Generating Solutions for Periodic Orbits in the Circular Restricted Three-Body Problem

Abstract

This paper proposes a method for calculating the periodic orbits in the Circular Restricted Three-Body Problem. The solution is based on the continuation method with a parameter. The parameter is the angular velocity of rotation of the attracting masses around their barycenter. Analytical periodic solutions of the problem of two fixed attracting masses serve as generating solutions. Seven possible generating solutions are described. Numerical examples of the initial conditions are given for the Earth–Moon system, making it possible to calculate the described orbits and their graphic illustrations.

Full text

TECHNICAL NOTE
Generating Solutions for Periodic Orbits in the Circular
Restricted Three-Body Problem
Sergey Zaborsky
1
#American Astronautical Society 2020
Abstract
This paper proposes a method for calculating the periodic orbits in the Circular
Restricted Three-Body Problem. The solution is based on the continuation method
with a parameter. The parameter is the angular velocity of rotation of the attracting
masses around their barycenter. Analytical periodic solutions of the problem of two
fixed attracting masses serve as generating solutions. Seven possible generating solu-
tions are described. Numerical examples of the initial conditions are given for the
Earth–Moon system, making it possible to calculate the described orbits and their
graphic illustrations.
Keywords Periodic orbits .Generating solution .Continuation method
Introduction
Numerous studies of periodic orbits in the circular restricted three-body problem
(CRTBP) are found in the literature ([1–20]; etc.). The simplest form of the spatial
periodic orbits are halo orbits around the collinear libration points, first studied by
Farquhar and Kamel [1] and Breakwell and Brown [21]. From these results, additional
families were calculated for various mass ratios by Howell [4]. Analytical solutions of a
third-order [14] and higher of the linearized equations of motion in the vicinity of the
collinear points were used as generating solutions for the periodic orbits. The most
famous of these are Lyapunov’s planar orbits, the halo orbits and the near-rectilinear
halo orbits (NRHO). Hénon and Bruno developed a systematic theory for the study of
periodic orbits in the CRTBP. The works of Hénon and Bruno are limited to the
problem of Hill [22], which is a special case of the CRTBP where the mass parameter
of the system tends to zero with fixed the angular velocity of rotation of the attracting
bodies around their barycenter. Analytical solutions of the Hill equations of motion
The Journal of the Astronautical Sciences
https://doi.org/10.1007/s40295-020-00222-3
*Sergey Zaborsky
Sergey.Zaborsky@rsce.ru
1
RSC Energia, Korolev, Moscow Region 141070, Russia

were used as generating solutions for the periodic orbits around the Moon. In that case,
the motion occurs around a smaller attracting body. The most famous of these are the
distant retrograde orbits (DRO). Also generating solutions can be Keplerian resonant
orbits [23]. In this case, the parameter is one of the mass of the attracting bodies. As of
now, three kinds of generating analytical solutions are better known: for the motion in
the vicinity of the libration points, for the motion occurs around a smaller attracting
body and Keplerian resonant orbits. A differential correction method, together with a
continuation method, is usually exploited to find the initial conditions for each orbit in
the CRTBP.
This paper shows how periodic orbits are generated and evolve when the centrifugal
is added to multi-body gravity (assuming static attracting bodies) and suggests a
method for determination of periodic spatial and in-plane orbits families for the
CRTBP. The periodic solutions (from the viewpoint of Poincaré, H. [24]) are obtained
on the continuation method with a parameter. The parameter is the angular velocity of
rotation of the attracting bodies around their barycenter. Analytical periodic solutions of
the problem of two static attracting bodies [25,26] serve as generating solutions. Seven
possible generating solutions are described. This paper also describes the calculation
conditions and provides graphic examples of periodic orbits. In this case, as an
example, Lyapunov’s planar orbit, NRHO in the vicinity of L1and the orbit in-plane
around the Earth-Moon system [23] correspond to different generating solutions.
Problem Statement
The task is to obtain the motion parameters at the true periapsis about the Moon rπthat
ensure the periodicity of the spacecraft’s motion around the Moon in the CRTBP. Here,
the radius of the true periapsis about the Moon is fixed. The motion is described in a
rectangular coordinate system with the origin Oin the barycenter of two bodies of
masses m1(Earth) and m2(Moon) and the distance 2cbetween them, the abscissa Ox in
the direction m2and the axis Oz directed along the angular velocity of rotation ωof
these masses around their shared barycenter. The Oy- axis completes the right-handed
set. The mathematical model of the spacecraft’s motion in this coordinate system takes
the following form:
d2x
dt2−2ωdy
dt −ω2x−∂U
∂x¼0
d2y
dt2þ2ωdx
dt −ω2y−∂U
∂y¼0
d2z
dt2−∂U
∂z¼0
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:
ð1Þ
where.
U=γ1/r1+γ2/r2- total potential of the attracting bodies,
γ1,γ2- gravitational parameters of bodies m1and m2,
The Journal of the Astronautical Sciences

ω- angular velocity of the circular motion attracting masses around their
barycenter,
r1¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x−e
x1
ðÞ
2þy2þz2
q,r2¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x−e
x2
ðÞ
2þy2þz2
q- distances from the space-
craft to the attracting bodies m1and m2,
e
x1¼−2cγ2=γ1þγ2
ðÞ,e
x2¼2cγ1=γ1þγ2
ðÞ-xcoordinates of the attracting
bodies m1and m2.
Based on the formulation of the problem, the coordinates and the velocity projections
of the spacecraft in the coordinate system Oxyz at the initial time t= 0 can be written in
the form depending on two parameters xπand vπ:
xt¼0ðÞ¼xπyt¼0ðÞ¼0zt¼0ðÞ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r2
π−xπ−e
x2

2
rð2Þ
vxt¼0ðÞ¼0vyt¼0ðÞ¼vπvzt¼0ðÞ¼0ð3Þ
The orbital period Tis determined from the condition of the integer number Nof
revolutions around the axis Ox:
wt¼TðÞ¼2Nπð4Þ
where wis the angle between the axis Oz and the r2projection to the plane yOz (i.e.,
w(t= 0) = 0).The invariance of the system of Eq. (1) with respect to the mappings t→
−tand y→−ymakes it possible to record the conditions of the orbital periodicity:
vxt¼T=2ðÞ¼0vzt¼T=2ðÞ¼0ð5Þ
Thus, the task is reduced to the calculation of the parameters xπand vπin Eqs. (2)and
(3), so that the conditions of Eq. (5)withw(t=T/2) = Nπfrom Eq. (4) are satisfied.
Generating Solutions
It is proposed to solve the problem by the continuation method with a differ-
ential correction scheme [27] with the parameter ω.Atω=0,theEq.(1)are
transformed into a system of differential equations of motion of the material
point of zero mass in the gravitational field of two fixed masses having a
solution in quadratures:
The Journal of the Astronautical Sciences

d2x
dt2−∂U
∂x¼0
d2y
dt2−∂U
∂y¼0
d2z
dt2−∂U
∂z¼0
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ð6Þ
We shall consider the solutions to the boundary value problem of Eqs. (2)–(6)with
respect to the parameters xπand vπas generating solutions for the problem. Its numerical
continuation is carried out by solving the series of boundary value problems in Eqs.
(1)–(5) with an angular velocity ωthat is successively changed by a step Δω. Each
solution serves as an initial approximation for the next. The value Δωis determined by
the area of convergence of the solution. The process ends after solving the boundary
value problem of Eqs. (1)–(5)atω¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
γ1þγ2
ðÞ=2cðÞ
p=2cðÞ.
To obtain the generating solutions, we express x,y,zin terms of the coordinates λ,μ
and w, first introduced by Euler [25] and Lagrange [26]:
x¼cλμ þ1
2e
x1þe
x2

y¼cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
λ2−1

1−μ2
ðÞ
qsin wðÞ
z¼cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
λ2−1

1−μ2
ðÞ
qcos wðÞ
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ð7Þ
Herewith,
λ¼1
2cr1þr2
ðÞμ¼1
2cr1−r2
ðÞ ð8Þ
The coordinate system (λ,μ,w) corresponds to the following coordinate surfaces (Fig.
1):
Fig. 1 Coordinate systems Oxyz and (λ,μ,w)
The Journal of the Astronautical Sciences

at μ=const - bipolar hyperboloid of rotation (this is evident from μ=(r1−r2)/(2c)
and Eq. (7)), the axis of symmetry of which coincides with the axis Ox,andthefociare
located at points m1and m2:
x−1
2e
x1þe
x2

2
c2μ2−y2þz2
c21−μ2
ðÞ
¼1ð9Þ
at λ=const - confocal hyperboloid Eq. (9) elongated ellipsoid of rotation (this is
evident from λ=(r1+r2)/(2c)andEq.(7)):
x−1
2e
x1þe
x2

2
c2λ2þy2þz2
c2λ2−1

¼1ð10Þ
at w=const - the plane turned around the axis Ox at the angle wfrom the vertical plane
(w=±π/2 corresponds to the xOy plane):
y−tan wðÞz¼0ð11Þ
The first generating solution is obtained when μ=const,i.e.,themotionoccursonthe
surface of the two-cavity hyperboloid of rotation Eq. (9). Taking this into account, after
substituting Eq. (7) into the system of Eq. (6) and excluding the coordinates w,we
obtain
μd2λ
dt2¼−1
c3
γ1λμ þ1ðÞ
λþμðÞ
3þγ2λμ−1ðÞ
λ−μðÞ
3
!
λ2−μ2
λ2−1
d2λ
dt2−λ1−μ2

dw
dt

21
λ2−1

2
dλ
dt =dw
dt

2
þ1
!
¼−1
c3
γ1
λþμðÞ
2−γ2
λ−μðÞ
2
!
d
dt λ2−1

dw
dt

¼0
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:ð12Þ
From the third equation of Eq. (12), it follows that
dw
dt ¼Cλ
λ2−1ð13Þ
where Cλ=const. Eliminating d2λ/dt2from Eq. (12), keeping in mind (dλ/dt)/(dw/dt)=
dλ/dw and using Eq. (13), we get
λ2−μ2

21
λ2−1

2
dλ
dw

2
þ1
!
¼−1
C2
λc3μλ2−1

λ−μðÞ
2γ1−λþμðÞ
2γ2

ð14Þ
The Journal of the Astronautical Sciences

From Eq. (14), it follows that
1þ1−μ2
λ2−1
 dλ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
−1
C2
λc3μλ2−1

λ−μðÞ
2γ1−λþμðÞ
2γ2

þλ2−μ2

2

s¼dw ð15Þ
Since the integral of the left-hand side of Eq. (15) is elliptic, the value, determined by
the integer number 2Mλof its periods, under periodic motion must meet the condition
Eq. (4):
2Mλ
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þγ1−γ2
C2
λc3μ
r∫
λ3
λ4
1þ1−μ2
λ2−1
 dλ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
λ−λ1
ðÞλ−λ2
ðÞλ−λ3
ðÞλ4−λðÞ
p¼wTðÞð16Þ
where λ4>λ3>λ2>λ1are the algebraic roots (Descartes–Euler or Ferrari solutions
(Korn and Korn 1961)) of the radical expression in Eq. (15). The lower bound of the
integral Eq. (16)λ3corresponds to the rπ. Therefore, λ3and μshould be related by the
definition in Eq. (8) by the equation
μ¼λ3−rπ
cð17Þ
Since at t= 0 the derivative dλ/dw = 0, then from Eq. (14) it follows that Cλis
expressed through λ3and μ
Cλ¼1
cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
−λ2
3−1
cμ
γ1
λ3þμðÞ
2−γ2
λ3−μðÞ
2
!
v
u
u
tð18Þ
Table 1 Initial conditions Eq. (33) (the third case) for the NRHO in the vicinity of L1,T= 7.83 days
xπ−ex2;km zπ,km vπ,km/s
Generating solution −50.140 4499.721 −1.04380
Problem solution 444.595 4453.318 −1.45788
Table 2 Initial conditions Eq. (21) (the first case) for a periodic orbit at Nλ=2, Mλ=3, T= 34.3810 days
xπ−e
x2;km zπ,km vπ,km/s
Generating solution −4317.755 1267.672 −1.461104
Problem solution −1664.101836 −4181.000488 −1.5041647424
The Journal of the Astronautical Sciences

After integrating Eq. (16), factoring in Eq. (4), we get
2Mλ
1þμ2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þγ1−γ2
C2
λc3μ
rKαλ
ðÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
λ4−λ2
ðÞλ3−λ1
ðÞ
pþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1−nλ
ðÞ1−sin2αλ
nλ

sПnλnαÞðÞ¼2Nλπ
0
B
B
@ð19Þ
where Nλis an integer, K(αλ)andП(nλ\αλ) are full elliptic integrals of the second and
third kind, respectively [28–31],
sin2αλ¼λ2−λ1
ðÞλ4−λ3
ðÞ
λ4−λ2
ðÞλ3−λ1
ðÞ
nλ¼−λ2−λ1
ðÞλ4−λ3
ðÞ
1þλ1λ4
ðÞ1þλ2λ3
ðÞ
ð20Þ
After solving Eqs. (17)and(19), factoring in (18)forthegivenMλand Nλ,wegetλ
and μ.FromEqs.(7)and(13), the values xπand vπincluded in Eqs. (2)and(3)are
obtained:
xπ¼cλ3μþ1
2e
x1þe
x2

vπ¼zdw
dt

w¼0
¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
cμ−1
μ
γ1
λ3þμðÞ
2−γ2
λ3−μðÞ
2
!
v
u
u
tð21Þ
The second generating solution is obtained when λ=const, i.e., the motion occurs on
the surface of the prolate ellipsoid of revolution Eq. (10). Then, after substituting Eq.
(7) into the system of Eq. (6) and excluding the coordinates w,weget
Table 3 Initial conditions Eq. (21) (the first case) for a periodic orbit at Nλ=3, Mλ=5, T= 57.78 days
xπ−ex2;km zπ,km vπ,km/s
Generating solution −4435.499 −759.175 −1.471054
Problem solution −1846.554545 −4103.685698 −1.5070263301
Table 4 Initial conditions Eq. (31)(thesecondcase)foraperiodicorbitatNμ=2,Mμ=1,T= 17.20 days
xπ−e
x2;km zπ,km vπ,km/s
Generating solution 2802.133 3521.086 −1.332120
Problem solution 2289.451353 3874.069244 −1.4436509763
The Journal of the Astronautical Sciences

Table 5 Initial conditions Eq. (34) (the fourth case) for a periodic orbit at T= 10.143 days
xπ−ex2;km zπ,km vπ,km/s
Generating solution 4500.0 0.0 −1.4756507
Problem solution 4500.0 0.0 −2.4845708
Fig. 2 NRHO in the vicinity of L1(the third case)
The Journal of the Astronautical Sciences

λd2μ
dt2¼−1
c3
γ1λμ þ1ðÞ
λþμðÞ
3þγ2λμ−1ðÞ
λ−μðÞ
3
!
λ2−μ2
1−μ2
d2μ
dt2þμλ
2−1

dw tðÞ
dt

21
1−μ2
ðÞ
2
dμ
dt =dw
dt

2
þ1
!
¼−1
c3
γ1
λþμðÞ
2þγ2
λ−μðÞ
2
!
d
dt 1−μ2

dw tðÞ
dt

¼0
8
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:ð22Þ
From the third equation of Eq. (22), it follows that
dw
dt ¼Cμ
1−μ2ð23Þ
where Cμ=const. Eliminating d2μ/dt2from Eq. (22), keeping in mind that (dμ/dt)/(dw/
dt)=dμ/dw and using Eq. (23), we get
Fig. 3 Generating periodic orbit at Nλ=2, Mλ=3 (the first case)
The Journal of the Astronautical Sciences

λ2−μ2

21
1−μ2
ðÞ
2
dμ
dw

2
þ1
!
¼1
C2
μc3λ1−μ2

λ−μðÞ
2γ1þλþμðÞ
2γ2

ð24Þ
From Eq. (24), it follows that
1þλ2−1
1−μ2
 dμ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
C2
μc3λ1−μ2

λ−μðÞ
2γ1þλþμðÞ
2γ2

−λ2−μ2

2
s¼dw ð25Þ
Fig. 4 Periodic orbit in the inertial frame at Nλ=2,Mλ=3,T= 34.3810 days (the first case)
The Journal of the Astronautical Sciences

Since the integral of the left-hand side of equation Eq. (25) is elliptic, the value,
determined by the integer number 2Mμof its periods, under periodic motion must
meet the condition Eq. (4):
2Mμ
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þγ1þγ2
C2
μc3λ
s∫
μ3
μ4
1þλ2−1
1−μ2
 dμ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
μ−μ1
ðÞμ−μ2
ðÞμ−μ3
ðÞμ4−μðÞ
p¼wTðÞ ð26Þ
where μ4>μ3>μ2>μ1are the algebraic roots (Descartes–Euler or Ferrari solutions
(Korn and Korn 1961)) of the radical expression in Eq. (25). The upper bound of the
integral Eq. (26)μ4corresponds to the rπ.Therefore,μ4and λshould be related by the
definition in Eq. (8) by the equation
Fig. 5 Periodic orbit at Nλ=2, Mλ=3, T= 34.3810 days (the first case)
The Journal of the Astronautical Sciences

λ¼μ4þrπ
cð27Þ
Since at t= 0 the derivative dμ/dw = 0, then from Eq. (24) it follows that Cμis
expressed through μ4and λ
Cμ¼1
cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
cλ1−μ2
4
γ1
λþμ4
ðÞ
2þγ2
λ−μ4
ðÞ
2
!
v
u
u
tð28Þ
After integrating Eq. (26), factoring in Eq. (4),
2Mμ
1þλ2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þγ1þγ2
C2
μc3λ
sKαμ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
μ4−μ2
ðÞμ3−μ1
ðÞ
pþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1−nμ

1−sin2αμ
nμ

sПnμnαÞ

¼2Nμπ
0
B
B
B
B
@ð29Þ
where Nμis an integer, K(αμ)andП(nμ\αμ) are full elliptic integrals of the second and
third kind, respectively [28–31],
Fig. 6 Generating periodic orbit at Nλ=3, Mλ=5 (the first case)
The Journal of the Astronautical Sciences

sin2αμ¼μ2−μ1
ðÞμ4−μ3
ðÞ
μ4−μ2
ðÞμ3−μ1
ðÞ
nμ¼−μ2−μ1
ðÞμ4−μ3
ðÞ
1þμ1μ4
ðÞ1þμ2μ3
ðÞ
ð30Þ
After solving Eqs. (27)and(29), factoring in Eq. (28)forthegivenMμand Nμ,weget
μ4and λ.FromEqs.(7)and(13), the values xπand vπincluded in Eqs. (2)and(3)are
obtained:
xπ¼cλμ4þ1
2e
x1þe
x2

vπ¼zdw
dt w¼0
¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
cλ−1
λ
γ1
λþμ4
ðÞ
2þγ2
λ−μ4
ðÞ
2
!
v
u
u
tð31Þ
The third generating solution for NRHO in the vicinity of L1is obtained when rα(the
distance to the Earth) and rπare fixed (consequently, λ=const and μ=const). In this
case, the motion follows a circular orbit with constant angular velocity Ω¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
γ1=r3
αþγ2=r3
π
pin the parallel yOz plane, and the following condition is satisfied
Fig. 7 Periodic orbit in the inertial frame at Nλ=3,Mλ=5,T= 57.78 days (the first case)
The Journal of the Astronautical Sciences

∂U/∂x=0:
r2
α−r2
πþ4c2

γ1
r3
α
þr2
α−r2
π−4c2

γ2
r3
π
¼0ð32Þ
After solving Eq. (32)withrespecttorα,wehave.
xπ¼r2
α−r2
π
2cþ1
2e
x1þe
x2

vπ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r2
π−xπ−e
x2

2
rΩð33Þ
The fourth generating solution (ellipse with the foci located at points m1and m2)for
planar orbits around the Earth–Moon system is obtained when λ=1+rπ/cand w=±π/
2:
Fig. 8 Periodic orbit at Nλ=3, Mλ=5, T= 57.78 days (the first case)
The Journal of the Astronautical Sciences

xπ¼e
x2þrπvπ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
cλ−1
λ
γ1
1þλðÞ
2þγ2
λ−1ðÞ
2
!
v
u
u
tð34Þ
The fifth generating solution (segment of a hyperbola with the foci located at points m1
and m2) for Lyapunov’s planar orbits in the vicinity of L1is obtained when μ=1−rπ/c
and w=±π/2 (the hyperbola representing the passage near the point m2):
xπ¼e
x2−rπvπ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
c
1
μ−μ
γ1
1þμðÞ
2−γ2
1−μðÞ
2
!
v
u
u
tð35Þ
The sixth generating solution for planar orbits around the Earth–Moon system [32]is
obtained when w=±π/2, λ4=1+rπ/cand μ4= 1 (in this case λ≠const and μ≠const,
λ4and μ4correspond to the rπ). For the given integers Mμand Nλthe solution is
obtained after solving the following equation for μ3:
MλMKαλ
ðÞ−NμKαμ

¼0ð36Þ
Fig. 9 Generating periodic orbit at Nμ=2, Mμ=1(the second case)
The Journal of the Astronautical Sciences

where M¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þμ3
ðÞ1−kλ4
ðÞ
2λ4−kμ3
ðÞ
q,sin
2αλ¼λ4−1ðÞ1þkμ3
ðÞ
2λ4−kμ3
ðÞ
,sin
2αμ¼1þkλ4
ðÞ1−μ3
ðÞ
1−kλ4
ðÞ1þμ3
ðÞand
k¼γ1
λ4þμ3
−γ2
λ4−μ3

=γ1
λ4þμ3þγ2
λ4−μ3

xπ¼e
x2þrπvπ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
c1−μ3
ðÞ−γ1
λ4þμ3
ðÞ1þλ4
ðÞ
þγ2
λ4−μ3
ðÞλ4−1ðÞ

sð37Þ
The seventh generating solution for planar orbits around the Earth–Moon system [32]
is obtained when w=±π/2, λ3= 1 and μ3=1−rπ/c(in this case λ≠const and μ≠
const,λ3and μ3correspond to the rπ). For the given integers Mμand Nλthe solution is
obtained after solving Eq. (36)forλ4:
xπ¼e
x2−rπvπ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
cλ4−1ðÞ γ1
λ4þμ3
ðÞ1þμ3
ðÞ
þγ2
λ4−μ3
ðÞ1−μ3
ðÞ

sð38Þ
Fig. 10 Periodic orbit in the inertial frame at Nμ=2, Mμ=1,T= 17.20 days (the second case)
The Journal of the Astronautical Sciences

Fig. 11 Periodic orbit at Nμ=2,Mμ=1, T= 17.20 days (the second case)
Fig. 12 Periodic orbit at T=10.143 days (the fourth case)
The Journal of the Astronautical Sciences

Graphic Examples of Periodic Orbits around the Moon
Initial conditions for different orbits and their generating orbits are given in Tables 1,2,
3,4,and5. Graphic examples are shown in Figs. 2,3,4,5,6,7,8,9,10,11,and12 for
the generating periodic orbits and for the periodic orbits around the Moon of the Earth–
Moon system in the CRTBP. The equations of motion (1) were integrated using the 4th
order Runge–Kutta method with a fixed step size of 100 s (in the given examples the
integration step length within 10–100 s has a weak influence on the orbit shape). It is
necessary to explore more precise integration methods for more complex orbits. The
orbital period Tis determined from the condition of the integer number of revolutions
around the axis Ox. When solving the problem by the continuation method with the
differential correction scheme (Chap. 3), a slow convergence is found with a sequential
increase in the parameter ω.Herewith,
γ1¼3:98600436233km3=s2γ2¼0:004902800076km3=s22c¼384;400km
rπ¼4;500km
Conclusion
This paper proposes a method for calculating the family of periodic spatial orbits in the
CRTBP. The solution is based on the continuation method with a parameter. The
parameter is the angular velocity of rotation of the attracting bodies around their
barycenter. Periodic solutions of the problem of two fixed attracting masses serve as
generating solutions. Seven possible generating solutions are described. Obtained
periodic orbits are not similar to Kepler orbits (other than NRHO) and have a complex
shape because.
- complex form of generating orbits in the gravitational field of two attracting
bodies;
- orbits beyond the moon’s gravitational sphere of influence (~66,000 km). The
farther and more frequent these exits are, the more complex the orbit is.
Also it is necessary to explore more precise integration methods for more complex
orbits.
Acknowledgments I would like to acknowledge Yuri Ulybyshev of the Space Ballistics Department at the
Rocket-Space Corporation “Energia”for his input and fruitful discussions during paper preparation. I also
acknowledge Maksim Shirobokov, Keldysh Institute of Applied Mathematics, Moscow, for constructive
comments. I would like also to thank Evgeniia Bonnet, Scientific Assistant at the LEARN Center (EPFL,
Switzerland), for her contribution on the design of the figures.
The Journal of the Astronautical Sciences

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