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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 (derived using the graphically obtained linear ephemeris).

1st Figure.: KOI-806.03 TTV, P = 29.16 days [Plot avg. error bars = ± 8.91 min. (smaller than symbols)]
TTV_minimum: 359.80 ± 4.08 days, Amp_ttv_minimum: -1214.27 ± 21.64 min.
TTV_maximum: 855.80 ± 5.41 days, Amp_ttv_maximum: 1685.93 ± 21.64 min.
TTV_minimum: 1351.80 ± 7.22 days, Amp_ttv_minimum: -1214.27 ± 21.64 min.
P_ttv: 992.01 ± 4.54 days.
Amp_ttv: 2900.20 ± 30.60 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 990.49 days; Power: 20.72; FAP: 0.
Linear ephemeris (this work): Tc = 29.16319753(Tc#) + 83.71026895

2nd Figure.: Residuals of TTV of 806.03. [Plot avg. error bars = ± 23.40 min.]
Binomial Equation: (O-C) = 0.000102(Tc^2) - 0.01584(Tc) - 71.454308.

3rd Figure.: KOI-806.02 TTV, P = 60.33 days [Plot avg. error bars = ± 1.11 min. (smaller than symbols)]
TTV_maximum: 380.31 ± 14.46 days, Amp_ttv_maximum: 25.05 ± 1.50 min.
TTV_minimum: 866.98 ± 18.36 days, Amp_ttv_minimum: -40.50 ± 1.50 min.
TTV_maximum: 1353.65 ± 23.77 days, Amp_ttv_maximum: 25.05 ± 1.50 min.
P_ttv: 973.34 ± 14.14 days.
Amp_ttv: 65.55 ± 2.12 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 984.25 days; Power: 9.23; FAP: 1.684 x 10^-8.
Linear ephemeris (this work): Tc = 60.32503546(Tc#) + 176.89446110

4th Figure.: Residuals of TTV of 806.02. [Plot avg. error bars = ± 23.40 min.]
TTV_maximum: 190.80 ± 18.60 days, Amp_ttv_maximum: 4.22 ± 0.63 min.
TTV_minimum: 384.93 ± 19.64 days, Amp_ttv_minimum: -4.50 ± 0.63 min.
TTV_maximum: 579.06 ± 21.25 days, Amp_ttv_maximum: 4.22 ± 0.63 min.
TTV_minimum: 773.19 ± 23.33 days, Amp_ttv_minimum: -4.50 ± 0.63 min.
TTV_maximum: 967.32 ± 25.76 days, Amp_ttv_maximum: 4.22 ± 0.63 min.
TTV_minimum: 1161.45 ± 28.45 days, Amp_ttv_minimum: -4.50 ± 0.63 min.
TTV_maximum: 1355.58 ± 31.33 days, Amp_ttv_maximum: 4.22 ± 0.63 min.
P_ttv: 388.26 ± 7.29 days.
Amp_ttv: 8.72 ± 0.89 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 388.06 days; Power: 8.42; FAP: 2.060 x 10^-6.

5th Figure.: KOI-806.01 TTV, P = 143.22 days [Plot avg. error bars = ± 2.39 min. (smaller than symbol)]
TTV_maximum: 98.00 ± 23.86 days, Amp_ttv_maximum: 59.65 ± 15.89 min.
TTV_minimum: 286.83 ± 25.15 days, Amp_ttv_minimum: -79.50 ± 15.89 min.
TTV_maximum: 475.65 ± 27.53 days, Amp_ttv_maximum: 59.65 ± 15.89 min.
TTV_minimum: 664.47 ± 30.74 days, Amp_ttv_minimum: -79.50 ± 15.89 min.
TTV_maximum: 853.29 ± 34.56 days, Amp_ttv_maximum: 59.65 ± 15.89 min.
TTV_minimum: 1042.11 ± 38.80 days, Amp_ttv_minimum: -79.50 ± 15.89 min.
TTV_maximum: 1230.93 ± 43.35 days, Amp_ttv_maximum: 59.65 ± 15.89 min.
P_ttv: 377.64 ± 11.14 days.
Amp_ttv: 139.15 ± 22.47 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 374.89 days; Power: 3.28; FAP: 0.0236.
[LSP minor periodicity: P_ttv: 602.17 days; Power: 2.17; FAP: 0.333 (not very credible; but see below).]
Linear ephemeris (this work): Tc = 143.20781303(Tc#) + 87.22997966

6th Figure.: Residuals of TTV of 806.01 [Plot avg. error bars = ± 16.07 min.]
TTV_minimum: 229.75 ± 59.32 days, Amp_ttv_minimum: -20.75 ± 8.98 min.
TTV_maximum: 538.31 ± 67.58 days, Amp_ttv_maximum: 24.89 ± 8.98 min.
TTV_minimum: 846.87 ± 80.37 days, Amp_ttv_minimum: -20.75 ± 8.98 min.
TTV_maximum: 1155.44 ± 95.88 days, Amp_ttv_maximum: 24.89 ± 8.98 min.
P_ttv: 617.13 ± 41.05 days.
Amp_ttv: 45.64 ± 12.69 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 613.44 days; Power: 2.61; FAP: 0.157.

References:
• For Q1-6 TTV data of 806.03, 806.02, and 806.01 (thru ~ 563 (BJD-2454900) days), see: Fabrycky et al., 2012, arXiv-1201.5415 [ApJ, 750, 114 (2012)]
• For Q1-6 TTV data of 806.03, 806.02, and 806.01 (thru ~ 563 (BJD-2454900) days), see: Ford et al., 2012, arXiv-1201.1892.
• For Q1-12 TTV data of 806.03, 806.02, and 806.01 (thru ~ 1116 (BJD-2454900) days), see: Mazeh et al., 2013, arXiv-1301.5499.
• For Q1-12 Calc. properties from TTVs of 806.03 and 806.02, see: Hadden & Lithwick, 2013, arXiv-1310.7942.
• For Q1-12 TTV data of 806.03, 806.02, and 806.01 (thru ~ 934 (BJD-2454900) days), see Rowe et al., 2014, arXiv-1402.6534.

23 Apr 2014
KOI-806 (Kepler-30, KIC-3832474) 4?-(or more?)-Planets

Discussion:
In anticipation of KAFO observations beyond Kepler, transit timing variations (TTVs) were determined for the three confirmed planets of the KOI-806 system using Kepler data from Q1 through the first-third of Q17 (when Kepler lost its required exquisitely-stabilizing third reaction wheel).  Several reports have appeared previously (see references below) describing TTVs and derived properties through either Q6 or Q12.

806.03 (Kepler-30b), the innermost planet in this system, shows a TTV periodicity (P_ttv; 1st Figure below) of 992.01 days (errors tabulated below and on the figures) and an enormous overall total amplitude (Amp_ttv) of 2900.20 minutes (or ~ 48.3 hours).  This is supported by a strong, highly credible, Lomb-Scargle periodicity (LSP) of 990.49 days.  The ratio of periods of the first two planets, 806.03 and 806.02, is just wide of a mean motion resonance (MMR) of 2 (actually, 2.07:1.00).  The large amplitude is consistent with the huge difference in the masses of the pair (reported by Haddon & Lithwick): M_806.03_earth: 76.6 and M_806.02_earth: 1562.8.  The residuals from the TTV plot showed no clear periodicity although it they do indicate a gradual drift (see 2nd Figure below), potentially resulting from orbital eccentricities and/or precessions.

806.02 (Kepler-30c)'s TTV is extremely well anti-correlated with that of 806.03 (compare 1st and 3rd Figures below), typical of strongly-interacting planets.  It's P_ttv is 973.34 days (LSP: 984.25 days) which matches closely with that for 806.03.  Its Amp_ttv, again as expected from the relative masses reported for 806.02 and 806.03, is considerably smaller: 65.55 minutes.  The residuals from the TTV plot (4th Figure below) also show a clear "secondary P_ttv" of 388.26 days (LSP: 388.06 days) with an Amp_ttv of 8.72 minutes; this is consistent with the interaction of 806.02 with the third planet, 806.01.

806.01 (Kepler-30d) also shows a strong TTV (see 5th Figure) not perfectly anti-correlated (offset for the maxima and minima, ranging from 98-125 days, probably due to relative eccentricities) with the "secondary P_ttv" of 806.02, and with a similar P_ttv of 377.64 days (LSP: 374.89 days); its Amp_ttv is 139.15 minutes.  The residuals from 806.01 (see 6th Figure) also describe a clear sinusoidal array, another "secondary P_ttv", that could be arising through the interaction with a fourth, as yet unseen (non-transiting) planet ("806.xx") having a P_ttv of 617.13 days (LSP: 613.44 days) and an Amp_ttv of 45.64 minutes.