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-4571.01, P = 47.31 days [Plot avg. error bars = ± 12.62 min. (~ size of symbols)]
TTV_minimum: 112.65 ± 16.93 days, Amp_ttv_minimum: -65.95 ± 20.82 min.
TTV_maximum: 212.11 ± 17.27 days, Amp_ttv_maximum: 58.11 ± 20.82 min.
TTV_minimum: 311.58 ± 17.80 days, Amp_ttv_minimum: -65.95 ± 20.82 min.
TTV_maximum: 411.04 ± 18.51 days, Amp_ttv_maximum: 58.11 ± 20.82 min.
TTV_minimum: 510.50 ± 19.38 days, Amp_ttv_minimum: -65.95 ± 20.82 min.
TTV_maximum: 609.97 ± 20.39 days, Amp_ttv_maximum: 58.11 ± 20.82 min.
TTV_minimum: 709.43 ± 21.51 days, Amp_ttv_minimum: -65.95 ± 20.82 min.
TTV_maximum: 808.89 ± 22.74 days, Amp_ttv_maximum: 58.11 ± 20.82 min.
TTV_minimum: 908.36 ± 24.05 days, Amp_ttv_minimum: -65.95 ± 20.82 min.
TTV_maximum: 1007.82 ± 25.43 days, Amp_ttv_maximum: 58.11 ± 20.82 min.
TTV_minimum: 1107.28 ± 26.88 days, Amp_ttv_minimum: -65.95 ± 20.82 min.
TTV_maximum: 1206.75 ± 28.37 days, Amp_ttv_maximum: 58.11 ± 20.82 min.
TTV_minimum: 1306.21 ± 29.91 days, Amp_ttv_minimum: -65.95 ± 20.82 min.
TTV_maximum: 1405.68 ± 31.49 days, Amp_ttv_maximum: 58.11 ± 20.82 min.
P_ttv: 198.93 ± 3.77 days.
Amp_ttv: 124.06 ± 29.45 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 198.46 days; Power: 4.41; FAP: 0.0551.
Linear ephemeris (this work): Tc = [47.31414952 ± 0.00117241](Tc#) + [79.33337645 ± 0.01913445]


Figure 2.: Residuals of Figure 1. [Plot avg. error bars = ± 24.35 min.]

TTV_minimum: 119.06 ± 44.89 days, Amp_ttv_minimum: -47.10 ± 15.20 min.
TTV_maximum: 366.34 ± 48.56 days, Amp_ttv_maximum: 42.14 ± 15.20 min.
TTV_minimum: 613.61 ± 55.24 days, Amp_ttv_minimum: -47.10 ± 15.20 min.
TTV_maximum: 860.88 ± 63.99 days, Amp_ttv_maximum: 42.14 ± 15.20 min.
TTV_minimum: 1108.15 ± 74.08 days, Amp_ttv_minimum: -47.10 ± 15.20 min.
TTV_maximum: 1355.42 ± 85.03 days, Amp_ttv_maximum: 42.14 ± 15.20 min.
P_ttv: 494.54 ± 26.45 days.
Amp_ttv: 89.25 ± 21.49 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 492.43 days; Power: 4.39; FAP: 0.0782.


Figure 3.: Residuals of Figure 2.; Residuals-of-the-Residuals of Figure 1. [Plot avg. error bars = ± 28.70 min.]

TTV_maximum: 677.65 ± 245.87 days, Amp_ttv_maximum: 33.25 ± 13.69 min.
TTV_minimum: 1332.17 ± 372.22 days, Amp_ttv_minimum: -31.11 ± 13.69 min.
P_ttv: 1309.05 ± 318.95 days.
Amp_ttv: 64.36 ± 19.37 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 1260.26 days; Power: 3.91; FAP: 0.164.


Figure 4.: Added combination of Figures 1., 2., and 3. [Plot error bars = ± 39.70 min.]


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

30 Nov 2014

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                Kepler KOI-4571 (KIC-6761777) 4-(or more?)-Planet System

Discussion:
In the example of 4571.01, after a sinusoidal curve-fit of the (O-C) vs. Time data showed (Figure 1. below) a periodicity (P_ttv) of 198.93 ± 3.77 days (198.46 days was observed in the Lomb-Scargle Periodogram (LSP) of the same data), a plot (Figure 2.) of the Residuals also gave a good sinusoidal curve-fit with a periodicity of 494.54 ± 26.45 days (LSP: 492.43 days).  Similarly, the Residuals-of-the-Residuals also gave a good sinusoidal curve-fit (Figure 3.) with a periodicity of 1309.05 ± 318.95 days (LSP: 1360.26 days).  The added combination of all three sinusoidal curves, arrayed in Figure 4. 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 4 planets (with only 4571.01 transiting) in near-circular orbits are mutually-interacting to give the TTV distribution observed.