Key Words.:Planets and satellites: detection, dynamical evolution and stability, fundamental parameters, individual: Kepler-9b, Kepler-9c
Aims:Kepler-9 was the first case where transit timing variations have been used to confirm the planets in this system. Following predictions of dramatic TTVs - larger than a week - we re-analyse the system based on the full Kepler data set.
Methods:We re-processed all available data for Kepler-9 removing short and long term trends, measured the times of mid-transit and used those for dynamical analysis of the system.
Results:The newly determined masses and radii of Kepler-9b and -9c change the nature of these planets relative to the one described in Holman et al. 2010 (hereafter H10) with very low, but relatively well charcterised (to better than 7%), bulk densities of 0.18 and 0.14 g cm (about 1/3 of the H10 value). We constrain the masses (45.1 and 31.0 M, for Kepler-9b and -9c respectively) from photometry alone, allowing us to see possible indications for an outer non-transiting planet in the radial velocity data. At Kepler-9d is determined to be larger than suggested before - suggesting that it is a low-mass low-density planet.
Conclusions:The comparison between the H10 analysis and our new analysis suggests that small formal error in the TTV inversion may be misleading if the data does not cover a significant fraction of the interaction time scale.
Transit timing variations (TTVs) are deviations from strict periodicity in extra solar planetary transits, caused by non-Keplerian forces – usually the interaction with other planets in the system. These TTVs are particularly important in multi-transiting systems since they can allow learning about dynamics in the system, which in turn can confirm the exoplanetary origin of the transit signals with no further observations (e.g. Holman et al. 2010, H10 hereafter, or Xie 2013), and sometimes even allow deriving the planets’ mass from photometry alone (Kepler-87, Ofir et al., 2014). For these reasons TTVs had attracted a lot of attention since they were first predicted by Holman & Murray (2005) and Agol et al. (2005), and especially since they where first observed by H10 in the prototypical Kepler-9 system.
Kepler-9 is prototypical not just because it was the first object detected with TTVs, but also since it is a textbook-like example of TTVs: exhibiting very large TTVs on very deep transits, making the effect abundantly clear. The first study of the Kepler-9 system also included a prediction for the expected TTVs during the following few years (their Figure S4) which included dramatic TTV spanning up to about relative to the nominal ephemeris, accumulated over long interaction times scales (e.g. from first maximum to first minimum TTV excursion of Kepler-9c). These very large TTVs are easy to compare to the observed ones in later Kepler data. Indeed by the time we re-analysed this object much more data were available, revealing that the actual TTVs, while still large, were much less extreme than initially predicted. We observed TTV spans of about for the same features as above, and TTV time scale about half as long as predicted. These prompted us to revisit the analysis of Kepler-9.
This paper is therfore divided in the following way: in sections §2 and §3 we describe the input data and TTV analysis procedures we used. In §4 we made sure we are able to recover the H10 results when using only the data that was available at the time, showing consistent analysis, which then allowed us to perform a full analysis using the full data set in §5, before discussing the updated analysis in §6.
2 Photometry and light curve modeling
We processed quarters 1 through 16 of Kepler-9 long cadence photometry which spans 1426d, more than six times longer than the original study of H10. We processed it similarly to the processing of Kepler-87 (Ofir et al., 2014), fitting for Kepler-9b and Kepler-9c’s individual times of mid-transit, and fitting Kepler-9d using linear ephemeris since we detected no TTVs for it. In short, this processing used a light curve that was corrected for short-term systematic effects using the SARS algorithm (Ofir et al., 2010; Ofir & Dreizler, 2013), which was then iteratively: fitted for long-term trends, modeled for its transit signals, and corrected for the model by dividing it out – till convergence. The resultant photometric model parameters are given in Table 1.
3 TTV modeling
We did not include Kepler-9d in the TTV modeling since it is dynamically decoupled from the outer two planets (see also Section 5). For the modeling of the TTVs we use the mercury6 code (Chambers & Migliorini, 1997; Chambers, 1999). The integration of the planetary orbits is done using the Bulirsch-Stoer integrator implemented in mercury6 starting from a set of initial values for the orbital elements for the planetary system. For integration of orbits in the Kepler-9 system, we use a time step of 0.5 days, i.e. about one 40 of the orbital period of planet b. The duration of the integration is limited to the duration of the Kepler mission. From the osculating orbital elements at each time step we calculate the next transit time. The final transit times are then calculated from spline interpolations. These calculated and the observed mid transit times are used to run a Levenberg-Marquardt optimization resulting in an optimized parameter set for the planetary system.
Since the fit may depend on the choice of the initial values, we use the best fit parameters as well as the formal fit errors from the covariance matrix for a second extended fit. Within the 3- limit we randomly vary the start parameters, however obeying parameter limits, e.g. positive eccentricity, if required. This procedure probes the -landscape around the initial best fit value, it typically finds a better best fit and we use the distribution of parameters as an estimate of the error bars.
As a final check, we also integrate the best fit orbital solution over 5 Gyr using the hybrid-symplectic integrator of mercury6 at a time step of 0.8 days. Only a long-term stable solution is accepted.
4 Recovery of previous results
In a first step, we use the TTV data from H10 in order to demonstrate that we can recover their solution. Given the rather low number of TTV measurements, we restrict the orbits to coplanar orbits, given the low dispersion in measured inclinations that seems not to be a restrictive constraint. The free parameters therefore are the orbital period, eccentricity, argument of periastron, and mean anomaly at the beginning of the integration for each of the two planets. We take the mass of the central star as input with a distribution according to Havel et al. (2011). During each fit, the stellar mass is fixed. Instead of using the planetary masses as parameters, the mass of planet b is given relative to the stellar mass, the mass of planet c relative to that of planet b. We use 2500 random starting values for the Levenberg-Marquardt optimization as described in Sect. 3. As also discussed by H10, the planetary masses can only be weakly constrained from the partial TTV data set. We therefore also included the radial velocity (RV) measurements from H10 in our fit for the partial data set.
|Parameter||this work||Holman et al.|
|best fit||best fit|
|this work||Havel et al.|
|derived, i.e. not fitted parameters|
|note that H10 use and|
Our best fit parameters for the partial data set are summarized in Table 2 and compared to pervious results. Our error bars are taken from the distribution of parameters as shown in Figures 1 and 2. The best fit model compared to the RV data is shown in Figs. 3. The comparison with Fig. (3) of H10 shows an nearly identical situation: The observed RV variations can be matched reasonably well, however, the deviation between the model and observation is up to 8 m/s and larger then expected given the error bars. Like in H10, we also conclude that more RV observations would be necessary to check for the influence of additional planets or stellar activity jitter. In Fig. 4 we show the TTV data together with our best fit of the partial data set and in Fig. 5 we show the residuals to that fit. The reduced of 2.4 is dominated by the deviations from the RV measurements while the TTV measurements can be matched within their error bars.
Basically, we can recover the literature values from H10 for the planetary masses and the mass ration which agree well within the error range. Also, the eccentricities and arguments of periastron agree within the error range. In the other orbital parameters, we find some discrepancies. We find a significantly lower orbital period for planet b but a higher for planet c.
The fit to the partial data is also extended towards later measurements (Fig. 4) and compared with the actual transits observed by Kepler. About 250 days after the last transit reported by H10 the deviation is already 0.1 days for planet b and 0.3 days for planet c. Given the small error bars, it is clear that the planetary parameters have to be re-determined using the full Kepler data set.
5 TTV analysis using the full data set
We repeated the analysis using the full data set in the same way as described in the previous section, except that we now allow for non coplanar orbits and increased the number of random starting values to 3000 (see Figures 6, and 7). The full data set also allows to constrain the planetary masses without including the RV data in the fit. As expected from the poor agreement between the extrapolated solution from the partial data set, the planetary parameters changed. The main change is a significant reduction of the planetary masses in our new fit, while the mass ratio agrees within the error range. In Table 3 we compare the new parameters to those we obtained from the partial data set (Table 2). The increase in the orbital period for planet b and the decrease for planet c are expected from the deviations seen in Figure 4.
Combining the fitted parameter from Table 1 with the orbital semi major axis from the TTV fit (Table 3), we derive the stellar radius. Using this in combination with (Table 1) provides the planetary radius, which then can be combined with the planetary masses (Table 3) to the mean planetary densities (Table 3, lower block).
Comparing the observed and calculated transit times in Fig. 8 now shows a good match over the whole observing period. This can also be seen from the residuals (Fig. 9) as well as from the reduced of 1.81. The lower planetary masses, however, lead to a less good match of the RV variations (Figures 10 and 11). We now have residuals of up to 12 m/s. Note, however, that the H10 masses were determined using the RVs in the fit, minimizing the RVs residuals by construction, whereas our full-dataset model did not fit for the RVs data.
|Parameter||this work||this work|
|full data set||partial data set|
|without RV||with RV|
|best fit||best fit|
The discrepancy of the observed and calculated RV variations as well as possible slight systematic residuals in the TTVs of planet c raised the question whether or not we can find evidence for a forth, possibly non-transiting planet in the system. We therefore first searched for additional transiting planets in the system in the photometric model’s residuals using Optimal BLS (Ofir, 2014) but found none. We also repeated the dynamical analysis adding an outer, co-planer, plane to the system. Since the parameter range for an outer planet is huge, we restricted our search to orbits of the test planet in 3:2, 2:1, 5:2, and 3:1 mean motion resonances to planet c. No solution with a better reduced could be found. We note that the short time span of the RV data – less than one orbit of Kepler-9c – severly limit the orbits that one can hope to fit to such a test outer planet.
Additionally, we have also checked our assumption that planet d has no impact on our results: We included planet d in the dynamical model by fixing its orbital period at the measured value, assuming a co-planar and circular orbit, the latter motivated by the short circularization time scale at the small orbit distance, and determined the mean anomaly at the beginning of the integration to match the measured transit time . We then refitted the full TTV data set for planet d in the mass range of 1 to 30 Earth masses letting the parameters of planet b and c re-adjust. We find a very shallow -minimum around 10,M but with insignificant improvement compared to the tested mass range for planet d. The parameters of planet b and c are unaffected within their error bars. We conclude that we cannot derive any meaningful constraints on the mass of planet d and find our solution sumarized in Table 3 unaffected.
We conclude that neither an additonal outer planet in a low-order resonant orbit nor the inclusion of planet d can improve on the very systematic residuals of the RV signal. A more extended RV follow-up would be necessary in order to come to a more conclusive result.
6 Discussion and conclusion
6.1 Partial vs. full dataset
We re-analysed the Kepler-9 system using both the partial Kepler data set that was available to H10 and the full data set available today. The comparison between the previous and new results show, that a very good fit to a planetary system in first order mean motion resonance can be misleading if only a fraction of the interaction time scale is covered. Even the much longer currently available Kepler data set might not be sufficiently long for that. We therefore follow H10 and extrapolated our best fit model into the future (Fig. 8 and Table 4). Given the large TTVs, ground based observations even with a marginal detection of the transit should be able to check the solution proposed in this work.
H10 could confirm the Kepler-9b and Kepler-9c as planets from photometry alone, but could only place weak constraints on their masses without using RV data. They therefore included a few RV measurements in their fit, and it comes as no surprise that the RV fit is good since the partial photometry of the time did not have the constraining power to match the RV data. They also predicted that future Kepler data would be more constraining of the planetary masses, and indeed our results have smaller formal error bars on both planets’ masses from photometry alone. We note, however,that the systematic residuals shown in Fig. 9, and especially Fig. 11 cause us to warn of unmodeled phenomena, such as other planets in the system or longer time-scale interaction between the planets or stellar activity.
6.2 The revised planets
The scaled radii r we determined are slightly larger than the ones obtained by H10 by and for Kepler-9b and Kepler-9c, respectively. The new values are much more constrained with formal errors 5 to 8 times smaller. Actually, Kepler’s data allows in principle to determine the planets’ mass to and the planets’ radii to better than 0.2% – but those are limited by our knowledge of the host star properties. Furthermore, Kepler-9 was measured in short cadence mode (1 minute sampling instead of the regular 30-minute sampling) starting from Quarter 7, which allows for an even better timing precision (and thus mass determination). While we did not use short cadence data, using this data would have had little effect on the global uncertainty which is dominated by stellar parameters errors.
The newly determined masses and radii of Kepler-9b and -9c change the nature of these planets relative to the one described in H10. Both planets are now detetmined to have size similar to Jupiter’s but they are 7 to 10 times less massive than Jupiter, i.e. have densities about 1/3 of the density given in H10. Consequently, both planets have very low derived densities of and – among the lowest known. H10 specifically excluded coreless models for the planets, but the more abundant data we have today forces us to reconsider that Kepler-9b and -9c may not have cores at all. This result is of special interest in the context of the core accretion theory (Pollack et al., 1996): with masses of 30.6 and 44.5 these planets have apparently just started their runaway growth when it stopped at this relatively rare intermediate mass.
Figure 12 shows the masses and radii of lower-mass () planets that have both mass and radius known to better than 111Extracted from the NASA Exoplanet Archive (http://exoplanetarchive.ipac.caltech.edu/) on January 21, 2014. It is evident that the new locations of Kepler-9b and -9c put them at the edge of the mass-radius distribution, with very low density and in a mass range that is very poorly sampled, and yet – both planets are now among the best-characterized exoplanets known with bulk densities known to or better. The recent successful launch of the GAIA mission further highlights that last point: the knowledge about both Kepler-9b and -9c in both radius and mass is limited by the knowledge about their host star. GAIA’s observations will fix Kepler-9’s properties to high precision, allowing to use other data (such as the available short cadence data) to further reduce the uncertainty on the physical parameters of Kepler-9b and -9c, and significantly so.
Finally, we note that Kepler-9d is now determined to have a radius of , an increase relative to H10. The increased size, together with the low metal content of its neighboring planets, suggest that Kepler-9d may not be rocky, or at least that it may have a significant volatiles fraction, again unlike the initial suggestion by H10. If this is true, then Kepler-9d is perhaps similar to the new and exciting subgroup of low-mass low-density planets (e.g. Kepler-87c or GJ 1214 Ofir et al., 2014; Charbonneau et al., 2009; Fortney et al., 2013)
A.O. acknowledges financial support from the Deutsche Forschungsgemeinschaft under DFG GRK 1351/2.
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