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     Kepler KOI-282 (Kepler-130, KIC-5088536) 7(?)-(or more?)-Planet System

Discussion:
KOI-282.02 (Kepler-130b), the closest-in known transiting planet, shows no credible periodicity (see Figure 1.).

KOI-282.01 (Kepler-130c) shows a credible periodicity potentially due a non-transiting planet in an adjacent orbit (see Figure 2.).  The P_ttv in this case is different from each of those found for the more complex KOI-282.03 (below) case (suggesting that 282.01 is not one of the perturbers of 282.03).

For KOI-282.03 (Kepler-130d, the farthest-out of the known transiting planets), after a sinusoidal curve-fit of the (O-C) vs. Time data showed (Figure 3.) a periodicity (P_ttv) of 259.36 ± 4.39 days (258.38 days was observed in the Lomb-Scargle Periodogram (LSP) of the same data), a plot (Figure 4.) of the Residuals also gave a good sinusoidal curve-fit with a periodicity of 201.47 ± 6.21 days (LSP: 202.23 days).  Similarly, a plot (Figure 5.) of the Residuals of the Residuals gave a good sinusoidal curve-fit with a periodicity of 956.13 ± 54.00 days (LSP: 970.49 days).  The added combination of these three sinusoidal curves is arrayed in Figure 6. 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.

TTVs are derived from Q0-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-282.02, P = 8.46 days [Plot avg. error bars = ± 6.36 min. (smaller than symbols)]
Linear ephemeris (this work): Tc = [8.45744375 ± 0.00004119](Tc#) + [72.07012849 ± 0.00402970]


Figure 2.: KOI-282.01, P = 27.51 days [Plot avg. error bars = ± 1.62 min.]
TTV_maximum: 266.44 ± 45.13 days, Amp_ttv_maximum: 2.20 ± 0.72 min.
TTV_minimum: 501.40 ± 49.84 days, Amp_ttv_minimum: -2.16 ± 0.72 min.
TTV_maximum: 736.35 ± 56.60 days, Amp_ttv_maximum: 2.20 ± 0.72 min.
TTV_minimum: 971.31 ± 64.78 days, Amp_ttv_minimum: -2.16 ± 0.72 min.
TTV_maximum: 1206.27 ± 73.91 days, Amp_ttv_maximum: 2.20 ± 0.72 min.
TTV_minimum: 1441.23 ± 83.68 days, Amp_ttv_minimum: -2.16 ± 0.72 min.
P_ttv: 469.92 ± 23.38 days.
Amp_ttv: 4.35 ± 1.01 minutes.
Lomb-Scargle periodogram, periodicity (P_ttv): 472.79 days; Power: 5.68; FAP: 0.0513.
Linear ephemeris (this work): Tc = [27.50864103 ± 0.00002081](Tc#) + [115.10593065 ± 0.00060284]


Figure 3.: KOI-282.03, P = 87.52 days [Plot avg. error bars = ± 6.50 min.]
TTV_maximum: 132.86 ± 15.59 days, Amp_ttv_maximum: 57.83 ± 10.03 min.
TTV_minimum: 262.54 ± 16.05 days, Amp_ttv_minimum: -45.88 ± 10.03 min.
TTV_maximum: 392.22 ± 16.79 days, Amp_ttv_maximum: 57.83 ± 10.03 min.
TTV_minimum: 521.90 ± 17.77 days, Amp_ttv_minimum: -45.88 ± 10.03 min.
TTV_maximum: 651.58 ± 18.95 days, Amp_ttv_maximum: 57.83 ± 10.03 min.
TTV_minimum: 781.26 ± 20.31 days, Amp_ttv_minimum: -45.88 ± 10.03 min.
TTV_maximum: 910.94 ± 21.80 days, Amp_ttv_maximum: 57.83 ± 10.03 min.
TTV_minimum: 1040.62 ± 23.40 days, Amp_ttv_minimum: -45.88 ± 10.03 min.
TTV_maximum: 1170.30 ± 25.09 days, Amp_ttv_maximum: 57.83 ± 10.03 min.
TTV_minimum: 1299.98 ± 26.85 days, Amp_ttv_minimum: -45.88 ± 10.03 min.
TTV_maximum: 1429.66 ± 28.67 days, Amp_ttv_maximum: 57.83 ± 10.03 min.
P_ttv: 259.36 ± 4.39 days.
Amp_ttv: 103.72 ± 14.19 minutes.
Lomb-Scargle periodogram, periodicity (P_ttv): 258.38 days; Power: 4.94; FAP: 0.00121.
Linear ephemeris (this work): Tc = [87.51834027 ± 0.00177602](Tc#) + [134.15950101 ± 0.01602209]


Figure 4.: Residuals of Figure 3. [Plot avg. error bars = ± xxx min.]
TTV_maximum: 137.88 ± 26.41 days, Amp_ttv_maximum: 17.24 ± 6.91 min.
TTV_minimum: 238.61 ± 27.08 days, Amp_ttv_minimum: -17.85 ± 6.91 min.
TTV_maximum: 339.35 ± 28.09 days, Amp_ttv_maximum: 17.24 ± 6.91 min.
TTV_minimum: 440.08 ± 29.39 days, Amp_ttv_minimum: -17.85 ± 6.91 min.
TTV_maximum: 540.82 ± 30.94 days, Amp_ttv_maximum: 17.24 ± 6.91 min.
TTV_minimum: 641.55 ± 32.72 days, Amp_ttv_minimum: -17.85 ± 6.91 min.
TTV_maximum: 742.29 ± 34.69 days, Amp_ttv_maximum: 17.24 ± 6.91 min.
TTV_minimum: 843.02 ± 36.81 days, Amp_ttv_minimum: -17.85 ± 6.91 min.
TTV_maximum: 943.76 ± 39.07 days, Amp_ttv_maximum: 17.24 ± 6.91 min.
TTV_minimum: 1044.50 ± 41.43 days, Amp_ttv_minimum: -17.85 ± 6.91 min.
TTV_maximum: 1145.23 ± 43.89 days, Amp_ttv_maximum: 17.24 ± 6.91 min.
TTV_minimum: 1245.97 ± 46.42 days, Amp_ttv_minimum: -17.85 ± 6.91 min.
TTV_maximum: 1346.70 ± 49.03 days, Amp_ttv_maximum: 17.24 ± 6.91 min.
P_ttv: 201.47 ± 6.21 days.
Amp_ttv: 35.09 ± 9.78 minutes.
Lomb-Scargle periodogram, periodicity (P_ttv): 202.23 days; Power: 3.37; FAP: 0.123.


Figure 5.: Residuals of Figure 4.; Residuals of Residuals of Figure 3. [Plot avg. error bars = ± 13.81 min.]
TTV_minimum: 581.16 ± 61.44 days, Amp_ttv_minimum: -18.50 ± 3.79 min.
TTV_maximum: 1059.22 ± 79.22 days, Amp_ttv_maximum: 14.49 ± 3.79 min.
P_ttv: 956.13 ± 54.00 days.
Amp_ttv: 32.99 ± 5.36 minutes.
Lomb-Scargle periodogram, periodicity (P_ttv): 970.49 days; Power: 4.43; FAP: 0.0121.


Figure 6.: Summed combination of Figures 3., 4., and 5. [Plot error bars = ± 19.38 min.]


References:
• Numerous additional pertinent literature and major Tc#, Tc, and TTV tabulations can be found on my "Summary" webpage following the table.

6 Feb 2015