by Joachim Schubart,
retired staff member of Astron. Rechen-Institut,
ZAH, University of Heidelberg, Heidelberg, Germany
The present text is another addition to the author's theoretical studies about orbits of the three-body problem at resonance. Several preceding articles that refer to these subjects are available by an index. Especially the title Massive Circumbinary Bodies refers to the 1/5 resonance.
This text is a continuation of the article
Triple Stellar Systems at the 1/6 Resonance, now called Part 1. Again I
study the evolution of orbits of the planar general three-body problem. A binary
consisting of two equal masses is encircled by an outer body with a mass
that equals 1.5 times the mass of a component of the binary. The
orbital period of this process is about equal to six times the period of
revolution of the binary. All the bodies revolve in the same direct way.
Since the variation of the eccentricity of the binary
reaches the vicinity of zero a special method of study is necessary that makes
use of digital filtering to remove the short-period effects. I have introduced
this method in Part 1. I am mainly interested in orbits with an evolution that
is ruled by a single period after removal of the short-period effects.
However in the neighborhood of some of these orbits I find orbits that show
an interesting type of libration that affects two-dimensional features.
Table 3 of Part 1 already shows results about orbits of the problem considered
in this text. Some of these results are due to an application of the special
method. Suitable angular elements show a process of libration of small or
vanishing amplitude. If this amplitude is equal to zero and the short-period
effects are removed only effects by a single period remain. This is demonstrated
by graphical methods. Now I am interested in orbits of this kind that start
with a comparatively small eccentricity of the orbit of the outer body. For this
I have to leave the domain where libration of the usual way occurs. In the last
section I apply a graphical method that I have earlier used in studies of the
restricted three-body problem.
A method of numerical integration by Schubart and Stumpff gives the basis of the computations. Short-period effects are removed by digital filtering in the way proposed by Schubart and Bien. References appear in Part 1. The method of filtering is applicable to variables that are nearly equal to a constant plus a series of periodic terms. The following results do not contradict this. I retain the designations of Part 1, the way of integrating in barycentric rectangular coordinates and of deriving osculating elements of relative orbits. I refer to the components of the binary by their masses, m1 and m2, and to the third body by m3. Using the special value of solar mass, mS, of Part 1 I put m1 = m2 = mS and m3 = 1.5 mS so that these values are equal to the ones of Table 3 of Part 1.
Again I shall use the designations:
a = semi-major axis, e = eccentricity, lp = longitude of pericenter,
lm = mean longitude
are the osculating orbital elements of m3.
a2, e2, lp2, and lm2 are the respective elements of m2. At the start
I put a2, the semi-major axis of the relative orbit of m2 with respect to m1,
equal to 1 au. All the listed orbits start with lp = lp2 = lm = lm2 = 180 deg.
Dlp = lp - lp2 and the slow argument s = lm2 - 6 lm + 4 lp + lp2 are well
defined in general, but not in this study, since e2 can be almost equal to zero
so that lp2 is useless. Following the respective way of Part 1 I remove the
terms of short period from e2 cos Dlp, e2 sin Dlp, e2 cos s, and e2 sin s and
call the listed result : eps cos DEL, eps sin DEL, ez cos s2,and ez sin s2,
or xi, eta, x2, and y2, respectively. Then DEL = arctg eta/xi,
s2 = arctg y2/x2. I use DEL and s2 instead of Dlp and s. Since m1 = m2 the
mean period of revolution of DEL/2 is one of the main periods that rule the
evolution of the orbits. I call this period TPh. Tma designates the mean period
of revolution of the mean anomaly of m3, lm - lp .
The orbits in the lower part of Table 3 of Part 1 are examples of a variation of e2 that leads to very small values and I have removed the short-period terms and used DEL and s2 to study them. The resulting variations are ruled by one or two periods. Now I allow circulation of s2 but I find analogous types of evolution. The variation of xi and eta is ruled by two periods. By a variation of the starting value of a I find examples ruled by only one period. Then a plot of eta versus xi, now called plot A, shows points that appear on a closed curve. A plot of y2 versus x2, now called plot B, produces another closed curve. eps and ez depend on time in a similar way. The period TPh rules the variations.
In this section I report on results that appear in orbits after removal of the short-period effects by filtering. At first I consider orbits that are ruled by one period. Orbit No. 1 of Table 1 is equal to orbit No. 11 of Table 3 of Part 1. I have selected this orbit to demonstrate that it is possible to find orbits with smaller starting values of e and with variations ruled by one period. In Table 1 the numbers without an attached letter refer to such orbits. In Plots A and B of these orbits the points on the curves revolve with the periods two times TPh and TPh, respectively. The curves of Plot A are symmetric with respect to both the xi and eta axis. The plots show an interesting evolution from orbit No. 2 to 14. The changes are due to an increasing influence of a second frequency on the figure of the curve of plot A. This frequency is equal to three times the frequency of the period of revolution and causes variations of eps and of ez in a similar way.
Table 1. Starting values and results
No. e2 e a TPh TA D Tma
1 0.15 0.20 3.93780 2.26 2.73 o.14853045
2 0.15 0.19 3.93498 1.19 2.76 0.14824878
3 0.15 0.18 3.95196 1.26 2.81 0.1490930
4 0.15 0.17 3.962364 1.35 2.85 0.1497137
4a 0.15 0.17 3.9618 1.34 38.3 2.85
4b 0.15 0.17 3.9611 1.34 64.8 2.84
4c 0.15 0.17 3.9609 1.37 46.1 2.85
4d 0.15 0.17 3.9600 1.39 22.0 2.85
4e 0.15 0.17 3.9582 1.43 11.8 2.85
5 0.15 0.16 3.96854 1.41 2.89 0.15016467
6 0.15 0.15 3.972258 1.46 2.92 0.1505093
7 0.15 0.14 3.97450 1.50 2.96 0.1507843
7a 0.15 0.14 3.9743 1.50 90 2.95
8 0.15 0.13 3.97586 1.54 2.99 0.15101297
9 0.15 0.12 3.976731 1.56 3.02 0.1512109
9a 0.15 0.12 3.97668 1.56 213 3.02
10 0.15 0.11 3.977368 1.59 3.06 0,15138848
11 0.15 0.10 3.977932 1.61 3.09 0,15155235
12 0.15 0.09 3.978524 1.63 3.12 0.1517068
13 0.15 0.08 3.979196 1.65 3.16 0.15185418
14 0.15 0.07 3.9799691 1.67 3.19 0.1519956
Notes to Table 1. The numbers in the left column refer to the listed orbits.
Numbers with a letter refer to orbits that show two long periods. The letters
a and b appear at orbits that evolve with libration of a special type. The
following columns show the starting values of e2, e, and a. The next values
refer to the period TPh and to a second long period, TA, in special cases.
D is the minimum distance of m3 to a component of the binary. The values of
Tma allow the application of a special method in Section 4. a and D are
given in au. The unit of time is equal to 10**4 days, or to about 27.4 yr.
** indicates exponentiation.
Plot A of orbit No. 1 shows an approximately circular curve centered at the origin. The points of the curve revolve according to two times TPh. eps remains close to 0.0415 . The corresponding plot B consists of points concentrated at the positive x2 axis so that ez is close to 0.0436 and s2 close to 0 . A magnification shows points that encircle the point x2 = 0.0436, y2 = 0 on a narrow curve according to the period TPh. Now I proceed to orbits 2,3, and 4. The curve of Plot A develops inward deformations near the eta axis and beginning with orbit 3 loops appear that stretch inward from top and bottom. These loops grow and beginning with orbit 4 they encircle the origin and intersect each other. The curve of Plot B that encircles a point on the positive x2 axis grows in extension. At first x2 remains positive and s2 is restricted. Beginning with orbit 4 this curve surrounds the origin so that s2 circulates according to the period TPh. The orbits 5 to 14 show a continuous increase of the dimension of the loops so that finally the figure of Plot A consists of three nearly circular arcs of similar dimension. These arcs almost coincide in case of orbit 14. The approximately circular curve of Plot B grows in extension as well and is centered at a point on the positive x2 axis. This point is close to the origin if the starting value of e is small.
If I vary the starting value of a of orbit 4 downward by small amounts effects by a second long period, TA, apppear. In case of orbits 4a and 4b the shape of the symmetric curve with the intersecting loops of Plot A of orbit 4 appears to be maintained but the originally horizontal axis of symmetry shows periodic turns according to TA about the origin with respect to the xi axis of Plot A. It turns in both directions to a maximiun angle of 26 or 75 deg, respectively. Evidently this is a process of libration that acts on a two-dimensional feature. In case of orbits 4c to 4e the libration has changed to circulation: the originally horizontal axis rotates with variable angular velocity according to the listed values of the period TA. Again the shape of the non-rotating curve appears to be maintained. This is indicated by a suitable rotation of the resulting curves in the opposite direction. The described process of libration is active at orbits 7a and 9a as well but proceeds slowly. The turning leads to maximum angles of 25 and 15 deg, respectively. The periods TA cause periodic variations in the plots of type B. The curves deviate from a nearly circular shape so that the points finally fill a ring.
Now I restrict myself to the orbits with an evolution that is ruled by one period after removal of the short-period effects. If the filtering is omitted a graphical method indicates an evolution that is ruled by two basic periods. Such an evolution is suggested by the following remarks and results. If e2 is greater than zero s equals (lm2-lm) -5(lm-lp) - Dlp. If s does not circulate the mean frequency of the first term equals 5 times the one of the mean anomaly of m3 plus the one of Dlp. If DEL replaces Dlp this relation remains valid according to tests. Therefore the mean period of revolution of the first term is a function of two other mean periods, Tma and TPh, that can be the basic periods.
I have described this graphical method in the text Long-period evolution of resonant orbits. Here I store xi and eta or x2 and y2 at intervals that equal the length of the shorter period, Tma, so that the phase of this period is fixed. If I plot the stored values of eta versus the ones of xi in general the points are densely positioned on a closed curve. In special cases most points coincide with the ones of a beginning sequence but a closed curve is indicated. The analogous use of x2 and y2 produces another closed curve. Other closed curves appear if the first plotted point does not correspond to the beginning of the computation but to an optional later moment of time. Apparently only one additional basic period leads to this distribution of the points. In general all the phases of this period are active in the generation of the closed curve.
I have applied this method to the orbits of Table 1 that are listed with a value of Tma. In all but two of these cases the plotted points are well positioned and densely distributed on closed curves. In case of orbit No. 6 a sequence of 97 isolated points indicates the shape of a closed curve. Most points coincide with the first 97 points, since 97 Tma closely equals 10 TPh. In case of orbit No. 10 there are 21 short arcs since 21 Tma is close to 2 TPh. Closed curves are indicated again. I have applied the method to some orbits with a vanishing amplitude of libration that are listed in Table 3 of Part 1 and found closed curves as well.