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TTVs are derived from Q8-Q17 Kepler data (no data was available for Q0-Q7).  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-3581.01, P = 10.99 days [Plot avg. error bars = ± 0.53 min.]
TTV_maximum: 714.74 ± 17.77 days, Amp_ttv_maximum: 0.45 ± 0.10 min.
TTV_minimum: 777.13 ± 18.23 days, Amp_ttv_minimum: -0.46 ± 0.10 min.
TTV_maximum: 839.52 ± 18.73 days, Amp_ttv_maximum: 0.45 ± 0.10 min.
TTV_minimum: 901.91 ± 19.25 days, Amp_ttv_minimum: -0.46 ± 0.10 min.
TTV_maximum: 964.29 ± 19.79 days, Amp_ttv_maximum: 0.45 ± 0.10 min.
TTV_minimum: 1026.68 ± 20.35 days, Amp_ttv_minimum: -0.46 ± 0.10 min.
TTV_maximum: 1089.07 ± 20.93 days, Amp_ttv_maximum: 0.45 ± 0.10 min.
TTV_minimum: 1151.46 ± 21.53 days, Amp_ttv_minimum: -0.46 ± 0.10 min.
TTV_maximum: 1213.85 ± 22.14 days, Amp_ttv_maximum: 0.45 ± 0.10 min.
TTV_minimum: 1276.23 ± 22.77 days, Amp_ttv_minimum: -0.46 ± 0.10 min.
TTV_maximum: 1338.62 ± 23.41 days, Amp_ttv_maximum: 0.45 ± 0.10 min.
TTV_minimum: 1401.01 ± 24.06 days, Amp_ttv_minimum: -0.46 ± 0.10 min.
TTV_maximum: 1463.40 ± 24.73 days, Amp_ttv_maximum: 0.45 ± 0.10 min.
P_ttv: 124.78 ± 1.68 days.
Amp_ttv: 0.91 ± 0.14 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 125.13 days; Power: 11.51; FAP: 0.00709.
Linear ephemeris (this work): Tc = [10.99382793 ± 0.00000203](Tc#) + [71.80552214 ± 0.00019288]


Figure 2.: Residuals of Figure 1. [Plot avg. error bars = ± 0.54 min.]

TTV_maximum: 730.27 ± 45.52 days, Amp_ttv_maximum: 0.32 ± 0.08 min.
TTV_minimum: 864.71 ± 48.17 days, Amp_ttv_minimum: -0.35 ± 0.08 min.
TTV_maximum: 999.14 ± 51.10 days, Amp_ttv_maximum: 0.32 ± 0.08 min.
TTV_minimum: 1133.58 ± 54.25 days, Amp_ttv_minimum: -0.35 ± 0.08 min.
TTV_maximum: 1268.02 ± 57.60 days, Amp_ttv_maximum: 0.32 ± 0.08 min.
TTV_minimum: 1402.46 ± 61.10 days, Amp_ttv_minimum: -0.35 ± 0.08 min.
P_ttv: 268.88 ± 9.15 days.
Amp_ttv: 0.67 ± 0.12 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 267.17 days; Power: 9.53; FAP: 0.0502.


Figure 3.: Added combination of Figures 1. and 2. [Plot error bars = ± 0.76 min.]


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

18 Nov 2014
                 Kepler KOI-3581 (KIC-8263746) 3-(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 3581.01, after a sinusoidal curve-fit of the (O-C) vs. Time data showed (Figure 1. below) a periodicity (P_ttv) of 124.78 ± 1.68 days (125.13 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 268.88 ± 9.15 days (LSP: 267.17 days).  The added combination of these two sinusoidal curves is arrayed in Figure 3. and produces a complex overall curvature consistent with the initial data points.  While it is certainly possible (see recent work of Lithwick and others) that some of this unusual curvature obtains from eccentric orbits (and perhaps even precessing eccentric orbits) of planetary objects in this system, it is also possible that at least 3 planets in near-circular orbits are mutually-interacting to give the TTV distribution observed.