Modeling radial velocities of HD 240210 with the Genetic Algorithm
More than 450 extrasolar planets are known to date. To detect these intriguing objects many photometric and radial velocity surveys are in progress. We developed the Keplerian FITting code, to model published and available radial velocity data. This code is based on a hybrid, quasi-global optimizati...
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| Zitieren: | Modeling radial velocities of HD 240210 with the Genetic Algorithm / A. Rozenkiewicz, K. Gozdziewski, C. Migaszewski // Advances in Astronomy and Space Physics. — 2011. — Т. 1., вип. 1-2. — С. 84-88. — Бібліогр.: 6 назв. — англ. |
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Rozenkiewicz, A. Gozdziewski, K. Migaszewski, C. 2017-06-03T20:10:58Z 2017-06-03T20:10:58Z 2011 Modeling radial velocities of HD 240210 with the Genetic Algorithm / A. Rozenkiewicz, K. Gozdziewski, C. Migaszewski // Advances in Astronomy and Space Physics. — 2011. — Т. 1., вип. 1-2. — С. 84-88. — Бібліогр.: 6 назв. — англ. 987-966-439-367-3 https://nasplib.isofts.kiev.ua/handle/123456789/119088 More than 450 extrasolar planets are known to date. To detect these intriguing objects many photometric and radial velocity surveys are in progress. We developed the Keplerian FITting code, to model published and available radial velocity data. This code is based on a hybrid, quasi-global optimization technique relying on the Genetic Algorithms and simplex algorithm. Here, we re-analyse the radial velocity data of evolved K3III star HD 240210. We found three equally good solutions which might be interpreted as signals of twoplanet systems. Remarkably, one of these best-fits describes long-term stable two-planet system, involved in the 2:1 mean motion resonance. It may be the first instance of this strong mean motion resonance in a multi-planet system hosted by evolved star, as the 2:1 mean motion resonance configurations are already found around a few sun-like dwarfs. This work is supported by Polish Ministry of Science, Grant 92/N-ASTROSIM/2008/0. en Advances in astronomy and space physics Advances in Astronomy and Space Physics Modeling radial velocities of HD 240210 with the Genetic Algorithm Article published earlier |
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Modeling radial velocities of HD 240210 with the Genetic Algorithm |
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Modeling radial velocities of HD 240210 with the Genetic Algorithm Rozenkiewicz, A. Gozdziewski, K. Migaszewski, C. |
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Modeling radial velocities of HD 240210 with the Genetic Algorithm |
| title_full |
Modeling radial velocities of HD 240210 with the Genetic Algorithm |
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Modeling radial velocities of HD 240210 with the Genetic Algorithm |
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Modeling radial velocities of HD 240210 with the Genetic Algorithm |
| title_sort |
modeling radial velocities of hd 240210 with the genetic algorithm |
| author |
Rozenkiewicz, A. Gozdziewski, K. Migaszewski, C. |
| author_facet |
Rozenkiewicz, A. Gozdziewski, K. Migaszewski, C. |
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2011 |
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English |
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Advances in Astronomy and Space Physics |
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Advances in astronomy and space physics |
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More than 450 extrasolar planets are known to date. To detect these intriguing objects many photometric and radial velocity surveys are in progress. We developed the Keplerian FITting code, to model published and available radial velocity data. This code is based on a hybrid, quasi-global optimization technique relying on the Genetic Algorithms and simplex algorithm. Here, we re-analyse the radial velocity data of evolved K3III star HD 240210. We found three equally good solutions which might be interpreted as signals of twoplanet systems. Remarkably, one of these best-fits describes long-term stable two-planet system, involved in the 2:1 mean motion resonance. It may be the first instance of this strong mean motion resonance in a multi-planet system hosted by evolved star, as the 2:1 mean motion resonance configurations are already found around a few sun-like dwarfs.
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987-966-439-367-3 |
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https://nasplib.isofts.kiev.ua/handle/123456789/119088 |
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Modeling radial velocities of HD 240210 with the Genetic Algorithm / A. Rozenkiewicz, K. Gozdziewski, C. Migaszewski // Advances in Astronomy and Space Physics. — 2011. — Т. 1., вип. 1-2. — С. 84-88. — Бібліогр.: 6 назв. — англ. |
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| first_indexed |
2025-11-25T21:07:27Z |
| last_indexed |
2025-11-25T21:07:27Z |
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1850550682260602880 |
| fulltext |
Modeling radial velocities of HD 240210 with the Genetic Algorithms
A. Rozenkiewicz, K. Gozdziewski, C. Migaszewski
Toru« Centre for Astronomy, Nicolaus Copernicus University, Gagarin Str. 11, 87-100 Toru«, Poland
{a.rozenkiewicz,k.gozdziewski,c.migaszewski}@astri.umk.pl
More than 450 extrasolar planets are known to date. To detect these intriguing objects many photometric
and radial velocity surveys are in progress. We developed the Keplerian FITting code, to model published
and available radial velocity data. This code is based on a hybrid, quasi-global optimization technique relying
on the Genetic Algorithms and simplex algorithm. Here, we re-analyse the radial velocity data of evolved
K3III star HD 240210. We found three equally good solutions which might be interpreted as signals of two-
planet systems. Remarkably, one of these best-�ts describes long-term stable two-planet system, involved
in the 2:1 mean motion resonance. It may be the �rst instance of this strong mean motion resonance in a
multi-planet system hosted by evolved star, as the 2:1 mean motion resonance con�gurations are already
found around a few sun-like dwarfs.
Introduction
Most of the discovered exoplanets are found around main sequence stars, perhaps because the determina-
tion of stellar parameters is much easier than, for instance for giant or active stars. The formation theory of
planets hosted by such stars is still developing and not understood well. For today, there are only about 10
exoplanets known orbiting stars with masses greater than 2 M¯ (see, e.g., Extrasolar Planets Encyclopaedia,
http://exoplanet.eu). Some recent radial velocity (RV) surveys focus on detecting planets around the red
giants, which are evolved main sequence stars. Unfortunately, the giants and sub-giants are di�cult targets
for the RV technique. Usually, they are chromospherically active, pulsating, surface-polluted by large spots,
and rotating slowly. The giants produce small number of sharp spectral lines and their chromospheric activ-
ity and spots may change the pro�les of spectral lines. This intrinsic RV variability (also known as stellar
�jitter�) is signi�cantly larger than instrumental errors, and may be ∼ 20− 30 m/s and larger. Actually, the
stellar activity may even mimic planetary signals (see, e.g. [1]). Hence, when interpreting the RV variability,
one may expect that di�erent uncertainties and errors (usually, of unknown origin) may signi�cantly shift
the best �ts from the �true� solutions in the parameters space. In such a case, possibly global exploration
of the parameter space and the dynamical stability of multiple systems as an additional, implicit observable
may help us to correct and verify the derived best-�t solutions for these factors, and to conclude on the
architecture of interacting systems, even when limited data are available [4].
Keplerian model of the RV and optimization method
We recall the kinematic RV model for the N -planet system, as the �rst order approximation of the N -body
model:
Vr(t) =
N∑
i=1
Ki [cos(ωi + ν(t)) + ei cos(ωi)] + V0, (1)
where, for each planet in the system, Ki is the semi-amplitude of the signal, ei is the orbital eccentricity, ωi
is the argument of pericenter, νi(t) is the true anomaly, which depends on the orbital period Pi, the time of
periastron passage τi and eccentricity, and V0 is a constant instrumental o�set. The N -planet system is then
characterized by Np = 5N + 1 free parameters, to be determined from observations. Let us note, that the
inclinations and longitudes of nodes are not explicitly present in Eq. (1) and cannot be determined directly
from the RV data alone, at least in terms of the kinematic model in Eq. (1).
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Advances in Astronomy and Space Physics A. Rozenkiewicz, K. Gozdziewski, C. Migaszewski
To �t model Eq. (1) to the RV data, the Gaussian Least-Squares method is commonly used. To estimate
the best-�t model parameters, we seek for the minimum of the reduced χ2 function, min
√
χ2
ν(p), which
is computed on the basis of synthetic signal Vr(ti,p), where ti are moments of observations V obs
r,i with
uncertainties σi (i = 1, . . . , NRV is the number of data), and p ≡ (Kp, Pp, ep, ωp, τp, V0), and p = 1 ≡ b, 2 ≡
c, . . . , N are for the model parameters of the N -planet system.
Clearly, even in the kinematic formulation,
√
χ2
ν(p) is a non-linear function. It is well known that it may
exhibit numerous local extrema. Hence, the exploration of multi-dimensional parameter space p requires
e�cient, possibly global numerical optimization. In the past, we found that good results in this problem may
be achieved by an application of the Genetic Algorithms (GAs) [2], which mimic the biological evolution.
GAs search for the best �t solutions (best adapted population members) to the model (to the environment)
through �breading� an initial population (parameters set), under particular genetic operators (e.g., cross-
over, mutation), and through a selection of gradually better adapted members. Although the GAs start
with purely random initial �population�, the search converges to the best �t solutions [2] by deterministic
way. Of course, the best �t solutions may be not unique. This is very common in the case of modeling RV
data. Overall, GAs are robust and quasi-global optimization technique, although they not provide e�ciency
and accuracy of fast local methods, like the Levenberg-Marquardt algorithm. GAs are used in our hybrid
Keplerian FITting (KFIT) code, developed for a few years [3], which also makes use of the local, and accurate
simplex algorithm [6]. It helps to re�ne quickly the �nal population of the best �ts, �grown� by the GAs.
The results for the RV of HD 240210
In a recent work, Niedzielski et al. [5] detected a planet hosted by evolved dwarf HD 240210. They found
an excess in the rms of Keplerian 1-planet model, and interpreted this as possible signal of an additional
planet. We reproduced their 1-planet solution (see Fig. 1, the top-left panel). The rms has signi�cant scatter
of 39 m/s, compared to the mean instrumental accuracy of ∼ 8 m/s. This rms excess is di�cult to explain by
the internal variability of the star, hence we tried to �nd a better 2-planet model. At �rst, we reproduced (see
Fig. 1, the top-right panel) a tentative 2-planet solution given in the discovery paper. However, we notice
that the orbital periods are very similar ∼ 440 and ∼ 530 days, respectively, indicating strong 4:3 mean
motion resonance (MMR) or 1:1 MMR. In such a case, the kinematic model is not proper anymore, due to
signi�cant mutual planetary interactions. Moreover, such con�gurations could be hardly explained by the
planetary formation theory.
Still, having in mind that this solution is not unique, we performed an extensive search for the local minima
of
√
χ2
ν with the KFIT code. The statistics of gathered �ts is shown in Fig. 2, through their projection onto
orbital periods plane. In the range of orbital periods ∈ [60, 3600] days, we found three, equally good best-�t
models, which correspond to di�erent orbital con�gurations, and may be resolved at the 2σ con�dence level.
In the parameter maps, the mentioned 2-planet model is labeled as Fit I. Two additional �ts with orbital
periods ratio close to 3:2 and 2:1 are labeled as Fit II and Fit III, correspondingly (see Table 1). Their
synthetic RV curves, with the RV of 1-planet model (thin curve) and observations overplotted, are shown in
subsequent panels of Fig. 1. Note, that the alternative 2-planet �ts reduce the rms signi�cantly, to ∼ 25 m/s
(i.e., by 1/3), that is consistent with a conclusion in [5].
Table 1: Keplerian model parameters of 2-planet best �t solutions to the RV of HD 240210. Formal measurement
errors are rescaled by adding the stellar jitter of σj = 20 m/s in quadrature. T0 ≡ 53, 000 days, Nv = NRV −Np = 27.
the best �t I II III
parameters b c b c b c
P [day] 540± 29 441±27 484±14 667±54 485± 18 994±97
K [m/s] 131±29 85±30 147±16 82±36 129± 30 63±23
e 0.05±0.09 0.20±0.14 0.18±0.14 0.74±0.38 0.30±0.13 0.58±0.35
ω [deg] 204±51 356±73 261±41 36±76 302±35 163±87
τ − T0 [days] 352±65 610±109 484±45 502±72 536±31 326±156
V0[m/s] -3±7 12±9 22±12√
χ2
ν 1.35 1.36 1.36
rms [m/s] 25.2 25.2 25.2
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Advances in Astronomy and Space Physics A. Rozenkiewicz, K. Gozdziewski, C. Migaszewski
As may be seen in Table 1, Fits II and III have large eccentricities. A question remains, whether inferred
orbital con�gurations are dynamically stable. In fact, all these Fits I�III, transformed to osculating elements
at the epoch of the �rst observation, lead to self-disruption of 2-planet systems. Nevertheless, remembering
that stable solutions may still be found in their neighborhood, we did dynamical analysis with the N -body,
self-consistent GAMP code [4] (which also relies on the GAs), trying to re�ne Fits I�III with the requirement
of the long-term stability (the edge-on, coplanar models are tested). Certainly, at most one of these models
might correspond to the real system. We did not found any stable orbits in the vicinity of Fit I. In a tiny
neighborhood of Fit II, there is a stable con�guration with
√
χ2
ν ∼ 1.50 and an rms ∼ 27 m/s which, as
the direct numerical integrations show, is stable over 1 Gyr. The best result is found for Fit III as a stable
solution, corresponding to the 2:1 MMR with
√
χ2
ν ∼ 1.36 and an rms of ∼ 25 m/s, i.e., the same as in the
kinematic Fit III. The dynamical map [4] around this solution (Fig. 3, the right-hand panel) reveals extended
island of stability (∼ 0.15 au). This �t has moderate semi-amplitude librations (∼ 15 − 30◦) of the critical
angles θ1 = 2λb − λc −$b (around 0◦), θ2 = 2λb − λc −$c (around 180◦), and θ3 = $b −$c (around 180◦).
The numerical integrations con�rmed that its stability is preserved at least over 1 Gyr.
Figure 1: The RV observations of HD 240210 [5] and synthetic curves for the best �t solutions found in this
paper. The left-upper panel: the best 1-planet �t. Subsequent �gures are for the best �ts labeled with I, II, III
in Fig. 2. For a reference, all these plots are accompanied by the model curve of the 1-planet �t. See Table 1
for orbital elements.
The 2:1 MMR Fit III seems to be the most promising planetary model explaining the RV variability of
the HD 240210. The 2:1 MMR is quite frequent in the sample of ∼ 40 known extrasolar systems with jovian
planets, because 5�6 con�gurations were reported (see, http://exoplanet.eu). Hence, this new system,
which could be the �rst one around evolved star, is likely. We stress that solution III is found in relatively
extended stability zone, unlike Fit II, which lies in a tiny, isolated area (∼ 0.01 au, Fig. 3, the left-hand
panel). These two maps almost overlap in the ac-range, hence other, relatively extended stable islands are
rather excluded in this region.
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Advances in Astronomy and Space Physics A. Rozenkiewicz, K. Gozdziewski, C. Migaszewski
Figure 2: Statistics of the Keplerian best �ts gathered with the KFIT code, projected onto
the (Pb, Pc)�plane of orbital periods. Solutions marked with white circles and labeled by I,
II, III are for the best �ts in di�erent �islands� of the parameters space. Black �lled circles
are for all �ts within formal 3σ-level of Fit I; for a reference contours with 3, 2, 1σ-level of
this solution, found through extensive, systematic scanning [4] of the (Pb, Pc)-plane with the
Levenberg-Marquardt algorithm, are plotted as curves of increasing thickness. See Table 1 for
model parameters.
Figure 3: Dynamical maps in terms of the MEGNO indicator [4] around coplanar, edge-on,
GAMP N -body �ts (crossed circle): Fit II (3:2 MMR,
√
χ2
ν = 1.50, rms ∼ 27 m/s, the left-
hand panel), and Fit III (2:1 MMR,
√
χ2
ν = 1.36, rms ∼ 25 m/s, the right-hand panel). White
color is for unstable con�gurations, black is for stable solutions. Orbits at the epoch of the
�rst RV of Fit III in terms of (m [mJ ], a [au], e, ω [deg],M [deg]) are the following: (4.11, 1.14,
0.284, 304.8, 101.8)b, (2.27, 1.83, 0.592, 162.3, 304.2)c for planets b, and c, correspondingly,
V0 = 21.57 m/s, stellar mass is 0.82 M¯ [5]; Fit II is: (4.50, 1.143, 0.217, 249.24, 163.87)b,
(1.94, 1.506, 0.562, 53.09, 243.39)c, V0 = 8.98 m/s.
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Advances in Astronomy and Space Physics A. Rozenkiewicz, K. Gozdziewski, C. Migaszewski
Conclusions
Extrasolar planets hosted by giant or evolved stars bring important border conditions for the planet
formation theory. In this work, we re-analysed the literature RV data for evolved dwarf HD 240210, with
our KFIT code relying on quasi-global GAs. In the reasonable range of orbital periods less than 3600 days,
we found three Keplerian solutions, which have the same
√
χ2
ν and an rms. By further dynamical analysis
of these best-�t models, we selected the most likely, stable solution, which corresponds to 2:1 MMR, and is
located in relatively extend zone of dynamical stability. Overall, if the 2-planet con�guration is assumed, the
dynamical constraints seem rule out other two models, but only new observations may con�rm the 2:1 MMR
hypothesis.
Acknowledgements.
This work is supported by Polish Ministry of Science, Grant 92/N-ASTROSIM/2008/0.
References
[1] Berdyugina S. V. Living Reviews in Solar Physics, V. 2(8) (1995)
[2] Charbonneau P. Astrophys. J. Suppl., V. 101, p. 309 (1995)
[3] Gozdziewski K., Migaszewski C. Astron. & Astrophys., V. 449, pp. 1219-1232 (2006)
[4] Gozdziewski K., Migaszewski C., Musielinski A. Proc. IAU Symp., V. 249, pp. 1219-1232 (2008)
[5] Niedzielski A., Nowak G., Adamow M., Wolszczan A. Astrophys. J., V. 707, pp. 768-777 (2009)
[6] Nelder J. A., Mead R. Comp. J., V. 7, pp. 308-313 (1965)
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