All transit areas were approximated by the so-called "method of narrow trapezoids" where the number of trapezoids was large compared to the width of the transit. For example, a transit with a duration of 0.122 days was broken into 180 narrow trapezoids, the areas of each were calculated, and then all were summed.
Figure 8. shows the area results for the transits defined during a 676.01 Exofast evaluation; those that overlap with one of the 676.02 transits are separated as red symbols. Each series approximates a near-zero-sloped line; error bars (see below) are smaller than the symbols. The average areas (units: days • normalized flux • 10^6) are:
• difference (area of 676.02 within this group): 105.5 ± 22.4.
Figure 9. shows the area results for the transits defined during a 676.02 Exofast evaluation; those that overlap with one of the 676.01 transits are separated as red symbols. The non-overlapping group defines a near-zero-sloped line; the overlapping group seems to define a slighly-positive-sloped line*; in this case, error bars (see below) are only smaller than the symbols for the non-overlapping case. In reasonable agreement with the values above, the average areas (units: days • normalized flux • 10^6) are:
• difference (area of 676.01 within this group): 305.9 ± 28.5.
Bottom Line: These area evaluations are consistent with KOI-676.01 and KOI-676.02 being on their own individual chords...from our (Kepler's) point of view...as they transit the star.
* Assuming this Figure 9. slope is caused by some astronomical reality, one wonders if it could be the result of the gradual change of the Sun's (and Kepler's) vantage point relative to an orbital plane(s) of the KOI-676 system…leading to an observed decrease in impact parameter(s) with a lengthening of transit durations (along paths that cross wider and wider parts of the star), thereby resulting in increasingly larger transit areas?
Table 1. and Table 2. summarize data for other transit parameters and other Exofast-mode determinations. Note that a prominent periodicity was observed for several parameters at ~ 31.89 days, in accordance with this repeatedly-transit-overlapping two-planet system. Correspondingly, for 676.01, 4 times its Period = 31.890176 days; for 676.02, 13 times its Period = 31.892042 days.
Figures 3A, 3B, 4A, and 4B show the effects of the periodic overlap on T_FWHM and on Depth.
Figures 5A, 5B, 6A, and 6B correspond to Figures 1A, 1B, 2A, and 2B, in that order, but for only Q7-Q17 (the only available) short-cadence data; note how closely these compare to the earlier corresponding Q1-Q17 long-cadence results.
The TTVs were determined for each planet after removal of overlapping transits. For 676.01 (Figure 2A), the P_ttv of 749.47 ± 38.46 days derived from the sinusoidal best curve-fit curve agrees very well with that obtained from the Lomb-Scargle periodogram (LSP): 740.459 days (power: 13.184; FAP: 0.002292). Additional relevant data is given below. This suggests the presence of a third unseen (non-transiting) planetary object in this system (see the detailed conclusions reached by Ioannidis, et al. (reference below) on the characteristics of such a third planet).
For 676.02 (Figure 2B), there appears to be, at most, only a relatively insignificant TTV. Contrary to initial visual inspection, a periodicity was found, initially via its LSP, embedded in the [(O-C) vs. Time] data at 41.37 days (LC data); this is also consistent with a subsequently determined sinusoidal curve-fit of 41.38 days. (Short-cadence data, Q7-Q17, gave slightly different corresponding values of 45.11 and 45.09, respectively.) If this periodicity were arising from some stellar characteristic, one might have expected it to be present in the LSP's of both planets. However, no such periodicity is present at all for 676.01. Further, stellar rotation was separately determined from an LSP of Q1-Q17 PDCSAP_Flux data, presumably due to long-enough-lived starspots, to be 12.29 days (Power: 9442; FAP: 0) and a literature value has been reported to be 12.20 days (Walkowicz & Basri's reference below). It thus appears to be associated only with 676.02. Although the underlying cause of these observations might point to a weak TTV, it is reminiscent of the same (e.g.: potential exomoon?, orbit precession, etc.) effect disclosed for 774.01 and 139.02 (see corresponding pages). This periodicity also does not correspond to the periodic transit overlapping with 676.01 which occurs every ~ 31.89 days (see below).
For all plots on this web page, error bars that are not visible are smaller than the size of the symbols used.
TTVs are derived from Q1-Q17 (LC cases) or Q7-17 (SC cases) 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 2A: KOI-676.01, long-cadence data, Q1-17, P = 7.97 days [Plot avg. error bars = ± 1.26 min.]
TTV_minimum: 198.73 ± 47.94 days, Amp_ttv_minimum: -1.46 ± 0.31 min.
TTV_maximum: 573.46 ± 55.32 days, Amp_ttv_maximum: 1.47 ± 0.31 min.
TTV_minimum: 948.20 ± 67.54 days, Amp_ttv_minimum: -1.46 ± 0.31 min.
TTV_maximum: 1322.93 ± 82.48 days, Amp_ttv_maximum: 1.47 ± 0.31 min.
When the transits of two exoplanets in a given extra-solar system occasionally overlap, mid-transit time determinations (etc.) by transit-curve-fitting algorithms can result in significant errors. It is for this reason that all TTVs reported on this website were determined only after removing all overlapping transits. A notable example of frequently recurring transit overlap is found for the two known planets in the Kepler KOI-676 system.
The ratio of the Periods of 676.01 and 676.02 is almost exactly 13:4, namely:
• this work: 13.00000:4.00022;
• NASA Exoplanet Archives (NEA): 13.00000:4.00023.
As luck would have it, the initial transit observed by Kepler of 676.01, later designated as T_(-5), overlapped with the initial one for 676.02, later designated as T_(-16). The values calculated from the NEA ephemerides are 64.720622 and 64.639698 days, respectively, a difference of only 0.080924 days. Given the transit durations (T_1,4) of these two planets, 0.1227 and 0.0725 days respectively, overlap occurs whenever their mid-transit times are within 0.0976 days…as for the above first-observed transits. Also, in this case, one would expect significant distortion since the respective transit depths, 3220 and 1761 ppm, are both substantial. (In numerous other cases involving large differences in depths and durations, the overlap distortion can often be minimal.)
Since the Period ratio was so close to 13:4 (see above), it was found that every 4th transit of 676.01, throughout the entire Q1-Q17 Kepler observation period, overlapped with one of 676.02. And, correspondingly, overlap occurred with every 13th transit of 676.02. Because the ratio was not exactly 13:4, a gradual drift in that overlap was clearly noted. This is illustrated below.
For the 676.01 case, a "TTV plot" [Tc vs. (O-C); EXOFAST run using LC data for Q1-Q17] was prepared (Figure 1A.) before removing the overlapping transits; for emphasis, the latter are shaded in green. In addition, inspection of several short-cadence [time vs. flux] plots of overlapping cases (Figures 7a.-f.), the EXOFAST-algorithm-calculated "overlapping transit times", and the individual NEA-ephemeris-calculated transit times indicate that 676.02's mid-transit times occur just prior to those of 676.01 during most of the observing time. The resulting early-weighted-distortion then causes the algorithm to produce "early-appearing" ("Observed", in the (O-C) expression) mid-transit times for 676.01…hence the separated series of negative (O-C) values in most of Figure 1A. "Calculated" times, in the (O-C) expression, are those obtained from the ephemeris, derived from the plot of Tc# vs. Tc. (Of course, inclusion of the overlapping transits leads to an erroneous ephemeris for 676.01; see below.)
For the 676.02 case, a "TTV plot" [Tc vs. (O-C); EXOFAST data] was similarly prepared (Figure 1B.) before removing the overlapping transits; for emphasis, the latter are again shaded in green. Now, however, with respect to 676.02, the distortions are "late-weighted" and cause the Exofast algorithm to produce late-appearing ("Observed") mid-transit times…hence the separated series of positive (O-C) values in most of Figure 1B. (Again, inclusion of the overlapping transits leads to an erroneous ephemeris for 676.02; see below.)
Within the Q14-Q15 time-frame, the "non-overlapping-Tc" and "overlapping-Tc" curves cross. That is, the transits for both planets overlap nearly exactly. An example of this is shown in Figure 7e.
Thereafter, in the overlapping cases, the transits for 676.02 begin to appear slightly after those of 676.01. An example of this is shown in Figure 7f.
The sinusoidally-curved lines in the 676.01 case (Figure 1A.) and the virtually linear ones in the 676.02 case (Figure 1B.) are consistent with the prominent and minimal TTVs observed, respectively, for the two planets (see below).