Bifurcation of frozen orbits in a gravity ﬁeld with zonal harmonics Irene Cavallari1and Giuseppe Pucacco2 1Dipartimento di Matematica Università di Pisa

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Bifurcation of frozen orbits in a gravity ﬁeld with zonal harmonics

Irene Cavallari,1and Giuseppe Pucacco2

1Dipartimento di Matematica, Università di Pisa

2Dipartimento di Fisica and INFN – Sezione di Roma II, Università di Roma “Tor Vergata”

Abstract

We propose a methodology to study the bifurcation sequences of frozen orbits when the 2nd-

order fundamental model of the satellite problem is augmented with the contribution of octupolar

terms and relativistic corrections. The method is based on the analysis of twice-reduced closed

normal forms expressed in terms of suitable combinations of the invariants of the Kepler problem,

able to provide a clear geometric view of the problem.

1 Introduction

Among the manifold versions of the perturbed Kepler problem, the investigation of the gravity ﬁeld

expanded in multipole terms has traditionally received great attention for its relevance in applications.

Therefore, several analytical tools have been developed to highlight the most important phenomena.

Perturbation theory with the construction of normal forms is the standard method since the ﬁrst

pioneering studies (Brouwer, 1959; Kozai, 1962). The case in which only zonal terms are included in one

of the settings in which we can obtain explicit approximations of the regular dynamics since the normal

form is integrable. However, the presence of several parameters, both dynamical (or “distinguished” in

the language of the theory of integrable systems) and physical like the multipole coeﬃcients, hinders

a global description of the dynamics. More eﬃcient geometric and group-theoretic tools have been

exploited to study the bifurcation of invariant objects when these parameters are varied (Coﬀey et al.,

1994, 1986; Cushman, 1983; Palacián, 2007).

Here we study the bifurcation sequences of frozen orbits when the 2nd-order fundamental model of the

satellite problem is augmented with further features of a typical planetary gravity ﬁeld. We consider

the contribution of the octupolar term (Coﬀey et al., 1994; Vinti, 1963) and the relativistic correction

due to the quadrupolar term (Heimberger et al., 1990). We implement a twice-reduced normal form

(Cushman, 1988; Pucacco, 2019; Pucacco and Marchesiello, 2014) which allows us to obtain in an

eﬃcient way the conditions for relative equilibria corresponding to the family of periodic orbits with

ﬁxed eccentricity and inclination. The method is tested in the 2nd-order J2-problem in which known

results are reproduced (Palacián, 2007) and then applied to the above-mentioned perturbations. For the

J4-problem, interesting features around the parameter values of the “Vinti problem” are highlighted

with an additional family of stable frozen orbits. For the relativistic J2-correction, the treatment

extends and completes several results obtained by Jupp and Brumberg (1991).

The plan of the paper is as follows: in Section 2 we recall the model problem based on the normal

from obtained after averaging with respect to the mean anomaly; in Section 3 we review the reduction

methods adapted to the symmetries of the present model, discuss the version adopted here to cope

with the structure of the Brouwer class of hamiltonians and show how it works in locating relative

equilibria; in Section 4 we illustrate the results in concrete cases; in Section 5 we conclude with some

hint for possible developments and future works.

arXiv:2210.12182v1 [math-ph] 21 Oct 2022

2 The model in closed normal form

We are discussing some aspects of the general problem described by a Hamiltonian of the form

H(L, H, G, `, g, h) = ∞

j=0

jHj(L, H, G, `, g, h),(1)

where H0is the Kepler Hamiltonian and the canonical Delaunay variables have the following expression

in terms of the standard Keplerian elements (a, e, i, `, ω, Ω)

L=√µa, G =√µap1−e2, H =√µap1−e2cos i, (2)

`=M, g =ω, h = Ω.(3)

In the above equation, is a formal parameter, called book-keeping parameter, suitably chosen to order

the hierarchy of perturbing terms (see Efthymiopoulos, 2012). Therefore, we have a perturbed Kepler

problem.

Speciﬁcally, in the even zonal artiﬁcial satellite problem, we assume to start with the “original Hamil-

tonian”

H(q,p) = 1

2p2+VCGF −1

c2p4

8−V2

CGF

2−3

2VCGF p2(4)

in standard Cartesian form, where q={x, y, z},p={˙x, ˙y, ˙z},p=|p|,VCGF is the classical gravity

ﬁeld and cis the speed of light. We include the classical gravity ﬁeld VCGF expanded in terms of the

zonal harmonics of even degree1

VCGF =−µ

r"1−∞

k=1

J2k

R2k

r2kP2k(sin θ)#,(5)

where µ=GMPis the product of Newton constant and the mass of the “planet”, RPis its radius and

the Pkare the Legendre polynomials with

sin θ=z

r, r =px2+y2+z2.

We also add the ﬁrst-order relativistic corrections following e.g. Weinberg (1972).

To simplify the structure of the Hamiltonian, we then perform a closed-form normalisation like in

(Coﬀey et al., 1994) and (Heimberger et al., 1990). This method, inspired by works of Deprit (1981,

1982), has the advantage of avoiding expansions in the eccentricity and inclination (Cavallari and

Efthymiopoulos, 2022; Palacián, 2002). The model in (4) is rich enough to convey several interesting

dynamical features keeping the closed form structure at the lowest level of complexity. In fact, after

the Delaunay reduction and the elimination of the ascending node, we deal with a secular Hamiltonian

in closed form which depends on only one degree of freedom, corresponding to the pair Gand g(the

argument of the perigee):

K(L, H, G, g) = X

jKj(L, H, G, g),(6)

with Land Hformal integrals of the motion. The zero-order term is clearly

K0=H0=−µ2

2L2.(7)

The ﬁrst-order term is

K1=µ4J2R2

P(G2−3H2)

4G5L3−µ4

c2L43L

G−15

8.(8)

1In this work, we focus on the even zonal problem. Thus, only the even zonal harmonics are considered in the

expansion of the gravitational potential. The complete expansion, including also tesseral terms, can be found in (Kaula,

1966).

The second-order term K2consists of two contributions:

K2=T2+hH2i.

The ﬁrst is related to the propagation at second order of the J2term in the normalising transformation

(Deprit, 1969; Efthymiopoulos, 2012),

T2=3µ6J2

2R4

128L5G11 −5G6−4G5L+ 24G3H2L−36GH4L−35H4L2+G4(18H2

+ 5L2)−5G2(H4+ 2H2L2) + 2(G2−15H2)(G2−L2)(G2−H2) cos 2g

−3µ6J2R2

4c2L5G7(G2−3H2)(4G2−3GL −5L2)+(L2−G2)(G2−H2) cos 2g.

(9)

The second is associated directly with the average of the H2term:

hH2i=1

2πZ2π

0H2d` =µ6J2R2

8c2L5G7(G2−3H2)(6L2−5G2)

−3(L2−G2)(G2−H2) cos 2g+3µ6J4R4

128L5G11 (3G4−30G2H2

+ 35H4)(5L2−3G2)−10(G2−7H2)(L2−G2)(G2−H2) cos 2g

(10)

In this work, we do not consider terms of order higher than j= 2. Hamiltonians of this type are

generally denoted as “Brouwer’s” ones (Brouwer, 1959; Cushman, 1983). They are characterised by the

independence on the mean anomaly `and the longitude of the node h(with corresponding conservation

of the actions Land H), whereas the argument of perigee appears only with the harmonic cos 2g. These

symmetries will all be exploited in the geometric approach described in the following.

The two relativistic terms proportional to J2/c2appearing in (9) and (10) have the same structure.

However, in the literature (Heimberger et al., 1990; Schanner and Soﬀel, 2018), they are usually kept

separate and are respectively referred to as the indirect and direct term related to the non-trivial

relativistic contribution of the quadrupole of the gravity ﬁeld of the central body. The ordering of

the perturbing terms is performed by assuming (with a certain degree of arbitrariness) the J2and c−2

terms to be of order and the J4term of order 2, like the J2

2and J2×c−2terms.

We remark that, with a slight abuse of notation, we have denoted with the same symbols the Delaunay

variables appearing in (1) and (6). We have to recall that actually they are respectively the original

and the new variables related by the normalising transformation. In the present work, we are not

interested in the explicit construction of particular solutions. Therefore, we will not detail the back-

transformation from the new to the original coordinates. Moreover, we are not going to investigate

any issue connected with the convergence of the expansions. We rely on the asymptotic properties of

these series and their ability to provide reliable approximations, especially in the cases of Earth-like

gravity ﬁelds.

For sake of completeness, the diﬀerent parts of the normalised Hamiltonian K=K0+K1+(T2+hH2i)2,

expressed in terms of the orbital elements (a, e, i, ω), are given by

K0=−µ

2a,

K1=1

µJ2R2

a3η31−3 cos2i−3

µ2

c2a21

η−5,

T2=3µJ2

2R4

128 a5η7h−5η2+ 36 η+ 35sin4i+ 8 −η2+ 6 η+ 10sin2i

+ 8 η2−2η−52 sin2i1−η21−15 cos2icos 2ωi

−3µ2J2R2

4c2a4η5h4η4−3η−5(1 −3 cos2i) + sin2i(1 −η2) cos 2ωi,

hH2i=µ2J2R2

8c2a4η5(6 −5η2)(1 −3 cos2i)−3 sin2i(1 −η2) cos 2ω+

3µJ4R4

128a5η7(5 −3η2)(35 sin4i−40 sin2i+ 8)

−10 sin2i(1 −η2)(1 −7 cos2i) cos 2ω,

with η=√1−e2.

3 Geometric reduction

The secular Hamiltonian in closed form in (6), while computed with an ingenious combination of tools

based on the Lie transform method (Deprit, 1969; Efthymiopoulos, 2012) and the elimination of the

parallax (Deprit, 1981), is nonetheless standard in being essentially an average with respect to the mean

anomaly (Deprit, 1982; Palacián, 2002). However, it is liable to be treated with a group theoretically

approach. It can be interpreted as a suitable combination of the invariants generating the SO(3)

symmetry of the Kepler problem. In fact, the dynamics ensues from the reduction of the Hamiltonian

deﬁned on the space of the trajectories having, for the unperturbed Kepler problem with negative

energy, the structure of the direct product of two spheres. The additional symmetries of the closed

form of the perturbed problem are exploited to identify a regular reduced phase space with the topology

of the 2-sphere. In practice, we will use a further transformation leading to a singular reduction on a

surface with equivalent topology, which produces a clearer geometric view of the bifurcation sequence

of frozen orbits. Here, we provide a quick reminder of the invariant theory of the Kepler problem and

then apply the reduction process to perturbed Kepler problems described by Brouwer’s Hamiltonians.

3.1 Invariants of the Kepler problem

Let us call Gthe angular momentum and Athe Laplace-Runge-Lenz vector, given by

G=G



sin isin h

−sin icos h

cos i

,A=r1−G2

L2



cos gcos h−sin gsin hcos i

cos gsin h+ sin gcos hcos i

sin gsin i

,

with i= arccos (H/G)the orbital inclination. By deﬁning

x=G+LA,y=G−LA,(11)

we get the Poisson structure of the generators of SO(3)

{x1, x3}=x2,{x3, x2}=x1,{x2, x1}=x3,

{y1, y3}=y2,{y3, y2}=y1,{y2, y1}=y3,

and phase-space deﬁned by the direct product of the two 2-spheres

1+x2

2+x2

3=L2, y2

1+y2

2+y2

3=L2.(12)

It can therefore be imagined as the invariant space of the states characterised by given eccentricity,

inclination, and arguments of perigee and node, but nonetheless equivalent for what pertains to the

mean anomaly. In the unperturbed problem, the state is a given still point of the invariant space. The

state point is kept moving on it by the action of the perturbation.

3.2 Reduction of the axial symmetry

Perturbed Kepler problems described by Hamiltonians of the form (6) are characterised by axial sym-

metry with Has formal third integral. In Cushman (1983) and Coﬀey et al. (1986) it is shown that, if

0<|H|< L, the two-dimensional phase space of such problems is still diﬀeomorphic to a sphere. Two

diﬀerent sets of variables, both functions of the Keplerian invariants xk, yk(k= 1,2,3) and suitable

to analyse the dynamics, are proposed. The variables (π1, π2, π3)are deﬁned as

π1=1

2(x3−y3) = L(A·k),

π2=x1y2−x2y1= 2L(A×G)·k,

π3=x1y1+x2y2=|G×k|2−L2|A×k|2,

where k= (0,0,1)T(see Cushman, 1983). The phase-space is then

P=(π1, π2, π3)∈R3:π2

2+π2

3= ((L+π1)2−H2)((L−π1)2−H2).(13)

Instead, in Coﬀey et al. (1986), the variables (ξ1, ξ2, ξ3)are introduced, deﬁned as

ξ1=L(G×A)·k, ξ2=L|G|(A·k), ξ3=1

2|G×k|2−L2|A|2,

or, in terms of Delaunay variables,

ξ1=p(G2−H2)(L2−G2) cos g,

ξ2=p(G2−H2)(L2−G2) sin g,

ξ3=G2−L2+H2

(14)

In this case, the phase-space is the sphere of radius (L2−H2)/2:

S=(ξ1, ξ2, ξ3)∈R:ξ2

1+ξ2

2+ξ2

3=(L2−H2)2

4.(15)

The relation between the πkand the ξkis

π1=√2ξ2

p2ξ3+L2+H2,

π2=−2ξ1,

π3=2ξ3+2ξ2

2ξ3+L2+H2.

The advantage of both these sets of variables with respect to the Delaunay variables is well explained

in Coﬀey et al. (1986) with an imaginative metaphor. In simpler words, we can say that the Kepler

reduction allows us to translate the closed form dynamics in terms of the invariants of the unperturbed

problem (formal conservation of L) and the further reduction generated by the invariants ξkis readily

apt to account for the axial symmetry associated with the formal conservation of H. Recalling the

description of the states of the space deﬁned in (12), we now have that the states of (15), given a

value of H, are characterised by the eccentricity and the perigee but are nonetheless equivalent for

what concerns h. The dynamical evolution of the system is then determined by the intersections of the

reduced phase-space Swith the Hamiltonian expressed in terms of the invariants, e.g. K(ξ1, ξ2, ξ3).

Whenever one uses the (G, g)chart to analyse the dynamics of the closed form for given values of L

and H, one excludes circular and equatorial orbits. Indeed, when either the orbital eccentricity or the

orbital inclination is zero, the argument of the perigee gis not deﬁned, thus the Delaunay variables

result unsuitable to evaluate the stability of such orbits, if they are periodic as typically happens in

the artiﬁcial satellite problem. Following Cushman (1983), in Iñarrea et al. (2004) it is shown that

when Kpossesses independent symmetries of the type

R1:(π1, π2, π3)→(−π1, π2, π3),

R2:(π1, π2, π3)→(π1,−π2, π3),

R3:(π1, π2, π3)→(−π1,−π2, π3),

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BifurcationoffrozenorbitsinagravityeldwithzonalharmonicsIreneCavallari,1andGiuseppePucacco21DipartimentodiMatematica,UniversitàdiPisa2DipartimentodiFisicaandINFNSezionediRomaII,UniversitàdiRomaTorVergataAbstractWeproposeamethodologytostudythebifurcationsequencesoffrozenorbitswhenthe2nd-orderfund...

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Bifurcation of frozen orbits in a gravity ﬁeld with zonal harmonics Irene Cavallari1and Giuseppe Pucacco2 1Dipartimento di Matematica Università di Pisa

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