7 Hill's Restricted Three-Body Problem
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8.7 Hill’s Restricted Three-Body Problem
181
problem (Hill 1878). This is a problem of practical importance both in celestial
mechanics and astrodynamics. The most important feature of Hill’s restricted threebody problem compared to the original restricted three-body problem is that the
dynamics of the former are not dependant on the masses, and can hence be used
to any celestial or astrodynamical system in which two masses are much smaller
than the remaining one. We will present some interesting periodic orbits around the
secondary mass that emerge in Hill’s restricted problem.
8.7.1 Equations of Motion
Let m denote the mass of the secondary and M be the mass of the primary. In
previous sections, a barycentric rotating coordinate system, the origin of which was
located at the center of mass of m and M, was used (Fig. 8.1). Here we will use a
similar coordinate system; the only difference compared to the previous one is that
the origin is shifted to the center of m, as shown in Fig. 8.7. This coordinate system
was used by Hill in his lunar theory (1878).
Let r D Œx; y; zT be the position vector of point P.x; y; z/ relative to m, R be the
position of m relative to M, and be the position of P.x; y; z/ relative to M, as shown
in Fig. 8.7. Also we assume that m moves on a circular orbit about M. The equations
of motion of the point P are
d2
D
dt2
k2 mr
krk3
k2 M
k k3
(8.76)
̂
( , , )
̂
̂
Fig. 8.7 Coordinate system for Hill’s restricted three-body problem
182
8 The Restricted Three-Body Problem
Substituting
D R C r into Eq. (8.76) yields
d2 r
D
dt2
k2 mr
krk3
k2 M.R C r/
kR C rk3
d2R
dt2
(8.77)
The gravitational acceleration of m relative to M is given by
d2R
D
dt2
k2 MR
kRk3
(8.78)
We now substitute Eq. (8.78) into Eq. (8.77), and write R D ŒR; 0; 0T where R
is the constant orbital radius of m. The angular velocity is ! D Œ0; 0; nT , where
n2 D k2 .M C m/=R3 . Using Eq. (8.1) and substituting for the gravitational terms
yields
xR
2Py
n2 x D
C
yR C nPx
n2 y D
zR D
k2 mx
k2 M.x C R/
Œx2 C y2 C z2
3
2
3
Œ.x C R/2 C y2 C z2 2
k2 M
R2
(8.79a)
k2 my
k2 My
Œx2 C y2 C z2
3
2
3
(8.79b)
3
(8.79c)
Œ.x C R/2 C y2 C z2 2
k2 mz
k2 Mz
Œx2 C y2 C z2
3
2
Œ.x C R/2 C y2 C z2 2
To obtain a normalized set of equations, we divide the position components by R,
so that
D
y
z
x
; ÁD ; D
R
R
R
(8.80)
and time is normalized by n. A new dimensionless gravitational parameter
defined as
D
m
mCM
is
(8.81)
Thus, the velocity and the acceleration are normalized as
P D xP ; R D xR
nR
n2 R
(8.82)
8.7 Hill’s Restricted Three-Body Problem
183
We now rewrite Eq. (8.79a) into
R n2 R
2Án
P 2R
k2 m C k2 M
R
R3
. 2 C Á2 C 2 /
n2 R D
3
2
.1
Ä
1/2 C Á2 C
.
C
/. C 1/
1
3
2
2
k2 m C k2 M
R
R3
.k2 m C k2 M/
R2
(8.83)
If we substitute n2 D k2 .m C M/=R3 into Eq. (8.83), we obtain the normalized
equation. In the same manner, we can manipulate the other two equations and get a
set of normalized equations,
.1
R D 2ÁP C
Œ
2
C Á2 C
2
3
2
/. C 1/
Œ. C 1/2 C Á2 C
3
2 2
C1
ÁR D
RD
(8.84a)
2P C Á
Œ
2
C
Á2
Á
Œ
C
2
.1
C Á2 C
3
2 2
2
3
2
Œ. C
/Á
Œ. C 1/2 C Á2 C
.1
/
1/2
C Á2 C
3
2 2
2 32
(8.84b)
(8.84c)
8.7.2 Hill’s Equations of Motion
As noted previously, m
M and hence
1 and moreover,
fact, apply the following transformation to Eqs. (8.84),
D1
C
1
3
x
! 0. To use this
(8.85a)
ÁD
1
3
y
(8.85b)
D
1
3
z
(8.85c)
When we substitute Eqs. (8.85) into Eqs. (8.84) and let
equations (Hénon 1969),
xR
2Py
3x D
@W
@x
! 0, we obtain Hill’s
(8.86a)
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8 The Restricted Three-Body Problem
yR C 2Px D
@W
@y
(8.86b)
zR C z D
@W
@z
(8.86c)
where
WÁ
1
1
D p
2
r
x C y2 C z2
(8.87)
Obviously, Eqs. (8.86) are independent of . Another interesting observation is
the location of the Lagrangian collinear points; In Hill’s model L3 ! 1 and
1
.L1 ; L2 / D ˙. 13 / 3 D ˙0:69336. The Jacobi constant is given by
D 3x2 C z2 C
2
r
.Px2 C yP 2 C zP2 /
(8.88)
8.7.3 Families of Periodic Orbits
Periodic orbits about the smaller primary m can be found using a numerical search.
The question is which initial conditions to choose in the six-dimensional state space.
To simplify the problem, we can eliminate some of the parameters that need to be
found.
One possibility is to search for orbits only in the xy plane. This reduces the
problem into a four-dimensional search. We can simplify the search further if we
choose to find only symmetrical orbits with respect to the x-axis. In this case we
know that the orbit must intersect the x-axis, so we can define the initial condition in
the intersection point, x0 . Also, symmetry implies that the velocity at the intersection
point will only be in the y direction. Thus, looking for symmetric orbits simplifies
the problem to finding x0 and yP 0 . It is generally more convenient to use the Jacobi
integral instead of yP 0 .
In order to find orbits in the planar case, we need to know the value of the Jacobi
constant, , and the initial condition x0 , where the orbit crosses the x axis with
yP > 0. For that point we have y0 D 0; xP0 , and yP0 are found from Eq. (8.88).
It is convenient to represent an orbit by a point in the .; x0 / plane. In Fig. 8.8,
we depict the characteristic families of periodic orbits, which are denoted by
a; c; f ; g; g0 ; g02 ; g03 (Hénon 1969, 1970).
The hatched areas in Fig. 8.8 are “forbidden”, i.e. areas in which yP 0 < 0. All the
orbits found are symmetric with respect to the x axis because xP 0 D 0.
We can use the Jacobi constant as an energy measure, where near m we have
! C1. From Fig. 8.8, it is evident that at the energy level of the L1 ; L2 points,
there are periodic orbits are closer to the secondary than L1 . To find larger orbits, we
8.7 Hill’s Restricted Three-Body Problem
185
Fig. 8.8 General map of periodic orbits in Hill’s problem
need to change the velocity of P, and thereby change the value of to the value of
the desired orbit.
Families a and c
Families a and c include periodic orbits about the collinear points L1 and L2 . Some
orbits of Family a can be seen in Fig. 8.9. Table 8.2 displays the Jacobi constant ,
the initial condition x0 and the orbital period. The orbits of Family c are symmetric
with respect to Family a as can be seen in Fig. 8.10 and in Table 8.3.
Family f
Family f includes distant retrograde orbits about the secondary. These orbits are
stable and symmetric with respect to the x axis (Fig. 8.11). From Fig. 8.8 it is seen
that these orbits can be found for any < 0. For ! 1 the orbits become
ellipses centered at the secondary with a major to minor axis ratio of 2 as can be
seen in Fig. 8.12. Orbits that are very close to the secondary become circles due to
the decreasing gravitational effect of the primary. Figure 8.11 shows some of the
characteristic orbits in this family. The orbit parameters are given in Table 8.4.
186
8 The Restricted Three-Body Problem
3
2
y
1
0
−1
−2
−3
−4
−3
−2
−1
0
1
2
3
4
5
x
Fig. 8.9 Family a in Hill’s problem
Table 8.2 Family a: Jacobi constant and initial conditions
4:327
4:2
4
3:5
3
2:5
2
1:5
x0
0:69336
0:62698
0:5802
0:4958
0:42585
0:36181
0:30114
0:24307
T
3:0513
3:084
3:172
3:288
3:44
3:464
3:928
4:32
1
0.5
0
0.5
1
1.5
2
x0
0:18797
0:13756
0:09515
0:06402
0:04383
0:03121
0:02314
T
4:88
5:6
6:3504
6:95
6:95
7:37
7:656
8.7 Hill’s Restricted Three-Body Problem
187
1.5
1
y
0.5
0
−0.5
−1
−1.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
x
Fig. 8.10 Family c in Hill’s problem
Table 8.3 Family c: Jacobi constant and initial conditions
4.327
4.2
4
3.5
3
2.5
2
1.5
x0
0:69336
0:74757
0:77522
0:81245
0:83714
0:8597
0:88586
0:92204
T
3.0513
3.084
3.172
3.288
3.44
3.646
4.32
4.32
1
0.5
0
0.5
0.5
1
1.5
2
x0
0:9778
1:0677
1:2082
1:3992
1:3992
1:615
1:8304
2:0352
T
4.88
6.3504
6.3504
6.95
6.95
6.95
7.37
7.656
Family g
Family g can be divided into several groups according to the value of . For
4:5, we get a group of stable distant prograde orbits. They can be seen in Fig. 8.13.
These orbits are almost circular in form and resemble the orbits of the two-body
problem.
For 2 Ä < 4:5 we obtain unstable distant prograde orbits. Figure 8.14 shows
that the orbits become ellipses, and in the critical value D 2 the orbit resembles a
parabolic orbit, in which the velocity reduces to zero at infinity.
When < 2, we obtain unstable orbits as seen in Fig. 8.15. These orbits can
be used as transfer orbits. In the WIND mission, these orbits were used to get to
188
8 The Restricted Three-Body Problem
6
4
2
y
0
−2
−4
−6
−6
−4
−2
0
2
4
6
x
Fig. 8.11 Family f in Hill’s problem
2.2
2
Axis Ratio
1.8
1.6
1.4
1.2
1
0.8
−30
−25
−20
−15
−10
Γ
Fig. 8.12 Family f major to minor axis ratio
−5
0
5
8.7 Hill’s Restricted Three-Body Problem
Table 8.4 Family f : Jacobi
constant and initial conditions
6
5.5
5
4.5
4
3.5
3
2.5
2
1.5
189
x0
0:14779
0:15888
0:17169
0:18661
0:20421
0:22523
0:25071
0:28212
0:32163
0:37252
T
0.3394
0.3794
0.422
0.474
0.537
0.616
0.714
0.84
1.02
1.222
0
0.1
1
0.5
0
0.5
1
1.5
2
2.5
3.5
x0
0:43991
0:53182
0:65966
0:83185
1:034
1:2341
1:4168
1:5817
1:8705
T
1.52
1.938
2.526
3.292
4.08
4.7
5.12
5.38
5.694
0.25
0.2
0.15
0.1
y
0.05
0
−0.05
−0.1
−0.15
−0.2
−0.25
−0.3
−0.2
−0.1
0.2
0.3
x
Fig. 8.13 Family g with
4:5: Stable distant prograde orbits
and from a distant retrograde orbit.1 As mentioned previously, in Hill’s problem the
Lagrangian points L1 and L2 are located at x D ˙0:69336, so the orbits of family
g may reach to about five times this value. The initial conditions of these orbits are
displayed in Table 8.5.
Family g0
The orbits in family g0 pass close to the primary; they are unstable. The orbits are
shown in Figs. 8.16 and 8.17. There are two branches of the Family that split from
family g at the critical point D 4:5.
1
See http://wind.nasa.gov/.
190
8 The Restricted Three-Body Problem
0.8
0.6
0.4
y
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−0.8
−0.6 −0.4 −0.2
0
x
0.2
0.4
0.6
0.8
Fig. 8.14 Family g with 2 Ä < 4:5: Unstable prograde orbits
2.5
2
1.5
1
y
0.5
0
−0.5
−1
−1.5
−2
−2.5
−3
−2
−1
0
x
Fig. 8.15 Family g with < 2: Unstable prograde orbits
1
2
3
1
8.7 Hill’s Restricted Three-Body Problem
Table 8.5 Family g: Jacobi
constant and initial conditions
191
6
5.5
5
4.75
4.5
4.25
3.75
3.5
3
2.5
2
x0
0.19489
0.21788
0.247
0.26435
0.2835
0.30343
0.33178
0.33173
0.3069
0.26679
0.22168
−0.4
−0.3
T
0.62
0.734
0.94
1.06
1.226
1.728
2.054
2.4
3.02
3.6
4.22
1.5
1
0.5
0
0.5
1
1.5
2
2.5
3
x0
0.17545
0.13032
0.08902
0.05587
0.03392
0.02104
0.01356
0.00909
0.006313
0.004523
T
4:95
5:9
7:094
8:44
9:52
10:26
10:72
11:02
11:26
11:42
0.3
0.2
y
0.1
0
−0.1
−0.2
−0.3
−0.7
−0.6
−0.5
−0.2
−0.1
0
0.1
x
Fig. 8.16 Family g0 : Unstable orbits
The trajectory close to the secondary is equivalent to a hyperbolic flyby as seen
in the two-body problem. Figure 8.17 demonstrates a “slingshot” effect. A particle,
e.g. a spacecraft, will be diverted into an escape orbit with respect to a planet due to
this effect.
The orbits in this family display a new kind of motion. The orbits revolve around
the secondary only once, and can be characterized by two points x01 and x02 along
the orbit as noted in Table 8.6. The two branches of this family can be seen in
Fig. 8.8, where the branch of Family g0 splits from Family g at the critical point
D 4:5.