The Evolution of Long-Period Comets

Home , Hills cloud

The Evolution of LongPerio d Comets

Paul Arnold Wiegert

A Thesis submitted in conformance with the requirements

for the Degree of Do ctor of Philosophy

Graduate Department of Astronomy

University of Toronto

ABSTRACT OF THE DISSERTATION

The Evolution of LongPerio d Comets

Paul Arnold Wiegert

Do ctor of Philosophy in Astronomy

University of Toronto

Toronto Canada

The observed distribution of longp erio d yr comet orbits has proved dicult

to reconcile with theory Among the discrepancies is the fading problem the fraction of

comets in the observed sample which are presumed to have made more than one p erihelion

passage since leaving the Oort cloud is much smaller than that predicted by simple dy

namical mo dels of the Solar System This may indicate that the lifetime of the longp erio d

comets is signicantly shorter than exp ected from purely dynamical considerations This

in turn p oints to the imp ortance of comet losses through volatile depletion

We examine the evolution of longp erio d comets through a direct numerical integration

a more realistic approach than the Monte Carlo metho ds previously used to study this

problem Our mo del follows the individual tra jectories of thousands of comets from the

Oort cloud to their nal demise The comets evolve within a mo del solar system consisting

of the Sun the four giant planets and the Galactic tide and to which nongravitational

forces and a solar companion ob ject or circumsolar disk may b e added We also consider

the eects of the heliopause solar wind and radiation pressure and drag on the nucleus

None of these inuences are capable of pro ducing a distribution of longp erio d comet orbits

matching observations In particular the comets dynamical lifetimes are to o long

We also investigate the eects of fading ie the reduction of comet brightness over time

due to volatile loss which may lead to a shortening of comets observable lifetimes A

numb er of simple fading laws are explored One in which the fraction of comets remaining

observable go es like m where m is the apparition numb er provides a reasonable

match with observations and may imply a dierential p owerlaw mass distribution dN

M dM A twop opulation mo del in which approximately of comets live for only a

short time orbits and the remainder indenitely also matches observations reasonably

well and could b e explained physically by a division of the Oort cloud p opulation on the

basis of their internal cohesiveness into fragile and robust ob jects ii

ACKNOWLEDGEMENTS

I would like to thank my sup ervisor Scott Tremaine for his help and patience

my parents and Kathy for their unagging supp ort and understanding and all

my fellow students at the University of Toronto Astronomy department for their

friendship over the years I also gratefully acknowledge the nancial supp ort of

the National Science and Engineering Council of Canada the Ontario govern

ment and the University of Toronto iii

Contents

Intro duction

The nucleus

The gas coma

The dust coma

The tail

Jets and streamers

Observing longp erio d comets

Research goals

Observations

The catalogue of cometary orbits

Orbital elements uncertainties

Comet families

Orbital elements

Semima jor axis

Perihelion distance

Inclination

Longitude of the ascending no de

Argument of p erihelion

Aphelion directions

Summary

Dynamics

The planets

Energy iv

The Gamblers Ruin problem

Distant planetary encounters

Angular momentum

The loss cylinder

Planet X

The Galactic tidal eld

The Galactic reference frame

Nongravitational forces

Passing stars

Energy

Angular momentum

Comet showers

Molecular clouds

A massive circumsolar disk

Miscellaneous p erturbations

Radiation pressure and the solar wind

Drag

Comet lifetimes

The Oort cloud

Problems in longp erio d comet dynamics

The fading problem

The ratio of prograde to retrograde comets

The clustering of aphelion directions

The source of shortp erio d comets

Hyp erb olic comets

The present state of the eld

Algorithm

Comparison with observations

Numerics

The integration algorithm

Regularisation v

Error tolerances

Random numb ers

Chaos

Time requirements

Planetary encounters

Initial conditions

The entrance surface

The ux of comets into the entrance surface

The endstates of comets

Mo del implementation and testing

Integration tolerance

The twob o dy problem

The planets

The Galactic tide

Nongravitational forces

Massive circumsolar disk

Results

Original elements

The newly visible comets

The longestlived comets

Dynamically evolved longp erio d comets

Element distribution parameters

Evolved longp erio d comets

The current Oort cloud p opulation

The original Oort cloud p opulation

Discovery probability function

Shortp erio d comets

Planetary encounter rates

Nongravitational forces

Two simple cases

More realistic nongravitational forces vi

Discovery probability function

Other scenarios

Massive circumsolar disk

Massive solar companion

Heliopause

Fading

Determining the fading function directly

One parameter fading functions

Two parameter fading functions

Summary

Conclusions

A Celestial Mechanics

A Orbital elements

A Galactic elements

A Keplers third law

A Radius in the orbit

A Twob o dy energy

A Energy in multib o dy systems

A Gauss equations

B Error tolerances

C The Flux of LongPerio d Comets

C The ux across the entrance surface

C The ux into the visibili ty cylinder vii

Chapter

Intro duction

Hast thou neer seen the comets aming ight

The illustrious stranger passing terror sheds

On gazing nations from his ery train

Of length enormous takes his ample round

Through depths of ether coasts unnumb erd worlds

Of more than solar glory doubles wide

Heavens mighty cap e and then revisits earth

From the long travel of a thousand years

Edward Young

Night Thoughts

Comets are sources of much information ab out the origin of our Solar System They

provide insight into the physical and chemical pro cesses underlying stellar and planetary

formation b ecause they are b elieved to contain the condensed remnants of the solar nebula in

relatively unpro cessed form As well the present distribution of cometary orbital elements

may reect the dynamics of the early stages of planetary formation Comets also serve as

prob es of the interplanetary medium and the solar wind

The nucleus

At the heart of the comet is the nucleus a solid b o dy typically a few kilometers in diameter

and with a mass of kg Earth masses Inferred densities range from to

g cm Mendis suggesting a volatilerich andor p orous makeup This is reected

in the generally accepted mo del of the comet nucleus Whipples dirty snowball

which depicts the nucleus as a single solid conglomerate of refractory eg silicates and

CHAPTER INTRODUCTION

volatile eg H O CO CO materials Interplanetary prob es sent to meet comet PHalley

during its p erihelion passage returned pictures of the nucleus which conrmed it was

a single solid ob ject and was releasing b oth gas and dust AHearn The released

material is inuenced by the solar wind the interplanetary magnetic eld and the Suns

gravity to form the coma and tail asso ciated with cometary apparitions

The gas coma

The nucleus b ecomes increasingly heated by sunlight if it approaches the Sun The comets

volatiles b egin to sublimate dragging solid particles along with them This mixture of gas

and dust is called the coma the comets bright fuzzy head A comet typically develops a

coma or b ecomes active at a cometSun distance r b etween and AU though signicant

outgassing from more distant b o dies has b een observed For example the minor planet

Chiron which never approaches closer to the Sun than AU has b een observed b oth

with and without an attendant gas cloud Meech and Belton Thus the distinction

b etween comets and asteroids the latter traditionally characterised by a complete lack of

coma and outgassing may b e to some degree articial

Solid H O sublimates appreciably in interplanetary space at r AU Delsemme

Spinrad in the region where coma pro duction typically b egins and p ointing

to H O as a p ossible constituent of the nucleus This hyp othesis is supp orted by sp ectro

scopic evidence including the detection of water and its photolysis pro ducts eg OH H

H O H O in the coma In fact it is estimated that as much as by mass of the

comas gas phase is derived from H O Festou et al b The detection of comae

at distances signicantly b eyond AU may b e attributable to p o ckets of solid CO in the

nucleus This molecules lower vap our pressure allows it to sublimate up to AU from the

Sun Delsemme The presence of CO in the nucleus has b een inferred from the sp ec

troscopic detection of it in the coma and tail though photolysis remains a p ossible source

One of its ions CO dominates the visible emission of the comets gas tail Other less

abundant volatiles that are seen directly or inferred to exist from their photolysis pro ducts

include NH CN CO S CH and N among others Mendis

The prex P indicates a p erio dic comet dened to have an orbital p erio d of less than years or to

have conrmed observations at more than one apparition and C indicates a comet which is not p erio dic

in the ab ove sense Minor Planet Circulars

CHAPTER INTRODUCTION

The coma can b e divided into three concentric overlapping layers Whipple and Huebner

The innermost layer is the molecular or inner coma Its size is determined by the

sublimating molecules lifetimes against photodisso ciation in the solar radiation

eld Jackson calculated at AU for the more abundant cometary volatiles

water is fairly typical with s The neutral coma gases expand away from

the nucleus at roughly constant velo city v km s The resulting size of the

molecular coma v is km consistent with observations A typical gas pro duction

rate Q of s AHearn and Festou yields a mass ux of kg s and

a mean numb er density of cm if we assume that the mean molecular mass of

the coma is that of a water molecule

Outside the molecular coma is the radical coma where the comp osition of the

outowing gas b ecomes dominated by radicals molecular fragments pro duced from

their parents by photodisso ciation This region is also called the visible coma and

pro duces prominent uorescence lines including those of CN OH NH C C and

NH AHearn and Festou Festou et al b The OH radical has a lifetime

s at AU Whipple and Huebner The theoretical radius of the

radical coma is thus roughly km consistent with the typical observed size of a

few times km

The exosphere is also called the hydrogen coma b ecause it is visible primarily in

Lyman emission This region extends out into the interplanetary medium ending

where the coma gases are swept away by the solar wind and radiation pressure

A neutral groundstate hydrogen atom of mass m has an absorption crosssection

dominated by the Lyman transition The acceleration v imparted to the molecule

by radiation pressure is thus

v L F L m a

p H

where F L is the momentum ux of the radiation eld in the Lyman line The

absorption crosssection of hydrogen in L L is given by e f m c eg Spitzer

where f is the Lyman transitions oscillator strength The mo

mentum ux is related to the energy ux F through F F c Equation a can

E p E

CHAPTER INTRODUCTION

thus b e rewritten

e f R

v r F L R b

m r c

m s c

where R is the radius of the Sun and F L R is the Lyman energy ux at

its surface approximately erg cm s Noyes and Avrett The

distance D from the nucleus at which the radiationinduced change in velo city is

comparable to the gases initial velo city D v v constitutes the outer b oundary

of the exosphere D The hydrogen atoms having absorb ed kinetic energy during

exo

the photobreakup of their parent molecules are now travelling with typical velo cities

of km s AHearn and Festou putting the edge of the exosphere at

D km from the nucleus This simple theory is consistent with observations

exo

Lyman emission has b een detected out to a few tens of millions of kilometers from

some comets Whipple and Huebner

The molecules mean free paths are less than their distance from the nucleus inside the

collisional radius of the coma D which denes the b oundary b etween hydro dynamic

coll

and collisionless ow The neutral coma gases are thought to expand freely away from the

nucleus thus their density n go es as Qv D ignoring disso ciation which will add a factor

of The collisional radius such that nD D where is now the collisional

coll coll

crosssection implying

coll

Q D v v

coll

A typical value for is cm AHearn and Festou from which a collisional

radius of a few times km can b e deduced putting the collisional radius inside the visible

coma

The total mass of the gaseous coma M is roughly QmD v where m is the mean

exo

molecular mass of the coma constituents taken to b e that of a water molecule When

AU from the Sun the comas total mass M g negligible next to that of the

nucleus

CHAPTER INTRODUCTION

The dust coma

An active comet also pro duces a dust coma consisting of submicron to centimetersized

solid particles ero ded from the nucleus This dust is dragged along by the expanding

gases decoupling from the gaseous coma at ab out km The dusts dynamics are then

dominated by solar gravity with radiation pressure and the PoyntingRob ertson eect also

playing some role for the smaller comp onents The dust coma may have a radius of km

at r AU Grun and Jessb erger

The dust grains may consist of solid H O or other volatiles which continue to sublimate

or refractory materials which are mo died only slowly eg by solar wind and cosmicray

sputtering The dusttogas mass ratio of comets is dicult to determine dep ending

critically on the numb er of large cmsized particles but is estimated to b e of order unity

Grun and Jessb erger Thus the mass of the dust coma is also small compared to

that of the nucleus

The tail

The ow of gas within the coma is complicated by the solar wind and the interplanetary

magnetic eld A b ow sho ck forms ahead of the nucleus near the p oint where solar and

cometary mass ows M and M balance each other Whipple and Huebner

sw comet

Given that

M Qm a

comet

M D n m v b

sw sw sw sw

where n m and v are the numb er density molecular mass and velo city of the solar

sw sw sw

winds constituents resp ectively then the b ow sho ck is exp ected near

bow

n v m

sw sw sw

At the Earths orbit n cm v cm s and m m g

sw sw sw pr oton

Lang implying that the b ow sho ck is approximately half a million km ahead of the

nucleus in accord with more sophisticated calculations and spacecraft observations Galeev

et al

CHAPTER INTRODUCTION

In Alfven theorised that interplanetary magnetic eld lines would drap e them

selves over the cometary ionosphere a prediction which has b een conrmed by spacecraft

measurements of PGiacobiniZinner and PHalley This draping arises b ecause the mag

netic eld lines are frozen in the solar plasma The b oundary b etween the solar and

cometary plasmas is called the discontinuity surface or cometopause The details of

the comet ionosphere are to o complex to treat here see Festou et al b for a review

but one result of the ionospheric structures and magnetic eld is to deect cometary plasma

into a gas tail p ointing in the antisunward direction This structure also called a plasma

or typ e I tail is visible in the sp ectral lines of its ions primarily CO with contributions

from H O N CO CH and OH Though not all comets develop detectable gas tails

Antrack et al emission from CO has b een detected over km AU from

the nucleus in the tails of the most sp ectacular comets Brandt Saito Gas

tails may b e km wide with CO densities reaching to cm Brandt

At the surface of the nucleus the solar gravitational acceleration exceeds the comets

AU Thus dust particles once decoupled from own gravity at helio centric distances r

the gas orbit the Sun indep endently of the nucleus with those particles of small micron

or less size b eing strongly inuenced by radiation pressure The dust that comets shed

creates the dust coma and the dust or typ e I I tail Visible in scattered sunlight this tail

is typically curved and shorter than the gas tail though dust has b een detected up to

km from the nucleus Brandt Comets generally show b oth typ e I and typ e I I tails

though comets which have displayed only one or neither are known

Jets and streamers

In general the nucleus will b e aspherical and inhomogeneous and the sublimation of

volatiles will b e nonuniform Evidence for asymmetric outgassing includes dust jets and

streamers fountainlike structures commonly visible in the coma and indicative of strong

lo calised dustgas release Images of PHalley taken by the Giotto spacecraft eg Keller

reveal a highly irregular distribution of active regions across the comets surface

Sublimation is thus likely to result in a net reaction force commonly termed the non

gravitational NG force which contributes to the comets dynamical evolution x

CHAPTER INTRODUCTION

Observing longp erio d comets

of a comet nucleus is very low The ESA Giotto spacecraft The visual geometric alb edo

measured a value of to for for PHalley Mendis For comparison Ctyp e

asteroids o ccasionally have as low as though some Etyp e asteroids have alb edos

as high as Morrison The planets have surfaceaveraged alb edos ranging from

Mercury the Mo on to Venus with their satellites reaching greater extremes

as low as to for Jupiter V and VI Amalthea Himalia with Saturn I I I Tethys

reaching Weast et al

The very low alb edo of the nucleus makes it dicult to observe comets b efore they

b ecome active A comet nucleus has an apparent visual magnitude m given by

m m log F F

where m is the Suns apparent visual magnitude and F and F are the visual

uxes received at the Earth from the Sun and the comet resp ectively The ux received

from the comet is the reected ux attenuated by the inversesquare law

r r R F

F R F

D r D r

where R is the radius of the nucleus and r r and D are the Suncomet SunEarth and

the Earthcomet distances resp ectively Substituting Equation into Equation and

taking D r yields

m m log

R r

A large bare comet nucleus R km at Saturns orbit r AU

c V

thus has a visual magnitude of This value increases to if the comet is moved

to AU The Hubble Space Telescop e WFPC camera can reach magnitudes of

to in the V and I bands with long exp osures eg Groth et al and provides

the practical observational limit for the nearfuture Thus a comet is almost undetectable

with present technology unless it approaches the Sun closely enough to develop a coma It

should b e noted however that larger b o dies km p ossibly cometary in nature but

The geometric alb edo is dened as the ratio of the ux received to that exp ected from a p erfectly

reecting p erfectly diusing disk of the same radius and distance measured at zero phase angle Hopkins

CHAPTER INTRODUCTION

lacking comae have b een detected by the Hubble Space Telescop e around AU from the

Sun Co chran et al

After coma pro duction has b egun the comets brightness increases rapidly The visual

magnitude m of active comets is traditionally describ ed by the equation

m H log D n log r

where D and r are the Earthcomet and Suncomet distances in AU The parameter n

which usually ranges b etween and describ es the comets increase in brightness with r

The value of n is generally smaller for longp erio d comets than shortp erio d ones the latter

tending to have brightness proles which vary more strongly with r H is the comets

absolute magnitude dened to b e its apparent magnitude were it to b e placed AU

from b oth the Sun and Earth The observed distribution of H p eaks at The intrinsic

distribution however is exp ected to increase monotonically through values of or more

though the faint end of the luminosity function is p o orly known Everhart b

Research goals

The goal of this research is to test our current understanding of the dynamical evolution

of longp erio d comets against the observed distribution of their orbits Limited investiga

tions along these lines have prevously b een done eg Weissman revealing signicant

discrepancies b etween the exp ected and observed orbital distributions But until recently

restrictions in computing sp eed have prevented the numerical integration of a large ensem

ble of Oort cloud comets thus it has b een unclear whether the gap b etween theory and

observations is the result of oversimplications in the mo dels used to predict the comets

distribution or a real gap in our understanding of the Solar System

Here the results of the rst large scale numerical integration of longp erio d comets are

presented In Chapter the observed sample of longp erio d comets is discussed The

distributions of orbital elements is used to supp ort the hyp othesis that the Solar System

is likely surrounded by a spherical cloud of comets the Oort cloud and that the tidal

eld of the Galaxy is an imp ortant mechanism for p erturbing comets in such a cloud onto

orbits which pass through the inner Solar System In Chapter the dynamics of long

p erio d comets are detailed and the imp ortance of the Galactic tide and the giant planets

is demonstrated from theory In Chapter the algorithm used here to simulate comet

CHAPTER INTRODUCTION

tra jectories is describ ed including testing and error control In Chapter the results of

the simulations are detailed and the gap b etween theory and observations discussed along

with an examination of p ossible reasons b ehind the mismatch In Chapter conclusions

are presented and an overview of future research p ossibilities is outlined

Chapter

Observations

The catalogue of cometary orbits

Marsden and Williams Catalogue of Cometary Orbits lists apparitions of

individual comets observed b etween BC and AD though with p o or complete

ness at early times This compilation includes where p ossible the comets osculating or

instantaneous elements with resp ect to the FK J system The orbital elements of

comets are traditionally quoted at an osculation ep o ch at or near p erihelion but if the

comets aphelion distance is large the elements of the orbit on which the comet approached

the planetary system called the original elements are also of interest These are likely to

b e dierent from those measured at p erihelion b ecause of the gauntlet of planetary p ertur

bations the comets must run In this context original will mean corrected for planetary

p erturbations during its most recent passage through the planetary system The original

elements can b e calculated from the orbit determined near p erihelion by integrating the

comets tra jectory backwards until well outside the planetary system and are tradition

ally quoted in the centre of mass frame Marsden and Williams include such a list for

those comets with large aphelia for which orbits of sucient accuracy are known This list

contains a total of ob jects observed b etween and AD

Orbital elements uncertainties

Marsden and Williams do not provide error estimates for elements in their catalogue but do

sub divide the orbits into classes IA IB I IA and I IB in descending order of accuracy These

CHAPTER OBSERVATIONS

classes are based on the estimated error in the determination of orbital energy the time

span during which the comet was observed and the numb er of planets whose p erturbations

were taken into account These classes are describ ed in more detail in Marsden et al

The distribution of the comets among these orbits is and resp ectively

Comet families

Comets can b e group ed usefully on the basis of their orbital p erio ds the divisions of Carusi

and Valsecchi will b e used here though there are others in the literature Figure

plots the values of the semima jor axis a versus the cosine of the ecliptic inclination i for

all comet apparitions Note that a statistically uniform distribution of angular momentum

vectors up on the celestial sphere called a spherically symmetric or SS distribution will

have a at distribution in cos i The division of comets into families is based largely on the

clustering seen in this plot

Shortp erio d comets

The shortp erio d or SP comets are those on orbits with p erio ds less than years

A subset of this class the Jupiter family is comprised of those comets with less than

years The designation Jupiterfamily arises from the clustering of their aphelion

distances Q around Jupiters orbit as shown in Figure and the consequent domination of

their dynamics by this giant planet Marsden and Williams catalogue records p erihelion

passages by memb ers of the Jupiter family all on prograde orbits lying near the ecliptic

Largely b ecause of their low inclinations these ob jects are b elieved to have b een transferred

relatively recently into the planetary system from a ring of material b eyond Neptune known

as the Kuip er b elt x

Also counted among the shortp erio d comets are the Halleytyp e yr

yr comets which have a wider distribution of inclinations Figure Over of the

apparitions of Halley family comets listed in Marsden and Williams have retrograde

orbits though PHalley yr i itself contributes apparitions dating back

to BC The upp er b oundary of the Halley family corresp onds through Keplers third

The orbital elements used here along with some celestial mechanics results imp ortant to this pro ject are outlined in App endix A

CHAPTER OBSERVATIONS

Figure The cosine of the ecliptic orbital inclination i plotted against inverse semima jor

axis a for all observed comet apparitions The two vertical lines indicate the family

b oundaries at orbital p erio ds of and years Data taken from Marsden and Williams

Figure Aphelion distance Q versus the cosine of the ecliptic orbital inclination i for

the Jupiter family comets The horizontal lines mark the semima jor axes of Jupiter and

Saturns orbits Data taken from Marsden and Williams

CHAPTER OBSERVATIONS

law to a semima jor axis a AU and thus the shortp erio dlongp erio d b oundary pro

vides a useful distinction b etween comets whose aphelia lie within or close to the planetary

system and those that venture signicantly b eyond

Longp erio d comets

The longp erio d or LP comets have p erio ds exceeding years and their orbits extend

outside those of the giant planets These comets typically have p erio ds of tens of millions

of years and semima jor axes of tens of thousands of astronomical units AU Figure

reveals that LP comets are not conned to the ecliptic plane These facts suggest that the

LP comets are at a dierent stage of dynamical evolution than the SP comets or as is

thought more likely are a dynamically dierent p opulation from the SP comets In any

case the LP comets will b e the fo cus of our interest here

Orbital elements

Semima jor axis

The orbital energy E p er unit mass of a b ound Keplerian orbit is simply GM M a

where a is measured in the centre of mass frame and M and M are the two b o dies

masses For a test particle orbiting the Sun this expression reduces to GM a These

expressions are not strictly valid in a multib o dy system but nevertheless provide a useful

measure of a comets binding energy For simplicity the inverse semima jor axis a is

used here as a measure of the comets orbital energy diering from the Keplerian energy

only by a simple constant factor see App endix A

The b oundary b etween SP and LP comets is at a yr AU

Figure displays histograms of a for the LP comets with known original orbits

at two dierent magnications

From Figure b it is clear that relatively large numb ers of comets travel on orbits with

yr By way of comparison Plutos semima jor axis is only AU AU a

yr Also notable is a lack of strongly hyp erb olic original orbits Comets entering

the Solar System from interstellar space would b e exp ected to have velo cities comparable to

Unless otherwise stated the error bars on histograms are standard deviation assuming Poissonian

statistics N

CHAPTER OBSERVATIONS

150

100

0 -0.01 0 0.01 0.02 0.03

1/a (1/AU)

a b

Figure Distribution of original inverse semima jor axes of longp erio d comets at two

dierent magnications Data taken from Marsden and Williams

the velo city disp ersion of disk stars roughly km s Mihalas and Binney This

velo city is equivalent to an inverse semima jor axis of approximately AU imp ossible

to reconcile with the most hyp erb olic original orbit observed CSato I which had

a AU The few weakly hyp erb olic orbits in Figure may b e due to

observational error or the inuence of nongravitational forces x The sharp p eak in

the a distribution was interpreted by Oort as evidence for a p opulation of comets

orbiting the Sun at large a AU distances a p opulation which has come to b e

known as the Oort cloud

It is useful to consider here the distribution of original energies of comets with p erihelia

inside AU for the purp oses of comparison with later results These distributions shown

in Figure are similar to those in Figure but the spike is not as high due to a

tendency for Oort cloud comets to b e brighter than other comets and thus visible at larger

distances

Perihelion distance

A histogram of the numb er N of LP comets versus p erihelion distance q is shown in Fig

ure There is a strong p eak near AU due to observational biases comets app ear

brighter when nearer b oth the Sun and the Earth Everhart b concluded that the

CHAPTER OBSERVATIONS

100

0 0 -0.0004 -0.0002 0 0.0002 0.0004 -0.01 0 0.01 0.02 0.03

1/a (1/AU) 1/a (1/AU)

a b

Figure Distribution of original inverse semima jor axes of longp erio d comets with

p erihelion distances less than AU at two dierent magnications Data taken from Mars

den and Williams

intrinsic distribution ie the distribution which includes all LP comets observed and un

observed has a slop e dN dq q inside the Earths orbit but that the distribution

at larger distances is p o orly constrained probably lying b etween a at prole and one in

0 0 2 4 6

q (AU)

a b

Figure Numb er N versus p erihelion distance q for longp erio d comets Data taken

from Marsden and Williams

CHAPTER OBSERVATIONS

creasing linearly with p erihelion distance Kresak and Pittich also found the intrinsic

distribution of q to b e largely indeterminate at q AU but consistent with dN dq q

over the range q AU

There are two estimates in the literature of the numb ers of comets which pass unobserved

through the inner Solar System Everhart estimates that only of all comets approaching

the Sun to within AU are observed Kresak and Pittich estimate are observed at

q AU dropping to only at q AU Though not directly comparable these

estimates are roughly consistent in that they indicate that a large fraction of comets passing

near the Sun likely go unnoticed It will b e assumed here that p erihelion distance is not

strongly correlated with the comets semima jor axis or angular elements and thus that any

selection eects acting on q do not aect the observed distributions of the other elements

Figure The cumulative probability distribution as a function of p erihelion distance q

for the shortp erio d comets and for the longp erio d comets with computed original orbits

Data taken from Marsden and Williams

It is interesting to compare the cumulative distributions of SP and LP comets as a

function of q displayed in Figure Comets of all typ es are rarely observed if their

p erihelia are b eyond AU but those that are seen are more likely to b e LP than SP This

CHAPTER OBSERVATIONS

dierence can b e explained if the SP comets have typically undergone more apparitions

than their longp erio d counterparts and hence have smaller volatile inventories and pro duce

fainter comae The discrepancy b ecomes even more striking when one considers that there

are more chances to discover SP comets due to their more frequent returns The reduction in

cometary brightness with rep eated apparitions is imp ortant to our understanding of comet

dynamics and will b e discussed more fully in x

Inclination

Figure shows the distribution of the cosine of the LP comet inclinations For comparison

a spherically symmetric distribution is indicated by the heavy line Everhart b showed

that selection eects due to inclination should only aect the distribution at the level

well b elow the statistical noise The data matches the at line fairly well by eye the and

KolmogorovSmirnov KS tests return probabilities that the distribution is consistent with

spherical symmetry of roughly and resp ectively The distribution examines the

match at each p oint and is thus more sensitive to high frequencies in the data set than the KS

test which works with the cumulative distribution Thus a high probability of atness as

indicated by the KS test along with a low probability according to the test is consistent

with smallscale clumpiness but little or no lowfrequency signal Discrepancies b etween

KS and tests o ccur for a numb er of the distributions to follow but as their atness is

not central to the discussion strong interpretations will not b e imp osed on the and KS

results

Long p erio d comets unlike those with shorter p erio ds are not conned to the ecliptic

and are equally likely to b e on prograde or retrograde orbits The ratio of prograde to

retrograde comets is The and KS tests conict returning probabilities of

and that the ecliptic distributions are at The distribution is less at to the eye in the

Galactic frame There may b e a gap near zero inclination p ossibly due to the inuence of

the Galactic tide x or to selection eects resulting from the confusion of comets with

other ob jects in the Galactic plane

Longitude of the ascending no de

The distribution of longitudes of the ascending no des is plotted in Figure The at

line again indicates a SS distribution The two curves match fairly well consistent with

CHAPTER OBSERVATIONS

Everharts ab conclusion that there are unlikely to b e any selection eects based on

over time scales long compared to one Earth year assuming the intrinsic distribution

is azimuthally symmetric The and KS tests indicate probabilities of and

resp ectively that the observed longitudes of the ascending no des are drawn from an intrin

sically at distribution When applied to the Galactic distribution the and KS tests

yield probabilities of and that the intrinsic distributions are at again the low

value determined by the test may either b e due to noise in the sample or indicate a real

deviation of the distribution from uniformity on small scales

Argument of p erihelion

Figure shows the distribution of the arguments of p erihelion for the LP comets The

test reveals a probability less than that is drawn from a at distribution but

the KS test puts it at over Comets with less than outnumb er those with

greater than by a factor of This is probably due to an observational selection eect

Everhart a Kresak comets with pass p erihelion ab ove the ecliptic

and are more easily visible to observers in the northern hemisphere The lack of observed

apparitions with is a result of the smaller numb er of comet searchers in the southern

a b

Figure The distribution of the cosine of the inclination for the longp erio d comets in

a ecliptic co ordinates i and b Galactic co ordinates A spherically symmetric sample

is indicated by the at line Data taken from Marsden and Williams

CHAPTER OBSERVATIONS

a b

Figure The distribution of the longitude of the ascending no de of the longp erio d comets

in the a ecliptic frame and b in the Galactic frame Data taken from Marsden

and Williams

hemisphere until very recent times The distribution in the Galactic frame has a slight

excess of comets with orbits in the range sin of the total numb er and the

distribution has a probability of b eing at of less than and over according to the

and the KS test resp ectively

Aphelion directions

Figure shows the distribution of the aphelion directions of the LP comets in the ecliptic

and Galactic references frames Unfortunately Marsden and Williams do not provide

the complete set of elements for the original orbits and thus Figure was calculated

from the orbital elements at p erihelion It will b e shown that the angular elements are

typically only weakly p erturb ed during a single passage within the planetary system x

so the errors in the aphelion p ositions are likely to b e small

Claims have b een made for a clustering of aphelion directions around the solar antap ex

eg Tyror Oja but newer analyses with improved catalogues eg Lust

have shed doubt on this hyp othesis The presence of complex selection eects such as the

uneven coverage of the sky by comet searchers render dicult the task of unambiguously

determining whether or not clustering is present Attempts to avoid selection eects end up

CHAPTER OBSERVATIONS

a b

Figure The distribution of the argument of p erihelion in a the ecliptic frame and

b in the Galactic frame for the longp erio d comets Data taken from Marsden and

Williams

sub dividing the samples into subsamples of such small size as to b e of dubious statistical

value

Whipple has shown that it is unlikely that there are many large comet groups

ie comets related through having split from the same parent b o dy in the observed sample

though the numerous observed comet splittings makes the p ossibility plausible A

comet group would likely have spread somewhat in semima jor axis the resulting much

larger spread in orbital p erio d a makes it unlikely that two or more memb ers of

such a split group would have passed the Sun in the years for which go o d observational

data exist The Kreutz group of sungrazing comets is the only generallyaccepted exception

A feature of the plot of aphelion directions in the Galactic frame Figure b is their

concentration at Galactic latitudes b Figures a and b show histograms of

comet numb er versus the sine of the ecliptic latitude and the Galactic latitude b The

ecliptic latitudes deviate only weakly from a SS distribution and this deviation is likely due

to the lack of southern hemisphere comet searchers The Galactic distribution shows two

broad p eaks centred roughly on sin b It will b e shown that this is likely due to

the inuence of the gravitational tidal eld of the Galaxy x which acts most strongly

when the Suncomet line makes a angle with the Galactic p olar axis though the gap

CHAPTER OBSERVATIONS

180 E 90 E O 90 W 180 W

a S

180 E 90 E O 90 W 180 W

Figure Longp erio d comet aphelion directions on a ecliptic and b Galactic equal

area maps The crossed circle is the solar ap ex Data taken from Marsden and Williams

CHAPTER OBSERVATIONS

30 30

20 20

10 10

0 0 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1

sin b

a b

Figure The sine of the aphelion latitudes of longp erio d comets in the ecliptic a and

Galactic b reference frames Data taken from Marsden and Williams

near b may b e a selection eect resulting from the increased diculty of sp otting

comets against the more crowded skies of the Galactic plane The weak selection eects in

the ecliptic frame are unlikely to signicantly aect the distribution in the Galactic frame

the two frames b eing tilted at a large angle to each other

Summary

The angular orbital elements in b oth the ecliptic and Galactic frame may or may not b e

consistent with a spherically symmetric distribution The test typically pro duces a low

probability of the distribution b eing uniform while the KS test which examines the cumu

lative distribution generally pro duces a much higher probability This implies that there is

high frequency noise in the sample but no strong low frequency signal However the

distribution of Galactic latitudes do es app ear to have a doublyp eaked distribution p ossibly

due to the Galaxys tidal eld

The p erihelion distribution is fraught with selection eects and only its gross features are

useful for comparison with theory at this p oint Fortunately the orbital energy distribution

has a distinctive signature It will provide the primary diagnostic when comparisons with

simulations are p erformed though the other distributions also provide useful information

Chapter

Dynamics

The equations of motion of the comet can b e written as

r F F F F F F F F F F

planets tide star s clouds disk j et r p sw dr ag

where the dierent terms on the righthand side represent the dierent accelerations to

which the comet is sub ject Considering initially the helio centric frame r is then the vector

from the Sun to the comet The rst term of Equation represents the Suns gravitational

pull

r F

where G is the gravitational constant and M is the mass of the Sun The second term

of Equation represents the gravitational inuence of the planets F and the re

planets

maining terms the accelerations due to the Galaxys tidal eld F individual close

tide

encounters with stars F and molecular clouds F a hyp othetical disk of matter

star s clouds

outside the planetary orbits F and nongravitational forces resulting from outgassing

disk

F solar radiation pressure F solar wind pressure F and drag F resp ec

j et r p sw dr ag

tively These eects will b e considered separately

The planets

The functional form of F dep ends as do all the terms on the reference frame in which

planets

it is expressed The frames of interest here are the helio centric and barycentric frames In

CHAPTER DYNAMICS

the barycentric frame F can b e expressed simply as

planets

F r

planetsbar y

where M is the planetary mass and r is the distance vector p ointing from the planet to

p pc

the comet Complications arise when considering the helio centric frame b ecause it is non

inertial the Sun orbits the Solar Systems centre of mass The additional terms needed to

account for the solar motion are called the indirect terms and serve as corrections to the

principal terms Equation when working in the helio centric frame

X X

GM GM

p p

F r r

pc p

planetshelio

r r

pc p

p p

where r is the Sunplanet radius vector

The planets may strongly inuence a comets path but the comet is not massive enough

to have a detectable eect on any of the planets a typical nucleus has a mass only

that of Pluto and only that of Jupiter

Energy

The motion of the comet in the eld of even one planet and the Sun has no analytic

solution and may b e quite complicated However if the comets aphelion is well outside the

planetary system ie it is a longp erio d comet then the planets inuence is concentrated

near p erihelion and can b e approximated for some purp oses by an instantaneous kick in

the comets orbital energy

The energy kick E and the corresp onding change in the inverse semima jor axis a

are dicult to calculate analytically eg van Wo erkom but have b een determined

from numerical exp eriments Everhart Fernandez For a single planet dimen

sional considerations show that

jE j GM r a

p p

jaj M r b

p p

where in the second equation M is in solar masses The values of M and r for the planets

p p p

are listed in Table

CHAPTER DYNAMICS

Planet M r M r M r

p p p p p

Mercury

Venus

EarthMo on

Mars

Jupiter

Saturn

Uranus

Neptune

Pluto

Table Quantities related to the mass M and semima jor axis r of the planets of the

p p

Solar System M r is indicative of the size of the energy p erturbation a comet receives

p p

p er p erihelion passage Equation a M r of the torque due to the planets orbital

quadrup ole Equation Units are M and AU Data taken from Lang

In comparison given the same conditions as ab ove simple theory predicts that the

angular orbital elements i and and the p erihelion distance q receive p erturbations

i q q M M

For a longp erio d comet with a AU and q inside Jupiters orbit aa

while the fractional change in the angular elements and p erihelion distance is only of order

Thus the energy of LP comets on high eccentricity orbits evolves on a shorter time

scale than i and q

The kicks due to each individual planet are uncorrelated so the total change in E is

given by the squarero ot of the sum of the squares of the individual kicks

X X

jE j E GM r

p p p

p p

If the comets p erihelion is inside the orbits of all the giant planets Jupiter dominates the

summation having M r over six times greater than the next largest contributor Saturn

p p

see Table column The contributions of the inner planets and Pluto together

constitute less than of Jupiters contribution Equation is constant within the

constraint q r and implies a constant

J up

AU jaj

r M

J up

p er orbit as well Of course these values are only rough estimates the actual changes

in the orbital elements b eing sensitive functions of the initial conditions Nevertheless

CHAPTER DYNAMICS

Equation provides a useful simple mo del called the diusion mo del of the evolution

of Oort cloud comets with p erihelia within the planetary system

Under the diusion mo del a nearparab olic comet which makes a series of passages

within the planetary system receives an energy kick each time The kicks are symmetrically

distributed ab out zero and are uncorrelated and identically distributed as long as the

comets orbital p erio d is long compared to that of the planets The evolution of such

a comet can thus b e approximated by a random walk in energy space with step size

given by Equation The region of energy space LP comets inhabit has two absorbing

b oundaries

At a the comet leaves the Solar System on an unb ound orbit

As a the comets orbit contracts bringing it into collision with the Sun In

reality comets do not reach such a state the diusion approximation b eing invalid

where a r Instead some upp er limit a is dened b elow which the diusion

p sp

mo del is no longer valid It is useful to take this cuto to b e the b oundary b etween

long and shortp erio d comets ie where yr corresp onding to a AU

or a AU This b oundary is not truly absorbing as there is nothing

to prevent a SP comet from evolving back into an LP comet However only a small

numb er of LP comets survive to b ecome SP Equation b and later Table

hence the p ossibility of SP comets returning to the LP domain is small and can b e

neglected

The Gamblers Ruin problem

The random walk of a LP comet under the diusion approximation is very similar to the

wellknown Gamblers Ruin problem with the endstates of ejection and b ecoming short

p erio d corresp onding to bankruptcy and breaking the house resp ectively

Consider a comet randomwalking on an integer lattice of energies Let b e the initial

numb er of steps the comet is from ejection and let b e its initial distance in steps from

the shortp erio d barrier For a typical visible Oort cloud comet and

See eg Kannan for a more complete description of the Gamblers Ruin problem

CHAPTER DYNAMICS

The probabilities p and p of the comet reaching the ejecting or shortp erio d barriers

ej sp

resp ectively are simply

p a

ej sp ej sp

p b

sp ej ej sp

If m is the numb er of orbits a comet survives b efore crossing one of the absorbing barriers

its exp ectation value m is

ej sp

However it should b e noted that the distribution of lifetimes b eing very broad as would

b e exp ected for a diusion pro cess is not wellcharacterised by Equation

In the case of no shortp erio d barrier ie the numb er N of LP comets

remaining on orbit m is given by Everhart Yabushita

N m N m

where N is the initial numb er of comets This implies a probability p of ejection at each

orbit of

p m m

Distant planetary encounters

Comets with p erihelia outside the planetary system do not have close encounters with

the planets and the resulting p erturbations are signicantly decreased Heggie

calculated the change in energy of a binary star system when approached by an interlop er

on a nearparab olic orbit His results provide a useful approximation to the situation in

question though he made the assumptions that the three b o dies were roughly equal in

mass that the interlop er was approaching on a nearparab olic orbit and that q r

among others With the Sun and Jupiter playing the role of the binary the change in their

binding energy is through conservation of energy just the energy absorb ed by the comet

From Equation of Heggies pap er the energy kick is

jE E j exp

r p

CHAPTER DYNAMICS

Though Equation was derived based on assumptions not always strictly valid in the

case of comets the conclusion that the energy p erturbation drops exp onentially as q

is certainly correct

Angular momentum

In the case of LP comets with p erihelia outside the planetary system changes in the an

gular momentum J induced by the planets are dominated by the torques resulting from

the quadrup ole moments of the timeaveraged planetary orbits These torques aect the

p erihelion distances q related to J through

J GM a e GM q where e

Approximating the planet orbits by coplanar circles the total timeaveraged quadrup ole

moment of the planets Q is the sum of the planets individual moments Q M r Ta

p p

ble column

X X

Q Q M r M AU

p p

and the asso ciated torque J on the comet is

J sin cos for q r

where is comets ecliptic latitude given by sin sin i sin f and is the ecliptic

azimuthal unit vector The rate of change of angular momentum J is related to the torque

through

J J

sin i cos f J J

j J j

The absolute change in angular momentum p er orbit jJ j assuming jJ j jJ j is given

jJ j J dt a

GQ sin cos sin i cos

dt b

q Z

k GQ

e cos f sin cos k sin d c

aJ e

CHAPTER DYNAMICS

where f and k sin i The planetary quadrup oles pro duce no net change in the

cometary p erihelion distance regardless of their relative orientation The change in angular

momentum is zero b ecause the quadrup ole p otential and hence the torque go es like r

this is not necessarily the case for p otentials with arbitrary dep endences on r

The loss cylinder

A comet with a semima jor axis greater than AU that comes close enough to the Sun

to b ecome visible is likely to receive an energy kick jE j comparable to its orbital energy

E Equation Such a relatively large kick results in the comet taking on either an

unb ound or a much more tightly b ound orbit dep ending on the sign of E In either case

the comet is no longer a memb er of the Oort cloud

The orbit of Saturn is a rough outer limit to the p erihelion distance at which an Oort

jE j Thus the region of phase space where a cloud comet typically receives jE j

AU is called the loss cylinder b ecause it is swept clear of Oort cloud AU and q

comets by the giant planets in roughly one comet orbit

The loss cylinder gets its name from its geometry in a particular threedimensional

velo city space one axis of which denotes the radial velo city v and the others the tangential

comp onents v and v with v v v Any xed orbital angular momentum

t t t

t t

J r v

corresp onds to a cylindrical surface in this space As the angular momentum is related to

p erihelion distance q through Equation the loss cylinder can b e dened equivalently

by a xed q if e The b oundary of the loss cylinder is denoted J or by the asso ciated

p erihelion distance q A similar surface called the visibility cylinder represents the range

of p erihelia for which comets pro duce comae its size will b e taken to b e AU here

The existence of the loss cylinder implies that visible comets which approach the plane

tary system on orbits with a AU are probably making their rst p erihelion passage

close to the Sun Such comets are referred to as dynamically new Dynamically new

comets in the loss cylinder must have recently had their p erihelia displaced inwards from

greater distances by some mechanism

The loss cylinder is emptied on a time scale comparable to the comets orbital p erio d

yr for a AU and must b e relled if a steadystate distribution is to

CHAPTER DYNAMICS

b e maintained If the Oort cloud is the source of new comets a mechanism must exist for

reducing their p erihelia and bringing them into the loss cylinder

A change in p erihelion distance implies a change in angular momentum If some mech

anism pro duces a change in orbital angular momentum p er orbit J which is much smaller

than J LP comets make their rst p erihelion passage inside the loss cylinder close to its

b oundary and the loss cylinder is said to b e empty In this case the comets do not b ecome

part of the observed sample They are removed from the loss cylinder b efore their p erihelia

can evolve inward suciently for comatail development Such comets are sometimes said

to encounter the Jupiter barrier b ecause they are typically removed when their p erihelia

approach Jupiters orbit

Oort cloud comets may hurdle the Jupiter barrier and b ecome visible if a mechanism

exists to pro duce

q q

which could push comets deep into and p ossibly even through the loss cylinder Under

these conditions the cylinder is said to b e full

Due to the lack of net change in angular momentum the giant planets pro duce in comets

with q r Equation d some other mechanism is required to draw in the Oort cloud

comets which are observed Though the ma jor planets may pro duce larger changes in J in

comets with p erihelia near their orbits such encounters would strongly aect the comets

energies as well The narrow spike in the observed distribution of comet inverse semima jor

axes Figure arges against the giant planets b eing the dominant injectors of Oort cloud

comets

Planet X

There is little evidence for a massive solar companion b eyond Pluto and dynamical con

siderations set an upp er limit to its mass of roughly Jupiter masses and probably much

less Tremaine Hogg et al Nevertheless the presence of such a companion

could strongly aect the evolution of longp erio d comets which may b e useful prob es of the

existence of such an ob ject and will b e discussed further in x

CHAPTER DYNAMICS

The Galactic tidal eld

The Solar System resides within an extended mass distribution namely the Galaxy This

distribution pro duces a tidal eld in our vicinity which is referred to as the Galactic tidal

eld or the Galactic tide Morris and Muller Torb ett Heisler and Tremaine

Matese and Whitman

The eect of the Galactic tide is distinct from that of individual close stellar encounters

They constitute two dierent parts of the Galaxys gravitational eld the overall smo oth

eld and the clumpy eld due to the concentration of mass into stars Heisler and

Tremaine have shown that individual stellar encounters are the source of the variance

of the changes in comet velo city while the Galactic tide is the source of the mean change

The Galactic reference frame

Consider a set of mutually p erp endicular unit vectors fe e e g with their origin at the

x y z

Sun and rotating with it ab out the Galactic centre Let e b e directed radially outward

from the Galactic centre let e b e directed tangentially to the Galaxy in the direction of

its rotation and let e b e directed towards the North Galactic Pole These vectors form

the Galactic reference frame see also App endix A

The acceleration term due to the Galactic tide in Equation has the form Heisler

and Tremaine

F A B A B x e A B ye G B A z e

tide x y z

where is the mass density in the solar neighb ourho o d and A and B are the usual Oort

constants The numerical values of the Oort constants are A km s kp c

and B km s kp c Kerr and LyndenBell The lo cal mass density

is less wellknown Observable matter stars and gas contributes ab out M p c but

the amount of dark matter present in the solar neighb ourho o d if any is controversial

Recent calculations based on dynamical arguments allow the totalobserved mass ratio P

to b e b etween and Bahcall Kuijken and Gilmore Kuijken A

recent determination by Bahcall et al nds P and a constant value for of

M p c will b e adopted here It should b e noted that is probably not constant but

mo dulated somewhat by the Suns excursions ab ove and b elow the Galactic plane during

its orbit around the Galaxy Matese et al

CHAPTER DYNAMICS

Co ordinate Momentum

f L GM a

J GM a e

J J cos

Table A set T of canonical co ordinatemomentum pairs useful for the orbitaveraged

Hamiltonian of a comet orbiting the Sun in the presence of the Galactic tide

Given the ab ove values of A B and the G term of Equation exceeds the

others by an order of magnitude This dominant comp onent of the tidal acceleration can

b e expressed as

F G ze G r sin b e

tide z z

where b is the comets Galactic latitude sin b sin sin f This dominant comp onent

is along the Galactic p olar axis and corresp onds to a gravitational p otential of the form

V G z

tide

The Hamiltonian H of a b o dy orbiting the Sun under the inuence of the Galactic tide

provides a complete description of the b o dys motion However this description is more

comprehensive than is required for some investigations If the changes in the orbit due

to the tidal p erturbation are small it is reasonable to average H over a full orbit and

consider the resulting simpler Hamiltonian H The orbitaveraged Hamiltonian provides

a useful description of the evolution of the comets orbital elements under the tide though

at a loss of short time scale t information

Following the example of Heisler and Tremaine the set T of canonical co ordinate

momentum pairs listed in Table will prove useful in the discussion of the orbitaveraged

Hamiltonian The symb ols and represent the inclination longitude of the ascending

no de and argument of p erihelion measured in the Galactic frame f is the true anomaly

which is indep endent of the reference frame The momentum J is the usual orbital angular

momentum p er unit mass of the comet and J is its comp onent along the zaxis L is a

measure of the twob o dy orbital energy through the semima jor axis a

Expressed in these canonical variables the orbitaveraged Hamiltonian H has the form

Heisler and Tremaine

h i

L GM

J J J L J sin a H

L GM J

CHAPTER DYNAMICS

which can b e expressed in terms of the standard orbital elements as

H G a sin e e sin b

The canonical variables f and are absent from Equation a so the corresp onding

momenta L and J are conserved The conservation of L implies that of a as well hence the

semima jor axis and the orbital energy are conserved under H However do es app ear in

Equation a implying that the angular momentum J and hence the p erihelion distance

q are not constants of the motion

The comets angular momentum oscillates with time with the eccentricity reaching

minimum and maximum values e If C e e sin sin is greater than then

K C K e C K

z z z

where K e j cos j If C then the limiting eccentricities are given by

C a e

K e C K b C K

z z z

Equations and ab can b e used in conjunction with the conservation of the semima

jor axis to compute the minimum and maximum p erihelion distances a comet will oscillate

b etween under the tides inuence

To determine whether or not the tide can ll the loss cylinder consider the orbit

averaged rate of change of angular momentum J which can b e obtained from H through

Hamiltons canonical equations

J a

J J L J sin b

e L sin sin c

from which it can b e deduced that

jJ j e L

Though no notational distinction is made here the orbitaveraged co ordinates and momenta in H

ie L J J and are not in general equal to their instantaneous values in the unaveraged system except

in the limit 0

CHAPTER DYNAMICS

J dtj jJ j is given by The change in angular momentum over a single orbit jJ j j

jJ j e L e L

GM GM G M

Equation can b e solved to determine the conditions under which the tide can ll the

loss cylinder ie pro duce J J These conditions are assuming e that

q M

a a

AU b

M p c

The Galactic tide thus provides a mechanism by which Oort cloud comets may b ecome

observable but only if the comets semima jor axes exceed AU

Nongravitational forces

The asymmetric sublimation of cometary volatiles results in a net acceleration of the nucleus

These nongravitational NG forces are limited to times of signicant outgassing ie

coma pro duction and remain small even then For example as PHalley passed p erihelion

in the nucleus was sub jected to a radial NG acceleration only times that of

the Suns gravity The transverse and normal comp onents were over times weaker still

Rickman NG forces are small but not negligible acting in the same direction

over many p erihelion passages they may pro duce signicant changes in a comets orbit In

fact the need for NG correction terms in comet orbit calculations has long b een known

As early as Encke noted that some comets orbits deviated from purely gravitational

ones which he attributed to a resisting medium through which the comets passed

Nongravitational forces are dicult to mo del Their strength dep ends on the comets

distance from the Sun but displays less regular variability as well Gas pro duction may vary

by a factor of or more b etween the pre and p ostp erihelion legs of the orbit Sekanina

Festou and jets and streamers are observed to evolve on time scales of less

than a day Festou et al b suggesting that NG forces change on similar time scales

Traditionall y the term nongravitational forces has b een reserved for the reaction forces resulting from

the uneven sublimation of cometary volatiles and it will b e used here in that manner It will b e shown in

x that the other forces of a nongravitational nature eg radiation pressure are negligibl e in comparison to the outgassing forces

CHAPTER DYNAMICS

Further complications arise from the rotation of the nucleus which is dicult to measure

through the coma and which may b e quite complicated due to precession Wilhelm

Our inability to measure or predict the eects of outgassing with condence makes a precise

treatment of these accelerations dicult

A simple and naive mo del of NG accelerations which is all the data allows assumes that

the short time scale comp onents of the NG forces are uncorrelated and cancel out leaving

only fairly regular longer time scale comp onents as dynamically imp ortant A simple and

widelyused mo del called Style I I parameters was devised by Marsden et al The

in Equation is written as NG acceleration term F

j et

F F e F e F e

j et

where the three orthogonal comp onents are radial F p ositive outward from the Sun

transverse F in the orbital plane p ositive along the direction ahead of the Sun

comet line and normal F p erp endicular to the orbital plane parallel to e e The

Style II mo del assumes that the accelerations are symmetric ab out p erihelion and can b e

represented by

F r A g r F r A g r F r A g r

where fA A A g are indep endent constants and g r is a nonnegative function describing

the increase in activity with decreasing cometSun distance r The form of g r is based on

an empirical water sublimation curve by Delsemme and Miller

m n

r r

g r

r r

o o

where m k n r AU and the normalisation parameter

is chosen to b e so that g AU Note that g r is roughly prop ortional to

r r for r r At r r g r drops much faster than the simple inverse square

o o

that describ es the incident solar ux Figure

The constants A A and A are calculated by Marsden et al for each comet

by a tting pro cess the constants are assigned the values which minimise the dierence

b etween the observed and mo delled p ositions of the comet If the residuals calculated from

a mo del including NG forces are signicantly smaller than those predicted from a purely

The symb ol F is again used here to represent the acceleration and not the force to maintain consistency with the literature

CHAPTER DYNAMICS

1000 g(r)

100

0.1

0.01

0.001 0 1 2 3

r (AU)

Figure The Style I I nongravitational acceleration function g r and a r curve

gravitational description of the comets motion then NG forces may play a signicant role

Note that determinations of fA A A g sometimes assume that their values are constant

over one or more apparitions despite the fact that some comets show changes in these

values in b oth sign and amplitude from apparition to apparition Marsden As

the p eak outgassing o ccurs on the sunward side of the cometary nucleus the sign of A is

always p ositive The signs of A and A are determined by the rotation of the nucleus and

the nonsymmetrical nature of the gas release which cause the acceleration to deviate from

the precisely sunward direction

Despite the uncertainties involved the calculated values of the NG constants give us an

idea of the order of magnitude of the forces involved The value of A is typically to

AU day to m s day with error estimates variable but in the

range Marsden et al Comets with shorter p erio ds tend to have smaller

values of A which do es suggest that their volatile supplies have b een depleted by their

more frequent passages near the Sun The values of jA j are typically only of jA j with

similar errors The tangential comp onent is presumed to b e due to the displacement of the

CHAPTER DYNAMICS

most active outgassing region away from the subsolar p oint due to rotation of the nucleus

If this is the case a ratio of A A implies a lag angle of roughly sin The

inclusion of a normal comp onent in the mo dels do es not reduce the residuals signicantly

and so A is generally taken to b e zero Marsden et al Marsden

The eect of the NG forces can b e deduced from Gausss planetary equations describ ed

in App endix A Note that the absence of a normal comp onent F would mean that Equa

tions Ac and Ad are identically zero and that the comets orbital plane is constant

Gausss equations allow us to estimate the impact of NG forces on cometary dynamics

Assuming the accelerations imparted are given by the Style I I function g r Equation

Equations Aaf yield exp ected changes in inverse semima jor axis and p erihelion of

q g r

a A df a

GM e cos f

cos f q

A df b g r q

GM e cos f

where e has b een assumed and df J dtr which can b e deduced from Keplers

second law has b een used The median value of the semima jor axis in these simulations

will b e shown to b e around AU x Taking a typical visible longp erio d comet to

have q AU and a AU though the result is insensitive to the exact value of a as

long as e the integrands in Equations a and b can b e numerically integrated

using the expression r a e e cos f to transform g r to a function of the true

anomaly We nd

AU a a

AU day

q AU b

AU day

Typical values of A are AU day Marsden et al and thus acting alone NG

forces could move a comet out of the visibili ty cylinder only on a time scale of hundreds of

thousands of orbits However the energy change imparted is only ab out a factor of sixty

less than that due to the planets Equation thus a comet could conceivably b e moved

from an orbit with a semima jor axis of AU to an unb ound one in a thousand orbits

Equation a and b do not dep end on A b ecause the radial NG forces eect on

the energy and p erihelion distance averages to zero over a full orbit However the radial

CHAPTER DYNAMICS

acceleration may pro duce variations in the elements on shorter time scales eg from aphelion

to p erihelion These shortterm variations might b e imp ortant if say the NG forces were

strong enough to push comets into unb ound orbits during the outb ound leg of the orbit

The typical kick during the p erihelionaphel i on leg is

q g r sin f

a df a A

GM e cos f

AU day

which is to o small to unbind orbits with semima jor axes less than AU Larger

values of A impart larger kicks and an order of magnitude increase would provide an

energy change of order that of the planets thus radial nongravitational forces may have

some role to play in certain highly active comets

When a comets no des cross the orbit of a planet a close encounter b ecomes much more

likely The no dal distances r are given by

a e

e cos

where the plus in the denominator refers to the ascending no de the minus to the descending

no de The rate of change of the no dal distances under NG forces is

q sin q

cos cos

where e has b een used Using Gausss equations the change in is found to b e

q g r cos f

A df

GM e cos f

and the change in r p er orbit can then b e deduced to b e

sin A A

AU r

cos cos

AU d AU d

where again a AU and q AU have b een used

If r r a close encounter is very likely if one of the no des crosses a planets

n p

orbit The numb er of orbits m required for a no de to migrate near to a planets orbit is of

order r r or for Jupiters orbit and the values used ab ove m orbits Thus the

p n

motion of the no des is to o slow to appreciably shorten the lives of LP comets

CHAPTER DYNAMICS

Passing stars

A longp erio d comet passing aphelion far from the Sun may have its orbit p erturb ed by

stars travelling through the solar neighb ourho o d The comets velo city at aphelion is of

order m s much less than the velo city disp ersion of stars in the Galactic disk which

is km s Mihalas and Binney Thus to a rst approximation the comet can b e

considered stationary during a stellar encounter and the impulse approximation used

The net impulse v due to a passing star of mass M and velo city v is the dierence

b etween the impulses imparted to the Sun v and the comet v

GM GM

v v v D D

c c

v D

where D and D are the vectors directed to the p oint of closest approach from the comet

and the Sun resp ectively In the case of a star passing very close to the Sun D D

and Equation can b e approximated by

a jv j

v D

If the encounter is a distant one D and D are nearly parallel and the impulse reduces to

GM r cos

jv j b

v D

where is the angle b etween D and the Suncomet vector r

Consider the case of the Sun and its attendant comet cloud moving with velo city v

through a homogeneous and isotropic distribution of static stars of mass M and numb er

density n The stars transfer kinetic energy to the comet cloud the average rate of change

in the square of a comets velo city hv i is given by cf Bailey Equation

r G M n

ln ln hv i

v d b

min min

thus the change p er orbit v is

G M n r

hv i ln ln a

v d b

min min

r G M n a

ln b

d b

min min

v M

n a v M t

G M n a

ln c

v M

CHAPTER DYNAMICS

where b n v t is the minimum impact parameter exp ected during the Solar

min SS

Systems lifetime t d M GM r M v is the distance within which a single

SS min

encounter would result in the comet escaping from the Solar System and r hr i

a has b een used

Energy

The change in a comets inverse semima jor axis a v GM caused by passing

stars can b e obtained from Equation c Taking v km s the lo cal stellar velo city

disp ersion n p c and M M Bahcall and Soneira the change in

a is

n km s M a

a AU

v M AU

p c

p er orbit where the logarithmic term has b een taken to b e constant This result is consistent

with other derivations cf Fernandez Fernandez and Ip and from it one

deduces that stellar p erturbations have only a very small eect on comet orbital energies

over a single orbit Over time however the net transfer of energy to the cloud unbinds

its memb ers and may signicantly deplete its numb ers over the age of the Solar System

Bailey

Angular momentum

The analytic determination of the change in cometary angular momentum due to passing

stars is complex and b eyond the scop e of this pro ject The sub ject was treated thoroughly

by Heisler and Tremaine A result of interest is that the loss cylinder is only lled

at semima jor axes

a AU

AU Equation b l c Equation well outside the Galactic tides value of a

We will see that the numb er density of comets in the Oort cloud drops sharply with

distance x with the result that the tide dominates the overall ux into the loss

cylinder This result allows the injection of comets into the loss cylinder by passing stars

to b e neglected when constructing a theoretical mo del of LP comet evolution

CHAPTER DYNAMICS

Comet showers

Every yr on average a very close stellar encounter D AU may cause a

comet shower enhancing the comet inux rate by up to a factor of twenty Hills

Heisler et al Duncan et al Because the loss cylinder is cleared on time scales

of order yr the o dds are against a shower currently b eing in progress Heisler

estimates that the comet ux signicantly exceeds its background rate only of the time

The correlation of comet aphelion directions with the Galactic plane along with the lack

of strong clustering asso ciated with any other p oints in the sky Figure also suggest

that the present comet ux is at its quiescent level though the p ossibility of a weak shower

b eing in progress has b een advanced Heisler The p ossibility of comet showers will

b e ignored here

Molecular clouds

A p enetrating encounter b etween the Solar System and an interstellar molecular cloud with

velo city v and impact parameter D applies an impulse of

r GM D

jv j j cos j

v D

to a comet where is the angle b etween r and D and the cloud is assumed to b e spherical

and of uniform density with mass M and radius R Biermann Fernandez and Ip

cl cl

Assuming for simplicity that D R e and that the comet is at aphelion

during the encounter the corresp onding change in angular momentum J is

GM a sin

J r v sin

v D

The semima jor axis ab ove which the molecular cloud lls the loss cylinder is

M D q v

GM sin

v q M D

cl cl

p c AU

M km s

Such encounters stir the Oort cloud to great depths and result in large increases in the

M cometary inux but their frequency is unknown giant molecular clouds M

R p c may b e encountered as rarely as every yr Bailey Torb ett cl

CHAPTER DYNAMICS

but smaller clouds M to M could b e times more common Drapatz and

Zinnecker It will b e assumed here that the current ux of comets is unaected

by a recent encounter with a molecular cloud b ecause of the rarity of such encounters

and the Galactic tides strong signature in the distribution of cometary aphelion directions

Figure b

A massive circumsolar disk

If the Sun were to have a matter disk a p ossibility suggested by the presence of disks around

Pictoris and other stars the dynamics of longp erio d comets would b e aected A disk

p otential can b e approximated by a MiyamotoNagai p otential expressed mathematically

as eg Binney and Tremaine

disk

i h

z b x y a

cm cm

where M is the disk mass r x y z is the distance to the Solar Systems

cm cm cm

barycentre and a and b are parameters describing the disks characteristic radius and

d d

thickness resp ectively This disk will b e taken here to b e centred on the Solar Systems

barycentre with the disk plane coinciding with the ecliptic

The resulting acceleration F rV is

GM a z

d d

F r

disk cm

h i

z b

z x y a b

cm cm

The disk p otential is conservative and axisymmetric and thus conserves the z comp onent

of the angular momentum

The disk around Pic is observed in the infrared out to a least AU Smith and

Terrile It is seen nearly edgeon allowing its axis ratio a b to b e estimated at ve

d d

a few hundred AU from the central star Smith and Terrile Paresce and Burrows

The mass of the Pic disk is p o orly known estimates of H column density based

on observations of CO predict values b etween r kg and r kg where r

disk

disk disk

is the gas distance from the central star in AU For values of r of AU this yields

disk

values of to Jupiter masses Early estimates of the mass in dust yield results of

to one Jupiter mass Smith and Terrile

As the planets have b een assumed to b e coplanar and in the ecliptic z z cm

CHAPTER DYNAMICS

There is little or no evidence for a substantial disk in our own Solar System A study

of planetary residuals limits the mass in a to AU disk to less than a few Jupiter

masses A study of PHalley sets much more stringent limits around Jupiter masses

though the inclusion of nongravitational corrections could b e masking the eects of a disk

Tremaine Hogg et al

Observational evidence also puts relatively strong limits on the mass of such a disk

Imaging with the Hubble Space Telescop e puts a preliminary limit of less than Jupiter

masses in km sized ob jects within AU of the Sun Co chran et al Mo dels

of the infrared emission exp ected from dust generated by collisions in a b elt of comets

distributed over AU from the Sun puts a similar limit Backman et al A

more distant AU b elt could have an upp er mass of roughly one Jupiter mass

Backman

Thus it is unlikely that a signicant amount of mass resides in an unseen disk within

AU of the Sun but that at larger distances AU much larger masses M

J up

could b e present

Miscellaneous p erturbations

Radiation pressure and the solar wind

The acceleration F imparted to the nucleus by the solar wind or radiation pressure is given

F R F M

p c

where M and R are the comets mass and radius F is the momentum ux to which the

c c p

comet is sub jected and is a co ecient describing the eciency of the momentum transfer

For a radiation eld F is related to the energy ux F through F F c For the solar

p E p E

wind whose parameters are given in x the momentum ux is n m v

sw sw

These accelerations are small and since they are always directed radially outward the

p erturbations arising during the inward and outward legs tend to cancel However the

acceleration during the outb ound leg could p otentially serve to eject comets Using Gausss

equations for the e case Equation Aa one nds the p erturbation incurred during

CHAPTER DYNAMICS

the outb ound leg to b e

a a dt a

r R F r

GM M q

where r is the radius of the Earths orbit The numerical values of these p erturbations

for a small nucleus R km M kg with p erihelion q AU are

c c

q R M

c c

a AU a

r p

AU km

q R M

c c

a AU b

AU km

where the solar constant F r is taken to b e W m and inelastic collisions

are assumed

During times of close approach to the Sun the eective solar wind crosssection of the

nucleus is increased by the draping of solar magnetic eld lines over the coma However the

nucleus itself has no substantial magnetic eld any back reaction can only b e transmitted

back to the nucleus through the comas gases Thus instead of the solar wind impacting

the nucleus directly a pressure gradient is set up across the coma This pressure will b e of

order the solar wind pressure and thus will not result in p erturbations signicantly larger

than those calculated ab ove

The singleleg solar wind and radiation pressure p erturbation are small and unlike

outgassing accelerations the contributions on the inward and outward b ound legs tend to

cancel These p erturbations will thus b e assumed to b e negligible

Under a radial acceleration there is no change in a comets angular momentum a fact

which is implicit in the calculation of Equations a and b But the ow direction

of the solar wind is not precisely radial and the resulting transverse comp onent of the

acceleration do es not have opp osite signs on the inward and outward legs However the

angle by which the solar wind deviates from radial is less than Foukal The

resulting oneorbit p erturbations are for the orbital energy

a dt a ja j

CHAPTER DYNAMICS

r R F r j sin j

GM M q

j sin j R M q

c c

AU c ja j

AU km

and for the p erihelion distance

jq j a e ae dt a

q r R F r

cos f e cos f

sin df b

GM M e cos f

R M j sin j q

c c

AU c jq j

AU km

These p erturbations are also small compared to those resulting from outgassing Thus the

solar wind and radiation pressure have negligible eects on cometary orbital dynamics

Drag

The solar wind unlike solar radiation do es not reach distances arbitrarily far from the Sun

but is halted by the pressure of the interstellar medium at the heliopause This b oundary

lies b etween and AU from the Sun in the direction of the solar ap ex and further in

other directions Hall et al When outside this b oundary comets are sub jected to

drag from the interstellar medium ISM Longp erio d comets sp end most of their orbital

p erio ds outside the heliopause and for the purp oses of computing the drag from the ISM

on comet nuclei it will b e assumed they sp end all their time there

The drag acceleration exerted on the nucleus is

R C n m v

D ism ism

dr ag

where n and m are the numb er density and mass of the interstellar medium particles

ism ism

and C is the drag co ecient of the nucleus of order unity for spheres in high Reynolds

numb er ie low viscosity uids Streeter and Wylie The lo cal interstellar medium

has n cm and m kg Baranov

ism ism

The drag force is always opp osite to the comets direction of motion and thus the use of

Gausss equations b ecomes quite complicated Nonetheless the eect of drag on Keplerian

CHAPTER DYNAMICS

orbit is wellundersto o d eg Roy and the resulting change in a p er orbit is

R C n m e cos u

D ism ism

a du a

e cos u

R n M

c ism c

AU b

kg cm

where u is the eccentric anomaly and e has b een assumed eg a AU

and q AU This result is indep endent of the semima jor axis but small in any case

The change in p erihelion distance for this same comet is

e cos u R C n m aq

D ism ism

q cos u du a

M e cos u

n R M

ism c c

AU b

kg cm

Both the change in energy and angular momentum due to drag by the ISM are negligible

and will b e ignored

Comet lifetimes

Continued loss of volatiles ultimately transforms comet nuclei into inert b o dies containing

only the leftover refractory elements of their initial inventories Given a p erihelion distance

of AU a typical comets volatiles might b e depleted after a thousand orbits Weissman

There is evidence that sp ent comets may either remain a single solid b o dy or break into

a collection of fragments

Some regular meteor showers have b een asso ciated with the orbits of comets eg the

Aquarids and Orionids with PHalley

A few asteroids have elliptical orbits strongly resembling those of Jupiterfamily

comets eg XA which has a p erihelion distance of AU and an aphelion

distance of AU Kresak Marsden and Williams

Whether or not a dead comet breaks up probably dep ends on various factors including its

internal cohesiveness and the patterns of thermalgravitational stress to which it is sub ject

CHAPTER DYNAMICS

Comets may also b e destroyed or b ecome unb ound from the Sun b efore their volatiles

are exhausted Approximately of comets entering the planetary system on nearpara

b olic orbits will b e transferred to hyp erb olic orbits by p erturbations from the giant planets

esp ecially Jupiter after their rst p erihelion passage a further fraction will b e lost on

each subsequent p erihelion passage as the comets diuse through the available energy

space Gamblers Ruin x

The nucleus may also break into one or more large pieces b efore complete loss of volatiles

o ccurs After such a splitting event a comet is often not observed at its next exp ected

return A comet stands a roughly chance of b eing disrupted on its rst close p erihelion

passage the probability drops to less than p er p erihelion passage for shortp erio d comets

Weissman Kresak

A splitting probability p p er revolution yields a halflife m against splitting of

m ln ln p

Thus splitting may signicantly reduce a comets lifetime

Comets may b e destroyed by collision with the Sun or a planet but this is unlikely If

the collision probability is simply taken to b e the ratio of the planets crosssection to the

area of a sphere of the same radius as its orbit then the probability of a comet passing

within the Ro che limit of the Sun or a planet is only of order p er p erihelion passage

Weissman

The Oort cloud

The existence of the Oort cloud is now generally accepted see Lyttleton for a dis

senting viewp oint based primarily on the observed distribution of a Figure but

the mechanism of its formation as well as its present characteristics remain the sub ject of

debate

The Oort cloud may either b e primordial ie formed from the solar nebula or have

b een captured or pro duced by the Solar System at a later time In the latter case the Oort

cloud may have a survival time short compared with the age of the Solar System However

the pro duction of cometary b o dies within the Solar System after the dissipation of the solar

nebula almost certainly can b e ruled out for lack of a viable mechanism though the origin

of comets from the breakup of a planet in the presentday asteroid b elt has b een p ostulated

CHAPTER DYNAMICS

van Flandern It has also b een p ostulated that a nonprimordial Oort cloud could b e

captured from a passing molecular cloud Club e and Napier Yabushita and Hasegawa

but it remains unclear whether comet nuclei exist in such clouds In addition the

Solar System has a very low capture crosssection for interstellar comets owing to the high

encounter velo cities involved

If the Oort cloud is primordial its formation through in situ accretion seems unlikely

the condensation of cometary b o dies from the solar nebula at Oort cloud distances is dif

cult to explain due to the low density of matter exp ected there Opik though

radiation pressure Hills or windp owered shells Bailey have b een prop osed

as mechanisms by which the required density enhancements could b e pro duced The most

widely accepted mo del of the origin of the Oort cloud holds that comet nuclei are planetes

imals that accreted in or near the planetary region r AU at the same time as the

planets The growing planets esp ecially Uranus and Neptune would have scattered some

planetesimals from their initial nearcircular orbits onto highly elliptical ones Safronov

Tremaine Those protocomets nding themselves on orbits with large semi

AU could have their p erihelia rapidly increased by the Galactic ma jor axes a

tide The removal of their p erihelia from the planetary system eectively decouples the

planetesimals from the planets and at this p oint the comets are said to have reached the

Oort cloud

As cometary isotop e abundances are consistent with solar values Krankowsky et al

Eb erhardt et al the current understanding of the Solar System and its

formation is consistent with a primordial origin for the Oort cloud Fernandez oers a

more complete review of the primordial vs captured question The question of the origin

of the Oort cloud is only of secondary interest here except insofar as it aects the steady

state nature of the Oort cloud on the basis of the clouds likely primordial origin it will b e

assumed that the Oort cloud is in a quasisteady state ie the clouds dynamical evolution

time scale is comparable to the age of the Solar System

The present distribution of comets in the Oort cloud cannot b e observed directly but

Duncan et al have derived a theoretical distribution based on the assumption that

these comets formed in the outer planetary region and were scattered out into the Oort cloud

through the combined p erturbations of the tide and planets They found the clouds inner

edge to b e near AU with a space numb er density of comets roughly prop ortional to

CHAPTER DYNAMICS

r from to AU This p ower law is consistent with Baileys analytical

treatment of the Oort cloud l c Equation Hills rst p ointed out that the

cloud might extend further inwards than indicated by the minimum semima jor axis in the

spike thus the inner region a AU is often referred to as the Hills cloud

Though the orbits of the comets would have initially b een near the ecliptic the inclina

tions of orbits with semima jor axes greater than ab out AU are randomised by passing

stars on a time scale of yr This mixing results in the Oort cloud comets o ccupying a

spherical rather than a attened distribution

External inuences strip comets with large orbits from the Solar System thus truncating

the Oort cloud at some distance from the Sun The last closed Hills surface provides

a useful measure of the maximum p ossible size of the Oort cloud Antonov and Latyshev

calculated the Hills or zerovelo city surface for a comet moving in the eld of

the Sun and the Galaxy On such a surface the Jacobi integral W x is constant

Expressed in the Galactic frame W is

W AA B x B A G z

where A and B are the Oort constants A particle having zero velo city relative to the

Sun inside a closed Hills surface cannot leave the enclosed volume in the absence of other

GM AA B Substituting p erturbations The last closed surface is at W

this value into Equation and solving yields semiaxes for the Hills surface

x p c AU a

y p c AU b

z p c AU c

The last closed Hills surface is triaxial and resembles a prolate ellipsoid In this work

the outer b oundary of the Oort cloud is taken to b e simply spherical and at an aphelion

distance of AU a AU for comets with e rather than AU

The prolate nature of the cloud and the exact lo cation of the b oundary is unlikely to b e

relevant here due to the rapid drop o in comet numb er density with r

The steep r radial density prole deduced by Duncan et al provides an

answer to the question of why the aphelion directions of Oort cloud comets are crowded at

midGalactic latitudes The ability of the tide to ll the loss cylinder at smaller distances

CHAPTER DYNAMICS

than passing stars cf Equation b to allows it to reach into regions of higher

comet density and makes the tide the dominant injector of Oort cloud comets Heisler and

Tremaine have shown that the ux due to the tide exceeds that due to passing stars

by a factor of to The tides maximum injection eciency is at midGalactic latitudes

a signature which can b e seen in Figure b

In the absence of a recent close encounter with a star or a molecular cloud the loss

cylinder is lled only at distances b eyond AU yet the inner edge of the Oort cloud

may b e as close to the planetary system as AU Comets in this inner region never

b ecome visible even if their p erihelia are evolving inwards under the tide b ecause they hit

the Jupiter barrier These comets may however provide a source from which the outer

Oort cloud is replenished Encounters with stars and molecular clouds may scatter some

of the comets in this inner Oort cloud into more lo osely b ound orbits pumping them up

into the outer Oort cloud and may also give rise to o ccasional rare comet showers

The p opulation of the Oort cloud is exp ected to b e ero ded over time scales comparable

to the age of the Solar System as comets are ejected into interstellar space or captured

into smaller orbits Between Duncan et al and Weissman of the

original Oort cloud may have b een lost over the lifetime of the Solar System leaving

comets totalling M in the presentday comet cloud Weissman These numb ers

are p o orly known and estimates of the current Oort cloud p opulation range from

Opik to Marsden ob jects

Problems in longp erio d comet dynamics

The fading problem

The energy kick received by a visible comet AU Equation is larger than the

width of the main spike in the a distribution of longp erio d comets Figure From

this it has b een concluded that the spike consists of dynamically new comets and that

older comets diusing in energy space over many p erihelion passages p opulate the tail

The spike will b e taken here to b e the region where the original inverse semima jor axis

of the comets is less than AU This value is chosen b ecause of the width of the

spike in the observed distribution Figure b All remaining LP comets are considered to b e part of the tail

CHAPTER DYNAMICS

Dene to b e the ratio of the numb er of longp erio d comets in the spike to the total

numb er

N a AU

Then is an estimate of a comets life exp ectancy in p erihelion passages A more precise

measure of comet life exp ectancy is where is the ratio of dynamically new to the

total numb er of LP comets

N m

where m is the numb er of apparitions a comet has made Theory and observations can

b e compared through these quantities let the prime symb ol denote the relevant quantity

derived from observations eg is the ratio of the numb er in the spike to the total numb er

for the observed sample and let If then observations and theory match

on this p oint

The value of computed from the observed sample is where

comets with p erihelion b eyond AU have b een excluded and the quoted error is based on

n noise Note that the denition of the spike includes the seventeen comets Poissonian

with original a in Figure on the assumption that they are coming from the Oort

cloud rather than interstellar space The outright exclusion of the comets on hyp erb olic

original orbits yields a value of

The observations provide a value of the Gamblers Ruin problem predicts

m Equation that is Thus the Gamblers Ruin

implies that only in of the p erihelion passages exp ected to b e made by older LP comets

are observed Why this large discrepancy

Observations and theory have proved dicult to reconcile on this p oint Though more

sophisticated analytical treatments than the Gamblers Ruin narrow the gap signicantly

the problem p ersists eg Kendall Exp erimental results show the same discrep

ancy for example Everhart found using a straightforward Monte Carlo

simulation that included Jupiter Saturn and passing stars

The gap b etween theory and observation is known as the fading problem since it can

b e resolved if dynamically new comets fade drastically in brightness after their rst p erihe

lion passage near the Sun This fading makes them less likely to b e observed at subsequent

p erihelion passages thus reducing their apparent lifetimes and thus increasing

CHAPTER DYNAMICS

Weissman using a Monte Carlo scheme similar to Everharts was able to

increase to unity but not without adding such a fading law

The standard explanation prop osed for such fading go es along the following lines comets

in the Oort cloud may never have approached the Sun to within more than a few tens of

astronomical units since their condensation from the solar nebula and thus may contain

particularly volatile ices eg CO CO that cannot survive the comets rst p erihelion

passage close to the Sun These volatiles create a large bright coma for the new comet

but are substantially or completely depleted in the pro cess When the comet subsequently

returns assuming it has avoided ejection and the other loss mechanisms it will b e much

fainter and may escap e detection The decrease in brightness is required to b e largest over

the comets rst few p erihelion passages levelling o as the most volatile comp onents of the

comets inventory are lost Thus the fading problem may simply b e caused by selection

eects

However a comets failure to reapp ear at its next p erihelion passage could b e the result

of other p ossibly unsusp ected loss mechanisms Any phenomena which results in a decrease

in the life span of LP comets would tend to increase The reduction in brightness of

the comet due to a depletion of readily vap ourised volatiles will b e referred to as standard

fading Determining whether or not standard fading is required to solve the fading problem

or if some other dynamical mechanism is involved is a central goal of this research Some

key p oints p ertaining to the fading problem are listed b elow

Pre and p ostp erihelion brightnesses The evidence against the fading hyp othesis in

cludes the lack of observed large decreases in brightness as LP comets pass p erihelion

decreases which might b e exp ected if their volatile inventory is b eing exhausted Fes

tou Though no collection of Oort cloud comet light curves seems available in

the literature those few published show brightness variations typically no larger than

those of dynamically older comets Whipple Ro ettger et al

Shortp erio d comet fading The reduction in brightness of comets over many p erihelion

passages remains controversial even for SP comets Sekanina claims PEncke

has faded by magnitudes p er orbit over the last century but Kresak

has argued that this is an artifact of instrumental and selection eects and

that random variations in a comets brightness dominate any secular trend In either

CHAPTER DYNAMICS

case the brightnesses of SP comets have not b een observed to change drastically and

p ermanently over a few orbits except for o ccasional splittings

Splitting The physical breakup of the nucleus may provide a comet sink but a halflife of

m or a splitting probability p would b e required to pro duce the

required tailspike ratio see Equation Such a high rate of splitting would not

allow comets to survive long enough to diuse in a up to the large values seen in

the tail of Figure However dynamically young comets have higher splitting rates

than older ones Weissman showed that over the p erio d to long

p erio d comets had a chance p er p erihelion passage of splitting the shortp erio d

comets only In addition the probability was higher for new comets than

older LP comets

The cause of splitting is not well understo o d though some are caused by passages near

the giant planets many are not asso ciated with such encounters Studies of cometary

outbursts during which the comet may brighten by up to a factor of for of order

a week show that impacts by interplanetary b oulders and chemical andor phase

changes in the nucleus are not capable of fully explaining the distribution of events

Hughes The splitting events app ear not or only weakly correlated with the

ecliptic plane asteroid b elt and cometary p erihelion p oints Pittich but rather

are randomly distributed Splitting events are however more likely to o ccur p ost

p erihelion by a factor of or so though b etter observational coverage at this time

may b e a factor Smoluchowski

Cratering rates If comets do fade drastically rather than b eing ejected or otherwise de

stroyed then their cores may still b e present in the Solar System but b e to o faint

to b e observed These dead comets should however contribute to the cratering

rate Sho emaker compared the cratering rate determined from numb er counts

of impact basins on the Earths surface to that exp ected from the observed ux of

p otential impactors and found them to b e consistent within a factor of two Both

rates are dicult to compute and are based on the extrap olation of relatively p o orly

determined data but there is no evidence from cratering rate studies for an additional source of Earthimpacting ob jects

CHAPTER DYNAMICS

The ratio of prograde to retrograde comets

In the Gamblers Ruin problem the lifetime m is prop ortional to the initial distance to the

shortp erio d barrier see Equation A more careful determination of the energy

kicks imparted by the planets reveals that retrograde comets receive smaller a on

average and hence should have lifetimes three times longer on average

If the lifetime of retrograde comets is three times that of prograde comets the obser

vations should reect this fact through a ratio of retrograde to prograde of three to one in

the absence of other imp ortant comet loss mechanisms But the observations plotted in

Figure show no such bias Why is this the case

The clustering of aphelion directions

A numb er of researchers Tyror Oja Lust have rep orted that the

aphelion directions of LP comets are clustered in sp ecic directions on the sky However

due to the presence of strong selection eects the results are not comp elling There is a clear

concentration towards midGalactic latitudes Figure b an eect which is exp ected

since the Galactic tide is most ecient when the Suncomet line is at to the plane of

the Galaxy

However even comets with semima jor axes greater than AU show concentrations

at midGalactic latitudes Fernandez and Ip shown in Figure At these distances

stellar p erturbations should also b e able to ll the loss cylinder Equation so the

distribution of aphelion directions should b e isotropic The test indicates only a

chance of the Galactic latitudes b eing drawn from a spherically symmetric distribution

However the sample size is small and thus the concentration at midlatitudes may b e a

result of sampling noise

The source of shortp erio d comets

Some LP comets may survive long enough to diuse into shortp erio d orbits The incli

nations of Halleytyp e orbits are at rst glance consistent with a spherically symmetric

source see Figure is this source the Oort cloud

The Jupiter family of comets have inclinations which are clearly not uniformly dis

tributed but rather concentrated in the ecliptic Though Everhart showed that

CHAPTER DYNAMICS

Figure The distribution of Galactic latitudes of the aphelion directions of the long

p erio d comets in Marsden and Williams with original semima jor axes greater than

AU The heavy line indicates the distribution exp ected for a spherically symmetric

distribution of aphelion directions

Jupiter is most ecient at capturing the lowest inclination comets from a spherical source

the capture rate is still to o low by a factor of to account for the numb er of Jupiterfamily

comets seen to day Joss The orbital elements of the Jupiterfamily comets are most

consistent with capture from a lowinclination b elt of material at to AU Whipple

Fernandez and Ip Quinn et al Kuip er prop osed in that material

might remain in this region as leftovers from the formation of the planets so this ring is

commonly known as the Kuip er b elt Though the existence of the Kuip er b elt has b een

conrmed observationally eg Jewitt and Luu some fraction of the SP comets are

almost certainly dynamically very old Oort cloud comets Is the Oort clouds contribution

to the p opulation of shortp erio d comets imp ortant

CHAPTER DYNAMICS

Hyp erb olic comets

Some comets app ear to b e approaching the Solar System on weakly hyp erb olic orbits Fig

ure and treatments of nongravitational forces are unable to fully explain this phe

nomena Are there p erhaps unexplored dynamical mechanisms which might explain such

orbits

The present state of the eld

Most research on comet dynamics to date has b een limited to analytical approximations and

Monte Carlo simulations which may not capture all the dynamically imp ortant physics

Analytical investigations of cometary b ehaviour can b e p erformed through p erturba

tional and averaging techniques These metho ds treat a simple twob o dy problem with an

additional necessarily weak p erturbation For example the eects of the Galactic tide

Heisler and Tremaine Torb ett Matese and Whitman and of a single

planet Quinn et al on a comets evolution have b een examined over restricted re

gions of phase space These metho ds usually examine only a single facet of comet evolution

and break down if the interactions are to o strong

Monte Carlo metho ds allow long time scale investigations to b e made relatively cheaply

by evolving comets within a phase space with fewer dimensions than the full problem The

simulation advances in xed discrete time steps at each one the comets are redistributed

throughout the phase space based on precomputed transition probabilities determined

by averaging over the omitted dimensions However Monte Carlo metho ds require sig

nicant simplications of the problem and may prove to o coarsegrained to reveal all the

dynamics of interest see Fro eschle and Rickman for a review

Weissman has completed the most extensive Monte Carlo inves

tigation of longp erio d comets dynamics to date His mo del included the planets non

gravitational forces the eects of passing stars fading and splitting However his simula

tions had some restrictions they did not include the tide the initial semima jor axis was

always AU the eects of the planets were represented by a Gaussian distribution of

energy kicks and comets were run for only returns

His mo del can pro duce to with reasonable agreement b etween the incli

nation and intrinsic p erihelion distributions His mo dels include ad hoc assumptions eg

CHAPTER DYNAMICS

some fraction of indestructible comets increased fading for comets with small p erihelia and

a xed disruption probability for new and old comets but none that contradict cometary

physics as now understo o d

Weissman had to add very strong fading of his sample of comets fade to his

simulations in order to match observations Because his Monte Carlo simulations were fairly

coarsegrained and there is little or no other evidence for strong fading the p ossibility that

the fading required is simply an unmo delled facet of the dynamics remains strong

The most direct approach to the study of cometary dynamics is the numerical inte

gration of the comets equations of motion including all signicant p erturbations Until

recently computational restrictions have placed severe limitations on the sophistication and

time scale of p ossible investigations For example Dvorak and Kribb el numerically

integrated the tra jectories of ve Halleytyp e comets in the presence of the Sun Jupiter and

Saturn for years orbits Manara and Valsecchi used similar metho ds

to follow shortp erio d comets for revolutions in low inclination orbits in the outer

Solar System

As technological development has eased computational restrictions direct integrations

of cometary dynamics have increased in complexity and length Levison and Duncan

numerically integrated the known SP comets under the inuence of all the planets except

Mercury and Pluto for years However the research presented here represents the most

sophisticated direct integration of the longp erio d comets yet published

Chapter

Algorithm

The purp ose of this research is to examine the dynamical pro cesses imp ortant in the evo

lution of longp erio d comets and to investigate the discrepancies b etween the observed

distribution of orbital elements and simple theoretical mo dels To accomplish this a mo del

has b een created to simulate the imp ortant dynamical eects inuencing a longp erio d

comets tra jectory as it travels from the Oort cloud to its destruction or departure from the

Solar System

The mo del is emb o died by a computer co de called LOCI which has the following basic

framework

Each comet is represented by a massless test particle The test particles are followed

indep endently one at a time thus interactions b etween comets are neglected Each

comet is started in the Oort cloud and its evolution is followed analytically until it

approaches the Sun close enough for planetary p erturbations to b ecome imp ortant

The comet is subsequently followed by numerically integrating its equations of motion

expressed in regularised co ordinates x until lost from the Solar System

The mo del Solar System in which the test particles evolve consists of the Sun and

four planets representing Jupiter Saturn Uranus and Neptune Their physical char

acteristics are listed in Table The terrestrial planets and Pluto are omitted for

the following reasons

The orbitaveraged quadrup ole moments of their orbits Q M R are each at

least two orders of magnitude less than that of any giant planet Equation

CHAPTER ALGORITHM

Name M M r AU R km

p p p

Jupiter

Saturn

Uranus

Neptune

Table The values of the recipro cal masses M M orbital r and physical R

p p p

radii used for the giant planets Newhall et al Lindal US Naval Observatory

and column of Table and thus the smallest planets inuence on cometary

p erihelia is negligible compared to that of the giant planets

The energy p erturbations imparted by the smallest planets are unimp ortant if

the comets p erihelion distance is within the planets orbit a M R

p p

Equation b and the p erturbation is dominated by Jupiter and Saturn col

umn of Table

The planets orbits are mo delled as circular and coplanar There is no reason to

exp ect that the small eccentricities and inclinations of the planets play signicant

roles in longp erio d comet dynamics Mutual planetary p erturbations are ignored It

should b e noted that the planets are represented consistently throughout the numerical

integration No further approximations eg putting the planets mass into the Sun

when the comets are far outside the planetary system are made

The mo del includes the dominant comp onent of the Galactic tidal eld as describ ed

by Equation The Solar Systems orbit ab out the Galactic centre is taken to b e

circular and in the Galactic plane Deviations from this idealised orbit are ignored

though they may result in some temp oral variation in the tidallyinduc ed cometary

p c scale height in ux if there is a large amount of dark matter with small

our Galaxy Matese et al The Galactic centre and p oles are oriented so as to

match their current p ositions relative to the ecliptic

The mo del do es not account for encounters with passing stars In the absence of close

stellar encounters the transfer of comets from the Oort cloud to the loss cylinder

is dominated by the Galactic tide b ecause the tide lls the loss cylinder at smaller

semima jor axis than passing stars page The omission of stellar encounters has the

CHAPTER ALGORITHM

considerable b enet of yielding a deterministic mo del in which the comets evolution

is completely determined by the systems initial conditions

The mo del includes the eects of nongravitational forces of arbitrary magnitude and

direction These parameters can b e varied to observe their eects on the simulation

output

The mo del also contains provisions for investigating hyp othetical phenomena such as

a circumsolar disk or a solar companion

Comparison with observations

The sample of known comets includes ob jects of varied and unknown dynamical ages ob

served to varying degrees of completeness over a relatively indenite p erio d This is in

sharp contrast to the data available from simulations where dynamical ages are known and

selection eects are nonexistent Thus a basis for comparing the two sets must rst b e

constructed

Let O b e the set of comets for which Marsden and Williams calculated original

orbital elements and which have p erihelion distances less than AU It will b e assumed that

the distributions of orbital elements of O are free of selection eects except in the p erihelion

distance q With this exception O is representative of a complete sample of longp erio d

comets passing within AU of the Sun over some xed though unknown interval of time

This assertion assumes that the ux of LP comets has not varied signicantly in rate or

functional form over the observation interval yr and is based on the fact that no LP

comet has a p erio d short enough to have made more than one app earance in the observed

sample

Consider the exp ected ux of LP comets into the loss cylinder This ux can b e derived

x and constitutes the probability distribution from which the simulations initial

conditions are chosen Any sample of such initial conditions represents the LP comets

injected into the entrance surface over some xed p erio d of time Let S b e the sample of al l

apparitions ie visible p erihelion passages p erformed by the comets represented by such

a set of initial conditions Then comparisons b etween the mo del and observations can b e

made by directly contrasting S and O

That O and S are directly comparable is due to our choice of the ux into the entrance

CHAPTER ALGORITHM

surface as the initial conditions for the simulations as well as the steadystate distribution

of LP comets In the Solar System there are comets of all ages but given a steadystate one

comet dies eg is ejected destroyed etc for every one that arrives from the Oort cloud

This continuous distribution of ages is repro duced by the simulations by following individual

ob jects throughout their entire lifetimes As the lifelong evolution of the ensemble of

simulated comets is statistically equivalent to a snapshot of the steadystate distribution

S and O are directly comparable with each other

Numerics

The integration algorithm

LOCIs primary integration algorithm is the BulirschSto er metho d The implementation

used is bsstep from Press et al The routine bsstep has automatic stepsize

control achieved by monitoring the lo cal truncation error ie the error due to the omission

of higher order terms by the integrator A fourthorder RungeKuttaFehlb erg algorithm a

RungeKutta variant designed for ecient stepsize control Burden and Faires was

used for testing the BulirschSto er routine

Regularisation

Integrating the equations of motion of a comet on a highly elliptical orbit is dicult in

Cartesian co ordinates due to the very small step sizes required to maintain accuracy near

p erihelion The KustaanheimoStiefel or KS transformation replaces the six Carte

sian co ordinates of p osition r fx y z g and velo city v fx y z g with ten regularised

co ordinates It also replaces the indep endent variable the physical time t with the cti

tious time s where dt r ds The advantage of the regularised co ordinates is that the

unp erturb ed twob o dy equations of motion b ecome those of a harmonic oscillator thus

the acceleration do es not blow up as r go es to zero and the ensuing small stepsizes and

numerical diculties are avoided Regularised co ordinates are used in all the simulations

discussed here

The distinction b etween the KustaanheimoStiefel transformation and the KolmogorovSmirnov test

though usually clear from the context will b e made by using the abbreviations KS and KS resp ectively

CHAPTER ALGORITHM

Eight of the regularised co ordinates represent the particles phase space p osition u

fu u u u g and velo city u fu u u u g The ninth regularised co ordinate is the

negative of the orbital energy h of the particle given by

GM GM ju j

h jv j V V

r juj

where V represents any p otentials b esides the Suns The tenth co ordinate is the physical

time t Both h and t are dep endent variables and must b e integrated along the particles

tra jectory in the same manner as the eight p osition and velo city co ordinates

The regularised p osition and velo city are obtained from the Cartesian co ordinates by

means of the KS transformation see Stiefel and Scheifele for a fuller exp osition of the

KS formalism Because the regularised space has two more dimensions than Cartesian

space the KS transformation is not onetoone each p osition in physical space corresp onds

to a onedimensional manifold in KS space as do the velo cities

A requirement of KS regularisation is that the frame origin must coincide with the

primary force centre This fact dictates the reference frame in which the comet integration

can b est b e p erformed

Reference frames

The equations of motion dier dep ending on the reference frame in which they are expressed

Two frames prove useful

The barycentric frame is an inertial frame However the barycentre do es not coin

cide with the primary force centre the Sun Regularisation requires the force centre

and frame origin to coincide if its sup eriority in handling highly eccentric orbits near

p erihelion is to b e eective

The helio centric frame is nonrotating and centred on the Sun Its advantage is

that the origin and central force coincide thus allowing regularisations b enets to b e

fully exploited during cometary passages close to the Sun Its primary disadvantage

is that it is a noninertial frame it suers accelerations as the Sun orbits the Solar

Systems barycentre and thus the indirect terms Equation are not identically

zero These terms do not go to zero as r b ecomes large and result in extremely small

The prime symb ol here represents a derivative with resp ect to s ie u du ds etc

1 1

CHAPTER ALGORITHM

stepsizes at large distances To understand why consider a comet far outside the

planetary system whose p erio d is large compared to those of the planets On time

scales much less than the comet is eectively travelling in a straight line relative to

the Solar Systems barycentre But the Sun continues its orbit around the centre of

mass a complicated motion which is reected in the helio centric frame by the comet

taking on a tortuous lo oping tra jectory like a telephone cord This complicated path

results in very small step sizes in the integration algorithm with corresp onding slow

progress and numerical diculties

The two frames complement each other the b enets of b oth can b e obtained by p er

forming the integration in the helio centric frame near p erihelion and switching to the

barycentric frame at large radii Regularised co ordinates are used in b oth frames regular

isation provides little b enet and some increase in complexity over Cartesian co ordinates

in the barycentric frame but its use eliminates the need for duplicate Cartesian and regu

larised co ordinate subroutines The switch b etween the two frames is accomplished by the

mo del at a constant distance from the Sun normally taken to b e at AU

Error tolerances

The error in a single integration step is dictated by a set E of ten error limits

E E E fE E E E g

u u h t

u u

4 1

1 4

one for each of the regularised co ordinates The BulirschSto er routine compares its own

estimate of the lo cal truncation error against E in order to adjust the stepsize and to keep

the singlestep error b elow those limits

The value of the error limits is controlled by means of a single parameter called the tol

erance which is translated into an error limit through a pro cess describ ed in App endix B

The tolerance is typically chosen to b e for reasons describ ed in x with an upp er

limit set by the machine precision of roughly

Random numb ers

A sequence of random numb ers is required to initialise the simulations In this research

approximately random numb ers are required ten for each simulated comet Six are

CHAPTER ALGORITHM

required for the six initial orbital elements and four for the initial phases of the planets

The mo del uses ran from Press et al mo died for use with double precision

bit numb ers to pro duce the required values This routine uses BaysDurham shuing

to avoid serial correlations and has a p erio d of over calls

Chaos

The motion of Halleys comet has b een shown to b e chaotic Chirikov and Vecheslavov

as has that of comets on nearparab olic orbits with p erihelia near the giant planets

Petrosky Sagdeev and Zaslavsky It is likely that the motions of most comets

in these simulations will b e chaotic as well Thus the numerically integrated tra jectory

provided by LOCI is exp ected to diverge exp onentially from the true one as a result of

truncation and roundo errors eg Miller

This problem is inherent in all Nb o dy simulations arbitrary precision mathematical

routines are b ecoming available but remain much to o slow However it is reasonable to

supp ose that the simulated results still provide an accurate reection of reality The errors

intro duced by nite precision arithmetic roughly in p er step are likely similar

in order of magnitude to those intro duced into the real Solar System by such weak and

neglected eects as the nonuniform distribution of matter in the solar neighb ourho o d

Thus one may hop e that statistically the simulations continue to reect reality

Time requirements

The question of where most of the CPU time is likely to b e sp ent can b e addressed as

follows The probability of a comet b eing ejected from a simple Sunplanet system on the

m orbit is given by a p ower law Equation If the probability of a comet reaching an

endstate remains a p ower law under the addition of the tide and the other relevant physics

and this p ower law has the form

pm m

where is some constant then the exp ectation value m of m is just

X X

m m pm m

m m

Equation diverges if A Sunplanet system has Equation and

though the extra physics in the mo del is exp ected to increase the rate at which comets evolve

CHAPTER ALGORITHM

and hence lower it seems likely that m will ultimately prove to diverge ie that most

of the time will b e sp ent following a few very longlived comets This problem is handled

by putting on hold any comets which prove to have very long lifetimes and reexamining

them at a later date However such cases are relatively rare and only a few dozen of the

hundreds of thousands of comets simulated here are not eventually integrated throughout

their full lifetimes

Planetary encounters

Close encounters b etween comets and planets including collisions are of interest for de

termining b oth current and past cratering rates These simulations track such events A

close encounter with a planet is dened here to b e a passage through a planets sphere of

inuence R The sphere of inuence is dened to b e surface around a planet at which the

p erturbation of the planet on the twob o dy helio centric orbit is equal to that of the Sun on

the twob o dy planeto centric orbit If the planets mass is much less that that of the Sun

this surface is roughly spherical and is given by

R r

I p

where r is the planets orbital radius The Sun has no sphere of inuence in this sense so

a close encounter with the Sun is instead dened to b e passage within solar radii

Each crossing from outside the sphere of inuence to within is counted as one encounter

the simulation do es not check for multiple close approaches while the comet is within the

sphere of inuence However it will b e seen that captures dened here to b e a close

encounter with a planet during which the eccentricity relative to the planet at closest

approach is less than unity are extremely rare events

Initial conditions

yr stellar encounters the injection of new longp erio d In the absence of recent t

comets from the Oort cloud is dominated by the Galactic tide Equation c reveals that

a comets p erihelion decreases under the tides inuence when sin that is when

CHAPTER ALGORITHM

Comets with outside this region have increasing p erihelia and therefore are not of imme

diate interest

LOCI randomly selects the comets initial conditions from the ux of comets with

decreasing p erihelia The ux is measured at a phase space b oundary called the entrance

surface

The entrance surface

The entrance surface denes the angular momentum at which LOCI b egins the simulation

of a comet Taking the entrance surface J to b e a cylinder ie a xed p erihelion distance

proves to o restrictive for our purp oses and thus J is p ermitted to b e a function of comets

initial orbital elements

There are two main criteria for the selection of an appropriate entrance surface Firstly

the corresp onding p erihelion distance q should b e far enough outside the planetary sys

tem that the orbitaveraged approximation for the tide is valid for q q Secondly q

E E

should b e close enough to the planetary system that CPU cycles are not wasted by numer

ically integrating the comets tra jectories in the regions where they can b e handled well

analytically

To calculate J consider the change in angular momentum p er orbit J given by

Equation If J is taken to b e a constant Z times J then the orbitaveraged ap

A value of for Z will b e used here proximation is correct outside J as long as Z

The expression for J thus b ecomes through Equation

J L Z e L

G M

or in terms of the entrance p erihelion distance q

e a

a q a a Z

a where e b Z

There are three restrictions on this expression

The initial p erihelion distance q must b e suciently far outside the planetary system

that the typical energy p erturbation p er orbit is small Equation b has no minimum

CHAPTER ALGORITHM

Figure The base logarithm of the fractional ro otmeansquare change in energy

hE E i p er p erihelion passage caused by the four giant planets as a function of

p erihelion distance q

p erihelion distance so a lower limit q on the p erihelion distance must b e imp osed

The exp onential decrease of E with r see Equation suggests that q need

not b e much greater than the size of the planetary system The value of q was

chosen so that the ro otmeansquare of the fractional change in energy hE E i

after one orbit aphelion to aphelion in a mo del Solar System containing only the

Sun and the four giant planets would b e less than for typical Oort cloud comets

Figure shows the distribution of a in such a system as a function of q all

orbits have an initial semima jor axis of AU and the values of cos i and

were selected from uniform probability distributions The energy change drops b elow

p er orbit at ab out AU this distance is taken to b e q

The semima jor axis a where Equation b is equal to q is easily shown to b e

M q

a a

CHAPTER ALGORITHM

Z q

AU b

M p c

It is clear from Table that J is always less than L which reects the requirement

q a Equations and a b violate this condition at large L as the analytical

treatment of the tide breaks down where J J The condition J L is violated

where

a a

AU b

M p c

This is outside but near the Oort cloud outer b oundary chosen here and which is given

by the maximum initial semima jor axis a AU Equation a provides a

for our chosen value restriction on our choice of Z conning it to values of Z

of a

As the entrance surface expands towards larger semima jor axes q a and the frac

tion of comets which evolve entirely within the entrance surface b ecomes imp ortant

In fact at q a all comets are already inside or just on the entrance surface

The assumed uniform distribution in angular momentum assumed implies that at any

given semima jor axis the fraction of comets already inside the entrance cylinder

and which can cross the Jupiter barrier is J L or from Equation

Z a

L Z

G M AU

M p c

Thus a substantial fraction of comets at the outer edge of the Oort cloud are missed

but this fraction decreases rapidly with decreasing L Integrating the phasespace

density describ ed later in x of comets b oth inside and outside the entrance

surface shows that for the aforementioned values approximately of comets

reside outside the entrance surface Thus only a small fraction of comets are missed

through our choice of the entrance surface

In a similar vein the approximation e b ecomes invalid as q a Equation a

yields a value for q at the outer Oort cloud edge of approximately AU E

CHAPTER ALGORITHM

implying e Thus the large eccentricity approximation remains accurate in this

limit under Z

Equations and ab reveal that not all cometary p erihelia are eventually drawn

into the planetary system by the tide Many comets reach a minimum p erihelion

distance far outside the planetary system and thus do not b ecome sub ject to signif

icant planetary p erturbations In the absence of stellar p erturbations such comets

remain in the Oort cloud and never b ecome visible To avoid wasting CPU cycles

on such comets the tra jectories of comets whose initial conditions would result in a

tidallyinduced minimum q greater than AU are not numerically integrated but

terminated immediately From numerical exp eriment it is found that approximately

ve of every six comets entering our entrance surface have p erihelia which will not

cross the q AU b oundary thus the exclusion of such orbits nets a signicant

savings in pro cessor time

The entrance surface as detailed ab ove can b e summarised as

q where a a a

Z a where a a a

sub ject to the condition Z Z where

and where a and a are the outer and inner limits of the Oort cloud resp ectively Plugging

in the previously discussed numerical values Equation b ecomes

AU where AU a AU

noting that our choices of Z and a AU satisfy Z Z Equation

The ux of comets into the entrance surface

Let the phase space density of comets expressed in the canonical system T of Table b e

given by g L J J f The phase space density may in principle b e a function of all

the co ordinates and momenta However if the Oort cloud is collisionless and in a quasi

steady state Jeans theorem states that g is a function only of the integrals of motion If no

CHAPTER ALGORITHM

p erturbations are present the integrals of motion include the energy and the comp onents

of the angular momentum

The Oort cloud is certainly collisionle ss with a twob o dy relaxation time of over

years Binney and Tremaine Equation For a quasisteady state in which L J

and J are integrals requires rstly that any p erturbing forces b e weak and secondly that

the loss cylinder b e small

The requirement that the p erturbing forces b e weak is fullled if the tidal forces are

weak as the tide is the dominant p erturb er of the Oort cloud The ratio of the tidal

to the solar force cf Equations and given by

F jz jr r

tide

M M

is less than unity where

p c AU r

AU the assumption of weak tidal forces is valid Thus for comets with a

The next strongest p erturbing forces are those of the planets but they are imp ortant

only near the loss cylinder As the entrance surface q is chosen to b e everywhere well

outside the loss cylinder and the planetary system the assumption of weak planetary

p erturbations is valid outside q

The loss cylinder must b e small b ecause losses are irreversible and detract from the

steadystate The crosssectional area of the loss cylinder J r v Equation

J GM q

r r

An upp er limit to the b ound transverse velo city v p ossible at any radius is the

parab olic velo city v GM r yielding a total crosssectional area v

GM r The ratio of the loss cylinder area to the total phase space area is

q q r

r AU AU

indicating that the loss cylinder is indeed small

CHAPTER ALGORITHM

Having ascertained that the distribution of Oort cloud orbits outside and near the en

trance surface is collisionless and in a quasisteady state Jeans theorem now assures us that

the distribution function g can b e a function only of L J and J The Oort cloud phase

space density g used in these simulations is based on the results of Duncan et al

who determined the Oort clouds numb er density n to b e a function only of radius r and

to b e of the form

nr r

where within a range of r from to AU Integrating Equation

over r yields the numb er of comets N r within a given shell with its inner edge lo cated

AU from the Sun

N r r r dr r

The Oort cloud b eing in a quasisteady state r can b e replaced by its timeaverage hr i

a e yielding

N a a where AU a AU a

N L L b

Duncan et al also found cometary inclinations to b e randomised for a AU

indicating that g is not a function of the inclination i and hence J at large radii It will b e

assumed here that the angular momenta are randomised and hence that g is not a function

of J This last assumption rests on the mixing of the Oort cloud by passing stars which is

also presumed to replenish any regions of phase space depleted by the actions of the tide

and the loss cylinder

Taking g L L where is some constant the total numb er of comets can b e expressed

in terms of L as

L L L J

Z Z Z Z Z Z Z

N L df d d g LL dL L dL dJ g dJ

Comparison of Equations b and shows that and hence that

g L The nal form chosen for the phase space density is

g L g L AU a AU

CHAPTER ALGORITHM

where g is a constant factor related to the total numb er of comets in the cloud The inner

edge of the Oort cloud was chosen to b e a AU instead of to AU

to conserve CPU time by neglecting comets which cannot pass the Jupiter barrier as our

interest here is restricted to comets that can b ecome visible The outer edge of the Oort

cloud is taken to b e at a AU

Let L J J f dL dJ d d df b e the average ux p er unit time of comets

E z z

crossing into the entrance surface J at a given p oint then

J J

L J J f g L J J f dJ

E z z

where is the comet orbital p erio d and J is the usual time derivative of the angular

momentum and here represents the phasespace velo city normal to the entrance surface

p ositive outwards Note that Equation holds whether or not the entrance surface is

lled ie whether or not J J

Substituting Equation b and Equation into Equation yields the ux into

the entrance surface

g L

L J J sin L J J J

E z

E z E

GM J

The sign of Equation is determined by sin Since only the ux into the surface is

of interest the quantity is dened to b e

where that is where or

otherwise

such that is nonzero only where the tide is moving cometary p erihelia inwards

In these simulations the comets initial elements are drawn at random from A

practical implementation of this selection pro cess requires the cumulative probability dis

tribution P w of comets drawn into the entrance surface by the tide as a function of each

relevant variable w If P w is known one can generate an appropriately distributed set of

w s by solving the equation w P where is a uniform random deviate and

P is the inverse transform of P mapping w P w is related to through

Z Z Z Z

P w dw w x y z dx dy dz

where w x y and z are the canonical variables and it has b een assumed that is separable E

CHAPTER ALGORITHM

Combining Equations and and converting to orbital elements the expressions

for the cumulative probabilities are found to b e

P a

cos

P b

cos

P sin c

d P

Note that the mean anomaly is chosen from a uniform random distribution The orbit

averaged approximation by its very nature tells us nothing ab out the role of or the

true anomaly f in the dynamics The ux into the entrance cylinder could in principle b e a

function of f but our choice of initial conditions should b e insensitive to this fact unless the

comet can cross the entrance cylinder in less than one orbit ie J J Our denition

of J Z J with Z sp ecically precludes this p ossibility so may b e chosen at

random

The cumulative probability function of the semima jor axis is more complicated its

calculation b eing left to App endix C Between the inner edge of the Oort cloud a and a

h i

a a a a q e P a

and where a a a

a a Z a a

f P a P a

M P M

where P is a constant such that P a The eccentricity e is determined by our choice

of q and a

e g

These seven equations a through g are used by LOCI to cho ose the comets initial

orbital elements

CHAPTER ALGORITHM

The endstates of comets

A comets dynamical lifetime is nite any of a numb er of pro cesses may transfer it to an

unb ound orbit or destroy it outright A comet is said to have reached an endstate if its

orbital parameters allow it to b e classied into one of nine categories listed b elow Entering

an endstate may indicate the loss or destruction of a comet or simply an intermediate

stopping p oint from which the simulation can subsequently b e restarted The nine end

states are

Collision The distance b etween the comet and a Solar System memb er ie the Sun or one

of the giant planets is less than that ob jects physical radius Simply comparing the

planetcomet distance to the planets radii at each time step is insucient to detect

all collisions the integrators nite stepsize could result in the integration stepping

over the planet and failing to record a collision Collision detection is ensured

by calculating the Keplerian ie twob o dy orbit of the comet around the nearest

planet and using the p ericentric distance of this orbit as the minimum planetcomet

separation

Ejection The comet is leaving the Solar System on an orbit which is unb ound ie

parab olic or hyp erb olic with resp ect to the Solar Systems barycentre The simulation

is not terminated until the comet is at least AU from the Sun to allow for the

p ossibility that subsequent p erturbations will result in the comet losing energy and

returning to a b ound state

Escap e Though still b ound to the barycentre in the twob o dy sense ie e the comet

has ventured b eyond the last closed Hills surface Equations a b and c and is

considered stripp ed from the Solar System by the action of passing stars molecular

clouds etc The distinction b etween escape and ejection is somewhat arbitrary

Exceeded age of Solar System The comet lifespan has exceeded the present age of

the Solar System taken to b e yr Such a comet would given time reach

one of the other endstates but has not yet had time to b ecome part of the observed

sample

Exceeded time limit The comet has completed more than orbits without entering

one of the other endstates The simulation is terminated and saved and will b e re

CHAPTER ALGORITHM

examined at a later date This is merely a safeguard that prevents extremely longlived

comets from consuming large amounts of CPU time as discussed in x

Faded The comet is considered to have faded through loss of volatiles splitting or other

mechanisms and is no longer bright enough to b e observed even if its orbit should

carry it close to the Sun The probability of a comet fading is largely unknown and

the determination of this fading function is a primary goal of this research The

fading endstate is not allowed as an endstate in any simulations unless explicitly

mentioned in the accompanying text

Perihelion to o large The comets p erihelion q has evolved b eyond some limit usually

taken to b e AU and is moving outwards under the inuence of the tide ie sin

Such a comet is unlikely to b ecome visible in the nearfuture Though the comets

p erihelion may b egin to decrease once again these new orbital parameters should

already b e represented within the ux of comets into the entrance surface and thus

need not b e considered further

ShortPerio d The comets orbital p erio d has decreased b elow yr it has b ecome a

shortp erio d comet The p ossibility exists that such a comet will diuse back out to the

LP regime but the small numb er of comets which reach this endstate see Tables

and later make the consideration of a return from this endstate unnecessary

Visible The comet has passed within AU of the Sun Such comets continue to evolve

after their rst apparition however it will b e seen in x that the rst appari

tion provides a useful intermediate stopping p oint for the simulations Though some

comets do b ecome visible with p erihelia greater than AU the observed sample is

almost certainly not complete b eyond this p oint

Mo del implementation and testing

The mo del was implemented as a computer co de written in ANSI C with all oating

p oint values in double bits digits precision The primary testing platform was

a Silicon Graphics IRIS DS the primary simulation platforms were two DEC Alpha AXP s

CHAPTER ALGORITHM

Integration tolerance

Testing was accomplished by examining each segment of included physics eg planets non

gravitational forces etc in isolation and comparing these restricted simulations with known

analytical results First however a set of criteria is needed to cho ose the tolerance while

balancing sp eed and accuracy The full simulation has no conserved quantities suitable for

determining the tolerance so was chosen such that the typical fractional error p er orbit

is less than for the relevant conserved quantities in each of the restricted simulations

describ ed b elow After exp erimentation the value of the tolerance was chosen to b e

The typical errors describ ed in the sections b elow are summarised in Table The

fractional energy change p er orbit for a typical LP comet in the complete mo del is of order

unity so an integration tolerance of one part in is more than sucient for this problem

The twob o dy problem

Consider the Sun and a single test particle This simple system tests the implementation

and b enets of regularising the co ordinates A hundred particles were run on orbits with

a AU and q AU e The other elements were chosen randomly

from a spherically symmetric distribution and the particle tra jectories were integrated for

orbits yr

The angular elements i and the angular momentum b oth showed a roughly

linear secular trend in error growth with orbit numb er with a fractional change of roughly

orbit A certain amount of scatter was also present with a fractional ro otmean

square RMS value p er orbit of

The errors in orbital energy showed no net growth but did show a fractional scatter of

ab out ab out the correct value However this scatter was not due to error in the regu

larised energy co ordinate h x which was p erfectly conserved but to slight roundo

errors in the transformation from the regularised co ordinates to Cartesian co ordinates to

orbital elements Also jE j is small to b egin with only AU so this large error is

not a concern

When the particles equations of motion are expressed in Cartesian co ordinates a much

lower integration accuracy is achieved Given the same initial conditions the angular ele

ments are typically conserved to only in over orbits b oth the energy and angular

CHAPTER ALGORITHM

Test a AU q AU E E J J W W

Twob o dy

Jupiter

Tide

Disk

Table Typical p erorbit errors for dierent test scenarios Initial semima jor axes a and

p erihelion distances q are shown on the left The errors are in the orbital energy E E

angular momentum J J and Jacobi constant W W

momentum drift by factors of ten or more and particles frequently reach unb ound orbits

The regularised co ordinates are thus to b e preferred for the integration of higheccentricity

orbits

The planets

In this section the gravitational inuences of the Sun and Jupiter alone on a test particle

will b e considered Collisions are ignored Jupiters orbit is taken to b e circular and the

only allowed endstate is ejection This simplied system will b e termed the SunJupiter

mo del or a Sunplanet mo del if an arbitrary planet is to b e considered

Precession

The planets p erturb comets in ways other than a simple energy kick Quinn et al

showed that the planets pro duce a precession of the comets orbits which can b e pictured

as a tra jectory in q space This precession is due to the quadrup ole moment of the

planets timeaveraged orbit A comparison of the results obtained with LOCI with those

of Quinn et al under the same initial conditions is shown in Figure

The phasespace tra jectories show the same general b ehaviour and the dierences ob

served are exp ected Quinn et al orbitaveraged the systems Hamiltonian and thus their

results lack the realistic sto chasticity present in our integration As well quantities which

are integrals of the motion of the averaged Hamiltonian are not strictly conserved under

the unaveraged Hamiltonian Figure a reects this fact through the slow wandering of

tra jectories away from their averaged ideals eg two of the tra jectories in Figure a cross

the q line due to drift in the z comp onent of the angular momentum Though the

existence of the ab ovementioned dierences b etween LOCIs and the orbitaveraged results

CHAPTER ALGORITHM

1.5

0.5

0 0.5 1 1.5

a b

Figure Evolution in q space of a test particle in a SunJupiter system a LOCIs

integration and b orbitaveraged results from Quinn et al The arrows in a

indicate the direction of evolution with time

are due to the more realistic treatment used the quantitative correctness of these deviations

remains unveried by this test

Lifetime against ejection

The Gamblers Ruin problem x predicts that of an initial sample of parab olic orbits

the probability of a particle remaining b ound to a Sunplanet system at the m p erihelion

passage is prop ortional to m Equation LOCIs duplication of this result app ears

in Figure the straight line indicates the leastsquares b est t the slop e of which matches

the exp ected result to within The initial cometary p erihelia are uniformly distributed

inside Jupiters orbit

Jacobis integral

The circular restricted threeb o dy problem deals with a massless particle in a Sun

planet system The only known integral of motion of this system is Jacobis integral W

Expressed in a nonrotating barycentric co ordinate system with the xaxis increasing along

the SunJupiter line and the y axis p erp endicular to x and in the orbital plane Jacobis

CHAPTER ALGORITHM

Figure Numb er N of test particles remaining in the SunJupiter system as a function of

orbit numb er m for a set of initially parab olic comets The particles initial p erihelia

were distributed linearly in q inside Jupiters orbit The straight line is a leastsquares t

to the p oints

integral can b e expressed as

M M

J up

W v y x xy G

J up

r r

J up

where is Jupiters orbital p erio d and v x y z is the test particles velo city

J up

Consider the test particles in the random walk describ ed in the previous section

Most particles are ejected by the one thousandth orbit but twenty remain These survivors

displays errors of hW W i or p er orbit

The error in the Jacobi integral is large compared to say the energy error in the

twob o dy problem x The diculty arises from close planetary encounters the

co ordinates are regularised ab out the Sun not the planet and the large accelerations which

o ccur during encounters degrade the simulations accuracy despite the reductions in step

size Though one could switch to a co ordinate system regularised ab out the planet in

question during close approaches the added complexity has b een judged excessive for our

purp oses The Jacobi error is the exception to the rule that the integrals of motion should

CHAPTER ALGORITHM

b e constant within p er orbit x

The distribution of energy p erturbations

The energy p erturbations imparted to parab olic comets making a single p erihelion passage

through a Sunplanet system were examined numerically by Everhart Figure

shows a comparison of our results for a SunJupiter system with Everharts functional t

n error bars the solid curve is Everharts result The The p oints are LOCIs results with

initial conditions for the comets used in our simulation were

Parab olic orbits started AU from the Sun

Perihelion distances distributed uniformly b etween and times Jupiters orbital

radius

Angular elements i chosen from a spherically symmetric distribution

The initial conditions for Everharts result were spherically symmetric angular elements and

p erihelia at Jupiters orbit The distributions show some slight dierences but these are to

b e exp ected Everharts function is chosen empirically to match the observed distributions

shap e rather than from any consideration of the physics and b ecause the shap e of the

distribution dep ends strongly on inclination it would b e surprising if an empirical function

could accurately represent all the phase space in question

In fact the function shown in Figure is not the function Everhart claims as rep

resentative of the distribution averaged over all inclinations but rather the sum of the

separate ts he pro duced each for a small range of inclinations b etween and Everharts

allinclin ations function diers markedly from our results and from the sum of his separate

determinations esp ecially for small p erturbations jaj AU The cause of

this discrepancy seems to b e that the empirical function Everhart chose though providing a

reasonable match when tted to the distribution asso ciated with a small inclination range

it is rather p o orly designed to match the sum of these distributions

The Galactic tide

Heisler and Tremaine analytically derived the orbitaveraged evolution of a comet

moving under the Galactic tide The comet moves along a tra jectory in K space where

e K J L and is the usual argument of p erihelion relative to the Galactic

CHAPTER ALGORITHM

1000

100

-0.01 -0.005 0 0.005 0.01 -0.001 -0.0005 0 0.0005 0.001

a b

Figure The a distribution for parab olic comets making a single orbit with p eri

helion near Jupiters orbit at two dierent magnications The heavy line is a comp osite

of Everharts empirical ts

frame The allowed curves are parametrised by K K cos the comp onent of K along

the Galactic zaxis Figure a displays LOCIs results for K Figure b those

of Heisler and Tremaine The data in Figure a which is sampled ab out once

p er orbit shows some scatter around the analytical tra jectory b ecause our mo del do es not

ignore the highfrequency comp onents that have b een averaged out of Heisler and Tremaines

result

The quantities J K and L are all conserved in the orbitaveraged approximation but

z z

only J is an integral of the unaveraged motion Our results display a net error in J of less

z z

than part in p er orbit over the to orbits required to travel from and

back again along the curves plotted in Figure The RMS error p er orbit was roughly

in the source of this error again b eing the accumulated error in the transformation

of co ordinates from regularised to Cartesian to orbital elements to J and is thus not of

concern

Nongravitational forces

Nongravitational forces can b e handled by KS regularisation despite the fact that they

are not conservative and thus cannot b e represented by a p otential The accuracy of the

CHAPTER ALGORITHM

0.9

0.8

C=1.40

0.7 C=1.20

C=1.01

C=0.75

0.6 C=0.42

0.5

0 1 2 3

a b

Figure Orbital evolution in K space under the dominant comp onent of the Galactic

tide a LOCIs results and b analytical treatment by Heisler and Tremaine The

curves lab elled a b d f and g in b corresp ond to the curves with C

and in a resp ectively

NG mo dule of the simulation was evaluated by comparing its results with a parallel but

indep endent integration of Gausss equations Equations Aaf

The test orbits had semima jor axes of AU p erihelion distances of AU and

random spherically symmetric angular elements The test duration was orbits Each

of the three NG parameters radial tangential and normal was tested individuall y with

values of AU day

The two metho ds showed dierences less than in on all elements on which the

NG forces in question should have no eect eg semima jor axis in the case A A

Otherwise the metho ds typically diered by rad orbit in the angular co ordinates

The fractional dierences in the energy and angular momentum were typically in and

in or and p er orbit resp ectively

Indep endent here means that though all dierential equations are integrated by the same routines

the simulation and the parallel integration cannot see each others co ordinate values

CHAPTER ALGORITHM

Massive circumsolar disk

The eects of a hyp othetical disk of material orbiting the Sun at distances of to AU

were mo delled through the addition of a MiyamotoNagai disk p otential describ ed in x

The test case had a AU b AU and M M with the test particle

d d d

orbits having a AU q AU and spherically symmetric angular elements The

test particles were followed for orbits The z comp onent of the angular momentum

J is an integral of the motion and was conserved to b etter than in p er orbit The

Keplerian orbital energy a is not conserved in the presence of the disk p otential but

the total energy Keplerian disk p otential is in the simulations it varied by only in

p er orbit as well

A more comprehensive test involved the cointegration of Gausss equations as was done

for the NG forces In this case the two schemes typically diered by rad orbit in the

angular co ordinates The fractional dierences in the angular momentum and the Keplerian

energy were in and in p er orbit resp ectively

Chapter

Results

The dynamical evolution of an Oort cloud comet can b e conveniently divided into two

separate stages for our purp oses The rst is the comets so journ in the outer reaches of

the Solar System b efore rst b ecoming visible visibili ty taken here to mean passage within

AU of the Sun The second stage encompasses the remaining evolution up to the p oint

where the comet is ejected or otherwise p ermanently excluded from the sample of observable

comets

Let the set of those LP comets which are making their rst visible p erihelion passage b e

called the V comets and those making their m visible apparition the V comets The

union of all visible LP comets will b e called the dynamically evolved visible LP comets The

lifetime of a comet m can b e measured by the numb er of p erihelion passages since and

including its rst apparition The numb er of visible p erihelion passages a comet makes in

its lifetime is denoted m The comets lifetime in terms of physical time t is also useful

v x

but is not clearly dened in all cases eg the exact moment of ejection may b e dicult to

determine

Original elements

The original elements of observed comets are computed by integrating their orbits backwards

from p erihelion and are typically quoted at distances of to AU from the Sun

However hyp othetical structures with radii of to AU are intro duced into some

of our simulations and complicated artifacts might b e intro duced into the elements if they

are calculated near a massive p erturb er To avoid this p ossibility the original elements

CHAPTER RESULTS

of our simulations are measured at aphelion instead Adjustments must thus b e made to

the simulated distributions of orbital elements in order to compare them with the observed

distributions but these adjustments are usually small The original inverse semima jor axis

is aected somewhat by the comets climb out of the Galactic p otential well Equation

but the resulting change in a is only of order

z z

a AU

M AU

which is small and can b e ignored However the shift in energy pro duced by a disk or

Planet X may b e more pronounced of order

r M M

X X X

r M M AU

a M M

d d d

a M M AU

where the companion has mass M and orbital radius r and the disk has a p otential

X X

describ ed by Equation The numerical values in Equation corresp ond to the max

imum change in a among the simulations to b e describ ed here In most cases the eect

is smaller and will b e neglected

It is interesting to note that that the mean excess velo city of observed hyp erb olic comets

roughly equivalent to a AU is not well explained by the presence of such dark

matter A shift of AU in inverse semima jor axis would pull the outer edge of the

Oort cloud inside AU and thus preclude the lling of the loss cylinder either by the

tide or passing stars Equations and b Nevertheless a few high darkmatter

simulations will b e p erformed for completeness

The newly visible comets

The newly visible or V comets constitute the injection sp ectrum from which the observed LP

distribution has evolved The V set can b e used as a starting p oint for any investigation of

phenomena that only aect the comet after its rst apparition eg nongravitational forces

fading The elements of the V comets are here measured in the barycentric frame at the

aphelion immediately preceding their rst apparition

The set of V comets pro duced in the course of this research has memb ers and was

obtained from a set of initial conditions within the Oort cloud Of these new

comets rst b ecame visible while still Oort cloud memb ers ie when a AU

CHAPTER RESULTS

Endstate Ejection Escap e Exc Limit Large q Short p d Visible Total

Numb er

Fraction

Minimum t

Median t

Maximum t

Minimum m

Median m

Maximum m

Table The distribution of endstates of the Oort cloud comets with minimum

p erihelia under the tide of less than AU The minimum median and maximum lifetimes

m and t are shown in in orbital p erio ds and millions of years resp ectively No comets

x x

suered collisions or survived for the lifetime of the Solar System

The V comets were pro duced over twentyseven separate runs on a DEC Alpha requiring

over thirteen weeks of real time and roughly eight weeks of CPU time at a tolerance

On average a new Oort cloud comet was started every seconds and a visible

comet pro duced every seconds

Of the Oort cloud comets simulated were determined to have minimum

p erihelion distances under the Galactic tide Equations and ab greater than

AU to o far outside the planetary system to suer appreciable planetary p erturbations

Figure These simulations were terminated immediately and counted as part of the

Perihelion too large endstate The remaining were integrated numerically and

of these only in b ecame visible Table shows the distribution of these comets

among the various endstates

During this previsibili ty stage of LP comet dynamical evolution there were close

encounters with the giant planets by individual comets distributed as shown in Ta

ble There were no captures or collisions though some comets did pass within the

planets satellite families

Only of the comets which were numerically integrated triggered the Exceeded

Time Limit ag see x set at revolutions These comets constitute only of

all initial conditions and are not included in the discussions to follow but will b e addressed

again in x

CHAPTER RESULTS

Planet Jupiter Saturn Uranus Neptune Total

Numb er of comets

Numb er of encounters

Encounterscomet

Collisions

Captures

R R

I p

Min distance R

Outer satellite R

Table Planetary close encounter data for the initial conditions which had initial

p erihelia within AU of the Sun Encounters for the comets in the Exceeded Time

Limit endstate are included only up to their orbit The size of planets sphere of

inuence R Equation and that of the orbit of the planets outermost satellite are also

given

Perihelion distance

The distribution of p erihelion distances of the V comets is shown in Figure a The

distribution is slop ed slightly upwards towards larger q This is to b e exp ected a full

loss cylinder should have a at distribution in q but empty or partially full loss cylinders

will have a prep onderance of orbits at larger p erihelia This conclusion is strengthened by

an analysis of Figure b which plots rst apparition p erihelion distance versus original

0 0 1 2 3

q (AU)

a b

Figure The V comets a their numb er distribution versus p erihelion distance q and

b their rst apparition p erihelion distance q versus original semima jor axis a

CHAPTER RESULTS

AU semima jor axis and which conrms the lling of the loss cylinder at a

Equation b

a b

Figure Distribution of orbital energies for the V comets a at the aphelion b efore

their rst apparition ie original energies and b at the aphelion immediately following

rst apparition The energies of unb ound orbits are measured at r AU

Orbital energy

The distribution of V orbital energies is shown in Figure b oth at the aphelion b efore

the comets b ecome visible and at the aphelion immediately after their rst apparition The

cuto at AU in Figure is articial a result of our choice of the Oort clouds

outer edge at AU Note the concentration of original orbits at very small but p ositive

energies In fact all but are in the region of the spike a AU The

orbits within the spike have a mean a AU This result is in go o d accord

with Heislers Monte Carlo simulations which indicated that outside of showers

the energy distribution of new comets is exp ected to p eak at AU Heislers

simulations also included passing stars indicating that our omission of these p erturb ers

do es not strongly aect the distribution of original semima jor axes

The p ostp erihelion energy distribution Figure b is much broader due to the plane

tary p erturbations discussed in x The distribution is highly symmetric ab out zero with

a median a of AU its full width at half maximum is AU

CHAPTER RESULTS

in reasonable agreement with the exp ected size of planetary p erturbations Equation

200

150

100

0 0

a (AU)

Figure Distribution of original semima jor axes a of the V comets The curve is

an analytical approximation to the exp ected distribution derived in App endix C

Despite the decrease in comet numb er density with radius in the Oort cloud comets

are injected primarily from its outer p ortion The distribution of original semima jor axes

is shown in Figure The median original semima jor axis is roughly AU and the

distribution displays a doublep eaked shap e with one p eak near and a smaller one

at AU An analytical approximation to the exp ected ux is shown by the curve it

is valid only where the loss cylinder is full and is derived in App endix C

Our choice of the edge of the Oort cloud at AU is the cause of the sharp decrease

in simulated comets b eyond this p oint in Figure The sharp dropo at semima jor axes

smaller than ab out AU is a result of the emptying of the loss cylinder as semima jor

axis decreases x The p osition and steepness of this dropo supp orts our choice

of the inner b oundary of the Oort cloud at AU rather than closer in Oort cloud

comets on orbits smaller than ab out AU are unable to enter the visibili ty cylinder

in appreciable quantities and thus their exclusion from our simulated Oort cloud should not bias the results

CHAPTER RESULTS

The nature of the smaller p eak at AU is unclear if the sample is split into two

parts it app ears only in one half and thus may b e a statistical uke despite its relatively

large size It will assumed to b e unimp ortant for two reasons Firstly only a few p ercent

of the V comets are involved and secondly the subsequent planetdominated evolution of

the V comets is relatively insensitive to the comets original semima jor axis as all Oort

cloud comets approach the Sun on essentially parab olic orbits

Angular elements

The aphelion directions of the V comets measured at the aphelion passage immediately

preceding their rst apparition are shown in Figure The most striking feature is the

concentration towards midGalactic latitudes a result of the Galactic tide However the

real distribution is exp ected to b e much clumpier due to the injection of comets by passing

stars

The inclinations longitudes of the ascending no des and arguments of p erihelion in the

ecliptic and Galactic frames are shown in Figures and The p eak in the cos

distribution near zero is exp ected the ux is prop ortional to J J J sin

E E

Equation Thus the ux is exp ected to increase towards high Galactic inclination ie

The p eaks in Figure b are also exp ected the regions where sin are the

regions where the p erihelia are moving inwards under the tide The distributions of cos i

and are relatively uniform exp ected b ecause the Galactic tide is indep endent of

these quantities

The longestlived comets

Only a small fraction of comets survive for more than orbits after their initial apparition

but this remnants extremely long lifetimes make it dicult to follow their evolution to

completion as noted in x

The numb er of comets remaining in the simulations as a function of orbit numb er is

shown in Figure a The amount of CPU time needed to follow all remaining comets for

the previous orbits is shown in Figure b As the numb er of orbits increases neither

the numb er of comets remaining nor the CPU time approach zero rapidly if at all Thus

for all practical simulation lengths some numb er of comets will always remain These long

lived comets are not necessarily the oldest in terms of physical time though they tend to

CHAPTER RESULTS

180 E 90 E O 90 W 180 W

a S

180 E 90 E O 90 W 180 W

Figure Equal area plots of the aphelion directions of the V comets in the a eclip

tic and b Galactic frames measured at the aphelion immediately preceding their rst apparition

CHAPTER RESULTS

a b

Figure Distribution of the cosine of the orbital inclinations for the V comets a at

p erihelion in the ecliptic frame and b at the previous aphelion in the Galactic frame

a b

Figure Distribution of the longitudes of the ascending no des for the V comets a at

p erihelion in the ecliptic frame and b at the previous aphelion in the Galactic frame

CHAPTER RESULTS

a b

Figure Distribution of the arguments of p erihelion for the V comets a at p erihelion

in the ecliptic frame and b at the previous aphelion in the Galactic frame

b e The nite age of the Solar System provides no relief as the dynamically oldest comets

in our simulations are signicantly less than billion years old Column of Table As

it seems imp ossible to follow all comets to completion it b ecomes necessary to arbitrarily

terminate the simulations at some p oint chosen here to b e at their orbit

The longestlived comets may provide clues to particularly stable regions in our Solar

System in terms of survival for large numb er of p erihelion passages rather than for long

times The p erihelion distances and semima jor axes of these comets on their orbit

are indicated in Figure These comets tend to b e on small orbits with semima jor axes

less than AU and there is an excess of prograde comets unexp ected

due to the shorter diusion times of prograde comets x These comets p erio ds

are typically much longer than the planets and only a few comets are near mean motion

resonance

Dynamically evolved longp erio d comets

Given a set of V comets the next logical step is to evolve them forward in time The

set of all apparitions made by the LP comets throughout their evolution consitutes the

dynamically evolved LP comets

CHAPTER RESULTS

a b

Figure Numb er of a longp erio d comets remaining in the Solar System and b CPU

seconds required to simulate all surviving comets for the previous orbits b oth plotted

as a function of age m measured in p erihelion passages since their to initial apparition

Figure Perihelion distance q versus semima jor axis a for the comets which survived

orbits after their initialisation within the Oort cloud

CHAPTER RESULTS

Element distribution parameters

It is useful to dene a few simple parameters which describ e the distributions of various

orbital elements

The ratio of the numb er of comets in the spike a AU to the total numb er

of longp erio d comets was intro duced in x

The inverse semima jor axis range available to LP comets runs from zero unb ound

to AU shortp erio d Let the ratio of the numb er of comets in the inner half

of this range to AU to the total numb er b e providing a measure

of the tail of the energy distribution

Let the ratio of prograde comets to the total numb er b e

For each of these parameters the ratio of the theoretical value to the observed indicated

by a prime value will b e called The following values of are adopted to b e used

for the determination of

These numb ers are based on those comets in Marsden and Williams catalogue with p erihe

lion distances less than AU The error estimates are based on Poisson noise

Evolved longp erio d comets

The simulation that evolves the V comets throughout their lifetimes but which do es not

include any p erturb ers except for the giant planets and the tide will b e called the standard

case The distribution of endstates for this case is shown in Table The visible end

state is disabled as it is in all further simulations describ ed in this chapter and all comets

are evolved until destroyed or lost The mean comet lifetime is orbits p er comet a

factor of two less than for a simple diusion pro cess x but signicantly less than

required to solve the fading problem x The standard case has

and and fails to match the observations The maximum allowed numb er

CHAPTER RESULTS

Endstate Ejection Large q Short p d Total

Numb er

Fraction

Minimum t

Median t

Maximum t

Minimum m

Median m

Maximum m

Table The distribution of endstates of the V comets simulated from initial apparition

until all are either lost or destroyed The minimum median and maximum lifetimes t of

these comets are measured from their rst apparition Of the subsequent p erihelion

passages are made by the comets which survive for or more orbits

after their rst apparition No comets escap e suer collisions with the planets or Sun or

survive for the age of the Solar System

of orbital p erio ds b efore the Exceeded time limit endstate x is invoked is

orbits for the standard case simulations but no comets reach this endstate

The planets are the dominant p erturb ers of dynamically older comets and ejection is the

most common endstate of V comets meet this fate The details of close encounters

b etween comets and the giant planets after initial visibility are detailed in Table Perhaps

most surprising is the high frequency of multiple encounters with a giant planet by a single

comet This do es not indicate a capture by the planet but typically arises when a very

distant encounter often near the comets p erihelion and which leaves the comets orbit

relatively undisturb ed is followed by one or more subsequent encounters in many cases

resulting in the ultimate ejection of the comet

Ab out of comets move back out to large p erihelion distances Most of

these remain memb ers of the Oort cloud the median a of these comets is AU

a AU Only have a AU with the smallest orbit having a semima jor

axis of ab out AU

N errorbars are no longer appropriate for histograms of the numb er distri Poissonian

butions as the individual apparitions are no longer uncorrelated one comet may contribute

hundreds or thousands of p erihelion passages The appropriate error bars in this case are

b o otstrap estimators and subsequent gures show the one standard deviation uncertainties

estimated by this metho d Efron Press et al

CHAPTER RESULTS

Planet Sun Jupiter Saturn Uranus Neptune Total

Numb er of comets

Numb er of encounters

Encounterscomet

Collisions

Captures

Min distance R

Outer satellite R

Table Planetary close encounter data for the dynamically evolved longp erio d comets

The distance to each planets outermost satellite is given in the last row

Perihelion distance

The p erihelion distribution of the dynamically evolved visible comets is shown in Figure

The structure visible is partly due to the strongly correlated contributions of very long lived

comets of the visible p erihelion passages over are made by the

comets which survive for or more orbits after their rst apparition Comparison

with the observed distribution Figure reveals some sup ercial similarities but the

strong observational selection in favour of ob jects near the Sun or the Earth makes drawing

conclusions dicult

Let the total numb er of comets with p erihelia inside q b e N Then a linear least

squares t to the simulated distribution yields dN dq roughly prop ortional to q for q

AU similar to Everharts b earlier estimate of the intrinsic p erihelion distribution

However the simulations could arguably b e consistent with any numb er of slowly varying

functions of p erihelion over q AU p ossibly including dN dq q as prop osed

by Kresak and Pittich The estimates of the intrinsic p erihelion distribution of LP

comets published by Everhart and by Kresak and Pittich are indicated on Figure by

the solid and dashed curves

Orbital energy

The original energy distribution of the visible comets is shown in Figure at two dierent

magnications for all p erihelion passages and for all visible passages The

fraction of comets in the spike obtained from these simulations is

all p erihelion passages with AU of the Sun b eing deemed observed Thus the simulations

pro duce visible LP comets for each comet in the spike whereas the observed sample has

CHAPTER RESULTS

4000

3000

2000

1000

0 0 1 2 3

q (AU)

Figure Distribution of p erihelion distances q for the dynamically evolved LP comet

p opulation Error bars are b o otstrapbased one estimates The curves are Everharts

b dashed lines and Kresak and Pittichs solid line estimates of the intrinsic

p erihelion distribution

them in only a to ratio Figure

This disagreement is at the heart of the fading problem how can the loss of of the

dynamically older longp erio d comets b e explained This question will b e addressed in the

up coming sections of this chapter

These simulations allow us to estimate the contamination of the spike by dynamically

older comets There are V comets of which have a AU but a total of

apparitions are made by comets with a AU Thus roughly of comets in the

spike are not dynamically new and of comets coming from the Oort cloud do not make

their rst app earance within the spike However these numb ers ignore the p ossibility of

signicant reductions in the brightness of LP comets over time and are thus only upp er

limits x

CHAPTER RESULTS

1000

4000

500 2000

0 0 -0.01 0 0.01 0.02 0.03 0 0.0002 0.0004 0.0006 0.0008

1/a (1/AU) 1/a (1/AU)

a b

5000

1000 4000

3000

500 2000

1000

0 0 -0.01 0 0.01 0.02 0.03 0 0.0002 0.0004 0.0006 0.0008

1/a (1/AU) 1/a (1/AU)

c d

Figure Distribution of original orbital energies for the dynamically evolved comets

a b for all p erihelion passages and c d for those p erihelion passages which are visible

q AU

CHAPTER RESULTS

Angular elements

The inclination distributions in the ecliptic and Galactic frames are shown in Figure

There is a noticeable excess of comets in ecliptic retrograde orbits the fraction on prograde

orbits is This is inconsistent with observations as Marsdens comets have

a ratio near a half Figure a The simulated Galactic inclinations are consistent with

a at distribution as exp ected from theory but whether or not the observed distribution Figure b is at is less clear

5000

6000

4000

3000 4000

2000

1000

0 0 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1

cos i

a b

Figure Distribution of the cosine of the inclination for the visible LP comets a taken

at p erihelion in the ecliptic frame and b at the previous aphelion in the Galactic frame

Figure shows the distribution of the longitude of the ascending no de and Fig

ure that of the argument of p erihelion The planets the dominant p erturb ers are not

exp ected to pro duce strong signatures in these angular elements and our results seem mo d

erately consistent with this exp ectation There are a few p eaks in the gures particularly

in and which may b e statistically signicant These bumps are the result of several

longlived comets clustered together in the phase space in question but whether this

is chance or a systematic eect is unclear

The standard mo del provides only a p o or t to the observed distributions in particular

as regards the orbital energies and inclinations Assuming that our simulations correctly

p ortray the intended physics the next question is how do our simulations dier from

CHAPTER RESULTS

4000

2000

a b

Figure Distribution of the longitude of the ascending no de of the visible LP comets

a taken at p erihelion in the ecliptic frame and b at the previous aphelion in the Galactic frame

6000 6000

4000 4000

2000 2000

0 0

a b

Figure Distribution of the argument of p erihelion of the visible LP comets a taken

at p erihelion in the ecliptic frame and b at the previous aphelion in the Galactic frame

CHAPTER RESULTS

reality Before going on to examine this question we will consider the implications of these

simulations regarding the p opulation of the Oort cloud

The current Oort cloud p opulation

The ux into the entrance surface Equations Cab can b e integrated over all semima jor

axes or all L in order to relate the constant g to the total numb er of simulated comets

L a

+ +

Z Z

L J dL N a q da t t

E E E E s s

where t is the length of real time represented by our simulation Equation can easily b e

integrated numerically to yield N g t for the usual values of the parameters

in question For the comets in our Oort cloud simulation x this implies

g t yr The value of t dep ends on N and the total numb er of comets in

the Oort cloud

Before calculating the value of t consider rst the total numb er of Oort cloud comets

As the Oort cloud is assumed to b e in a steadystate r can b e replaced in Equation

by its time average hr i a Then the total numb er N of comets in the Oort can b e

total

obtained through

a L

+ + L J

Z Z Z Z Z Z Z

n a a da N df d dL d dJ g L dJ a

total z

L J

g a a b

year

where has b een assumed and from which the numerical value for the density

co ecient can also b e obtained

n g

year

If one knows the numb er N of longp erio d comets crossing within AU p er unit time

v o

in our Solar System the value of t and hence the total numb er of comets in the Oort

cloud can b e estimated from Equation c If the numb er of simulated comets which enter

the visibili ty cylinder is N for our simulations then t N N The timescale

v t s v t v o

CHAPTER RESULTS

over which Marsden and Williams catalogue is complete if any is unclear but rough

estimates of comet uxes can b e made Everhart b estimated that comets passed

within AU of the Sun within a year p erio d implying roughly yr Kresak and

Pittich deduced yr within Jupiters orbit Taking yr within AU as an

estimate and assuming one in three of these is dynamically new Festou et al b yields

t yr

This value implies an Oort cloud p opulation of roughly comets b etween

and AU Equation c However this metho d b ecomes an increasingly p o or prob e

of the Oort clouds p opulation as distances b ecome large and the visible ux falls to zero

Figure

The original Oort cloud p opulation

Ab out of Oort cloud comets crossing the entrance surface are removed from the cloud

x The ux of comets across the entrance surface is prop ortional to the total numb er

of comets in the cloud Equations and a thus b oth decay exp onentially with time

ignoring other loss mechanisms Roughly comets are removed during the years

corresp onding to these simulations implying that approximately ob jects have b een

lost since the Solar Systems formation As these gures ignore other loss mechanisms

particularly stripping by passing stars they likely to b e much to o low Nevertheless it

seems likely that the Oort cloud originally had at least twice its current p opulation

Discovery probability function

The previous simulations assumed that all comets passing within AU of the Sun would

b e detected by astronomers however the strong bias in the observed distribution towards

comets near either the Sun or the Earth Figure indicates that this is unlikely to b e

true

Everhart a examined the LP comets which b ecame part of the observed sample

and concluded that comets which have the same excess magnitude S have equal a pri

ori chances of b eing discovered The excess magnitude is dened by

S H H H H dt

an an

CHAPTER RESULTS

where is the comets p erio d H is its visual magnitude at the Earth Equation H

is some lower limiting magnitude and x is a step function which is unity where x

and zero where x Comets with large values of S are bright and visible for a relatively

long time and thus more likely to b e observed

Everhart found that the discovery probability p increases roughly linearly with S

Here a form for p S of

S magweeks if S magnitudeweeks

p S

if S magnitudeweeks

will b e adopted The excess magnitude dep ends sensitively on the p osition of the Earth

but for simplicity the Earth will b e taken to b e at its average p osition ie at the Sun for

these calculations In this case Equation b ecomes

H H n log r

recalling that r must b e measured in AU The values of n H and H are taken to b e

and resp ectively Everhart ab These values imply that the excess magnitude

b ecomes nonzero at q AU

Given the previous assumptions the excess magnitude is simply a function of q The

tra jectories of LP comets near the Sun approximate parab olas and thus r q cos f

will b e adopted The excess magnitude is then

S H H n log r H H dt

an an

q log q cos f

cos f

where f is the true anomaly at which the comets visual magnitude exceeds the limit H

an an

given by

H H

5+25n

j cos q j q cos f

A plot of S versus q app ears in Figure The excess magnitude is magnitude

weeks at q AU and drops to zero at q AU with a roughly linear relationship

CHAPTER RESULTS

Figure The simplied excess magnitude S versus p erihelion distance q as expressed

in Equation

b etween these two p oints As an approximation the discovery probability is taken to b e

if q AU

p q

q AU if q AU

if q AU

The application of this discovery probability to our simulations is shown in Figure

This addition changes the agreement with observations very little as the other orbital

elements are only weakly correlated with p erihelion In some simulations where this is

not the case the comet discovery probability proves to b e imp ortant but it do es little to

improve the standard mo del

Shortp erio d comets

Before going on to consider other p ossible dynamical eects lets consider rst those short

p erio d comets which originate at the Oort cloud During the standard simulations only

Oort cloud comets eventually b ecome shortp erio d comets of them after having made

CHAPTER RESULTS

3000

2000

1000 1000

0 0 -0.01 0 0.01 0.02 0.03 0 1 2 3

1/a (1/AU) q (AU)

a b

Figure Distribution of a original orbital energies a and b p erihelion distances q

for the visible longp erio d comets sub ject to the discovery probability function given by

Equation

one or more apparitions as LP comets Only of these have p erihelia less than AU ie

are visible on their rst p erihelion passage as SP comets

The aphelion directions of all of these SP comets are shown in Figure Though

small numb ers make the data relatively noisy there may b e some concentration of comets

in the ecliptic plane

The distributions of inverse semima jor axis p erihelion distance and inclination are

shown in Figure None of the SP comets arrive directly from the Oort cloud The

one with the largest orbit at the previous aphelion has a semima jor axis of only AU

and was on its sixth orbit since crossing the entrance surface The p erihelion distribution

tends to increase towards the Sun but the error bars are to o large to draw any rm con

clusions from this trend There is a distinct concentration of orbits near zero inclination

as exp ected from studies of captures of comets by Jupiter from spherical sources Everhart

but much less than that of shortp erio d comets in our Solar System Figure

The prograde fraction is

In x the standard case simulations are found to corresp ond to approximately

years of real time Equation The standard mo del thus implies that

shortp erio d comets p er year arrive indirectly from the Oort cloud though this numb er

CHAPTER RESULTS

180 E 90 E O 90 W 180 W

Figure Shortp erio d comet aphelion directions standard case on an ecliptic equal

area map Aphelion directions are measured on the aphelion previous to their rst p erihelion

passage as a shortp erio d comet

is really an upp er limit as fading has not yet b een considered As an average of ve new

SP comets are discovered each year Festou et al a one deduces that the Oort cloud

must contribute less than of the p opulation of SP comets and another source for these

comets is required Our results thus are consistent with the the primary source of SP comets

b eing the Kuip er b elt

Planetary encounter rates

The length of time represented by these simulations will b e shown to b e roughly yr

x and from this the rate of close encounters b etween the LP comets and the giant

planets can b e calculated A total of encounters by ob jects were recorded for Jupiter

by for Saturn by for Uranus and by for Neptune These numb ers

translate to total rates of and comets p er year passing through the spheres

of inuence Equation of Jupiter through Neptune resp ectively

If these numb ers are naively taken to represent a uniform ux F across the sphere of

inuence the rate n of impacts b etween LP comets and the giant planets can b e deduced

to b e

R M

p p

n F

M r

CHAPTER RESULTS

20 30

0 0 0 0.01 0.02 0.03 0.04 0 0.01 0.02 0.03 0.04 1/a (1/AU) 1/a (1/AU)

0 0 0 2 4 6 8 10 -1 -0.5 0 0.5 1

q (AU) cos i

Figure The distribution of the inverse semima jor axis a p erihelion distance q and

cosine of the inclination i for the shortp erio d comets pro duced in the standard mo del

The distribution of a on the left is measured at the aphelion previous to and the other

distributions measured at the initial p erihelion passage as a SP comet for the standard case

CHAPTER RESULTS

where r and R are the planets orbital and physical radii and M their mass The

p p p

resulting collision rates are p er year for Jupiter through

Neptune resp ectively It should b e noted that Comet Sho emakerLevy which collided

with Jupiter in July of was not a LP comet when captured by that giant planet but

rather a SP Jupiterfamily comet Benner and McKinnon

Nongravitational forces

In the previous simulations only the Galactic tide and giant planets were included The

exp ected eects of outgassing forces on the dynamics of comets were detailed in x Non

gravitational forces are sp ecied by two parameters x The radial comp onent A is

always p ositive as the outgassing force always accelerates the comet away from the Sun

The tangential comp onent A is generally less than A jA j jA j will typically b e

adopted here and may b e of either sign dep ending on the comets rotation

Two simple cases

Consider two simple cases each with nongravitational forces A AU day and

jA j A In the rst case A will b e taken to b e p ositive and in the second negative

These choices imply a constant rotation vector for the nucleus which is unlikely but test

the cases in which the NG forces are maximally ecient

In the rst case A AU day and A AU day The distributions of

orbital elements pro duced are shown in Figure and are characterised by the parameters

and The orbital energy and p erihelion distance are exp ected

to decrease secularly under the NG forces Equations a and b Thus the NG forces

act to unbind comets b oth by reducing a and by drawing the comets p erihelia inwards

to where the NG forces are more eective This results in an increase in to from

in the standard case insucient to pro duce a match with observations

There is only a increase in the numb er of comets ejected over the standard case

but there is a signicant reduction in comet lifetimes to a mean of from orbits

This is due to a decrease in the lifetime of ejected comets to orbits from The

p erihelion distribution Figure shows a strong erosion of the comet p opulation at small

p erihelion distances This indicates that even mo dest nongravitational forces likely play a

CHAPTER RESULTS

4000

3000

2000

1000 1000

0 0 -0.01 0 0.01 0.02 0.03 0 1 2 3

1/a (1/AU) q (AU)

a b

Figure Distribution of the a original inverse semima jor axis a and b the p eri

helion distance q for the visible LP comets under nongravitational forces characterised by

A AU day and A AU day

signicant role in shaping the p erihelion distribution of the LP comets

The second case with A AU day A AU day is identical to

the previous one except for A b eing of the opp osite sign In this case the NG forces act

to increase the p erihelion distance and a of comets Equations a and b tending

to circularise their orbits The increase in a causes comets to evolve into more tightly

b ound and hence longlived orbits average lifetime orbits There is also an increase

in shortp erio d comet pro duction versus in the standard case The distribution

parameters are and which show no improvement over the

standard case

The p erihelion distribution shows little or no erosion near the Sun as there is no pref

erential unbinding of comets with small q The distribution do es not increase noticeably

towards larger p erihelion despite the secular increase in q presumably b ecause the rate of

increase of p erihelion is quite small

More realistic nongravitational forces

Neither of the two simple cases presented ab ove match the observed distributions particu

larly well But realistically Oort cloud comet nuclei are likely to have randomly oriented

CHAPTER RESULTS

5000

4000 4000

3000 3000

2000 2000

1000 1000

0 0 -0.01 0 0.01 0.02 0.03 0 1 2 3

1/a (1/AU) q (AU)

a b

Figure Distribution of the a original inverse semima jor axis a and b the

p erihelion distance q for the LP comets under nongravitational forces characterised by

A AU day and A AU day

axes of rotation with a corresp onding random value of A While a complete investigation of

the available parameter space is b eyond this pro ject the inadequacy of NG forces to resolve

the fading problem completely can b e demonstrated with only a few sample simulations

It will b e assumed that the value of A is always ten times that of jA j Two distributions

of the sign of A will b e considered

Half the comets have p ositive values of A half negative and the sign of A is constant

throughout a comets lifetime This corresp onds to the axis of rotation of the nucleus

remaining essentially unchanged throughout the comets dynamical lifetime

The sign of A is randomised after each p erihelion passage with a chance of

b eing either p ositive or negative This corresp onds to a rapid changing of the axis of

rotation due to precession of the nucleus

Eight dierent simulations were p erformed four values of A are examined from

to AU day in factor of ten increments and each is run for b oth distributions

of A discussed ab ove The sets containing the weakest NG forces ie A and

AU are reasonably consistent with the NG forces calculated for known LP comets

Marsden et al The two largest values for A are probably unrealistic in the

CHAPTER RESULTS

strongest case the a due to NG forces over one orbit exceeds that due to the planets

by a factor of Nonetheless these exp eriments provide useful test cases

There are a numb er of biases intro duced into the simulations when very strong NG forces

are intro duced Firstly NG forces may b ecome nonnegligible for comets with q AU

which is not accounted for in our simulations Secondly though some NG p erturbations av

erage to zero over a full orbit their eects over fractions of an orbit may b ecome dynamically

imp ortant for very large NG forces eg Equation a

Figure illustrates the results of the simulations including the more realistic NG

forces and Table lists some asso ciated quantities It is found that increasing the NG

forces do es decrease the numb er of dynamically older LP comets in the system ie in

creasing decreasing but also ero des the p opulation of comets at small p erihelion

distances Note that even excessive NG forces cannot bring the distribution of inverse semi

ma jor axes into line with observations and result in an extreme depletion of comets with

small p erihelia in contradiction with observations Figure The failure of the NG forces

can b e summarised as follows

Perturbations due to radial NG forces average to zero over a full orbit and thus have

no longterm eect on LP comet evolution assuming outgassing is symmetrical ab out

p erihelion

Positive values of the tangential acceleration A reduce the tail of the p opulation

resulting in an increase in towards unity and improving the match with observa

tions but ero de the p opulation at small p erihelia an eect which is not seen in the

observed sample

Negative values of A preserve a reasonable p erihelion distribution but increase the

numb er of comets in the tail of the energy distribution thus reducing and degrading

the match of the a distribution

Though one might b e able to conco ct a mixture of NG forces which will result in a b etter

match with the observations nongravitational forces seem unable to decisively resolve the fading problem

CHAPTER RESULTS

Figure Distributions of the inverse semima jor axis a and p erihelion distance q for the visible

LP comets On the left constant values for A half p ositive half negative On the right the sign

of A is randomised for each p erihelion passage The values of jA j used are from the top down

2 2

9 8 7 6 2

and AU day with A jA j The b ottom section is for comparison

1 2

and includes the standard case left side and the observations right side The observed p erihelion

distribution includes curves indicating the estimated intrinsic distribution x

CHAPTER RESULTS

Figure Distributions of the inverse semima jor axis a and p erihelion distances q for

the LP comets when a discovery probability Equation is applied The simulations

are otherwise identical to those in Figure

CHAPTER RESULTS

A A Total Spike Tail Prograde m

Table The parameters of the visible LP comet orbits under dierent nongravitational

forces Total is the total numb er of apparitions ie q AU Spike is the numb er of

these with original semima jor axes greater than AU and Prograde the numb er with

ecliptic inclinations less than The lifetime in orbits m includes all p erihelion passages

regardless of q after the initial apparition The sup erscript indicates that half the sample

have p ositive A half negative indicates A has a random sign assigned for each p erihelion

passage

A A Total Spike Tail Prograde Undisc

Table The parameters of the LP comets for the same conditions as in Table but

with the inclusion of the discovery probability as given by Equation The rightmost col

umn lists the fraction of comets with q AU that go undiscovered under this probability

function

Discovery probability function

The application of a discovery probability to these simulations is shown in Figure the

nongravitational forces are identical to those in Figure but apparitions are given a

weight prop ortional to their discovery probability Equation

The addition of the discovery probability to the simulations improves the match with

the observations to some degree and all tend towards unity as one moves down

CHAPTER RESULTS

Table However unrealistically strong NG forces are still required to match the inverse

semima jor axis distribution and result in an unacceptable p erihelion distribution Thus it

seems that NG forces are unable to solve the fading problem though they likely contribute

to the shaping of b oth the a and q distributions

Other scenarios

The fading problem refers to the discrepancy b etween theory and observations in the numb er

of comets with semima jor axes b etween a hundred and a few thousand AU see Figure

Thus the presence of a p ossibly unsusp ected mechanism which preferentially removes

such comets could explain the discrepancy Such a removal mechanism might arise from

structures with size scales comparable to those of the orbits which they are to aect most

strongly Some real and hyp othetical structures with hundred AU size scales include

The heliopause where the solar wind meets the interstellar medium

A massive circumsolar disk p erhaps related to the to kilometer sized ob jects

that have b een discovered in the Kuip er b elt b eyond Neptune Jewitt and Luu

Co chran et al

A massive solar companion ob ject at to AU

The existence of the heliopause is well established eg Kurth et al Linsky

and Wo o d however it will b e shown that its dynamical inuence is small and no

simulations were p erformed to examine its eects on LP comets x

The two other hyp otheses are more controversial Though little if any evidence supp orts

the existence of undiscovered disks or planetary ob jects in our Solar System they cannot

yet b e excluded and longp erio d comets may prove to b e the most sensitive to ols we have

for constraining their prop erties For this reason these two scenarios will b e examined here

In order to reduce the computational cost of these investigations the V comets are used

as a starting p oint ie the eect of the disk or companion is ignored previous to the comets

rst visible apparition More precisely V comets are restarted at the aphelion previous

to their initial apparition in order to correctly calculate any p erturbations o ccurring on

the inb ound leg immediately preceding their rst apparition Though the addition of the

p erturbations due to a disk or companion only after comets b ecome visible is unrealistic

CHAPTER RESULTS

it provides a rst lo ok at whether or not these features act in the right direction to resolve

the fading problem

Massive circumsolar disk

A circumsolar disk is represented by means of a MiyamotoNagai p otential Equation

centred on the Solar Systems centre of mass The disk lo oks like a p oint mass at distances

r from the centre of mass which are large compared with the disks characteristic radius

a thus it might b e exp ected to inuence the dynamical lifetimes of comets with a a

d d

most strongly

Comets coming in from the Oort cloud falling through the disks p otential are sub ject to

an apparent decrease in their original inverse semima jor axis This oset AU

for a M disk with radius AU and less for larger or less massive disks Equation

is omitted from the gures to follow The original inverse semima jor axes are measured at

aphelion as discussed in x

Three disk masses were examined and Jupiter masses The results are displayed

in Figures and with a discovery probability given by Equation used in the

latter In x it was noted that disk masses ab ove one Jupiter mass violate various

dynamical and observational constraints and that the upp er limits might b e even lower

thus our chosen disks include some of unrealistically high mass Two disk shap es were

investigated b oth with axis ratios of a b The rst had a characteristic size a

d d d

AU corresp onding to a Kuip er b eltlike disk the second had a AU similar in

size to the Pictoris disk

The evolution of comets with p erihelia outside the planetary system is more complicated

in the presence of a disk Of most concern here is the validity of the Perihelion Too

Large endstate x This endstate assumes that comets with p erihelia at AU and

with sin are unlikely to b ecome visible x This assumption is only correct if

the torque is dominated by the Galactic tide and this may not b e the case when a disk is

present However to keep the simulation times reasonable it was necessary to retain the

Perihelion Too Large endstates threshold at AU The a posteriori justication

is that in simulations which include disks this endstate is similarly p opulated in fact

ordinarily a bit underp opulated relative to the standard case Table indicating that

this shortcut is not grossly aecting the simulation results

CHAPTER RESULTS

The threshold for the Exceeded Time Limit endstate x was again set at

orbits Only one comet reached this state in the simulation with the largest and most

massive disk It was on a highinclin ation large p erihelion orbit a AU q AU

i Up on reexamination it was found to b e ejected orbits later after con

tributing visible p erihelion passages These apparitions are included in all the relevant

gures

The disk applies a torque to the comets resulting in a change in p erihelion distance As

with the Galactic tide the disk torque can pro duce an oscillatory motion of the cometary

p erihelion the frequency of which increases with increasing disk mass This eect ordinarily

results in an increase in the comets average lifetime as the risk of ejection is much reduced

when the comets p erihelion is outside Saturns orbit but also signicantly reduces the

numb er of apparitions comets make in a given numb er of p erihelion passages

The net eect of the disks is shown in Figure The p erihelion distribution of visible

comets is not strongly aected remaining more or less at The disk do es decrease the

p erihelion distance of some comets suciently to collide with the Sun The numb er of such

incidents is noted in Table

The values of the parameters for this mo del are also listed in Table The values of

are far smaller than unity even for the most massive disks and it is clear from Figure

that the simulated a distributions remain much broader than the observations though

with some improvement as disk mass is increased

M a b Numb er Oort Min a Med a Max a

d d d

Table Characteristics of the inverse semima jor axis distributions for the Perihelion

Too Large endstate when the Solar System contains a massive circumsolar disk Num

b er is the numb er of comets which entered this endstate and Oort the numb er with

semima jor axes greater than AU Units of a are AU The standard case no

disk is indicated on the rst line and all simulations start from the set of V comets

M a and b are the parameters of the disk measured in Jupiter masses and AU Equa

d d d tion

CHAPTER RESULTS

Figure Distribution of the inverse semima jor axis a and p erihelion distance q for

the visible LP comets when the Solar System contains a massive circumsolar disk All

simulations have disk axis ratios a b The simulations on the left have characteristic

d d

disk widths a AU those on the right a AU The disk masses increase from

d d

the top down with values of and Jupiter masses The b ottom line of graphs is for

comparison and includes the standard case left side and the observations right side The

lower rightmost graph includes Everharts and Kresaks estimates of the intrinsic p erihelion

distribution x shown as the dashed and solid curves resp ectively

CHAPTER RESULTS

Figure Distributions of the inverse semima jor axis a and p erihelion distance q for

the visible LP comets when the Solar System contains a massive circumsolar disk and the

discovery probability function is given by Equation The disk characteristics are the same as in Figure

CHAPTER RESULTS

M a b Total Spike Tail Prograde m R

d d d

Table Parameters of the distributions of the visible longp erio d comets when the Solar

System contains a circumsolar disk The units of M are Jupiter masses those of a and

d d

b AU The rightmost column indicates the numb er of comets which collided with the Sun

The sup erscript indicates that the discovery probability from Equation has b een

applied The denitions of the other columns are the same as in Table

M M M M

d J up J up J up

a AU AU AU AU AU AU

Numb er of ob jects

Numb er of encounters

Encountersob ject

Collisions

Captures

Table Planetary close encounter data for the dynamically evolved longp erio d comets

under a massive disk All collisions listed are with the Sun however close encounters with

the Sun are not included in this table

Increasing the disk mass also tends to improve and for the AU disk though

it acts in the opp osite direction for the smaller disk There is no set of disk parameters

that comes close to pro ducing a match with observations More massive disks might b e

able to do b etter but these would violate even more strongly the mass constraints on such

an ob ject x Thus it seems unlikely that the fading problem is the result of a massive

circumsolar disk

Massive solar companion

A massive unseen companion to the Sun such as a Planet X or Nemesis ob ject can b e

mo delled as a fth planet For simplicity only circular orbits lying in the ecliptic plane

are considered The companion was added to the simulations at the aphelion immediately

CHAPTER RESULTS

preceding the comets rst visible apparition as was done for the disk x

Companions of and Jupiter masses were simulated on orbits with radii of

and AU The orbital p erio ds of these ob jects are and years resp ectively

The companion masses used are based on Figure in Tremaine and chosen so that

the companion would not violate or violate only weakly the constraints on its mass arising

from considerations of the prop erties of Oort cloud comets and the planets Note that as in

the previous section the original semima jor axes of the comets are measured at aphelion

and thus do not include the energy oset caused by their fall through the companions

gravitational p otential Equation

Nine comets reached the Exceeded Time Limit endstate here set at orbits

many more than in the standard case where it was not reached at all Of these nine ve

survived for another orbits without b ecoming visible these will b e neglected The

other four do eventually b ecome visible contributing a total of apparitions which are

included in the following gures

The simulation results are presented in Figure and in Figure with the addition

of a discovery probability Equation The p erihelion distribution takes on a variety of

forms from those concentrated at smaller AU distances to those concentrated further

out There is no clear trend with mass or companion orbit size The large error bars seen

on some of the histogram bars indicate a noisy distribution ie one where a few individual

comets contribute a signicant fraction of the total numb er of apparitions In particular

the p erihelion distribution for the AU Jupiter mass disk shows a sharp spike in

the smallest bin but with an error bar roughly twothirds its height This is due to a single

comet b ecoming trapp ed for a long time in highinclin ation orbit and do es not seem to

b e indicative of a real clustering of ob jects in that vicinity

The parameters for this mo del are listed in Table As the companion mass is

increased the fraction of prograde to retrograde comets improves esp ecially in the case

of the larger companion orbit The companion also reduces the height of the tail of the

a distribution increasing slightly with disk mass but remaining b elow unity The

numb er of dynamically oldest comets remains high without any clear trend with increasing

mass Overall the presence of a companion pro duces little improvement in the match with

observations thus we conclude that the fading problem is unlikely to arise due to such a

companion ob ject at least of the typ e examined here

CHAPTER RESULTS

Figure Distribution of the inverse semima jor axis a and p erihelion distance q for

the visible LP comets when the Solar System contains a massive solar companion The

simulations on the left have companion semima jor axes a AU those on the right

a AU The companion masses increase from the top down with values of

and Jupiter masses The b ottom line of graphs is for comparison and includes the

standard case left side and the observations right side The lower rightmost graph

includes Everharts and Kresaks estimates of the intrinsic p erihelion distribution x

shown as the dashed and solid curves resp ectively

CHAPTER RESULTS

Figure Distribution of the inverse semima jor axis a and p erihelion distance q for

the visible LP comets when the Solar System contains a massive solar companion and the

discovery probability function is of the form of Equation Conditions are otherwise

identical to those in Figure

CHAPTER RESULTS

M a Total Spike Tail Prograde m R

X X

Table Parameters of the visible longp erio d comets when the Solar System contains

a massive solar companion The companions mass M is in Jupiter masses the size of

its orbit a in AU The rightmost column indicates the numb er of comets which collided

with the Sun The sup erscript indicates that the discovery probability Equation

has b een applied The other columns are the same as in Table

M M M M

X J up J up J up

a AU AU AU AU AU AU

Numb er of ob jects

Numb er of encounters

Encountersob ject

Collisions

Captures

Table Planetary close encounter data for simulations including a solar companion

Close encounters with the Sun and the companion are excluded All collisions o ccurred

with the Sun

Heliopause

The outowing solar wind encounters the interstellar medium at the heliopause Sho cks form

b oth inside and outside this interface as the lowdensity highvelo city solar wind meets the

higher density but slower moving ISM These sho cks heat the gas to temp eratures of up

to K throughout a shell roughly AU thick This shell is lo cated approximately

AU from the Sun in the upstream direction Hall et al Both the radio and

ultraviolet signatures of these sho cks have b een detected Kurth et al Linsky and

Wo o d and the heliopauses existence is wellestablished

If the sho cks asso ciated with the heliopause pro duced conditions which signicantly

ero ded comet nuclei they could signicantly reduce the lifetime of LP comets However a

CHAPTER RESULTS

rst glance seems to indicate that there is neither enough mass nor energy in these structures

to do so The density at the b oundary is not exp ected to exceed that of the ISM and the gas

velo city is not exp ected to exceed that of the solar wind Steinolfson et al Baranov

and Zaitsev thus the plasma density remains low and the conditions of drag do not

appreciably dier from those discussed in x

The gas temp erature at the heliopause is roughly K exceeding that of the interstellar

medium by roughly two orders of magnitude An upp er limit to the amount of heat E

dep osited in the nucleus through direct contact with this hot medium can b e set by assuming

complete transfer to the comet of all the kinetic energy in any molecule coming striking the

nucleus Then E is just the energy ux due to gas thermal motions multiplied by the time

the comet is in the heated region

E R v nk T a

h g

R nw k T b

GM m

R w n r T

c h h

J c

km AU AU

cm K

where R is the comets radius n is the gas numb er density m is the gas molecular mass

c g

here assumed to b e protons T is the gas temp erature v k T m is the gas thermal

g g

velo city w is the width of the heated zone r is its distance from the Sun and v is the

h h

comets velo city there The latent heat of sublimation L for H O is ab out kJ mol

Keller and the dep osition of E thus results in the release of a mass M of gas such

that

E N m

h A H

M a

L E

kg b

J kJ mol

where N is Avogadros numb er The loss of this small amount of material is unlikely to

signicantly aect the nucleus

Other p ossibilities remain the sho cked gas emits UV photons which might dep osit sig

nicant energy in the nucleus however the photon energy density is unlikely to exceed the

thermal energy density by the several orders of magnitude required More sp eculative p os

sibilities include thermal andor acoustic sho cks to the nucleus during its passage through

CHAPTER RESULTS

this region However it seems most likely that the heliopause has little or no eect on the

evolution of longp erio d comets

Fading

In the previous sections a numb er of dynamical mechanisms were explored with a view to

resolving the fading problem However none of them are capable of bringing the simulations

into agreement with observations These mechanisms share a common defect the tail

of the simulated inverse semima jor axis distribution is overp opulated relative to that of

the observed distribution Since the comets diuse in a after leaving the Oort cloud

x the overp opulation of the tail app ears to b e a result of longer comet lifetimes

in the simulations than in reality Thus one is led to consider fading ie the physical

degradation of the nucleus either into an increasingly faint ob ject through loss of volatiles

or through its breakup into less easily detectable pieces

Our interest here lies not in attempting to mo del in detail the physical pro cesses by

which comet nuclei fade over time but rather in determining the mathematical relationship

b etween a comets brightness and its age the fading function The distributions of all

the orbital elements ecliptic and Galactic are available to help with this construction

However only the a i and distributions display signicant changes over the rst

several apparitions the others change more slowly or not at all

The ecliptic and Galactic inclination are closely related but the relationship of i to the

planets the dominant p erturb ers of visible longp erio d comets is clearer and thus only

the ecliptic inclination will b e used here to constrain the fading function Though our

assumption that the Galactic tide is the dominant injector of comets x is correct the

contribution of passing stars is likely to heavily contaminate the observed distribution of

these elements Thus is likely to b e of little usefulness in the determination of the fading

function The inverse semima jor axis and the ecliptic inclination will serve as our primary

fading b enchmarks through the values of and x

As it seems likely that nongravitational forces play a role in determining the orbital

distribution of evolved LP comets their contribution along with that of the discovery

probability function will b e investigated The set of NG parameters chosen as typical for our

purp oses has the following characteristics A AU day A AU day

CHAPTER RESULTS

A with a random sign for A at each p erihelion passage Simulations containing

these NG forces will b e referred to as the standard NG case and were investigated earlier

Determining the fading function directly

The most direct approach to the construction of the fading function would b e to break

down the simulated data set into individual distributions one for each p erihelion passage

ie fV V V V g and then use a tting pro cedure to determine the fractional

amount of each distribution required such that the union of them matches the observed

distributions sub ject to the restriction that Unfortunately this problem is

i i

p o orly conditioned The inclination and inverse semima jor axis distributions change only

slowly after the rst few apparitions creating a degeneracy ie V V when n

n n

The only feature reliably extracted via the direct approach is the need for a fairly rapid

fading over the rst few orbits The numerical complications asso ciated with the

direct approach lead us instead to exp eriment with a few simple fading laws with clear

physical bases

One parameter fading functions

Consider a numb er of simple oneparameter fading functions In each case a weight func

tion ranging b etween one no fading and zero completely faded is applied to each

apparition This weight function represents the probability that any given apparition will

b e observed Let m b e the numb er of p erihelion passages and m b e the numb er of appari

tions since a comets rst apparition inclusive and let t b e the time in Myr since the rst

apparition The fading functions examined here are

A Constant lifetime Each comet is assigned a xed lifetime measured either in

apparitions m m m otherwise

v v

p erihelion passages m m m otherwise

x x

time t t t otherwise

x x

B Constant fading probability Comets are assigned a xed probability of fading measured either

CHAPTER RESULTS

p er apparition

p er p erihelion passage

p er million years e

Note that there is no fading previous to the rst visible p erihelion passage For time

based fading the equivalent exp onential decay has b e used where the decay time

t ln

C Power law The fraction of comets remaining go es like a p ower law based on either

numb er of apparitions m

p erihelion passages m

where is constant and greater than zero Note that this implies that the comets life

times m are distributed such that m ddm m If lifetime is prop ortional

x x

to comet mass as might b e exp ected if each apparition releases an approximately equal

amount of volatiles then the comet mass M has a dierential numb er distribution

such that

dN M dM

To determine the eects of each of these fading functions the three quantities are

plotted as a function of the asso ciated parameter If all three are unity for a particular

value of the parameter then the fading function provides a go o d match to our observed

sample at least in terms of a and i

Fading by orbit numb er

The fading laws based on apparition and orbit numb er will b e considered together as they

pro duce very similar results shown in Figures to

The rst two of these gures display the parameters assuming longp erio d comets

have constant lifetimes mo dels A and A The spiketotal ratio matches observations

at m or m but the tailtotal ratio is to o low at that p oint Given a longer lifetime

the numb er in the tail increases matching the observed tail at m but is now

to o low The progradetotal ratio is typically near but b elow unity The match with

observations Figure is p o or

CHAPTER RESULTS

4 4

2 2

1 1 0.8 0.8

0.6 0.6

0.4 0.4 SPIKE SPIKE TAIL TAIL PROGRADE PROGRADE 0.2 0.2

0.1 0.1 0.08 0.08

1 10 100 1000 1 10 100 1000

4 4

2 2

1 1 0.8 0.8

0.6 0.6

0.4 0.4 SPIKE SPIKE TAIL TAIL PROGRADE PROGRADE 0.2 0.2

0.1 0.1 0.08 0.08

1 10 100 1000 1 10 100 1000

Figure The values of given a xed lifetime m in apparitions mo del A The

i v

graphs on the right include a discovery probability those on the left do not The upp er two

graphs are based on the standard case the lower two on the standard NG case

CHAPTER RESULTS

4 4

2 2

1 1 0.8 0.8

0.6 0.6

0.4 0.4 SPIKE SPIKE TAIL TAIL PROGRADE PROGRADE 0.2 0.2

0.1 0.1 0.08 0.08

1 10 100 1000 1 10 100 1000

4 4

2 2

1 1 0.8 0.8

0.6 0.6

0.4 0.4 SPIKE SPIKE TAIL TAIL PROGRADE PROGRADE 0.2 0.2

0.1 0.1 0.08 0.08

1 10 100 1000 1 10 100 1000

Figure The values of given a xed lifetime m in p erihelion passages mo del A

i x

The graphs on the right include a discovery probability those on the left do not The upp er

two graphs are based on the standard case the lower two on the standard NG case

CHAPTER RESULTS

4 4 SPIKE SPIKE TAIL TAIL

2 PROGRADE 2 PROGRADE

1 1 0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0.1 0.1 0.08 0.08

0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1

4 4 SPIKE SPIKE TAIL TAIL

2 PROGRADE 2 PROGRADE

1 1 0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0.1 0.1 0.08 0.08

0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1

Figure The values of given a xed fading probability p er apparition mo del B

The graphs on the right include a discovery probability those on the left do not The upp er

two graphs are based on the standard case the lower two on the standard NG case

CHAPTER RESULTS

4 4 SPIKE SPIKE TAIL TAIL

2 PROGRADE 2 PROGRADE

1 1 0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0.1 0.1 0.08 0.08

0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1

4 4 SPIKE SPIKE TAIL TAIL

2 PROGRADE 2 PROGRADE

1 1 0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0.1 0.1 0.08 0.08

0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1

Figure The values of given a xed fading probability p er p erihelion passage

mo del B The graphs on the right include a discovery probability those on the left do

not The upp er two graphs are based on the standard case the lower two on the standard NG case

CHAPTER RESULTS

4 4

2 2

1 1 0.8 0.8

0.6 0.6

0.4 0.4

SPIKE SPIKE 0.2 TAIL 0.2 TAIL PROGRADE PROGRADE

0.1 0.1 0.08 0.08

0 0.5 1 1.5 2 0 0.5 1 1.5 2

4 4

2 2

1 1 0.8 0.8

0.6 0.6

0.4 0.4

SPIKE SPIKE 0.2 TAIL 0.2 TAIL PROGRADE PROGRADE

0.1 0.1 0.08 0.08

0 0.5 1 1.5 2 0 0.5 1 1.5 2

Figure The values of given a p ower law lifetime based on apparition numb er and

with exp onent mo del C The graphs on the right include a discovery probability

those on the left do not The upp er two graphs are based on the standard case the lower

two on the standard NG case

CHAPTER RESULTS

4 4

2 2

1 1 0.8 0.8

0.6 0.6

0.4 0.4

SPIKE SPIKE 0.2 TAIL 0.2 TAIL PROGRADE PROGRADE

0.1 0.1 0.08 0.08

0 0.5 1 1.5 2 0 0.5 1 1.5 2

4 4

2 2

1 1 0.8 0.8

0.6 0.6

0.4 0.4

SPIKE SPIKE 0.2 TAIL 0.2 TAIL PROGRADE PROGRADE

0.1 0.1 0.08 0.08

0 0.5 1 1.5 2 0 0.5 1 1.5 2

Figure The values of given a p ower law lifetime based on orbit numb er and with

exp onent mo del C The graphs on the right include a discovery probability those

on the left do not The upp er two graphs are based on the standard case the lower two on the standard NG case

CHAPTER RESULTS

100

0 -0.01 0 0.01 0.02 0.03

1/a (1/AU)

Figure The distribution of the inverse semima jor axis a p erihelion distance q and

cosine of the inclination i for the observed comets Marsden and Williams

CHAPTER RESULTS

1500 800

600

1000

400

500

200

0 0 -0.01 0 0.01 0.02 0.03 0 0.0001 0.0002 0.0003 0.0004 1/a (1/AU) 1/a (1/AU)

200 300

150

200

100

0 0 0 1 2 3 -1 -0.5 0 0.5 1

q (AU) cos i

Figure The distribution of the inverse semima jor axis a p erihelion distance q

and cosine of the inclination i given a p ower law fading function with mo del

C These simulations include standard NG forces and the discovery probability given by Equation

CHAPTER RESULTS

Figures and display the b ehaviour of the parameters given a xed fading

probability mo dels B and B This fading law do es not allow for a simultaneous

matching of the requirements on b oth the spike and tail and must b e considered unlikely

In Figures and the eects of a p owerlaw lifetime are shown mo dels C and

C Though the match is not p erfect note in particular that the prograde fraction

remains to o low for all explored it is b etter than in the previous two cases A value of

of seems to provide the b est match The distributions of orbital elements under this

fading law are shown in Figure For comparison the observed distributions are shown

in Figure The C mo dels can provide a reasonable match with observations a prograde

to retrograde ratio is near unity and a p erihelion distribution decreasing somewhat towards

the Sun

If the C mo dels represent reality correctly then some information ab out the LP comets

mass distribution may b e extracted If comets give up approximately the same amount

of mass p er apparition and mass is prop ortional to lifetime then the dierential mass

distribution of the LP comets go es roughly like M Equation The actual mass

distribution of LP comet nuclei is dicult to obtain owing to the obscuring eects of the

coma but is estimated at M for the brightest LP comets and at M for the fainter

ones Weissman Measurements of the sizes of mainb elt asteroids imply dierential

mass distributions prop ortional to M at large km diameters Hughes and Harris

There is also evidence for shallower slop es M at smaller sizes but these

measurements still only include diameters greater than km Cellino et al Thus

it seems that a fading law is not inconsistent with our knowledge of the size

distributions of small ob jects in the Solar System

Fading based on time since rst apparition

A cometary lifetime t measured in physical time since rst apparition seems a priori less

likely than one based on the numb er of apparitions Such a lifetime might b e exp ected if

the rst apparition removes a protective layer p ossibly of insulating refractory material

from the nucleus and thus starts the clo ck on some kind of timebased decay pro cess

A constant lifetime t mo del A Figure provides a remarkably go o d match with

observations if t yr This scenario works b ecause one hundred thousand years

provides enough time for the relatively few comets captured directly into tight orbits a

CHAPTER RESULTS

AU to ll out the tail while not providing enough time for those on larger a

AU orbits to return as frequently and ll out the middle part of the a distribution

Even typically to o low when the fading function is based on orbit numb er here reaches

unity near the lifetime in question The orbital element distributions for this case are shown

in Figure Neither the p erihelion nor inclination distributions match observations very

well

The timebased exp onential decay law mo del B shown in Figure provides only

fairly p o or matches when NG forces are excluded but is improved by their inclusion The

orbital elements distributions for the case t Myr is shown in Figure The

inclination distribution shows an excess of prograde comets and the p erihelion distribution

is concentrated towards smaller values of q This fading law provides a fair but far from

ideal match

Baileys fading law

One other fading law prop osed by Bailey will b e considered here though it has

no free parameters Bailey derived a fading function by using an analytical treatment

to calculate the exp ected a distribution The resulting fading law has a p errevolution

probability p of a comet fading completely and p ermanently given by

p a

where a is measured in AU Note that this fading law dep ends solely on the size of the orbit

Though not derived from physical principles the mechanism prop osed a posteriori to explain

this fading law is thermal sho ck comets with large aphelia have lower temp eratures T

as they approach p erihelion than those with shorter p erio ds The thermal diusivity of

the nucleus is prop ortional to its thermal conductivity which go es like T and inversely

prop ortional to the sp ecic heat which go es like T and thus is a strongly decreasing

function of temp erature The resulting deep er and more rapid heating is prop osed to disrupt

the nucleus p ossibly by mechanisms similar to those which pro duce cometary outbursts and

splittings x

The results of the application of this fading function to the simulations are listed in

Table The b est match is provided when standard NG forces are included but not a

discovery probability however the p erihelion distribution shows a sharp spike at q AU

CHAPTER RESULTS

4 4

2 2

1 1 0.8 0.8

0.6 0.6

0.4 0.4 SPIKE SPIKE TAIL TAIL PROGRADE PROGRADE 0.2 0.2

0.1 0.1 0.08 0.08

0.1 1 10 0.1 1 10

4 4

2 2

1 1 0.8 0.8

0.6 0.6

0.4 0.4 SPIKE SPIKE TAIL TAIL PROGRADE PROGRADE 0.2 0.2

0.1 0.1 0.08 0.08

0.1 1 10 0.1 1 10

Figure The values of given a constant lifetime t in physical time mo del A The

i x

graphs on the right include a discovery probability those on the left do not The upp er two

graphs are based on the standard case the lower two on the standard NG case

CHAPTER RESULTS

4 4

2 2

1 1 0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2 SPIKE SPIKE TAIL TAIL PROGRADE PROGRADE 0.1 0.1 0.08 0.08

0 0.05 0.1 0.15 0.2 0 0.05 0.1 0.15 0.2

4 4

2 2

1 1 0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2 SPIKE SPIKE TAIL TAIL PROGRADE PROGRADE 0.1 0.1 0.08 0.08

0 0.05 0.1 0.15 0.2 0 0.05 0.1 0.15 0.2

Figure The values of given an exp onential decay with decay constant t Myr mo del

i x

B The graphs on the right include a discovery probability those on the left do not The

upp er two graphs are based on the standard case the lower two on the standard NG case

CHAPTER RESULTS

800

1000

600

400

500

200

0 0 -0.01 0 0.01 0.02 0.03 0 0.0001 0.0002 0.0003 0.0004 1/a (1/AU) 1/a (1/AU)

400 250

200 300

150

200

100

0 0 0 1 2 3 -1 -0.5 0 0.5 1

q (AU) cos i

Figure The distribution of the inverse semima jor axis a p erihelion distance q and

inclination i given a xed lifetime t yr mo del A Simulation includes standard

NG forces and a discovery probability

CHAPTER RESULTS

800

1000

600

400

500

200

0 0 -0.01 0 0.01 0.02 0.03 0 0.0001 0.0002 0.0003 0.0004 1/a (1/AU) 1/a (1/AU)

250

200 200

150 150

100 100

50 50

0 0 0 1 2 3 -1 -0.5 0 0.5 1

q (AU) cos i

Figure The distribution of the inverse semima jor axis a p erihelion distance q and

inclination i given a exp onential decay constant t Myr mo del B Simulation

includes standard NG forces and a discovery probability

CHAPTER RESULTS

NG forces Discovery

No No

No Yes

Yes No

Yes Yes

Table Parameters under Baileys fading function for cases which may

include NG forces and a discovery probability

which is not present in the observations or the exp ected intrinsic distribution The case

which includes b oth NG forces and a discovery probability is barely consistent within the

error bars with the observations but provides a more reasonable p erihelion distribution

The orbital element distributions for this case are shown in Figure The match with

observations Figure is go o d except for the p erihelion distribution which increases

near the Sun There is also a small excess of comets on retrograde orbits Baileys fading

law provides a go o d but not ideal match with observations

Summary

The b est overall match by the oneparameter fading laws considered is provided by the

p ower law mo del based on apparition or p erihelion passage numb er mo dels C and C

which alone provide a go o d match to the p erihelion distribution Figure Fair matches

are provided by a timebased exp onential decay B with decay constant t Myr

a constant lifetime t Myr A and Baileys fading law whereas the other mo dels

pro duce only p o or matches or can b e ruled out entirely

Two parameter fading functions

Though the available function space b ecomes increasingly large a few twoparameter fading

functions will b e examined here Fading laws based on orbit numb er will not b e considered

due to the similarity of the results for apparition numb er and orbit numb er

D Two p opulations Let the Oort cloud consist of two p opulations of comets distin

guished by their internal strength The rst and more fragile set of ob jects have

a nite lifetime while the other ob jects comprising a fraction f of the total are

unaected by fading The fragile p opulations lifetimes are either

a xed numb er of apparitions m m m otherwise f

v v

CHAPTER RESULTS

600

1500

400

1000

200

500

0 0 -0.01 0 0.01 0.02 0.03 0 0.0001 0.0002 0.0003 0.0004 1/a (1/AU) 1/a (1/AU)

150

100

0 0 0 1 2 3 -1 -0.5 0 0.5 1

q (AU) cos i

Figure The distribution of the inverse semima jor axis a p erihelion distance q and

inclination i given Baileys fading law Simulations include NG forces and a discovery

probability

CHAPTER RESULTS

a xed time t from rst apparition t t otherwise f

x x

Such a fading law might b e appropriate if the Oort cloud contains b oth relatively

weak snowy nuclei and stronger icy nuclei

E Fixed fading probability plus survivors In this case one p ortion of the LP comets

p opulation has a xed fading probability while some fraction f survive indenitely

The fading functions are based either on

apparition numb er m f f

time t since rst apparition f e f

This is a more sophisticated treatment of the two p opulation mo del

F Power law variant The fading function is a variant of the oneparameter p ower law

and is based either on

apparition numb er m m

time t in Myr since rst apparition t

where and are p ositive constants Note that is constructed so as to b e unity at

t or m

Fading based on apparition numb er

The results of mo del D are shown in Figure The t is generally worse than the one

parameter case A shown by the heavy line b ecause the prograde fraction is lower when

some fraction of comets are allowed to live indenitely The b est match among the families

of curves with b oth NG forces and a discovery probability o ccurs near m orbits for a

survival fraction of roughly This case corresp onds to roughly of Oort cloud comets

b eing fragile with lifetimes against disruption or fading of approximately six orbits while

the remaining are longerlived p erhaps more similar in nature to the fadingresistant

SP comets The orbital element distributions for this scenario are in Figure and

match observations Figure well Weissman also found that a twop opulation

Monte Carlo mo del in which a large fraction of LP comets had signicant fading

probabilities while the reminder survived indenitely matched the observations b est Note

that the observed splitting probability for dynamically new Oort cloud comets of p er

CHAPTER RESULTS

4 4

2 2

1 1 0.8 0.8

0.6 0.6

0.4 0.4 SPIKE SPIKE TAIL TAIL PROGRADE PROGRADE 0.2 0.2

0.1 0.1 0.08 0.08

1 10 100 1000 1 10 100 1000

4 4

2 2

1 1 0.8 0.8

0.6 0.6

0.4 0.4 SPIKE SPIKE TAIL TAIL PROGRADE PROGRADE 0.2 0.2

0.1 0.1 0.08 0.08

1 10 100 1000 1 10 100 1000

Figure The values of given a twop opulation mo del based on apparition numb er

mo del D The graphs on the right include a discovery probability those on the left do

not The upp er two graphs are based on the standard case the lower two on the standard

NG case The fraction f which survive past the lifetime m are

and b eginning with the heavy line

CHAPTER RESULTS

4 4

2 2

1 1 0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

SPIKE SPIKE 0.1 TAIL 0.1 TAIL 0.08 PROGRADE 0.08 PROGRADE

0 1 2 3 0 1 2 3

4 4

2 2

1 1 0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

SPIKE SPIKE 0.1 TAIL 0.1 TAIL 0.08 PROGRADE 0.08 PROGRADE

0 1 2 3 0 1 2 3

Figure The values of given a a xed fading probability p er apparition together

with a fraction f of unfading survivors mo del E The graphs on the right include a

discovery probability those on the left do not The upp er two graphs are based on the

standard case the lower two on the standard NG case The values of f are

b eginning with the heavy line

CHAPTER RESULTS

4 SPIKE 4 SPIKE TAIL TAIL PROGRADE PROGRADE

2 2

1 1 0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0.1 0.1 0.08 0.08

0 1 2 3 0 1 2 3

4 SPIKE 4 SPIKE TAIL TAIL PROGRADE PROGRADE

2 2

1 1 0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0.1 0.1 0.08 0.08

0 1 2 3 0 1 2 3

Figure The values of given a twoparameter p ower law lifetime based on apparition

numb er mo del F The graphs on the right include a discovery probability those on the

left do not The upp er two graphs are based on the standard case the lower two on the

standard NG case The values of are and b eginning with the heavy line

CHAPTER RESULTS

orbit results in a halflife of seven orbits Equation suggesting splitting may b e the

source of fading if there is indeed a fragile p opulation of Oort cloud comets

The fading law involving a xed fading probability for one segment of the p opulation

and an indenite lifetime against fading for the rest mo del E pro duces results as shown

in Figure The match is usually no b etter than the no survivor case B shown by

the heavy line and typically gets worse as the survival fraction f increases

The twoparameter p ower law mo del F displayed in Figure ordinarily do es no

b etter than its oneparameter counterpart Though the prograde fraction shows little

variation the intersection of the and curves is typically b elow the value of one required

to match observations when Though an eventual return to near unity as

is not excluded by these gures the extremely rapid fading required in such a case seems

unlikely

The twoparameter fading mo dels based on apparition numb er typically do no b etter

job of matching the observed distribution than oneparameter mo dels with the exception

of mo del D which can provide a go o d match to observations when most comets have short

orbits lifetimes while a small fraction live indenitely

Fading based on time since rst apparition

When the twop opulation fading mo del is based on the time since rst apparition mo del D

Figure the t is typically only slightly b etter than the corresp onding oneparameter

mo del The normalised tail to total ratio provides the most stringent restriction on the

survival fraction Only a small fraction roughly of comets could survive indenitely

and still pro duce a match as long as the remaining comets have a lifetime against fading

t yr The distributions of orbital elements for this case are displayed in Figure

The p erihelion distance and inclination distributions are very similar to those of the one

parameter A mo del Figure and match the observations only p o orly

The case of exp onential decay with time plus a small fraction of nonfading comets

pro duces the results displayed in Figure mo del E Only a very small fraction

can survive in either of the four cases if a match is to b e obtained and the b est match

app ears to remain with the case of no survivors mo del B

Figure displays the eects of mo del F The intersection of and o ccurs ab ove

unity for all curves in the families plotted approaching it more closely as increases The

CHAPTER RESULTS

800

1500

600

1000

400

500 200

0 0 -0.01 0 0.01 0.02 0.03 0 0.0001 0.0002 0.0003 0.0004 1/a (1/AU) 1/a (1/AU)

250

300

200

150 200

100

0 0 0 1 2 3 -1 -0.5 0 0.5 1

q (AU) cos i

Figure The distribution of the inverse semima jor axis a p erihelion distance q and

inclination i given a two p opulation mo del D with m apparitions and a survival

fraction f of Simulations include NG forces and the discovery probability

CHAPTER RESULTS

4 4

2 2

1 1 0.8 0.8

0.6 0.6

0.4 0.4 SPIKE SPIKE TAIL TAIL PROGRADE PROGRADE 0.2 0.2

0.1 0.1 0.08 0.08

0.01 0.1 1 10 0.01 0.1 1 10

4 4

2 2

1 1 0.8 0.8

0.6 0.6

0.4 0.4 SPIKE SPIKE TAIL TAIL PROGRADE PROGRADE 0.2 0.2

0.1 0.1 0.08 0.08

0.01 0.1 1 10 0.01 0.1 1 10

Figure The values of given a two p opulation mo del where a fraction f survive

indenitely and the remainder have a lifetime t based on time since rst apparition mo del

D The graphs on the right include a discovery probability those on the left do not The

upp er two graphs are based on the standard case the lower two on the standard NG case

The values of f are and b eginning with

the heavy line

CHAPTER RESULTS

4 4

2 2

1 1 0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

SPIKE SPIKE TAIL TAIL 0.1 PROGRADE 0.1 PROGRADE 0.08 0.08

0.01 0.1 1 0.01 0.1 1

4 4

2 2

1 1 0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

SPIKE SPIKE TAIL TAIL 0.1 PROGRADE 0.1 PROGRADE 0.08 0.08

0.01 0.1 1 0.01 0.1 1

Figure The values of given an exp onential decay plus survivors fading mo del with

a decay time t based on time since rst apparition mo del E The graphs on the right

include a discovery probability those on the left do not The upp er two graphs are based

on the standard case the lower two on the standard NG case The survivor fractions are

and b eginning with the heavy line

CHAPTER RESULTS

4 4

2 2

1 1 0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0.1 0.1 0.08 0.08

0 1 2 3 4 5 0 1 2 3 4 5

4 4

2 2

1 1 0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0.1 0.1 0.08 0.08

0 1 2 3 4 5 0 1 2 3 4 5

Figure The values of given a two parameter p ower law with exp onent and

based on time since rst apparition mo del F The graphs on the right include a discovery

probability those on the left do not The upp er two graphs are based on the standard case

the lower two on the standard NG case The parameter runs from Myr to Myr in

steps of in the base logarithm b eginning with the heavy line solid line dotted

dashed

CHAPTER RESULTS

800

1000

600

400

500

200

0 0 -0.01 0 0.01 0.02 0.03 0 0.0001 0.0002 0.0003 0.0004 1/a (1/AU) 1/a (1/AU)

300 250

200

150

100 100

0 0 0 1 2 3 -1 -0.5 0 0.5 1

q (AU) cos i

Figure The distribution of the inverse semima jor axis a p erihelion distance q and

inclination i given a two p opulation mo del with t yr and a survival fraction of

mo del D Simulations include NG forces and the discovery probability

CHAPTER RESULTS

value of also remains close to unity particularly in the case which includes NG forces and

a discovery probability Thus the b est matches are provided as but the increasing

steepness of this fading law makes it an unlikely candidate

Summary

The twoparameter fading laws examined here typically do only slightly b etter than the one

parameter laws at matching the observed distributions if at all The b est oneparameter

fading functions based on the values of the parameters and an examination of the

orbital elements distributions are C and C p ower law based on apparitionorbit numb er

The b est match among the twoparameter families is obtained from mo del D two

p opulations of comets live only orbits the remainder indenitely In particular

C C and D can provide go o d matches to the p erihelion distributions particularly in

regards to the decrease in numb ers observed at q close to the Sun

Chapter

Conclusions

The dynamical evolution of longp erio d comets has b een simulated from its b eginning in

the Oort cloud to its end with the comets loss or destruction A numerical integration

algorithm was used to follow each comets tra jectory individuall y under the inuence of

the giant planets and the dominant comp onent of the Galactic tide though the eects of

passing stars were ignored Additional simulations studied the eects of outgassing reaction

forces a hyp othetical circumsolar disk or solar companion and the disruption or fading of

the nucleus Solar wind and radiation pressure the heliopause molecular clouds and drag

were also examined but proved to b e either negligible or inapplicable Various conclusions

can b e drawn from this research

The concentration of aphelion directions of the observed longp erio d comets at mid

Galactic latitudes Figure b is due to the action of the Galactic tide and is not simply

an artifact of statistical noise It may b e p ossible to use the observed distribution of the

Galactic argument of p erihelion to estimate the relative comet injection rates of the tide

and passing stars as injection due to the tide should b e restricted to the range sin

while that due to stars will b e uniform over However the observed sample of dynamically

new longp erio d comets is to o small to disentangle these comp onents as yet

The current Oort cloud contains roughly ob jects orbiting b etween and

AU from the Sun x assuming the Oort cloud is in a near steadystate with

a numb er density prop ortional to r and that the ux of dynamically new longp erio d

comets through a sphere of radius AU around the Sun is currently yr As many as

Oort comets may have entered the loss cylinder since the Solar Systems formation

CHAPTER CONCLUSIONS

indicating that the original Oort cloud had over memb ers and probably more

All but a small fraction of comets in the Oort spike are making their rst apparition In

the absence of fading roughly of observed comets with a AU have made

one or more previous apparitions similarly of comets originating in the Oort cloud do

not make their rst visible p erihelion passage within the spike but on more tightly b ound

orbits x As these numb ers ignore the eects of fading they provide only upp er

limits and the actual fractions are likely to b e much lower

The Oort cloud provides only a small fraction of the shortp erio d comets up to

if fading is ignored Thus another source or sources must b e providing the bulk of the

shortp erio d comets

Longp erio d comets pass through each giant planets sphere of inuence at a rate of

approximately one every few years x Collision rates may b e as high as once p er

million years for Jupiter and Saturn dropping to roughly once p er few hundred million

years for Uranus and Neptune

The purely dynamical lifetimes of simulated longp erio d comets are to o long to repro

duce the observed distributions of these comets with discrepancies in particular in the

original inverse semima jor axis and inclination distributions Thus the fading problem is

not simply a result of incomplete theoretical treatments but represents a real gap in our

understanding of the evolution of longp erio d comets and p ossibly in our knowledge of the

inventory of the Solar System

The observed prograde fraction of longp erio d comets is near as is that of the

dynamically new V comets simulated here However is exp ected to decrease as the

comet p opulation ages due to preferential ejection of prograde comets by the giant planets

in the standard mo del is x The prograde fraction thus provides a measure

of the age of the longp erio d comet p opulation The observed value of near implies that

longp erio d comet lifetimes are much shorter than their dynamical lifetimes and indicates

that a fastacting fading mechanism of some kind is at work

Nongravitational forces play a signicant role in shaping the distributions of the long

p erio d comet orbital elements in particular by shortening their lifetimes against ejection and

by sculpting the distribution at small p erihelia NG forces reduce the dynamical lifetimes

of comets but are to o small by roughly two orders of magnitude to resolve the fading the

problem of themselves

CHAPTER CONCLUSIONS

The fading problem probably do es not reect the presence of a massive circumsolar

disk or solar companion ob ject at least as far as can b e determined from the somewhat

simplied treatment given these features here Low mass disks and companions are unable

to pro duce orbital element distributions which match observations while at higher masses

these features prevent the lling of the loss cylinder x

No clear explanation for the existence of the observed visible comets on hyp erb olic

original orbits is provided by this research The excess velo cities are small corresp onding

to roughly AU in inverse semima jor axis but are larger than those pro duced by the

Galactic tide AU Equation or nongravitational forces AU

Equation a over a single comet orbit A circumsolar disk or solar companion might

pro duce a change a of this magnitude but would strongly inuence the Oort cloud

and prevent the lling of the loss cylinder x Other eects such as rapid variations in

outgassing or asymmetrical outgassing ab out p erihelion might pro duce hyp erb olic comets

but such eects were not examined here

The observed distributions of orbital elements can b e matched by the addition of fading

to the simulations though of a fairly restricted form a large fraction of comets must fade

orbits while a smaller fraction must survive much longer times fairly quickly

orbits The fact that the cratering rate is near that exp ected from the current known

p opulations of comets and asteroids x implies that fading results in the complete

disruption of the comet nucleus This hyp othesis is supp orted by the lack of observed sharp

decreases in the brightnesses of longp erio d comets as they pass p erihelion decreases which

might b e exp ect if fading were due to a rapid loss of volatiles which left the comets inert

but intact x

A oneparameter p ower law fading function mo del C and C with exp onent

provides a go o d match b etween our simulations and observations This fading law might b e

exp ected if each apparition results in the loss of approximately equal amounts of volatiles

and the dierential mass distribution of dynamically new longp erio d comets N M dM

M dM x

A twop opulation fading mo del D in which approximately of comets survive for

roughly six orbits and the remainder indenitely also provides go o d agreement with obser

vations and could b e explained by a division of the Oort cloud p opulation into ob jects with

low and high internal cohesiveness Such a fading mo del would b e roughly consistent with

CHAPTER CONCLUSIONS

the observed splitting probabilities of dynamically new longp erio d comets approximately

p er orbit Equation

The fading problem remains partly unresolved Though it seems likely to b e asso ciated

with the disruption of the nucleus rather than a dynamical eect the exact nature of

this decay pro cess remains unclear The fading pro cess is likely to b e sensitive to the each

comets particular prop erties and to the pattern of thermal and other stresses to which they

are sub ject and thus very dicult to predict Future progress will likely require improved

observational data rather than more sophisticated theoretical treatments in particular more

information on the physical characteristics of comet nuclei as well as greater observational

coverage of their orbits is needed

App endix A

Celestial Mechanics

A Orbital elements

The six standard elements of a twob o dy orbit are semima jor axis a eccentricity e incli

nation i longitude of the ascending no de argument of p erihelion and true anomaly

f eg Roy The reader is assumed to b e familiar with these elements but a brief

sketch of the angular orbital elements is presented in Figure A

The elements are usually measured in the helio centric reference frame but can also b e

taken in the barycentric frame in which case the centre of mass is the origin ab out which

the elements are computed instead of the Sun The notation used here for the helio centric

and barycentric elements is the same with the context indicating which is b eing used

A Galactic elements

The angular orbital elements as measured in the Galactic frame are also of interest here

x In this case the Sun is at the origin but the vernal equinox is directed instead

towards the Galactic centre and the ecliptic p ole is directed towards the Galactic p ole

The Galactic angular elements are denoted by a tilde ie and The Galactic argument

of p erihelion should not b e confused with the commonlyused symb ol for the longitude

of p erihelion as the longitude of p erihelion will not b e used here

APPENDIX A CELESTIAL MECHANICS

Ecliptic pole Orbiting body

Perihelion Vernal equinox f ω Sun i

Ω

Ascending node Ecliptic

Orbital plane

Figure A The inclination i longitude of the ascending no de argument of p erihelion

and true anomaly f in the ecliptic frame Adapted from Roy

A Keplers third law

The orbital p erio d of a comet can easily b e related to its semima jor axis a through Keplers

third law

where G is the gravitational constant M is the mass of the Sun and the mass of the

orbiting b o dy has b een assumed to b e negligible In the case where a is measured in AU

M in solar masses and in years this expression reduces to

APPENDIX A CELESTIAL MECHANICS

A Radius in the orbit

The distance r of the orbiting b o dy from the origin is given

a e

r A

e cos f

The true anomaly f can b e determined from the time t since p erihelion passage by solving

the transcendental Keplers equation but this equation will not b e required for these

discussions

A Twob o dy energy

The energy p er unit mass E of a particle on a b ound orbit in a twob o dy system is simply

GM a where M is the total mass of the system and a is the semima jor axis measured

in the barycentric or centre of mass frame For a massless test particle orbiting the Sun

M M and the barycentric and helio centric frame coincide In this case the inverse

semima jor axis a provides an unambiguous measure of the test particles energy

A Energy in multib o dy systems

If p otentials other than that of the Sun are present the semima jor axis a of a test particles

orbit may vary due to the p erturbing presence of these other elds In the presence of

massive planets the Solar Systems helio centric and barycentric frames no longer coincide

owing to the accelerations imparted to the Sun by the planets Nevertheless a useful

snapshot of a test particles energy is provided by the osculating value of a The

osculating elements are those which would b e measured if the particle were travelling with

its instantaneous p osition and velo city in a simple twob o dy system rather than a p erturb ed

one For our purp oses if is convenient to measure a in the barycentric frame though its

accuracy as a measure of energy is degraded while the comets travel within the planetary

system the barycentric a reduces to the correct twob o dy value as the particle moves

towards innity For this reason the inverse semima jor axis of comets will b e measured at

aphelion here

APPENDIX A CELESTIAL MECHANICS

A Gauss equations

If a particle orbiting the Sun is sub ject to a small p erturbing force F Gausss planetary

per t

equations can b e used to deduce the resulting change in its orbital elements A small

p erturbing force is taken here to mean that the fractional change in any orbital element is

small over a single orbit

Let the p erturbing force b e written as

F F e F e F e A

per t

where the three orthogonal unit vectors are radial e p ositive outward from the Sun

transversee in the orbital plane p ositive along the direction ahead of the Suncomet

line and normale p erp endicular to the orbital plane parallel toe e Then in terms

of these comp onents Gausss planetary equations are

da a

F e sin f F e cos f Aa

dt GM e

de a e

F sin f F cos u cos f Ab

dt GM

di F r cos f

GM a e

d F r sin f

GM a e sin i

a e e cos f d d

F cos f F cos i Ae sin f

dt e GM e cos f dt

d i e d F r d d

e sin Af

dt dt dt dt

e GM a

where u is the eccentric anomaly such that cos u a r ae and is the mean longitude

at t ie the mean longitude is

dt A

where ddt the rate of change of the mean anomaly is just the mean Keplerian angular

velo city ddt

The median eccentricity of Marsdens comet catalogue is for the long

p erio d comets with computed original orbits the median e is Thus it is convenient

APPENDIX A CELESTIAL MECHANICS

to consider Gausss equations under the approximation e or equivalently q a

da a

F sin f F cos f Aa

dt GM q

q de

F sin f F cos E cos f Ab

dt GM

di q cos f

Ac F

dt GM cos f

d q sin f

F Ad

dt GM cos f sin i

cos f q d d

cos i Ae F cos f F sin f

dt GM cos f dt

q d d q F i d

sin Af

dt dt a dt

GM a cos f

These equations will prove useful in determining the eects of outgassing and other p ertur bations on comets

App endix B

Error tolerances

The error tolerance is converted to an error limit for each individual regularised co ordinate

through a combination of relative and absolute terms the rst is dep endent on the

comets instantaneous orbit the second is not

Because the unp erturb ed regularised equations are those of a simple harmonic oscillator

x the error limits for the p osition and velo city co ordinates are taken simply to b e

times their instantaneous amplitudes For the energy h and time t the relative term is

based on the instantaneous values of h and and the p erio d resp ectively

The absolute term is required to avoid excessively stringent error limits on co ordinates

that happ en to have nearzero amplitudes The absolute terms are based on an arbitrary

Suncentred reference orbit usually taken to have a AU with corresp onding regu

larised energy h The error limit equations are shown in Equations Babc and d b elow

with the relative term rst the absolute term second The error tolerance for t has no

absolute term as comets with small orbits are terminated in our simulations b efore their

p erio d b ecomes to o small

u j Ba u a E

j j

h h a

E u u j Bb

j j

E h h Bc

E Bd

App endix C

The Flux of LongPerio d Comets

C The ux across the entrance surface

The orbitaveraged inwards ux p er unit time due to the Galactic tide across any surface

of constant angular momentum J J p er unit L is the integral of Equation over the

all other canonical co ordinates

J J J

E E

Z Z Z Z Z

df d d dJ g L J J f J dJ C L J dL

z z E E

J J

E E

where J is a step function unity when J and zero otherwise which removes the

outwards ux from the integral Using the indep endence of the orbitaveraged J on f and

Equation b and assuming the phase space density is of the form g g L

Equation we easily integrate over f and to get

J J J

E E

Z Z Z

g L dL

L J dL d dJ J dJ C

E z

J J

E E

Integrating this equation with resp ect to J simply yields

Z Z

L J dL g L dL J J dJ C d

E z

The expression for J is given by Equation b Up on substitution Equation C b ecomes

Z Z

J J L J sin J dJ Ca L J dL g L dL d

z E

E z E

GM J

APPENDIX C THE FLUX OF LONGPERIOD COMETS

Z Z

g L

L J dL J J dJ Cb sin sin d

E E z

GM J

where the dep endence of the sign of J on sin has b een explicitly acknowledged The

integrals over and J are easily done yielding

L J L J dL C L J dL

E E

The total ux across the entrance cylinder p er unit L is the integral of Equation C

over the entrance cylinder which is given by Equation and which is here expressed in

our chosen canonical co ordinates as

J where L L L

J L C

Z k L where L L L

where L and L are the minimum and maximum values of L in the Oort cloud corre

sp onding to the minimum and maximum semima jor axes a and a k G M

Equation and L is the p oint at which J Z k L ie L J Z k Equa

E E

tion a

Equation C must b e integrated along the path in LJ space corresp onding to the

entrance cylinder and may require an extra factor measuring the arclength along this path

However the ux is always parallel to J and so is reduced by a factor of the cosine of the

angle b etween J and the normal to the entrance cylinder It easy to conclude that these

contributions cancel out and deduce that

L J L J dL Ca L J dL

E E

g k Z

L J dL L Z k L dL Cb

where the sup erscripts i and o indicate the inner and outer regions of the entrance surface

resp ectively This ux can b e expressed in terms of the semima jor axis a L GM by

GM ada using the expression dL

a q da g G M q a a q da Ca

E E

a g Z G M a a q da da Cb

APPENDIX C THE FLUX OF LONGPERIOD COMETS

Figure C The ux of longp erio d comets into the entrance cylinder as a function of

semima jor axis a

GM q has b een made in the inner region The ux where the approximation J

is plotted in Figure C assuming values of M p c Z q AU

and with g normalised so that at a This function is the basis for the

probability function used to compute comet initial elements Equation f

C The ux into the visibili ty cylinder

The ux exp ected into the visibili ty cylinder can b e deduced from the ux into the entrance

cylinder under the assumption that the tide remains the dominant p erturb er until the comet

reaches the visibili ty cylinder this is true in the outer Oort cloud where the loss cylinder

is lled

Consider Figure C which shows a crosssection of the entrance and visibili ty cylinders

J in J J space where J J such that J J J The angular momenta J

z xy xy v

z x y

x y

and J are those at which the comets p erihelion is within the visibili ty cylinder and the

entrance surface resp ectively

APPENDIX C THE FLUX OF LONGPERIOD COMETS

Figure C Diagram of the entrance cylinder

Since J is conserved under the tide but J is not x the evolution of an LP comet

z xy

is a horizontal tra jectory in Figure C The sequence of dots moving inwards from the right

side of the entrance cylinder represents the angular momentum values of a particular comet

at a series of p erihelion passages The ux into the visibility cylinder is just the fraction

of p erihelion passages made within the entrance surface to that made within the visibili ty

cylinder

The step size J is not generally constant from orbit to orbit thus the ux into the

visibility cylinder is reduced by a factor J J J J over that into the entrance surface

xy v xy E

Since J must b e small if the comet is to enter the visibili ty cylinder it is easy to show

thatJ J J J J J J J The ux into the visibility cylinder is also reduced

xy v xy E v E

by a factor of J J due to the smaller crosssection of the visibility cylinder

v E

The visible ux is also reduced by the p ossibility that a comet passing through the visi

bility cylinder will do so in less than one orbit If J J the comets angular momentum

could move through and out the other side of the visibility cylinder b etween p erihelion

passages and the comet would thus fail to b ecome visible Ignoring the dep endence of this

APPENDIX C THE FLUX OF LONGPERIOD COMETS

phenomenon on J the fraction of comets b ecoming visible is just the area of the visibili ty

cylinder divided by the rectangle J J where J J J Assumed that J J

v v v E

certainly true for Oort cloud comets the probability of a comet passing within J b ecoming

visible is J J J J

v v

Combining these three factors one nds that the ux into the visibility cylinder is

J J J J

v v

v E

J J J J J

E v v

J J J

E E

AU the expres Considering only the region where the loss cylinder is full a

sions J Z k L Equation C J e k L Equation and J GM q can

E E v

E v

b e used express Equation Cb as

q M

E E

e e Z a k Z L

E E

where the sup erscript indicates the ux in the region a a Using the expression

e L J L the ux can b e reduced to

g q GM a da C a q da

v v

Note that this expression do es not dep end on the characteristics of the entrance surface

as would b e hop ed It is also indep endent of removing the p ossibility of measuring the

lo cal matter density directly from the ux of visible comets

For an Oort cloud with the ux into the visible cylinder from the outer cloud

will fall as a This decrease is due in part to the increasing likeliho o d of a comet jumping

over the visibility cylinder b etween p erihelion passages as J b ecomes large The ux into

the visibility cylinder is plotted in Figure along with a plot of the distribution of the

original inverse semima jor axes of the V comets

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