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TTVs are derived from Q1-Q17 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-564.01, P = 21.06 days [Plot avg. error bars = ± 7.02 min. (smaller than symbols)]
TTV_minimum: 801.56 ± 265.40 days, Amp_ttv_minimum: -627.21 ± 790.41 min.
P_ttv: 2603.41 ± 609.90 days.
Amp_ttv: 2982.80 ± 1117.80 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 1475.14 days; Power: 25.97; FAP: 3.010 x 10^-9.
Linear ephemeris (this work): Tc = 21.08614279(Tc#) + 104.38957405

Figure 2.: Residuals of Figure 1. [Plot avg. error bars = ± 790.44 min., large dotted-line error bars]
TTV_maximum: 556.23 ± 26.93 days, Amp_ttv_maximum: 170.99 ± 14.56 min.
TTV_minimum: 1050.28 ± 35.34 days, Amp_ttv_minimum: -172.09 ± 14.56 min.
P_ttv: 988.12 ± 25.37 days.
Amp_ttv: 343.08 ± 20.59 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 1003.74 days; Power: 22.79; FAP: 7.364 x 10^-8.

Figure 3.: Residuals of Figure 2.; Residuals-of-the-Residuals of Figure 1. [Plot avg. error bars = ± 790.57 min., large dotted-line error bars]
TTV_maximum: 366.23 ± 38.76 days, Amp_ttv_maximum: 49.29 ± 11.25 min.
TTV_minimum: 659.82 ± 44.82 days, Amp_ttv_minimum: -52.93 ± 11.25 min.
TTV_maximum: 953.40 ± 52.97 days, Amp_ttv_maximum: 49.29 ± 11.25 min.
TTV_minimum: 1246.99 ± 62.40 days, Amp_ttv_minimum: -52.93 ± 11.25 min.
P_ttv: 587.17 ± 24.09 days.
Amp_ttv: 102.22 ± 15.90 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 584.43 days; Power: 10.53; FAP: 0.0154.

Figure 4.: Added combination of Figures 1., 2., and 3. [Plot error bars = ± 1117.96 min.]

Figure 5.: KOI-564.02, P = 127.90 days [Plot avg. error bars = ± 3.12 min. (smaller than symbols)
]TTV_maximum: 252.37 ± 151.00 days, Amp_ttv_maximum: 31.20 ± 10.15 min.
TTV_minimum: 826.70 ± 200.33 days, Amp_ttv_minimum: -39.63 ± 10.15 min.
P_ttv: 1148.66 ± 192.08 days.
Amp_ttv: 70.82 ± 14.36 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 1346.65 days; Power: 3.29; FAP: 0.0222.
Linear ephemeris (this work): Tc = 127.90901178(Tc#) + 179.45323038

Figure 6.: Residuals of Figure 5. [Plot avg. error bars = ± 10.62 min.]
TTV_minimum: 314.79 ± 79.76 days, Amp_ttv_minimum: -12.66 ± 5.35 min.
TTV_maximum: 669.33 ± 95.69 days, Amp_ttv_maximum: 17.21 ± 5.35 min.
TTV_minimum: 1023.86 ± 118.18 days, Amp_ttv_minimum: -12.66 ± 5.35 min.
P_ttv: 709.07 ± 63.47 days.
Amp_ttv: 29.86 ± 7.56 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 689.96 days; Power: 2.76; FAP: 0.114.

Figure 7.: Added combination of Figures 5., and 6. [Plot error bars = ± 11.07 min.]

Figure 8.: KOI-564.03, P = 6.22 days [Plot avg. error bars = ± 9.03 min. (smaller than symbols)]
Linear ephemeris (this work): Tc = 6.21709491(Tc#) + 64.73050187

Literature Tc#, Tc, and TTV Tabulations/References:
• For Q0-6 TTV data (thru ~ 563 (BJD-2454900) days): Ford et al., 2012, arXiv-1201.1892.
• For Q0-10 TTV data (thru ~ 934 (BJD-2454900) days): Rowe et al., 2014, arXiv-1402.6534.
• For Q0-12 TTV data (thru ~ 1116 (BJD-2454900) days): Mazeh et al., 2013, arXiv-1301.5499.
• For Q0-12 Calc. properties from TTVs: Hadden & Lithwick, 2013, arXiv-1310.7942.
• For Q0-13/14 TTV data, Xie: http://www.astro.utoronto.ca/~jwxie/TTV/TTV_Home.html
• Two ground-based Kepler-TTV-follow-up programs are in place at the moment: --- Gary, "KAFO" project: http://brucegary.net/kafo/ --- von Essen, "KOINet": http://koinet.astro.physik.uni-goettingen.de/

9 May 2014

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KOI's (Blue)  or
K2 Objects (Green)
Kepler KOI-564 (KIC-6786037) 3-(or more?)-Planet System

NOTE:For KOI-564.01, significant errors are propagated via its so-called "sinusoidal" [(O-C) vs. Time] plot (and undoubtedly in its Lomb-Scargle Periodogram (LSP)) because considerably less that one sinusoidal cycle was observed during Kepler's 4-year lifetime.  {But for now, rather than pondering how much we don't know about > 4-year cycles of Kepler planets, we can celebrate all we've learned (and are yet to learn) about the < 4-year ones.}

Discussion:
After a "sinusoidal" best-fit curve of the initial [(O-C) vs. Time] data for 564.01 showed (Figure 1. below) a periodicity (P_ttv) of ~ 2603 days (a ~ 1475 day periodicity was observed in the LSP of the same data), a plot (Figure 2.) of the Residuals gave a good sinusoidal curve-fit with a periodicity of 988.12 ± 25.37 days (LSP: 1003.73 days).  Similarly, the Residuals-of-the-Residuals also gave a good sinusoidal curve-fit (Figure 3.) with a periodicity of 587.17 ± 24.09 days (LSP: 584.43 days).  The added combination of the three sinusoidal curves is arrayed in Figure 4. and reasonably reproduces a complex overall curvature consistent with the initial 564.01 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 3 planets are interacting with this one to produce the observed TTV behavior.

Similarly for the outermost planet, KOI-564.02, after a sinusoidal curve-fit of the initial [(O-C) vs. Time] data showed (Figure 5.) a P_ttv of 1148.66 ± 192.08 days (1346.65 days was observed in the LSP of the same data), a plot (Figure 6.) of the Residuals also gave a good sinusoidal curve-fit with a periodicity of 709.07 ± 63.47 days (LSP: 689.96 days).  (The Residuals-of-the-Residuals here showed no credible periodicity in the LSP.)  The added combination of these two sinusoidal curves is arrayed in Figure 7. and reasonably reproduces an overall curvature consistent with the initial 564.02 data.

Lastly, the [(O-C) vs. Time] plot for the innermost planet, KOI-564.03 (Figure 8.), shows no periodicities (in agreement with its LSP), which may be partially due to the relative weakness of its transits; depth is 1/4th that of 564.01 and 1/26th that of 564.02.