TTVs are derived from short-cadence data from Kepler Q3-Q17.  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-70.03, P = 77.61 days [Plot avg. error bars = ± 1.64 min.]
TTV_minimum: 299.84 ± 53.27 days, Amp_ttv_minimum: -5.22 ± 1.71 min.
TTV_maximum: 516.91 ± 57.12 days, Amp_ttv_maximum: 5.65 ± 1.71 min.
TTV_minimum: 733.97 ± 62.56 days, Amp_ttv_minimum: -5.22 ± 1.71 min.
TTV_maximum: 951.03 ± 69.22 days, Amp_ttv_maximum: 5.65 ± 1.71 min.
TTV_minimum: 1168.10 ± 76.77 days, Amp_ttv_minimum: -5.22 ± 1.71 min.
TTV_maximum: 1385.16 ± 84.99 days, Amp_ttv_maximum: 5.65 ± 1.71 min.
P_ttv: 434.13 ± 21.26 days.
Amp_ttv: 10.87 ± 2.42 minutes.
Lomb-Scargle periodogram, periodicity (P_ttv): 436.50 days; Power: 3.81; FAP: 0.0441.
Linear ephemeris (this work): Tc = 77.61146686(Tc#) + 97.73080969


Figure 2: Residuals of Figure 1 [Plot avg. error bars = ± 2.37 min.]

TTV_minimum: 283.02 ± 14.69 days, Amp_ttv_minimum: -2.99 ± 0.97 min.
TTV_maximum: 368.67 ± 15.06 days, Amp_ttv_maximum: 4.35 ± 0.97 min.
TTV_minimum: 454.32 ± 15.52 days, Amp_ttv_minimum: -2.99 ± 0.97 min.
TTV_maximum: 539.97 ± 16.06 days, Amp_ttv_maximum: 4.35 ± 0.97 min.
TTV_minimum: 625.63 ± 16.67 days, Amp_ttv_minimum: -2.99 ± 0.97 min.
TTV_maximum: 711.28 ± 17.34 days, Amp_ttv_maximum: 4.35 ± 0.97 min.
TTV_minimum: 796.93 ± 18.07 days, Amp_ttv_minimum: -2.99 ± 0.97 min.
TTV_maximum: 882.58 ± 18.85 days, Amp_ttv_maximum: 4.35 ± 0.97 min.
TTV_minimum: 968.23 ± 19.67 days, Amp_ttv_minimum: -2.99 ± 0.97 min.
TTV_maximum: 1053.89 ± 20.53 days, Amp_ttv_maximum: 4.35 ± 0.97 min.
TTV_minimum: 1139.54 ± 21.42 days, Amp_ttv_minimum: -2.99 ± 0.97 min.
TTV_maximum: 1225.19 ± 22.34 days, Amp_ttv_maximum: 4.35 ± 0.97 min.
TTV_minimum: 1310.84 ± 23.29 days, Amp_ttv_minimum: -2.99 ± 0.97 min.
TTV_maximum: 1396.49 ± 24.27 days, Amp_ttv_maximum: 4.35 ± 0.97 min.
TTV_minimum: 1482.14 ± 25.26 days, Amp_ttv_minimum: -2.99 ± 0.97 min.
P_ttv: 171.30 ± 2.42 days.
Amp_ttv: 7.34 ± 1.38 minutes.
Lomb-Scargle periodogram, periodicity (P_ttv): 170.24 days; Power: 3.95; FAP: 0.0229.


Figure 3: Residuals of Figure 2; Residuals of Residuals of Figure 1 [Plot avg. error bars = ± 2.56 min.]

TTV_minimum: 366.73 ± 104.28 days, Amp_ttv_minimum: -1.71 ± 0.64 min.
TTV_maximum: 755.81 ± 126.30 days, Amp_ttv_maximum: 1.61 ± 0.64 min.
TTV_minimum: 1144.90 ± 156.68 days, Amp_ttv_minimum: -1.71 ± 0.63 min.
P_ttv: 778.18 ± 83.91 days.
Amp_ttv: 3.32 ± 0.91 minutes.
Lomb-Scargle periodogram, periodicity (P_ttv): 777.64 days; Power: 3.21; FAP: 0.180.


Figure 4.: Summed combination of Figures 1., 2., and 3. [Plot error bars = ± 3.85 min.]


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

31 Jan 2015
           Kepler KOI-70 (Kepler-20, KIC-6850504); 7- or 8-(?)-Planet System

Discussion:
KOI-70.03, the farthest-out known transiting planet in this system:
After a sinusoidal curve-fit of the (O-C) vs. Time data showed (Figure 1.) a periodicity (P_ttv) of 434.13 ± 21.26 days (436.50 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 171.30 ± 2.42 days (LSP: 170.24 days).  Similarly, a plot (Figure 3.) of the Residuals of the Residuals gave a good sinusoidal curve-fit with a periodicity of 778.18 ± 83.91 days (LSP: 777.64 days).  The added combination of these three sinusoidal curves is arrayed in Figure 4. and reasonably reproduces a complex overall curvature consistent with the initial data.  While the sinusoidal behaviors observed could be partially due to orbital eccentricities, it is also possible that 3 additional as yet unseen (perhaps non-transiting) planets are present in the system.
No significant TTVs were noted for the other four known, closer-in planets in this system (see the 4 smaller plots following those for KOI-70.03).

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