eccentricity.


These results are actually not too bad.


This shows that when the particles are not in


mean motion resonance with any planet, the secular perturbation theory does describe

the evolution of asteroidal-type particles to a certain accuracy. However, when small

particles are spiralling towards the Sun due to Poynting-Robertson and corpuscular


drag, this classical secular perturbation theory does not apply any more.


chapter we use the theory of Dermott et al.


In the next


(1992) to include the effect of drag.






































































1 0
26
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18





z
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cd&

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rt1-4

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-,^ ~ ^^^ -^ ^^.c S^r










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1 m I' i i 11) 1 I' '' 1 11"i' l I' o



:0 0






S*** D m O N m a
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n0 '0 I
Cr,
0 a N 0 N 0 0 C Cf
-g "\ -g \ -g C \ 8 B














Sc

oE
/ / -





*M
*e 5..


SE
1.7..
I ( / ^ 3











20















S
0

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8 O O
oo _


13~~- c'F
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Co





aggg

I< I< ^^ ~ I s^ "* 1f '* a .^.
P2 I C







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aE
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(M~i)
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(U)uis

































-r 0


|o ea


*. *..* *


(m)uis


aa
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p~Jaojq















CHAPTER


ORBITAL EVOLUTION OF


ASTEROIDAL PARTICLES


Radiation Pressure, Poynting-Robertson Drag, and the Corpuscular Drag


The orbital evolution of micron-sized dust particles in the solar system is affected

by several forces in addition to the gravitational forces due to the Sun and planets.


These include


radiation pressure,


Poynting-Robertson


(PR) drag,


and solar wind


corpuscular drag.

The radiation pressure force is the force exerted on a dust particle due to the

interception of sunlight. It varies as the inverse square of the distance from the Sun

and depends on the properties of the particle (size, chemical composition, etc.).


When a micron-sized dust particle


is moving around the Sun, it absorbs the


sunlight and reradiates the energy away isotropically in its own reference frame.

However, when viewed from the rest frame this particle radiates more energy along


the same direction as its velocity vector.


This will cause a net loss in its orbital


energy and angular momentum.


It acts like a retarding force on the particle.


This


deceleration drag causes


the particle


to spiral


towards the


Sun.


It is called


Poynting-Robertson drag (Poynting,


1903


, Robertson,


1937).


This effect is present


even if the particle scatters the sunlight instead of


absorbing and reradiating the


.. .








26

The solar wind corpuscular drag is due to the interaction between the solar wind


protons and the dust particles.


drag.


The physical interpretation is similar to that of the PR


The classical paper dealing with radiation pressure, PR drag, and corpuscular


drag is the paper by Burns et al.


(1979).


The equation of motion of a dust particle with geometrical cross section A and

mass m moving under the influence of the gravitational forces of the Sun, planets,

and the radiation force from the solar radiation can be written as


mv =


rGMm


GMnm


(3.1)


A/c)Qpr


(1-


r\
- C
c/


where v and r are the velocity and position vectors of the particle in the heliocentric


coordinate system,


while Mn, rn


are the mass of the nh planet and the position


vector of the particle with respect to that planet, respectively.


S, c, and Qpr are the


solar energy flux density, the speed of light, and the radiation pressure coefficient,


respectively.


The particle's radial velocity and the unit vector of the incident radiation


are represented by t and S, respectively.


The non-velocity dependent part of the third


term at the right hand side of Equation 3.1 is the radiation pressure force term, while


velocity


dependent parts


are the drag terms.


Qpr depends on


the properties


(density, shape, size,


etc.) of the particle.


It can be calculated from the Mie theory


(Burns et al., 1979, Gustafson, 1994).


Traditionally, a dimensionless quantity, P/, is


introduced to specify the effect of radiation pressure and PR drag. It is defined as


radiation
"1 1I


force
r-


(3.2)







27

The major topic in this chapter is to study the dynamical evolution of asteroidal


dust particles which contribute most to the 25 pm waveband IRAS observations.


we adopt the solar wind drag as being 35% of the PR drag, the major contributors to


pm waveband are particles with y3 equal to 0.05037, corresponding to 9 pm


in diameter.


These results are obtained by assuming 2.7 gm/cm3 perfectly spherical


astronomical


silicate


and using


theory


to integrate


over the


solar spectrum


(Gustafson,


1994).


The corpuscular drag is


35%


that of the


PR drag according


to the latest average solar wind data from Helios 1 and 2 (Schwenn, 1990).

When a dust particle is released from a large parent body, it immediately "sees"


a less massive Sun, due to the radiation pressure.


Because the dust particle has


identical position and velocity vectors as its parent body (assuming it leaves with

zero relative velocity), this reduction in the centripetal force will change some of its


orbital elements right away.


Since radiation force is a radial force, it has no effect


on I and fl of the dust particle; only the semimajor axis, eccentricity, and longitude

of pericenter need to be recalculated.

The new semimajor axis, an, and eccentricity, en, are given by


an=a


- 2a'3/r


(3.3)


(1 2a/3/r)(1


-p3)2


(3.4)


X-


where a and e are the semimajor axis and eccentricity of the parent body and r is

the radial distance from the sun at which the dust particle is released. From the first

equation we can see that the new semimamir axis of the dust article is always larger


- e2


en










than that of its parent body. However, this is not true for the new eccentricity.


new semimajor axis and eccentricity of the dust particle also depend on where it is

released (i.e., r). It is not too difficult to show that both an and en have their maxima


at perihelion release and have their minima at aphelion release.


When the quantity


X in Equation 3.3 becomes negative, the new eccentricity is the square root of the

absolute value of X, and there is an 1800 shift in the longitude of pericenter of the

dust particle compared with that of its parent body.


Static


Theory and Dynamical Theory


The classical


secular perturbation


theory mentioned in


Chapter


2 deals


particles having constant semimajor axes.


This is the so-called static theory. Micron-


sized dust particles in the solar system will spiral in towards the Sun due to the drag


force in about 50,000 years (Wyatt and Whipple, 1950, Burns et al., 1979).


Thus, it


is necessary to develop a new secular perturbation theory which includes the effect


of drag.


We use the theory developed in Dermott et al.


1992.


We test the theory


on asteroidal dust particles in this section.

In the low-order static theory, the rates of change of p, q, h, and k are


N
E vj cos(gjt + yj)
j=l


(3.5)


-Ah +


-Aq +


I = An -


vy sin(gyt + yj)


PJ cos(fyt + yj)


(3.6)


"' u." fsin( f;t 4-+ 'y


h = Ak-









where A, v gj, fj, Pj, and 7j, are defined in Chapter


When there is no drag,


the precession rate, A, is

spirals in towards the Sun.


a constant.


When there is a drag force, a dust particle


Its semimajor axis becomes a function of time.


So, A,


vY, and pj (through their dependence on semimajor axis) all become functions of


time.


The eigenfrequencies and phases, gj, fj, and 7j, are constants regardless


of the presence of drag force.


In order to solve the problem,


we define a complex equation to combine the


equations involving p and q (or h and k), namely


z=q


+ zp .


(3.7)


By using


z, Equation 3.6 becomes


N
z = -iAz+i ei(ft+ .
j=l

In the static theory, the solution of the above equation is


(3.8)


pei(-At+Y) +


Sj ei(tft+1+)
= A+ f
j~l


(3.9)


where j and


7 are the proper inclination and node, respectively, of the particle at


time equals to zero.

When A and /j become functions of time, Equation 3.8 becomes a standard first


order inhomogeneous differential equation.


Its solution is


z(t) = e


e'rdt


(3.10)










where


iAdt


(3.11)


13jei(fut+rY)


(3.12)


The constant C in Equation 3.10 depends on the initial conditions of the dust particle.


Before


we find the solution of Equation 3.10,


we need to know the rates of


change of semimajor axis and eccentricity due to PR drag.


They are (Wyatt and


Whipple,


1950, Burns et al.,


1979)


-(yr/a)Qpr(


+- 3e2)


(3.13)


1 e2)3/2

-5'Qpre


(3.14)


2a2(1


e_ 2)1/2


where 17 = 2.53 x l01 /ps for spherical particles in heliocentric orbits with density

p, radius s in (c.g.s.) units, and Qpr is the radiation pressure coefficient.


Now


let us take a look at how the drag force affects


the forced and proper


eccentricities.


We first consider a group of dust particles from a given family having


a distribution in the (h,k) phase space as shown in Figure 3.1.


circle is


The equation of the


given


2
= epO


(3.15)


where he and kc define the forced elements (i.e., the center of the circle) at time equal


to 0 and epo is the initial proper eccentricity.


The total eccentricity of a given particle


r = i


da
dt)


de
dtJ


+ (h hc)2










center of the circle moves to ni


forced eccentricity changes from efo to en.


while the radius of the circle changes to ep1.


The change in the proper eccentricity


the radius of the circle) is due to drag, while the change in the forced eccentricity


and longitude of pericenter is due to both gravitational perturbation and drag. From


Equation 3.14 we


particle


see that the drag force changes the total eccentricity of any given


by the amount


6e=


-Cleto


(3.16)


where CI is a positive constant.


in the circle, etl


The new total eccentricity of the previous particle


then becomes


= (1 C1)eto


The equation of the new circle after S

[(1 C1)k (1 1 C1) kcj


(3.17)


becomes


2 + [(1 C1)h -(1- C)h]2


(3.18)


= [(1 Ci)epo]


This means that the drag force moves the center of the circle (which is defined by

the forced eccentricity and longitude of pericenter) towards the origin by the factor

of (1-C1) and, at the same time, makes the circle (defined by its radius, the proper


eccentricity) shrink by the same


factor.


This factor, can of course


be calculated


analytically from


Equation 3.14.


The overall picture of the dust evolution thus becomes very clear.


The gravita-


tional perturbation changes the forced eccentricities of the dust particles while having


Sl- .*


m i I i .


+ h l








32

towards the Sun and changes their proper and forced eccentricities the same way it

changes their total eccentricities.

By combining Equations 3.13 and 3.14 we can obtain the following equation:


5e(1


-e2)


(3.19)


2a(2 + 3e2)


For asteroidal dust particles we can drop the e2 terms.

log(ep) plot during their evolution is then 1.25. This v


The slope in a log(a) vs.


vill be verified numerically


the next section.


Numerical Comparisons


In this section we first test the dynamical theory by placing dust particles at 1.8


AU initially in a (Sun-Jupiter-Saturn) system.


This is to avoid the passage through


mean


motion


resonances


associated


with Jupiter.


Then


we place dust


particles


initially at 3 AU and observe how they are affected when they do go through the

Kirkwood Gaps (to be defined later).

Drag Effect


We place 254 Eos-type dust particles initially at 1.8 AU to start the numerical


integration. Again, we use the "particles in a circle"


set up their initial orbital elements.

and inclinations as those of Eos.


method (Dermott et al., 1992) to


The particles have the same proper eccentricities

Their initial forced elements are calculated using


the eigenfrequencies, amplitudes, and phases from the numerical integration of the









(a, e, and w) due to the releasing process.


The reason for this is to simplify the


problem and study the effect of drag on the orbital evolution alone.


The dust particles are integrated for 20,000 years.

We plot their (p,q) and (h,k) once every 2,200 year


Figure 3.3.


They end up around 0.5 AU.


s as shown in Figure 3.2 and


The fact that they remain in nice circles in both (p,q) and (h,k) phase


spaces during their evolution proves that the method of using forced and proper

elements to describe their evolution in the presence of drag force is valid. Figure 3.4

shows that the proper inclinations of the dust particles are unchanged during their


evolution from


1.8 AU


to 0.5


In Figures 3.5 through 3.8 we compare the results from the numerical integration


(shown as dots) and the prediction from the dynamical theory (solid lines).


The forced


inclination and node will be shown in Chapter 5 to be very important parameters


when we construct the zodiacal cloud model.


difference


After 20,000 years of evolution, the


the forced inclination between the dynamical theory and the actual


numerical integration is about 0.02 degree.


less than 1 degree.


The difference in the forced node is


This shows that the dynamical theory can describe properly the


evolution of dust particles under the influence of both gravitational perturbations and

PR drag as long as the effects of mean motion resonance and point mass scattering

are not important.


In Figure 3.9 we plot the forced and proper eccentricity in a log(a) vs.


frame.


log(e)


Two reference lines with slopes equal to 1.25 are also shown in the diagram.


A uxe^ mentirnndP intbn th lcl etrtln rQn frrst ;0 ts etnwr fou-tr tlhat h-hanao c








34

the proper eccentricity. It changes the proper eccentricity according to Equation 3.19,


which predicts an 1.25 slope in the log(a) vs.


log(e) plot.


The forced eccentricity


does not approach the reference line until the particles are very close to the Sun.


This


is because both the gravitational perturbation and drag act on the forced eccentricity.

When particles are far away from the Sun, the gravitational perturbation dominates

the change in the forced eccentricity. It is only when the particles are very far away

from the planets (Jupiter and Saturn), that the drag becomes the dominant factor and

the forced eccentricity in the diagram approaches the straight line.


Passage


Through Resonance


In the real solar system, asteroid dust particles produced from the main asteroid

belt will go through several Kirkwood Gaps and secular resonance regions while


they are spiralling towards


the Sun.


Kirkwood Gaps are the regions at which a


particle is in mean motion resonance with Jupiter (e.g. Dermott and Murray, 1983).


For example,


resonance


is at a=2.46


resonance at


AU for particles having /=0.05037


. The locations of the Gaps are slightly shifted


due to the radiation force.


Secular resonance regions are


the places at which a


particle's orbital precession rate matches one or a simple combination of several


of the eigenfrequencies of the solar system.


The major secular resonance region


is located near


Williams, J.


1969,


Scholl et al.,


1989).


Passage


through different Kirkwood Gaps has some complicated effects on the orbits of the








35

is beyond the scope of this dissertation. However, one can observe their effects from

direct numerical integration, as we have done here.


Again,


this is


a Sun-Jupiter-Saturn


system.


The initial


forced


elements


calculated similar to the methods mentioned in the previous section.

and fifty four Eos dust particles are placed at 3.015 AU at t=0. The


Two hundred


ir other orbital


elements are generated by the "particles in a circle" method (Dermott et al., 1985,


1992).


The changes in orbital elements due to the releasing process are not included.


These dust particles are integrated for 50,000 years.


They end up at about 0.7


The distributions in


the (p,q) and


(h,k) phase spaces of these particles while


they are moving towards the Sun are shown in Figure 3.10 and Figure 3.11.


time interval between two consecutive panels is 6,000 years.


The first impression


from the figures is that they remain in a well-defined circle in the inclination space

while going through large scattering in the eccentricity space. In Figure 3.12 we plot


the proper inclination vs.


semimajor axis of the particles.


The proper inclination


jumps up and down somewhat (particularly at the 3:1 resonance) during the particles'

passage through Kirkwood Gaps even though the magnitude of change is not very


significant (from


the observational point of view).


Figure


and Figure


show the variation in the forced inclination and longitude of ascending node from


the numerical integration (dots) and the results from


the dynamical


theory


(solid


line).


The difference is quite small.


This implies the dynamical theory works well


in inclination even in the case of mean motion resonance passage.


The variations


z C .... r- .. -..! L- ji 1 2- C A ------------------- r -" -_-- F, -- 2 1^ -__-_-










3.16.


The sudden drop at the 3:1 resonance region is very obvious.


The difference


between the numerical results and the dynamical theory is around 30% near the end

of the integration.

In all the diagrams there is no dramatic change that can be seen when particles

pass through the secular resonance region.


We have also performed another test run here.


We changed the masses of Jupiter


and Saturn to 10% of their real masses and repeated the same numerical integration.

This was to reduce the strength of each mean motion resonance (Dermott and Murray,


1983).


The difference in the forced eccentricity was dramatically reduced this time


as we can see


in Figure


. This also supports


what we have


just suggested,


namely that the discrepancies in Figures 3.15 and 3.16 are actually due to passage

through mean motion resonance zones.

From the above results we can conclude that passage through Kirkwood Gaps


does produce a significant effect on particles'


eccentricities and longitudes of peri-


center.


Orbital Evolution of Dust Particles from the Major Hirayama Asteroidal Families


The orbital evolution of dust particles in the real solar system is more complicated


than the cases presented previously.


First of all, even if the parent bodies which


produce dust particles constitute a well-defined group (i.e.


they form nice circles


in both (p,q) and (h,k) spaces), the releasing process changes some of the orbital








37

the dust particles from a well-defined family do not form a well-defined circle in


the eccentricity space from the very beginning.


This makes the attempt to study


analytically the variation in forced and proper eccentricities of dust particles coming


from the same family very difficult. Secondly, inside the orbit of Jupiter, there are

three important planets, Mars, Earth, and Venus. When dust particles pass right by


these planets, there are singularities (due to zero distance to the planet) which cannot


be handled analytically.


The third reason is that some of the dust particles will be


trapped in the near-Earth mean motion resonance (and to a lesser extent, near-Mars


and near-Venus resonances).


Due to these difficulties, the practical way to study the


orbital evolution of micron-sized dust particles is direct numerical integration.


In this section we perform the numerical integration for dust particles coming


from three major Hirayama asteroid families, Eos,


Themis, and Koronis.


All the


planetary perturbations


(Mercury and Pluto are


not included),


radiation


pressure,


Poynting-Robertson drag, and corpuscular drag are included in the calculation.


changes in


the orbital


elements due


to the


process of release


are also included.


Particles are assumed to leave their parent bodies with zero relative velocity. In order


to utilize the vector facility on the IBM ES/9000,


7 planets for each integration.


we set up 249 dust particles and


The solar system is wound back in time numerically


to an epoch such that dust particles being released at that epoch will reach a chosen


heliocentric distance


1983.


initial


forced elements of


particles


calculated from the orbital elements of the 7

nrnnsrar nrnAsC anrl nrinti'h1A f narlrn^a tr


planets at the beginning epoch.


Anrra n r n n n,- A ,,1.i,


/^i-ic'i nnr









between 00 and 3600


Dermott et al.


The proper eccentricities and inclinations of families are from


(1985).


9 um Diameter Particles


We first integrate one wave of particles from each family.


Each wave ends up


inside 1 AU in 1983. For Eos particles, the variations in (p,q) and (h,k) spaces during


their evolution are shown in Figures 3.18 and 3.19.


The first panel in each figure


is the distribution of the parent bodies which produce the dust particles.


The time


interval between two consecutive panels is 6,500 years.


The label above each panel


is the average semimajor axis of the dust particles at that time.

In the inclination space, Eos dust particles remain in a circle even after they pass


the Earth while


Themis and Koronis dust particles have large scatterings when they


approach the Earth.


This is because


Themis and Koronis dust particles have much


smaller inclinations compared to that of Eos dust particles.


The probability of low


inclination objects being scattered by terrestrial planets is higher than that of high


inclination objects.


In the eccentricity space, Poynting-Robertson drag reduces the


eccentricities of dust particles while they are moving towards the Sun.


However,


when they approach the Earth some of them are trapped in the near-Earth mean


motion resonances.


These particles gain energy and lose angular momentum during


the process. Eventually they will escape from the resonance zone and continue their


journey towards the Sun.


This phenomenon will be discussed in more detail in the








39

Obtaining the particle distribution at different heliocentric distances in 1983 (for

the purpose of constructing the zodiacal cloud model) is different from studying the


dynamical evolution of the particles. Different waves (each "wave"


is the numerical


integration of the orbits of 249 dust particles) of particles must be sent out at different


epochs such that they end up at different heliocentric distances in


1983.


This task


requires considerable CPU


time.


data for three families


took about 3


months on an IBM ES/9000 supercomputer.


For Eos dust particles,


we sent in 23


different waves dating back to the year 57,500 years before 1983.


between each wave is 2,500 years.

integration and obtained the orbital


distances.


The time interval


Then we analyzed the data at the end of each

elements distribution at different heliocentric


Thus, we have the forced elements as a function of heliocentric distance


from around 0.5 AU to 3.2 AU


The forced inclinations and nodes for Eos particles


are shown in Figures 3.20 and 3.21.


Themis and Koronis


particles have similar


forced inclinations and nodes until they reach around 1 AU where they no longer


remain in a circle due the gravitational scattering by the Earth.


In the (h,k) space,


particles are scattered even from the very beginning as we can see in Figure 3.19.


Hence,


we did not attempt to find the corresponding forced elements by fitting a


circle to the points in eccentricity space.


Instead,


we used a tabular form from the


data to generate the eccentricities and longitudes of pericenter for the dust particles

when we constructed the zodiacal cloud model.


Once the orbital element distributions of dust particles from a known family have
l ^ ^ ^ ^ J ^ -- ._ -_ _- _- l.. i* 1. 1 i i










cloud from the results.


We did just that for Eos dust particles.


Figure 3.22 is the


vertical cross-section picture of the Eos cloud.


This is the cloud which may


responsible for the


dust band.


Actually the so-called "solar system dust band"


is not a band.


Because of the drag effects, dust particles go all the way from their


origin to the Sun (the reason there is a hole in the figure inside 0.5 AU is that we


terminate the integration when dust particles reach there).


The actual "band" is the


enhancement of the observed flux at certain ecliptic latitudes due to the concentrations


of dust particles at the edges.


This concentration can be seen in Figure 3.22. In the


real solar system, this cloud is embedded in the zodiacal cloud which includes dust

particles from many different sources. However, this concentration is strong enough

such that when the whole zodiacal cloud was scanned by IRAS, this "band" (more

precisely, a tiny peak on top of the background) at around 100 was observed.


The forced inclination and longitude of the ascending node of the dust particles


determine the plane of symmetry of the zodiacal cloud.


This plane is a warped plane


as the forced inclination and longitude of the ascending node vary with heliocentric


distances.


When


we observe the zodiacal cloud,


the latitudes of the peak fluxes


around the


are determined


warped plane.


These


positions


are very


well-defined in the IRAS observations.


One of the most important achievements


of the present work is


being able


to use


the forced inclination and longitude of


the ascending node from the dynamical study and the SIMUL model to predict the

latitudes of the peak fluxes as observed by IRAS. Comparison of those predictions







41

In Figure 3.20 the data show that the forced inclination increases towards small


heliocentric distances.


This increase had been reported from observations prior to


the launch of IRAS (e.g. Misconi and Weinberg, 1978, Leinert et al., 1980).


Is this


increase due to terrestrial planets, especially Venus (which has a very high inclination,

3.40) as some early papers suggested (Misconi and Weinberg, 1978, Gustafson and

Misconi, 1986)? We examine this by taking out the terrestrial planets one at a time


and integrating the orbits of Eos dust particles in each case.


inclinations from these cases are in Figure 3.23.


can be seen clearly.


The resulting forced


The effect of the terrestrial planets


The increase in the forced inclination has nothing to do with the


existence of the terrestrial planets.


It is the passage through the secular resonance


zone that increases the forced inclination.



4 pm and 14 pm Diameter Particles


We assume that 9 pm diameter dust particles are the major contributor to the 25

Cpm waveband observation of IRAS. So far we have studied the dynamical evolution

of 9 pm diameter dust particles and their distribution in the solar system. In Chapter

5 we will show a zodiacal cloud model based on particles of this one size. However,


particles of


different sizes do contribute to the 25


waveband observations as


well.


Eventually the model zodiacal cloud must include results from particles with


a range of sizes.


In this section we present the dynamical study for particles with


diameters equal to 4 pm and 14 pm.


This covers a range of 3.5 in sizes (a factor


nf 43 in m ~ p. if the.v have the same den itv.










The integration procedures here are similar to those in the previous section.


corresponding CPU time for 14 pm diameter dust particles of course is much longer


than that of 9


pm diameter dust particles.


The forced inclinations and nodes as functions of semimajor axes for the three

different size particles are shown in Figures 3.24 and 3.25. Among them, the 14 pm


diameter dust particles have the smallest drag rate.


They pick up the most increase


in the forced inclination while they pass through secular resonance regions and end


up with the largest forced inclination in the inner solar system.


The difference in


inclination at 1 AU between these three different size particles is about 1.5 degree.

Obviously this difference must be considered when a better zodiacal cloud model

(comprising particles of 4, 9, and 14 pm in diameters) is to be constructed. How to

combine these results is very important in constructing a successful zodiacal cloud


model in


the future.


Near-Earth Dust Grains


In this section we will describe some properties of asteroidal dust particles when


they are in the vicinity of the Earth.


These include seasonal variations in the number


of particles that intersect the Earth, how to identify the origins of near-Earth dust

particles, and the phenomenon of trapping in near-Earth mean motion resonances.

The forced inclination and longitude of ascending node determine the plane of


symmetry of the zodiacal cloud. They also produce another interesting phenomenon










collected in near-Earth orbit.


The distributions in inclination space for 4 pm and


9 pm dust particles are shown in Figure 3.26.


These are particles in the Earth-


crossing region in 1983 found from our numerical integration. Notice the off-center


distribution for


Themis and Koronis particles.


There are more


Themis and Koronis


particles in the first and fourth quadrants than in the second and third quadrants of


the diagram.


When the Earth moves around its orbit over one year, it will encounter


more


Themis and Koronis dust particles at certain places.


These places correspond


to the areas in the inclination space where there are more particles (first and fourth


quadrants).

Figure 3.27


This kind of seasonal variation for 4 and 9 pm particles is shown in

Here we have a histogram showing the variation in the relative number


of particles as a function of the time of the year.


The Earth encounters more Themis


particles at their ascending nodes in November than in April.


The difference is more


than a factor of 8.


There is no significant seasonal variation for particles with large proper inclina-


tions, like Eos.


In the vicinity of the Earth the proper inclination of Eos particles


is much larger than the forced inclination.


Consequently, even though the circle is


shifted from the origin in the inclination space (see Figure 3.26), the difference in

terms of number of particles in each quadrant is not very large. Hence the seasonal

variation for Eos particles in Figure 3.27 is not as dramatic as that of Themis particles.


Themis, Koronis, and Eos are the most prominent families in the asteroid belt.

The ratio of dust particles produced from these three families to non-family asteroids


'ten~~ 4.l a n 4..l .~ -r A Ct r. e


4 t fl a *- n 1t I ..


Sc


*n~ nA


*L--r










Themis


: Eos : Koronis =


0.61


: 0.12 (Durda,


1993).


This means


Themis


and Eos


produce


17% and


10% of all


the asteroidal dust particles, respectively.


This, together with the facts that (1) particles from certain families do not arrive


uniformly in time (e.g.


most Themis particles with ascending nodes encounter the


Earth around November) and (2) particles from one family have a well-defined range

of their inclinations, makes it possible to identify the origins of some family dust

particles. In Figure 3.28 we show the relative numbers of particles as a function of


inclination.


These are 4 and 9 pm dust particles at their ascending nodes which will


encounter the Earth in November.


Themis and Eos particles are represented by bold


lines in the histogram while the whole non-family asteroidal particles are shown as


thin lines.


Much useful information can be obtained from this figure.


First of all,


the low inclination dust particles in November are dominated by

Half of the particles around 120 inclination are of Eos origin.


Themis members.

If we utilize these


results to interpret the samples from an Earth-orbiting dust-collection probe, we can

obtain detailed information (e.g. chemical composition) about some asteroid families

without having to go to the asteroid belt. For example, from this dynamical study we

know that half of the dust particles being collected in November with 12 inclination


are Eos particles.


Then,


we can collect, say, 20 particles with inclinations around


and analyze their composition.


Half of them


will have similar compositions


(if we assume particles from the same family have the same compositions).


must be Eos particles.


family.


These


Their chemical compositions are those of the Eos asteroid


Also, most of the low inclination (around 30) dust particles being collected









in November must have similar compositions.


These are particles from the


Themis


family.

An important objective for any future dust collection mission is to establish a


relationship between the dust particles being collected and their parent bodies.


results here have shown that it may be possible to achieve such a goal.


With modem


techniques, intact capture of particles with velocities less than 9 km/sec is possible

(these are most likely to be asteroidal dust particles). From our study, we can provide

not only the best time but also the orientation of the orbits of the dust particles that

can be collected and related to several known asteroidal families.


When dust particles approach the


Earth from outside,


some of them


will be


trapped in the near-Earth mean motion resonances.


on many factors.


The trapping probability depends


These include the drag rate, the orbital elements of the particles,


the strength of the resonance, and the geometry of how a particle approaches the


resonance.


This trapping in mean motion resonance will produce some interesting


observational consequences (e.g. Jackson and Zook,


1989


, Reach, 1991, Dermott et


1993).


Here we will describe qualitatively the behavior of particles when they


approach the Earth based on our numerical results. Similar phenomena happen near

Venus and Mars, but to a lesser extent.

The orbital evolution of particles being trapped in a mean motion resonance with


a planet is, actually, very easy to understand.


When a particle is spiralling towards the


Sun, the drag force takes away its energy and angular momentum.


When a particle


10 fronnaA in a oa nnai n no anno ono m Crrn-n *tTfl annnnnnn fal nn,.ndaa *n i *n 4










loss due to drag force.


Whether a particle will be trapped or not depends on how


much energy it gains when it passes through a given resonance zone.


If it does


not obtain enough energy, it simply passes right through (with some minor changes


in its orbital elements).


Otherwise it will get trapped.


However, this particle will


not be trapped forever.


Eventually it will have a series of close encounters or a


very deep close encounter with the planet.


particle from


Then, the close encounter removes this


the resonance.


In our numerical integration,


diameters equal to 4, 9, and 14 pm.

than that for the big particles. Hard


we have studied the evolution of particles


The drag rate for small dust particles is larger


ly any of the 4 pm dust particles are trapped in


any resonances. A few of the 9 pm dust particles are trapped.

show a wave of 9 pm Eos particles moving towards the Earth.


show the semimajor axes vs.


In Figure 3.29 we

In Figure 3.30 we


time for 50 of them when they approach the Earth.


Some of them do get trapped,


but the overall


trapping in resonance is not very


significant.


14 pm dust particles, the trapping percentage is higher because of


their lower drag rate.




























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CHAPTER


ORBITAL EVOLUTION OF COMETARY PARTICLES


major


difficulty


studying the


dynamical


evolution


cometary


particles

sources.


and their contribution to the zodiacal cloud is that of identifying their

Unlike asteroidal dust particles, which are mostly from the main asteroid


belt, cometary dust particles have no well-defined parent bodies.


We do not know


whether they are from a highly active comet, a group of comets, or all short period


comets.


This makes the attempt to find the contribution of cometary particles to the


zodiacal cloud seem impossible.


Nevertheless,


we feel that, even if we don't know


the exact source of cometary dust particles, by studying the characteristics of the


model zodiacal cloud produced from a typical cometary source,


we will gain some


insight into the overall problem.


The typical cometary source


we choose is comet Encke.


In this chapter we


study the orbital evolution of Encke-type dust particles numerically and find their


distribution in the solar system.


We also show that comets cannot be the source of


the solar system dust bands observed by IRAS.


Orbital Evolution of Encke-type Dust Particles


Encke


has long been


proposed to be the major source of the zodiacal cloud


*^ I 4 4 S S S C C S S C C C C


xI


4 ^\ /^j-rv








78

against the gravitational perturbation of Jupiter (Opik, 1963) and because of its strong

correlation with several meter showers and the so-called Taurid Complex (e.g. Steel,


et al,


1991a,b, Hartung,


1993).


It is thus natural to study the orbital evolution of


Encke-type dust particles


when there are no other better choices.


Our definition


for Encke-type dust particles is those with initial semimajor axes, eccentricities and


inclinations similar to those of Encke.


The mean semimajor axis, eccentricity, and


inclination for Encke are 2.2 AU


0.85, and 12, respectively.


The initial longitudes


of ascending nodes, longitudes of pericenters, and mean longitudes of dust particles


are randomly chosen


between 00


and 360


. This means these micron-sized dust


particles are not released by means of direct sublimation from the parent body when


it is near its pericenter.


The PR drag lifetime for


Encke-type micron-sized dust


particles is about 5,000 years.


If they are produced from a comet at its pericenter,


they will not have time to process and spread out their orbits to produce the axial


symmetry structure of the zodiacal cloud.


We assume that it is the disruption of


large Encke-type cometary meteoroids (large enough such that their longitudes of


ascending nodes and longitudes of pericenters have been dispersed between 00


3600) that produces these dust particles.


Griin et al.


(1985a,b) concluded that this is


the most efficient way to deposit cometary dust particles into the zodiacal cloud.


We first set up the initial condition for five hundred 9 pm diameter dust particles


by the


"particles in a circle"


method (Dermott et al.,


1985,


1992).


We start the


integration at five thousand years before 1983. Dust particles are then integrated for








79

variations in the (h,k) and (p,q) spaces when the particles are spiralling towards the


Sun are shown in Figures 4.1 and 4.2.

panels is six hundred years. The first


The time difference between two consecutive


: panel is the distribution of the parent bodies


at time equal to zero.


The major difference between Figure 4.2 and Figure 3.18 is that cometary dust

particles do not remain in a well-defined circle in inclination space right after they are

released. If we examine the inclination of each individual particle, we will find that

it goes through huge variations as the particle moves towards the Sun. As a whole,

cometary particles form a doughnut-like distribution with well-defined maxima and


minima in the inclination space.

of cometary type dust particles.


These features are the most important characteristics

They may play crucial roles in combining with the


asteroidal-type dust particles to form the zodiacal cloud.


detail in Chapter 5.


This will be discussed in


The physical interpretation for these inclination variations is


discussed in the next section.


The evolution of Encke-type dust particles in (a,e) space is shown in Figure


These five groups correspond to data from one thousand (large filled circles)


to five thousand (crosses) years after they were released. Very few particles are

trapped in mean motion resonances with Jupiter or the Earth. This is due to their


high eccentricities which give them much higher drag rates than asteroidal particles


with the same pf and thus very much lower probabilities of capture.


We demonstrate,










particle.


The drag rate of the dust particle is given by (see Equation 3.13)


const.(2 + 3e2)


const.
a


(4.1)


where a and e are the semimajor axis and eccentricity of the dust particle, respec-


tively.


With the same starting semimajor axis,


the drag rate of the dust particle


increases dramatically towards the high eccentricity range. For example, da/dt of an


eccentricity 0.85 particle is nearly


particle.


14 times larger than that of an eccentricity 0.1


Comparing Encke-type dust particles with typical asteroidal dust particles,


we notice that this high drag rate makes them move much faster through the mean

motion resonance zone in such a way that they do not have time to gain enough

energy from the resonance to counterbalance the loss due to PR drag orbital decay.


Gauss


Equations


The reason why Encke-type dust particles go through huge variations in their


inclinations can be understood from Gauss'


equations.


These equations provide a


better intuitive understanding of the gravitation perturbations from planets (see, for


example,


Danby,


1988) than the Lagrange equations.


The rate of


change of the


inclination of the particle is given by


- 1Wcos(w + f)
na2-vTC-


where r is the heliocentric distance of the dust particle,


(4.2)


w and f are the argument


pericenter and


anomaly


the dust particle,


respectively.


W is the


--e2










of the particle.


When we consider the long-term average perturbation, it is obvious


that as the eccentricity of a dust particle'


W increase.


orbit increases, both the factor 7 and


This will lead to an increase not only in the rate of change of inclination


but also in the maximum and minimum inclinations that particle can reach. Particles

starting with the same orbital elements will reach the same maxima and minima in

their inclinations.




Can Comets be the Source of the Solar System Dust Bands?


IRAS discovered several dust


bands during its all-sky survey in


1983.


central band (later it was found that there are actually two central bands) and the 100


band are very clear in all four wavelengths.


Several theories have been proposed


to explain their origins.


Dermott et al.


(1984,


1985) first pointed out the possible


correlation between these bands and three major Hirayama asteroidal families and

later confirmed it from their dynamical study and modeling results (Dermott et al.,


1990, 1992,


1993).


The central bands are probably due to the


Themis and Koronis


families while the Eos family may be responsible for the


100 band.


However, the


possibility still exists that some comets may produce these bands (Dermott et al.,


1984, Clube and Asher, 1990).


Among all the possible candidates, Encke has been


proposed to be the source of the


100 band.


This is primarily due to the fact that


Encke's


mean inclination is very close to


10 degrees.


Here we will discuss, from










Before we show why there is no cometary origin of dust bands,


we must first


understand how to form a dust band.


In the


(p,q)


phase


space,


the osculating inclination


a dust particle


is the


vectorial


sum


two components,


forced and


proper components.


proper inclinations of asteroidal dust particles remain unchanged during their orbital


evolutions


Figure


3.18,


a key


feature


to form


a dust


band)


while


their proper nodes process about a point in inclination space defined by the forced


inclination and node.


From the geometric point of view, this means that the orbits


of asteroidal dust particles process about the axis of a common plane, the plane


of symmetry,


which is defined by the forced inclination and node.


Dust particles


produced from the same family will form a doughnut-like structure if the family

members are old enough such that their nodes have been spread all over the sky.

This doughnut-like structure is tilted to the ecliptic with the inclination and node


defined by the forced inclination and node.


When we observe this structure from


the Earth, we will see two maxima in flux at the two edges of the doughnut at any


given ecliptic longitude.


This is the so-called "dust band"


. These maxima still exist


even when we consider the fact that dust particles are spiralling towards the Sun and


fill up the space between their origin and the Sun (see Figure 3.22).


With this band-


forming mechanism in mind, we can now examine the orbital evolution of cometary

dust particles and see if they are capable of forming a dust band.


As we mentioned previously,


while asteroidal dust particles remain in a well-










variations in their inclinations.


From the geometric point of view, this means there


will be no well-defined doughnut like structure, hence no well-defined edges.


Thus,


it is impossible for cometary particles to form a dust band.

Another observational consequence from this huge variation in inclination is that


cometary dust particles will contribute more (relatively speaking,


when compared


with asteroidal dust particles)


to the observed flux at high ecliptic


latitudes.


Chapter 5,


we will show that this may be the key feature needed to account for the


observed shape of the zodiacal cloud.


Distribution of Encke-type Dust Particles in the Solar System


In order to find the contribution of Encke-type dust particles


to the zodiacal


cloud we must have an idea of the distribution of these particles in the solar system


at the time of the IRAS observations.


We need the distributions of all the orbital


elements.


We have constructed two models, to be described in this section.


The only


difference between these two models is in their initial inclinations. Tl

used to obtain the dust distributions are similar to those in Chapter


ie procedures


We send


out waves of particles once every


300 years starting from


year 5,400 years


before


1983.


Most of the particles


being released before the


year (1983-5,400)


will have semimajor axes less thai

contributions to the IRAS flux data

orbits are integrated numerically.


AU by the year


1983 and thus make no


Each wave contains 500 dust particles.


All the planetary perturbations from


Their


7 planets,









the numerical integration.


The 3 for the dust particles is assumed to be 0.05037


The corpuscular drag is assumed to be 35 % that of PR drag.


The initial semimajor


axes and proper eccentricities of particles are 2.2 AU and 0.85, respectively. In the

first model, the initial proper inclinations of all the dust particles are set equal to


Encke's mean inclination, 120.


In the second model, the initial proper inclinations


of the particles are generated according to Encke's


past.


variation in inclination in the


The longitudes of ascending node, the longitudes of pericenter, and the mean


longitudes of the dust particles are generated randomly between 0


and 3600


. The


forced elements at 2.2 AU are calculated according to the orbital elements of the

planets at the time of particle release. All the calculations are performed on an IBM


ES/9000.


We exclude particles when their semimajor axes become less than 0.1 AU


or when their orbits become unbounded. For each integration, all six orbital elements


(a, e, I, fl,


w, A) of the dust particles are recorded every


100 years until the end of


the integration.


Finally we combine the results from the end of each integration to


get the orbital elements of particles at different heliocentric distances in 1983.

The final distributions of dust particles from the second model (nearly 8,000


of them) in (a,e),


(a,I), and (f2,w) phase spaces are shown in Figures 4.5 to 4.7


Few particles are found to be trapped in mean motion resonances with Jupiter or

the Earth, as we expected.

Figure 4.6 shows the distribution in inclination as a function of semimajor axis


in 1983. Particles are distributed quite uniformly between 20 and 350


Particles with


1*;-U ~ ..11 ;*,^1:-./ ..,-, ,;1 LL, t!,t -_l 1*- -^ _~( -^ l ^ *- ^ i -