Smithsonian

Contributions to Astrophysics

VOLUME 3, NUMBER 7

PERIODIC ORBITS OF A PLANETOID PASSING CLOSE TO

TWO GRAVITATING MASSES

by ROBERT R. NEWTON

SMITHSONIAN INSTITUTION

Washington, D. C.

1959
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Periodic Orbits of a Planetoid Passing Close

to Two Gravitating Masses1

By ROBERT R. NEWTON

Introduction

This paper concerns a special case of the

restricted problem of three bodies: Two masses

(E and M) move in circular orbits under their

mutual gravitation, while a third body (P,

called the planetoid) of negligible mass (that

is, it is acted upon by E and M, but does not

perturb their motion) moves in the same plane.

We shall assume that the mass of E is much

greater than the mass of M. We are interested

in periodic motions of the planetoid, meaning

that the position and velocity of P relative to

E and M are periodic functions of time, but not

requiring that the motion be periodic as seen

from a stationary coordinate system.

It is clear that the motion will be periodic in

the stationary system if the period of the relative motion is a rational multiple of the rotational period of E and M. Further, with a motivation which is probably obvious, we are seeking

periodic orbits in which P passes close to both

E and M.

Periodic solutions of the restricted threebody problem have been studied so extensively

that we cannot attempt a thorough citation of

the literature. Stromgren (1933) presents

many periodic orbits, determined by numerical

integration for equal masses of E and M and

classified according to the type of their motions

about E, M, and the Lagrangian points.

Among these are orbits in which P passes close

to both E and M, but these are of a quite

different sort from those studied here. Darwin

(1911, pt. 1) also gives many samples of periodic

orbits determined for a mass ratio of 10 be1

 This work was supported by the Department of the Navy, Bureau

of Ordnance, under Contract NOrd 7388.

1

 Applied Physics Laboratory, The Johns Hopkins University, Silver

Spring, Md.

tween E and M; among these is one that we

should describe as of type #, class direct.

However, his interest was mainly in satellite

and planetary theory, and he does not seem to

have studied these orbits further.

Our study concerns those periodic orbits that

involve close passage to both E and M and that

are generated in a particular way. Among

other distinguishing features, these orbits all

have periods close to the rotational period of

E and M. As viewed from a stationary coordinate system, they resemble highly eccentric

ellipses, some of which periodically develop

cusps or loops at the end of one axis, and all

of which precess slowly. Other periodic orbits

passing close to E and M of course exist; some

of these have been studied by Egorov (1958).

Also presented are a fuller qualitative description of the orbits, a classification of their types,

and methods of calculating approximate initial

conditions for the orbits. The final section of

this report gives examples of these orbits,

which were determined with the aid of a largescale digital computer.

Zero-order determination of the orbits

It is convenient to choose the unit of mass as

the total mass of E and M, letting n be the mass

of M, and 1—M the mass of E. We take the

unit of length to be the distance between E

and M, and the unit of time to be such that

the rotational period of E and M is 2x. This

choice of units makes the gravitational constant unity, and leaves n as the only parameter

in the Hamiltonian of P.

If M is small, E is nearly stationary, and the

motion of P is dominated by E except when P

is quite close to M, so that neglecting the mass

of M entirely gives an approximate description

69
70 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

of the motion.* Thus, if P is to pass close to

both M and E, its orbit must be nearly an

ellipse focussed on E, with major axis nearly

unity. Since Kepler's Third Law is, in our

units,

(1)

anda«.5 , we must have !Fp« 2T/3. Clearly,

then, if P is in an orbit about E with period

(2)

where a and /3 are relatively prime positive

integers, and a//3 ~ %, the motion of P will be

periodic relative to E and M, and also in a

stationary coordinate system. 2va will be the

complete period of the motion, and /S will be

the number of times P describes its ellipse about

E during one period. If we arrange the phase

of the motion so that P is sometimes close to

M when it is farthest from E, we have a periodic

orbit of the desired sort when JU=O.

If a is precisely .5,!TP=2X/2V2«.3536(2T).

If a//3<.3536, the orbit of P will pass inside

of M; if a/j8>.3536, P may pass either outside

or inside of M, as viewed from E. In the coordinate system rotating with E and M, it is

clear that the orbit has /S lobes, and has a /3-fold

axis of rotational symmetry about the orgin.

If n is small, but not zero, we expect to find

a system of periodic orbits of P, each still corresponding to a fraction aj(3, which we shall

call the type of the orbit, and which describes

the orbit in the limit /t=0. /3 will still give the

number of lobes of the orbit in the rotating

coordinate system. However, the orbit will no

longer have a /3-fold rotational axis, because that

lobe of the orbit in which P passes close to M

will be more perturbed than the other lobes, so

that the lobes will not all have the same shape.

Further, the perturbation of M will cause the

orbit of P to rotate in a stationary coordinate

system, so that the period will not be 2va, and

will depart farther from 2va the more closely

P approaches M.

We consider explicitly only orbits of type }£,

and for which P is outside of M at its closest

approach to M. Orbits corresponding to frac-

* The argument of this section is a special case of an argument used by

Poineare (1892, chap. 3). The orbits we are seeking, while apparently

not studied explicitly by Poincare, represent extreme examples of what

he calls periodic orbits of the first kind.

tions other than ){ can presumably be studied

by similar methods. For this type, the unperturbed major axis of the orbit of P about E

is 1.26, so that 0.26 is the sum of the "miss

distances." Closer approaches can be achieved

by other types. Also we consider only orbits

for which the distance E-P is a relative maximum when the distance M-P is a minimum.

In studying the orbits, we find two coordinate

systems of most interest. One of these, xy, has

its origin at the center of mass, and rotates

with E and M (see figure 1). M is taken to lie

on the x-axis at the point 1—M, 0. while E lies

at the point — n, 0. The other coordinate

system XY is fixed, and coincides with xy at t=

0. Since the rotational period of E and M is

2T, xy has an angular velocity of unity with

respect to XY.

We shall choose t—0 at the instant when P

is closest to M, and hence at a relative maximum4

 from E for the orbits being studied. We

have therefore a standard set of initial

conditions:

x(0)=xQ,

2/(0)=0,

x(0)=0, 1

(3)

-I

M ,

FIGURE 1.—The rotating coordinate system xy.

* We say a relative maximum because P attains the same distance

from E on the other lobe of the orbit when ji=0, and, as we shall see,

attains an even greater distance on the other lobe when we Include the

perturbation of M.
PERIODIC ORBITS OF A PLANETOID 71

Only one of x0 and y0 is independent if the

orbit is to be periodic; we choose this one to

be x0, and write t/0 as a function of z0.

For the chosen value of a/0, there are two

classes of orbit when n=0, and hence y0 is a

double-valued function of x0. For one of these

classes, the sense of rotation of P with respect

to XY is direct (that is, with the same sense

as M); for the other class, the rotation of P is

retrograde. We shall call these classes "direct

about E" and "retrograde about E." There is

one member of each class for each value of

£0*0.26, and it is a trivial exercise to calculate

yo(%o) for each class when /L*=0. When n^O,

the "direct about E" class divides into three

subclasses (p. 73).

The members of each class having the same

x0 are identical in XY, except for the sense of

rotation, when /x=0. However, they present

quite different appearances in xy, as shown in

figure 2. One point should be noted particularly in figure 2: both orbits are retrograde in

xy when P is close to M. If the motion of P

were direct in xy at this point, the velocity of

P in XY would be greater than that of M, and P

could not come closer to E than M does.

(With HT*O, so that P could lose energy to M,

it might go into an orbit close to E. We shall

not pursue this possibility.)

In the next section we give a crude method of

allowing for the perturbation produced by M,

which does not allow us to calculate the alteration in period from 2v, but is sufficient to show

the subdivision of the class "direct about E."

Following that, we classify the orbits and give

a more accurate perturbation method, which

allows estimating the period and affords an

accurate calculation of the function yo(xo).

Crude perturbation calculation

The Hamiltonian of the planetoid P is known

 6

to be

(4)

In this, pB and pM are the distances from P to E

and M, respectively. H does not contain the

time explicitly, therefore it is an integral of the

motion, and is generally called Jacobi's integral.

However, H is not the total energy of P. The

total energy of P is not a constant of the

motion, but can be changed by interaction with

EorM.

If we change the sign of y and simultaneously

change the sign of t, the Hamiltonian is unaltered. Since our initial conditions (3) are

unaltered by the same transformation, it follows

that, for our orbits, x(—t)=x(t) and y(—t) =

—y(t). Our orbits are therefore symmetrical

about the x-axis, and we need study the orbits

only from t=0 until x has gone through its full

range of values.

With this background remark, the method of

the present section is based upon the following

observation: The perturbation of M is significant only over a short distance, and hence only

for small values of t, if |yo| is reasonably large.

* See, for example, Whittaker (1944), sect. 162, which summarizes all

the earlier work on the restricted three-body problem needed for the

purposes of this paper.

FIGURE 2.—General appearance of the periodic orbits in the

rotating system. Line E-M rotates counterclockwise as

seen from an inertial system, a, "Direct" orbit; b, "retrograde" orbit.
72 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

During this small time interval, the motions of

M and P in AT are nearly rectilinear, parallel,

and oppositely directed, and the speed of M is

constant. Therefore we can, during this time,

estimate the perturbation produced by M by

taking A/ to be the origin of an inertial coordinate system. Because of the symmetry of the

motion, we can start the calculation at the

nearest approach of P to M, that is, at t=0.

If Po.m denotes the initial speed of P with

respect to M, and vf-m its speed after "escape

from M," we have

(5)

where pjfo=£o+M—1> and denotes the initial

distance of P from M. Letting Yf denote the

value of Y after "escape," and using (5), the

reader can readily verify that y0 and Yf are

related by the equation

This curve (II in fig. 3) is given approximately

by the condition ^o . m=nlpuo or

t/o= (6)

We can calculate the motion of P subsequent

to escape by assuming that it travels only in

the field of E, with initial conditions:

(7)

Before using this method to find yo(^o), let

us first use it to draw curves in the fa, y0) plane

which delimit the regions containing various

classes of periodic orbit. These curves will

apply to orbits with any value of a/P, but we

shall continue, for simplicity, the restriction

that aro>l — MFirst, there is the curve of "collision with

E": for any Zo, there is a value of y0 such that P

will collide with E. We can find an approximation to this curve by letting Yf=0 in equation

(6). The resulting curve, if we use the value

M= 1/82.45, is shown as I in figure 3. (This

value of n corresponds approximately to a

planetoid moving in the earth-moon system.)

The curve of "collision with E" divides orbits

direct about E from those that are retrograde

about E.

Next, there is the curve of "escape from M,"

which divides orbits permanently encircling

only M from those that escape the vicinity of M.

— 2/o= PM o (8)

—y0 must exceed this value for "escape from

M."

Next, there are the curves of "escape from

E and M." These curves are not as accurately

defined as the preceding ones; they are found

by assuming, contrary to fact, that the energy

of P is conserved, and requiring that this

energy, as measured in XY, be zero. Since the

energy acquired by P in its first near passage

to Mi s small, these curves give approximate

limits to the condition that P shall not escape

the system, at least during the first circumnavigation of the system. More precisely

defined curves could be used, but these have

an intuitive value and suffice for our purposes.

The curves in question are given by the

equation

-2/0= P » ± V2 (M/PJTO) + [2 (1 -M)/P*O] • (9)

(The curve using the minus sign is curve III

in fig. 3; the other, curve IV, has no significance

within the range of values in fig. 3.)

Finally, there is the curve of "zero initial

velocity" (curve V in fig. 3). If

—2/o—*-o> v1"/

it is clear that Fo is zero; since X is also zero

at this time, P has initially no velocity in XY.

Orbits that lie (i.e., for which the initial

conditions lie) between curves I and III in

figure 3 are "retrograde about E." Since I lies

above V everywhere, orbits that are "retrograde about E" also have a retrograde motion

when near M, and this class has no subdivisions.

Orbits that lie between I and II are "direct

about E." Since curve V passes through this

region, this class of orbit has three subdivisions,

namely, those that lie below, on, and above

curve V.

Classification of orbits

In the full classification that follows, the terms

"retrograde" and "direct" refer to the sense

of the motion as seen in the XY coordinates;

the sense of motion about M is always retro-
PERIODIC ORBITS OF A PLANETOID 73

-Yc

0

\MJX

I

s^_ . (1/2

^ ^ (1/

nr

*==*-—

•

Retrogn

• — .

,

11

Direct

^ = =

w

O.I 0.2 0.3

FIGURE 3.—Regions in which initial conditions must lie for various classes of periodic orbit, pm is

the minimum value of the distance P-M, and j/o is the velocity in rotating coordinates at the closest

approach to M.

grade in xy, and about E is always the same

in xy and XY.

1. Orbits that are retrograde about both E and M. This

class can be designated "retrograde" for brevity.

2. Orbits that are direct about E. These are divided

into three subclasses:

(a) Orbits that are direct when nearest M. This

class can be briefly designated "direct"

without ambiguity.

(b) Orbits of zero velocity when nearest M. This

class can be designated "zero initial

velocity" for brevity.

(c) Orbits that are retrograde when nearest M.

This class can be designated "mixed"

for brevity.

We have not shown yet that orbits of all these

classes exist; however, we shall present examples

of all of them subsequently.

We can now make estimates of yofao) by the

following procedure: We assume that P will

execute an orbit in the field of E, after escape

from M, of semi-major axis a=.63 (for orbits of

type ()0), and passing through the point

(xo, 0). This allows us to calculate the velocity that P must have relative to XY at this

point, and we use this value of velocity for

Yf in equation (6). Yf is taken as positive

for orbits direct about E, and negative for orbits

retrograde about E. This process gives the

two curves marked "00 , a" in figure 3.

These curves terminate at one end where

Pjro=-26, as shown. They do not terminate at

the other end, but we did not calculate them for

PMO^H, because they are almost surely useless

for smaller puoSince the curve for orbits direct about E

crosses the curve of zero initial velocity, we

expect to find orbits of all the classes listed
74 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

above, unless the error in this method is too

large.

A more accurate perturbation calculation

If we transform to polar coordinates r, 6 in the

xy plane, we find (Whittaker, 1944) that the

Hamiltonian becomes:

(11)

The momenta pT and p$ are:

pT=t, p9=r*d[t&n-1(Y/X)]/dt. (12)

That is, pT is the usual radial component of

momentum, while p$ is the angular momentum

with respect to XY, not to xy.

The transformation

'-Pr 2, Pe=P2,

is a contact transformation (Poincare, 1892,

chap. 1; see also Whittaker, 1944, sec. 162),

with Pi, P2 as the momenta conjugate to

Qi, Q2 respectively. P2 is still the angular

momentum with respect to XY, while the other

quantities are related to the ellipse that P

would describe if launched with its instantaneous position and velocity in the field of a unit

mass fixed at the origin. Pi2 is the semi-major

axis of this ellipse, Qt is the mean anomaly of

P in this ellipse, and Q2 is the longitude of the

line of apses, measured from the z-axis. (Hence,

Q2-\-t is the longitude of the line of apses measured from the X-axis.) Since equations (13)

cannot be solved explicitly for r and 6, we cannot write the Hamiltonian explicitly in terms of

the new canonical variables.

At <=0, the line of apses is along x or X,

which coincide at that time. At the end of a

(13)

period, the line of apses must again lie along x,

which now differs from X. The angle between

x and X is the amount of rotation of the apses,

taken with respect to X, that occurs during

one period. This amount of rotation is

A(Q2-\-t) over one period, and is also clearly

the difference between the perturbed and unperturbed periods. That is:

(14)

By Hamilton's equations,

(dr/5P2) +

We find dH/dr, etc., by differentiating equation (11). We find <5r/dP2, etc., by taking differentials in equations (13) and solving for dr,. ..,

dpe. By tedious manipulation, we find for

these differentials:

dr = ^{[P\P\(r-Pl) +2r»(Pf-P|)]/rP1 (P?- •)/r(P?-PI) }dP2,

r)lr{P\-Pl)}dP2,

{pTP\PHr*{P\-Pl)}dP2,

(15)

The result for Q2 is:

<22=[(l-M)P2/rPi(P?-P§)] [P2(PI-r)(r+M cos e)+m(Pt+r) sin d) +[/iP?/rPlf(P?-P2

2)]

X [P2(P2

2-r)(r-(l-(jL) cos d)-{l-n)rpT(Pl+r) sin d] -[P2

P2 (Pi-r)/r 3(P2-Pf)]-l. (16)
PERIODIC ORBITS OF A PLANETOID 75

When n=0, Q2 reduces to —1 , as it should.

Q2-\-l is large only when P is near M, when

both pT and sin 6 are small. Neglecting terms

of the order of (P and 6pr in the numerators of

the various terms, we obtain:

X [(1— — r~

2

].

When this is expanded in powers of n, the term

independent of n vanishes, of course. Of the

terms proportional to n, the one involving

pM dominates when pM is small, so we obtain

as a good approximation:

&+l = (MPMO/PL) [P\P2(Pl-r)lr{P\-Pl)).

(17)

When /u=0, Pi and P2 are constants of the

motion; during the period of close approach to

M, r is nearly constant and nearly equal to

x0. For a first-order calculation, we can thus

take the bracketed quantity in equation (17)

to be constant, and we shall use for Pi and P%

the values which they assume after escape

from M. In terms of the elliptic elements a

and e, Px=-)[a, P2=Ja(l-e2), and r»x0^

a(l+c), so that the bracketed quantity reduces

to — P2/e. Since y and y are zero when 0=0,

we can approximate pM by [pm+yU2]*- With

this approximation for pM, it is valid to

integrate from t= — <» to £=+<», so, upon

integrating equation (17) to find A(Q2-\-t),

and substituting the result into equation (14),

we obtain:

T (18)

In the derivation of this result, there occurs a

sign ambiguity which we must resolve by remembering that y0 is negative for the orbits being

considered.

For orbits direct about E, PC>0, so that

T<2TT. For orbits retrograde about E, P2<0,

and T^>2T. The precession of the line of apses

is seen to have always the opposite sense to the

motion of P near E.

By a set of iterative calculations, we can estimate T and y0 for each x0, as follows: UsiDg

TP=TC, we calculate a from equation (1), and

use x0 and this value of a to estimate y0 from

equation (6), as we did on page 73, and then

substitute into equation (18) to estimate T.

Then we use TP=y2T, and repeat the sequence

of calculations until the process converges.*

The result of this process is the pair of curves

labeled "(}0,b" in figure 3.

Perturbations in the other canonical variables

can be obtained by a process similar to that used

for $}, but we shall not need them for our purposes. We note only that the value of pm can

be shown to be less than the value of ps when P

is farthest from M, as commented upon in footnote 4 (p. 70).

Orbits determined by numerical integration

Sample periodic orbits, for the value M= 1/82.45

already used, have been determined by numerical integration by use of a large-scale

digital computer. The integration was performed with use of the coordinates xy. The

only point in the integration needing comment

is the choice of the time interval. When either

pM or PB is small, it is necessary to use small intervals, but using a small interval over the entire orbit lengthens the calculations unduly and

increases the rounding errors. Strdmgren

(1933) met this problem by using, instead of t,

an independent variable ^ defined by $=1/PMPB,

so that increments in ^ can be approximately

equal. We chose instead to use for the time interval the smallest of the quantities

.01, .025p|/2, .25pM

2

.

This choice relates the time interval to the period of an unperturbed circular orbit about

either E or M, when the instantaneous value of

Ps or pM is used as the radius of the orbit. The

value .01 allows the use of a standard interval

when pB and pM are reasonably large.

The accuracy of the integration can be estimated by comparing the values of Jacobi's

integral at the beginning and end of a run. In

the worst case encountered, with xo=l-2, pg

became as small as .035, and the value of Jacobi's integral shifted by almost 1 percent. In

the other cases, the shift was typically about one

part in 106. In order to follow P through a

close passage to E, one should use a time interval based upon both position and velocity.

The purpose of the calculations was to find

the correct values of yQ for both periodic orbits

• When Put is too small, Pi and Pi vary too rapidly for this elementary

estimate to be accurate; it is even possible to obtain the wrong sign for

A(Qi-M). However, for the range of Pm used here, the estimate is

reasonably accurate.
76

SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

FIGURE 4.—A typical retrograde orbit as seen in inertial coordinates. The small circle is the orbit of E, the segment of

the large circle is part of the orbit of M. The position of P

at conjunction with M is indicated by the cross-mark on the

orbit of P. Direct orbits resemble this one except for the

sense of rotation of P.

of type K, for each of five values of x0. For

each XQ, we started the calculations with the

values of y0 found by the method of the preceding section. The principal problem was to find

an automatic process for changing y0 toward

the correct value.

Because the periodic orbits are symmetric

about the x-axis, there must be some time, other

than t=0, when x and y vanish simultaneously.

For orbits of type %, this must occur at the

third time when y=0, not counting the initial

value, as we can see by reference to figure 2.

For orbits with y0 near the correct value, the

value of x, call it i3, when t/=0 for the third

time, will not be zero, and x3 will be a linear

function of y0 near the correct value.

Accordingly, the computer was programmed

to count the changes of sign of y; after the third

change of sign, it interpolated between the last

two sets of values to find x3. By calculating i3

for two neighboring values of y0, the derivative

dx3/dy0 was estimated, and the zero of the function £3(2/0) was found by the Newton-Raphson

method. When this zero was found to the

assigned accuracy of 10~4 in y0, the orbit was

calculated and printed out.

In addition to finding the orbit for five specified values of x0, we also wished to find the orbit

of zero initial velocity, for which the value of x0

was not known a priori. This orbit was found

by determining periodic orbits for several

values of xQ near the predicted value, and interpolating to find the periodic orbit for which

the initial velocity is zero.

The values of pM0 &nd 2/o for the 11 periodic

orbits found (not counting the ones which were

ancillary to finding the orbit of zero initial

velocity) are plotted in figure 3. The values

agree rather well with the curves marked

"(30,b".

Various important characteristics of the orbits

are summarized in table 1. The distance of

TABLE 1.—Characteristics of periodic orbits for ix = l 182.45

MO

. 012129

. 062129

. 112129

. 162129

. 212129

. 012129

.03211

. 062129

. 112129

. 162129

. 212129

2. 3314

1. 7739

1. 6604

1.5842

1.498

1. 5364

1. 0200

.8475

.8303

.9124

1.049

Perigee

.5207

.2624

. 1756

. 1547

.052

. 1815

. 1945

. 1732

. 1289

.0818

.035

Apogee

1. 3911

1. 1022

1. 1531

1. 1968

1. 242

1. 1966

1. 1408

1. 1458

1. 1780

1. 2206

1.263

Period

7. 8925

6. 5227

6. 3841

6. 3335

6. 302

5. 4292

5. 5718

5. 7574

5. 9750

6. 1081

6. 192

Class

retrograde

retrograde

retrograde

retrograde

retrograde

mixed

zero initial velocity

direct

direct

direct

direct
PERIODIC ORBITS OF A PLANETOID 77

closest approach to M is the initial distance

pM0- The distance pE at perigee is the minimum

ps, and occurs twice each period. The distance

pE at apogee means the absolute maximum of pB,

and occurs at the mid-point of the period, when

P is in opposition to M. The precession of the

line of apses per period, counted positive if in

a

FIGURE 5.—Portions of two orbits, as seen in inertial coordinates, when P is near M. Cross-marks indicate simultaneous positions of M and P. a, Mixed orbit, direct near E

and retrograde near M. b, Orbit of zero initial velocity

(i. eM when nearest M).

the sense of rotation of E and M, is the algebraic difference between the period and 2r.

Values given are believed correct within one

unit in the last figure.

We conclude with a few remarks about the

appearance of the orbits. In the xy plane there

are only two characteristic shapes, corresponding to orbits direct about E or retrograde about

E. These characteristic shapes are similar to

those shown in figure 2, except that the lobes

in the left-half plane should be larger than

those in the right-half plane.

In the XY plane, the retrograde orbits resemble precessing ellipses that contract and

expand their major axes on alternate revolutions. A typical retrograde orbit, for pMo=

.112129, is shown in figure 4. Direct orbits

resemble this one except for the sense of rotation. For the orbit of zero initial velocity, the

orbit develops a cusp at the point of closest approach to M, and for mixed orbits, the cusp

opens out, converting the orbit, on the revolution passing close to M, into a figure-eight.

The orbit of zero initial velocity, and the mixed

orbit for pMo= .012129, are shown as they appear

in inertial coordinates, wheD near M, in figure 5.

When well away from M, these resemble precessing ellipses, as in figure 4.

Acknowledgment

The author expresses gratitude to Miss Joan

Billerbeck of the Applied Physics Laboratory,

The Johns Hopkins University, for preparing

the computing programs and supervising the

computations used in this work.

References

DARWIN, G. H.

1911. Scientific papers. Vol. 4, 592 pp.

EGOROV, V. A.

1958. In Blokhintsev et al., eds., The Russian

literature of satellities. Pt. 1, p. 115.

PoiNCARE, H.

1892. Les me'thodes nouvelles de la me'canique

c61este. Vol. 1, 385 pp.

STROMGREN, E.

1933. Bull. Astron., vol. 9, p. 87.

WHITTAKER, E. T.

1944. A treatise on the analytical dynamics of

particles and rigid bodies. 456 pp.
78 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS VOL. 3

Abstract

This paper presents a study of periodic orbits in the restricted three-body problem in which the small mass passes

close to each of the large masses. The orbits are classified according to the types of motion involved and methods

are given for estimating initial conditions for the various classes of orbit. Some sample orbits, as determined by

a digital computer, are presented.

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