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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-2681.01, P = 135.50 days [Plot avg. error bars = ± 3.91 min.]
TTV_maximum: 296.63 ± 62.24 days, Amp_ttv_maximum: 16.64 ± 5.34 min.
TTV_minimum: 648.81 ± 72.00 days, Amp_ttv_minimum: -15.30 ± 5.34 min.
TTV_maximum: 1000.99 ± 86.43 days, Amp_ttv_maximum: 16.64 ± 5.34 min.
TTV_minimum: 1353.18 ± 103.59 days, Amp_ttv_minimum: -15.30 ± 5.34 min.
P_ttv: 704.37 ± 44.18 days.
Amp_ttv: 31.94 ± 7.55 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 702.33 days; Power: 3.13; FAP: 0.0312.
Linear ephemeris (this work): Tc = 135.49767529(Tc#) + 166.69674709


Figure 2.: Residuals of Figure 1. [Plot avg. error bars = ± 6.62 min.]

TTV_minimum: 521.22 ± 125.61 days, Amp_ttv_minimum: -8.52 ± 3.02 min.
TTV_maximum: 1239.70 ± 184.79 days, Amp_ttv_maximum: 9.98 ± 3.02 min.
P_ttv: 1436.97 ± 173.15 days.
Amp_ttv: 18.50 ± 4.26 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 1434.84 days; Power: 3.17; FAP: 0.0683.


Figure 3.: Summed combination of Figures 1. and 2. [Plot error bars = ± 7.69 min.]

4th figure: KOI-2681.02, P = 22.25 days
Linear ephemeris: Tc = 22.25179864(Tc#) + 82.40182083 [this work].


• Fabrycky et al., arXiv-1201.5415; P_ttv = 1/|(nmmri_a/P_a - nmmri_b/P_b)|.
• Numerous literature and major Tc#, Tc, and TTV tabulation references can be found on my "Summary" webpage following the table.

21 Oct 2014
              Kepler KOI-2681.01 (KIC-6878240) 3-(or more?)-Planet System

Discussion:
In this example, after a sinusoidal curve-fit of the (O-C) vs. Time data showed (Figure 1. below) a periodicity (P_ttv) of 704.37 ± 44.18 days (702.33 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 1436.97 ± 173.15 days (LSP: 1434.84 days).  The summed combination of these two sinusoidal curves is arrayed in Figure 3. and reasonably reproduces a complex overall curvature 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 of planetary objects in this system, it is also possible that at least 2 planets in near-circular orbits are interacting with 2681.01 to give the TTV distribution observed.  And, of course, it is certainly possible that one of those other 2 planets is 2681.02 (see next paragraph); noisy data may be responsible for the lack of a clear TTV for 2681.02.

The ratio of periods for 2681.01 and 2681.02 is 6.0895, which corresponds to a mean motion ratio (MMR) of ~ 6/1.  Using this MMR and the equation for P_ttv of Fabrycky, et al. (shown below), an approximate P_ttv of 1513.32 days is predicted.  Interestingly, this corresponds closely to the P_ttv of the sinusoidal array of the TTV-Residuals (Figure 2.): 1436.97 ± 173.15 days.