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 3 refers to the same type of
motion and ratio of the periods but the orbit of the planet is inclined with
respect to this plane. For details about the method of study and the basic
computer programs go to the
Introduction of Part 2.
Again I neglect the mass of the planet and study the motion on the
basis of the elliptic or circular restricted three-body problem.
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 = 1/5. 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 about 1000 revolutions of m3 or to 5000 revolutions of the binary system.
According to graphs all these orbits appear to evolve in a quasi-periodic way
in the considered interval.
At first I present examples of libration that result as a continuation from
orbits listed in Part 2 to orbits with a small non-zero inclination. This is
possible if the amplitude A due to the period of libration, TL, in the critical
argument of a planar orbit is sufficiently small. Tables 1 and 2 of Part 2 show
values of A and TL and present examples with A = 0. I can continue such
examples to orbits that are inclined by about 5 deg. I have started the
integration of these orbits with an argument of pericenter, arp, equal to
90 deg so that the pericenter of the orbit is turned off from the plane of the
binary that is the plane of reference. Together with the process of libration
such a value of arp can be helpful to avoid too close an approach of m3 to m2.
In Section 3 I present examples of large inclination. Most of these refer
to the circular restricted three-body problem. Special critical angular
arguments that depend on the longitude of node allow the detection of cases of
libration at values of inclination up to 140 deg. Some of these cases show
libration of arp in addition. It can happen that both the amplitudes due to
the libration of the critical argument and of arp vanish. One of the orbits of
m3 starts with a vanishing eccentricity. The last example refers to an elliptic
orbit of the binary and shows a simultaneous libration of two critical
arguments.
a = semi-major axis, e = eccentricity, i = inclination,
lm = mean longitude, lp = longitude of pericenter, ln = longitude of node,
with arp = lp - ln 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 zero inclination and with e1 = e2,
lp2 - lp1 = lm2 - lm1 = 180 deg.
Dlp = lp - lp2 is defined, if e2 is greater than zero. Dlp can revolve in
both directions. mr = m2/m1 is equal to 1/5.
I put the semi-major axis of the relative orbit of m2 with respect to m1 equal
to 1 au, so that a1 = 1/6 and a2 = 5/6 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 Dlp or of lp.
TO is either the period of revolution of arp or in case of libration of arp
about a mean value the period that rules this libration.
TI is a period that rules the main oscillations of i in case of some orbits.
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.
s11 = lm2 - 5*lm + 3*lp + lp2 and s12 = lm2 - 5*lm + 4*lp + 180 deg
are arguments that can show libration in case of planar and inclined orbits of m3.
s13 = lm2 - 5*lm + 2*lp + 2*ln + 180 deg and s14 = lm2 - 5*lm + 4*ln + 180 deg
can librate in case of large inclination.
beta = s13 - s11 = 2*ln - lp - lp2 + 180 deg is a special argument.
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.
Table 1. Starting values
No. e2 lp2 lm2 i e a ln lp lm s D
1 0.2 180 180 5 0.18 3.0463 90 180 180 s11 1.81
2 0.1 180 180 5 0.2 2.96841 90 180 180 s11 1.63
3 0.1 180 180 8 0.2 2.93 90 180 180 s11 1.56
4 0.0 0 0 4 0.2 2.935221 90 180 180 s12 1.64
5 0.0 0 0 3 0.2 2.93 90 180 180 s12 1.64
6 0.0 0 0 30 0.2 2.93 180 180 180 s13 1.68
7 0.0 0 0 30 0.2 2.94767 180 180 180 s13 1.70
8 0.0 0 0 30 0.2 2.89 180 180 180 s13 1.60
9 0.0 0 0 35 0.2 2.94 180 180 180 s13 1.71
10 0.0 0 0 45 0.2 2.94 180 180 180 s13 1.89
11 0.0 0 0 60 0.2 2.94 180 180 180 s13 1.98
12 0.0 0 0 90 0.2 2.94 180 180 180 s13 1.91
13 0.0 0 0 120 0.2 2.99 180 180 180 s13 1.77
14 0.0 0 0 135 0.2 3.007 180 180 180 s13 1.78
15 0.0 0 0 140 0.2 3.009 180 180 180 s13 1.74
16 0.0 0 0 38.5 0.22 2.96 195 255 0 s13 1.69
17 0.0 0 0 38 0.2228 2.96 195 249.8 357.4 s13 1.75
18 0.0 0 0 38 0.22425 2.950825 195 249.666 356.6289 s13 1.76
19 0.0 0 0 30 0 2.967 180 180 180 s14 2.17
20 0.05 0 0 40.1 0.2 2.94 180 180 180 s13 1.73
Notes to Table 1. The numbers in the left column refer to the listed orbits.
The following columns show the starting values of the orbital elements. The
designations appear above in the text. The last two columns show the symbol
of the respective librating angle and D, the minimum distance between m3 and
m2 during the integration. a and D are given in au, the angles in degree.
I have succeeded in a continuation of planar orbits with small effects by TL
to orbits with small values of inclination. Orbits 1 to 5 of Table 1 represent
such cases. Orbits 1 and 2 correspond to planar orbits with vanishing effects
of the amplitude by TL, A, but here small changes in the starting values of a
are necessary so that the effects by TL vanish again. The new orbits start with
i = 5 and ln = 90 deg and with unchanged values of the remaining elements.
s11 again librates about 0 deg with a maximum amplitude of the unfiltered
results, Amx, of 52 and 32 deg, respectively. In case of orbit 1 the inclination
varies between 3.8 and 5.6 deg, mainly due to an oscillation according to a
period TI = 22. Similar to the respective planar orbit the filtered results of
s11 oscillate with a mean amplitude of 30 deg according to TP = 3.8. Now small
variations of this amplitude follow TI. arp rotates backward with a period
TO = 6.6 .
In case of orbit 2 arp librates about 90 deg with maximum deviations of 63 deg
so that the pericenter is always turned off from the plane of the binary. This
process is ruled by a period TO = 57. The variation of i between 5 and 9 deg
depends on the periods TO and TP = 9.4 . s11 librates about zero, the filtered
values vary according to TP and TO and can reach 16 deg of deviation. Changes
in a and i at the start lead to orbit 3. Then arp does not deviate from 90 deg
by more than 24 deg, mainly due to oscillations by TP = 9.5. TO = 47 only
contributes a variation of 4 deg. Apparently due to this the evolution of the
orbit still appears to be quasi-periodic, although now i varies between 7.7 and
11 deg. This is not true, if I start with i = 10 deg. Due to the variation of
a the libration of s11 about 0 shows effects A = 31 deg by TL = 0.72
in the filtered results of orbit 3 and Amx equals 56 deg.
Orbits 4 and 5 show libration of s12 about zero with maximum deviations Amx of
97 and 92 deg, respectively. I had to start these orbits with comparatively
small values of i and the range of variation of i does not exceed one degree.
Periods TI near 20 rule these variations. arp revolves forward with TO = 5.2
in both cases.
Orbis 6 to 15 refer to the circular restricted problem and show libration
of s13 about 0 deg. This process avoids the unfavourable constellation with
lm = lp = lm2 and lm - ln at 0 or 180 deg and therefore too close an approach
of m3 to m2. Orbit 6 starts with i = 30 deg and is a typical example. The
libration of s13 according to TL = 0.9 proceeds with Amx = 33 and A = 20 deg.
arp rotates forward with TO = 13. The following two orbits depend on a variation
of a. A is very small in case of orbit 7, but Amx = 78 and
A = 63 deg correspond to orbit 8. Orbits 9 to 15 represent an increasing
sequence of inclinations. Values of Amx between 14 and 28 deg and of A between
1 and 12 deg result. Comparatively large values of TL = 1.8 and 2.4 occur at
orbits 14 and 15, respectively. arp rotates forward in case of orbits 6 to 9
and at orbits 14 and 15, but backward at orbits 10 to 13. This indicates that
arp rotates slowly at special starting values of i and possibly librates.
Indeed, starting with i = 38.5 deg I find orbit 16 that shows libration of arp
about 54.5 deg with a filtered amplitude of 7 deg together with libration of s13
about 2.5 deg with Amx = 16 and A = 7 deg. The periods are TO = 24, TL = 0.62 .
Orbits 17 and 18 start with i = 38 deg and with special values of e or of a
and e so that in the first case the filtered value of arp has a very small
amplitude, but s13 librates about 2.5 with A = 7 deg again with TL = 0.66 .
The filtered values of s13 and arp of orbit 18 remain extremely
close to the values s13 = 2.667 and arp = 53.77 deg. Then effects by TL and TO
are negligible in graphs and the basic periods of lm-lm2 and lm-lp are linked
due to the relation s12 = s13 + 2*arp = const. According to this a point that
corresponds to an object moving along orbit 18 will describe a closed curve
in rotating barycentric x-y-z coordinates that keep m1 and m2 in fixed positions
on the x-y plane. lm-lm2 rotates backward and faster so that the curve closes
after four revolutions around the z axis. I have demonstrated this by a
projection of the moving point on the x-y and x-z planes.
Orbit 19 starts with e = 0 and i = 30 deg. s14 is an argument that does not
depend on lp and shows libration about 0 with a filtered mean amplitude of 6
deg and maximum deviations of 11 deg in this case. If the variations of e and
lp are studied in coordinates e*cos lp and e*sin lp and filtered results are
plotted, the point describes a nearly circular curve with radius = 0.0025,
but the center of this curve rotates backward around the origin at a distance
of 0.005 with a period equal to 86.5 .
Orbit 20 starts with e2 > 0 and shows libration of s13 about 0, but this is not
enough to save the orbit from close approaches of m3 to m2. However in this
special case s11 librates in the same way and the simultaneous libration of the
two arguments leads to a sufficiently large minimum distance D. s13 varies with
an amplitude of 19 deg due to TL = 0.55 plus effects by TP = 5.8 and effects of
short period so that Amx = 35 deg. The variation of s11 is very similar.
Therefore I study the difference beta = s13 - s11 = 2*ln - lp - lp2 + 180 deg.
beta oscillates about zero with maximum deviations of 6 deg or of 3.6 deg, if
the filtered values are considered. Then TP and TP/2 are the dominant periods.
The contribution by TL is very small and points to the similarity of the
libration of s13 and s11.