Orbit 27894
Dhara Lyford
HD 27894 b is a gas giant with a mass at least two thirds that of Jupiter, or twice that of Saturn. The distance from the planet to the star is one third compared that of Mercury from the Sun, and it takes almost exactly 18 days to complete one roughly circular orbit.[1]
Context. In radial velocity (RV) observations, a pair of extrasolar planets near a 2:1 orbital resonance can be misinterpreted as a single eccentric planet, if data are sparse and measurement precision insufficient to distinguish between these models.
Aims. Using the Exoplanet Orbit Database (EOD), we determine the fraction of alleged single-planet RV detected systems for which a 2:1 resonant pair of planets is also a viable model and address the question of how the models can be disentangled.
Methods. By simulation we quantified the mismatch arising from applying the wrong model. Model alternatives are illustrated using the supposed single-planet system HD 27894 for which we also study the dynamical stability of near-2:1 resonant solutions.
Results. Using EOD values of the data scatter around the fitted single-planet Keplerians, we find that for 74% of the 254 putative single-planet systems, a 2:1 resonant pair cannot be excluded as a viable model, since the error due to the wrong model is smaller than the scatter. For 187 EOD stars χ2-probabilities can be used to reject the Keplerian models with a confidence of 95% for 54% of the stars and with 99.9% for 39% of the stars. For HD 27894 a considerable fit improvement is obtained when adding a low-mass planet near half the orbital period of the known Jovian planet. Dynamical analysis demonstrates that this system is stable when both planets are initially placed on circular orbits. For fully Keplerian orbits a stable system is only obtained if the eccentricity of the inner planet is constrained to < 0.3.
Conclusions. A large part of the allegedly RV detected single-planet systems should be scrutinized in order to determine the fraction of systems containing near-2:1 resonant pairs of planets. Knowing the abundance of such systems will allow us to revise the eccentricity distribution for extrasolar planets and provide direct constraints for planetary system formation.
With radial velocity (RV) measurements of the stellar reflex motion caused by orbiting companions a pair of planets in low-eccentricity orbits near a 2:1 mean motion resonance (MMR) can be misinterpreted as a single planet with moderate eccentricity (Anglada-Escud et al. 2010; Wittenmyer et al. 2013). This is, in particular, possible when the available data are sparse and have large errors, when the stellar RV amplitude induced by the inner planet is smaller than that induced by the outer planet, and when the overall scatter of the data around either model is too large to make it possible to distinguish them.
In the present paper we investigate the possibility that some fraction of the extrasolar planets in eccentric orbits found by the RV technique are actually pairs of planets in orbits that are close to circular and near a 2:1 resonance. By simulating and comparing the respective orbits as a function of their model parameters (Sects. 2 and 3) we explore the circumstances under which the two models are indistiguishable. This enables us to identify those of the known eccentric planets in the literature for which suitable follow-up observations could determine which model is correct. To illustrate this, we study one such example (Sect. 4). In the following discussion we provide recommendations for the strategy for such new observations that may uncover a so far unknown additional planet near the 2:1 resonant orbit (Sect. 5). Finally, we summarize our main conclusions (Sect. 6).
In order to compare the fits, their rms deviations σmodel from the simulated 2:1-RCO data were employed to indicate the quality of the fit. This is justified since our simulations are based on idealized data without errors, and we compare the optimum fits of either type without any uncertainties in the fit parameters. However, whenever we draw any conclusions from real data (Sects. 4 and 5) we instead compare the probability of chi-square p(χ2) of the fits. We express rms values for both the Keplerian fits and the sine fits in units of the semi-amplitude K obtained for the Keplerian fit with P = 1.
In order to compare the results from our idealized densely sampled data sets with those that are obtained from real observations, we created rather sparsely sampled data sets consisting of only 20 data points sampled at the temporal pattern of the observations of the planet host star HD 27894 (Moutou et al. 2005). These observations are quite unusual due to the small number of data points that led to the discovery of this planet.
With this sparse sampling grid we simulated again 2:1-RCO RV data with the same 17 values for the amplitude ratio κ2/κ1 and the same 12 phase shift values Δφ as for the densely sampled data set described in Sect. 2.1. We then fitted these data, but restricted ourselves for the purpose of this comparison to only the Keplerian model with period P = 1. Due to the larger number of free parameters compared with the sinusoidal fits the strongest discrepancies are expected for the Keplerian fits.
Figure 1 shows examples of single-planet model fits to the densely sampled simulated input data with different amplitude ratios and two selected values of the phase shift Δφ = 0 and 90, while Table 1 shows the mean rms deviation σmodel of these model fits from the data where the mean has been taken over all 12 phase shift values (see Sect. 3.2 on the relatively small variation of the individual values of σmodel as a function of the phase). Actually, as can be seen from Fig. 2 (left panel), the single Keplerian always yields the best fit, i.e. the smallest σmodel, for , while for the fit with P = 0.5 is better.
Specifically, for an amplitude ratio (Fig. 1, left panels), the data (thick solid line) and the Keplerian fit with the period of the outer planet (thin solid line) become indistiguishable in our plot, whereas the single sinusoidal fit with the same period (dotted line) can be seen to systematically deviate from the data, and the fit with the period of the inner planet (dashed line) is completely inadequate. For an amplitude ratio of 1/2 (Fig. 1, second column of panels), the single Keplerian with the period of the outer planet deviates systematically from the data, but could still be a reasonable match to real data, if signal-to-noise ratios are moderate. For this amplitude ratio, the sinusoidal fit with the period of the outer planet deviates much more from the data, and the fit with the period of the inner planet deviates even more (but less than for the amplitude ratio of 1/8). For an amplitude ratio of 1/1 (Fig. 1, third column of panels), all single-planet fits are inadequate, but the Keplerian fit with the period of the outer planet is still the formally best fit. For this amplitude ratio, the sine fits with P = 1 and P = 0.5 are of equally poor quality since they have the same rms deviation (see Table 1). If the amplitude ratio of the inner vs. the outer planet is reversed then for the fit with the period of the inner planet is the best one, followed by the Keplerian fit with the period of the outer planet, and then the sine fit with the period of the outer planet (see Fig. 1, right panels which are for ).
We also find that in the case of the Keplerian models with period P = 0.5 all fits to the densely sampled data sets yield an eccentricity e = 0 and an amplitude and rms equal to that of the sine fits with the same period value. That is why we will henceforth no longer distinguish between Keplerian fits and sine fits for P = 0.5. As noted above, we have not made any fits with P = 0.5 to the sparsely sampled data sets. In this case, the Keplerian fit with P = 0.5 will normally show differences from the circular fit, since the sparse sampling does no longer guarantee the symmetric distribution of the RVs which is naturally present for a densely sampled sine wave. However in this paper, we will largely concentrate on cases of small deviations from the single sinusoidal model, such as low-eccentricity Keplerians or small-amplitude second circular planets with half the period of the (circular) first one. Consequently, we are exploring the regime where P = 0.5 fits alone are never a good match anyway.
This behaviour can also be deduced from Fig. 2 which shows the dependence of the rms of the different types of fits on the amplitude ratio of the input data (left panel). Figure 2 also shows the variation of the eccentricity e (middle panel) and the semi-amplitude K of the Keplerian fit (right panel) with this ratio.
Fig. 3Rms deviation σmodel of the Keplerian fit with P = 1 from the 2:1 resonant input model (solid line with points and labels). This deviation is expressed in units of the RV semi-amplitude K and plotted as a function of the eccentricity e determined by the fit. The labels indicate the amplitude ratio of the input sinusoids. Thin dashed line: amplitude of the first harmonic in the Fourier expansion of the Kepler equation.
Figure 3 shows the rms deviation of the Keplerian fit with P = 1 from the simulated 2:1-RCO data as a function of the pertinent value for the eccentricity obtained by the same fit. This is based on the densely sampled data. We will use this relation as a tool to identify those of the known RV determined planetary systems for which single objects on eccentric orbits have been fitted, but a pair of 2:1 resonant planets is also a possible model.
Looking, in observed data (with not too sparse sampling and moderate amplitude ratios; see Sect. 3.5), at the scatter σKep of the RV measurements around the fitted Keplerian model (expressed in units of the RV semi-amplitude K of the fit), Fig. 3 allows us to determine the expected (typical, average) contribution σmodel to the scatter arising from the wrong model fit. This means that all systems that we can identify in the literature which have a planet in a putative eccentric orbit as well as a scatter of the data around the eccentric model larger than the one given by the solid curve in Fig. 3 are potential candidates for a 2:1 resonant pair of planets.
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