Families of periodic orbits around asteroids: From shape symmetry to asymmetry

Abstract

In Karydis et al. (2021) we have introduced the method of shape continuation in order to obtain periodic orbits in the complex gravitational 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 introduction 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 orbits, 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. Numerical simulations, using an ellipsoid and a mascon model of 433-Eros, verify the existence of these schemes.

Full text

Multi-scale (time and mass) dynamics of space objects
Proceedings IAU Sy mpo si um No . 364, 2022
A.Celletti,C.Gale¸s, C. Beaug´e, A. Lemaitre, eds.
doi:10.1017/S174392132100123X
Families of periodic orbits around asteroids:
From shape symmetry to asymmetry
G. Voyatzis ,D.KarydisandK.Tsiganis
Section of Astrophysics, Astronomy and Mechanics, Dept. of Physics,
Aristotle University of Thessaloniki,
GR 54124, Thessaloniki, Greece
email: voyatzis@auth.gr,dkarydis@auth.gr,tsiganis@auth.gr
Abstract. In Karydis et al. (2021) we have introduced the method of shape continuation in order
to obtain periodic orbits in the complex gravitational 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 introduction 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
orbits, 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. Numerical simulations, using an ellipsoid and a mascon model of 433-Eros,
verify the existence of these schemes.
Keywords. Asteroids, Orbital mechanics, Periodic orbits
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 irregu-
lar 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 escape paths (Scheeres (2012)). In such complex gravitational fields, which can be suf-
ficiently 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 & Baoy i n
(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 character-
istic curves of the families and their stability. In the present study, we use a theoretical
©The Author(s), 2022. Published by Cambridge University Press on behalf of International Astronomical
Union. This is an Open Access article, distributed under the terms of the Creative Commons Attribution
licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and
reproduction in any medium, provided the original work is properly cited.
https://doi.org/10.1017/S174392132100123X Published online by Cambridge University Press

Periodic orbits around irregular-shaped asteroids 247
analysis and numerical simulations in order to show how families are affected by asym-
metric 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),(2.1)
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)=X0
implies a periodic orbit of period T. Supposing that the orbit intersects a Poincar´e
section, say x=0with ˙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 (2.2)
for the first time. In this case, t∗and period Tcoincide and mdenotes 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 variational equations
of system (2.1), namely
˙
ξ=A(t)ξ⇒ξ=Φ(t)ξ(0).(2.3)
Matrix Ais computed along a periodic orbit and thus, it is also periodic. Φ(t)isthe
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)fromM, then we obtain
the reduced monodromy matrix Mof size 4 ×4 and the unit eigevalues are removed.
The periodic orbit is stable if the two reciprocal pairs of eigenvalues of Mlie on the
complex unit circle. In computations, we use the Broucke’s stability indicies b1and b2,
which are computed from the elements of Mand their stability implies that they are
real and |bi|<2(Broucke (1969)).
When Mis 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
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248 G. Voyatzis, D. Karydis & K. Tsiganis
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).
(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(ε)=U0+εU1,0εε0,(3.1)
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(ε0)=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 Othe planar orbits of Fpare of the
same horizontal stability type. In the present study, we consider that they are stable so,
the eigenvalues of Mhare 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
ε.Asεincreases smoothly towards value ε0, 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 ±1asε→ε0. On the other hand,
the critical eigenvalues λ3,4,asεvaries, 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 ε, we obtain the set of families F(ε), with F(0) = Fp. All orbits of F(ε)withε=0
are spatial and asymmetric and family Fast =F(ε0) 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 O∈Fast .Whenscheme I takes place, Fast should consist of stable
orbits at least near O. Instead, in scheme II the orbit Ois unstable and there should
exist a continuous segment on Fast near Oconsisting of unstable orbits.
Let us consider the family, F3D(0), of three dimensional orbits that bifurcates from O.
Similarly to the planar family, analytic continuation with respect to εcan be also applied
providing the set of families F3D(ε). All orbits should be asymmetric for ε= 0 and family
F3D(ε0) is the asteroid’s family of periodic orbits associated to the family F3D(0) of the
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Periodic orbits around irregular-shaped asteroids 249
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.ThepointsBmindicate 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.
symmetric ellipsoid model. When scheme I takes place, the families Fand F3D,which
for ε= 0 intersect at O, should be detached for ε>0 since no bifurcation point exists on
F(ε) (whole family near Ois stable). However, in scheme II the edges of the unstable
segment formed on F(ε0) may be bifurcation points for the family F3D(ε0). 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 maximum 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) circular 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 bialong 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
presented in Fig. 2with 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 B2and 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 L
24. Such family breaks are discussed also in Paper
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250 G. Voyatzis, D. Karydis & K. Tsiganis
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.
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 indicates 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.
I. In the same paper, where family 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
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Periodic orbits around irregular-shaped asteroids 251
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)weobtainthati)
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
Broucke, R. 1969 Stability of periodic orbits in the elliptic, restricted three-body problem AIAA
Journal, 7:1003-1009
H´enon, M. 1973 A&A, 28, 415
Jiang, Y., Schmidt, J., Li, H., Liu, X. & Yang, Y. 2018 Astrodynamics,2,69
Karydis, D., Voyatzis, G., & Tsiganis, K. 2021 Advances in Space Research (to appear)
Meyer, K.R., Hal, G.R., & Offin, D. 2009 Introduction to Hamiltonian dynamical systems and
N-body problem (Springer)
Ni, Y., Jiang, Y., & Baoyin, H. 2016 ASS, 361, 170
Scheeres, D.J. 2012 Orbital motion in strongly perturbed environments (Springer, Berlin
Heidelberg)
Scheeres, D.J. 2012 Acta Astronautica, 72, 1
Soldini, S., Takanao, S., Ikeda, H., Wada, K, Yuichi, T., Hirata, N, & and Hirata, N. 2020
Planetary and Space Science 180, 104740
Yu, Y., & Baoyin, H. 2012 MNRAS, 427, 872
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