###### Figure 5.: KOI-351.01, P = 331.60 days [Plot error bars average = ± 1.02 min., smaller than the symbols.]

###### TTV_maximum: ~ 364.62 days, Amp_ttv_maximum: ~ 35.70 min.

###### TTV_minimum: ~ 673.24 days, Amp_ttv_minimum: ~ -25.34 min.

###### TTV_maximum: ~ 981.86 days, Amp_ttv_maximum: ~ 35.70 min.

###### TTV_minimum: ~ 1290.48 days, Amp_ttv_minimum: ~ -25.34 min.

###### P_ttv: ~ 617.24 days.

###### Amp_ttv: ~ 61.04 minutes.

###### [With only 3 points, approximate (~) indicators are needed for these data since the software (Kaleidograph) was able to develop a perfect (R = 1) best-fit curve.]

###### [Lomb-Scargle periodogram: not able to calculate since at least 4 points are needed.]

###### Linear ephemeris (this work): Tc = 331.6005(Tc#) + 73.5021

###### (Linear ephemeris (NEA reported): Tc = 331.6430(Tc#) + 73.4772)

###### References:

###### Two landmark papers dealing with the first Kepler septet of exoplanet candidates, KOI-351.0x, have recently appeared:

###### • (a) Schmitt, Wang, Fischer, Jek (P.H.), Moriarty, Boyajian, Schwamb, Lintott, Smith, Parrish, Schawinski, Lynn, Simpson, Omohundro (P.H.), Winarski (P.H.), Goodman (P.H.), Jebson (P.H.), & Lacourse (P.H.), arXiv:1310.5912v1, 22 Oct 2013.

###### • (b) Cabrera, Csizmadia, Lehmann, Dvorak, Gandolfi, Rauer, Erikson, Dreyer, Eigmüller, & Hatzes, arXiv:1310.6248v1, 23 Oct 2013.

###### Both groups uncovered the seventh planet ("351.07", "11442793-f", P = ~ 124.9 days) from within the Kepler data for this system, and both discussed what appear to be highly unusual TTV results for another (351.02, "11442793-g", P = ~ 210.6 days).

**-------------------------------------------------------------------------------------------**

###### • The Home page for the KAFO project: see Bruce Gary, http://brucegary.net/kafo/; KAFO suggested list of Kepler follow-up targets: Table "Kepler198v3808": P > 7 days; Depths from 3-40 mmag; Magnitude < 16. Also see the KAFO page within this website.

###### • The Data and Plots page for the KAFO project:

###### • KOINet, Principal Investigator: Carolina von Essen, http://koinet.astro.physik.uni-goettingen.de/; the site contains plotted TTV data for 351.01 and 351.02 through Kepler Q16.

###### • For Q1-6 TTV data of 351.02 and 351.03 (thru ~ 563 (BJD-2454900) days, see Ford2012arXiv-1201.1892.

###### • For Q0-12 TTV data of 351.01, 351.02, and 351.03 (thru ~ 1116 (BJD-2454900) days), see: Mazeh et al., 2013, arXiv-1301.5499.

###### 28 Mar 2014

###### TTVs are derived from Q1-Q16 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-351.02, P = 210.72 days [Plot avg. error bars = ± 1.60 min. (smaller than symbols)]

###### TTV_minimum: 372.04 ± 121.52 days, Amp_ttv_minimum: -530.00 ± 287.81 min.

###### TTV_maximum: 704.19 ± 148.32 days, Amp_ttv_maximum: 573.50 ± 287.81 min.

###### TTV_minimum: 1036.33 ± 183.55 days, Amp_ttv_minimum: -530.00 ± 287.81 min.

###### P_ttv: 664.29 ± 94.48 days.

###### Amp_ttv: 1103.50 ± 407.03 minutes.

###### Lomb-Scargle periodogram, candidate P_ttv: 658.59 days; Power: 1.88; FAP: 0.234.

###### Linear ephemeris (this work): Tc = 210.71963214(Tc#) + 79.87683690

###### Figure 2.: Residuals of Figure 1. [Plot avg. error bars = ± 287.82 min.]

###### TTV_maximum: 425.95 ± 133.22 days, Amp_ttv_maximum: 267.74 ± 142.24 min.

###### TTV_minimum: 967.76 ± 173.46 days, Amp_ttv_minimum: -288.49 ± 142.24 min.

###### P_ttv: 1083.63 ± 138.53 days.

###### Amp_ttv: 556.23 ± 201.16 minutes.

###### Lomb-Scargle periodogram, candidate P_ttv: 1081.32 days; Power: 1.95; FAP: 0.280.

###### Figure 3.: Residuals of Figure 2.; Residuals-of-the-Residuals of Figure 1. [Plot avg. error bars = ± 321.05 min.]

###### TTV_maximum: 282.18 ± 26.10 days, Amp_ttv_maximum: 132.53 ± 20.56 min.

###### TTV_minimum: 571.32 ± 29.70 days, Amp_ttv_minimum: -132.81 ± 20.56 min.

###### TTV_maximum: 860.46 ± 34.92 days, Amp_ttv_maximum: 132.53 ± 20.56 min.

###### TTV_minimum: 1149.61 ± 41.15 days, Amp_ttv_minimum: -132.81 ± 20.56 min.

###### P_ttv: 578.29 ± 16.51 days.

###### Amp_ttv: 265.34 ± 29.07 minutes.

###### Lomb-Scargle periodogram, candidate P_ttv: 578.32 days; Power: 2.43; FAP: 0.00443.

###### Figure 4.: Added combination of Figures 1., 2., and 3. [Plot error bars = ± 431.18 min.]

###### Discussion:

###### Continuing further with this extraordinary extrasolar system, the largest of the known exoplanets, 351.01, seems to also show a significant TTV. The respective periods of 351.01 to 351.02 are just wide of a 3:2 near Mean Motion Resonance (actually 3.15:2) and the best-fit sinusoidal curve for 351.01 (albeit based only on 3 valid transits to date*) suggests a P_ttv of ~ 617.24 days, i.e. within experimental error of the above-reported (with very large amplitude variation) 664.29 ± 94.48 days P_ttv for 351.02. Note also that the anti-correlated relationship of this pair of TTV curves (see Figures 1. and 5.)for these two exoplanets is also strongly suggestive of their mutual interactions. By comparison with our Solar System, one might have anticipated such a relatively strong gravitational interactions between these two exoplanets since 351.01's semi-major axis (0.98 AU) is about the same as Earth's and it is nearly the size of Jupiter (R_jup = 0.89), while 351.02's (0.72 AU) is the same as Venus's and it is nearly the size of Saturn (R_sat = 0.75).* It is hoped that further observations within ground-based programs, such as the KAFO and/or the KOINet programs(reference below) will uncover additional transits of these two large planets to further constrain of their TTVs.

**KOI-351 (Kepler-90, KIC-11442793) 7-(or more?)-Planet System**

__Discussion__:

**For the known second-most-outer exoplanet candidates of this system, 351.02 (P = 210.60 days), after a sinusoidal curve-fit of the initial Time vs. (O-C) data showed (Figure 1. below) a periodicity (P_ttv) of 664.29 ± 94.48 days (a 658.59 days periodicity 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 1083.63 ± 138.53 days (LSP: 1081.32 days). Similarly, the Residuals-of-the-Residuals also gave a good sinusoidal curve-fit (Figure 3.) with a periodicity of 578.29 ± 16.51 days (LSP: 578.32 days). Lastly, the "linear" re-combination of all three sinusoidal curves is arrayed in Figure 4. and reasonably reproduces a complex overall curvature consistent with the initial data of Figure 1. 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 of the other planets in near-circular orbits are mutually-interacting to give the TTV distribution observed.**