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                 Kepler KOI-730 (KIC10227020) 4-(or more?)-Planet System
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: KOI-730.01, P = 14.79 days [Plot avg. error bars = ± 10.36 min. (smaller than symbols)]
TTV_minimum: 250.29 ± 64.42 days, Amp_ttv_minimum: -55.49 ± 12.60 min.
TTV_maximum: 895.41 ± 91.48 days, Amp_ttv_maximum: 92.32 ± 12.60 min.
(TTV_minimum: 1540.52 ± 131.68 days, Amp_ttv_minimum: -55.49 ± 12.60 min.)
P_ttv: 1290.23 ± 97.48 days.
Amp_ttv: 147.81 ± 17.81 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 1670.36 days; Power: 13.11; FAP: 0.00108.
Linear ephemeris (this work): Tc = [14.78707483 ± 0.00022521](Tc#) + [109.79918614 ± 0.01101942]


Figure: KOI-730.03, P = 19.72 days [Plot avg. error bars = ± 12.09 min.]
(TTV_maximum: 41.85 ± 212.89 days, Amp_ttv_maximum: 71.50 ± 23.75 min.)
TTV_minimum: 831.63 ± 296.02 days, Amp_ttv_minimum: -57.74 ± 23.75 min.
(TTV_maximum: 1621.42 ± 454.35 days, Amp_ttv_maximum: 71.50 ± 23.75 min.)
P_ttv: 1579.57 ± 391.16 days.
Amp_ttv: 129.24 ± 33.58 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 1462.11 days; Power: 9.70; FAP: 0.000202.
Linear ephemeris (this work): Tc = [19.72405301 ± 0.00032409](Tc#) + [68.05642367 ± 0.01364920]


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

3 Oct 2015
Discussion:
A sinusoidal curve-fit of the [(O-C) vs. Time] data for KOI-730.01, the 3rd planet in this system, showed (Figure 1. below) a periodicity (P_ttv) of 1290.23 ± 97.48 days; a Lomb-Scargle Periodogram (LSP) of the same data gave a high-credibility periodicity of 1670.36 days.

Similarly, a sinusoidal curve-fit of the [(O-C) vs. Time] data for KOI-730.03. the 4th planet in this system, showed (Figure 2. below) a periodicity (P_ttv) of 1579.57 ± 391.16 days; a Lomb-Scargle Periodogram (LSP) of the same data gave a high-credibility periodicity of 1462.11 days.

Despite (a) substantial errors due to weak planetary transits and a noisy star and
(b) there being only about one TTV cycle, the two sinusoidal arrays are reasonably well anti-correlated (compare the plots), a good indication of the strength of the two planets’ mutual gravitational interactions.  Their period ratio also suggests they are in a mean motion resonance relationship very close to 4 to 3 (or, precisely, 4.0000 to 2.9989).