On the orbital structure of the HD 82943 multi-planet system
aa r X i v : . [ a s t r o - ph . E P ] O c t Complex Planetary SystemsProceedings IAU Symposium No. 310, 2014Z. Knezevic & A. Lemaˆıtre c (cid:13) On the orbital structure of the HD 82943multi-planet system
Roman V. Baluev , and Cristian Beaug´e Central Astronomical Observatory at Pulkovo of Russian Academy of Sciences, Pulkovskojeshosse 65, St Petersburg 196140, Russia Sobolev Astronomical Institute, St Petersburg State University, Universitetskij prospekt 28,Petrodvorets, St Petersburg 198504, Russiaemail: [email protected] Instituto de Astronom´ıa Te´orica y Experimental, Observatorio Astron´omico, UniversidadNacional de C´ordoba,Laprida 854, (X5000BGR) C´ordoba, Argentina
Abstract.
HD 82943 hosts a mysterious multi-planet system in the 2:1 mean-motion resonancethat puzzles astronomers for more than a decade. We describe our new analysis of all radialvelocity data currently available for this star, including both the most recent Keck data and theolder but more numerous CORALIE measurements.Here we pay a major attention to the task of optimal scheduling of the future observationof this system. Applying several optimality criteria, we demonstrate that in the forthcomingobservational season of HD 82943 (the winter 2014/2015) rather promising time ranges can befound. Observations of the near future may give rather remarkable improvement of the orbitalfit, but only if we choose their time carefully.
Keywords. stars: planetary systems - stars: individual: HD 82943 - techniques: radial velocity- methods: data analysis - methods: statistical
1. Introduction
This paper can be treated as an addition to our recent work (Baluev & Beaug´e 2014)devoted to a reanalysis of the radial velocity (RV) data for a unique multi-planet host starHD 82943. Here we only briefly summarize the most important of our previous results(Sect. 2), and present new ones, related to seeking the optimal observation dates for thisstar (Sect. 3).
2. Main results of the RV data analysis
In our work we used the entire set of the RV data currently available for HD 82943in the public literature. These include the old CORALIE (Mayor et al. 2004) and therecent Keck (Tan et al. 2013) data. The Keck data were separated in two independentsubsets that were acquired before and after a hardware upgrade. The primary resultsconcerning our re-analysis of these data are described in (Baluev & Beaug´e 2014). Thuswe do not duplicate this discussion here, except for a brief summary of the conclusions:( a ) The Keck and CORALIE data are not in a good agreement with each other: fittingthe entire data set plainly leads to a severely unstable orbital configuration of the twomajor planets b and c .( b ) One of the reasons for this mutual inconsistency is the likely presence of an ad-ditional systematic variation in the CORALIE (but not Keck) data with a period close1 R.V. Baluev & C. Beaug´e Table 1.
Prescribed observations scheduling goals for HD 82943. scheduling goal critical parameters to refine their numbergoal 1 parameters of all three planets 14goal 2 common orbital inclination 1goal 3 location relatively to the two-planet ACR (see text) 4goal 4 parameters of the third planet 3 to a year. Likely, this variation appeared due to some imperfections of the spectrumprocessing algorithm used for CORALIE.( c ) After removal of the CORALIE annual variation, the RV data still contain a sig-nificant periodicity with a period of ∼ d ) An RV fit implying a stable planetary configuration can be obtained only by in-cluding both the third planet and the CORALIE annual term in the RV model. Withoutthese terms, the nominal (best fitting) solution appears unstable due to an antialignedinitial apsidal state of the two major planets, and forcing this configuration to be stablewould infer an unsuitably large shift of the fit from its nominal position.( e ) The planets in the best fitting configuration lie near the three-planet resonancewith the periods ratio P c : P b : P d ≈
3. Optimal planning of the future RV observations
Clearly, the orbital and dynamical structure of the HD 82943 system is still ratheruncertain. In view of this, it may be useful to apply some optimal planning routines, inorder to predict the time segments in future in which the new RV observations wouldimprove or knowledge about the system, as well as to identify the time ranges where thenew observations would be almost useless.We solve this task by means of the optimal planning approaches described in (Baluev2008a). In this method we should select the entire set of the fitted parameters, or anytheir subset, or even a set of some other quantities expressed by smooth functions ofthe original parameters. Our goal is to find an optimal time for a new observation inthe future, in order to achieve a maximum reduction of the uncertainties in the targetedquantities. Here we adopt the so-called D-optimality criterion, in which the “reduction”of a multi-dimensional uncertainty is treated in terms of the volumes ratio for the relevantuncertainty ellipsoids (or determinants of the relevant covariance matrices).In this work we consider the three-planet fit with the eccentricity of the third planetalways fixed at zero. Otherwise this eccentricity is ill determined and generates dramaticnon-linearity effects, which are not desirable. Four sets of target quantities to refinewere considered in this work, defining four scheduling “goals”. These goals are describedin Table 1. The goal 3 from this table is defined in a rather complicated manner. Itspurpose is to refine our knowledge about the position of the dynamical system relativelyto the ACR of the two major planets. This information is important for the long-termdynamics and the stability of the system (Beaug´e et al. 2003). In this case the set of
D 82943 p r ed i c t ed e ff i c i en cy ( pe r a deg r ee o f f r eedo m ) JD-2450000 refining parameters of all three planets p r ed i c t ed e ff i c i en cy ( pe r a deg r ee o f f r eedo m ) JD-24500001.001.201.401.601.802.00 6000 7000 8000 9000 100002011 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022 2023 2024 p r ed i c t ed e ff i c i en cy ( pe r a deg r ee o f f r eedo m ) JD-2450000 refining common inclination p r ed i c t ed e ff i c i en cy ( pe r a deg r ee o f f r eedo m ) JD-24500001.001.101.201.301.401.50 6000 7000 8000 9000 100002011 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022 2023 2024 p r ed i c t ed e ff i c i en cy ( pe r a deg r ee o f f r eedo m ) JD-2450000 refining relationship with an apsidal corotation p r ed i c t ed e ff i c i en cy ( pe r a deg r ee o f f r eedo m ) JD-24500001.001.021.041.061.081.10 6000 7000 8000 9000 100002011 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022 2023 2024 p r ed i c t ed e ff i c i en cy ( pe r a deg r ee o f f r eedo m ) JD-2450000 refining planet d parameters p r ed i c t ed e ff i c i en cy ( pe r a deg r ee o f f r eedo m ) JD-2450000
Figure 1.
The predicted Keck observations efficiency for scheduling goals from Table 1.
R.V. Baluev & C. Beaug´e
Table 2.
Optimal observation dates for HD 82943. scheduling goal 1 scheduling goal 2 scheduling goal 3 scheduling goal 4JD-2450000 max. eff. JD-2450000 max. eff. JD-2450000 max. eff. JD-2450000 max. eff.observational season of 2014/2015begin – 6988 1 .
038 7024 ±
27 1 .
016 begin – 6985 1 .
123 7055 ±
32 1 . ±
15 1 .
116 7182 ±
17 1 .
018 7113 ±
17 1 .
343 7140 – end 1 . .
035 - - 7201 – end 1 .
095 - -observational season of 2015/20167334 ±
10 1 .
090 7313 ± .
049 7334 ±
10 1 .
187 begin – 7315 1 . ±
33 1 .
051 7351 ± .
054 7393 ±
32 1 .
175 7357 ± . ±
16 1 .
133 7454 ±
26 1 .
022 7551 ±
21 1 .
394 - -observational season of 2016/2017begin – 7743 1 .
047 7755 ± .
082 begin – 7739 1 .
145 7748 ±
11 1 . ±
10 1 .
102 7790 ± .
080 7773 ±
10 1 .
200 7793 ± . ±
33 1 .
057 7891 ±
28 1 .
032 7831 ±
31 1 .
174 - -
4. Conclusions
A few interesting matters can be noticed in Fig. 1 and Table 2:( a ) The peaks of the efficiency functions are rather narrow, meaning that allocatingobservation time randomly is not the best course of actions for HD 82943.( b ) The task of refining the orbital inclination looks antagonistic to refining the mostother parameters. But nonetheless the relevant optimal time ranges tend to stick togetherside-by-side.( c ) We have good chances to refine the accuracy of the usual planetary parameters byup to 30 −
40% in the forthcoming observing seasons. But the orbital inclination, which isonly constrained thanks to the gravitational planet-planet perturbations, is an exception.It looks unrealistic to drastically improve the accuracy of this inclination before 2020s,when the orbital apsidal lines make a larger fraction of a secular revolution.( d ) The refining of the parameters of the third planet seems rather difficult both inthe near and distant future. It seems that to reach this goal we should just patientlyaccumulate more and more observations. The work was supported by the President of Russia grant for young scientists (MK-733.2014.2),by the Russian Foundation for Basic Research (project 14-02-92615 KO a), and by the pro-gramme of the Presidium of Russian Academy of Sciences “Non-stationary phenomena in theobjects of the Universe”.
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