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Figure 5.: KOI-3531.01, Average Duration: 0.369149 ± 0.000400 days = 531.58 ± 0.58 min. [Avg. error bars = ± 0.58 min.]
TDV_minimum: 365.59 ± 89.68 days, Amp_tdv_minimum: -0.91 ± 0.38 min.
TDV_maximum: 730.56 ± 109.34 days, Amp_tdv_maximum: 0.96 ± 0.38 min.
TDV_minimum: 1095.54 ± 135.91 days, Amp_tdv_minimum: -0.91 ± 0.38 min.
TDV_maximum: 1460.51 ± 166.11 days, Amp_tdv_maximum: 0.96 ± 0.38 min.
P_tdv: 729.94 ± 72.18 days.
Amp_tdv: 1.87 ± 0.54 minutes.
Lomb-Scargle periodogram, candidate P_tdv: 723.23 days; Power: 3.48; FAP: 0.136.


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

18 Nov 2014
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-3531.01, P = 70.58 days [Plot avg. error bars = ± 0.14 min. (smaller than symbols)]
TTV_maximum: 216.19 ± 11.74 days, Amp_ttv_maximum: 1.39 ± 0.21 min.
TTV_minimum: 364.92 ± 12.32 days, Amp_ttv_minimum: -1.54 ± 0.21 min.
TTV_maximum: 513.64 ± 13.15 days, Amp_ttv_maximum: 1.39 ± 0.21 min.
TTV_minimum: 662.36 ± 14.18 days, Amp_ttv_minimum: -1.54 ± 0.21 min.
TTV_maximum: 811.09 ± 15.37 days, Amp_ttv_maximum: 1.39 ± 0.21 min.
TTV_minimum: 959.81 ± 16.70 days, Amp_ttv_minimum: -1.54 ± 0.21 min.
TTV_maximum: 1108.54 ± 18.12 days, Amp_ttv_maximum: 1.39 ± 0.21 min.
TTV_minimum: 1257.26 ± 19.63 days, Amp_ttv_minimum: -1.54 ± 0.21 min.
TTV_maximum: 1405.98 ± 21.19 days, Amp_ttv_maximum: 1.39 ± 0.21 min.
P_ttv: 297.45 ± 3.78 days.
Amp_ttv: 2.93 ± 0.30 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 297.85 days; Power: 7.75; FAP: 5.081 x 10^-6.
Linear ephemeris (this work): Tc = [70.58285551 ± 0.00003175](Tc#) + [123.50164074 ± 0.00035283]


Figure 2.: Residuals of Figure 1. [Plot avg. error bars = ± 0.25 min.]

TTV_maximum: 157.49 ± 49.12 days, Amp_ttv_maximum: 0.45 ± 0.14 min.
TTV_minimum: 452.73 ± 53.95 days, Amp_ttv_minimum: -0.56 ± 0.14 min.
TTV_maximum: 747.96 ± 62.37 days, Amp_ttv_maximum: 0.45 ± 0.14 min.
TTV_minimum: 1043.20 ± 73.14 days, Amp_ttv_minimum: -0.56 ± 0.14 min.
TTV_maximum: 1338.44 ± 85.39 days, Amp_ttv_maximum: 0.45 ± 0.14 min.
P_ttv: 590.47 ± 31.03 days.
Amp_ttv: 1.01 ± 0.19 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 587.83 days; Power: 5.26; FAP: 0.0116.


Figure 3.: Residuals of Figure 2.; Residuals-of-the-Residuals of Figure 1. [Plot avg. error bars = ± 0.29 min.]

TTV_maximum: 144.89 ± 47.14 days, Amp_ttv_maximum: 0.26 ± 0.11 min.
TTV_minimum: 336.46 ± 49.73 days, Amp_ttv_minimum: -0.27 ± 0.11 min.
TTV_maximum: 528.02 ± 54.08 days, Amp_ttv_maximum: 0.26 ± 0.11 min.
TTV_minimum: 719.59 ± 59.79 days, Amp_ttv_minimum: -0.27 ± 0.11 min.
TTV_maximum: 911.15 ± 66.52 days, Amp_ttv_maximum: 0.26 ± 0.11 min.
TTV_minimum: 1102.72 ± 74.00 days, Amp_ttv_minimum: -0.27 ± 0.11 min.
TTV_maximum: 1294.28 ± 82.01 days, Amp_ttv_maximum: 0.26 ± 0.11 min.
P_ttv: 383.13 ± 19.99 days.
Amp_ttv: 0.53 ± 0.15 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 380.98 days; Power: 3.59; FAP: 0.164.


Figure 4.: Added combination of Figures 1., 2., and 3. [Plot error bars = ± 0.41 min.]


         
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KOI-3531.01: a Grazing Planet…additionally showing a Transit Duration Variation (TDV)

EXOFAST data for T_(FWHM), tau, and T_(1,4) were precisely consistent with 3531.01 being a grazing planet:
         
Avg. T_(FWHM) = Avg. T_(Full Width @ Half Maximum) = 0.184575 ± 0.000200 days;
                                             Avg. tau = Avg. [Ingress + Egress] = 0.184575 ± 0.000347 days;
                                 Avg. T_(1,4) = Avg. T_(1st to 4th Contact) = 0.369149 ± 0.000400 days.

This grazing-evidence-data is consistent with an NEA-reported impact parameter of 0.901.

In addition, a sinusoidal pattern describes the variation in its transit durations times.  It's periodicity (P_tdv) was 729.94 ± 72.18 days (LSP: 723.23 days).  The TDV data is plotted in Figure 5. and elaborated a bit below.
                Kepler KOI-3531 (KIC-6947164) 4-(or more?)-Planet System

TTV Evidence:
The presence of both a credible periodicity in the Lomb-Scargle Periodogram (LSP) and a corresponding pronounced one (P_ttv) in a sinusoidal distribution of [(O-C) vs. Time] ("TTVxy") data points is strongly suggestive of the presence of a second (possibly unseen, non-transiting) gravitationally-perturbing planetary object in a system.
In the example of 3531.01, after a sinusoidal curve-fit of the (O-C) vs. Time data showed (Figure 1. below) a periodicity (P_ttv) of 297.45 ± 3.78 days (297.85 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 590.47 ± 31.03 days (LSP: 587.83 days).  Similarly, the Residuals-of-the-Residuals also gave a good sinusoidal curve-fit (Figure 3.) with a periodicity of 383.13 ± 19.99 days (LSP: 380.98 days).  The summed combination of all three sinusoidal curves, arrayed in Figure 4. shows 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 3531.01 transiting) in near-circular orbits are mutually-interacting to give the TTV distribution observed.