###### Key points from Table 1.:

###### 1. Both the "Fit1" and "Fit2" equations (see just below Table 1.), in all cases, gave nearly identical curve-fits as measured by their P_ttv's and Amp_ttv's, and each agreed quite well with the Lomb-Scargle periodogram results. Possibly one difference is that the "Fit2" approach seemed to give lower errors for P_ttv's than the "Fit1" one.

###### 2. 152.03: This innermost planet candidate shows only one clearly perturbing object; the residuals show no significant sinusoidal periodic curvature (and a Lomb-Scargle periodogram showed only "by chance" periods). The P_ttv values (LSP/Fit1/Fit2: 835.34/830.87/828.06) match quite closely with that calculated (852.87 days; see Table 2.) for the combined innermost two planets (152.03 and 152.02) using the equation from the Fabrycky reference.

###### 3. 152.02, the second planet candidate, shows a strong TTV with a P_ttv of about 1040 days; its residuals also show two sinusoidal "best-fits" with P_ttv's at about 595 and 161 days, respectively. Interestingly, one of these is not too different from the value of 526 days calculated for the 152.02/152.01 planet pair (lower table) while the origin of the other is obscure at this time.

###### 4. 152.01, the third known planet candidate, similarly shows a strong TTV with a P_ttv at about 580 days. The residuals similarly produce two more "best-fit" sinusoidal arrays with P_ttv's of about 840 and 453 days, respectively. The P-values for the latter (0.107 and 0.154) suggest it could simply be a "by chance" period, while the former is not too different from the value calculated (721.09 days) for the interaction of 152.01 and 152.04 in the absence of other perturbers (Table 2.).

###### 5. 152.04, the furthest-out known planet candidate, shows one main perturbing object and a P_ttv of about 840 days along with possibly another one with a 273 day period. While it seems likely that the first results from the interaction with 152.01 (see above), it is possible that the second arises from an interaction with another further-out planet yet to be observed (estimated by Bovarid and Lineweaver). It is also possible that the 273-day period might not be real since its P-values are a little high (0.1340 and 0.1210).

###### References:

###### • Bovaird & Lineweaver, 2013, arXiv-1304.3341: "Exoplanet Predictions Based on the Generalized Titius-Bode Relation".

###### • P_ttv equation: P_ttv = 1/|(nmmri_a/P_a - nmmri_b/P_b)|…(see Fabrycky, et al., 2012, arXiv-1201.5415)

###### • For Q1-Q6 TTV data of 152.01, 152.02, and 152.03 (thru ~ 563 (BJD-2454900) days), see: Ford2012arXiv-1201.1892.

###### 14 August 2013

###### 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_152.03/P_152.02: 1/2;**

**P_152.02/P_152.01: 1/2;**

**P_152.01/P_152.04: 2/3.**

**The TTVs of the innermost two exoplanet candidates (152.03 and 152.02) are quite well anti-correlated. That this is much poorer for the outer two planetary candidates (152.01 and 152.04) could reflect differences in eccentricities of their orbits, precession therein, or simply the effects of another as yet unseen (perhaps non-transiting) planet further out ("152.05"). Bovaird and Lineweaver, using the Titius-Bode approximation, have recently estimated such an additional planet (P: 160 ± 16 days; a = ~ 0.60).**

###### 1st: KOI-152.03, P = 13.48 days

###### TTV_minimum: 402.92 ± 57.98 days, Amp_ttv_minimum: -8.90 ± 2.45 min.

###### TTV_maximum: 828.35 ± 73.13 days, Amp_ttv_maximum: 10.69 ± 2.45 min.

###### TTV_minimum: 1233.79 ± 93.20 days, Amp_ttv_minimum: -8.90 ± 2.45 min.

###### P_ttv: 830.87 ± 51.99 days.

###### Amp_ttv: 19.59 ± 3.46 minutes.

###### Lomb-Scargle periodogram, candidate P_ttv: 835.34 days; Power: 9.70; P-value: 0.0530.

###### Linear ephemeris (this work): Tc = 13.48455388(Tc#) + 69.62108828

###### Estimated P_ttv = 1/|(nmmri_a/P_a - nmmri_b/P_b)| = 852.87 days

###### 2nd: KOI-152.02, P = 27.40 days

###### TTV_maximum: 341.13 ± 51.26 days, Amp_ttv_maximum: 14.60 ± 3.05 min.

###### TTV_minimum: 862.63 ± 69.25 days, Amp_ttv_minimum: -20.71 ± 3.05 min.

###### TTV_maximum: 1384.13 ± 94.03 days, Amp_ttv_maximum: 14.60 ± 3.05 min.

###### P_ttv: 1042.99 ± 61.29 days.

###### Amp_ttv: 35.31 ± 4.31 minutes.

###### Lomb-Scargle periodogram, candidate P_ttv: 1064.74 days; Power: 12.06; P-value: 0.00000233.

###### Linear ephemeris (this work): Tc = 27.40236028(Tc#) + 66.62693697

###### Estimated P_ttv = 1/|(nmmri_a/P_a - nmmri_b/P_b)| = 526.00 days

###### 3rd: KOI-152.01, P = 52.09 days

###### TTV_minimum: 291.10 ± 42.24 days, Amp_ttv_minimum: -7.59 ± 2.22 min.

###### TTV_maximum: 582.65 ± 48.35 days, Amp_ttv_maximum: 9.08 ± 2.22 min.

###### TTV_minimum: 874.19 ± 57.10 days, Amp_ttv_minimum: -7.59 ± 2.22 min.

###### TTV_maximum: 1165.73 ± 67.47 days, Amp_ttv_maximum: 9.08 ± 2.22 min.

###### TTV_minimum: 1457.28 ± 78.82 days, Amp_ttv_minimum: -7.59 ± 2.22 min.

###### P_ttv: 583.09 ± 27.18 days.

###### Amp_ttv: 16.67 ± 3.14 minutes.

###### Lomb-Scargle periodogram, candidate P_ttv: 576.47 days; Power: 5.98; P-value: 0.00356.

###### Linear ephemeris (this work): Tc = 52.09097800(Tc#) + 91.74603039

###### Estimated P_ttv = 1/|(nmmri_a/P_a - nmmri_b/P_b)| = 721.09 days

###### 4th: KOI-152.04, P = 81.07 days

###### TTV_minimum: 410.01 ± 88.79 days, Amp_ttv_minimum: -37.95 ± 15.63 min.

###### TTV_maximum: 833.02 ± 110.39 days, Amp_ttv_maximum: 44.43 ± 15.63 min.

###### TTV_minimum: 1256.02 ± 139.34 days, Amp_ttv_minimum: -37.95 ± 15.63 min.

###### P_ttv: 846.00 ± 76.52 days.

###### Amp_ttv: 82.38 ± 22.10 minutes.

###### Lomb-Scargle periodogram, candidate P_ttv: 827.36 days; Power: 3.55; P-value: 0.0942

###### Linear ephemeris (this work): Tc = 81.06449079(Tc#) + 72.53104886

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__Analysis of Best-Fit-Sinusoidal-Curve RESIDUALS__

**Best-fit sinusoidal curves were obtained for each of the exoplanet candidates (see the graphs above). From these, residuals (i.e.: differences between observed y-values and those calculated from the equation of the sinusoidal curves) were determined and plotted vs. time to see if they showed any regular patterns. Indeed, each of these appeared to show additional periods in their Lomb-Scargle periodograms and "secondary" curve-fit sinusoidal arrays. The overall results are summarized in Table 1.**

**KOI-152 (Kepler-79, KIC-8394721) 4-(or more?)-Planet System**