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**KOI's (Blue)**** or**

**K2 Objects (Green)**

###### Additional References:• Kepler Community Follow-up Observing Program (CFOP): https://cfop.ipac.caltech.edu/home/

###### 15 Oct 2013

**Jeff Coughlin's (P. H. blog, see above) description of Event 4**: First, this event is at the start of a quarter, and the decreasing flux trend and jump from the previous quarter are systematics not fully corrected by the Kepler pipeline. But, 60 days after Event 3, we see that stars B and C are again transiting star A. This time however, unlike Event 2, stars B and C are also lined up to eclipse each other. Thus, stars B and C are going in and out of eclipse while at the same time transiting star A, resulting in this very unusual looking event.

**Jeff Coughlin's (P. H. blog, see above) description of Event 3**: 238 days after Event 1, we see another occultation.This time stars B and C are not in eclipse, but instead well-separated, and thus each occult separately. The occultation depths agree with Event 1, yielding flux contributions of ~2.4% and 2.2% for stars B and C respectively.

**Jeff Coughlin's (P. H. blog, see above) description of Event 2**: About 60 days after Event 1, star A now starts to pass behind stars B and C. As opposed to Event 1, this time, as we see from the 10.7 day eclipse pattern, stars B and C are well separated and out of eclipse. Thus, stars B and C transit star A individually, resulting in the round-bottom transit curves. From the transit depths we know, approximately, that star B is ~31.5% the radius of star A, and star C is ~27.7% the radius of star A. This is really approximate though, because we don't know whether they are central transits, or grazing, or the mutual inclination between the B-C orbital planet and the A-BC orbital planet. But, they should be fairly close.

__Jeff Coughlin's (P. H. blog, see above) description of Event 1__: As we see from the regular 10.7 eclipses, stars B and C are going into eclipse at the same time that star A, which is larger, passes in front of them. We get the very deep, flat-bottomed occultation as both B and C disappear behind A. As stars B and C are still occulted behind star A, they start to come out of eclipse. Star B becomes visible to us as it pops out from behind A, while C is still behind star A, resulting in the half-as-deep occultation. Star C then comes out of occultation, and we see all three stars again. From this event, and depth of the deepest occultation, we know stars B and C together comprise about 4.2% of the total system light, with star C comprising ~2% and star B comprising ~2.2%. Thus star A comprises 95.8% of the system light.

**BC Eclipses by A and BC Transits of A EVENTS:**

###### Lastly, all 10 of the major [BC-binary-pair-eclipse-by, or BC-binary-pair-transit-of A] "Events" observed by Kepler through Q16 are shown below.

**While, as mentioned above, 1351.01 is probably destined to become a "False Positive" in Kepler "exoplanet-speak", there is nothing "False" about this truly remarkable, Earth-aligned, complex star system which will probably be studied for some time to come.**

**Time Variation Between Eclipses of BC Binary Pair:**

###### In addition, the eccentricity of the BC binary was probed by measuring the time between sequential pairs of eclipses (where equal times would have meant zero eccentricity). Surprisingly, these comparative times were far from simple (see Figure below). At the start of Q4, the time between the B-to-C eclipses was less than that for the C-to-B eclipses by about 0.14 day (or about 3.4 hours). With time, this gradually reversed, and by the end of Q16, the time between the B-to-C eclipses was about 0.35 day (or about 8.4 hours) more than that for the C-to-B eclipses. Also, relatively abrupt changes in these times were particularly noted as the BC-pair was eclipsed by the larger star A, while comparatively less changes followed BC-pair transits of A. (This probably implies that the BC-pair eclipses by A occurred much closer to A (nearer to periastron) than the BC-pair transits of A.) Additional "fine-structure" was also noted after each BC-pair's transit of A in that the differences in sequential times between B and C paired eclipses seemed to progress through slightly "curved plateaus" until the next eclipse-by-A Event. One cannot help wondering if these overall trends will eventually reverse or if the system is in an evolutionary process that will continue to alter it indefinitely.

###### (Plot error bars (smaller than symbol) = ± 0.00136)

###### ETVs are derived from Q4-Q16 Kepler data. x-axes: “Observed Eclipse” (Mid-Eclipse Time): EXOFAST’s best-fits from Kepler light flux vs. time data. y-axes: “(O – C)”: difference between Observed Eclipse and the Calculated Eclipse from the graphically obtained linear ephemeris.

###### A: major star; B and C: minor stars, components of a binary pair orbiting the major star A.

**Star B:**

###### P = 10.724 days [Plot error bars (smaller than symbol) = ± 1.35 min.]

###### ETV_minimum: 387.40 ± 4.15 days, Amp_etv_minimum: -80.89 ± 3.55 min.

###### ETV_maximum: 506.80 ± 4.36 days, Amp_etv_maximum: 79.73 ± 3.55 min.

###### ETV_minimum: 626.19 ± 4.61 days, Amp_etv_minimum: -80.89 ± 3.55 min.

###### ETV_maximum: 745.59 ± 4.90 days, Amp_etv_maximum: 79.73 ± 3.55 min.

###### ETV_minimum: 864.98 ± 5.21 days, Amp_etv_minimum: -80.89 ± 3.55 min.

###### ETV_maximum: 984.38 ± 5.56 days, Amp_etv_maximum: 79.73 ± 3.55 min.

###### ETV_minimum: 1103.77 ± 5.92 days, Amp_etv_minimum: -80.89 ± 3.55 min.

###### ETV_maximum: 1233.17 ± 6.30 days, Amp_etv_maximum: 79.73 ± 3.55 min.

###### ETV_minimum: 1342.56 ± 6.69 days, Amp_etv_minimum: -80.89 ± 3.55 min.

###### ETV_maximum: 1461.96 ± 7.10 days, Amp_etv_maximum: 79.73 ± 3.55 min.

**P_etv: 238.79 ± 0.98 days.**

**Amp_etv: 160.62 ± 5.02 minutes.**

**Lomb-Scargle periodogram, candidate P_etv: 238.49 days; Power: 41.27; P-value: 0.**

**Linear ephemeris (this work): Tc = 10.72427045(Tc#) + 290.13483714**

**Star C:**

###### P = 10.727 days [Plot error bars (smaller than symbol) = ± 1.42 min.]

###### ETV_minimum: 391.15 ± 5.89 days, Amp_etv_minimum: -93.13 ± 6.23 min.

###### ETV_maximum: 510.62 ± 6.19 days, Amp_etv_maximum: 99.56 ± 6.23 min.

###### ETV_minimum: 630.10 ± 6.54 days, Amp_etv_minimum: -93.13 ± 6.23 min.

###### ETV_maximum: 749.57 ± 6.95 days, Amp_etv_maximum: 99.56 ± 6.23 min.

###### ETV_minimum: 869.04 ± 7.40 days, Amp_etv_minimum: -93.13 ± 6.23 min.

###### ETV_maximum: 988.52 ± 7.89 days, Amp_etv_maximum: 99.56 ± 6.23 min.

###### ETV_minimum: 1107.99 ± 8.40 days, Amp_etv_minimum: -93.13 ± 6.23 min.

###### ETV_maximum: 1227.47 ± 8.94 days, Amp_etv_maximum: 99.56 ± 6.23 min.

###### ETV_minimum: 1346.94 ± 9.50 days, Amp_etv_minimum: -93.13 ± 6.23 min.

###### ETV_maximum: 1466.42 ± 10.07 days, Amp_etv_maximum: 99.56 ± 6.23 min.

**P_etv: 238.95 ± 1.38 days.**

**Amp_etv: 192.69 ± 8.81 minutes.**

**Lomb-Scargle periodogram, candidate P_etv: 238.13 days; Power: 39.21; P-value: 0.**

**Linear ephemeris (this work): Tc = 10.72677194(Tc#) + 295.41057304**

**Star B, Residuals of Sinusoidal ETV:**

###### (Plot error bars (smaller than symbol) = ± 3.18 min.)

###### ETV_Res.minimum: 300.18 ± 33.63 days, Amp_etv_Res.minimum: -23.27 ± 1.71 min.

###### ETV_Res.maximum: 360.04 ± 40.33 days, Amp_etv_Res.maximum: 23.87 ± 1.71 min.

###### ••••••• (several others) •••••••

###### ETV_Res.minimum: 1377.63 ± 154.33 days, Amp_etv_Res.minimum: -23.27 ± 1.71 min.

###### ETV_Res.maximum: 1437.49 ± 161.03 days, Amp_etv_Res.maximum: 23.87 ± 1.71 min.

**P_etv: 119.72 ± 0.41 days.**

**Amp_etv: 47.14 ± 2.42 minutes.**

**Lomb-Scargle periodogram, candidate P_etv: 119.98 days; Power: 35.11; P-value: 5.16 x 10^-13.**

**Star C, Residuals of Sinusoidal ETV:**

###### (Plot error bars (smaller than symbol) = ± 5.29 min.)

###### ETV_Res.minimum: 296.68 ± 51.20 days, Amp_etv_Res.minimum: -35.94 ± 4.34 min.

###### ETV_Res.maximum: 356.36 ± 61.50 days, Amp_etv_Res.maximum: 33.54 ± 4.34 min.

###### ••••••• (several others) •••••••

###### ETV_Res.minimum: 1370.95 ± 236.59 days, Amp_etv_Res.minimum: -35.94 ± 4.34 min.

###### ETV_Res.maximum: 1430.63 ± 246.89 days, Amp_etv_Res.maximum: 33.54 ± 4.34 min.

**P_etv: 119.36 ± 0.69 days.**

**Amp_etv: 69.48 ± 6.14 minutes.**

**Lomb-Scargle periodogram, candidate P_etv: 119.39 days; Power: 24.92; P-value: 1.51 x 10^-8.**

**Eclipse Depth & Duration Variations:**

###### It was also found that the eclipsing BC pair showed relatively large variations in their eclipse depths ("**EPV**"s) and small but distinct variations in their eclipse durations ("**EDV**"s). That those of B are larger than those of C is in complete agreement with Jeff Coughlin's assessments of the relative sizes and luminosities of B and C. Scrutiny of the plotted eclipse depths vs. time, in addition to showing the overall Q4-Q16 increases and decreases, also seems to show some repeat "plateau-ing" along the way. This, of course, would be a further constraint-detail to be accounted for in any future model of this system. The following two Figures summarize these observations.

###### (Star B **EPV**, Plot error bars (smaller than symbol) = ± 0.00033; Star C **EPV**, Plot error bars (smaller than symbol) = 0.00034.)

###### (Star B **EDV**, Plot error bars = ± 0.00374; Star C **EDV**, Plot error bars = 0.00395.)

**Kepler KOI-1351 (KIC-6964043) Trinary Star System**

**The current (11-Oct-2013) disposition of KOI-1351.01 in the NASA Exoplanet Archive (NEA) and the Kepler Community Follow-up Observing Program (CFOP) is as an exoplanet candidate. As the Kepler light curve of the KOI-1351 system was being collected during Q4-Q16, it was being evaluated and discussed in detail by the "Planet Hunters" (P. H.)( http://talk.planethunters.org/discussions/DPH101e6cv?object_id=APH52300057&page=1&per_page=10 ), under a blog heading "KID 6964043 Trinary" which included contributions from numerous people (Sean Babin, Mike Barrett, Jeff Coughlin, Abe Hoekstra, Tony Jebson, Kian Jek, Mark Omohundro, Trevor Poile, Hans Martin Schwengeler, Johann Sejpka, and others with P.H. user names: arvintan, ggccg, nighthawk_black, planetsam, and troyw.). Pretty convincingly, it appears to be what they suggest: an eclipsing trinary star system. In fact, one of the discussions (by Jeff Coughlin) explains the first 4 of the most unusual light-curve "Events" in detail with great clarity; these were the first two observed eclipses of the BC binary pair by the larger star A, and the first two observed transits by the BC pair of star A. (At the same time, he likens them to the KOI-126 trinary system (ref: Carter, et al., arXiv-1102.0562, 2011) and its corresponding video ( http://www.youtube.com/watch?v=9fowFudwdHQ ) that wonderfully portrays this triple star's orbital patterns through its Q3-Q5 Kepler light curve.)**

**To date, KOI-1351 has treated us to 5 such eclipse and 5 such transit "Events" in addition to the frequent eclipses about every 5.36 days within the BC binary pair itself. While it seems likely that 1351.01 will eventually be removed from the exoplanet-[with transit timing variations (TTVs)]-candidate category and re-dispositioned as a "False Positive" eclipsing-[with eclipse timing variations (ETVs)]-trinary system, it is included in this [mostly TTVs] website to highlight some recent findings that update and augment what others have reported on this extremely interesting system.**

**As was depicted by Kian Jek for Q4-Q13, the repeating skewed-sinusoidal ETV behavior of the BC binary pair continues through Q16 (see below). Furthermore, the best-fit, non-skewed-sinusoidal curve of all the data for each of these seems to yield:#1: accurate values for the periodicity of the eclipse-timing variations (P_etv's)),#2: reasonably accurate values for the amplitude of those periodicities (Amp_etv's), and#3: fairly accurate times for each minimum and maximum amplitudes (Amp_etv_minimum's and Amp_etv_maximum's).Also found is a clearly repeating relationship between the actual ETV skewed-sinusoidal behavior and the the non-skewed-sinusoidal best-fit curve, for both B and C, from the fact that the Residuals therefrom also gave excellent sinusoidal curve-fits (see below). (It is assumed that this would have a strong bearing on the validity of #1 above, but an expert's opinion would certainly be appreciated on this point.)In addition, the periodicities thus determined for B, C, B-Residuals, and C-Residuals were each substantiated by corresponding (and credible, i.e.: very low "by chance" probabilities) Lomb-Scargle-Periodogram periodicities. The following Figures and Data summarize all of these results.**