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TTVs are derived from Q1-Q16 Kepler data.  x-axes: “Observed Tc” (Mid-Transit Time): EXOFAST’s best-fits from Kepler (Normalized PDCSAP_FLUX) 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-372.01, P = 125.63 days [Plot avg. error bars = ± 2.25 min. (smaller than symbols)]
TTV_minimum: 389.32 ± 131.28 days, Amp_ttv_minimum: -19.66 ± 10.61 min.
TTV_maximum: 809.17 ± 158.13 days, Amp_ttv_maximum: 19.67 ± 10.61 min.
TTV_minimum: 1229.02 ± 195.50 days, Amp_ttv_minimum: -19.66 ± 10.61 min.
P_ttv: 839.71 ± 104.35 days.
Amp_ttv: 39.33 ± 15.01 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 844.48 days; Power: 2.205; FAP: 0.392.
Linear ephemeris (this work): Tc = 125.62786045(Tc#) + 186.36394721

Figure 2.: Residuals of Figure 1. [Plot avg. error bars = ± 10.85 min.]
TTV_minimum: 518.78 ± 96.07 days, Amp_ttv_minimum: -24.99 ± 6.90 min.
TTV_maximum: 1186.17 ± 141.13 days, Amp_ttv_maximum: 14.91 ± 6.90 min.
P_ttv: 1334.78 ± 129.36 days.
Amp_ttv: 39.91 ± 9.76 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 1433.90 days; Power: 2.55; FAP: 0.356.

Figure 3.: Residuals of Figure 2.; Residuals-of-the-Residuals of Figure 1. [Plot avg. error bars = ± 12.86 min.]
TTV_minimum: 230.94 ± 36.09 days, Amp_ttv_minimum: -9.51 ± 3.36 min.
TTV_maximum: 415.40 ± 38.34 days, Amp_ttv_maximum: 9.24 ± 3.36 min.
TTV_minimum: 599.85 ± 41.64 days, Amp_ttv_minimum: -9.51 ± 3.36 min.
TTV_maximum: 784.30 ± 45.75 days, Amp_ttv_maximum: 9.24 ± 3.36 min.
TTV_minimum: 968.75 ± 50.47 days, Amp_ttv_minimum: -9.51 ± 3.36 min.
TTV_maximum: 1153.21 ± 55.66 days, Amp_ttv_maximum: 9.24 ± 3.36 min.
TTV_minimum: 1337.66 ± 61.19 days, Amp_ttv_minimum: -9.51 ± 3.36 min.
P_ttv: 368.90 ± 13.84 days.
Amp_ttv: 18.74 ± 4.75 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 370.74 days; Power: 2.95; FAP: 0.113.

Figure 4.: Added combination of Figures 1., 2., and 3. [Plot error bars = ± 16.98 min.]

References:
• For Q0-Q6 TTV data of 372.01 (thru ~ 563 (BJD-2454900) days), see: Ford et al., 2012, arXiv-1201.1892.
• For Q0-12 TTV data of 372.01 (thru ~ 1116 (BJD-2454900) days), see: Mazeh et al., 2013, arXiv-1301.5499.
• For Q0-16 TTV data of 372.01 (plotted), see: VonEssen et al., "KOINet", see: http://koinet.astro.physik.uni-goettingen.de/

24 Mar 2014
Kepler KOI-372.01 (KIC-6471021) 4-(or more?)-Planet System

Discussion:
In this example, after a sinusoidal curve-fit of the initial Time vs. (O-C) data showed (Figure 1. below) a periodicity (P_ttv) of 839.71 ± 104.35 days (844.48 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 1334.78 ± 129.36 days (LSP: 1433.90 days).  Similarly, the Residuals-of-the-Residuals also gave a good sinusoidal curve-fit (Figure 3.) with a periodicity of 368.90 ± 13.84 days (LSP: 370.74 days).  The added combination of all three sinusoidal curves is arrayed in Figure 4. 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 4 planets in near-circular orbits are mutually-interacting to give the TTV distribution observed.

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