1 Nomenclature
1, 2 = gravitational constant of two
primary bodies
di3 = distance between the primary bodies
= common angular velocity
n = mean motion
(x, y) = position of mass-free body
i = ?1
U = potential in rotating frame in the
restricted three body problem
?U
= potential in rotating frame in the
rod-connected restricted three
body problem
(x, y) = perturbation around libration points
2 Introduction
In 2005, the hHayabusah spacecraft arrived at a
near-Earth asteroid hItokawah and provided us its
1
great important scientific results.
From this mission, it was found that Itokawa
has been many peculiar characteristics: low bulk
density, high porosity, rough surface with studded
boulders, etc. Those characteristics indicate that
Itokawa is a rubble pile asteroid. Also, a previous
paper showed that Itokawa was considered to
be a typical Near Earth Object [1]. It means that
studying the formation of Itokawa is great important
to uncover the planetary evolution scientifically.
Thus, in this paper, by using celestial mechanics,
this study focuses on how Itokawa has been
shaped.
According to another previous paper, it was
shown that Itokawa had been formed by the contact
of two core asteroids which had belonged to a
preexisting parent body [2]. Thus, the analysis is
divided into two processes: the case (1) before and
(2) after the contact of the two core asteroids. In
the case before the contact, it is considered many
large fragments to be re-aggregated by each gravitational
force. Since those large fragments assumed
to be mass particles, the motion is explained by
the Multi-Body Problem. In the case after the contact,
since the fragments are much smaller than the
core asteroids, they can be assumed to be mass-free
bodies. In this case, the core asteroids can be considered
to be a dumbbell shaped body, modeled as
two primary bodies which are connected with each
other by a rigid stick.
The case after the contact can be discussed more
easily than the case before the contact, because, in
the case after the contact, it is not necessary to discuss
the gravitational effects of each fragment. In
this case, to specify the motion of fragments, it is
necessary to discuss the stability of them. It is because
whether this system has stable points or not
causes to change the motion of fragments, and its
trait is an important factor to form Itokawa shape.
From these reasons, this paper focuses on the case
after the contact, and describes the position of equilibriums
and the stability of those positions around
the dumbbell shaped body.
In this problem, in addition to gravitational
forces from two primary bodies and centrifugal
forces, the force from the stick affects the motion
of two primary bodies. In other words, the force
can fix the distance between two primary bodies
regardless of any type of the common angular velocities.
Therefore, this problem has two degrees of
freedom. In this paper, the stability is specified in
any type of two non-dimensional parameters: the
non-dimensional angular velocity; and the mass ratio.
3 The restricted three-body
problem
Before indicating the Rod-Connected Restricted
Three-Body Problem, we introduce the normal Restricted
Three Body Problem (RTBP). This theory
makes it possible to discuss the stability in the Rod-
Connected Restricted Three-Body Problem. First,
the positions of equilibriums are discussed. Then,
the stability of motions around the equilibriums is
specified.
Figure 1: The libration points in the RTBP system;
Three equilibriums exist along the x axis and other
two equilibriums are at at the apex of an equilateral
triangle.
3.1 Equation of motion
Let us consider a position of a mass-free body
subject to the gravitational attraction from the primary
bodies which are rotating with a constant angular
velocity n, known as the mean motion. In this
paper, two-dimensional surface is discussed because
of its simplification of this discussion. The origin is
the center of mass, x axis is defined as a line passing
through both of the primary bodies, and y axis is
perpendicular to x axis. This system, rotating with
the primary bodies, is shown in Figure 1. Let us
define Ux as,
Ux = U
x
(a)
In the rotating frame, the equation of motions are
written by,
xN ? 2ny? = Ux (1.1)
yN+ 2nx? = Uy (1.2)
where,
U = 1
d13
+ 2
d23
+
1
2n2 (
x2 + y2)
(2)
d13 =
(x ? x1)2 + y2 (3.1)
d23 =
(x ? x2)2 + y2 (3.2)
2
subject to
1 + 2 = 1 (4.1)
d = 1 (4.2)
n = 1 (4.3)
In these equations of motion, there is only one
degree of freedom. Thus, once one variable is determined,
the motion of the mass-free body can be
described utterly.
3.2 Equilibriums
Because a mass-free body is not attracted by any
forces at equilibriums, the condition at equilibruims
is described as,
Ux = Uy = 0 (5)
According to this condition, there are five equilibriums
in this system. These points are known
as the libration points and each point is defined as
from L1 to L5. Three of them lie on the x axis,
known as the collinear libration points. Other two
points are at the apex of an equilateral triangle with
a base formed by the line joining the primary bodies,
known as the triangular libration points. These
are shown in Figure.1.
Let us define two non-dimensional parameters as,
= d23
d
, = 2
1
(6)
Speaking generally, the position of the collinear libration
points can be expressed only by using sum
of series. However, by using these parameters,
the relational expressions which are satisfied at the
collinear libration points can be obtained.
The relational expressions of the collinear libration
points are described as,
2
(1 ? )2
(1 ? )3 ? 1
3 ? 1
= (7.1)
2
(1 ? )2
(1 + )3 ? 1
3 ? 1
= ? (7.2)
2
( ? 1)2
( ? 1)3 ? 1
3 ? 1
= ? (7.3)
Eq.(7.1) shows the position of L1, Eq.(7.2) shows
the position of L2, and Eq.(7.3) shows the position
of L3.
On the other hand, the position (x0, y0) of L4 and
L5 is written by,
(x0, y0) =
(
1 ?
2(1 + ) ,}
3
2
)
(8)
3.3 Stablity of the libration points
First, we difine Uxx as,
Uxx = 2U
x2 (b)
In the vicinity of the libration points, the linearized
equations of motion is described as,
xN ? 2ny? = Uxx0x + Uxy0y (9.1)
yN+ 2nx? = Uxy0x + Uyy0y (9.2)
In these equations, Ths subscripts h0h means the
respect to the libration points. Although in our system
of units n = 1, we will retain n in the equations
to emphasize that all the terms in the equations of
motion are acceleration.
Suppose that x = Aet, y = Bet, and =
+ i in Eq.(9)s. Then, we obtain,
x? = Aet, y? = Bet (10.1)
Nx = A2et, Ny = A2et (10.2)
By substituting Eq.(10)s into Eq.(9)s, Eq.(9)s are
rewritten as,
[
2 ? Uxx0 ?2n ? Uxy0
2n ? Uxy0 2 ? Uyy0
] [
x
y
]
= 0 (11)
When the solution of Eq.(9)s (x, y) are not zeros,
the first matrix on the left-hand side of Eq.(11) is
satisfied as,
det
[
2 ? Uxx0 ?2n ? Uxy0
2n ? Uxy0 2 ? Uyy0
]
= 0 (12)
Then, Eq.(12) is rewritten by,
4 +
(
4n2 ? Uxx0 ? Uyy0
)
2+
Uxx0Uyy0 ? U2
xy0 = 0 (13)
Now let us define 4n2?Uxx0?Uyy0 and Uxx0Uyy0? U2
xy0 = 0 as,
a = 4n2 ? Uxx0 ? Uyy0 (14.1)
b = Uxx0Uyy0 ? U2
xy0 (14.2)
Thus, Eq.(13) is rewitten by,
4 + a2 + b = 0 (15)
Then, from Eq.(15), 2 is,
2 = ?a } a2 ? 4b
2
(16)
When the motion of the mass-free body always
stays in the vicinity of the libration points, 2 < 0
must be satisfied. By using a and b, this condition
can be described as,
a > 0 (17.1)
b > 0 (17.2)
a2 ? 4b > 0 (17.3)
By using the condition of stability, described as
Eq.(17)s, the stability of the libration points can
be obtained. The results are shown in Table 1.
These solutions are shown in [3], [4].
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Table 1: The stability of L1 through L5
Equilibrium L1CL2CL3 L4CL5
Stable region Not exist > 24.96
or < 0.0401
4 The Rod-Connected restricted
three-body problem
The Rod-Connected Restricted Three-Body
Problem (RCRTBP) is described as a model characterized
by three bodies; two primary bodies which
are connected with each other by a rigid and massfree
stick, and a mass-free bodies. This model is
shown in Figure.2.
Figure 2: RCRTBP system for the dumbbell shaped
body; There are also five equilibriums. Three equilibriums
exist along the x axis, and the other two
equilibriums are along the perpendicular bisector
between the two masses.
4.1 Equation of motion
In this problem, there are two degrees of freedom
because of the connection of the primary bodies. In
other words, even if only one parameter is determined,
the motion of the mass-free bodies can not
be described. In this paper, the angular velocity ,
which is different from the mean motion, is newly
defined. The way how to set the rotating frame in
this system is the same as that in the RTBP system.
The equations of motion are described as,
Nx ? 2 ? y = ?Ux (18.1)
Ny + 2 ? x = ?Uy (18.2)
where,
?U
= 1
d13
+ 2
d23
+
1
22 (
x2 + y2)
(19)
These equations of motion are also subject to
Eq.(4)s
The difference from the equations of motion in
the RTBP system is that the angular velocity is
altered from the mean motion n to because of
increasing the degree of freedom.
4.2 Equilibriums
Like the motion in the RTBP system, at the quilibriums,
the condition at the equilibriums is described
as,
?U
x = ?Uy = 0 (20)
From Eq.(20), there are five equilibriums. Three
of them lie on the line passing through the primary
bodies, and other two points are along the
parpendicular bisector between the primary bodies.
In this paper, these points are defined as the
pseudo libration points and are expressed by from
L1 to L5. In this paper, from L1 to L3 are defined
as the pseudo collinear libration points, and L4 and
L5 are described as the pseudo bisector libration
points. These points are shown in Figure.2.
Let us define a new parameter as follows,
= 2
n2 (21)
By using this parameter, we can obtain the relational
expressions of the pseudo collinear libration
points as,
2
(1 ? )2
(1 ? )3 ? 1
Ń3 ? 1
= (22.1)
2
(1 ? )2
(1 + )3 ? 1
Ń3 ? 1
= ? (22.2)
2
( ? 1)2
( ? 1)3 ? 1
Ń3 ? 1
= ? (22.3)
Eq.(22.1) shows the position of L1, Eq.(22.2) shows
the position of L2, and Eq.(22.3) shows the position
of L3. Also, the position of the pseudo bisector
libration points (x0, y0) can obtained as,
x0 =
1 ?
2(1 + )
(23.1)
y0 = }
[(
1
)2
3
?
1
4
]1
2
(23.2)
L4; y0 > 0, L5; y0 < 0
From Eq.(22)s and Eq.(23)s, it is found that the position
of the pseudo libration points are dependent
on two degrees of freedom.
4.3 The stability of the pseudolibration
points
Like the linearized equations of motion in RTBP
system, those in the RCRTBP system are described
as,
Nx ? 2փ ? y = ?Uxx0x + ?Uxy0y (24.1)
Ny + 2փ ? x = ?Uxy0x + ?Uyy0y (24.2)
4
From Eq(24), the stability conditions in this system
are written by,
42 ? ?Uxx0 ? ?Uyy0 > 0 (25.1)
?U
xx0?Uyy0 ? ?U 2
xy0 > 0 (25.2)
(
42 ? ?Uxx0 ? ?Uyy0
)2
?
4
(
?U
xx0?Uyy0 ? ?U 2
xy0
)
> 0 (25.3)
Those stability conditions are obtained by the same
processes which are mentioned in the RTBP system.
From Eq.(25)s, it is found that while L2 and L3
are always unstable in any type of parameters, L1,
L4, and L5 have stable regions under the certain
condition.
The details about L1, L4, and L5 are reported in
the following sections. First, the stability of L4 and
L5 is discussed. Then, that of L1 is specified.
4.3.1 The stability at L4 and L5
To begin with, let us define as the angle between
the line joining the two primary bodies and d23.
This is shown in Figure.3.
Figure 3: Definition of ; is the angle between the
line joining the two primary bodies and d23
Let us consider each equation in Eq.(25)s specifically.
First, the left-hand side of Eq.(25.1) is described
as,
4 ? (Uxx0 + Uyy0) = > 0 (26)
Eq.(26) indicates that Eq.(25.1) is always satisfied
at L4 and L5.
Second, the left-hand side of Eq.(25.2) is rewritten
by,
Uxx0Uyy0 ? U2
xy0 =
9ʃ
2
3
(1 + )2
((
1
)23
?
1
4
)
(27)
Then, by substituting the right-hand side of Eq.(27)
into Eq.(25.2), the condition of is described as,
< 8 (28)
Moreover, using Eq.(23.2), Eq.(28) is rewritten by,
= 8 cos3 (29)
Eq.(29) also means that
?= 0 (30)
At last, by substituting Eq.(29) into Eq.(25.3),
we can obtained the third condition defined as,
sin2 cos2 <
1
144
(1 + )2
(30)
Figure 4 shows the stability at the pseudo bisector
libration points. The stable regions are below
the boundary curve. In this region, the small fragments
can always stay in the vicinity of L4 and L5.
Figure 4: The stability condition at L4 and L5; The
stable regions are below the boundary curve.
4.3.2 The stability at L1
The condition of the stability at L1 is also obtained
by the same process mentioned in the above
section.
Let us define as,
=
1
1 +
(
3 +
1
(1 ? )3
)
(32)
By using this parameter, the condition at L1 can
be described.
First, from Eq.(22.1), is described as,
=
1
( + 1) ? 1
(
2 ?
1
(1 ? )2
)
(33)
By substituting Eq.(32) and Eq.(33) into Eq.(25)s,
Eq.(25)s are rewritten by,
2 ? > 0 (34-1)
? > 0 (34-2)
< 9/8 (34-3)
Eq.(34-1) is obtained from Eq.(25.1), Eq.(34-2) is
procured by using Eq.(25.2), and Eq.(34-3) is the
equivalent condtion of Eq.(25.3).
5
Therefore, from Eq.(34)s, the stability condition
is described as,
< <
9
8 (24)
Figure 5 shows the stability at L1. The stable regions
are between the two boundary curves. Also,
we indicated in this figure the stability condition in
RTBP system. The line of = 1 indicates it. According
to this figure, L1 in RTBP system is always
unstable.
Figure 5: The stability condition at L1; The stable
regions are between the two boundary curve
5 Conclusion
This paper focused on the stability in the Rod-
Connected Restricted Three Body Problem for the
dumbbell shaped body. The stability of this model
was specified by applying the Restricted Three
Body Problem. By using the fact that there are no
forces which affect the motion of the mass-free body
at the equilibriums, five pseudo libration points
were obtained. Three of them which were defined
as from L1 to L3 lie on the x axis, and other two
equilibriums which were expressed by L4 and L5
are along the perpendicular bisector between the
primary bodies. Then, by linearizing the equation
of motion, the linear stability of the motion of the
mass-free body in the vicinity of the pseudo libration
points was investigated. According to this investigation,
L2 and L3 are always unstable, L1, L4,
and L5 have stable regions. In this problem, because
there are two degrees of freedom in the Rod-
Connected Restricted Three Body Problem, the arbitrary
property of the position of those pseudo libration
points occurs. For example, L4 and L5 are
not necessary to be at the apex of an equilateral
triangle, mentioned in the Restricted Three Body
Problem. Thus, in this paper, the position of those
pseudo libration points and the stability at them
were indicated by three parameters: , , and .
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