14 Her: Search for 14 Her b and 14 Her c (November/2006)

Introduction

14 Her is a star with the following characteristics:

In November 2000, astrometry from Han et al (2000) claim a preliminary mass of 13 M J and inclination of 155.3 degrees. In 2002, a 14 Her b component discovered, further referenced by "The ELODIE survey for northern extra-solar planets. III. Three planetary candidates detected with ELODIE" (Naef, D. ; Mayor, M. ; Beuzit, J. L. ; Perrier, C. ; Queloz, D. ; Sivan, J. P. ; Udry, S.) in 2004, and "Long-Period Objects in the Extrasolar Planetary Systems 47 UMa and 14 Her" (Robert A. Wittenmyer , Michael Endl , William D. Cochran) 2006.

In 2005, a second possible planet (14 Her c), with 2 solutions for the RV fit of with the same chi 2 , is found by Gozdiewski et al (2005) with M = 6.289 or 2.086 M J , a = 8.911 or 5.81 AU and e = 0.1 or 0.004. (source: exoplanet.eu). Robert A. Wittenmyer, Michael Endl, William D. Cochran in "LONG-PERIOD OBJECTS IN THE EXTRASOLAR PLANETARY SYSTEMS 47 UMA AND 14 HER" write: "We also present new observations which clearly support a long-period companion in the 14 Her system. With a period of 6906±70 days, 14 Her c may be in a 4:1 resonance with the inner planet." Wittenmyer assumed K=24.5 +/- 1.4 m/s, resulting in minimum mass of 2.1 MJup and e~0. Tperi of 2449100.0 yields a good fit.

Below are the attempts to analyze all available RV data for 14 Her from CORALIE spectrograph (14Her_CORALIE04.vels), Harlan J. Smith Telescope (14Her_HJS06.vels) and B06K dataset (14Her_B06K.vels), obtained from Systemic and see if ASE will generate some common component besides the 14 Her b appearing throughout all the datasets as well as see if 14 Her c suggested by Wittenmyer and others appears throughout these datasets. Additionally, we combine all three data sets to produce a continuous "stitched" data set which will help us to find uncertainities for the long-period component 14 Her c.

Stitching three data sets

Instead of introducing two additional offset variables (CORALIE-HJS and CORALIE-B06K offsets), I suggest to stitch all the three data sets assuming CORALIE data set has zero offset and all other data sets will be offset in respect to it's data. The stitch is done by finding common intraday radial velocity measurements, calculating the difference and thus finding the mean offset between B06K-CORALIE and HJS-CORALIE.

Table A: Common points and offsets

In a new stitched dataset, CORALIE's sigma (instrumental error) remains intact, while B06K and HJS sigma values have been increased by the mean of CORALIE's instrumental error (mean sigma=7.23 m/s) to accomodate stitching mistakes. The result is a data set of 203 observations, covering 4083 days.

Single Keplerian Models (search for 14 Her b)

First, a Lomb-Scargle periodogram is created per each of the dataset and smoothened (blue line) using a 15-day moving average filter in ASE to detect strong periods: (click on the periodogram for full resolution view)

All four periodograms show the most significant peak at approx 1700 days which belongs to 14 Her b. At first, we shall find the best fit with a single 14 Her b companion model for each of the data set. As we already know the search window for the 14 Her b (1700-1800 days), we thus limit the search to 10 models to define the best fit of 14 Her b per each of the set. Results of the search are below (please note t0 varies per each data set as well as t0 and tlast vary, thus mean anomaly at t0 and Tperi are different). Each model was produced by Genetic Algorithm (GA) search with 1000 populations. After GA search was finished, the model was polished with Multiobjective Goal Attainment procedure built on the Sequential Quadratic Programming algorithm. We have used SQP because it seems to work faster than Levenberg-Marquardt which is traditionally used. More about SQP algorithm can be found on Argonne National Laboratory website.

Table 1: Single Keplerian fits for CORALIE data set (119 observations): (Partially shown table. Complete table available at Appendix)

Table 2: Single Keplerian fits for B06K data set (49 observations): (Partially shown table. Complete table available at Appendix)

Table 3: Single Keplerian fits for HJS data set (35 observations): (Partially shown table. Complete table available at Appendix)

Table B: Single Keplerian fits for Stitched data set (203 observations): (Partially shown table. Complete table available at Appendix)

Table 4: 14 Her b: Single Keplerian fits for all data sets

Obviously all data sets have a ChiSq/L > 1 with high value of required jitter (usually 5 m/s is maximum). Exclusion is B06K dataset which has moderately low required jitter, however very high ChiSq/L. All these factors indicate that there is an additional component in the 14 Her system besides 14 Her b. Below are the attempts to analyze residuals after fitting 14 Her b to the RV data.

Double Keplerian Models (search for 14 Her c)

Search Periods: Lomb-Scargle Periodograms of residuals.

First, periodograms are created for the RV data, substracting the best fit of 14 Her b per each data set. (click on the periodogram for full resolution view)

All four data sets residuals indicated a strong presence of periodicity, as seen above, with the following characteristics. An FWHM function has been applied to the central peak frequency to derive minimum and maximum period constraints.

Table 5: 14 Her c search periods

Wittenmyer's 14 Her c period assumption of 6907 days fits the FWHM min/max period constraints well, thus we are on the right track. A minimal period for the new component was derived to be the double of the known 14 Her b period derived in single Keplerian fits, in order to satisfy 2:1 resonance condition between components b and c.

To search 14 Her c, I have decided to start searching double Keplerian models within all the 14 Her datasets, without fixing 14 Her b at some particular position, despite we have derived it's characteristics in single Keplerian fits. The reason for this decision is I suppose that within a single Keplerian model, the best fit of 14 Her b may have "consumed" some amount of signal which potentially belongs to 14 Her c. If we release 14 Her b from the constraints set in the first search, we will allow 14 Her b and 14 Her c to, possibly, exchange their data points and find a better double Keplerian fit. A good evidence showing this problem exists in that if we will try to fix 14 Her b at the best single Keplerian fit parameters and search 14 Her c within the residuals; in this case the eccentricity of 14 Her c always -> 1, i.e. non-realistic models. If we include 14 Her b into search and allow 14 Her c to borrow RV data from 14 Her b and vice versa, 14 Her c eccentricity stabilizies at more or less acceptable levels.

For 14 Her c search, we find 100 double Keplieran models per each data set, with search window for 14 Her b set from 1500 to 1800 days for all data sets and search window for 14 Her c set as per table 5. Maximal eccentricities are limited to 0.8 for both planets. GA search of 1000 populations per each model.

Table 6: Double Keplerian fits for CORALIE data set (Partially shown table. Complete table available at Appendix)

In CORALIE double Keperian models, it's difficult to select the best fit because of great variance of 14 Her c. Additionally, notice that 14 Her c has borrowed some data points from 14 Her b, shifting it's period, semiamplitude and time of periastron passage if compared to the ones derived in single Keplerian fits of 14 Her b on the same data set. Notice there are zero eccentricity 14 Her c models which lie within ~0.02 ChiSq/L from the best fit with 0.8 eccentricity - these models are marked in red. Within these zero eccentricity models, if we fix the mass at Wittenmyer's ~2 MJup, 14 Her c period can be as low as 4875 days with ChiSq/L of the whole fit reaching 1.8725 and mass of 1.8435 MJup. This does not correlate with Wittenmyer's 6907 days period assumption. CORALIE data set analysis also reveals enormous 14 Her c mass variance - from ~1.5 to 16 MJup. Obviously CORALIE data alone is not enough to say anything definitive about 14 Her c.

Table 7: Double Keplerian fits for B06K data set (Partially shown table. Complete table available at Appendix)

With B06K we also see a great variance within 14 Her c, which has consumed data points from 14 Her b, making it thus different from the single Keplerian models derived previously. Notice mass of 14 Her c lowers at high ChiSq/L.

Table 8: Double Keplerian fits for HJS data set (Partially shown table. Complete table available at Appendix)

Conclusion: HJS data set, as well, shows high variance of 14 Her c characteristics, thus all three data sets do not produce enough data to be able to define certain limits for 14 Her c because 14 Her c period is well beyond the data frames captured in observationss in these data sets.

Table C: Double Keplerian fits for Stitched data set (Partially shown table. Complete table available at Appendix)

Within stitched data set, we see traces of similiar models with good ChiSq/L, moderate eccentricity (~0.28), approximately same mass (~2 MJup) and more or less the same period (~5200 days). We decided to continue to work with the best double Keplerian fit marked blue in the table C.

Deriving uncertainities of the best double Keplerian model for the Stitched data set (Quick bootstrap analysis)

Below we bootstrap the best double Keplerian fit for the Stitched data set. 1000 bootstrap trials are executed, each synthetic data set has 100% residuals scrambled after the best double Keplerian fit for the Stitched data set. Each synthetic model is fitted with the best fit from table C and then minimized with sequentive quadratic programming algorithm, i.e. this is a quick bootstrap run without GA search.

Values Distribution in Double Keplerian Fit Bootstrap Trial

Below are the parameter distribution histograms of 2nd bootstrap trial which possibly describe why there is so big difference in orbit parameters of 14 Her c within both bootstrap trials: (Click on the histogram for full resolution representation!)

General Model Parameters:

1st Component (14 Her b):

2nd Component (14 Her c):

Conclusions

Notice double peaks on period, eccentricity, mean anomaly at t0 and time of periastron passage distribution histograms of 14 Her c. Additionally, on the period distribution of 14 Her c we can see that there is a high peak on the maximum period constraint set, ~8000 days. By applying MERCURY6 Hybrid symplectic/Bulirsch-Stoer integrator N-body algorithm, high-eccentric low-period models have been penalized, leaving only those which comply with Wittenmyer's assumption of min period of ~6900 days and a circular orbit.

By repeating the experiment and broadening the period search window of 14 Her c to 18000 days, the situation has repeated with best stabile ChiSqL models having period of 18000 days. Thus, I agree with Wittenmyer's assumption that only the minimal period of 14 Her c can be determined basing on the amount of RV data we have up to date.

Contacts

E-mail: maxim.usatov@bcsatellite.net, polina.usatova@bcsatellite.net