# On the Evection Resonance and Its Connection to the Stability of Outer Satellites

Authors: Tadashi Yokoyama; Ernesto Vieira Neto; Othon Cabo Winter; Diogo Merguizo Sanchez; Pedro Ivo de Oliveira Brasil
Journal: Mathematical Problems in Engineering
 (2008)

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## Abstract

In terms of stability around the primary, it is widely known that the semimajor axis of the retrograde satellites is much larger than the corresponding semimajor axis of the prograde satellites. Usually this conclusion is obtained numerically, since precise analytical derivation is far from being easy, especially, in the case of two or more disturbers. Following the seminal idea that what is unstable in the restricted three-body problem is also unstable in the general N-body problem, we present a simplified model which allows us to derive interesting resonant configurations. These configurations are responsible for cumulative perturbations which can give birth to strong instability that may cause the ejection of the satellite. Then we obtain, analytically, approximate bounds of the stability of prograde and retrograde satellites. Although we recover quite well previous results of other authors, we comment very briefly some weakness of these bounds.


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## Body

## 1. Introduction

Among the  puzzling  questions
in the
solar system inventory, the problem of the irregular moons of the Jovian
planets is a crucial challenge and controversial topic. Their orbits are highly
tilted, very eccentric, and in opposition to inner satellites; these moons
orbit the mother planets at very large distances, being strongly disturbed by
the Sun. Recently, the number of these distant moons has increased at least one
order of magnitude than the pre-CCD era (Sheppard etal. [1], Holman etal. [2],
and references therein). These discoveries have provided a lot of new
ingredients indicating interesting clues to understand the origin, composition,
and evolution of such objects. In this work we basically analyze some dynamical
aspects  which govern the escape process of an irregular satellite. It is well
known that there are some remarkable differences in the stability domain
between retrograde and prograde moons
(Alvarellos  and  Dones [3], Hamilton and Krivov [4],
Vieira Neto etal. [5, 6],
Ćuk and Gladman [7], Domingos etal. [8], etc.).
Recently, a large number (about 60000) of irregular satellites were numerically
integrated by Nesvorný etal. [9]. Among
some interesting new results, they also confirm the role played by the
Kozai-Lidov and evection resonances which in general provoke the escape of
these objects. The evection resonance is caused by the 1:1 commensurability
between ϖ (longitude of the satellite pericenter)
and λ⊙ (Sun’s mean longitude) where index ⊙   refers to the Sun's elements. Nesvorný etal.
[9] show that once a prograde satellite is in an evection resonance, the
critical angle ϖ−λ⊙ can librate around 180°, resulting in a
cumulative perturbation which can cause the escape of the satellite. For the
retrograde satellites, the phenomenon is similar, however the definition of the
critical angle must be changed and the libration center is 90° or
270°.
Nesvorný etal. [9] show that both resonances occur in the vicinity of some fixed
values of the semimajor axes of the satellite. Some investigations on these
values were done by Alvarellos  and  Dones [3] and Hamilton and Krivov [4], using some
concepts of Jacob constant and generalization of the Tisserand constant for the
restricted three-body problem. In this work we derive an alternative and simple
way to obtain, theoretically, these semimajor values. Through the steps we
outline here, it becomes very clear the main idea which associates the
libration of the critical angle with the appearance of the instability.


## 2. Disturbing Function: Second-Order Expansion

Let us assume a Cartesian system
fixed on Jupiter. Initially, the reference plane is the equator of the planet.
Figure1 shows the
geometry of the problem. The disturbing function for the motion of a satellite
perturbed by the Sun is
(2.1)R⊙=k2M⊙r22r⊙3(3cos2(S)−1), where third-order terms in the ratio of the
distances (r/r⊙) are neglected; k2 is the constant
of the gravitation, M⊙ is the mass of the Sun, and r and r⊙ are the position vector of the satellite and
of the Sun, respectively. S is the angular distance between Sun and the
satellite. From the geometry we have(2.2)cos(S)=xrx⊙r⊙+yry⊙r⊙+zrz⊙r⊙.Figure 1
Geometry
of the problem.We adopt the classical notations:(a,e,I,l,ω,Ω,f)
stand for
semimajor axis, eccentricity, inclination, mean
anomaly, argument of pericenter, longitude of the node, and true anomaly,
respectively, for the elements of the satellite. The same variables with the index ⊙  are used for the Sun.Now let us choose the Sun-Jupiter
orbital plane as the reference plane, so thatz⊙=I⊙=0   and(2.3)cos(S)=[cos(f+ω)cos(Ω)−sin(f+ω)sin(Ω)cos(I)][cos(f⊙+ϖ⊙)]+[cos(f+ω)sin(Ω)+sin(f+ω)cos(Ω)cos(I)][sin(f⊙+ϖ⊙)],cos(S)=cos(θ1)cos(θ2)+sin(θ1)sin(θ2)cos(I),   where θ1=f+ϖ,  θ2=f⊙+ϖ⊙−Ω (ϖ, ϖ⊙ are the longitudes of
pericenter of satellite and of the Sun, resp.).Therefore we have(2.4)R⊙=k2M⊙a22r⊙3r2a2{−14+34[cos(2θ2)+cos(2θ2)cos(2θ1)+cos(2θ1)]+34cos2(I)×[1−cos(2θ2)−cos(2θ1)+cos(2θ2)cos(2θ1)]+32cos(I)sin(2θ1)sin(2θ2)}. Note that terms like
r2/a2, (r2/a2)cos(2f), (r2/a2)sin(2f),
and so forth can be averaged through simple formulae of the classical two-body
problem (Yokoyama etal. [10]). Let 〈⋅〉   indicates average with respect to the mean
anomaly of the satellite:(2.5)〈r2a2〉=12π∫02πr2a2dl. Omitting the details, we
easily get (2.6)〈r2a2〉=1+32e2,〈r2a2cos(2f)〉=52e2,〈r2a2sin(2f)〉=0. Note that up to now, in the
above calculations neither expansion in eccentricity nor in inclination was used.
This is an important point since we have to deal with high values of the
eccentricity and inclination. The expression of the averaged disturbing
function up to second order in the ratio r/a is(2.7)R⊙=k2M⊙a22r⊙3{−14(1+32e2)+34[(1+32e2)cos(2θ2)+52e2cos(2ϖ−2Ω)cos(2θ2)+52e2cos(2ϖ−2Ω)]+34cos2(I)[(1+32e2)−(1+32e2)cos(2θ2)−52e2cos(2ϖ−2Ω)+52e2cos(2ϖ−2Ω)cos(2θ2)]+154e2sin(2ϖ−2Ω)sin(2θ2)cos(I)}.


## 3. Prograde Satellites

Considering
this  disturbing function, we integrate the Lagrange variational equations
(Danby [11]). Figure 2(a)
shows the
behavior of the critical angle (ϖ−λ⊙) and the eccentricity. Note that if the
libration of this angle is centered at 180°, the apocenter of the satellite will always be close to Sun. In Figure 3 we show this situation which is a critical case when
Jupiter, Sun, and apocenter are aligned, that is, the amplitude of the libration
is zero.(a) Prograde satellite. Initial
conditions:a=395RJ, e=0.01, I=1.5°, and ϖ=180°. (b) Prograde satellite. Initial conditions:
a=400RJ, e=0.1, I=1°
and ϖ=180°.
(a)(b)Figure 3
Prograde case:ϖ−λ⊙=180°.In this case, no matter
the period of the satellite, each time it passes through the apocenterS we
have the Sun, satellite, and Jupiter aligned with the first two in their
closest approach.Therefore, it will occur a
cumulative perturbation. In particular, note that this is the worst situation
for the orbital stability of a massless object. Now suppose that this
configuration occurs repeatedly. After some passages, certainly, the
eccentricity and the semimajor axis of the satellite will be strongly
disturbed. They should increase approaching to some dangerous limit. This is
clearly shown in Figure2(b). Note, however, that while ϖ−λ⊙ remains in libration, the eccentricity is not
so high (~0.3). The significant increase of the eccentricity appears only when ϖ−λ⊙ enters in a circulation regime. This dynamics
will be discussed in Section 5. Of course in this model since we are using averaged
equations, by definition,
the semimajor axis is kept constant. Therefore, within the limits of our model
the escape does not necessarily occur during this short integration time. The
initial conditions we used were a=400RJ(0.538RH), e=0.1, I=1°, ϖ−λ⊙=180°, ϖ=180°, a⊙=5.202603 AU, e⊙=0.048497, ω⊙=0°, and Ω⊙=0°  (RJ means Jupiter’s equatorial
radius and RH is Hill’s
radius).Once we have this basic information
we can go  further and confirm very easily; some results shown in Hamilton and
Krivov [4]. Since  we have shown that the appearance of the evection resonance
causes large variation of the eccentricity, we assume that the corresponding
semimajor axis of the evection resonance has to do with the limit of stability of the satellite around the planet.
Therefore, we take this statement for granted and search the value of the
semimajor axis. To this end, in the averaged disturbing function (2.7), let us
consider only the secular and resonant terms due to the evection. All other
remaining terms can be neglected. In particular, since we showed a libration in ϖ−λ⊙=180°, we fix this angle at 180°. Again, from
Lagrange’s equation we easily obtain(3.1)ϖ˙≅k2M⊙(1−e2)1/22na⊙3{98+338cos2(I)+154cosI}+o(sin(I)). Now, if we consider
I=e=0, we have(3.2)ϖ˙≅92k2M⊙na⊙3. Since the apparent motion of
the Sun is Keplerian, we take n⊙=k2(M⊙+mJ)/a⊙3.Hill’s radius is defined asRH=a⊙(mJ/3M⊙)1/3.Therefore, equatingϖ˙=n⊙,
we get the resonant value of the semimajor axis: (3.3)a*=0.529RH,
which is in quite agreement with Hamilton
and Krivov
[4] and the numerical results of Nesvorný etal. [9].Recall that the semimajor axes we
used in Figures2(a), 2(b) were a=0.531RH(395RJ)  and a=0.538RH(400RJ),
respectively.A simple inspection in (2.7) shows that critical angle always
appears in the form  2(ϖ−λ⊙). This sugests that ϖ−λ⊙=0  is also another center of libration.We can confirm this very easily. Selecting only secular and
resonant terms in (2.7) we have(3.4)R⊙=k2M⊙a22a⊙3{12(1+32e2)+154e2cos(2ϖ−2λ⊙)}, where we assumed Jupiter in
a circular motion and I=0.The conservative Hamiltonian can easily be derived from the above simplified
relation:(3.5)H=k2M⊙a22a⊙3{34e2+154e2cos(2ϖ−2λ⊙)}−n⊙L⊙, where L⊙ is the canonical momentum conjugated to λ⊙. Considering the classical
Delaunay canonical variables: ϖ, G−L (Brouwer and Clemence [12]), and
writing e2 in terms of G and L, we proceed with
a new trivial canonical transformation:(3.6)(ϖ,λ⊙,G−L,L⊙)⟶(α1,α2,P1,P2), where α1=ϖ−λ⊙, P1=G−L, α2=λ⊙, P2=L⊙+(G−L).This allow us to
write a one-degree-of-freedom problem, sinceα2 becomes a kinosthenic variable;(3.7)H=k2M⊙a22a⊙3{34e2+154e2cos(2α1)}+n⊙G, where e2=(L2−(P1+L)2)/L2.
Note that with a
completely different way, we obtained (3.4) of Hamilton and Krivov [4].The level curves  of the above
Hamiltonian confirm that bothϖ−λ⊙=0°  and 180° are stable equilibrium points of
the system. In Figure 4 we show the level curves in the plane ecos(ϖ−λ⊙), esin(ϖ−λ⊙). This
clearly shows that the longitude of the pericenter of the satellite can remain
stably pointing to Sun direction or to the opposite direction. Of course the
net effect of this dynamics is to stretch the satellite orbit toward and away from the Sun
(Figure 4). We
also can say that the ϖ−λ⊙=180°  is a critical configuration, so that we
expect to have escape of the satellite mostly in this situation, not when ϖ−λ⊙=0°. In
Sections 5, 6 we confirm numerically
that escapes occur following these kind of behavior.Figure 4
Level curves of Hamiltonian (3.7) with a=396RJ.These mentioned two centers of libration play important role as
we can see in Section6.


## 4. Retrograde Satellites

For the retrograde satellites, the definition of the
longitude of the pericenter should be changed (Saha and Tremaine [13]). Let ϖI be the longitude of the pericenter for this
case. Therefore, we have(4.1)ϖI=Ω−ω. Let ϖ be the usual longitude of the prograde case.
Then, with a simple algebra we can relate both:(4.2)ϖI=2Ω−ϖ. As in the previous case, we
integrate again Lagrange's equations taking the disturbing function given by (2.7).In the precedent figures, we considered the following initial
condictions:ω=0°, Ω=90° (Figure 5(a)), and ω=90°, Ω=0°
(Figure 5(b)). For Jupiter (Sun) we
adopted a⊙=5.202603 AU, e⊙=0.048497, ω⊙=0°, Ω⊙=0°, and l⊙=0°.(a) Retrograde
satellite. Initial conditions:a=520RJ, e=0.03, I=176°, ω=0°, and Ω=90°.
(b) Retrograde
satellite. Initial conditions: a=520RJ, e=0.03, I=176°,
ω=90°, and Ω=0°.
(a)(b)This time we see that the critical
angleϖI−λ⊙ librates around 90° or 270°. The situation is
not so drastic as in the direct case. The schematic geometry given in
Figure 6
repeats each time the satellite passes through S.Figure 6
Retrograde case:ϖI−λ⊙=90°.From Lagrange’s variational equations we have(4.3)Ω˙=32GCna2,ϖ˙=92GCna,  where GC=k2M⊙a2/2a⊙3.
As before we fixed some values considering the current resonance: Ω−ω−λ⊙=90°, I=180°, and e=0.
Again equating: ϖ˙I=n⊙,
we get: a*=0.6933RH(≈515.3RJ)
which coincides again
with the results given in Hamilton and Krivov [4].Recall that the semimajor axis we
used in Figures5(a) and 5(b) was a=0.7RH(520RJ). It is worth noting that compared
to the previous direct case; the present resonance is not very strong since the
closest approach with the Sun is not like in the direct case. Even so, the
cumulative effect works quite efficiently in driving the eccentricity and
semimajor axis to critical values, sometimes causing ejection of the satellite.
This is clear in Figure 5(b).Again following the same steps
outlined for the prograde case, the conservative Hamiltonian can be written as(4.4)H=k2M⊙a22a⊙3{34(1−P12L2)+154(1−P12L2)cos(2σ1)}−n⊙P1, where
P1 and σ1  are the
canonical conjugated variables defined by P1=G=L(1−e2)1/2 and σ1=ϖI−λ⊙.Drawing level curves forH(P1,σ1), we can
easily check that σ1=90°,
270° are stable equilibrium solutions (Figure 7). This time, in this
approximation, the tendency of the orbits is to elongate perpendicularly to the
Sun-satellite direction, as suggested in Figure 6.Figure 7
Level curves of Hamiltonian (4.4) with a=516.4RJ.


## 5. Numerical Tests with Exact Equations

In
this section we show some simulations considering the exact differential
equations of a satellite of Jupiter disturbed by the Sun. In terms of the
radius of the planet, the resonant semimajor axis for the prograde case isa*=393RJ. Time variations of the eccentricity and ϖ−λ⊙ are shown in Figure 8. The initial conditions
were a=355RJ, e=0.001, I=1°,
ω=180°,
and Ω=0°. Initially, the critical angle remains
librating around zero when the eccentricity remains almost bounded and less than 0.4.Prograde
satellite. Initial conditions:a=355RJ, e=0.001, I=1°, ω=180°,
and Ω=0°.
(a)(b)In Figure9
we consider a=355RJ, e=0.011, I=1°, ω=180°, and Ω=0°. As before, initially,
libration is around 0° and in the beginning the dynamics is very similar to the
previous figure. A significant increase of the eccentricity is observed when ϖ−λ⊙ enters in the circulation regime and escape
occurs at about  t≈140 years.Prograde
satellite. Initial conditions:a=355RJ, e=0.011,
I=1°, ω=180°, and Ω=0°.
(a)(b)Figure10 shows a case when the libration is centered only in 180°.      As
before, the increase and escape occur when ϖ−λ⊙ changes to a circulation regime.Prograde
satellite. Initial conditions:a=344.088RJ,
e=0.001, I=1°, and ω=Ω=0°.
(a)(b)Figure4 is very useful to interpret the results of the previous simulations. Usually
the region deep inside the libration (near the center) is very regular and is
related to the existence of stable periodic orbits. For this reason, if ϖ−λ⊙ is trapped inside a libration region, the
eccentricity remains bounded, without suffering large excursions. On the other
hand, circulation regime allows large excursions of the eccentricity.That said,
we can analyze the dynamics of the three previous figures. In Figure8,
initially, the satellite librates around ϖ−λ⊙=0°  up to about 35 years when a
short transition to circulation appears for t~40 years, but soon, the system
again goes back to original libration regime up to about t~75 years when the
libration center moves to 180° at t~80 years. Next, a circulation appears, so
that the motion is no more trapped inside a region of bounded small
eccentricity, as discussed above. Outside of the libration curves the motion
can experience, very easily, higher variations mostly because now the complete
problem is not integrable and the domain of the regular region of the level
curves of Figure 4, certainly is very reduced (modified). Indeed, numerical
examples indicate that the librations of ϖ−λ⊙,
in general, are not permanent and the perturbations always cause transitions to
circulation. In Figure 8, the eccentricity remains below 0.4 and escape occurs only
after the resonant angle changes to circulation regime. The jump of the
eccentricity when ϖ−λ⊙ changes to circulation is best illustrated in
Figure 9. In Figure 10 the libration is always centered in 180°, but even so,
there are some brief transitions when ϖ−λ⊙ attains 0° or 360°. Again, as
predicted, the escape occurs after circulation appears. Although it is not
clear in these figures, we have checked that the escapes always occur in the
neighborhood of ϖ−λ⊙=180°.As pointed
out in Hamilton and Krivov, the limita=393RJ for the
resonant semimajor axis is rather overestimated. In some cases, escapes can occur for a=345RJ. One of the reasons of this discrepancy is the fact that our
simplified model in (2.1) considers Jupiter in circular orbit, however in the
numerical simulations Jupiter's eccentricity is eJ=0.048497. Another point which is important is related to the expansion of the
function given in (2.7), where terms of higher order in the ratio (r/r⊙) were neglected. The inclusion of higher order
terms is not difficult but laborious. We intend to investigate in a future work.Our
numerical simulations also show that sometimes the initial value of the
pericenter plays an important role in the stability. This seems to be more
salient for values not so close toa=a*. For
instance, in the case of a=350 we found stability if ω=180°, while if ω=0 the
satellite is ejected in less than 1000
years.For the retrograde case we havea*=515.31RJ. Figures
11 and 12 show two examples of escape,
where the center of libration changes several times, much more often than in
the prograde case. From Figure 6 we see
that the two centers of libration are completely symmetric in opposition to the
centers of the prograde case (Figure 3). Therefore, the behavior of the
eccentricity around these two centers is similar. As before, the change of the
center of the libration (90° to 270° and vice versa) is predicted in the
complete problem, due to the nonintegrability. Figure 7 suggests
that the occurrence of these changes is related to a chaotic motion in the
neighborhood of a separatrix. Again, each time the critical angle circulates,
the trajectory is in a region, where large excursions in eccentricity should
occur. Therefore sooner or latter this can result in an escape.Retrograde
satellite. Initial conditions:a=528.4RJ, e=0.15, I=179°, ω=90°, and
Ω=180°.
(a)(b)Retrograde
satellite. Initial conditions:a=516.8888RJ, e=0.01, I=179°, ω=90°, and
Ω=180°.
(a)(b)Indeed in Figures11-12, after several
changes of the center of the libration,
escape ocurs and in both cases we confirmed again the remarkable feature we
always have observed, that is, in general, escapes occur in the vicinity of the
center of the libration (ϖ−λ⊙=180°  for prograde satellites and ϖI−λ⊙=90°  or 270° for retrograde case).


## 6. Some Islands of Stability

As mentioned before, the bounds395RJ and 515RJ are approximate and overestimated.
Certainly, a model using higher order expansion of the disturbing function R⊙ would provide better determination of these
values. Since expansion in Legendre polynomials is crucial for large values of the ratio of the
distances, no wonder
about some discrepancies in these numbers since they were obtained taking
the simplest expansion of second order in (3.1). However, no matter the improvement in this
determination, we show, in this section, the existence of several islands of
stability beyond the values mentioned
above.In
Figures4 and 7 we found two stable equilibrium centers. Although the onset of
the resonance can cause large variations and sometimes escape, however if the
satellite is trapped deep inside the libration curve, this orbit can remain
very stable, free of dangerous variations in eccentricity. In general these are
periodic (quasi-) stable orbits and are not isolated. Our numerical experiments
have shown that there are some finite intervals of the semimajor axis even for a>395RJ, where the satellite survives for at least 5 myr.
Figures
13 and 14 show typical examples, where the satellite remains trapped in ϖ−λ⊙=0°  and 180°, respectively.Prograde
satellite. Initial conditions:a=340RJ, e=0.001, I=1°, ω=0°, and
Ω=0°.
(a)(b)Prograde satellite. Initial conditions:a=352RJ, e=0.001, I=1°, ω=180°,
and Ω=0°.
(a)(b)In the case
of retrograde orbits we found much more interesting intervals of stability. We
show only two Figures15 and 16, where although the resonant angle changes
sometimes from 90° to 270°, the eccentricity remains
quite safe from collision or escapes. Note that in these two figures, the
semimajor axis is much larger than a*. Recall
that this kind of stable regions is possible thanks to the two stable libration
centers predicted in our simplified model. In other words, the appearance of
the two stable centers is related to the existence of a family of stable
periodic orbits. In the complete problem, part of this region of stability is
still preserved. Therefore, we can numerically find some of these orbits even
for high  distances. Finally, we list some semimajor axis intervals (a>a*), where stability was
found for at least 5 million years, as follows:Retrograde satellite. Initial conditions:a=583RJ, e=0.001, I=179°, ω=0°, and
Ω=0°.
(a)(b)Retrograde satellite. Initial conditions:a=612RJ, e=0.001, I=179°, ω=180°,
and Ω=0°.
(a)(b)[604–640](l=180°,ω=l⊙=0°);[600–642](l=l⊙=0°,ω=180°);[564–676](l=l⊙=ω=180°);[568–677](l=ω=0°,l⊙=180°);[576–620], [652–680](l=0°,ω=l⊙=180°);[572, 618], [654–682](l=l⊙=180°,ω=0°);
where within parentesis, on
the right of the intervals, we indicate the initial values for l, l⊙,
and ω. For the remaining values we
considered e=0.001, I=179°, and Ω=Ω⊙=0°. Most probably some of these orbits are
the same as pointed out by Winter [14] for the Earth-Moon problem.


## 7. Conclusion

We have derived, analytically, the
values of the semimajor axis, where evection resonances can occur. These values
are important, since they define approximated limits, where direct and
retrograde orbits can remain stable around a planet. Through a simple model
based on the restricted three-body problem these values were obtained and
checked against exact numerical integration. Using a completely different
way, we confirm previous results of other authors. Our methodology is based on
the classical expansion of the disturbing function which can be improved much
more if we consider higher order terms. Therefore, we think that the current
values:395RJ and 515RJ perhaps can be improved. We
also showed that the existence of stable orbits beyond the above values is
related to the stability of the region in the vicinity of the libration centers
of the evection resonance, which still persist even for large values of the
semimajor axis.



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