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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-1599.02, P = 13.62 days [Plot avg. error bars = ± 9.65 min.]
TTV_maximum: 107.56 ± 49.21 days, Amp_ttv_maximum: 45.71 ± 8.03 min.
TTV_minimum: 652.02 ± 59.04 days, Amp_ttv_minimum: -58.59 ± 8.03 min.
TTV_maximum: 1196.48 ± 77.93 days, Amp_ttv_maximum: 45.71 ± 8.03 min.
P_ttv: 1088.92 ± 55.22 days.
Amp_ttv: 104.30 ± 11.36 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 1081.37 days; Power: 18.89; FAP: 5.984 x 10^-6.
Linear ephemeris (this work): Tc = 13.61650012(Tc#) + 73.93618582

Figure 2.: KOI-1599.01, P = 20.41 days [Plot avg. error bars = ± 9.68 min. (smaller than symbols)]
TTV_maximum: 604.29 ± 26.69 days, Amp_ttv_maximum: 144.64 ± 10.37 min.
TTV_minimum: 1165.52 ± 35.69 days, Amp_ttv_minimum: -117.62 ± 10.37 min.
P_ttv: 1122.46 ± 26.69 days.
Amp_ttv: 262.26 ± 14.66 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 1106.80 days; Power: 25.50; FAP: 5.623 x 10^-9.
Linear ephemeris (this work): Tc = 20.40828349(Tc#) + 73.14626855

Figure 3.: KOI-1599.01, Residuals of Figure 2. [Plot avg. error bars = ± 14.19 min.]
TTV_minimum: 82.67 ± 41.27 days, Amp_ttv_minimum: -32.69 ± 8.27 min.
TTV_maximum: 402.28 ± 44.78 days, Amp_ttv_maximum: 39.25 ± 8.27 min.
TTV_minimum: 721.90 ± 52.02 days, Amp_ttv_minimum: -32.69 ± 8.27 min.
TTV_maximum: 1041.52 ± 61.69 days, Amp_ttv_maximum: 39.25 ± 8.27 min.
TTV_minimum: 1361.13 ± 72.82 days, Amp_ttv_minimum: -32.69 ± 8.27 min.
P_ttv: 639.23 ± 28.23 days.
Amp_ttv: 71.94 ± 11.70 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 647.70 days; Power: 10.15; FAP: 0.0257.

Figure 4.: KOI-1599.01, Residuals of Figure 3.; Residuals-of-the-Residuals of Figure 2. [Plot avg. error bars = ± 16.42 min.]
TTV_minimum: 115.18 ± 31.54 days, Amp_ttv_minimum: -23.88 ± 7.14 min.
TTV_maximum: 306.14 ± 33.03 days, Amp_ttv_maximum: 25.72 ± 7.14 min.
TTV_minimum: 497.09 ± 35.70 days, Amp_ttv_minimum: -23.88 ± 7.14 min.
TTV_maximum: 688.05 ± 39.31 days, Amp_ttv_maximum: 25.72 ± 7.14 min.
TTV_minimum: 879.00 ± 43.63 days, Amp_ttv_minimum: -23.88 ± 7.14 min.
TTV_maximum: 1069.96 ± 48.46 days, Amp_ttv_maximum: 25.72 ± 7.14 min.
TTV_minimum: 1260.91 ± 53.67 days, Amp_ttv_minimum: -23.88 ± 7.14 min.
TTV_maximum: 1451.87 ± 59.17 days, Amp_ttv_maximum: 25.72 ± 7.14 min.
P_ttv: 381.91 ± 13.21 days.
Amp_ttv: 49.61 ± 10.10 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 382.95 days; Power: 7.44; FAP: 0.324.

Figure 5.: KOI-1599.01, added combination of Figures 2., 3., and 4. [Plot error bars = ± 23.76 min. (smaller than symbols)].
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The following two plots show the folded 96 LC-transits of 1599.02 and the folded 67 LC-transits of 1599.01.
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The following table depicts some of the properties calculated with EXOFAST and compares them to those reported in the NASA Exoplanet Archive (NEA).

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KOI's (Blue)  or
K2 Objects (Green)
References:
• For Q0-6 TTV data of 1599.01 (thru ~ 563 (BJD-2454900) days), see: Ford et al., 2012, arXiv-1201.1892.
• For Q0-10(part) TTV data of 1599.01 (thru ~ 869 (BJD-2454900) days), see: Mazeh et al., 2013, arXiv-1301.5499.
• For Q0-13 TTV data, which includes a discontinuity (likely arising from the low SNR of the 1599.01 system), see Ji-Wei Xie's website: http://www.astro.utoronto.ca/~jwxie/TTV/TTV_Home.html
• For a Q0-16 plot of the TTV data of 1599.01, see: vonEssen et al., "KOINet", see: http://koinet.astro.physik.uni-goettingen.de/
• Hadden & Lithwick, 2013, arXiv-1310.7942.

7 May 2014
Kepler KOI-1599 (KIC-5474613) 4(?)-(or more?)-Planet System

Discussion:
Because of the low signal-to-noise ratio of KOI-1599.01 and KOI-1599.02, the relatively few data points in the Q1-Q17 long-cadence flux vs. time, and the rather large TTV, good use of the versatility inherent in Online EXOFAST was successfully applied to these two exoplanet candidates.  Specifically, Online EXOFAST allows for preliminary graphical testing and then, if warranted, zeroing-in (making slight adjustments to the approximate Tc used as a Prior) to better center the transit on a preliminary fit {where the presence of TTVs had caused that transit to wander significantly from its ephemeris-predicted time}.  Thus, each transit of 1599.01 and 1599.02 was individually evaluated by:
(1) selecting an approximate T_(n) time by modifying the NASA Exoplanet Archive's (NEA's) ephemeris prediction with the time      difference found between the NEA's ephemeris predicted T_(n-1) and the observed T_(n-1),
(2) selecting a data range to be examined that included points ± 0.4 days on either side of the approximate T_(n),
(3) running the EXOFAST preliminary "Test Input" module to identify the transit and somewhat more accurately define it through     (as much as ~ ± 0.04 day) adjustments to the approximate T_(n) used as a Prior, and
(4) running the complete Online EXOFAST evaluation of T_(n).The ephemeris obtained for each candidate in this work is shown below.

As graphically shown in Figures 1. and 2. below, these two exoplanet candidates, with periods in a mean motion resonance ratio of 3:2 (actually 2.998:2), both show pronounced TTVs and, at the same time, are well anti-correlated (see Figures, and sinusoidal maxima and minima data below).  The sinusoidal curve-fit of the (O-C) vs. Time data for 1599.02 shows (Figure 1.) a periodicity (P_ttv) of 1088.92 ± 55.22 days with an overall amplitude of 104.30 ± 11.36 minutes; 1081.37 days was found for the credible periodicity in the Lomb-Scargle Periodogram (LSP) of the same data.  Importantly, in the case of this closer-in planet, the residuals from its TTV plot show no additional credible periodicities.

However, the case is more complex for the farther-out one, 1599.01.  After the sinusoidal curve-fit of the (O-C) vs. Time data showed (Figure 2.) a closely matching periodicity (P_ttv) of 1122.46 ± 26.69 days with an overall amplitude of 262.26 ± 14.66 minutes (1106.80 days was observed in the LSP of the same data), a plot (Figure 3.) of the Residuals also gave a good sinusoidal curve-fit with a periodicity of 639.23 ± 28.23 days (LSP: 647.70 days).  Similarly, the Residuals-of-the-Residuals also gave a good sinusoidal curve-fit (Figure 4.) with a periodicity of 381.91 ± 13.21 days (LSP: 382.95 days).  The added combination of all three of these latter sinusoidal curves is arrayed in Figure 5. and reasonably reproduces the complex overall curvature of the initial data (Figure 2.).  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 2 additional planets in near-circular orbits, external to 1599.01, are mutually-interacting to produce the unusual TTV distribution observed.