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Families of periodic orbits around irregular-shape
asteroids: From shape symmetry to asymmetry
G. Voyatzis, D. Karydis & K. Tsiganis
Department of Physics, Aristotle University of Thessaloniki, Greece,
voyatzis@auth.gr, dkarydis@auth.gr, tsiganis@auth.gr
October 29, 2021
Abstract
In Karydis et al (2021) we have introduced the method of shape
continuation in order to obtain periodic orbits in the complex gravita-
tional field of an irregularly-shaped asteroid starting from a symmetric
simple model. What’s more, we map the families of periodic orbits of
the simple model to families of the real asteroid model. The introduc-
tion of asymmetries in a gravitational potential may significantly affect
the dynamical properties of the families. In this paper, we discuss the
effect of the asymmetries in the neighborhood of vertically critical or-
bits, where, in the symmetric model, bifurcations of 3D periodic orbit
families occur. When asymmetries are introduced, we demonstrate
that two possible continuation schemes can take place in general. Nu-
merical simulations, using an ellipsoid and a mascon model of 433-Eros,
verify the existence of these schemes.
keywords Asteroids, Orbital mechanics, Periodic orbits
Proceedings of IAU symbosium No 346 ”Multiscale (time and mass)
dynamics of space objects”, October 2021
1 Introduction
Many space missions to small NEA have taken place recently or are planned
in the coming years. Close proximity operations around such small bodies,
which have irregular shape in general, demand sufficient knowledge of their
gravitational field and their dynamics. In orbital mechanics, periodic orbits
play an important role in understanding the dynamics and have been studied
widely in celestial mechanics and especially in the three body problem. In
addition, they can find direct applications in astrodynamics as parking orbits
for a spacecraft or the unstable ones may be used for computing landing or
1
escape paths (Scheeres (2012)). In such complex gravitational fields, which
can be sufficiently modeled e.g. by polyhedrals or mascons (see Scheeres
(2012)), the computation of periodic orbits is a challenge. The grid search
method introduced by Yu & Baoyin (2012) has been proved very efficient
and applied for various asteroids (e.g. Jiang et al (2018)).
In Karydis et al (2021), which will be referred in the following as ‘Paper
I’, we approach the potential of an irregular body by starting from the
symmetric potential of a simplified model (an ellipsoid), where the families
of periodic orbits can be easily computed and show particular structures and
types. Then, asymmetric terms are gradually introduced in the potential
and periodic orbits are continued along this procedure, which is called shape
continuation and ends when the ‘real’ potential of the target asteroid is
adequately approximated. In this way, we assign families of the simplified
model to families of the ‘real’ model and we can study the effect of the
symmetric perturbations in the characteristic curves of the families and their
stability. In the present study, we use a theoretical analysis and numerical
simulations in order to show how families are affected by asymmetric forces
when they are close to vertically critical orbits, where planar and 3D orbit
families intersect in the symmetric model.
2 Description of the orbital mechanics
We consider the motion of a mass-less body in the gravitational field of
an irregularly shaped asteroid which rotates with angular velocity ω. If
the center of mass of the asteroid is considered as the origin point of a
reference frame which rotates with the asteroid (i.e. it is a body-fixed frame),
and r= (x, y, z) is the position vector of the mass-less body, its motion is
described by the Hamiltonian
H(r,p) = 1
2p2−p(ω×r) + U(r),(1)
where the generalized momenta are given by p=˙r +ω×rand Uis the
gravitational potential of the asteroid. If ωis constant, which is the case
considered in this study, then His also constant (Hbeing the Jacobi integral
or, simply, the energy).
Let X= (x, y, z, ˙x, ˙y, ˙z) denote a phase space point and X=X(t;X0)
a trajectory with initial conditions X0. The system is autonomous and the
condition X(T;X0) = X0implies a periodic orbit of period T. Supposing
that the orbit intersects a Poincar´e section, say x= 0 with ˙x > 0 and energy
h, the orbits can be defined explicitly by a point in the 4Dspace of section,
called Π4, which is defined by vector Y= (y, z, ˙y, ˙z). Thus, the periodicity
conditions are reduced to
Y(t∗;X0) = Y0,(2)
2
where t∗is the time of the mth intersection of the orbit, with a section that
satisfies (2) for the first time. In this case, t∗and period Tcoincide and m
denotes the multiplicity of the section.
In general, in space Π4, periodic orbits are isolated and analytically
continued with respect to h, forming mono-parametric families (Meyer et al
(2009), Scheeres (2012)). In computations, we may consider a continuation
with respect to any variable but it is more convenient to continue the families
by using an extrapolation procedure and considering as parameter the length
sof the characteristic curve of the family in Π4(see Paper I). In this way,
the numerical continuation is still successful at energy extrema that may
exist along the family.
Let ξdenote a variation vector that satisfies the system of linear varia-
tional equations of system (1), namely
˙
ξ=A(t)ξ⇒ξ=Φ(t)ξ(0).(3)
Matrix Ais computed along a periodic orbit and thus, it is also peri-
odic. Φ(t) is the fundamental matrix of solutions and the constant matrix
M=Φ(T) is the monodromy matrix, which is symplectic. Therefore, two
eigenvalues are equal to unit and the rest four form reciprocal pairs. If we
remove the rows and columns that correspond to the variables which define
the Poincar´e section (e.g. xand ˙x) from M, then we obtain the reduced
monodromy matrix M0of size 4 ×4 and the unit eigevalues are removed.
The periodic orbit is stable if the two reciprocal pairs of eigenvalues of M0lie
on the complex unit circle. In computations, we use the Broucke’s stability
indicies b1and b2, which are computed from the elements of M0and their
stability implies that they are real and |bi|<2 (Broucke (1969)).
When M0is computed for a planar orbit, then it is decomposed in two
2×2 sub-matrices, Mhand Mvthat refer to horizontal stability (index b1)
and vertical stability (index b2), respectively. If b2= 2 then, the planar orbit
is called vertically critical orbit (v.c.o.) and signifies a bifurcation for another
family of 3D periodic orbits (H´enon (1973)). We note that b2may also take
the value of two when the planar orbit needs mtimes to complete a period
(multiplicity). Then, if Tis the period of the v.c.o., the 3D bifurcating orbit
close to the v.c.o. will be of period mT .
3 Continuation near a v.c.o. : from a symmetric
to an asymmetric model
Suppose that Uast is a potential model of the asteroid provided by a ‘real’
model (e.g. by mascons or a polyhedral model). Let us define a mono-
parametric set of potentials
U() = U0+U1,0≤≤0,(4)
3
where U0is the symmetric potential of the ellipsoid that approximates the
potential of the asteroid and U1includes the asymmetric part of the potential
such that U(0) = Uast with e0being sufficiently small.
Let Fpbe a symmetric planar family of periodic orbits with a potential
of U0and Oa v.c.o. of Fp. We suppose that in the neighborhood of O
the planar orbits of Fpare of the same horizontal stability type. In the
present study, we consider that they are stable so, the eigenvalues of Mh
are of the form λ1,λ2=e±iφ,φ∈(δ, π −δ), δ > 0. The eigenvalues of
Mvare critical for O, i.e. λ3,4= 1 when the appropriate multiplicity mis
taken into account. The distribution of λion the unit circle is shown in
the middle panel of Fig. 1. Suppose we perform an analytic continuation of
the v.c.o. Owith respect to parameter . As increases smoothly towards
value 0, the eigenvalues λ1,2should move smoothly on the unit circle due
to the analyticity (see Meyer et al (2009)) and if δis sufficiently large, the
eigenvalues do not reach the critical values ±1 as →0. On the other
hand, the critical eigenvalues λ3,4, as varies, may move either on the unit
circle or on the real axis. These cases are called scheme I and scheme II,
respectively, and are presented in Fig. 1. Which one of the two schemes will
take place, depends on the term U1, which represents the asymmetric part
of the asteroid’s potential.
Applying analytic continuation to all orbits of Fpin the neighborhood of
O,with respect to , we obtain the set of families F(), with F(0) = Fp. All
orbits of F() with 6= 0 are spatial and asymmetric and family Fast =F(0)
is the family of orbits of the real asteroid originating from the planar family
of the ellipsoid. The initial orbit O∈Fpis mapped to the orbit O0∈Fast.
When scheme I takes place, Fast should consist of stable orbits at least near
O0. Instead, in scheme II the orbit O0is unstable and there should exist a
continuous segment on Fast near O0consisting of unstable orbits.
Figure 1: Distribution of eigenvalues for a v.c.o. of the symmetric model
(center) and their displacement after introduction of asymmetry (scheme I
in the left panel and scheme II in the right panel)
Let us consider the family, F3D(0), of three dimensional orbits that bi-
furcates from O. Similarly to the planar family, analytic continuation with
respect to can be also applied providing the set of families F3D(). All
orbits should be asymmetric for 6= 0 and family F3D(0) is the asteroid’s
4
family of periodic orbits associated to the family F3D(0) of the symmetric
ellipsoid model. When scheme I takes place, the families Fand F3D, which
for = 0 intersect at O, should be detached for > 0 since no bifurcation
point exists on F() (whole family near Ois stable). However, in scheme
II the edges of the unstable segment formed on F(0) may be bifurcation
points for the family F3D(0). The above assumptions are verified by the
numerical computations presented in the next section.
4 Numerical computations : The asteroid 433-
Eros
In Paper I, we used the symmetric ellipsoid model (with normalized maxi-
mum semi-axis, a= 1, and angular velocity, ω= 1) to initially approximate
the potential of asteroid 433-Eros. Then, we applied shape-continuation to
identify families of periodic orbits for the ‘real’ gravitational potential of
433-Eros, implemented with a sufficient number of mascons (Soldini et al
(2020)). In the ellipsoid model, we consider the family of planar (z= 0) cir-
cular retrograde orbits, CR, which is fully stable. The family is also vertically
stable but there are v.c.o. for higher period multiplicities (m= 2,3,4, ..).
Their y0-position (where y0is the approximate radius of the orbit) is shown
in the left panel of Fig. 2. The right panel shows the stability indicies bi
along the family (dashed curves). The CRis continued when asymmetric
terms are added in the potential in order to simulate the potential of the
asteroid. The computed family for 433-Eros consists of orbits which are no
longer planar and symmetric but are almost circular. The family is pre-
sented in Fig. 2 with solid curves. The major part of CRof Eros consists
of stable orbits and this is also the case close to the radius of the v.c.o.
B3and B4. Therefore, such a situation implies scheme I for the 3D orbits
emanating in the symmetric model from these v.c.o.. However, it is evident
that the introduced asymmetries caused an unstable segment close to B2
and this implies scheme II. It should be noted that this instability has been
also mentioned in Ni et al (2016) who used a polyhedral model for 433-Eros.
Scheme I is shown by considering the 3D family L24 of the ellipsoid,
which bifurcates from the v.c.o. B4. The family near B4is stable but
becomes unstable when it becomes significantly inclined as shown in the left
panel of Fig. 3. For the asymmetric potential of 433-Eros the family is
represented by the characteristic curve in the right panel of Fig.3. We can
see that in the asymmetric asteroid case, family L24 does not intersect the
planar family CRand the two families are now separated. The stability type
of orbits is not affected by the asymmetry for orbits close to the plane z= 0.
However, a break of family L24 arises because of the irregular shape of Eros.
After this break, the family continues with the unstable segment L0
24. Such
family breaks are discussed also in Paper I. In the same paper, where family
5
Figure 2: (left) The characteristic curve of the circular family CRfor the
ellipsoid (dashed curve) and 433-Eros (solid curve) projected on the plane
y0−z0. The points Bmindicate the y0-position of the v.c.o. with the
subscript mbeing the multiplicity. The red segment indicates the part of
the family with unstable orbits. (right) The variation of the stability indices
b1and b2along the CR-family of ellipsoid and Eros.
L13 is studied, scheme I also holds true, with a change of stability at z≈0.
Scheme II holds true for the case of v.c.o. B2of the ellipsoid from which
the 3D families L02 and K02 originate (see Paper I). The two families are
equivalent because they consist of the same doubly symmetric periodic orbits
but their characteristic curves are presented in different spaces of initial
conditions. In the left panel of Fig. 4, we present the initial conditions
of the orbits in K02 family. As we have already mentioned, the CRfamily
of 433-Eros shows an unstable segment at B2, defined by the points B21
and B22. These points should be bifurcation points of other families. By
computing the families K02 and L02 in the asymmetric potential of 433-Eros
(see right panel of Fig. 4) we obtain that i) the two families are separated
and they now consist of different asymmetric periodic orbits ii) the families
pass from the points B21 and B22 and, therefore, the continuation scheme
II is valid here. K02 consists of unstable orbits and L02 of stable ones (at
least in the neighborhood of the bifurcation points). However, we cannot
claim that the appearance of a stable and an unstable family is a general
property for scheme II.
Acknowledgments The authors acknowledge funding support from the
European Unions Horizon 2020 research and innovation program under grant
agreement No. 870377 (project NEO-MAPP).
References
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6
Figure 3: (left) The characteristic curves of the planar family CRand the 3D
family L24 of the ellipsoid. Blue (red) color indicates stability (instability).
B4is the v.c.o. where the two families intersect. (right) The characteristic
curves for the corresponding families of 433-Eros potential. The transition
from the ellipsoid (left) to the mascon model of 433-Eros (right) indicates
that scheme I takes place.
H´enon, M. 1973 A&A, 28, 415
Jiang,Y., Schmidt, J., Li, H., Liu, X. & Yang, Y. 2018 Astrodynamics, 2, 69
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7
Figure 4: (left) The characteristic curves of the planar family CRand the
3D family L02 (and its equivalent K02 ) of the ellipsoid. Blue (red) color in-
dicates stability (instability). B2is the vco where the two families intersect.
(right) The characteristic curves for the corresponding families of 433-Eros
potential. L02 and K02 are families of different orbits. The transition from
the ellipsoid (left) to the mascon model of 433-Eros (right) indicates that
scheme II takes place.
8
