Libration of Arguments of Circumbinary Planets.
Part 4. The 1/6 and 1/7 Resonances

            by  Joachim Schubart, Astron. Rechen-Institut,
                       Center for Astronomy, University of Heidelberg
                                  Heidelberg, Germany
          
                schubart@ari.uni-heidelberg.de

Index

1. Introduction
2. The 1/6 Resonance
3. The 1/7 Resonance
4. General remarks
5. References

1. Introduction

Part 2 of the study on resonant motion of a small circumbinary planet has referred 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 in the orbital plane of the binary. The present Part 4 refers to the same type of motion but to the 1/6 and 1/7 ratios of the periods. In case of 1/6 both direct and retrograde motion of the planet is considered. For details about the method of study and the basic computer programs go to the Introduction of Part 2 or to the printed paper by Schubart (2017).
I neglect the mass of the planet and study the motion on the basis of the elliptic restricted three-body problem. The eccentricity of the orbit of the binary equals either 0.1 or 0.3. I refer to the components of the binary by their masses, m1 and m2, and to the planet by m3 = 0. Here I study only the ratio of the masses m2/m1 = 1/2. The simultaneous numerical integration proceeds in barycentric rectangular coordinates. m2 moves opposite to m1 on an analogous orbit and the distance from m1 to the barycenter is comparatively small. As before I put the sum m1 + m2 equal to a value that is very close to two times solar mass. m3 starts from positions that appear to be favourable in the present study of cases of libration of suitable critical arguments about a mean value. Strong effects of short period that appear in these and other arguments are removed by a process of digital filtering. I present orbits that show a permanent libration in a restricted interval of about 3500 yr. This interval corresponds to 5000 revolutions of the binary system. According to graphs all these orbits appear to evolve in a nearly quasi-periodic way in the considered interval.


In the following text I shall use the designations:
  a = semi-major axis, e = eccentricity, i = inclination,
  lm = mean longitude, lp = longitude of pericenter
                          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.
        m1 and m2 revolve with with e1 = e2 and lp2 - lp1 = lm2 - lm1 = 180 deg.
                     mr = m2/m1 is equal to 1/2
     i equals either 0 or 180 deg. If i=180 then both lm and lp increase in the
     backward or retrograde direction. All the longitudes are counted from a
     specific zero longitude.
 I put the semi-major axis of the relative orbit of m2 with respect to m1 equal
 to 1 au, so that a1 = 1/3 and a2 = 2/3 in this unit.
               * 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 lp.

 In the following sections the length of a period is given in 10**4 days.
 In general the integration of an orbit covers an interval of 129.1 in this
 unit. This corresponds to 5000 revolutions of the binary system.

 s6  = lm2 - 6*lm + 4*lp + lp2  =  lm2-lm  - 5*(lm-lp)  -  (lp-lp2)  and
 s61 = lm2 - 6*lm + 4*lp -3*lp2 =  lm2+lm  - 7*(lm-lp)  - 3*(lp+lp2)
 are arguments of the 1/6 resonance that show libration in case of i=0 and 180,
 respectively. Evidently they are independent of changes of the zero longitude.
 s7  = lm2 - 7*lm + 5*lp + lp2  corresponds to the 1/7 resonance.
      Amx gives the maximum amplitude of a librating argument,
     A refers to a filtered argument and shows the amplitude due to TL.
 These arguments are reduced to the interval -180 deg < s < 180 deg or to a
 larger interval if the amount of variation requires that.

In Table 1 I collect the starting values of the studied orbits and list results about observed amplitudes, basic periods, and minimum distances. The argument s6 refers to the first ten orbits, s61 to orbits 11-16. Both these arguments show a permanent libration about zero deg during the interval of integration in case of these orbits that represent the 1/6 resonance. The last four orbits belong to the 1/7 resoance. s7 refers to orbits 17-19 and shows libration about zero as well. Orbit 20 is a special case.
                  Table 1. Starting values and results

  No.   e2    i   e       a        lp    lm   Amx   A   TL    TP    D
   1   0.1    0  0.2    3.35      180   180  103   47   1.5  75    2.03
   2   0.1    0  0.2    3.38477   180   180   22    0    -   57    2.07
   3   0.1    0  0.2    3.43      180   180  192  121   3.1  10.8  2.14
   4   0.1    0  0.18   3.39        0     0   47   18   1.3   -    2.20
   5   0.1    0  0.183  3.39        0     0   47   19   1.3   -    2.21
   6   0.1    0  0.1875 3.404891    0     0   26    0    -    -    2.25
   7   0.3    0  0.2    3.52      180   180   80   17   0.9   6.2  2.23
   8   0.3    0  0.2    3.53      180   180   70    5   1.0   6.4  2.26
   9   0.3    0  0.2    3.532     180   180   67    0    -    6.5  2.27
  10   0.3    0  0.2    3.55      180   180  110   36   1.1   8.9  2.22
  11   0.1  180  0.2    3.4045    180   180  147  102  56     8.9  2.08
  12   0.1  180  0.2    3.406     180   180   62   14  46     8.9  2.09
  13   0.1  180  0.2    3.40625   180   180   50    1  45     8.9  2.09
  14   0.1  180  0.2    3.406273  180   180   49    0    -    8.9  2.09
  15   0.1  180  0.2    3.408     180   180  146   98  56     8.9  2.09
  16   0.3  180  0.2    3.405     180   180  164   27  14     8.2  2.12
 
  17   0.1    0  0.2    3.73      180   180   50    8   2.8  21    2.38 
  18   0.1    0  0.2    3.72695   180   180   41    0    -   21    2.38
  19   0.1    0  0.2    3.75      180   180  124   78   3.8  17    2.40 
  20   0.3    0  0.2    3.8407    180   180  172    1   2.2  27    2.61
 
 
  Notes to Table 1. The numbers in the left column refer to the listed orbits.
  Orbits 1-16 represent the 1/6 resonance, the remaining ones correspond to the
  1/7 resonance. For the different librating angular arguments see the text.
  The second column gives the eccentricity of the binary orbit and the next
  five columns show the starting values of the orbital elements of the planet.
  Both lp2 and the starting value of lm2 are equal to 180 deg. The accurate
  value of e of orbit 6 equals o.187498.
  Amx is the maximum amplitude of the unfiltered librating argument, A is the
  amplitude of the effects due to the period of libration, TL. The next values
  refer to TL and to the period of revolution of lp, TP. D is the minimum
  distance between m3 and m2 during the integration. a and D are given in au,
  the angles in degree.

2. The 1/6 Resonance

I have started the study of the 1/6 resonance with a search for examples of libration in case of e2 = 0.1 . At first I report on orbits 1-3 of Table 1. Graphs showing the filtered variation of s6 indicate a permanent libration of this argument. The libration is ruled by a period TL but the long period TP causes an additional oscillation that passes the process of filtering. In case of orbit 1 TP is very long and induces periodic variations of A and TL about mean values that appear in Table 1. Such an effect is not active in case of orbit 2 so that A remains at the value zero. The amplitude of the unfiltered variation of s6, Amx, arises from short-period effects as well and can exceed 180 deg. The large values of TP of orbits 1 and 2 point to a slow rotation of lp. It is especially slow, if lp passes the vicinity of 180 deg and the two orbits show an opposite direction of the revolution of TP. All this indicates that a suitable variation of the starting values can lead to orbits that show a libration of lp about zero deg.

Indeed, by changing the starting values of m3 I have found orbits 4-6 that show libration of both s6 and lp about zero deg. The starting values of lm and lp are equal to zero. The following results refer to filtered arguments. I call the period of libration of lp TP1. I find the amplitudes by TL in s6 and by TP1 in lp of orbit 4 to be 18 and 13 deg, respectively. TP1 equals 75. In case of orbit 5 the amplitude of effects by TP1 equals zero. At orbit 6 both these amplitudes in s6 and lp are extremely small and negligable. Then effects by TL and TP1 do not appear in graphs and the remaining periods of revolution of lm-lp2 and lm2-lp2 are linked due to the relation s6 = lm2-lp2 -6*(lm-lp2) +4*(lp-lp2). Now the filtered results show that one revolution of m3 is equal to 6 revolutions of the binary in non-rotating rectangular coordinates. Then the orbit of m3 is expected to close in the way of a periodic solution. I have confirmed this by using a step-length that divides the period of revolution to an integer number of equal parts and plotted the orbit of m3. During all the more than 800 revolutions the plot returns to the points of the first revolution.

Orbits 7-10 refer to e2 = 0.3 and start at larger values of a. They show a libration of the filtered argument s6 in analogy to orbits 1-3. The values of TP are smaller and large values of the amplitude A due to TL do not appear. Orbits 11-16 refer to i = 180 deg and to e2 = 0.1 or 0.3 . Now m2 and m3 revolve in opposite directions and s61 is the librating argument. All the values of the period TL of these orbits are large and the starting values of a are restricted to a small interval. Orbit 16 refers to the larger value of e2 and shows an approximately quasi-periodic evolution like all the preceding ones, but together with effects by TP = 8.2 and TL = 14 effects by an additional period of length 19.5 pass the filter that removes the short-period effects. The new period corresponds to the difference of the frequencies of TP and TL. Special filters isolate the effects of a period. The effects by TP are comparatively large. The amplitudes by TL and by the new period equal 27 and 19 deg, respectively. Both these amplitudes nearly vanish, if the starting value of a of orbit 16 is augmented to a = 3.4077 .

3. The 1/7 Resonance

The last four orbits of Table 1 refer to the 1/7 resonance and to i = 0. Orbits 17 - 19 start with e2 = 0.1 and show libration of s7 about zero or a vanishing amplitude A with respect to this argument. These three orbits evolve in an approximately quasi-periodic way according to the numerical results. Orbit 20 starts with e2 = 0.3 . At this eccentricity of m2 I have not found examples of libration of s7, but the argument s71 = s7 - 2*(lp - lp2) librates about zero with the small amplitude A = 1 deg in case of orbit 20. lp revolves with strong changes of speed and causes a large effect in this argument. I did not succeed in an attempt to find an example with an amplitude A = 0 of s71 by a variation of the starting value of a, since the observed amplitudes A of all the studied examples vary in an irregular way and the libration of s71 later on changes to circulation in several cases. According to this it appears that the libration of s71 is an unstable process. If I use larger variations of this starting value of a I find orbits that lead to an ejection of m3 during the interval covered by the integration, but with much larger variations orbits result that allow a long-lasting revolution of m3 in an apparently stable orbit.

Now I compare my results obtained with e2 = 0.3 with analogous results by Holman and Wiegert (1999). In Table 6 of their paper the authors demonstrate that orbits at the 1/7 resonance represent an island of instability within an otherwise stable region of orbits that refer to different starting values of a. Their results depend on e2 = 0.4 and on a ratio m2/m1 that is similar to the one used in my present study, and they start their examples at e = 0. My results with e2 = 0.3 indicate a range of instability or of at least an irregular evolution at the 1/7 resonance as well. Starting with e2 = 0.1 and suitable values of lp and lm I find an island of stable libration of s7 within a range of orbits that show an irregular change between libration and circulation of s7.

4. General remarks

The following remarks refer to the 1/5 resonance as well. In analogy to the respective critical argument used in Part 2 the libration of s6, s7, and s71 about zero is helpful to prevent too close approaches of m3 to one of the massive bodies. Due to the larger values of distance to the barycenter this is less important in case of the 1/7 resonance. In the typical cases of a resonance with libration of a critical argument the variation of the starting value of a leads from examples with continuous libration to cases with changes of the amplitude A and finally to an irregular interchange of libration and circulation that indicates a chaotic evolution of the orbit of m3.

Figure 14 of a paper by Doolin and Blundell (2011) gives valuable information about the stability of three-dimensional motion of circumbinary planets and the values i = 0 and i = 180 deg are included in the demonstration. It is visible that i = 180 deg allows stable motion at comparatively small orbital radii of m3 that lead to instability in case of i = 0. This is shown for many values of e2 and m2/m1. It appears that retrograde orbits have a better chance to be stable than direct ones. The above results about orbits 11-16 indicate this as well, since the large values of TL correspond to a weak influence of the 1/6 resonance and the range of the starting values of a covered by librating orbits that is surrounded by unstable orbits is small.

I am grateful to the directors of Astronomisches Rechen-Institut and to the University of Heidelberg for the possibility to continue with work on special problems of celestial mechanics.

5. References

  Doolin S. and Blundell K.M., 2011, The dynamics and stability of circumbinary
    orbits. Monthly Not R Astron Soc 418, 2656-2668
  Holman M.J. and Wiegert P.A., 1999, Long-term stability of planets in binary
    systems. Astron.J. 117, 621-628
  Schubart J., 2017, Libration of arguments of circumbinary-planet orbits
    at resonance. Celest Mech Dyn Astr 128, 295-301
Last modified: 2018 July 16