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DATA: Sinusoidal, Lomb-Scargle Periodogram, and Ephemerides:

TTVs are derived from Q3-Q17 Kepler short-cadence 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 ephemerides.

Figure 1.: KOI-157.06, Kepler-11b, P = 10.30 days [Plot avg. error bars = ± 4.88 min.]
TTV_maximum: 307.84 ± 21.26 days, Amp_ttv_maximum: 11.84 ± 3.48 min.
TTV_minimum: 422.39 ± 22.19 days, Amp_ttv_minimum: -11.29 ± 3.48 min.
TTV_maximum: 536.93 ± 23.36 days, Amp_ttv_maximum: 11.84 ± 3.48 min.
TTV_minimum: 651.47 ± 24.73 days, Amp_ttv_minimum: -11.29 ± 3.48 min.
TTV_maximum: 766.01 ± 26.27 days, Amp_ttv_maximum: 11.84 ± 3.48 min.
TTV_minimum: 880.55 ± 27.96 days, Amp_ttv_minimum: -11.29 ± 3.48 min.
TTV_maximum: 995.09 ± 29.76 days, Amp_ttv_maximum: 11.84 ± 3.48 min.
TTV_minimum: 1109.64 ± 31.66 days, Amp_ttv_minimum: -11.29 ± 3.48 min.
TTV_maximum: 1224.18 ± 33.64 days, Amp_ttv_maximum: 11.84 ± 3.48 min.
TTV_minimum: 1338.72 ± 35.69 days, Amp_ttv_minimum: -11.29 ± 3.48 min.
TTV_maximum: 1453.26 ± 37.80 days, Amp_ttv_maximum: 11.84 ± 3.48 min.
P_ttv: 229.08 ± 5.04 days.
Amp_ttv: 23.14 ± 4.92 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 228.17 days; Power: 7.17; FAP: 0.547.
Linear ephemeris (this work): Tc = 10.30405498(Tc#) + 71.49992065

Figure 2.: KOI-157.01, Kepler-11c, P = 13.02 days [Plot avg. error bars = ± 2.56 min.]
TTV_minimum: 297.68 ± 17.08 days, Amp_ttv_minimum: -6.41 ± 1.43 min.
TTV_maximum: 414.76 ± 17.83 days, Amp_ttv_maximum: 6.99 ± 1.43 min.
TTV_minimum: 531.84 ± 18.79 days, Amp_ttv_minimum: -6.41 ± 1.43 min.
TTV_maximum: 648.91 ± 19.91 days, Amp_ttv_maximum: 6.99 ± 1.43 min.
TTV_minimum: 765.99 ± 21.18 days, Amp_ttv_minimum: -6.41 ± 1.43 min.
TTV_maximum: 883.07 ± 22.57 days, Amp_ttv_maximum: 6.99 ± 1.43 min.
TTV_minimum: 1000.15 ± 24.06 days, Amp_ttv_minimum: -6.41 ± 1.43 min.
TTV_maximum: 1117.23 ± 25.63 days, Amp_ttv_maximum: 6.99 ± 1.43 min.
TTV_minimum: 1234.30 ± 27.27 days, Amp_ttv_minimum: -6.41 ± 1.43 min.
TTV_maximum: 1351.38 ± 28.96 days, Amp_ttv_maximum: 6.99 ± 1.43 min.
TTV_minimum: 1468.46 ± 30.70 days, Amp_ttv_minimum: -6.41 ± 1.43 min.
P_ttv: 234.16 ± 4.15 days.
Amp_ttv: 13.40 ± 2.03 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 233.08 days; Power: 11.73; FAP: 0.00585.
Linear ephemeris (this work): Tc = 13.02490533(Tc#) + 71.17709124

Figure 3.: KOI-157.06 and KOI-157.01: combined repeated best-fit sinusoidal curves from Figures 1. and 2., respectively.

Figure 4.: KOI-157.02, Kepler-11d, P = 22.69 days [Plot avg. error bars = ± 2.44 min.]
TTV_maximum: 308.18 ± 21.33 days, Amp_ttv_maximum: 5.54 ± 1.79 min.
TTV_minimum: 400.35 ± 21.98 days, Amp_ttv_minimum: -4.91 ± 1.79 min.
TTV_maximum: 492.52 ± 22.78 days, Amp_ttv_maximum: 5.54 ± 1.79 min.
TTV_minimum: 584.70 ± 23.70 days, Amp_ttv_minimum: -4.91 ± 1.79 min.
TTV_maximum: 676.87 ± 24.73 days, Amp_ttv_maximum: 5.54 ± 1.79 min.
TTV_minimum: 769.04 ± 25.87 days, Amp_ttv_minimum: -4.91 ± 1.79 min.
TTV_maximum: 861.21 ± 27.09 days, Amp_ttv_maximum: 5.54 ± 1.79 min.
TTV_minimum: 953.38 ± 28.39 days, Amp_ttv_minimum: -4.91 ± 1.79 min.
TTV_maximum: 1045.56 ± 29.76 days, Amp_ttv_maximum: 5.54 ± 1.79 min.
TTV_minimum: 1137.73 ± 31.18 days, Amp_ttv_minimum: -4.91 ± 1.79 min.
TTV_maximum: 1229.90 ± 32.66 days, Amp_ttv_maximum: 5.54 ± 1.79 min.
TTV_minimum: 1322.07 ± 34.18 days, Amp_ttv_minimum: -4.91 ± 1.79 min.
TTV_maximum: 1414.24 ± 35.73 days, Amp_ttv_maximum: 5.54 ± 1.79 min.
TTV_minimum: 1506.42 ± 37.32 days, Amp_ttv_minimum: -4.91 ± 1.79 min.
P_ttv: 184.34 ± 3.83 days.
Amp_ttv: 10.45 ± 2.53 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 184.18 days; Power: 4.93; FAP: 0.128.
Linear ephemeris (this work): Tc = 22.68723779(Tc#) + 81.45266322

Figure 5a.: KOI-157.03, Kepler-11e, P = 32.00 days [Plot avg. error bars = ± 1.63 min.]
TTV_maximum: 249.49 ± 17.27 days, Amp_ttv_maximum: 4.45 ± 1.52 min.
TTV_minimum: 333.28 ± 17.72 days, Amp_ttv_minimum: -5.15 ± 1.52 min.
TTV_maximum: 417.07 ± 18.29 days, Amp_ttv_maximum: 4.45 ± 1.52 min.
TTV_minimum: 500.86 ± 18.97 days, Amp_ttv_minimum: -5.15 ± 1.52 min.
TTV_maximum: 584.65 ± 19.73 days, Amp_ttv_maximum: 4.45 ± 1.52 min.
TTV_minimum: 668.45 ± 20.58 days, Amp_ttv_minimum: -5.15 ± 1.52 min.
TTV_maximum: 752.24 ± 21.50 days, Amp_ttv_maximum: 4.45 ± 1.52 min.
TTV_minimum: 836.03 ± 22.49 days, Amp_ttv_minimum: -5.15 ± 1.52 min.
TTV_maximum: 919.82 ± 23.53 days, Amp_ttv_maximum: 4.45 ± 1.52 min.
TTV_minimum: 1003.61 ± 24.62 days, Amp_ttv_minimum: -5.15 ± 1.52 min.
TTV_maximum: 1087.40 ± 25.76 days, Amp_ttv_maximum: 4.45 ± 1.52 min.
TTV_minimum: 1171.19 ± 26.93 days, Amp_ttv_minimum: -5.15 ± 1.52 min.
TTV_maximum: 1254.98 ± 28.13 days, Amp_ttv_maximum: 4.45 ± 1.52 min.
TTV_minimum: 1338.77 ± 29.36 days, Amp_ttv_minimum: -5.15 ± 1.52 min.
TTV_maximum: 1422.56 ± 30.62 days, Amp_ttv_maximum: 4.45 ± 1.52 min.
P_ttv: 167.58 ± 3.03 days.
Amp_ttv: 9.60 ± 2.14 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 167.53 days; Power: 5.32; FAP: 0.0471.
Linear ephemeris (this work): Tc = 31.99541891(Tc#) + 87.16359314

Figure 5b.: Residuals of Figure 5a. [Plot avg. error bars = ± 2.23 min.]
TTV_minimum: 352.16 ± 81.47 days, Amp_ttv_minimum: -3.04 ± 1.34 min.
TTV_maximum: 686.36 ± 95.22 days, Amp_ttv_maximum: 4.06 ± 1.34 min.
TTV_minimum: 1020.56 ± 114.27 days, Amp_ttv_minimum: -3.04 ± 1.34 min.
TTV_maximum: 1354.76 ± 136.44 days, Amp_ttv_maximum: 4.06 ± 1.34 min.
P_ttv: 668.40 ± 55.92 days.
Amp_ttv: 7.09 ± 1.89 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 666.47 days; Power: 4.21; FAP: 0.143.

Figure 5c.: Added combination of Figures 5a. and 5b. [Plot error bars = ± 2.76 min.]

Figure 6a.: KOI-157.04, Kepler-11f, P = 46.69 days [Plot avg. error bars = ± 3.95 min. (smaller than symbols)]
TTV_minimum: 203.71 ± 32.33 days, Amp_ttv_minimum: -30.71 ± 4.70 min.
TTV_maximum: 484.33 ± 35.29 days, Amp_ttv_maximum: 37.50 ± 4.70 min.
TTV_minimum: 764.95 ± 40.12 days, Amp_ttv_minimum: -30.71 ± 4.70 min.
TTV_maximum: 1045.57 ± 46.23 days, Amp_ttv_maximum: 37.50 ± 4.70 min.
TTV_minimum: 1326.19 ± 53.18 days, Amp_ttv_minimum: -30.71 ± 4.70 min.
P_ttv: 561.24 ± 18.09 days.
Amp_ttv: 68.21 ± 6.64 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 566.89 days; Power: 8.17; FAP: 3.001 x 10^-6.
Linear ephemeris (this work): Tc = 46.68578557(Tc#) + 158.04338262

Figure 6b.: Residuals of Figure 6a. [Plot avg. error bars = ± 6.14 min.]
TTV_maximum: 284.37 ± 29.82 days, Amp_ttv_maximum: 10.67 ± 3.91 min.
TTV_minimum: 398.30 ± 30.96 days, Amp_ttv_minimum: -9.38 ± 3.91 min.
TTV_maximum: 512.22 ± 32.42 days, Amp_ttv_maximum: 10.67 ± 3.91 min.
TTV_minimum: 626.15 ± 34.16 days, Amp_ttv_minimum: -9.38 ± 3.91 min.
TTV_maximum: 740.08 ± 36.14 days, Amp_ttv_maximum: 10.67 ± 3.91 min.
TTV_minimum: 854.00 ± 38.31 days, Amp_ttv_minimum: -9.38 ± 3.91 min.
TTV_maximum: 967.93 ± 40.66 days, Amp_ttv_maximum: 10.67 ± 3.91 min.
TTV_minimum: 1081.86 ± 43.15 days, Amp_ttv_minimum: -9.38 ± 3.91 min.
TTV_maximum: 1195.79 ± 45.75 days, Amp_ttv_maximum: 10.67 ± 3.91 min.
TTV_minimum: 1309.71 ± 48.46 days, Amp_ttv_minimum: -9.38 ± 3.91 min.
TTV_maximum: 1423.64 ± 51.24 days, Amp_ttv_maximum: 10.67 ± 3.91 min.
P_ttv (or just periodicity): 227.85 ± 6.81 days.
Amp_ttv: 20.05 ± 5.53 minutes.
Lomb-Scargle periodogram, candidate P_ttv (or just periodicity): 228.28 days; Power: 3.73; FAP: 0.126.

Figure 6c.: Residuals of Figure 6b.; Residuals-of-the-Residuals of Figure 6a. [Plot avg. error bars = ± 7.28 min.]
TTV_maximum: 236.65 ± 11.18 days, Amp_ttv_maximum: 9.40 ± 2.41 min.
TTV_minimum: 314.29 ± 11.41 days, Amp_ttv_minimum: -11.61 ± 2.41 min.
TTV_maximum: 391.94 ± 11.69 days, Amp_ttv_maximum: 9.40 ± 2.41 min.
TTV_minimum: 469.59 ± 12.03 days, Amp_ttv_minimum: -11.61 ± 2.41 min.
TTV_maximum: 547.24 ± 12.42 days, Amp_ttv_maximum: 9.40 ± 2.41 min.
TTV_minimum: 624.89 ± 12.86 days, Amp_ttv_minimum: -11.61 ± 2.41 min.
TTV_maximum: 702.54 ± 13.34 days, Amp_ttv_maximum: 9.40 ± 2.41 min.
TTV_minimum: 780.18 ± 13.85 days, Amp_ttv_minimum: -11.61 ± 2.41 min.
TTV_maximum: 857.83 ± 14.39 days, Amp_ttv_maximum: 9.40 ± 2.41 min.
TTV_minimum: 935.48 ± 14.96 days, Amp_ttv_minimum: -11.61 ± 2.41 min.
TTV_maximum: 1013.13 ± 15.56 days, Amp_ttv_maximum: 9.40 ± 2.41 min.
TTV_minimum: 1090.78 ± 16.18 days, Amp_ttv_minimum: -11.61 ± 2.41 min.
TTV_maximum: 1168.43 ± 16.83 days, Amp_ttv_maximum: 9.40 ± 2.41 min.
TTV_minimum: 1246.08 ± 17.49 days, Amp_ttv_minimum: -11.61 ± 2.41 min.
TTV_maximum: 1323.72 ± 18.16 days, Amp_ttv_maximum: 9.40 ± 2.41 min.
TTV_minimum: 1401.37 ± 18.85 days, Amp_ttv_minimum: -11.61 ± 2.41 min.
TTV_maximum: 1479.02 ± 19.56 days, Amp_ttv_maximum: 9.40 ± 2.41 min.
P_ttv: 155.30 ± 1.71 days.
Amp_ttv: 21.01 ± 3.41 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 155.61 days; Power: 6.11; FAP: 0.00223.

Figure 6d.: Added combination of Figures 6a., 6b., and 6c. [Plot error bars = ± 10.31 min.]

Figure 7.: KOI-157.05, Kepler-11g, P = 118.38 days [Plot avg. error bars = ± 2.84 min.]
TTV_minimum: 344.05 ± 7.25 days, Amp_ttv_minimum: -6.76 ± 0.40 min.
TTV_maximum: 505.81 ± 7.75 days, Amp_ttv_maximum: 4.51 ± 0.40 min.
TTV_minimum: 667.58 ± 8.39 days, Amp_ttv_minimum: -6.76 ± 0.40 min.
TTV_maximum: 829.34 ± 9.14 days, Amp_ttv_maximum: 4.51 ± 0.40 min.
TTV_minimum: 991.10 ± 9.97 days, Amp_ttv_minimum: -6.76 ± 0.40 min.
TTV_maximum: 1152.86 ± 10.87 days, Amp_ttv_maximum: 4.51 ± 0.40 min.
TTV_minimum: 1314.62 ± 11.83 days, Amp_ttv_minimum: -6.76 ± 0.40 min.
TTV_maximum: 1476.38 ± 12.82 days, Amp_ttv_maximum: 4.51 ± 0.40 min.
P_ttv: 323.52 ± 2.38 days.
Amp_ttv: 11.27 ± 0.56 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 322.83 days; Power: 3.67; FAP: 0.00281.
Linear ephemeris (this work): Tc = 118.37865600(Tc#) + 220.29082088

References:
• Lissauer, J. J., Fabrycky, D. C., et al., Nature, 470, 53-58, 3-Feb-2011; arXiv:1102.0291, 1-Feb-2011; Q1-Q6.
• Lissauer, J., Jontof-Hutter, D., et al., ApJ, 770:131 (15pp), 20-Jun-2013; arXiv:1303.0227, 14-Jun-2013; Q1-Q14.
• Eastman, J., Gaudi, B. S., & Agol, E., PASP, 125, No. 923, Jan 2013; arxiv:1206.5798, 25-Jun-2012.
• Lithwick, Y., Wu, Y., arxiv:1204.2555.
• Lithwick, Y., Xie, J.-W., Wu, Y., arxiv:1207.4192.
• Bovaird,T. & Lineweaver, C. H., arxiv:1304.3341, 27-May-2013.
• Fabrycky et al., arXiv-1201.5415; P_ttv = 1/|(nmmri_a/P_a - nmmri_b/P_b)|.


Additional recent Kepler-11 (KOI-157) references which also contain numerous additional key references:
• Hands,R.D., et al., MNRAS, 445, 749; arxiv:1409.0532, 1-Sep-2014.
• Mahajan,N. & Wu,Y., ApJ, 795, 32; arxiv:1409.0011, 29-Aug-2014.
• Borsato,F.M., et al., arxiv:1408.2844, 12-Aug-2014.
• Gelino,D.M. & Kane,S.R., ApJ, 787, 105; arxiv:1404.7538, 29-Apr-2014.
• Kislyakova,K.G., et al., A. & A., 562, A116; arxiv:1312.4721, 17-Dec-2013.

30 Oct 2014
Summary (also see Table below):
- Periods of 157.06 and 157.01 define a near MMR of 5:4 and their interaction produces a fairly pronounced P_ttv of ~ 231 days.
- The interactions of 157.02 and 157.03 generates a P_ttv of approximately ~ 175 days.
- Periods of 157.03 and 157.04 define a near MMR of 3:2 and their interaction produces a fairly pronounced P_ttv somewhere between 561 and 668 days; better definition is limited because of the small number of known periodic cycles.
- The Lineweaver-Bovaird (L-R) relationship (Figure 8.) is consistent with a seventh planet, "157.07", with a semi-major axis of ~ 0.308 AU and (from Kepler's 3rd Law) a period of ~ 63 days.  Comparing this with the period of 157.04 gives a near MMR of 4:3.  Using the observed P_ttv of 155.3 and the Fabrycky equation, a period of ~ 61.5 days would be predicted.
- The L-R relationship (Figure 8.) also predicts an eighth planet, "157.08", having a semi-major axis of ~ 0.815 AU and a period of ~ 271 days.  Although the near MMR, 9:4, of this one versus 157.05, not being of the [(j-1)/j] type, would not be expected to generate a strong TTV, a modest one was observed with a P_ttv of ~ 323 days and an Amp_ttv of 11.3 minutes.  As above, using this observed P_ttv and the Fabrycky equation, a period of ~ 293 days is predicted…quite close to the ~271 days obtained from the L-R relationship.
- The weakest periodicity found for 157.04, ~ 228 days, is not easily rationalized in the same way but may be an example of sinusoidal curvature associated with eccentricity of one or more of the planetary orbits.
Finally, for 157.05 (Kepler-11g; P = 118.38 days), the farthest-out of the known planets, an LSP of the [(O-C) vs. Tc] data showed a highly credible periodicity at 322.83 days and a best-fit sinusoidal curve (Figure 7.) with a P_ttv of 323.52 ± 2.38 days (Amp_ttv of 11.27 ± 0.56 minutes).  This periodicity does not correspond with any of those observed for its nearest known neighbor, 157.04 (see above).  This might suggest that another farther-out, thus-far unseen (perhaps non-transiting) planetary object is mutually-interacting to give this observed TTV. {See the section below on the Lineweaver-Bovaird predictions (an improvement on the Titius-Bode relation.)}
For 157.04 (Kepler-11f; P = 46.69 days), after a sinusoidal curve-fit of the [(O-C) vs. Time] data showed (Figure 6a.) a P_ttv of 561.24 ± 18.09 days (566.89 days was observed in the Lomb-Scargle Periodogram (LSP) of the same data), a plot of the Residuals also gave a good sinusoidal curve-fit (Figure 6b.) with a periodicity of 227.85 ± 6.81 days (LSP: 228.28 days).  Similarly, the Residuals-of-the-Residuals also gave a good sinusoidal curve-fit (Figure 6c.) with a periodicity of 155.30 ± 1.71 days (LSP: 155.61 days).  The summed combination of all three sinusoidal curves (Figure 6d.) reasonably reproduces a complex overall curvature consistent with the initial data points for this planet.  While it is certainly possible that some of this unusual curvature obtains from extant eccentric orbits, it is also possible that 4 planetary objects (or 3 such objects and some orbital eccentricity(ies)), not all of which are transiting, are mutually-interacting to give this complex TTV distribution. {See further discussion below on the Lineweaver-Bovaird predictions.}
For 157.03 (Kepler-11e; P = 32.00 days), an LSP of the [(O-C) vs. Tc] data showed a credible periodicity at 167.53 days and a best-fit sinusoidal curve (Figure 5a.) with a P_ttv of 167.58 ± 3.03 days (Amp_ttv of 9.60 ± 2.14 minutes).  The residuals from the TTV plot also gave an LSP showing a modestly-credible periodicity at 666.47 days and a best-fit sinusoidal curve (Figure 5b.) characterized by a P_ttv of 668.40 ± 55.92 (and an Amp_ttv or 7.09 ± 1.89 minutes).  Summing these two sinusoidal arrays gave a complex curve (Figure 5c.) that reasonably corresponds to the original set of data points.
The [(O-C) vs. Tc] data of 157.02 (Kepler-11d; P = 22.69 days) produced a modestly-credible periodicity in its LSP of 184.18 and gave a best-fit sinusoidal curve (Figure 4.) showing a P_ttv of 184.34 ± 3.83 (Amp_ttv of 10.45 ± 2.53 minutes).
Kepler KOI-157 (Kepler-11, KIC-6541920): Evidence for an 8-Planet System

Abstract:
A large number of publications have appeared on the KOI-157 (Kepler-11) system since its 2011 discovery (see Lissauer2011 and additional references below) discussing its six known transiting planets.  Two of these evaluated TTVs for Q1-Q6 (Lissauer2011) and for Q1-Q14 (Lissauer2013).  Bovaird & Lineweaver (2013), effectively improving the Titius-Bode relation and extending it to numerous Kepler multi-planet systems, were the first to suggest the presence of two additional unseen (probably non-transiting) planets in this system, one ["157.07"] between the 5th (157.04, Kepler-11f) and 6th (157.05, Kepler-11g) known ones and one ["157.08"] further out than the 6th.  On this webpage, in addition to reporting periodic frequencies (P_ttv) and amplitudes (Amp_ttv) associated with the TTVs for all quarters (Q3-Q17) for which short-cadence data is available, we also report the results of examining the residuals (and, in one case, residuals-of-residuals) for additional sinusoidal arrays that might imply additional mutual-gravitational influences.  Overall, these data offer evidence that support Bovaird & Lineweaver's prediction of (at least) eight (8) planets in the Kepler-11 system.
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Results and Discussion:
Mid-transit times (and other properties) were obtained using the EXOFAST code developed by Eastman, Gaudi, and Agol (reference below) and Kepler PDCSAP_FLUX Short-Cadence data from Q3-Q17.  All overlapping transits and those with significant data gaps were removed before determining the linear ephemerides and the TTVs.  Note that y-axes in the graphs below are chosen so that all data points and related descriptive information could be displayed.

For 157.06 (Kepler-11b; P = 10.30 days), the closest-in planet, a Lomb-Scargle Periodogram (LSP) of the [(O-C) vs. Tc] data showed a relatively-low-credibility periodicity at 228.17 days.  Fortunately, the same data gave a best-fit sinusoidal array (Figure 1.), with a P_ttv of 229.08 ± 5.04 days and an Amp_ttv of 23.14 ± 4.92 minutes, which turned out to be exquisitely anti-coordinated (see below) with the best-fit sinusoidal array of the second closest-in planet, 157.01; the ratio of the Periods for 157.01 vs. 157.06 is 1.264, or approximately in a Mean Motion Resonance of ~ (j+1)/j = 5/4.

For 157.01 (Kepler-11c; P = 13.02 days), an LSP of the [(O-C) vs. Tc] data showed a highly credible periodicity at 233.08 days and a best-fit sinusoidal curve (Figure 2.) with a P_ttv of 234.16 ± 4.15 days and an Amp_ttv of 13.40 ± 2.03 minutes.  Note that the relative Amp_ttv's of these first two planets (which ignores any perturbations due to orbital eccentricity differences (see, for example, Lithwick2013)) is in the order expected from their relative masses (Lissauer2013; see also exoplanet.eu).

The high anti-correlation between the two sinusoidal arrays above is graphically emphasized when both are plotted in the same graph (Figure 3.) or by comparing the maxima and minima in the data below.