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.
I study the motion of three massive bodies on the basis of the planar general three-body problem. All the bodies move in the direct way. Two bodies revolve around their common center of mass in the way of a binary. The third body encircles the orbits of the binary on a sufficiently wide orbit around the center of mass of the binary with a mean distance that equals about four times the mean distance between the components of the binary. The ratio of the orbital period of the outer body to that of the binary is close to 6/1 and causes effects of resonance. I am interested in orbits of the bodies that show libration of a special angular element due to the resonance. This element consists of a linear combination of the mean longitudes and longitudes of pericenter of the two orbits in most cases. If the mass of the third body is large the strong forces can produce extremely small values of the orbital eccentricity of the binary together with rapid changes of the respective longitude of pericenter so that the special element is useless. In Section 3 I introduce another definition of a librating angular element that is applicable to examples of this kind. This definition makes use of digital filtering.
I refer to the components of the binary by their masses, m1 and m2, and to the third body by m3. I use a special value of solar mass, mS. In case of mr = m2/m1 I consider the two values mr = 0.5 and 1.0 . Please remind that mr is equal to the ratio of two masses in this text. m1 + m2 is equal to two times mS. m3 equals 0.4, 1.0, or 1.5 mS. The way of study equals the one of the preceding article Massive cicumbinary bodies at the 1/5 resonance. The simultaneous numerical integration by use of the N-body program by Schubart and Stumpff (1966) proceeds in barycentric rectangular coordinates and covers about 5000 revolutions of the binary system. I use the value of the solar mass plus mass sum of the four inner planets recommended for this program to define the unit of mass, mS. The result of the integration consists in barycentric rectangular coordinates and velocities but I use different systems in the transformation from or to the osculating elements that appear in this article. I derive the elements of m2 from relative coordinates with respect to m1 that here represents the origin, and I use the sum m1+m2 in the transformation. The elements of m3 depend on relative coordinates with respect to the barycenter of only the two bodies m1 and m2 but the sum of all the bodies enters in the transformation. I apply the inverse transformations to the respective starting values of the osculating elements. All the results of interest are stored at equally spaced intervals.
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.
Dlp = lp - lp2 is defined, if e2 is greater than zero. Dlp can revolve in
both directions. The following results show a retrograde rotation in general.
mr = m2/m1 and the unit of mass, mS, are introduced above.
* is the multiplication sign, ** indicates exponentiation
TL is the period of libration that rules a process of libration.
TP is either the period of revolution of Dlp or is defined in a special way.
s = lm2 - 6*lm + 4*lp + lp2 is the special angular element that can show a
permanent process of libration. I am interested in orbits with libration of s
about 0 deg that show a small or vanishing amplitude of effects by TL.
Additional designations like xi, eta, eps, DEL appear in Section 3.
In this way I can find many orbits with libration of s about zero. I use the starting values lm2 = lp2 = lm = lp = 180 deg. Then s starts at 0 deg. I select pairs of starting values of e2 and e and vary the one of a to find orbits that show in s a small or vanishing amplitude A of the isolated effects due to TL. I call the maximum amplitude of the corresponding unfiltered effects of libration of s Amx. My studies refer to the two choices of the ratio mr. In this section I put m3 = 0.4 . Table 1 presents starting values and characteristic parameters of 24 orbits that show a permanent libration of s about zero during the interval of integration. In most cases there are no visible changes in A during this interval but proximity of TP/TL to a fraction given by two small integers can give rise to such changes. Variations of very long period can appear. In Table 1 an exclamation mark following A indicates such cases. mr = 1 means equivalence of m1 and m2. Then TP/2 is the dominant period with respect to the revolution of Dlp. All the 24 orbits show a retrograde revolution of Dlp. A starting value of e at or near 0.25 corresponds to comparatively small values of TL and Amx. The values of a and D are larger in the lower part of the table, if orbits with analogous starting values of e2 and e are compared. TP is largest, if e2 starts at a low value.
Table 1. The mass of the third body equals 0.4
No. mr e2 e a Amx A TL TP D
1 0.5 0.2 0.2 3.6302 32 0 - 4.4 2.36
2 0.5 0.2 0.2 3.64 47 12 1.04 4.5 2.35
3 0.5 0.25 0.2 3.686 47 5! 0.97 3.8 2.40
4 0.5 0.25 0.2 3.691 55 11 1.01 3.9 2.39
5 0.5 0.2 0.25 3.6448 20 0 - 4.9 2.20
6 0.5 0.2 0.25 3.65 24 4 0.58 5.0 2.20
7 0.5 0.24 0.24 3.6967 27 0 - 3.7 2.30
8 0.5 0.24 0.24 3.71 38 10 0.65 3.7 2.30
9 0.5 0.29 0.2 3.7195 52 0 - 3.5 2.42
10 0.5 0.29 0.2 3.729 69 10 0.99 3.6 2.41
11 0.5 0.15 0.2 3.5757 30 0 - 6.9 2.29
12 0.5 0.15 0.2 3.6 72 5 1.10 6.9 2.29
13 1.0 0.2 0.2 3.6632 28 0 - 4.4 2.53
14 1.0 0.2 0.2 3.675 47 15! 1.12 4.5 2.54
15 1.0 0.25 0.2 3.719 35 0 - 3.8 2.59
16 1.0 0.25 0.2 3.73 54 15 1.12 3.9 2.59
17 1.0 0.2 0.25 3.6866 16 0 - 5.2 2.36
18 1.0 0.2 0.25 3.695 23 6 0.61 5.2 2.37
19 1.0 0.25 0.25 3.7613 19 0 - 3.5 2.45
20 1.0 0.25 0.25 3.77 27 7! 0.58 3.6 2.45
21 1.0 0.3 0.2 3.77 46 0 - 3.5 2.62
22 1.0 0.3 0.2 3.78 62 11 1.08 3.7 2.61
23 1.0 0.15 0.2 3.6059 28 0 - 6.5 2.46
24 1.0 0.15 0.2 3.615 42 11 1.20 6.5 2.47
Notes to Table 1. The numbers in the left column refer to the listed orbits
with libration of s. The following columns show the values of mr and the
starting values of e2, e, and a. Amx is the maximum amplitude of the unfiltered
angle s, A is the amplitude of the effects due to the period of libration,
TL. The next values refer to TL and TP. D is the minimum distance of m3 to
a component of the binary. a and D are given in au, the amplitudes in degree.
The unit of time is equal to 10**4 days, or to about 27.4 yr.
Now I put m3 equal to 1.0 or 1.5 in a study of 22 or 16 orbits, respectively. In some cases it is necessary to use a special procedure, but in general I can determine the characteristic parameters in the way of the preceding section. Again I put the starting values of the longitudes equal to 180 deg and start a search for orbits that show libration about zero deg. In most cases I derive the amplitudes and periods of interest from plots of s or Dlp versus time. If the strong perturbations by m3 lead to almost vanishing values of e2 connected with rapid large changes of lp2 this is not possible. This occurs in case of some of the studied orbits. Then it is useful to study the variations of x=e2*cos lp2 and y=e2*sin lp2 by a plot of y versus x in rectangular coordinates. Curves appear that do not show effects of singularity. It turns out that the approach of e2 to zero is due to short-period effects of large amplitude. Therefore my special procedure consists in removing these effects from x and y by digital filtering and to use arctg of the quotient of the filtered values of y and x instead of lp2. Dlp and s depend on lp2 without a factor. This allows the same procedure with Dlp or s instead of lp2. The new values replace Dlp and s in a determination of TP, TL, and A, but there is no value Amx. If this is applied to an orbit without approaches of e2 to zero the three parameters agree with a determination by the original values of Dlp and s.
In an application to e2 with Dlp I call the filtered values of e2*cos Dlp and e2*sin Dlp xi and eta, respectively, and I introduce eps by eps**2 = xi**2 + eta**2 . eps shows the amount of the effects of eccentricity of body 2. DEL = arctg eta/xi replaces Dlp. In the cases with A = 0 deg studied in this way the evolution of eps only shows small variations so that an almost circular curve results in a plot of eta versus xi. In Table 2 the results about A, TL, and TP of the orbits started with e2 = 0.15 depend on an application of the special procedure. Small values of eps result for these orbits. Values near 0.065 appear at orbits No. 9 and 21 . Table 2 resembles Table 1 but the starting values of a are larger and smaller values of TP but larger ones of D appear. In Tables 2 and 3 again an exclamation mark following A indicates variations of very long period.
Table 2. The mass of the third body equals 1.0
No. mr e2 e a Amx A TL TP D
1 0.5 0.2 0.2 3.8271 40 0 - 1.8 2.53
2 0.5 0.2 0.2 3.835 50 7 1.12 1.8 2.53
3 0.5 0.25 0.2 3.909 43 0 - 1.6 2.60
4 0.5 0.25 0.2 3.92 60 11! 1.10 1.6 2.59
5 0.5 0.2 0.25 3.8163 43 0 - 3.2 2.33
6 0.5 0.2 0.25 3.825 48 5 0.63 3.3 2.33
7 0.5 0.24 0.24 3.8946 32 0 - 1.8 2.45
8 0.5 0.24 0.24 3.902 38 4 0.67 1.8 2.46
9 0.5 0.15 0.2 3.757 - 0 - 3.2 2.44
10 0.5 0.15 0.2 3.762 - 5 1.20 3.5 2.44
11 1.0 0.2 0.2 3.8612 41 0 - 1.8 2.69
12 1.0 0.2 0.2 3.87 54 9 1.18 1.8 2.70
13 1.0 0.25 0.2 3.9402 35 0 - 1.6 2.78
14 1.0 0.25 0.2 3.95 49 11 1.14 1.6 2.78
15 1.0 0.2 0.25 3.8636 43 0 - 4.8 2.50
16 1.0 0.2 0.25 3.87 48 4 0.68 5.0 2.50
17 1.0 0.25 0.25 3.9609 29 0 - 1.9 2.60
18 1.0 0.25 0.25 3.97 37 6! 0.65 1.9 2.60
19 1.0 0.3 0.2 4.0192 37 0 - 1.6 2.86
20 1.0 0.3 0.2 4.03 53 14 1.16 1.6 2.84
21 1.0 0.15 0.2 3.7927 - 0 - 3.4 2.61
22 1.0 0.15 0.2 3.80 - 8 1.26 3.7 2.61
Notes to Table 2. The way of presentation corresponds to Table 1. A missing
value of Amx indicates the application of a special procedure for the
determination of A and the periods.
All the orbits presented in Table 2 and most of the orbits of Table 3 show a retrograde revolution of Dlp or DEL. There are four cases of exception in Table 3 : Orbits No. 15 and 16 show a direct rotation of DEL. However in case of orbits 9 and 10 DEL is captured in a process of libration about 0 deg with an amplitude of 31 and 9 deg, respectively. The listed value of TP refers to this process. Small values of eps result in an application of the special procedure to the orbits in the lower part of Table 3 : eps is close to 0.083 at orbit No. 7, to 0.0164 at No. 13 but only close to 0.0016 in case of No. 15 . A comparison of orbits 3 - 8 of Table 3 with the anologous orbits of Table 2 again shows larger values of the starting values of a and of D and smaller values of TP due to the larger value of m3 of Table 3.
Table 3. The mass of the third body equals 1.5
No. mr e2 e a Amx A TL TP D
1 1.0 0.3 0.25 4.2083 38 0 - 1.1 2.81
2 1.0 0.3 0.25 4.22 49 7 0.68 1.1 2.81
3 1.0 0.3 0.2 4.1777 41 0 - 1.0 2.99
4 1.0 0.3 0.2 4.19 59 14 1.16 1.0 2.97
5 1.0 0.25 0.2 4.0852 51 0 - 1.1 2.89
6 1.0 0.25 0.2 4.095 67 10 1.18 1.1 2.90
7 1.0 0.2 0.2 3.9967 - 0 - 1.2 2.80
8 1.0 0.2 0.2 4.007 - 8! 1.26 1.3 2.80
9 1.0 0.2 0.25 3.997 - 0 - 6.6 2.59
10 1.0 0.2 0.25 4.007 - 5 0.60 6.2 2.60
11 1.0 0.15 0.2 3.9378 - 0 - 4.6 2.73
12 1.0 0.15 0.2 3.944 - 5 0.88 6.2 2.73
13 1.0 0.10 0.15 3.9063 - 0 5.1 2.87
14 1.0 0.10 0.15 3.912 - 6 1.07 8.1 2.87
15 1.0 0.06 0.106 3.998 - 0 0.8 3.10
16 1.0 0.06 0.106 4.005 - 10 0.47 0.8 3.11
Notes to Table 3. The way of presentation corresponds to Table 1. A missing
value of Amx indicates the application of a special procedure for the
determination of A and the periods. The four values of TP in italics
indicate either a libration or a rotation in the direct way of the argument
that results in the special method.
Orbit No. 9 of Table 3 with A = 0 shows a libration of DEL with an amplitude of 31 deg. I have succeeded in an attempt to find an orbit in the vicinity with A very close to zero and an almost vanishing amplitude of libration of DEL by a variation of the starting values of a and e to a = 3.99685 and e = 0.253415 . If these amplitudes are equal to zero there are no effects by the periods TL and TP in the evolution of the orbit. In the article Massive cicumbinary bodies at the 1/5 resonance I have shown by a study of analogous examples that in such cases a periodic evolution appears if the motion of the bodies is demonstrated in suitable rotating rectangular coordinates that are independent of the choice of the origin of counting the longitudes. Now I use a system of barycentric coordinates that rotates with variable speed in such a way that it follows the change of the direction from the origin to m3 so that this body only moves on the positive x axis with changing distance from the origin. Then the barycenter of m1 and m2 moves on the negative x axis and these bodies revolve about this barycenter. A plot of y versus x of m2 shows that after five revolutions of m2 the body evidently repeats these revolutions in the same way. m3 repeats the way of motion as well. This process continues during the interval covered by the integration in the way of a periodic solution.