Astron. Astrophys. 341, 928–935 (1999) ASTRONOMYAND

ASTROPHYSICS

The stable Kozai state for asteroids and comets

With arbitrary semimajor axis and inclination

G.F. Gronchi and A. Milani

Dipartimento di Matematica, Universita di Pisa, Via Buonarroti 2, I-56127 Pisa, Italy (e-mail: gronchi@mail.dm.unipi.it; milani@dm.unipi.it)

Received 11 June 1998 / Accepted 2 October 1998

Abstract. Semianalytical averaging is used to compute secularperturbations on the orbits of asteroids and comets; the methodis applicable even for planet-crossing orbits. We prove that forevery value of the asteroid/comet semimajor axis, and for anarbitrary number of perturbing planets, there is a stable regionof orbits free from node crossings; it corresponds to either cir-culation or libration of the argument of perihelion. This hasimplications on the possibility of collisions with the planets andalso, when encounters are possible, on the algorithms to com-pute the probability of collision.

Key words: celestial mechanics, stellar dynamics – comets:general – minor planets, asteroids

1. Introduction

The orbits of comets and asteroids with high enough eccentricitycan cross the orbits of some major planets. This results in theimpossibility to use the standard analytical theories to computethe long term dynamics of these orbits; on the other hand, theseorbits are especially worth studying, precisely because of theimpacts they can experience.

A method to provide at least a first approximation to the sec-ular evolution of high eccentricitye (and inclinationI) orbitshas been introduced in the pioneering work of Kozai (1962). Theadvantage of extending the theory to largee andI is balanced bythe disadvantage that the theory is expressed by means of func-tions not known by an explicit analytical expression, but onlyby means of integrals. We have recently shown (Gronchi andMilani 1998) that the Kozai type theories can be consistentlygeneralised to the case of planet-crossing orbits. Several authorshave published in graphical form the results of these computa-tions for different classes of planet-crossers (e.g. Thomas andMorbidelli 1996, Michel and Thomas 1996); the figures are veryinteresting, but every figure appears to tell a different story, todisplay a different and complicated dynamical structure.

Our goal was to find some general rules, applicable a priorito whatever combination of perturbing planets and to orbits witharbitrary semimajor axisa. Of course at this level of generality

Send offprint requests to: G.F. Gronchi

there cannot be too many applicable rules, but we have been ableto find one which we believe is significant, and is as follows.

For an arbitrary number of perturbing planets, whatever thevalues ofa and of the combination

√1 − e2 cos I of the small

body, in the approximation neglecting the eccentricities and mu-tual inclinations of the perturbing planets, there is a region fullof orbits whose secular evolution is stable; in particular, theorbits in this region cannot undergo close approaches with theperturbing planets, but remain in a permanent oscillation (withargument of perihelion either librating or circulating). This re-sult, and the arguments used in the proof, shed some light on thegeometry of the solution curves of the secular evolution equa-tions; in particular, a node crossing line appears as a crest inthe graph of the perturbing function, while the stable librationcenters are minima.

This is a rigorous mathematical theorem, which is formallyproven in this paper. The relevance of this result is not purelymathematical; indeed every computation of the long term fre-quency of close approaches, and of the probability of impact ofa planet-crossing body with some planet, including the Earth,uses some kind of averaging of the orbit, to replace time av-erages with space averages, as in every statistical mechanicsargument. As an example Bottke et al. (1994) have used anaveraging which assumes the argument of perihelionω of theEarth-crossing asteroids and comets to be circulating, with con-stante and I. The use of a Kozai type theory can give verydifferent results, and indeed Tables 3 and 4 of Kozai (1997)give two lists of Near Earth Asteroids for which the probabil-ity of collision is zero, but would be different from zero if thesecular perturbations were entirely neglected. Thus our resulthas the corollary that the computations of probability of impactperformed by neglecting the Kozai type secular perturbationsare inaccurate over a large portion of the orbital elements space,not only wrong in a few special cases.

This paper is organised as follows: in Sect. 2 we describethe geometry of crossing orbits, and the Kozai type averaging.In Sect. 3 we introduce the notion of stable Kozai librations,illustrated by examples from very different regions of the So-lar System, that is main belt asteroids, Near Earth Asteroids,and Centaurs. In Sect. 4 we formally state and prove the maintheorem on the existence of the stable Kozai states. In Sect. 5we discuss the implications of this result on the dynamics of

G.F. Gronchi & A. Milani: The stable Kozai state for asteroids and comets 929

asteroids and comets, and especially on the reliability of theimpact probability computations, illustrated with an example ofcometary long term dynamics.

2. Averaging on planet-crossing orbits

To describe the long term evolution of an asteroid (comet) or-bit we use theaveraging principle(Arnold 1976), by whichthe running time average of the complete solution can be ap-proximated by the phase space average over the faster movingvariables. The latter are the mean anomalies of both the asteroidand the perturbing planets. The first stage of this approximationprocedure was introduced by Kozai (1962), by assuming circu-lar and coplanar orbits of the perturbing planets.

The Hamilton functionH, describing the total energyof the asteroid orbit, is a function of all the orbital ele-ments(a, e, I,Ω, ω, `) (semimajor axis, eccentricity, inclina-tion, longitude of ascending node, argument of perihelion, meananomaly) and of time (through the mean anomalies of the per-turbing bodies). It is assumed that the inclinationI and the nodeΩ are with respect to the common orbital plane of all the perturb-ing planets. The averagedH is a function of(a, e, I, ω) only; bythe standard rules on cyclic variables, the momenta conjugate tothe angles andΩ are integrals of the “average” motion definedbyH. Thus botha and the combinationZ =

√a(1 − e2) cos I

(the component of the angular momentum normal to the refer-ence plane) are constant in the average motion.

It follows that, once the value of theZ integral is fixedfrom the initial conditions, both the averaged inclination andthe averaged eccentricity have maximum values

Imax =I∣∣∣e=0

= arccosZ√a

; emax =e∣∣∣I=0

=

√a − Z2

a.

When the eccentricity is large enough, the nodal distances

d+nod =

a(1 − e2)1 + e cos ω

− a′ ; d−nod =

a(1 − e2)1 − e cos ω

− a′ (1)

between the ellipse of the asteroid orbit and the circular orbit(with radiusa′) of some perturbing planet, can become zero:a node crossingoccurs. When this is the case, to computeHwe need to perform an integral, over both` and`′ (the meananomaly of the planet), of a function proportional to the inverseof the distance, which can become zero at a node crossing. Sincethis results in a polar singularity of order one, the improperintegral is anyway absolutely convergent, and the averageHis continuous even at exact node crossing. The averaging canbe performed numerically, e.g. by some adaptive quadraturealgorithm (Piessens 1983) which is suitable also for improperintegrals. The result can be displayed as level lines ofH as afunction of(e, ω), becauseI depends upone onZ = const; thenode crossing lines are just circles on the plane of coordinates(e cos ω, e sinω) (Fig. 1).

The averagedH is not differentiable at points on the nodecrossing lines, and also forI = 0, that ise = emax; thus it isnot obvious what the “averaged” orbit means for planet-crossingorbits. Recently we have shown (Gronchi and Milani 1998) that

−0.6 −0.4 −0.2 0 0.2 0.4 0.6

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0

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e cos(omega)

e si

n(om

ega)

Earth asc.Earth desc.

emax

Mars asc. Mars desc.

Fig. 1. Maximum eccentricity and node crossing lines (with the Earthand with Mars) for an asteroid witha = 1.5 AU and inclination up to41.

H has smooth one-sided derivatives even along the node cross-ing lines; therefore it is possible to define generalised averageorbits, piecewise smooth curves in the(e, ω) plane, correspond-ing to the level lines ofH which cross the node crossing lines.We have also shown thatH is differentiable forI = 0, even ifnode crossing occurs for allω; thus the boundarye = emax,whereH is constant for symmetry reasons, can be seen as anaverage solution withI = 0.

3. Kozai librations

To understand the qualitative behaviour of the average solutionsit is useful to follow the changes occurring for a fixed value ofa, as the other integralZ, and thereforeImax andemax, change.We represent this in the(e, ω) plane, withω in the range[0, π/2]because of the symmetriesH(ω) = H(−ω) = H(π + ω) =H(π − ω).

The example in Fig. 2 is an outer main belt asteroid, witha = 3.41 AU (like (1373) Cincinnati, the example originallyused by Kozai 1962). For lowImax, the value ofemax is solow that close approaches to Jupiter cannot occur, and the levellines ofH correspond to almost constante. As Imax increases,therefore alsoemax increases, the level lines bend (top left) insuch a way thate is maximum forω = π/2, 3π/2 and minimumfor ω = 0, π (e − ω coupling). For a critical value ofemax, abifurcation occurs at the pointe = 0 (top right): a maximumof the functionH splits into two symmetric maxima and a sad-dle; the maxima have to bifurcate at least in pairs, because ofthe symmetry, but they always bifurcate along theω = π/2andω = 3π/2 line, never along the other symmetry line (formain belt asteroids). Fora(1 + emax) > a′ (bottom left) the

930 G.F. Gronchi & A. Milani: The stable Kozai state for asteroids and comets

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Fig. 2. Level lines of the averaged Hamilto-nian H, in the (e, ω) plane, fora = 3.41AU; node crossing lines are also plotted,if present; the initial indicates the crossedplanet.Top left: for Imax = 25 there isa significante − ω coupling, but no libra-tion. Top right: for Imax = 31.33 theKozai bifurcation has occurred, and libra-tion of ω around90 is possible.Bottomleft: for Imax = 35 the libration region islarger, andemax is such that node crossingwith Jupiter can occur forω ' 0. Bottomright: for Imax = 50 the libration regionextends to large eccentricities, and is inter-sected by the node crossing line with Mars.Node crossing with the Earth is also possiblefor e > 0.7, but not in the libration region.

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Fig. 3. Level lines ofH for a = 1.1 AU;node crossing lines with Venus, the Earth,and Mars are also plotted, if present. ForImax = 8. node crossings with the Earthcan occur (top left), but the only maximumis for e = 0, and there is no libration. ForImax = 115 there is a symmetric libra-tion aroundω = 0 (top right). Note thatthe level lines are not uniformly spaced, toenhance the libration region. ForI = 23.5there is also the line of node crossing withVenus, and the size of the libration region de-creases (bottom left); the node crossing linewith Mars also appears in the top left cornerof the figure. ForImax = 37. the sym-metric libration region has been replaced bymuch smaller asymmetric libration regions(bottom right); the corresponding maximumis very shallow, to the point that enhancingthe libration region by denser level lines re-sults in graphic difficulties.

node crossing lines of an exterior planet (in this case, Jupiter)appear along thesinω = 0 axis (that is from the top near theω = 0, π line). Asemax further increases, theKozai separatrix,that is the level curve through the saddle, increases in size andtwo large regions oflibration, with ω permanently oscillatingaround eitherπ/2 or3π/2, appear (bottom left). For largee, thenode crossing line with Jupiter “pushes” the Kozai separatrix,squeezing the libration regions to higher eccentricity (bottomright); the libration regions are intersected by the node cross-

ing lines with Mars, but close approaches with the Earth do notoccur in the libration region.

The example in Fig. 3 refers to Earth-crossing orbits, witha = 1.1. The node crossing line with the Earth appear for com-paratively lowemax (top left), butω still circulates. For a largeremax, allowing one nodal point well inside1 AU, a Kozai li-bration appears aroundω = 0, π (top right). Asemax furtherincreases, the node crossing line with Venus appears (bottomleft), and the size of the Kozai libration region decreases. For

G.F. Gronchi & A. Milani: The stable Kozai state for asteroids and comets 931

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J

J

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JS

Fig. 4. Level lines ofH for a = 7 AU; nodecrossing lines with both Jupiter and Saturnare also plotted.Top left: for Imax = 25

a libration region appears aroundω = 0,between the node crossing with Jupiter (be-low) and with Saturn (above).Top right: forImax = 30 there is an an asymmetric li-bration region, and a symmetric one foreabove the Saturn crossing line.Bottom left:for Imax = 35 a minimum ofH occurs atthe point where the two node crossing lineswith Jupiter intersect; it is surrounded by aweb-like Russian roulettelibration region,where the average solutions undergo fournode crossings per period of libration ofω.Bottom right:for Imax = 40 a new libra-tion region appears forω ' 90 and largeeccentricity.

largeemax the symmetric Kozai libration bifurcates into asym-metric ones (bottom right); the stable libration regions are verysmall, and they correspond to very shallow maxima; level curvesfor very close values ofH are required, and a finer grid than theone used in this figure would be required to give them a smoothappearance. For lowe, circulation without node crossings isalways possible.

By comparing the two examples above, we can see that theKozai libration, be it around eitherω = π/2, 3π/2 or aroundω = 0, π, “protects” the secular evolution of the asteroid or-bit from node crossings and close approaches. When close ap-proaches can occur near the perihelion (and/or aphelion), theargument of perihelionω oscillates around values displacingas much as possible the perihelion (and the aphelion) from theplane of the perturbing planets; this results in a protection mech-anism preventing approaches to Jupiter for the outer main beltasteroids such as(1373) Cincinnati(Kozai 1962), and close ap-proaches to the Earth for Amor asteroids such as(3040) Kozai(Milani et al.1989). When close approaches could occur at dis-tances from the Sun near the value ofa, then the perihelion andthe aphelion are kept on the reference plane, so that the inter-mediate values of the heliocentric distance take place as muchas possible out of the plane (Michel and Thomas 1996).

The third example refers toCentaurtype orbits, witha = 7AU, that is a semimajor axis intermediate between those ofJupiter and Saturn (Fig. 4). For low values ofImax, emax issmall and the heliocentric distance remains within a range notallowing node crossings with both planets. For moderate valuesof Imax, the node crossing lines with Jupiter and Saturn appear(top left), and a bifurcation takes place, generating a couple ofsymmetric maxima alongω = 0, π; they correspond to orbits

with one nodal point inside the orbit of Saturn, but outside theorbit of Jupiter. For larger values ofImax, the line of node cross-ing with Saturn approaches to the libration region (top right),and two more bifurcations take place: the two libration centreswith ω = 0, π split into four asymmetric libration centres withone nodal point inside Jupiter and the other one between Jupiterand Saturn; a couple of symmetric libration centres appear, withω = 0, π and a nodal point beyond Saturn. For largerImax, theequationa

√1 − e2 = a′ has a solution for0 < e < emax,

and a double crossing point appears onω = π/2, 3π/2 (bot-tom left); this is a non-smooth minimum point, surrounded bya bizarreRussian roulette librationregion, with generalised so-lution of the averaged equations of motion undergoing as manynode crossings as possible. For even largerImax a new libra-tion region, protected from node crossings, appears beyond theRussian roulette region, that is with both nodal points inside theorbit of Jupiter (bottom right).

The weakness of the “protection mechanism” argument usedabove is apparent from more complicated examples, involvingpossible collisions with two (or more) planets, as in the exampleof Fig. 4. The apparition of a quartet of asymmetric maxima iseasily detected, and follows all the rules of bifurcation theory;however, it is hard to explain by a “protection” argument. Evenless acceptable would be the use of an argument about “suici-dal” mechanisms, such as the one apparently acting around theminimum where two node crossing lines intersect. From thisdiscussion we are led to conclude that there is a need for a ra-tional explanation for the existence of Kozai libration regions,without resorting to any anthropomorphic expression like pro-tected/suicidal.

932 G.F. Gronchi & A. Milani: The stable Kozai state for asteroids and comets

4. Existence theorem

In this section we are going to prove that someKozai state,around which stable oscillations are possible, always occurs.These Kozai states can appear, as in the examples of Figs. 2–4, in three forms: as circulations ofω around the single point(0, 0) in the (x = e cos ω, y = e sinω) plane; as couples ofsymmetric libration regions, with two libration centres (eitherwith ω = 0, π or with ω = π/2, 3π/2); as quartets of asym-metric librations aroundω = α,−α, π + α, π − α for someα.

Theorem 1 At least one Kozai state occurs for all value of theintegralsa, Z, for an arbitrary number of perturbing planets,and for all values of the parameters (semimajor axes and massesof the planets).

An outline of the proof follows. We consider the averagedHamiltonianH as a function of(x, y) for fixed a and fixedZ;it is continuous on the diskW of equationx2 + y2 ≤ e2

max,thus it must have some maximum. NowH = H0 − R, whereH0 is the unperturbed part (independent from(x, y)) andR isthe average of1/D (apart from some positive factor). ThusRmust have some minimum. If the minimum point is where thefunction R(x, y) is smooth, then a neighbourhood is a stableregion by the standard Lyapounov function argument (see e.g.Milani e Mazzini 1997, Sect. 3.3).

The real point of the proof is therefore to show that theminima of the functionR(x, y) can occur neither on the nodecrossing lines (step 1) nor on the boundary (step 2).

Step 1: let us concentrate on a single node crossing line, let ussayd+

nod = 0 for some planet with semimajor axisa′; let W+

be the open setd+nod > 0 (the part of the interior of the circle

contained inW ), andW− bed+nod < 0 (the part ofW outside

the circle). For any functionf smooth on bothW+ andW−

(but not necessarily onW ), let

∂+

∂xf ,

∂+

∂yf

be the derivatives computed inW+, for d+nod > 0, and let

∂−

∂xf ,

∂−

∂yf

be the same derivative computed inW−, whered+nod < 0.

Let us apply this definition to the absolute nodal distance|d+

nod|: by explicit computation from (1)

∂+

∂x|d+

nod| = ay2 − x2 − 2x − 1

(1 + x)2;

∂+

∂y|d+

nod| = − 2ay

1 + x

∂−

∂x|d+

nod| = −∂+

∂x|d+

nod| ;∂+

∂y|d+

nod| = −∂+

∂y|d+

nod|

These functions are limited and therefore can be extendedto the points withd+

nod = 0. However, the extensions do notcoincide on the node crossing line, but are opposite (and ingeneral non zero). The directional derivative along the exterior

normal tod+nod = 0 has an “interior” value, obtained as a limit

from values inW+

∂+

∂N|d+

nod| = − a

2(1 + x)3[1 + x2 + 2x − y2]2

− 2ay2

1 + x< 0

and an “exterior” value exactly opposite:

∂−

∂N|d+

nod| > 0 .

We need to use some results from Gronchi and Milani(1998), where we have shown that the integral, over the torusT : ` ∈ [0, 2π] ; `′ ∈ [0, 2π], of the inverse distance1/Drequired for the averaging can be decomposed as follows, withthe integral, of an approximate inverse distance1/d:∫ ∫

T

1D

d` d`′ =∫ ∫

T

1d

d` d`′ +∫ ∫

T

[1D

− 1d

]d` d`′

where the difference1/D−1/d is a limited function; the integralof the difference is a differentiable function, even on the nodecrossing lines.

The approximate distanced is obtained by a method derivedfrom Wetherill (1967), as the distance between points on straightlines tangent to the two orbits at the nodal points.d has a simpleanalytical expression, and this allows to reduce the integral tothe form:

I =∫ ∫

T

1d

d` d`′

= 2∆ ·[ 3∑

i=1

∫ θi

θi−1

√C + r2

i (θ) dθ − π√

C

](2)

where∆ is a function of(a, e, I, ω, a′) which is smooth (also onthe node crossing line) and always positive forx2 +y2 < e2

max;θi, ri(θ) are defined by a suitable change to polar coordinates,

and√

C is the distance between the two straight lines, withexpression

√C = |d+

nod| ·[1 − ∆2a2a′2e2 sin2 ω

(1 − e2)

] 12

.

The one variable integral appearing in (2) is an elliptic integral,which is a smooth function of all the orbital elements. Thus thenon differentiability of the averagedR at the node crossing line

d+nod = 0 arises only from−2π∆

√C.

This allows to conclude that minima ofR(x, y) cannot occuralongd+

nod = 0, because the normal derivative of the integralI is the sum of some smooth function plus a discontinuousfunction positive inW+ and negative inW−; the same appliestoR, which is the sum ofI plus a smooth function. In geometricterms, the graphic ofR has a crest with the edge towards highervalues, as is clearly visible in Fig. 5.

The same argument applies in the case of a descending nodecrossingd−

nod = 0. However, a descending node crossing canoccur simultaneously with an ascending node crossing, either

G.F. Gronchi & A. Milani: The stable Kozai state for asteroids and comets 933

020

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0

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0.81.6

1.65

1.7

1.75

1.8

1.85

x 10−4

omega

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eccentricity

Fig. 5. Graph of the averaged perturbingfunction R, for a Centaur type orbit. Thenode crossing lines with Jupiter appear ascrests, and the double crossing point is apyramid-like maximum. Fore → emax, thatis for I → 0, the slope of the graph goes to+∞, as shown in the step 2 of the proof.

with the same planet (as in Fig. 4, bottom left and right), orwith another one (as in Fig. 2, bottom right). This can result in apeak, where two crests meet, as in Fig. 5. The two node crossinglines being smooth, it is possible by a suitable smooth change tocoordinatesx′, y′ to describeR asf(x′, y′)−h|x′|−k|y′|, withf smooth andh, k > 0, and by elementary arguments it can beshown that this cannot be a minimum (it can be a maximum,and indeed it is in Fig. 5). Thus a minimum ofR cannot occuron any point belonging to one or more of the crossing lines.

Step 2:to exclude the possibility of a minimum along the bound-arye = emax we are going to prove that the directional deriva-tive along the exterior normal to the boundary, on a surfaceZ = const, has limit

lime→emax

∂R

∂e(e, ω)

∣∣∣Z=const

= +∞for almost allω. Fore = emax, henceI = 0, the averaged per-turbing function does not depend uponω for symmetry reasons,and its value is a local maximum, thus cannot be a minimum.

The computation of the normal derivative to the boundarydoes not coincide with the partial derivative∂R/∂e, becausethe eccentricity is related to the inclination by the equationZ =const, thus

∂R

∂e(e, ω)

∣∣∣Z=const

=∂R

∂e− cos I

sin I

e

1 − e2

∂R

∂I(3)

The derivative with respect to the inclination of the inversedistance can be computed by

1sin I

∂

∂I

(1D

)= − 1

sin I

∂D2/∂I

2D3 ; (4)

for I = 0, e = emax this expression has a polar singularity oforder≤ 3; we will show that it is indeed of order 3. This occurs

whenever there is orbit crossing; when there is no orbit crossingthe existence of a stable region is obvious.

The squared distance has a simple expression as a functionof the eccentric anomalyu of the asteroid and of the anomalyu′ of the planet (measured from the node); thus the derivative(4) can be computed as

1sin I

∂D2

∂I= 2aa′ sinu′[(cos u − e) sinω

+√

1 − e2 sinu cos ω]

.

For I = 0 the expression above needs to be computed at thecrossing points, that is at the values ofu, u′ such that

a(cos u − e) = a′ cos(u′ − ω) ;

a√

1 − e2 sinu = a′ sin(u′ − ω) (5)

(in the plane the argument of perihelionω and the anomalyu′

are measured from the same origin); at these crossing points

1sin I

∂D2

∂I= 2(a′)2 sin2 u′

is always positive, with the only exception of the caseu′ = 0, π,corresponding to the intersection of the node crossing lines withthe boundarye = emax (the argument could be extended alsoto this case, with a little more work). Thus the right hand sideof Eq. (4) has a polar singularity – occurring forI = 0 and forthe values ofu, u′ satisfying (5) – of order 3, moreover, it ispositive in a neighbourhood of the singularity.

The portion of Eq. (3) containing the partial derivative withrespect toe has a polar singularity of order two (explicit com-putations can be found in Gronchi 1997); thus the pole of order3 is the principal part of the right hand side of Eq. (3), and for

934 G.F. Gronchi & A. Milani: The stable Kozai state for asteroids and comets

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Fig. 6. Level lines ofH for a = 17.84AU; node crossing lines with the externalplanets are also plotted; the coordinates areω, in degrees, on the horizontal axis, and−√

1 − e2 on the vertical axis (this is usedto enhance the visibility of the region withe close to 1). For a minimum retrograde in-clination Imin = 130 (corresponding toe = 0; for retrograde orbitsemax corre-sponds toI = 180) there are two symmet-ric librations (top left). The level curves forImin = 120 (top right) and forImin =110 (bottom left) indicate a large numberof bifurcations, with Kozai states appearingand disappearing. For high retrograde incli-nationImin ' 104, the value correspond-ing to the real Halley comet, the Kozai stableregions have all moved to very highe, apartfrom a minute circulation region at very lowe (bottom right).

I = 0 the integral is positively divergent:∫ ∫T

∂

∂e

1D

d` d`′ = +∞ .

We can now conclude by using Fatou’s lemma (Brezis 1986,pp. 383-384): letPe(ω, `, `′) be the principal part of

∂R

∂e(e, ω, `, `′)

∣∣∣Z=const

for e → emax; then there is a neighbourhoodU of the singularitywhere the functionsPe are positive, and where

∂R

∂e(e, ω, `, `′)

∣∣∣Z=const

≥ 12

P (e, ω, `, `′) .

Then the Fatou lemma ensures that

lim infe→emax

∫ ∫U

12

Pe(ω, `, `′)d` d`′

≥∫ ∫

U

lim infe→emax

12

Pe(ω, `, `′)d` d`′

with the latter integral positively divergent; then

lim infe→emax

∫ ∫U

∂R

∂e(e, ω, `, `′)

∣∣∣Z=const

≥ lim infe→emax

∫ ∫U

12

Pe(ω, `, `′)d` d`′ = +∞ .

This argument has to be repeated for the other intersection pointoccurring forI = 0, and for its neighbourhoodU1. The integralson the remaining part of the torus,T − U − U1, are uniformlylimited and thus do not change the infinite limit. The derivativeof the integral, being the same as the integral of the derivativefor e < emax, has limit+∞.

This concludes the proof of the theorem.

5. Implications for the asteroids and comets dynamics

From the results of this paper, taking into account the state ofthe art in the understanding of the long term evolution of planet-crossing orbits, we would like to draw the following three con-clusions.

First, a stable Kozai state always exists in every case. Of-ten there are many such states: low eccentricity circulations,symmetric and asymmetric librations can occur together, forthe same values ofa andZ. This is often the case when theorbits of the asteroids/comets can cross the orbits of more thanone planet; as an example, Thomas and Morbidelli (1996) showthe complicated structure of the Kozai states for transneptunianobjects and intermediate period comets. In Fig. 6 we show thelevel lines ofH for the value of the semimajor axis of cometP/Halley: the sequence of bifurcations, appearances and dis-appearances of libration centres is impressive; there are stableKozai states with nodal points placed either inside or outsidethe orbit of each planet, in almost all possible combinations.

Second, some caution needs to be used before concluding,from the existence of a Kozai state in the averaged problem,that there is a region in the phase space of the complete problemfull of orbits stable over a long time span. The averaging overthe longitudes does not take into account the mean motion res-onances, which can change the orbital elements (especially theeccentricity) by large amounts. On the other hand, mean mo-tion resonances can protect from close approaches even whennode crossings occur (Milani and Baccili 1998). The averagingwith respect to circular, coplanar orbits of the perturbing plan-ets does not take into account secular resonances, and these canchange eccentricities and inclinations to extreme values (Mor-bidelli 1993). Even outside resonances, thee − $ coupling ofthe secular perturbations containing the planetary eccentrici-

G.F. Gronchi & A. Milani: The stable Kozai state for asteroids and comets 935

ties can significantly displace the nodal points and protect fromnode crossings (Milani and Nobili 1984). Moreover, the averag-ing principle is always an approximation of the first order in theperturbation parameters, and in case of very strong perturbationthe average solution might not be an accurate description of theaverage of the true solution.

Third, the Kozai type theory used in this paper is only a sim-plified approximate model, with the limitations outlined above.It is nevertheless a better approximation than simply assumingthat ω, Ω precess with constante, I; this is especially true ifthe purpose is to compute the probability and/or frequency ofclose approaches and collisions. As an example, there are as-teroids with perihelion descending below 1 AU, for which thefrequency of deep close approaches to the Earth is zero, becausethey belong to an high eccentricity Kozai state, as in Fig. 2, bot-tom right; these asteroids belong to theKozai class, accordingto the classification in Milani et al. (1989). The figures shownin this paper, as well as in Gronchi and Milani (1998) and inThomas and Morbidelli (1996), clearly indicate that the Kozaiclass is by no means the only case in which an oversimplifiedcomputation of the probability of collision is bound to be wrong.Only for orbits with low eccentricity circulation ofω it is pos-sible to havee andI almost constant, and a simple formula canbe used to estimate accurately the probability of collision; inmost of these low eccentricity cases, however, there are no nodecrossings and the encounters are impossible.

In conclusion we advocate the recomputation of the proba-bilities of collision for all kinds of planet-crossing orbits (NearEarth Asteroids, comets, Centaurs) with methods includingat least some kind of secular perturbation theory. This mightnot change in a very significant way the global probabilities ofimpact, e.g. on the Earth, of the entire population of potential im-

pactors, because the simplified computations can result in bothoverestimate and underestimate, and over a large number of bod-ies these can compensate each other (Milani et al. 1990). Moreaccurate computations can, however, change in a dramatic waythe estimated probabilities of collision for individual bodies.

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