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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 light flux vs. time data.  y-axes: “(O – C)”: difference between Observed Tc and the Calculated Tc from the graphically obtained linear ephemeris.  The plots are pictured in the order of orbital periods.

Discussion:
Near Mean Motion Resonance integers in this system:P_137.01/P_137.03: ~ 2/1; P_137.02/P_137.01: ~ 2/1.The two planets, 137.01 and 137.02, show pronounced and extremely well anti-correlated TTVs (see above plots and the detailed sinusoidal data below).  Sinusoidal best-fits give the respective P_ttv's as 266.53 and 265.92 days; the calculated value, assuming 2:1 MMR (see Fabrycky's (ref. below) equation), is 267.67 days; and the Lomb-Scargle periodogram well-defined periods are 265.42 and 266.71 days, respectively.The innermost planet (137.03), within the substantial scatter of the data, shows no clear TTV.

1st: KOI-137.03, P = 3.50 days
Linear ephemeris (this work): Tc = 3.50472402(Tc#) + 66.50939803

2nd: KOI-137.01, P = 7.64 days
TTV_minimum: 148.80 ± 4.40 days, Amp_ttv_minimum: -5.49 ± 0.33 min.
TTV_maximum: 282.07 ± 4.56 days, Amp_ttv_maximum: 5.55 ± 0.33 min.
TTV_minimum: 415.33 ± 4.82 days, Amp_ttv_minimum: -5.49 ± 0.33 min.
TTV_maximum: 548.60 ± 5.15 days, Amp_ttv_maximum: 5.55 ± 0.33 min.
TTV_minimum: 681.86 ± 5.55 days, Amp_ttv_minimum: -5.49 ± 0.33 min.
TTV_maximum: 815.12 ± 6.00 days, Amp_ttv_maximum: 5.55 ± 0.33 min.
TTV_minimum: 948.39 ± 6.48 days, Amp_ttv_minimum: -5.49 ± 0.33 min.
TTV_maximum: 1081.65 ± 7.00 days, Amp_ttv_maximum: 5.55 ± 0.33 min.
TTV_minimum: 1214.92 ± 7.55 days, Amp_ttv_minimum: -5.49 ± 0.33 min.
TTV_maximum: 1348.18 ± 8.11 days, Amp_ttv_maximum: 5.55 ± 0.33 min.
TTV_minimum: 1481.45 ± 8.69 days, Amp_ttv_minimum: -5.49 ± 0.33 min.
P_ttv: 266.53 ± 1.36 days.
Amp_ttv: 11.04 ± 0.47 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 265.42 days; Power: 55.74; P-value: 0.0000
Linear ephemeris (this work): Tc = 7.64156695(Tc#) + 68.40895771
Estimated P_ttv = 1/|(nmmri_a/P_a - nmmri_b/P_b)| = 267.66 days

3rd: KOI-137.02, P = 14.86 days
TTV_maximum: 156.37 ± 6.92 days, Amp_ttv_maximum: 4.06 ± 0.39 min.
TTV_minimum: 289.33 ± 7.18 days, Amp_ttv_minimum: -4.50 ± 0.39 min.
TTV_maximum: 422.29 ± 7.59 days, Amp_ttv_maximum: 4.06 ± 0.39 min.
TTV_minimum: 555.25 ± 8.11 days, Amp_ttv_minimum: -4.50 ± 0.39 min.
TTV_maximum: 688.21 ± 8.72 days, Amp_ttv_maximum: 4.06 ± 0.39 min.
TTV_minimum: 821.16 ± 9.42 days, Amp_ttv_minimum: -4.50 ± 0.39 min.
TTV_maximum: 954.12 ± 10.18 days, Amp_ttv_maximum: 4.06 ± 0.39 min.
TTV_minimum: 1087.08 ± 10.98 days, Amp_ttv_minimum: -4.50 ± 0.39 min.
TTV_maximum: 1220.04 ± 11.83 days, Amp_ttv_maximum: 4.06 ± 0.39 min.
TTV_minimum: 1353.00 ± 12.70 days, Amp_ttv_minimum: -4.50 ± 0.39 min.
TTV_maximum: 1485.96 ± 13.61 days, Amp_ttv_maximum: 4.06 ± 0.39 min.
P_ttv: 265.92 ± 2.11 days.
Amp_ttv: 8.56 ± 0.55 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 266.71 days; Power: 27.76; P-value: 6.95 x 10^-10.
Linear ephemeris (this work): Tc = 14.85892094(Tc#) + 61.15442473

References:
• Cochran, Fabrycky, et al., 2011, arXiv-1110.0820.
• P_ttv equation: P_ttv = 1/|(nmmri_a/P_a - nmmri_b/P_b)|…(see Fabrycky, et al., 2012, arXiv-1201.5415) where nmmri_a and nmmri_b are the near mean motion resonance integers a and b, and P_a and P_b are the respective periods.
• For Q1-Q6 TTV data of 137.01, 137.02, and 137.03 (thru ~ 563 (BJD-2454900) days), see: Ford2012arXiv-1201.1892.

17 Aug 2013
KOI-137 (Kepler-18, KIC-8644288) 3-Planet System