by Joachim Schubart,
retired staff member of Astron. Rechen-Institut,
ZAH, University of Heidelberg, Heidelberg, Germany
The present text is a third addition to the author's theoretical studies
about orbits of circumbinary planets. Long periods of a type mentioned in the
preceding Part 2 are attributed to linear combinations of basic frequencies.
Several preceding articles that refer to orbits of circumbinary planets
are
available by an index
I have continued my studies about the long-period evolution of resonant orbits of small circumbinary planets. I consider orbits of the planar elliptic restricted 3-body problem with orbital periods that are close to a 5/1 ratio with respect to the period of the binary system. The preceding Part 2 ot these studies is available in the internet: Evolution of orbits Part2. In Part 2 I have studied orbits first presented in P-type planets 2, now called the earlier study, and I have proposed a method of determination of the period of the process of libration, TL. In applications of the method I have noticed oscillations of very long period in case of orbits with a large amplitude of libration. Now I am interested in the process that gives rise to these long periods.
I have described the way of obtaining barycentric orbital elements in Part 2.
In the following text I retain the designations of Part 2. In analogy to TL I call
the mean period of revolution of the mean anomaly of the planet TM, and the one
of the longitude of pericenter TP. All the bodies revolve in the direct way but
in case of the orbits presented below TP refers to a retrograde revolution of the
longitude of pericenter of the planet.
The method of digital filtering introduced in Part 2 for example can remove the
short-period effects that appear in the orbital elements of the planet due to TM
or to linear combinations of several frequencies. I refer to the components
of the binary by their masses, m1 and m2, and to the planet by m3 = 0. The ratio
of the masses of the large bodies is given by mr = m2/m1 . I retain the definition
of the units of length, mass, and time of the earlier study. Therefore
the semi-major axis of the relative orbit of m2 with respect to m1 equals 1 au
and the sum m1 + m2 is slightly larger than two times solar mass.
The mean longitudes, lm and lm2, the longitudes of pericenter, lp and lp2, and
the eccentricities, e and e2, are elements of the planet and of m2, respectively.
M = lm - lp is the mean anomaly and a the semi-major axis of m3.
The librating angle s is given by
s = lm2 - 5*lm + 3*lp + lp2
or by s = lm2 - lp2 - 5*M - 2*(lp - lp2) .
I indicate multiplication and
exponentiation by * and **, respectively.
If TP is known the second equation
allows the computation of TM since s and lp2 do not revolve and the frequency of
lm2 is known. FM and FL are the frequencies that correspond to TM and TL,
respectively. The frequency FP corresponds to TP but is given by the revolution
of -(lp - lp2) so that FP is positive as well.
The method proposed in Part 2 makes use of plots of filtered results of s of
an integrated orbit versus time. I have plotted the results at intervals that
finally equal the mean length of TL. Due to the application of a narrow filter
such a sequence of points can only show effects by FL and by frequencies close
to it but finally effects by TL disappear. In this way I have noticed in the
plots variations of very long period that are due to linear combinations of FL
and at least one very small frequency.
Now I have realized that these long periods as well can appear in plots of
filtered results of both lp and M versus time. I use a
filter that removes effects of frequencies greater or about equal to FP.
Then an addtional long-period term can pass the filter. If I apply this to a
plot of lp - lp2 minus a term that represents the mean retrograde revolution
of this argument I find the very long periods considered before as an
addendum to the mean effect of rotation of lp. Due to the linear dependence
of s on lp and M the long periods found in this way in lp and M affect the
process of libration. For a study of the long periods I use such plots of
lp - lp2 versus time. Here I present my findings in case of a selected
number of orbits: In many cases the long periods
arise from terms that rotate with frequencies that are linear combinations
of FM, FL, and FP with integer coefficients. Results about such triples of
coefficients appear in Table 1.
The present
Table 1 is based on orbits that are related to some orbits of the earlier study.
The starting values are analogous to orbits No.1 and 2 or 11 and 12 shown in
Table 1 of that study. All the orbits start with e=0.2 and e2=0.1 and with
lm2 = lp2 = lm = lp = 180 deg. Again I vary the starting value of a to produce
a sequence of increasing values of A, the amplitude of the effects due to the
frequency FL in s.
Table 1. Periods and amplitudes
No. mr a A TM TL TP Per Amp Coeff
1 1.0 3.06 10 0.12777 0.56450 4.8164 11.159 0.0045 1 -4 -4
2 1.0 3.065 14 0.12782 0.58339 5.0005 18.878 0.0058 1 -5 +4
3 1.0 3.075 23 0.12794 0.62882 5.5388 7.382 0.625 1 -5 0
4 1.0 3.077 26 0.12798 0.63992 5.725 1 -5 0
5 1.0 3.0799 31 0.12808 0.66268 6.2269 17.124 1.88 1 -5 -2
6 1.0 3.08 33 0.12810 0.66725 6.386 1 -5 -2
7 0.2 2.946 15 0.12836 0.68386 8.4925 120.5 0.67 1 -5 -4
8 0.2 2.938 24 0.12839 0.67272 8.8561 56.8 0.65 1 -5 -3
9 0.2 2.937 25 0.12839 0.67169 8.9257 118 3.1 1 -5 -3
10 0.2 2.936 25 0.12839 0.66957 8.9302 66.8 1.2 1 -5 -3
11 0.2 2.93 29 0.12842 0.66047 9.2306 1 -5 -2
12 0.2 2.92 44 0.12854 0.65794 11.0930 1 -5 -2
Notes to Table 1. The numbers in the left column refer to the listed orbits with
libration of s. The following columns show the values of mr, of the starting
value of a, and of A, the amplitude of effects due to the frequency FL in s.
The next values refer to the periods TM, TL and TP. The three last columns show
results about observed long periods with length of period Per and amplitude Amp.
in the variations of lp.
Coeff lists the coefficients of the frequencies FM, FL, and FP, respectively,
of linear terms that give rise to the long period or to a secondary resonance
that is marked by blank spaces. a is given in au, A and Amp in degree.
The unit of time is equal to 10**4 days, or to about 27.4 yr.
Table 1 lists values of TM, TL, and TP. TL results in the way proposed in
Part 2 and FP is needed for finding the long periods. Then the second equation
given for s leads to a value of TM. The period, Per, refers to a long-period
term that revolves and causes oscillations of amplitude Amp in lp. No values
appear if the term is ruled by a secondary resonance. Coeff lists three
coefficients. If I call them c1, c2, c3 the long-period term varies according to
the frequency
c1*FM + c2*FL + c3*FP but the term equals c1*M - c3*(lp - lp2) + c2*FL*t , if
M and lp are restricted to a linear dependence on the time, t . The action of a
secondary resonance is indicated if the frequency of such a term is extremely
close to zero.
In Part 2 I have assumed that long-period terms with frequencies that are linear combinations of FM, FL, and FP will cause only small effects. Now the values of Amp indicate that this is only true if s librates with a small amplitude. In many cases secondary resonances affect the variation of such terms and can cause a chaotic evolution. In other cases it may be possible that the evolution of the respective orbit is ruled by the three basic frequencies although Amp is not small.