###### References:

###### • Nesvorny, Kipping, & Terrell, arXiv-1405.2060, 8 May 2014.

###### • For Q0-6 TTV data of 884.01 and 884.02 (thru ~ 563 (BJD-2454900) days), see: Ford et al., 2012, arXiv-1201.1892.

###### • For Q0-12 TTV data of 884.01, 884.02, and 884.03 (thru ~ 1116 (BJD-2454900) days), see Rowe et al., 2014, arXiv-1402.6534.

###### • For Q0-12 TTV data of 884.01, 884.02, and 884.03 (thru ~ 1116 (BJD-2454900) days), see: Mazeh et al., 2013, arXiv-1301.5499.

###### • For Q0-14 TTV data of 884.02, see Ji-Wei Xie's website: http://www.astro.utoronto.ca/~jwxie/TTV/TTV_Home.html

###### 1 Apr 2014; updated 9 May 2014.

**Light blockages by transits T_113 of KOI-884.01 and T_52 of KOI-884.02 occur at nearly identical times resulting in a deep **__"time-overlapping" transit__ with a very similar duration to those of each of the components (compare Figure A, C, and D). Since the approximate area (~ Depth x Duration), ~ 574 ± 20 ppm•days, is quite similar to the total area for one transit of each planet: ~ [(322 ± 53) + (309 ± 45)] = 631 ± 69 ppm•days, 844.01 and 844.02 must cross the star on two different chords so that each blocks its own full disk of light (i.e: no actual visual overlap).

**Another interesting case involves transits T_100 of 884.01 and T_46 of 884.02. Here they show a wide **__"time-abutting" transit__ with a depth similar to those of each of the individual planets (compare Figures B, C, and D) but about twice the duration. Their combined derived area (Figure B) is again very similar to those described above: ~ 610.20 ± 22.12 ppm•days. This is consistent with one planet exiting the star's disk simultaneously with the other entering, i.e.: the egress of one starting simultaneously with the ingress of the other. The insights of Denali Duhan Relles on this system are most gratefully acknowledged.

__EXOFAST Output Parameters__:

**The following Table lists a number of the output parameters calculated for the three known (transiting) planets.**** The average "approx. derived AREA = Depth x Duration" (in units of ppm•days) is included for reference in the following discussion of overlapping or near overlapping transits; near-trapezoidal shapes have been assumed for each transit in this approximation.**

###### Figure 5.: (O-C) vs. Time, KOI-884.01, P = 9.44 days [Plot avg. error bars = ± 2.41 min.].

###### Lomb-Scargle periodogram: no credible periodicities.

**Linear ephemeris (this work): Tc = 9.43946676(Tc#) + 110.18586463**

###### Figure 6.: (O-C) vs. Time, KOI-884.03, P = 3.33 days [Plot avg. error bars = ± 8.03 min.].

###### Lomb-Scargle periodogram: no credible periodicities.

**Linear ephemeris (this work): Tc = 3.33612868(Tc#) + 68.32093439**

###### xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

###### xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

**Multiple Orbital Chords for 884.01 and 884.02 (from "time-overlapping" transits):**

**[For direct comparative purposes, the x- and y-scales for Figures A - D are identical.]**

###### 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.: TTV of KOI-884.02, P = 20.48 days [Plot avg. error bars = ± 3.09 min. (within the plot symbols)].

###### TTV_maximum: 111.24 ± 21.47 days, Amp_ttv_maximum: 167.47 ± 11.72 min.

###### TTV_minimum: 558.56 ± 25.19 days, Amp_ttv_minimum: -152.33 ± 11.72 min.

###### TTV_maximum: 1005.87 ± 32.25 days, Amp_ttv_maximum: 167.47 ± 11.72 min.

###### TTV_minimum: 1453.19 ± 40.96 days, Amp_ttv_minimum: -152.33 ± 11.72 min.

**P_ttv: 894.64 ± 21.54 days.**

**Amp_ttv: 319.80 ± 16.58 minutes.**

**Lomb-Scargle periodogram, candidate P_ttv: 882.71 days; Power: 24.69; FAP: 1.137 x 10^-8.**

**Linear ephemeris (this work): Tc = 20.48395239(Tc#) + 111.61532546**

###### Figure 2.: Residuals of Figure 1. [Plot avg. error bars = ± 12.12 min. (just visible outside plot symbol)].

###### TTV_maximum: 377.26 ± 36.04 days, Amp_ttv_maximum: 43.54 ± 9.07 min.

###### TTV_minimum: 659.18 ± 41.20 days, Amp_ttv_minimum: -48.38 ± 9.07 min.

###### TTV_maximum: 941.11 ± 48.09 days, Amp_ttv_maximum: 43.54 ± 9.07 min.

###### TTV_minimum: 1223.03 ± 56.08 days, Amp_ttv_minimum: -48.38 ± 9.07 min.

**P_ttv: 563.84 ± 20.82 days.**

**Amp_ttv: 91.91 ± 12.82 minutes.**

**Lomb-Scargle periodogram, candidate P_ttv: 566.17 days; Power: 12.26; FAP: 0.00284.**

###### Figure 3.: Residuals of Figure 2.; Residuals-of-the-Residuals of Figure 1. [Plot avg. error bars = ± 15.14 min.]

###### TTV_minimum: 190.21 ± 177.72 days, Amp_ttv_minimum: -43.38 ± 6.46 min.

###### TTV_maximum: 1290.80 ± 298.70 days, Amp_ttv_maximum: 36.45 ± 6.46 min.

**P_ttv: 2201.18 ± 413.91 days.**

**Amp_ttv: 79.83 ± 9.13 minutes.**

**Lomb-Scargle periodogram, candidate P_ttv: 2134.70 days; Power: 18.13; FAP: 8.15 x 10^-6.**

###### Figure 4.: Summed combination of Figures 1., 2., and 3. [Plot error bars = ± 19.64 min.]

**KOI-884 (Kepler-247, KIC-7434875) 6(?)-(or more?)-Planets**

__TTV Discussion__:

**In the case of the outermost known planet, KOI-884.02 (Period: 20.48 days), after a sinusoidal curve-fit of the initial Time vs. (O-C) data showed (Figure 1. below) a periodicity (P_ttv) of 894.64 ± 21.54 days (882.71 days 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 563.84 ± 20.82 days (LSP: 566.17 days). Similarly, the Residuals-of-the-Residuals also gave a good sinusoidal curve-fit (Figure 3.) with a periodicity of 2201.18 ± 413.91 days (LSP: 2134.70 days). [Also, as has been seen in other systems with pronounced TTVs (e.g.: KOI-142.01), "chopping" of a portion of the TTV curve is also evident for 884.02 (see Figure 1a.).] Lastly, the summed combination of all three sinusoidal curves is displayed in Figure 4. and reasonably reproduces a complex overall curvature consistent with the initial (Figure 1.) data.**

**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 4 planets in near-circular orbits are mutually-interacting to give the TTV distribution observed. Neither of the other two known planets in this system (see Figure 5.: KOI-884.01 (P = 9.44 days) and Figure 6: KOI-884.03 (P = 3.33 days)) shows any appreciable TTV nor any credible periodicity for its (O-C) vs. Time data in a Lomb-Scargle periodogram. This is particularly surprising with respect to the former (884.01) since its orbital Period puts it just wide of a Near Mean Motion Resonance relationship (1:00 to 2.17) with that of 884.02 (with the sizable TTV…and overall amplitude of ~ 320 min. (~ 5.3 hours)…discussed above).**

**These observations are most consistent with "unseen" (non-transiting) planetary object(s), perhaps as many as three, with semi-major axis(axes) greater than that of 884.02, being responsible for the large TTV (and Residual) sinusoidal patterns observed.**