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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-190.01, P = 12.26 days [Plot avg. error bars = ± 0.87 min.]
TTV_minimum: 112.52 ± 6.76 days, Amp_ttv_minimum: -4.17 ± 0.39 min.
TTV_maximum: 247.67 ± 6.97 days, Amp_ttv_maximum: 4.37 ± 0.39 min.
TTV_minimum: 382.82 ± 7.32 days, Amp_ttv_minimum: -4.17 ± 0.39 min.
TTV_maximum: 517.98 ± 7.79 days, Amp_ttv_maximum: 4.37 ± 0.39 min.
TTV_minimum: 653.13 ± 8.36 days, Amp_ttv_minimum: -4.17 ± 0.39 min.
TTV_maximum: 788.28 ± 9.01 days, Amp_ttv_maximum: 4.37 ± 0.39 min.
TTV_minimum: 923.43 ± 9.73 days, Amp_ttv_minimum: -4.17 ± 0.39 min.
TTV_maximum: 1058.58 ± 10.50 days, Amp_ttv_maximum: 4.37 ± 0.39 min.
TTV_minimum: 1193.74 ± 11.32 days, Amp_ttv_minimum: -4.17 ± 0.39 min.
TTV_maximum: 1328.89 ± 12.16 days, Amp_ttv_maximum: 4.37 ± 0.39 min.
TTV_minimum: 1464.04 ± 13.04 days, Amp_ttv_minimum: -4.17 ± 0.39 min.
P_ttv: 270.30 ± 2.06 days.
Amp_ttv: 8.54 ± 0.55 minutes.
Figure 1-LSP.:
Lomb-Scargle periodogram, candidate P_ttv: 269.05 days; Power: 28.68; FAP: 2.953 x 10^-10.
Linear ephemeris (this work): Tc = 12.26487531(Tc#) + 72.30500745

Figure 2.: Residuals of Figure 1. [Plot avg. error bars = ± 0.95 min.]
TTV_maximum: 123.50 ± 5.73 days, Amp_ttv_maximum: 1.86 ± 0.31 min.
TTV_minimum: 190.91 ± 5.81 days, Amp_ttv_minimum: -1.78 ± 0.31 min.
:::::::::::::::::::::::::::::::
TTV_minimum: 1404.30 ± 10.87 days, Amp_ttv_minimum: -1.78 ± 0.31 min.
TTV_maximum: 1471.71 ± 11.26 days, Amp_ttv_maximum: 1.86 ± 0.31 min.
P_ttv: 134.82 ± 0.89 days.
Amp_ttv: 3.65 ± 0.44 minutes.
Figure 2-LSP.:
Lomb-Scargle periodogram, candidate P_ttv: 135.26 days; Power: 16.24; FAP: 7.414 x 10^-5.

Figure 3.: Residuals of Figure 2.; Residuals-of-the-Residuals of Figure 1. [Plot avg. error bars = ± 1.00 min.]
TTV_maximum: 86.04 ± 3.70 days, Amp_ttv_maximum: 1.16 ± 0.26 min.
TTV_minimum: 119.70 ± 3.72 days, Amp_ttv_minimum: -1.25 ± 0.26 min.
:::::::::::::::::::::::::::::::
TTV_maximum: 1432.38 ± 7.11 days, Amp_ttv_maximum: 1.16 ± 0.26 min.
TTV_minimum: 1466.04 ± 7.23 days, Amp_ttv_minimum: -1.25 ± 0.26 min.
P_ttv: 67.32 ± 0.29 days.
Amp_ttv: 2.41 ± 0.37 minutes.
Figure 3-LSP.:
Lomb-Scargle periodogram, candidate P_ttv: 67.38 days; Power: 11.78; FAP: 6.39 x 10^-3.

Figure 4.: Added combination of Figures 1., 2., and 3. [Plot error bars = ± 1.63 min.]

Literature Tc#, Tc, and TTV Major Tabulations and Other References:
• For Q0-6 TTV data (thru ~ 563 (BJD-2454900) days): Ford et al., 2012, arXiv-1201.1892.
• For Q0-10 TTV data (thru ~ 934 (BJD-2454900) days): Rowe et al., 2014, arXiv-1402.6534.
• For Q0-12 TTV data (thru ~ 1116 (BJD-2454900) days): Mazeh et al., 2013, arXiv-1301.5499.
• For Q0-12 Calc. properties from TTVs: Hadden & Lithwick, 2013, arXiv-1310.7942.
• For Q0-13/14 TTV data, Xie: http://www.astro.utoronto.ca/~jwxie/TTV/TTV_Home.html
• Two ground-based Kepler-TTV-follow-up programs are in place at the moment: --- Gary, "KAFO" project: http://brucegary.net/kafo/ --- von Essen, "KOINet": http://koinet.astro.physik.uni-goettingen.de/

18 May 2014
Kepler KOI-190 (KIC-5771719) 4-(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 270.30 ± 2.06 days (269.05 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 134.82 ± 0.89 days (LSP: 135.26 days).  Similarly, the Residuals-of-the-Residuals also gave a good sinusoidal curve-fit (Figure 3.) with a periodicity of 67.32 ± 0.29 days (LSP: 67.38 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 3 planets in near-circular orbits are mutually-interacting to give the TTV distribution observed.The above-mentioned LSP's are shown as Figures 1-LSP., 2-LSP., and 3-LSP., respectively.