by Joachim Schubart, Astron. Rechen-Institut,
University of Heidelberg, Heidelberg, Germany
schubart@ari.uni-heidelberg.de
This part of a study on resonant motion of a small circumbinary planet refers to an 1/4 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. Part 2 of the study refers to an 1/5 ratio of these periods. 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 2800 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. 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 4000 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; In general the effects by the period of libration are clearly visible in a plot of the librating angle against time; In many cases the amplitude of libration does not exceed a value that corresponds to unstable motion, but the horseshoe orbits of the 1/1 resonance pass both the stable libration points L4 and L5 and the unstable one L3; see e.g. the textbook by Morbidelli (2002) for details.
This suggests 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, now 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. Due to these large effects it can happen that a librating argument varies in an interval that exceeds 360 deg, but then the oscillations by the period of libration do not exceed a stability limit of, for instance, +/-180 deg. Examples appear in Section 3 in Table 2.
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.
s1 = lm2 - 4*lm + 2*lp + lp2 = lm1 - 4*lm + 2*lp + lp1 and
s2 = lm2 - 4*lm + 3*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 s1 is equal to 180 deg. Since the formula for s1 does not change, if m1 replaces m2, the same value of s1 results in case of a conjunction of m3 and m1 of this type. Therefore, a libration of s1 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 s1 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 s1. Indeed, in this way I find orbits with libration of s1 about zero for all three choices of mr. I use the starting values lm2 = 0, lp2 = 180 deg, lm = lp = 90 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 and I use three choices of e2 that are not too small or too large. Then I study the variation of s1 in a short interval of forward integration, using different starting values of a. If libration of s1 is indicated, an extended forward integration covers about 1000 revolutions of m3, or about 2800 yr. In Table 1 I collect results on orbits that show a permanent libration of s1 about zero without visible changes during this interval.
Table 1. Characteristic values
No. mr e2 a Amx A TL TP D
1 1.0 0.05 2.835 42 4 1.3 2.6 1.83
2 1.0 0.10 2.90 46 7 0.5 1.2 1.92
3 1.0 0.15 2.93 55 7 0.4 0.83 1.95
4 0.5 0.05 2.775 65 29 0.52 3.6 1.54
5 0.5 0.10 2.85 40 7 0.41 1.3 1.68
6 0.5 0.15 2.90 56 7 0.4 0.9 1.76
7 0.2 0.05 2.74 60 30 0.77 7 1.35
8 0.2 0.10 2.76 36 13 0.48 2.0 1.44
9 0.2 0.15 2.78 43 11 0.35 1.35 1.48
Notes to Table 1. The numbers in the left column refer to the listed orbits
with libration of s1. The following columns show the values of mr, e2, and of
the starting value of a. Amx is the maximum amplitude of the unchanged angle
s1, A is the amplitude of the effects due to the period of libration, TL.
Then follow the values of 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 s1 is only possible, if the unchanged values of s1 are restricted to an interval of 360 deg. This appears in contrast to the results on s2 presented 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 0.1 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 s2 is either equal to 180 deg or to 0. The latter case is unimportant, if mr = 0.2, since that causes a small distance of m1 to the barycenter. Therefore I try to find a process of libration of s2 that avoids the neighborhood of 180 deg, and in case of mr > 0.2, 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 s2 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 s2 about different centers during an interval of about 1000 revolutions of m3. In case of mr = 1.0 I start the first three orbits with angular elements that correspond to s2 = 90 deg, but the fourth orbit with values that lead to s2 = -90 deg. The respective values of s2 turn out to be the center of a process of libration. Table 2 demonstrates the large difference between the total amplitude Amx of s2 and of A, the amplitude caused by TL. In case of the first orbit the range of variation of s2 is equal to 382 deg, but since A is less than 90 deg, the process avoids a situation of instability. Orbits No. 12 and 13 lead to the same results. This is due to an approximately quasiperiodic evolution and to a property of symmetry of the equations of motion. This comment refers to orbits 16 and 17 as well.
Table 2. Characteristic values, if e2 = 0 and a12 = 1 au
No. mr lm2 lm lp e a Amx A TL D
10 1.0 0 90 90 0.14 2.6995 191 56 19 1.86
11 1.0 0 90 90 0.14 2.699 170 36 15 1.87
12 1.0 0 90 90 0.14 2.6974 123 0.3 12 1.88
13 1.0 0 -90 -90 0.14 2.6974 123 0.3 12 1.88
14 0.5 0 180 180 0.14 2.538 245 165 5.4 1.58
15 0.5 0 180 180 0.14 2.52 147 93 3.2 1.59
16 0.5 64 180 -60 0.19 2.61 33 1.6 0.48 1.54
17 0.5 -64 180 60 0.19 2.61 33 1.6 0.48 1.54
18 0.2 0 180 180 0.14 2.562 215 147 5.7 1.46
19 0.2 0 180 180 0.14 2.558 147 96 3.4 1.48
20 0.2 0 180 180 0.14 2.5336 29 0.0 - 1.59
Notes to Table 2. The listed orbits show libration of s2. 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
and the starting value of e precedes that of a. The angles are given
in deg. The units of length and time equal those of Table 1.
The transition from mr = 1 to 0.5 leads to the loss of symmetry with respect to the centers of libration between the situations of instability at 180 deg and 0 deg. The latter one is still present, but appears to be less important. Orbits No. 16 and 17 show libration of s2 about centers at 80.3 and -80.3 deg, respectively. In case of orbits 14 and 15 the center of libration is found at 0 deg and the amplitude of libration A exceeds 90 deg. Therefore, after the process of filtering, the variation of s2 passes both the stable centers in the neighborhood of 80 or -80 deg and the position of instability at 0 deg. Compare this with the variation at a horseshoe orbit mentioned in the Introduction. The value of A of orbit 14 is not far from 180 deg, in spite of the wide range of the total variation of s2.
The three last orbits of Table 2 depend on mr = 0.2 and show amplitudes A between zero and 147 deg and a center of libration at 0 deg. The last orbit evolves without significant changes of the total amplitude of s2 and demonstrates the stable condition at this center in case of mr = 0.2 . The minimum distance D of this orbit is larger in comparison with the two preceding orbits. This means that a vanishing amplitude A leads to a better separation of m3 and m2. This way of separating resembles a mechanism that works in case of the resonant system sun, Neptune, Pluto.
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 s2 is no longer permanent. In case of s1 the amplitude due to the period of libration is comparatively small. Therefore, and due to the restricted interval covered by the integrations, this study presents mainly theoretical results. Part 2 of this survey shows in two tables analogous results for arguments that librate in case of an 1/5 resonance.
I am grateful to Rudolf Dvorak and Nader Haghighipour for advice and comments with respect to the present study.
Dvorak R. (Ed.), 2008, Extrasolar Planets. Wiley Verlag, Weinheim
Haghighipour N. (Ed.), 2010, Planets in Binary Star Systems. Astrophysics
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Kley W. and Haghighipour N., 2014, Modeling circumbinary planets: The case
of Kepler-38. Astron. Astrophys. 564, A72, 1-14
Morbidelli A., 2002, Modern celestial mechanics, aspects of solar system
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Last modified: 2016 June 25