Numerical-Analytical Study of Linked Orbits in the

ISSN 0038-0946, Solar System Research, 2021, Vol. 55, No. 2, pp. 159–168. © The Author(s), 2021. This article is an open access publication, corrected publication 2021.Russian Text © The Author(s), 2021, published in Astronomicheskii Vestnik, 2021, Vol. 55, No. 2, pp. 182–192.

Numerical-Analytical Study of Linked Orbits in the Restricted Elliptic Doubly Averaged Three-Body Problem

M. A. Vashkov’yak*Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Moscow, 125047 Russia

*e-mail: [email protected] October 6, 2020; revised October 25, 2020; accepted November 6, 2020

Abstract—The restricted elliptic doubly averaged three-body problem is considered. The terms up to the sec-ond order inclusive with respect to the orbital eccentricity of the perturbing body are retained in the expansionof the perturbing function. The elements of the elliptical orbit for the perturbed body are deemed arbitrary.The special, so-called linked orbits of a negligible-mass body are investigated by numerically integrating theaveraged equations in Keplerian elements. For such orbits the points of their intersection with the orbitalplane of the perturbing body are on different sides of this orbit. The evolution of hypothetical and some realcometary orbits is described in the simplest Sun-Jupiter-comet model; their differences from the correspond-ing orbits in the circular problem have been revealed.

Keywords: restricted elliptic three-body problem, double averaging, linked orbits, numerical integrationDOI: 10.1134/S0038094621020064

INTRODUCTION AND FORMULATIONOF THE PROBLEM

Studies of the long-term evolution of orbits in therestricted elliptic three-body problem are generallycarried out in the doubly averaged formulation. Theintegrable case of a circular orbit of the perturbingbody is widely used. The studies of this case initiatedby the renowned scientists H. von Zeipel (von Zeipel,1910) and N.D. Moiseev (Moiseev, 1945) were elabo-rated significantly by M.L. Lidov (Lidov, 1961, 1962)and I. Kozai (Kozai, 1962). These studies aredescribed in detail in the monographs by Shevchenko(2017) and Ito and Ohtsuka (2019). Note that theextensive paper by H. von Zeipel has become deserv-edly famous and is reflected in the above scientific-historical study by Ito and Ohtsuka (2019) only due tothe reference to it in Baily and Emel’yanenko (1996) inconnection with the study of the evolution of one ofthe types of cometary orbits. The works by Moiseev,Lidov, and Kozai performed later were both the resultsof a qualitative study of the averaged three-body prob-lem and the necessity of investigating the orbitaldynamics of artificial satellites of planets and thedynamics of asteroids.

In his extensive paper Zeipel (1910) identified andqualitatively studied three main cases of the orbit loca-tion for the perturbed body in the doubly averaged cir-cular problem: the inner one, the outer one, and thecase of intersecting, in particular, the so-called linkedorbits. The spatial location of linked orbits is charac-terized by the fact that one of the points of intersection

of the orbit of a negligible-mass body with the plane ofthe orbit of the perturbing body is located inside it,while the other one is outside. Such a classification inthe restricted circular doubly averaged three-bodyproblem for uniformly close orbits, along with an anal-ysis of the conditions for their intersection, was pro-posed by Lidov and Ziglin (1974). The topology of twolinked and unlinked Keplerian orbits of all those typeswas described in detail by Kholshevnikov and Titov(2007).

Note that the phrase “like the rings of a chain” pro-posed in the monograph by Ito and Ohtsuka (2019) asthe English analog of the French term “comme lesanneaux d’une shaine” used by von Zeipel (1910) ismore sensible and geometrically understandable thanthe term “linked orbits”. The linked orbits in therestricted elliptic doubly averaged three-body problemare the subject of our study.

Consider the motion of particle Р of negligiblemass under the attraction of a central point S of massm and a perturbing point J of mass m1 m moving rel-ative to S in an elliptical orbit with a semimajor axis а1and eccentricity е1. Let us introduce a rectangularcoordinate system Oxyz with the origin at point Swhose reference plane xOy coincides with the orbitalplane of point J. Let the Ox axis be directed to thepericenter of the orbit of point J, the Oy axis be in thedirection of its motion from the pericenter in the ref-erence plane, and the Oz axis complements the coor-dinate system to a right-handed one. The perturbedorbit of point Р is characterized by osculating Keple-

!

159

160 VASHKOV’YAK

rian elements: the semimajor axis а, the eccentricity е,the inclination i, the argument of pericenter ω, and thelongitude of the ascending node Ω. In the chosencoordinate system the perturbing point J has coordi-nates x1, y1, and z1 = 0.

The secular part W of the complete perturbingfunction is used to investigate the orbital evolution ofpoint Р:

(1)

Here, isthe distance between the perturbed and perturbingpoints, λ and λ1 are the mean longitudes of thesepoints, and f is the gravitational constant. The absenceof low-order commensurabilities between the meanmotions of points J and Р is assumed. In the function Wa1 and e1 act as parameters of the evolution problem.

The first integrals of the equations of perturbedmotion in elements are

(2)

while one more first integral exists in the case of е1 = 0(Moiseev, 1945):

(3)

THE AVERAGED PERTURBING FUNCTION AND EVOLUTION EQUATIONS

Another expression, equivalent to (1), for the func-tion W via well-known formulas is also commonlyused in analytical studies:

(4)

where V is the attractive force function of the ellipticalGaussian ring simulating the averaged influence of theperturbing point and Е is the eccentric anomaly ofpoint Р.

Since for the orbits under consideration the dis-tance r can be both less than and greater than r1 as theyevolve, the commonly used expansions of the inversedistance 1/Δ into series in Legendre polynomials areinapplicable. Therefore, in this paper we will use theanalytical expression for the function V, though with a

limited accuracy up to inclusive, from Vashkov’yak(1986) for a nearly coplanar system of N of weaklyelliptical Gaussian rings, but valid for any relationbetween r and r1. Taking into account the orientationof the introduced coordinate system and assumingthat N = 1, i1 = ω1 = Ω1 = k1 = u1 = v1 = 0, and h1 = e1

in Eqs. (6)–(8) of this paper, we will obtain

(5)

and, for coherence, we will permit ourselves to alsoreproduce the basic simplified computational formu-las from this paper by supplementing them with newanalytical relations.

The function Ф dependent on the rectangularcoordinates x, y, z is expressed via hypergeometricGaussian functions F:

(6)

Functional series of various structures, dependingon the numerical value of the argument ζ ( ), areused for their calculation, so that

(7)

( )( )

π π

Ω = λ λΔ λ λπ

2 21

1 1 1210 0

1, , ,ω, , , .,4

fmW a e i a e d d

( ) ( )Δ = − = − + − +2 2 21 1 1x x y y zr r

= ω Ω =1 1const, , , , ,( ), , const,a W a e i a e

( )− = =2 211 cos const.e i c

( ) ( )

( )( )

π

π

= −π

λ=π Δ λ

2

02

1 1

10

1 1 cos ,2

,2 ,

W e E V E dE

fm dV EE

21e

( ) ( )= Φ+

12 21

, , ,fmV E x y za r

( ) ( )( ) ( )

Φ = − ε + μ ζ

+ ν − ε ζ

1 31 , ;1;4 4

1 5 3, ;2; ,2 4 4

F

F

ζ < 1

( )( )

( )[ ] ( ) ( )[ ] ( ){ } ( )

∞

=∞

=

+ ++ μ − ε + ν ζ ζ < ζ + + Φ = − − ζ + μ + + ν − + ε + ε − ν − ζ ζ > ζπ

0

0

3 2 1 4 1 *1 , ;2 1 1

1 *ln 1 1 4 4 1 8 3 8 2 1 , .2

nn

n

nn n

n

n n Bn n

H n n B

SOLAR SYSTEM RESEARCH Vol. 55 No. 2 2021

NUMERICAL-ANALYTICAL STUDY OF LINKED ORBITS 161

Here,

(8)

The constant coefficients Bn and Hn are defined bythe recurrence relations

(9)

while the empirical value of ζ* is taken to be 0.5.Remark. The function V depends only on the

squares of the coordinates y and z, while the coordi-nate x enters into this function both quadratically andlinearly (into the numerator of ε and via it into ζ). Theexistence of two planar particular solutions, y = 0 andz = 0, in the singly averaged (only over λ1) evolutionproblem follows from such a double symmetry of thefunction V with respect to y and z.

The rectangular coordinates x, y, z are expressed viaЕ by the well-known formulas for unperturbed Keple-rian motion

(10)

(11)

Below, it will be convenient to introduce a newindependent variable, a “dimensionless time” τ,according to the formula

(12)

( )

( )

( )( )

( )( )

+ε = μ = ε + − + ρ + ρ

ν = ε − = ρ ++

ρ = + ζ =+

× ρ + ε − ρ + + θ − ρ + + θ = − + ρ+

++ −ρ +

2 2 2 2 221 1 1 1 1

2 2 2 2 2 21 1

2 22 2 2 21 1

2 21

22 2 2 1

22 21

2 2 2 21

2 2 2 2 2 21 12 2 2 1

1 1 2 22 21

2 2 22 1 1

2 21

, 2 ,2

3 ,2 2

4

2

4

,

,

,

6

a e x a z a e ya r a r

a e r za r

ax ya r

a z

a a z a re a xa r

a z aya

2 .r

( ) ( )

( )( ) ( )

−

−

− −= =

−= + =− −

1 02

1 0

4 3 4 1, 1,

162 3 8

, 6ln2,4 3 4 1

n n

n n

n nB B B

nn

H H Hn n n

( ) = − + −

1 12

2 2

3 3

cos 1 sin ,x p qy a E e p e E qz p q

= ω Ω − ω Ω= ω Ω + ω Ω = ω

= − ω Ω − ω Ω= − ω Ω + ω Ω = ω

1

2 3

1

2 3

cos cos sin sin cos ,cos sin sin cos cos , sin sin ,

sin cos cos sin cos ,sin sin cos cos cos , cos sin .

p ip i p i

q iq i q i

( )τ = −10

1

,m a n t tma


where is the mean motion of point Р, and a

normalized perturbing function

(13)

To describe the evolution of orbits, we will use theLagrange equations in elements with the function wthat is their first and unique integral:

(14)

The existence of stationary solutions of these equa-tions is possible if the conditions

are fulfilled.

PARTIAL DERIVATIVES OF THE NORMALIZED FUNCTION w WITH RESPECT

TO THE ELEMENTS

Generally, for arbitrary orbits of point Р a solutionof Eqs. (14) can be found apparently only by a numer-ical method, while the process of calculations can becontrolled by the constancy of the function w alongthis solution. In Vashkov’yak (1986) the partial deriv-atives of the function w with respect to the elementswere calculated by a difference method. Here, we usea combined method, in which the quadratures

(15)

are found numerically by the Gaussian method, whilethe derivatives of the normalized function

(16)

are found analytically. For the completeness of the setof formulas, we give the necessary expressions to cal-culate the derivatives of the function with respect tothe orbital elements:

= 3/2fmn

a

= =1

1

const.aw Wfm

− ∂ ∂ ∂= − = −τ ∂ω τ ∂ω ∂Ω− −

ω − ∂ ∂= −τ ∂ ∂−

Ω ∂=τ ∂−

2

2 2

2

2

2

1 cot cosec, ,1 1

1 cot ,1

cosec .1

de e w di i w i wd e d e e

d e w i wd e e ie

d i wd ie

ω Ω= = = =τ τ τ τ

0de di d dd d d d

( )

( )

π

π

∂ ∂ = − − ∂ π ∂∂∂∂∂

∂ ∂= −∂ω π ∂ω∂ ∂∂Ω ∂Ω

2

0

2

0

1 1 cos cos ,2

1 1 cos2

w Ve E V E dEe e

Vwii

w Ve E dE

w V

( )= = Φ+

1 12 2

1 1

, ,a aV V x y zfm a r

V

162 VASHKOV’YAK

(17)

(18)

(19)

(20)

∂∂ ∂ ∂ ∂∂ ∂Φ∂∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂Φ Φ∂ ∂ ∂ ∂ = + + −

∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ + ∂ω ∂ω∂

=

ω∂ω ∂Φ∂ ∂ ∂∂∂ ∂ ∂∂Ω ∂Ω∂Ω∂Ω

+

2 21

2 211

, .

yV x zVe ee e

yV x z x x xV V V Vi i i i y

x y zV x y z y y a r zV

x z

a

a

z

r

yV z

( )

( ) ( )

∂ ∂∂ ∂ ∂ ∂ = − + = − ω + − ω ∂ ∂−

∂ ∂∂ ∂

∂ ∂∂ω ∂Ω − − ∂ ∂ = − − − = − + − ∂ω ∂Ω

∂ ∂∂ω ∂Ω

1 1 12

2 2 22

3 3 3

1 1 2 22 2

2 2 1 1

3 3

sin , cos sin 1 cos sin ,1

cos 1 sin , cos 1 sin ,0 0

x xe ip q ry ye Ea p q a E e e E re iep q rz ze i

x xq p p q

y ya E e q e E p a E e p e E qq pz z

= Ω = − Ω =1 2 3sin sin , sin cos , cos .r i r i r i

∂ζ ∂μ∂Φ ∂ε ∂ν∂ ∂∂ ∂ ∂∂ζ ∂μ∂Φ ∂Φ ∂ε ∂Φ ∂Φ ∂Φ ∂ν= + + +

∂ ∂ε ∂ ∂ζ ∂ ∂μ ∂ ∂ν ∂∂Φ ∂ε ∂ζ ∂μ ∂ν∂ ∂ ∂∂ ∂

,

x xx x x

y y y y y

z z zz z

( ) ( ) ( ) ( )

( ) ( ) ( )

( )

∂ε ∂ε ∂ε= − + + = − =∂ ∂ ∂+ + +

+∂μ ∂ε = ε + + × + + + − ε + ∂ ∂ ρ ρ +

+∂μ ∂ε= ε + + × + ∂ ∂ ρ ρ +

2 2 2 21 1 1 1 1 112 2 22 2 2 2 2 2

1 1 1

2 2 2 2 22 2 2 2 2 2 21 1 11 12 4 22 2

1

2 2 2 2 221 1 112 4 22 2

1

2 2, , ,

24 2 2 ,

24

a e a e xy a e xza x y zx y za r a r a r

a z a e yx a x y z a zx x a r

a z a e yy ay y a r

( ) ( ) ( )

( )

( )

+ + − ε + × + ρ

+∂μ ∂ε = ε + + ε + ∂ ∂ ρ ρ + ∂ν ∂ε∂ ∂∂ν ∂ε= ε +∂ ∂ +∂ν ∂ε∂ ∂

2 22 2 2 2 2 2 1 1

1 2 2 21

2 2 2 2 221 1 1

2 2 22 21

2 21 1

22 21

22 2 ,

24 ,

3 .

a e yx y z a za r

a z a e yzz z a r

x x xa e y

y y a r z

z z



(21)

(22)

Relations (7)–(11) and (15)–(22) represent a com-plete set of formulas to calculate the right-hand sidesof the evolution equations (14).

In view of the properties of the function V (see theremark in the previous section), system (14) has twoparticular solutions or integrable cases.

Case 1. If sini = 0, then the orbital plane of point Рcoincides with the orbital plane of the perturbing point J.By symmetry, it turns out that di/dτ = 0. However, inthis planar solution at any а, а1, and e1, in addition to

regular orbits, there exist irregular ones intersecting(but not linked) with the orbit of point J (Vashkov’yak,1982).

Case 2. If cos i = 0 and sinΩ = 0, then in the ellipticproblem under consideration the orbital plane of pointР is orthogonal to the orbital plane of the perturbingpoint J and passes through its line of apsides. By sym-metry, it turns out that di/dτ = 0 and dΩ/dτ = 0. Theabove formulas of the quadratic approximation in е1

( )( ) ( ) ( )

( )( ) ( ) ( )

( )( )

∂ζ ∂ε ε ∂θ= − ρ + − + − ε − ρ + θ + ∂ ∂ ∂+ + +

∂ζ ε∂ε ∂θ= − ρ + − + − ε − ρ + θ + ∂ ∂ ∂+ + +

∂ζ ∂ε ε= − ρ + −∂ ∂ ++

22 2 2 2112 2 2 2 22 2

1 11

22 2 2 2112 2 2 2 22 2

1 11

22 2 2112 2 22 2

11

4 4 42 2 1 2 ,

4 4 42 2 1 2 ,

4 42

a x xa z xx x xa r a ra r

a y ya z yy y ya r a ra r

a za zz z a ra r

( )

( )( )

( )( )

( )

( )( )

( )

∂θ+ ε − ρ + θ + ∂+

− − ρ + − + + + + +∂θ =∂

− − ρ + −ρ ρ

+∂θ = + + − ρ + + −∂ ρ

+ ++

22 21

2 222 2 2 2 21 112 2 2 22 2 2 2

12 1 11

2 22 2 214 2

2 2 2 222 2 2 2 2 21 11 1 14 2 2 2 2 22 2

1 11

44 ,

6 221 3 2

2

2 2

6 3 24

,

2

zzza r

a axa z x ya ra r a r

ex y xa z

a x x yxe a z a zy a r a ra

x

yr

( )( )

+

−∂θ = + − − ρ + ∂ ρ + +

2 22 22 2 2 2 2 21 11 12 2 2 22 2

11

,

.2 62 2 3a xy xe y x a zz a ra

zr

( ) ( )[ ]{ } ( )

( )( )

( ) ( )[ ] ( )( ) ( )[ ]{ } ( )

∞

=∞

=∞

−

=∞

−

=∞

=

+− ζ ζ < ζ +∂Φ = ∂ε − + − − ζ − ζ ζ > ζπ + ++ μ − ε + ν ζ ζ < ζ + +∂Φ = ∂ζ − + ε + μ + + ν − − − ζ − ε − ν − ζ ζ > ζπ

ζ∂Φ =∂μ

0

0

1

1

1

0

0

3 2 1 *, ,2 1

1 *8 8 3 ln 1 1 , .2

3 2 1 4 1 *1 , ,2 1 1

1 *1 8 3 4 4 1 1 ln 1 8 2 1 ,2

,

nn

n

nn n

n

nn

n

nn n

n

nn

n

n Bn

n H B

n n nBn n

n n n H n B

B

( )[ ] ( )

( ) ( )[ ]{ } ( )

∞

=∞

=∞

=

ζ < ζ

− − ζ − ζ ζ > ζπ + ζ ζ < ζ +∂Φ = ∂ν + − − ζ − − ζ ζ > ζπ

0

0

0

*,

1 *ln 1 1 , ,2

4 1 *, ,1

4 *4 1 ln 1 4 1 , .2

nn n

n

nn

n

nn n

n

H B

n Bn

n H B


164 VASHKOV’YAK

Table 1. Maximum changes of the elements e, i, ω, Ω in atime interval of 500000 years for i0 = 90°, 90° ± 5°

i0, deg Ω0, deg Δ1 Δ2, deg Δ3, deg Δ4, deg

90 60 0.001 24.3 1.0 21.990 120 0.002 20.8 1.1 20.385 0 0.0003 2.2 0.3 10.385 60 0.0004 24.0 0.8 12.385 120 0.0017 22.5 1.2 12.485 180 0.0001 2.0 0.2 7.595 60 0.0022 23.8 0.6 31.695 120 0.0028 18.7 1.4 28.2

Table 2. The same as Table 1, but for i0 = 90° ± 30°i0, deg Ω0, deg Δ1 Δ2, deg Δ3, deg Δ4, deg

60 0 0.010 1.4 9.8 60.060 60 0.003 19.8 7.3 35.160 120 0.004 29.0 8.3 26.560 180 0.005 15.1 7.9 42.6

120 60 0.013 9.1 9.6 74.3120 120 0.010 2.9 10.2 63.0

allow this to be also verified directly. Indeed, forcosi = 0 and sinΩ = 0 we have

(23)

In this case,

(24)

It is easy to verify that at y = 0 the derivatives of thefunctions ε, ζ, μ, and ν with respect to y become zero,

so that and

Equations (14) simplified for this case of orthogo-nal-apsidal orbits take the form

(25)

A general qualitative study of this case by takinginto account the possible intersections of the orbits ofpoints Р and J was carried out by a numerical-analyti-cal method in Vashkov’yak (1984) for arbitrary a, а1,and е1. In this paper greater attention is given to linkedorbits and, in particular, to the stationary solutions ofEqs. (25) existing at ω0 = 0, π. It can be shown that, in

this case, and the stationary eccentricities

themselves are determined as the roots of the tran-scendental equation

(26)

ON THE EVOLUTION OF SOME HYPOTHETICAL AND REAL COMETARY ORBITS IN THE SOLAR SYSTEM MODELFirst we will turn to the linked orbits of point Р

highly inclined to the reference plane. In the integra-

( )

( )

( )

( )π

δ = Ω = ±= δ ω = = ω= −δ ω = = ω

= = −δ = = δ − ω − ω −

= = − ω + ω −

∂ ∂∂ ∂= −∂ ∂π∂Ω ∂Ω

1 2 3

1 2 3

1 2 3

2

2

2

0

sign cos 1,cos , 0, sin ,sin , 0, cos ,

0, , 0,

cos cos sin 1 sin ,

0,

cos sin cos 1 sin ,

1 1 cos .2

p p pq q q

r r r

x a E e e E

y

z a E e e E

w Vi ie E dEw V

∂∂−δ −δ∂ ∂ ∂Φ∂ ∂= = =

∂∂ ∂ ∂ ∂+∂Ω ∂Ω

−δ ∂ζ ∂μ∂Φ ∂ε ∂Φ ∂Φ ∂Φ ∂ν= + + + ∂ε ∂ ∂ζ ∂ ∂μ ∂ ∂ν ∂ +

1

2 21

12 21

.

yVz zaV Vi i

y x xV y y ya r

zaxy y y ya r

∂ ∂= =∂ ∂Ω

0w wi

Ω= =τ τ

0.di dd d

− ∂ ω − ∂= − =τ ∂ω τ ∂

2 21 1, .de e w d e wd e d e e

∂ =∂ω

0w

( )∂ ω = π Ω = π =∂

1 1 0 0 0, , , , 0, ; 0,0.

w a a e ee

ble case of orthogonal-apsidal orbits, a numericalsolution of Eq. (26) makes it possible to find the sta-tionary values of the eccentricity е0 at given ω0 = 0°,180°, Ω0 = 180° and fixed parameters а, а1, е1. In addi-tion, it can be shown that the values of е0 depend on ω0and Ω0 only through the combinationδ1 = sign(cosω0cosΩ0) = ±1. However, the difference

in е0 at different signs of δ1 is fairly small and ~ .

At an inclination and longitude of the ascendingnode different from those adopted in case 2, all of theorbital elements will change with time. Our numericalintegration of the evolution system (14) by the Runge–Kutta method at а1 = 5.2 AU, е1 = 0.048, and the massratio m/m1 in the Sun–Jupiter system makes it possibleto estimate such changes for fictitious (or hypotheti-cal) cometary orbits. Tables 1 and 2 give such an esti-mate in the time interval t* = 500000 years for orbitswith a semimajor axis а = 10а1 = 52 AU, e0(δ1 = 1) =0.9890, е0(δ1 = –1) = 0.9905. These tables present

Table 1 was compiled for three values of i0 = 90°,85°, 95°. In the exact solution of Eqs. (14) i0 = 90°,Ω0 = 0°, 180° the deviations are zero and, therefore,the corresponding rows are omitted. At Ω0 = 0 and180° the results for i0 = 95° identical to the corre-sponding data for i0 = 85° are also omitted.

21e

( )Δ = Δ =Δ = ω ω Δ = Ω Ω

1 0 2 0

3 0 4 0

max * – , max ( *) – ,max ( *) – , max ( *) – .

e t e i t it t



Fig. 1. Polar diagram or projection of the phase trajec-tory onto the (еcosω, esinω) plane for case I (е0 = 0.55,i0 = 40°.78).

150

180

210

0.8

0.6

0.4

Fig. 2. Same as Fig. 1, but for case II (е0 = 0.4, i0 = 32°.31).

150

12090

180

210

240270

0.6

0.4

0.8

1.0

The initial deviations in i0 and Ω0 from their equilib-rium values at t = t* lead to insignificant changes in theshape of the orbit (Δ1, Δ3), but to a noticeable change inits orientation (Δ2 reaches 24°, Δ4 is about 32°).

Table 2 was compiled for more significant initialdeviations of the orbit from the orthogonal one, i0 =90° ± 30°. At Ω0 = 0° and 180° it also contains theresults for i0 = 120° identical to the corresponding datafor i0 = 60°.

In this case, the inclination also remains close tothe initial one, but the deviations of the remaining ele-ments are tens of degrees, reaching approximately 75°.

Below, we will consider the orbits of point Р linkedwith the orbit of point J, but with an arbitrary spatialorientation. The evolution of such orbits is usuallystudied using numerical methods even in the integra-ble doubly averaged circular problem (е1 = 0) due tothe absence of a rigorous analytical expression for theaveraged perturbing function. The families of phasetrajectories in the (еcosω, esinω) plane constructed asequipotential contours of the function W are presentedin Ito and Ohtsuka (2019, Section 5.8, Fig. 24). Thesefamilies correspond to the hypothetical linked orbits ofpoint Р for three pairs of values of the ratio a/a1 andthe constant of the integral с1. In all three cases con-sidered

there exist stationary center-type singular points andclosed periodic trajectories enclosing them in thephase plane.

The ellipticity of the orbit of the perturbing point J,naturally, leads to qualitative changes of the families oftrajectories in the circular problem. Since the equa-

= = = == =

1 1 1 1

1 1

0.9, 0.4 ; 0.7, 0.6 ; 0.8, 0

( ) ( ).31( )3

a a c a a ca a c

I IIIII


tions for е and ω are not decoupled from the remainingones, as is the case at е1 = 0, due to the absence of theintegral с1 in the elliptic problem, the evolution ofthese elements can be traced only in projection of thephase trajectory onto the (еcosω, esinω) or (ω, е)plane. In this paper we numerically integrated system (14)for а1 = 5.2 AU, е1 = 0.048, and the mass ratio m/m1 inthe Sun-Jupiter system. The difference of this ratiofrom the one adopted in the calculations by Ito andOhtsuka (2019) for the Sun–(Earth + Moon) systemdoes not affect the structure of the phase trajecto-riesне, but leads only to a change in the time scale.

For a comparison with the results of the circularproblem in cases I, II, and III, out of all families ofintegral curves in the circular problem we chose thetrajectories with ω0 = 180° and initial е0 = 0.55, 0.4,and 0.65, respectively. The initial inclinations were

calculated from е0 and с1 as , while

the initial longitude of the ascending node Ω0 wastaken to be zero.

Figures 1–3 show the trajectories for cases I, II,and III, respectively, and fragments of the polar dia-grams with plotted numerical values of the angles ωand radii е. The circles and triangles mark the initialand final points, respectively. The time intervals are100000 years for cases I, II and 500000 years for case III.The dashed lines are the so-called “separatrices”,which are not integral curves and correspond to theintersections of the orbits of points P and J. They aredefined by the equations

=−

10 2

0

arccos1

cie

( )( )

−± ω − =

±

2

1 1

1cos 1 0.

1

a ee

a e

166 VASHKOV’YAK

Fig. 3. Same as Fig. 1, but for case III (е0 = 0.65, i0 =42°.59).

150

180

210

0.6

0.8

1.0

Fig. 4. Example of a chaotic trajectory with a change of theregimes of variations in the argument of perihelion andwith intersections of the orbit of the perturbing point in atime interval of 55000 years for case I, but at е0 = 0.475 andi0 = 44°.052.

150

12090

180

210

240270

0.6

0.4

0.2

0.8

1.060

30

0

330

300

In all three cases, the closed periodic trajectories ofthe circular problem are modified and become non-periodic, nevertheless, retaining an oscillatory patternand remaining in the regions of linked orbits. Theamplitude of these oscillations can both decrease(Figs. 1, 3) and increase (Fig. 2) with time.

However, the eccentricity of Jupiter’s orbit can alsolead to qualitative changes in the behavior of the tra-jectory. Figure 4 shows a chaotic trajectory that beginsin the ω libration region at ω0 = 180°, then passes intothe circulation region, and returns to the librationregion, but relative to ω = 0. The initially linked orbitof point Р intersects the orbit of the perturbing point Jas it evolves, exiting from one linking region, then tra-verses the region of unlinked orbits and enters anotherlinking region.

Interestingly, real cometary orbits almost orthogo-nal to the ecliptic, with the above types of evolution,are also found within the model under consideration(Sun‒Jupiter‒comet). If we make a selection by peri-helion distance q < 2 AU and inclination 85° < i < 95°on the set of cometary orbits presented in the JPLdatabase (https://ssd.jpl.nasa.gov/sbdb_query.cgi#x),then only ten orbits remain in this sample. The evolu-tion of seven of them is reduced to successive intersec-tions of Jupiter’s orbit with passages of the linkingregions. Figures 5–7 give an idea of the evolution ofthe remaining three orbits. In contrast to the previousfigures, fragments of the (ω, е) plane are shown herenot in the polar coordinate system, but in the rectan-gular one, which is more convenient for highly ellipti-cal orbits. The up-to-date (initial) values of the ele-ments for our numerical integration were taken fromthe above JPL database. The initial and final values inthe (ω – е) plane are marked by the circles and trian-gles, respectively.

Figure 5 presents the variations in orbital elementsfor comet C/1955 L1 (Mrkos) in a time interval of1 Myr. The motion of the phase point begins in theregion of orbital linking and ω libration, while the cor-responding stationary value of the eccentricity calcu-lated by solving Eq. (26) is е0 = 0.9818. The phasepoint has no time to make one revolution relative tothe libration center, while the trajectory passesthrough the “separatrix” corresponding to the inter-section of the cometary orbit with Jupiter’s orbit. Afterthe relatively short time interval of ~200000 years,there occur the second intersection of these orbits andthe entry into another linking region with a ω librationcenter offset by 180° from the initial one. In this case,the initially prograde motion of the comet becomesretrograde already in 100000 years, while the inclina-tion, increasing monotonically, reaches about 115°.

Figure 6 presents the variations in orbital elementsfor comet C/1861 J1 (Great comet) in a time intervalof 3 Myr. The motion of the phase point begins andremains in the region of orbital linking and ω libration,while the corresponding stationary value of the eccen-tricity is е0 = 0.9820. In this case, the initially progrademotion of the comet becomes retrograde in 2 Myr,while the inclination reaches about 132° at a minimumof 31°.

Figure 7 presents the variations in orbital elementsfor comet 122P/de Vico in a time interval of 3 Myr.The motion of the phase point also begins and remainsin the region of orbital linking and ω libration, whilethe corresponding stationary value of the eccentricity



Fig. 5. Variations in the argument of perihelion and orbital eccentricity for comet C/1955 L1 (Mrkos) in a time interval of 1 Myrin a simplified model (Sun–Jupiter–comet).

1.000

0.965

0.960

0.955

0.950200150–150–200

e

ω, deg100500–50–100

0.995

0.990

0.985

0.980

0.975

0.970

Fig. 6. Same as in Fig. 5, but for comet C/1861 J1 (Great comet) and a time interval of 3 Myr.

1.000

0.965

0.9606040–40–60

e

ω, deg200–20

0.995

0.990

0.985

0.980

0.975

0.970

is е0 = 0.9630. The motion changes with time fromprograde to retrograde and vice versa. The extremeinclinations are 43° and 138°.

In conclusion, it should be recalled that all theabove examples of orbital evolution and, in particular,


the examples of libration of the argument of perihelionwere constructed within the adopted model(Sun‒Jupiter‒comet). However, the real motions ofcomets in the Solar System can differ from their modelmotions even qualitatively. These differences are dueto the neglect of the influence of the remaining plan-

168 VASHKOV’YAK

Fig. 7. Same as Fig. 6, but for comet 122P/de Vico.

1.00

0.95

0.94

0.932015–15–20

e

ω, deg1050–5–10

0.99

0.98

0.97

0.96

ets, except for Jupiter, and the averaged model of theproblem. For example, in the rigorous solution of thecomplete system of non-averaged differential equa-tions the libration of the argument of perihelion canchange to its circulation. The orbit of comet 122P/deVico is an example of chaotic orbital evolution in a realcometary medium, but this property is found only inthe non-averaged Solar System model including bothseveral perturbing bodies (Baily et al. 1992) and onebody (Ito and Ohtsuka, 2019; Section 5.8, Fig. 25).

Regarding the methodology of the work describedin this paper, note that owing to the studies performedrelatively recently and reflected in the monographsand the paper by Kondrat’ev (2007, 2012) andAntonov et al. (2008), the representation of the forcefunction for an elliptical Gaussian ring was obtained ina closed form without expansions in terms of anyparameters. However, its practical use involves well-known difficulties and may in future be the subject ofa special study.

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Translated by V. Astakhov


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