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

Publications of the Astrophysical Observatory

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Contents

Publications of the Astrophysical Observatory

Periodic Orbits of a Planetoid Passing Close to Two Gravitating Masses

Introduction

Zero-order determination of the orbits

Crude perturbation calculation

Classification of orbits

A more accurate perturbation calculation

Orbits determined by numerical integration

Acknowledgment

References

Abstract

Periodic Orbits of a Planetoid Passing Close to Two Gravitating Masses1

By ROBERT R. NEWTON2

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 three-body problem have been studied so extensively that we cannot attempt a thorough citation of the literature. Strömgren (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 between E and M; among these is one that we should describe as of type 1⁄2, class direct. However, his interest was mainly in satellite and planetary theory, and he does not seem to have studied these orbits further.

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1 This work was supported by the Department of the Navy, Bureau of Ordnance, under Contract NOrd 7388.

2 Applied Physics Laboratory, The Johns Hopkins University, Silver Spring, Md.

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 large-scale 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 μ be the mass of M, and 1—μ 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 2π. This choice of units makes the gravitational constant unity, and leaves μ as the only parameter in the Hamiltonian of P. 

If μ 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 of the motion.3 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,

and a ≈ .5, we must have Tp ≈  2π/3. Clearly, then, if P is in an orbit about E with period

where α and β are relatively prime positive integers, and α/β  ≈  1⁄3, the motion of P will be periodic relative to E and M, and also in a stationary coordinate system. 2πα will be the complete period of the motion, and β 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 μ=0.

If a is precisely  If α/β<.3536, the orbit of P will pass inside of M; if α/β>.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 β lobes, and has a β-fold axis of rotational symmetry about the orgin.

If μ is small, but not zero, we expect to find a system of periodic orbits of P, each still corresponding to a fraction α/β, which we shall call the type of the orbit, and which describes the orbit in the limit μ=0. β will still give the number of lobes of the orbit in the rotating coordinate system. However, the orbit will no longer have a β-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 2πα, and will depart farther from 2πα the more closely P approaches M. 

We consider explicitly only orbits of type 1⁄2, and for which P is outside of M at its closest approach to M. Orbits corresponding to fractions other than 1⁄2 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.

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3 The argument of this section is a special case of an argument used by Poincaré (1892, chap. 3). The orbits we are seeking, while apparently not studied explicitly by Poincaré, represent extreme examples of what he calls periodic orbits of the first kind.

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—μ, 0. while E lies at the point — μ, 0. The other coordinate system XY is fixed, and coincides with xy at t=0. Since the rotational period of E and M is 2π, 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:

FIGURE 1.—The rotating coordinate system xy. 

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4 We say a relative maximum because P attains the same distance from E on the other lobe of the orbit when μ=0, and, as we shall see, attains an even greater distance on the other lobe when we Include the perturbation of M. 

Only one of x0 and is independent if the orbit is to be periodic; we choose this one to be x0, and write  as a function of x0.

For the chosen value of α/β, there are two classes of orbit when μ=0, and hence  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 x0<1.26, and it is a trivial exercise to calculate (x0) for each class when μ=0. When μ≠0, 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 μ=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 μ≠0, 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 2π, 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 (x0). 

Crude perturbation calculation

The Hamiltonian of the planetoid P is known5 to be

In this, pE 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 E or M.

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5 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.

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 || is reasonably large.

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.

During this small time interval, the motions of M and P in XY 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 M 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 υ0.m denotes the initial speed of P with respect to M, and υf.m its speed after “escape from M,” we have

where ρM0=x0+μ—1, and denotes the initial distance of P from M. Letting Yf denote the value of after “escape,” and using (5), the reader can readily verify that  and f are related by the equation

We can calculate the motion of P subsequent to escape by assuming that it travels only in the field of E, with initial conditions:

Before using this method to find (x0), let us first use it to draw curves in the (x0, ) plane which delimit the regions containing various classes of periodic orbit. These curves will apply to orbits with any value of α/β, but we shall continue, for simplicity, the restriction that x0>l — μ.

First, there is the curve of “collision with E”: for any x0, there is a value of  such that P will collide with E. We can find an approximation to this curve by lettingf=0 in equation (6). The resulting curve, if we use the value μ=1/82.45, is shown as I in figure 3. (This value of μ 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. 

This curve (II in fig. 3) is given approximately by the condition  or

— 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 M is 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

(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

it is clear that 0 is zero; since 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 grade in xy, and about E is always the same in xy and XY. 

FIGURE 3.—Regions in which initial conditions must lie for various classes of periodic orbit. ρM0 is the minimum value of the distance P-M, and  is the velocity in rotating coordinates at the closest approach to M. 

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 (x0) 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 (1⁄2)), and passing through the point (x0, 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 f in equation (6).f is taken as positive for orbits direct about E, and negative for orbits retrograde about E. This process gives the two curves marked “(1⁄2), a” in figure 3. These curves terminate at one end where ρM0=.26, as shown. They do not terminate at the other end, but we did not calculate them for ρM0<μ, because they are almost surely useless for smaller ρM0.

Since the curve for orbits direct about E crosses the curve of zero initial velocity, we expect to find orbits of all the classes listed above, unless the error in this method is too large.

A more accurate perturbation calculation 

If we transform to polar coordinates r, θ in the xy plane, we find (Whittaker, 1944) that the Hamiltonian becomes:

The momenta pr and p0 are:

That is, pr is the usual radial component of momentum, while p0 is the angular momentum with respect to XY, not to xy.

The transformation

is a contact transformation (Poincaré, 1892, chap. 1; see also Whittaker, 1944, sec. 162), with P1, P2 as the momenta conjugate to Q1, 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. P12 is the semi-major axis of this ellipse, Q1 is the mean anomaly of P in this ellipse, and Q2 is the longitude of the line of apses, measured from the x-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 θ, we cannot write the Hamiltonian explicitly in terms of the new canonical variables.

At t=0, the line of apses is along x or X, which coincide at that time. At the end of a 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 Δ(Q2+t) over one period, and is also clearly the difference between the perturbed and unperturbed periods. That is:

By Hamilton’s equations,

We find , etc., by differentiating equation (11). We find  etc., by taking differentials in equations (13) and solving for dr,..., dpe. By tedious manipulation, we find for these differentials:

The result for  is:

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

+1 is large only when P is near M, when both pr and sin θ are small. Neglecting terms of the order of θ2 and θpr in the numerators of the various terms, we obtain:

When this is expanded in powers of μ, the term independent of μ vanishes, of course. Of the terms proportional to μ, the one involving ρM dominates when ρM is small, so we obtain as a good approximation:

When μ=0, P1 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 P1 and P2 the values which they assume after escape from M. In terms of the elliptic elements a and є,  and r≈x0≈a(1+є) so that the bracketed quantity reduces to —P2/є. Since y and ÿ are zero when θ=0, we can approximate ρM by  With this approximation for ρM, it is valid to integrate from t=—∞ to t=+∞, so, upon integrating equation (17) to find Δ(Q2+t), and substituting the result into equation (14), we obtain:

In the derivation of this result, there occurs a sign ambiguity which we must resolve by remembering that  is negative for the orbits being considered.

For orbits direct about E, P2>0, so that T<2π. For orbits retrograde about E,P2<0, and T>2π. 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  for each x0, as follows: Using TP=π, we calculate a from equation (1), and use x0 and this value of a to estimate  from equation (6), as we did on page 73, and then substitute into equation (18) to estimate T. Then we use TP=1⁄2T, and repeat the sequence of calculations until the process converges.6 The result of this process is the pair of curves labeled “(1⁄2),b” in figure 3.

Perturbations in the other canonical variables can be obtained by a process similar to that used for Q2, but we shall not need them for our purposes. We note only that the value of ρE0 can be shown to be less than the value of ρE 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 μ=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 ρM or ρE 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. Strömgren (1933) met this problem by using, instead of t, an independent variable  defined by=1/ρMρE, so that increments in  can be approximately equal. We chose instead to use for the time interval the smallest of the quantities

This choice relates the time interval to the period of an unperturbed circular orbit about either E or M, when the instantaneous value of ρE or ρM is used as the radius of the orbit. The value .01 allows the use of a standard interval when ρE and ρM 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 x0=l.2, ρEbecame 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  for both periodic orbits of type 1⁄2, for each of five values of x0. For each x0, we started the calculations with the values of  found by the method of the preceding section. The principal problem was to find an automatic process for changing toward the correct value.

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6 When ρM0 is too small, P1 and P2 vary too rapidly for this elementary estimate to be accurate; it is even possible to obtain the wrong sign for Δ(Q2+t). However, for the range of ρM0 used here, the estimate is reasonably accurate.

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. 

Because the periodic orbits are symmetric about the x-axis, there must be some time, other than t=0, when  and y vanish simultaneously. For orbits of type 1⁄2, 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  near the correct value, the value of , call it 3, when y=0 for the third time, will not be zero, and3 will be a linear function of  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 3. By calculating 3 for two neighboring values of , the derivative  was estimated, and the zero of the function 3() was found by the Newton-Raphson method. When this zero was found to the assigned accuracy of 10-4 in , 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 x0 near the predicted value, and interpolating to find the periodic orbit for which the initial velocity is zero.

The values of ρM0 and  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 “(1⁄2),b”.

Various important characteristics of the orbits are summarized in table 1. The distance of closest approach to M is the initial distance ρM0. The distance ρE at perigee is the minimum ρE, and occurs twice each period. The distance ρE at apogee means the absolute maximum of ρE, 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 the sense of rotation of E and M, is the algebraic difference between the period and 2π. Values given are believed correct within one unit in the last figure.

TABLE 1.—Characteristics of periodic orbits for μ = l/82.45 

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. e., when nearest M).

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 ρM0 = .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 ρM0 = .012129, are shown as they appear in inertial coordinates, when 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.

POINCARÉ, H.

1892. Les méthodes nouvelles de la mécanique céleste. Vol. 1, 385 pp.

STRÖMGREN, 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.

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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