by Joachim Schubart, Astron. Rechen-Institut,
University of Heidelberg, Heidelberg, Germany
schubart@ari.uni-heidelberg.de
This second part of a study on resonant motion of a small circumbinary planet
refers to an 1/5 ratio of the orbital periods of a binary system and of a
planet that revolves outside of the orbits of the components of the binary.
The preceding
Part 1 of the study refers to an 1/4 ratio of these periods. For the
convenience of the reader I repeat most of the introduction and large parts
of the sections of Part 1 with the necessary changes. This is possible, since
there are many similarities between the results of the respective sections of
the two parts.
I neglect the
mass of the planet and study the motion on the basis of the planar, elliptic
restricted three-body problem. Special examples depend on the circular
restricted problem. I refer to the components of the binary by their masses,
m1 and m2, and to the planet by m3 = 0. In case of mr = m2/m1 I consider the
three values mr = 1.0, 0.5, and 0.2. Please remind that mr is equal to the
ratio of the masses in this text. The simultaneous numerical integration
proceeds in barycentric rectangular coordinates. Then m2 moves opposite to m1
on an analogous orbit. All the bodies revolve in the same direct way.
m3 starts from positions that appear to be favourable in the present search for
cases of libration of suitable angular arguments about a mean value. Orbits
that show a permanent libration in a restricted interval of about 3500 yr
are listed. During more extended intervals the evolution of these orbits
can be completely different and can lead to an ejection of m3 from the system.
Otherwise it is possible that some of the studied processes of libration
continue during much more extended intervals of time. This is pointed out at
the end of this text. This search represents a general survey about the
possibility of libration at changing values of binary eccentricity and of mr.
Examples of libration have appeared in studies on the evolution of
circumbinary planets. In such work Kley and Haghighipour (2014) have found
an example that refers to an 1/5 resonance. References about extended numerical
studies on the evolution of orbits of circumbinary planets appear in the
respective chapters of volumes edited by Dvorak (2008) and Haghighipour (2010).
Since I have passed the age of 85 years, I have to be especially careful to avoid any systematical errors. Therefore I only use methods and computer programs that are well known to me since many years. For the numerical integration I use the N-body program by Schubart and Stumpff (1966). I have retained the value of the solar mass plus mass sum of the four inner planets recommended in this paper, and I put the sum m1 + m2 equal to two times this value. Then in case of mr = 1 the masses of the two components are very close to solar mass. Precise solutions result for the intervals of time that are covered by the integrations. In general the computation covers 5000 revolutions of the binary system, or about 1000 revolutions of m3. The result consists in barycentric rectangular coordinates and velocities of m3. These are transformed to osculating barycentric orbital elements by the assumption that the combined mass m1 + m2 is concentrated at the barycenter. The elements and angular arguments of interest are stored at equally spaced intervals.
For the present project it is helpful to remember some results on libration of critical arguments of the problem of sun, Jupiter, asteroid: Libration about a value that corresponds to stable motion of the asteroid can keep the object away from too close approaches to Jupiter; A period of libration rules the process of libration (see e.g. the textbook by Morbidelli (2002) for details). Here the perturbing components of the forces by m1 and m2 on the barycentric motion of m3 are strong and would be strongest in a close approach of m3 to one of the massive bodies, so that a lasting process of libration is improbable. These considerations suggest for the present project to start m3 on an orbit that avoids too close an approach to one of the massive bodies, and to select a process of libration that maintains this quality. However, the rapidly changing forces of the two revolving massive bodies cause large effects in a librating angle, so that one cannot estimate the amount of the effects by the period of libration in a plot of this angle versus time.
In the past I have applied digital filtering to approximately quasi-periodic processes using the method and formulas of Schubart and Bien (1984). This method is applicable, if the studied variable is very close to a quasi-periodic function of time in the restricted interval of time that is used by the method. Now I use this method to separate the effects by the period of libration in a librating angle from the large effects mentioned before. These effects have comparatively short periods that are about equal to the orbital period of m3 or smaller. A suitable filter eliminates these effects, but does not change constants and effects with smaller frequencies like those of a period of libration and of longer periods.
a = semi-major axis, e = eccentricity, lp = longitude of pericenter,
lm = mean longitude
are the barycentric osculating orbital elements of m3.
a1, e1, lp1, and lm1 are the respective elements of m1, and
a2, e2, lp2, and lm2 are the respective elements of m2. Then e1 = e2,
lp2 - lp1 = lm2 - lm1 = 180 deg.
a12 designates the semi-major axis of the relative orbit of m2 with respect to m1.
Dlp = lp - lp2 is defined, if e2 is greater than zero. Dlp can revolve in
both directions. mr = m2/m1 was introduced above.
* is the multiplication sign, ** indicates exponentiation
TL is the period of libration that rules the process of libration.
TP is the period of revolution of Dlp.
s11 = lm2 - 5*lm + 3*lp + lp2 = lm1 - 5*lm + 3*lp + lp1 and
s12 = lm2 - 5*lm + 4*lp + 180 deg are arguments that are tested for permanent
libration. These arguments are reduced to the interval -180 deg < s < 180 deg
or to a larger interval if the amount of variation requires that.
If all the bodies move on elliptic orbits, a conjunction in longitude of m3 and m2 with m2 at apocenter and m3 at pericenter leads to a comparatively small distance of the bodies and to strong forces, so that ejection of the planet can occur. If lm2 = lm, lp = lm, lp2 = lm2 + 180 deg = lm + 180 deg, then s11 is equal to 180 deg. Since the formula for s11 does not change, if m1 replaces m2, the same value of s11 results in case of a conjunction of m3 and m1 of this type. Therefore, a libration of s11 about zero can be helpful to prevent too close approaches of m3 to one of the massive bodies.
This suggests to start integrations of orbits with a set of angular elements that lets s11 be equal to zero, and with a suitable choice of the other starting values, but the one of a is allowed to vary among values that lead to a slow variation of s11. The range in a to be considered is not very large. Indeed, in this way I find orbits with libration of s11 about zero for all three choices of mr. I use the starting values lm2 = lp2 = lm = lp = 180 deg, and I let a12 be equal to 1 au. Then a1 = mr/(1+mr) and a2 = 1/(1+mr), in au. I put e = 0.2 or e = 0.18 and I use two choices of e2. Then I study the variation of s11 in a short interval of forward integration, using different starting values of a. At first I find orbits that lead to ejection of m3 from the system or to circulation of s11. If finally libration of s11 is indicated, an extended forward integration covers about 1000 revolutions of m3, or about 3500 yr. In Table 1 I collect results on orbits that show a permanent libration of s11 about zero without visible changes during this interval.
Table 1. Characteristic values
No. mr e2 e a Amx A TL TP D
1 1.0 0.1 0.2 3.09 77 41 0.73 7.8 2.02
2 1.0 0.1 0.2 3.05 25 2 0.54 4.6 1.99
3 1.0 0.1 0.2 3.0468 23 0 - 4.6 1.99
4 1.0 0.2 0.18 3.18 93 24 0.78 2.8 2.19
5 1.0 0.2 0.18 3.1605 40 <1 - 2.4 2.19
6 0.5 0.1 0.2 3.05 61 30 0.69 6.9 1.84
7 0.5 0.1 0.2 3.03 35 11 0.60 5.4 1.82
8 0.5 0.1 0.2 3.0154 23 0 - 5.1 1.81
9 0.5 0.2 0.18 3.105 64 17 0.56 2.6 1.97
10 0.5 0.2 0.18 3.12 53 4 0.63 2.6 2.01
11 0.2 0.1 0.2 2.92 60 44 0.66 11.1 1.55
12 0.2 0.1 0.2 2.95 35 15 0.70 8.4 1.59
13 0.2 0.1 0.2 2.9687 19 0 - 8.3 1.63
14 0.2 0.2 0.18 3.07 100 34 1.0 5.0 1.81
15 0.2 0.2 0.18 3.04746 50 0 - 3.9 1.83
Notes to Table 1. The numbers in the left column refer to the listed orbits
with libration of s11. The following columns show the values of mr, e2 and of
the starting values of e and a. Amx is the maximum amplitude of the unchanged
angle s11, 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 to one
of the massive bodies. 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.
Table 1 does not show large amplitudes A of the effects by TL. Somewhat larger values of A can result from a variation of the starting value of a, but it turns out that a permanent libration of s11 is only possible, if the unchanged values of s11 are restricted to an interval of 360 deg. This appears in contrast to results on s12 mentioned in Section 3. The values of A and TL of Table 1 depend on estimations from plots of the results of the filtering process. In addition to the effects by TL those of other long periods appear in a plot. Effects by TP can appear. Linear combinations of nearly resonant frequencies can give rise to long period effects. In the present case the orbital period of m3 is about equal to 0.13 in the unit of the table and some of the listed values of TL are small multiples of this period. The same occurs with TL and TP. Some of the values of A and TL are uncertain due to effects by a period of unknown origin that is similar in length to TL. The smallest values of the minimum distance correspond to mr = 0.2. They refer to an approach of m3 to m2.
Now the massive bodies revolve with a constant distance to the barycenter. A conjunction in longitude of m3 and one of these bodies with lm = lp can lead to a small distance. Then lp = lm and lm2 = lm or lm2 = lm + 180 deg, so that s12 is either equal to 180 deg or to 0. The latter case is less important, if mr < 1, since then m1 is closer to the barycenter than m2. Therefore I try to find a process of libration of s12 that avoids the neighborhood of 180 deg, and in case of mr = 1 of 0 deg as well. Surprisingly, this is possible if the process of avoiding is only required of the effects by the period of libration. The unchanged values of s12 can vary in a greater interval.
The success of my attempt is documented by results on 11 orbits that appear in Table 2. These orbits show a permanent libration of s12 about different centers during an interval of about 1000 revolutions of m3. In case of mr = 1.0 I start the three orbits with angular elements that correspond to s12 = 90 deg. This value turns out to be the center of a process of libration. The difference between the total amplitude Amx of s12 and of A, the amplitude caused by TL, is large. In case of the first orbit Amx is equal to 131 deg, but since A is less than 90 deg, the process avoids a situation of instability. In Section 3 of Part 1 it is pointed out that due to a property of symmetry of the equations of motion analogous orbits are expected, if the sign of the starting values of the angular elements is changed to minus. Then the sign of the center of libration changes as well. This comment refers to orbits 19 and 20 as well.
Table 2. Characteristic values, if e2 = 0 and a12 = 1 au
No. mr lm2 lm lp e a Amx A TL D
16 1.0 0 90 90 0.20 3.094 131 78 11 1.98
17 1.0 0 90 90 0.20 3.1 65 14 6.8 2.03
18 1.0 0 90 90 0.20 3.101577 51 0.0 - 2.05
19 0.5 0 180 105 0.20 2.97 54 18 5.0 1.85
20 0.5 0 180 104.5 0.20 2.96675 36 0.3 3.5 1.86
21 0.5 0 180 180 0.20 2.9 123 89 3.9 1.74
22 0.5 0 180 180 0.15 2.927 89 42 5.4 1.93
23 0.5 0 180 180 0.15 2.93414 48 0.1 5.6 2.01
24 0.2 0 180 180 0.20 2.869 188 161 2.5 1.42
25 0.2 0 180 180 0.20 2.93 40 19 3.2 1.75
26 0.2 0 180 180 0.20 2.935221 21 0.0 - 1.80
Notes to Table 2. The listed orbits show libration of s12. Most of the
designations on top of the columns correspond to those of Table 1.
In addition three columns show the starting values of lm2, lm, and lp
together with the starting values of e and a. The angles are given
in deg. The units of length and time equal those of Table 1.
The transition to orbits 19 and 20 with mr = 0.5 leads to the loss of symmetry with respect to the centers of libration between the values 180 deg and 0 deg. A situation of instability is still present at zero deg, but appears to be less important than the one at 180 deg. Orbit 19 and 20 show libration of s12 about centers at 32 and 39.5 deg, respectively. In case of the orbits from 21 to 26 the center of libration is found at 0 deg and the amplitude of libration A can exceed 90 deg. The value of Amx of orbit 24 is larger than 180 deg, but A is small enough to avoid a situation of instability. The minimum distance D of orbits 19 - 26 refers to an approach of m3 to m2. The smallest value of D appears at orbit 24. Larger values appear with the smaller values of A of the two last orbits as expected.
If the starting values given in Table 2 or similar values are used together with an eccentric orbit of the binary, it turns out that this orbit has to be almost circular. Otherwise the libration of s12 is no longer permanent. If the values e2 = 0.1 or e2 = 0.2 are considered and multiples of lp-lp2 are added to s11, other arguments with slow variation result. However I do not expect libration of these arguments within the range of the eccentricities that I have studied: If s11 librates, other arguments will circulate. If s11 circulates, this process is too fast to be compensated by the addition of a slow argument.
In case of s11 the amplitude due to the period of libration is comparatively small. Therefore, and due to the limited interval covered by the computations, most of the present results are mainly a contribution to theoretical celestial mechanics. However, I have extended the computations that refer to mr = 0.2 to larger intervals, that are equal to 17500 yr in case of orbits 11 - 13 and 24 - 26, and to one half of this length for orbits 14 and 15. This corresponds to 25000 or 12500 revolutions of the central binary. In all but one of these cases the permanent libration of s11 or s12 continues during the larger interval without visible changes. Orbit 14 is remarkable for the ratio TP/TL that is very close to 5. A resonance in the resonance occurs and apparently causes slow and somewhat irregular changes in the superposition of the effects by TP and TL, but even in this case there is no significant change in A and Amx in the extended interval. These results indicate the possibility that some of the studied processes of libration can continue during much more extended intervals of time.
I am grateful to Rudolf Dvorak and Nader Haghighipour for advice and comments with respect to the present study.
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Last modified: 2016 June 25