TTVs are derived from Q1-Q17 Kepler data.  x-axes: “Observed Tc” (Mid-Transit Time): EXOFAST’s best-fits from Normalized PDCSAP_FLUX Kepler light flux vs. time (BJD_tdb - 2454900) data.  y-axes: “(O – C)”: difference between Observed Tc and the Calculated Tc from the graphically obtained linear ephemeris.

Figure 1a.: OFFLINE, KOI-5086.01, P = 21.93 days [Plot avg. error bars = ± 0.52 min. (smaller than symbols)]
TTV_minimum: 219.26 ± 42.44 days, Amp_ttv_minimum: -17.46 ± 1.74 min.
TTV_maximum: 850.14 ± 57.84 days, Amp_ttv_maximum: 21.44 ± 1.74 min.
P_ttv: 1261.78 ± 60.36 days.
Amp_ttv: 38.90 ± 2.46 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 1323.09 days; Power: 23.67; FAP: 3.241 x 10^-8.
Linear ephemeris (this work): Tc = [21.92864124 ± 0.00007555](Tc#) + [68.52039236 ± 0.00284232]


Figure 2a.: OFFLINE, Residuals of Figure 1a. [Plot avg. error bars = ± 1.82 min.]
TTV_maximum: 373.27 ± 25.95 days, Amp_ttv_maximum: 8.72 ± 1.06 min.
TTV_minimum: 705.39 ± 30.91 days, Amp_ttv_minimum: -7.28 ± 1.06 min.
TTV_maximum: 1037.51 ± 37.56 days, Amp_ttv_maximum: 8.72 ± 1.06 min.
TTV_minimum: 1369.63 ± 45.17 days, Amp_ttv_minimum: -7.28 ± 1.06 min.
P_ttv: 664.24 ± 18.64 days.
Amp_ttv: 16.01 ± 1.50 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 674.85 days; Power: 17.81; FAP: 1.145 x 10^-5.


Figure 3a.: OFFLINE, Residuals of Figure 2a.; Residuals-of-the-Residuals of Figure 1a. [Plot avg. error bars = ± 2.11 min.]
TTV_maximum: 105.92 ± 13.05 days, Amp_ttv_maximum: 4.74 ± 0.59 min.
TTV_minimum: 305.78 ± 13.67 days, Amp_ttv_minimum: -5.74 ± 0.59 min.
TTV_maximum: 505.63 ± 14.84 days, Amp_ttv_maximum: 4.74 ± 0.59 min.
TTV_minimum: 705.49 ± 16.42 days, Amp_ttv_minimum: -5.74 ± 0.59 min.
TTV_maximum: 905.34 ± 18.31 days, Amp_ttv_maximum: 4.74 ± 0.59 min.
TTV_minimum: 1105.20 ± 20.43 days, Amp_ttv_minimum: -5.74 ± 0.59 min.
TTV_maximum: 1305.05 ± 22.71 days, Amp_ttv_maximum: 4.74 ± 0.59 min.
P_ttv: 399.71 ± 5.71 days.
Amp_ttv: 10.47 ± 0.84 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 400.29 days; Power: 19.81; FAP: 1.540 x 10^-6.


Figure 4a.: OFFLINE, Summed combination of Figures 1a., 2a., and 3a. [Plot error bars = ± 2.83 min.]

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Figure 1b.: ONLINE, KOI-5086.01, P = 21.93 days [Plot avg. error bars = ± 0.56 min. (smaller than symbols)]
TTV_minimum: 223.25 ± 42.02 days, Amp_ttv_minimum: -17.32 ± 1.74 min.
TTV_maximum: 851.27 ± 57.25 days, Amp_ttv_maximum: 21.38 ± 1.74 min.
P_ttv: 1256.05 ± 59.45 days.
Amp_ttv: 38.70 ± 2.46 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 1323.09 days; Power: 23.58; FAP: 3.571 x 10^-8.
Linear ephemeris (this work): Tc = [21.92864054 ± 0.00007524](Tc#) + [68.52039062 ± 0.00283069]


Figure 2b.: ONLINE, Residuals of Figure 1b. [Plot avg. error bars = ± 1.83 min.]

TTV_maximum: 374.12 ± 25.07 days, Amp_ttv_maximum: 8.87 ± 1.05 min.
TTV_minimum: 704.74 ± 29.82 days, Amp_ttv_minimum: -7.37 ± 1.05 min.
TTV_maximum: 1035.35 ± 36.18 days, Amp_ttv_maximum: 8.87 ± 1.05 min.
TTV_minimum: 1365.97 ± 43.45 days, Amp_ttv_minimum: -7.37 ± 1.05 min.
P_ttv: 661.23 ± 17.86 days.
Amp_ttv: 16.24 ± 1.48 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 674.85 days; Power: 18.19; FAP: 7.767 x 10^-6.


Figure 3b.: ONLINE, Residuals of Figure 2b.; Residuals-of-the-Residuals of Figure 1b. [Plot avg. error bars = ± 2.11 min.]

TTV_maximum: 107.35 ± 12.73 days, Amp_ttv_maximum: 4.70 ± 0.58 min.
TTV_minimum: 306.67 ± 13.35 days, Amp_ttv_minimum: -5.69 ± 0.58 min.
TTV_maximum: 506.00 ± 14.48 days, Amp_ttv_maximum: 4.70 ± 0.58 min.
TTV_minimum: 705.33 ± 16.03 days, Amp_ttv_minimum: -5.69 ± 0.58 min.
TTV_maximum: 904.65 ± 17.87 days, Amp_ttv_maximum: 4.70 ± 0.58 min.
TTV_minimum: 1103.98 ± 19.93 days, Amp_ttv_minimum: -5.69 ± 0.58 min.
TTV_maximum: 1303.30 ± 22.16 days, Amp_ttv_maximum: 4.70 ± 0.58 min.
P_ttv: 398.65 ± 5.57 days.
Amp_ttv: 10.39 ± 0.82 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 400.29 days; Power: 20.08; FAP: 1.174 x 10^-6.


Figure 4b.: ONLINE, Summed combination of Figures 1b., 2b., and 3b. [Plot error bars = ± 2.85 min.]

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Figure 5a.: OFFLINE Transit Duration vs. Time [Average error bars smaller than symbols].

Figure 6a.: OFFLINE Transit Depth vs. Time [Average error bars smaller than symbols].
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Figure 5b.:  ONLINE Transit Duration vs. Time [Average error bars smaller than symbols].
Figure 6b.:  ONLINE Transit Depth vs. Time [Average error bars smaller than symbols].
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Numerous literature and major Tc#, Tc, and TTV tabulation references can be found on my "Summary" webpage following the table.

7 Dec 2014

       Kepler KOI-5086 (KIC-4769799): Multiple Exoplanets or Multiple Stars

TTV and/or ETV Evidence:
The presence of both a credible periodicity in the Lomb-Scargle Periodogram (LSP) and a corresponding pronounced one in a sinusoidal distribution of [(O-C) vs. Time] data points is strongly suggestive of the presence of additional (including unseen, non-transiting) gravitationally-perturbing objects in a system.

For KOI-5086.01, uncertainty still exists as to whether the observed transits are due to an exoplanet or an eclipsing binary (EB) or higher system:
- NASA Exoplanet Archive (NEA): "planetary candidate".
- Community Follow-up Observing Program site (https://CFOP.ipac.caltech.edu): "planetary candidate or possible 'False Positive'", e. g., an EB.
- Kepler Eclipsing Binary Catalog v2 (http://keplerebs.villanova.edu): "eclipsing binary".
- Data Validation (DV) Report for Kepler ID 4769799, Quarters 1-16, 16-Aug-2013 (http://exoplanetarchive.ipac.caltech.edu):
--- pg. 2, Summary: "NOT ('false') a suspected EB".
--- pg. 102, Appendix, Table A-2, Average Odd vs. Even Transit Depths: 22435 ppm vs. 22495 ppm…inconclusive for an EB.

In the example of 5086.01 (regardless of its eventual definitive disposition), two extremely similar sets of EXOFAST-generated data were obtained, one from an OFFLINE run and one from an ONLINE run.  In each case, the (O-C) vs. Time data showed clear, and in good agreement, periodicity in both a Lomb-Scargle Periodogram (LSP) and a sinusoidal curve-fit.  Furthermore, in each case, the plots' residuals data and the subsequent plots' residuals-of-residuals data also showed clear, and in good agreement, periodicities in their LSPs and sinusoidal arrays.  Lastly, the summed combinations of sinusoidal curves, in each case, produced complex overall curves that were quite reasonably consistent with the originally-obtained data.

{In each of the Figures that follow, "TTV" is designated; the future will decide if these should be changed to "ETV".}
OFFLINE, TTV: 1323.09 days (LSP); 1261.78 ± 60.36 days (sinusoidal best-fit), Figure 1a.
 ONLINE, TTV: 1323.09 days (LSP); 1256.05 ± 59.45 days (sinusoidal best-fit), Figure 1b.
OFFLINE, TTV Residuals: 674.85 days (LSP); 664.24 ± 18.64 days (sinusoidal best-fit), Figure 2a.
 ONLINE, TTV Residuals: 674.85 days (LSP); 661.23 ± 17.86 days (sinusoidal best-fit), Figure 2b.
OFFLINE, TTV Residuals-of-Residuals: 400.29 days; 399.71 ± 5.71 days (sinusoid. best-fit), Figure 3a.
 ONLINE, TTV Residuals-of-Residuals: 400.29 days; 398.65 ± 5.57 days (sinusoid. best-fit), Figure 3b.
OFFLINE, SUM of 3 Sinusoidal Curves + original TTV data; see Figure 4a.
 ONLINE, SUM of 3 Sinusoidal Curves + original TTV data; see Figure 4b.


While it is certainly possible that some of this unusual curvature obtains from eccentric (and maybe even precessing eccentric) orbits in this system, it is also possible within the exoplanet scenario that at least 4 planets (with only 5086.01 transiting), in near-circular orbits, are mutually-interacting to give the TTV distribution observed.

Still within this scenario, 5086.01's near-isosceles-triangular-transit shapes are typical for a grazing planet and consistent with the following observed values:
OFFLINE, Avg. T_(FWHM): 0.207131 ± 0.000720 days; Avg. tau: 0.202938 ± 0.001247 days; Avg. T_(1,4): 0.410069 ± 0.001440 days; b: 1.05 ± 0.28.
 ONLINE, Avg. T_(FWHM): 0.202618 ± 0.000774 days; Avg. tau: 0.202618 ± 0.001341 days; Avg. T_(1,4): 0.405236 ± 0.001549 days; b: 1.13 ± 0.59.

The extent of this grazing reduces significantly with time.  This is evidenced by the large error in the impact parameters (b) above and, more clearly, in the observed trends in transit depths and durations; see Figures 5a, 6a, 5b, and 6b.  The apparent minima in each of these plots near 682.5 days, rather than being some sort of discontinuity, is probably (as in the case of the TTVs) associated with the effects of the various other interacting-planetary-objects in the system.

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