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     Kepler KOI-2433 (Kepler-385; KIC-11968463) 6-(or more?)-Planet System
                              (2 Confirmed Planets; 4 Candidate Planets)

Discussion:
KOI-2433.06:
P=6.06 days; no significant TTV observed (Figure 1)
KOI-2433.02 (Kepler-385b):
P=10.04 days; no significant TTV observed (Figure 2)
KOI-2433.01 (Kepler-385c):
P=15.16 days; TTV: Sinusoidal Fit: 910.31 days (Figure 3); LSP: 898.47 d.
KOI-2433.04:
P=27.90 days; TTV: Sinusoidal Fit: 133.89 days (Figure 4); LSP: 133.97 days.
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KOI-2433.03:
P=56.41 days; after a sinusoidal curve-fit of the (O-C) vs. Time data showed (Figure 5 below) a periodicity (P_ttv) of 247.11 ± 7.77 days (246.08 days was observed in the Lomb-Scargle Periodogram (LSP) of the same data), a plot (Figure 6) of the Residuals also gave a good sinusoidal curve-fit with a periodicity of 166.67 ± 2.96 days (LSP: 166.88 days).  The added combination of these two sinusoidal curves (arrayed in Figure 7) produces a complex overall curvature reasonably consistent with the initial data.  While it is certainly possible (see recent work of Lithwick and others) that some of this unusual curvature obtains from eccentric orbits (and maybe even precessing eccentric orbits) of planetary objects in this system, it is also possible that at least 2 of the other planets are mutually-interacting with 2433.03 to give the TTV distribution observed.
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KOI-2433.07:
P=86.43 days; TTV: Sinusoidal Fit: 238.59 days (Figure 8); LSP: 240.05 days.


Finally, note that the strongest P_ttv of KOI-2433.03 (Figure 5, ~ 247 days) is very similar and anti-correlated to that of KOI-2433.07 (Figure 8, ~ 239 days).  Further, the relative amplitudes of these two effects points to KOI-2433.03 as the more massive planet of the two.  Since the NASA Exoplanet Archive reports that they have essentially the same radii (Rp/R_earth ~ 2.46), the density of KOI-2433.03 must also be the higher of the two.

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 1.: KOI-2433.06, P = 6.06 days [Plot avg. error bars = ± 11.49 min.]
Linear ephemeris (this work): Tc = [6.06327695 ± 0.00006497](Tc#) + [69.18416010 ± 0.00966974]


Figure 2.: KOI-2433.02, P = 10.04 days [Plot avg. error bars = ± 10.11 min.]
Linear ephemeris (this work): Tc = [10.04370979 ± 0.00009396](Tc#) + [286.61617422 ± 0.00654360]


Figure 3.: KOI-2433.01, P = 15.16 days [Plot avg. error bars = ± 10.34 min.]
TTV_maximum: 728.44 ± 72.24 days, Amp_ttv_maximum: 57.27 ± 9.57 min.
TTV_minimum: 1183.59 ± 90.13 days, Amp_ttv_minimum: -44.88 ± 9.57 min.
P_ttv: 910.31 ± 52.60 days.
Amp_ttv: 102.14 ± 13.54 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 898.47 days; Power: 12.11; FAP: 0.00306.
Linear ephemeris (this work): Tc = [15.16208889 ± 0.00020331](Tc#) + [77.48586573 ± 0.01187111]


Figure 4.: KOI-2433.04, P = 27.90 days [Plot avg. error bars = ± 14.91 min.]
TTV_maximum: 338.79 ± 7.87 days, Amp_ttv_maximum: 70.21 ± 11.73 min.
TTV_minimum: 405.74 ± 8.04 days, Amp_ttv_minimum: -42.05 ± 11.73 min.
TTV_maximum: 472.69 ± 8.25 days, Amp_ttv_maximum: 70.21 ± 11.73 min.
TTV_minimum: 539.64 ± 8.48 days, Amp_ttv_minimum: -42.05 ± 11.73 min.
—(10 intervening additional maxima & minima)—
TTV_maximum: 1276.05 ± 12.15 days, Amp_ttv_maximum: 70.21 ± 11.73 min.
TTV_minimum: 1343.00 ± 12.55 days, Amp_ttv_minimum: -42.05 ± 11.73 min.
TTV_maximum: 1409.94 ± 12.96 days, Amp_ttv_maximum: 70.21 ± 11.73 min.
TTV_minimum: 1476.89 ± 13.38 days, Amp_ttv_minimum: -42.05 ± 11.73 min.
P_ttv: 133.89 ± 1.01 days.
Amp_ttv: 112.26 ± 16.59 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 133.97 days; Power: 6.36; FAP: 0.00547.
Linear ephemeris (this work): Tc = [27.90427221 ± 0.00044317](Tc#) + [87.92938724 ± 0.01427768]

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Figure 5.: KOI-2433.03, P = 56.41 days [Plot avg. error bars = ± 15.20 min.]
TTV_minimum: 408.43 ± 31.21 days, Amp_ttv_minimum: -27.46 ± 9.87 min.
TTV_maximum: 531.98 ± 33.00 days, Amp_ttv_maximum: 29.88 ± 9.87 min.
TTV_minimum: 655.53 ± 35.12 days, Amp_ttv_minimum: -27.46 ± 9.87 min.
TTV_maximum: 779.09 ± 37.53 days, Amp_ttv_maximum: 29.88 ± 9.87 min.
TTV_minimum: 902.64 ± 40.17 days, Amp_ttv_minimum: -27.46 ± 9.87 min.
TTV_maximum: 1026.19 ± 43.00 days, Amp_ttv_maximum: 29.88 ± 9.87 min.
TTV_minimum: 1149.75 ± 45.99 days, Amp_ttv_minimum: -27.46 ± 9.87 min.
TTV_maximum: 1273.30 ± 49.10 days, Amp_ttv_maximum: 29.88 ± 9.87 min.
TTV_minimum: 1396.85 ± 52.31 days, Amp_ttv_minimum: -27.46 ± 9.87 min.
P_ttv: 247.11 ± 7.77 days.
Amp_ttv: 57.34 ± 13.96 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 246.08 days; Power: 3.60; FAP: 0.0692.
Linear ephemeris (this work): Tc = [56.41645005 ± 0.00083252](Tc#) + [96.10645881 ± 0.01338826]


Figure 6.: Residuals of Figure 5. [Plot avg. error bars = ± 18.12 min.]
TTV_maximum: 360.66 ± 19.15 days, Amp_ttv_maximum: 21.29 ± 6.65 min.
TTV_minimum: 443.99 ± 19.69 days, Amp_ttv_minimum: -16.80 ± 6.65 min.
TTV_maximum: 527.33 ± 20.33 days, Amp_ttv_maximum: 21.29 ± 6.65 min.
TTV_minimum: 610.66 ± 21.05 days, Amp_ttv_minimum: -16.80 ± 6.65 min.
—(6 intervening additional maxima & minima)—
TTV_maximum: 1194.01 ± 27.83 days, Amp_ttv_maximum: 21.29 ± 6.65 min.
TTV_minimum: 1277.35 ± 28.97 days, Amp_ttv_minimum: -16.80 ± 6.65 min.
TTV_maximum: 1360.68 ± 30.14 days, Amp_ttv_maximum: 21.29 ± 6.65 min.
TTV_minimum: 1444.02 ± 31.34 days, Amp_ttv_minimum: -16.80 ± 6.65 min.
P_ttv: 166.67 ± 2.96 days.
Amp_ttv: 38.09 ± 9.41 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 166.88 days; Power: 3.60; FAP: 0.0877.


Figure 7.: Added combination of Figures 5., and 6. [Plot error bars = ± 23.65 min.]
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Figure 8.: KOI-2433.07, P = 86.43 days [Plot avg. error bars = ± 14.03 min.]
TTV_maximum: 388.57 ± 39.54 days, Amp_ttv_maximum: 43.12 ± 21.71 min.
TTV_minimum: 507.87 ± 41.69 days, Amp_ttv_minimum: -52.58 ± 21.71 min.
TTV_maximum: 627.16 ± 44.26 days, Amp_ttv_maximum: 43.12 ± 21.71 min.
TTV_minimum: 746.46 ± 47.19 days, Amp_ttv_minimum: -52.58 ± 21.71 min.
TTV_maximum: 865.75 ± 50.41 days, Amp_ttv_maximum: 43.12 ± 21.71 min.
TTV_minimum: 985.05 ± 53.87 days, Amp_ttv_minimum: -52.58 ± 21.71 min.
TTV_maximum: 1104.35 ± 57.52 days, Amp_ttv_maximum: 43.12 ± 21.71 min.
TTV_minimum: 1223.64 ± 61.34 days, Amp_ttv_minimum: -52.58 ± 21.71 min.
TTV_maximum: 1342.94 ± 65.29 days, Amp_ttv_maximum: 43.12 ± 21.71 min.
TTV_minimum: 1462.23 ± 69.35 days, Amp_ttv_minimum: -52.58 ± 21.71 min.
P_ttv: 238.59 ± 9.64 days.
Amp_ttv: 95.69 ± 30.70 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 240.05 days; Power: 2.59; FAP: 0.200.
Linear ephemeris (this work): Tc = [86.43258995 ± 0.00302268](Tc#) + [96.73049708 ± 0.03113259]


Numerous literature and major Tc#, Tc, and TTV tabulation references can be found on my "Summary" webpage following the table.

22 April 2015

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