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