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7 Hill's Restricted Three-Body Problem

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; zT 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; 0T where R

is the constant orbital radius of m. The angular velocity is ! D Œ0; 0; nT , 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







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







3

2



Œ. C







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)



184



8 The Restricted Three-Body Problem



yR C 2Px D



@W

@y



(8.86b)



zR C z D



@W

@z



(8.86c)



where





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.



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