Transit Timing Variation (TTV):
Figure 1-LC and 1-SC show TTVs for the transiting exoplanet KOI-142.01 (Kepler-142b) determined from Long Cadence and Short Cadence data, respectively.  Periodicities uncovered in Lomb-Scargle Periodograms (LSP) match reasonably well with Kaleidograph-software-derived sinusoidal best-fit curve periodicities (P_ttv’s).  That the latter do not match precisely with all the observed data points is probably the result (see: Hadden & Lithwick, 2013, arXiv-1310.7942.) of the level (see Nesvorny 2013) of eccentricity in the orbit of 142b and/or the unseen 142c.  Interestingly, residuals from Figures 1-LC and 1-SC show dual credible periodicities in their LSPs which were also well-matched by dual best-fitting sinusoidal curves (Figures 2-LC & 3-LC, and Figures 2-SC & 3-SC).  While it is likely that these latter curves could just be the consequence of the imperfect sinusoidal curve-fits (due to eccentricity distortions) in the 1-LC and 1-SC cases, the effects of additional gravitationally-interacting planetary objects in the KOI-142 system could also be at work.  Unfortunately, with Kepler's level of accuracy now significantly reduced, it may be a very long time before any such additional objects might be observed and/or verified.

• Figure 1-LC, TTV
[error bars = ± 1.37 min., within data symbol]
TTV_minimum: 329.44 ± 3.71 days, Amp_ttv_minimum: -634.24 ± 13.97 min.
TTV_maximum: 638.64 ± 4.29 days, Amp_ttv_maximum: 640.36 ± 13.97 min.
TTV_minimum: 947.84 ± 5.09 days, Amp_ttv_minimum: -634.24 ± 13.97 min.
TTV_maximum: 1257.04 ± 6.03 days, Amp_ttv_maximum: 640.36 ± 13.97 min.
P_ttv: 618.40 ± 2.42 days.
Amp_ttv: 1274.60 ± 19.76 minutes (21.24 ± 0.33 hours).
Lomb-Scargle periodogram, P_ttv: 623.01 days; Power: 58.23; FAP: 0.
Linear ephemeris (this work): Tc = 10.95305667(Tc#) + 54.68010276

• Figure 1-SC, TTV
[error bars = ± 0.99 min., within data symbol]
TTV_minimum: 325.72 ± 7.07 days, Amp_ttv_minimum: -630.82 ± 18.35 min.
TTV_maximum: 635.40 ± 8.01 days, Amp_ttv_maximum: 648.30 ± 18.35 min.
TTV_minimum: 945.08 ± 9.36 days, Amp_ttv_minimum: -630.82 ± 18.35 min.
TTV_maximum: 1254.77 ± 10.96 days, Amp_ttv_maximum: 648.30 ± 18.35 min.
P_ttv: 619.37 ± 4.28 days.
Amp_ttv: 1279.12 ± 25.96 minutes (21.32 ± 0.43 hours).
Lomb-Scargle periodogram, P_ttv: 625.14 days; Power: 48.48; FAP: 0.
Linear ephemeris (this work): Tc = 10.95366050(Tc#) + 54.62897127

• Figure 2-LC , TTV Residuals-1a
[error bars = ± 14.04 min.]
TTV_maximum: 216.64 ± 8.30 days, Amp_ttv_maximum: 95.86 ± 8.45 min.
TTV_minimum: 384.08 ± 8.75 days, Amp_ttv_minimum: -97.10 ± 8.45 min.
TTV_maximum: 551.52 ± 9.41 days, Amp_ttv_maximum: 95.86 ± 8.45 min.
TTV_minimum: 718.97 ± 10.23 days, Amp_ttv_minimum: -97.10 ± 8.45 min.
TTV_maximum: 886.41 ± 11.19 days, Amp_ttv_maximum: 95.86 ± 8.45 min.
TTV_minimum: 1053.85 ± 12.25 days, Amp_ttv_minimum: -97.10 ± 8.45 min.
TTV_maximum: 1221.29 ± 13.38 days, Amp_ttv_maximum: 95.86 ± 8.45 min.
TTV_minimum: 1388.73 ± 14.57 days, Amp_ttv_minimum: -97.10 ± 8.45 min.
P_ttv: 334.88 ± 2.93 days.
Amp_ttv: 192.95 ± 11.95 minutes.
Lomb-Scargle periodogram, P_ttv: 335.23 days; Power: 37.72; FAP: 0.

• Figure 2-SC , TTV Residuals-1a
[error bars = ± 18.38 min.]
TTV_minimum: 385.72 ± 15.37 days, Amp_ttv_minimum: -98.26 ± 13.07 min.
TTV_maximum: 553.94 ± 16.42 days, Amp_ttv_maximum: 107.34 ± 13.07 min.
TTV_minimum: 722.17 ± 17.74 days, Amp_ttv_minimum: -98.26 ± 13.07 min.
TTV_maximum: 890.39 ± 19.29 days, Amp_ttv_maximum: 107.34 ± 13.07 min.
TTV_minimum: 1058.62 ± 21.00 days, Amp_ttv_minimum: -98.26 ± 13.07 min.
TTV_maximum: 1226.84 ± 22.85 days, Amp_ttv_maximum: 107.34 ± 13.07 min.
TTV_minimum: 1395.06 ± 24.80 days, Amp_ttv_minimum: -98.26 ± 13.07 min.
P_ttv: 336.45 ± 4.88 days.
Amp_ttv: 205.60 ± 18.48 minutes.
Lomb-Scargle periodogram, P_ttv: 335.51 days; Power: 24.43; FAP: 2.542 x 10^-8.

• Figure 3-LC , TTV Residuals-1b
[error bars = ± 14.04 min.]
TTV_minimum: 78.46 ± 67.21 days, Amp_ttv_minimum: -56.01 ± 11.95 min.
TTV_maximum: 667.93 ± 83.13 days, Amp_ttv_maximum: 68.63 ± 11.95 min.
TTV_minimum: 1257.40 ± 114.39 days, Amp_ttv_minimum: -56.01 ± 11.95 min.
P_ttv: 1178.94 ± 86.95 days.
Amp_ttv: 124.63 ± 16.91 minutes.
Lomb-Scargle periodogram, P_ttv: 1240.85 days; Power: 15.86; FAP: 0.000159.

• Figure 3-SC , TTV Residuals-1b
[error bars = ± 18.38 min.]
TTV_maximum: 677.57 ± 50.08 days, Amp_ttv_maximum: 106.27 ± 12.51 min.
TTV_minimum: 1260.03 ± 68.05 days, Amp_ttv_minimum: -112.35 ± 12.51 min.
P_ttv: 1164.94 ± 50.52 days.
Amp_ttv: 218.62 ± 17.69 minutes.
Lomb-Scargle periodogram, P_ttv: 1184.60 days; Power: 27.45; FAP: 1.247 x 10^-9.
KOI-142 (Kepler-88, KIC-5446285) 2-(or More?)-Planet System

As with so many of the Kepler light curves, those of KOI-142 have treated us to a fascinating 4-year glimpse into billion years of gravitationally-determined behavior.  Reports by Nesvorny, et al. (arXiv1304.4283) and Barros, et al. (arXiv1311.4335) have done a good job of constraining properties of the unseen exoplanet, Kepler-142c, from its pronounced effect on the transits and RV behaviors of Kepler-142b (KOI-142.01) during this 4-year glimpse.
Toward the bottom of this webpage, I have listed stellar and planetary parameters determined using "Online" EXOFAST (Eastman, Gaudi, & Agol, PASP, 125, #923, pp. 83-112, 2013; arXiv1206.5798) for all LC (Q0-Q17) and SC (Q4-Q17) data.  First, however, are shown a variety of plots and their sinusoidal best-fit curves, values determined therefrom, and residuals that also show sinusoidal character; namely:
• Transit Timing Variation (TTV) (Figures 1,2, & 3)
• Transit Duration Variation (TDV) (Figures 4),
• Transit Time (of Full Width @ Half Maximum) Variation (T(T_fwhm)V) (Figures 5)
• Orbital Period Variation (OPV) (Figures 6 & 7), and
• Transit Depth Variation (TPV) (Figure 8).
Orbital Period Variation (OPV).
In addition to the pronounced P_opv seen for the transiting planet 142b in this system (Figures 6-LC & 6-SC), “chopping” (see: Fabrycky, et al., arXiv1201.5415; Holman, et al., Science, 2010, 330, 51.) is also clearly evident for the Periods derived from the EXOFAST-determined mid-transit times and especially in the residuals from these plots (Figures 7-LC & 7-SC).Again, the same discussion for the slight differences between the best-fit sinusoidal curves and the data points, as was given for the TTVs (vida supra), applies here.

Figure 6-LC, OPV
[error bars = ± 0.003809 day]
OPV_minimum: 184.90 ± 7.09 days, Amp_opv_minimum: -0.043531 ± 0.001925 day.
OPV_maximum: 491.09 ± 7.93 days, Amp_opv_maximum: 0.055619 ± 0.001925 day.
OPV_minimum: 797.28 ± 9.33 days, Amp_opv_minimum: -0.043531 ± 0.001925 day.
OPV_maximum: 1103.47 ± 11.07 days, Amp_opv_maximum: 0.055619 ± 0.001925 day.
OPV_minimum: 1409.66 ± 13.02 days, Amp_opv_minimum: -0.043531 ± 0.001925 day.
P_opv: 612.38 ± 4.79 days.
Amp_opv: 0.099150 ± 0.002722 day {= 142.78 ± 3.92 min.}.
Lomb-Scargle periodogram, P_opv: 618.19 days; Power: 52.31; FAP: 0.

Figure 6-SC, OPV
[error bars = ± 0.000982 day.]
OPV_maximum: 489.98 ± 10.90 days, Amp_opv_maximum: 0.049809 ± 0.002263 day.
OPV_minimum: 796.51 ± 12.66 days, Amp_opv_minimum: -0.049799 ± 0.002263 day.
OPV_maximum: 1103.03 ± 14.88 days, Amp_opv_maximum: 0.049809 ± 0.002263 day.
OPV_minimum: 1409.55 ± 17.39 days, Amp_opv_minimum: -0.049799 ± 0.002263 day.
P_opv: 613.05 ± 6.28 days.
Amp_opv: 0.099608 ± 0.003201 day {= 143.44 ± 4.61 min.}.
Lomb-Scargle periodogram, P_opv: 619.40 days; Power: 44.54; FAP: 0.

Figure 7-LC, OPV Residuals
[error bars = ± 0.004268 day.]
OPV_maximum: 151.29 ± 7.86 days, Amp_opv_maximum: 0.012183 ± 0.001160 day.
OPV_minimum: 316.41 ± 8.26 days, Amp_opv_minimum: -0.014763 ± 0.001160 day.
OPV_maximum: 481.54 ± 8.89 days, Amp_opv_maximum: 0.012183 ± 0.001160 day.
OPV_minimum: 646.66 ± 9.72 days, Amp_opv_minimum: -0.014763 ± 0.001160 day.
OPV_maximum: 811.78 ± 10.70 days, Amp_opv_maximum: 0.012183 ± 0.001160 day.
OPV_minimum: 976.90 ± 11.79 days, Amp_opv_minimum: -0.014763 ± 0.001160 day.
OPV_maximum: 1142.03 ± 12.96 days, Amp_opv_maximum: 0.012183 ± 0.001160 day.
OPV_minimum: 1307.15 ± 14.20 days, Amp_opv_minimum: -0.014763 ± 0.001160 day.
OPV_maximum: 1472.27 ± 15.48 days, Amp_opv_maximum: 0.012183 ± 0.001160 day.
P_opv: 330.24 ± 3.01 days.
Amp_opv: 0.026946 ± 0.001641 day {= 38.80 ± 2.36 min.}
Lomb-Scargle periodogram, P_opv: 329.76 days; Power: 37.42; FAP: 1.343 x 10^-13.

Figure 7-SC, OPV Residuals
[error bars = ± 0.002467 day.]
OPV_maximum: 474.01 ± 13.05 days, Amp_opv_maximum: 0.013325 ± 0.001339 day.
OPV_minimum: 642.02 ± 14.12 days, Amp_opv_minimum: -0.015213 ± 0.001339 day.
OPV_maximum: 810.02 ± 15.41 days, Amp_opv_maximum: 0.013325 ± 0.001339 day.
OPV_minimum: 978.03 ± 16.86 days, Amp_opv_minimum: -0.015213 ± 0.001339 day.
OPV_maximum: 1146.03 ± 18.43 days, Amp_opv_maximum: 0.013325 ± 0.001339 day.
OPV_minimum: 1314.03 ± 20.10 days, Amp_opv_minimum: -0.015213 ± 0.001339 day.
OPV_maximum: 1482.04 ± 21.85 days, Amp_opv_maximum: 0.013325 ± 0.001339 day.
P_opv: 336.01 ± 4.19 days.
Amp_opv: 0.028538 ± 0.001894 day {= 41.09 ± 2.73 min.}
Lomb-Scargle periodogram, P_opv: 335.74 days; Power: 32.01; FAP: 1.280 x 10^-11.
Transit {Time of Full Width @ Half Maximum} Variation (T(T_fwhm)V).
This also has a similar periodicity to that of the TTVs and TDVs.  As expected, scatter is somewhat greater for LC data case where each transit is defined by ~ 6-7 data points; for the SC cases, that number is closer to ~ 180-210.  One reason for the observed reduced scatter for T(T_fwhm)V compared to that for TDV is that the latter includes the considerable scatter observed for (the small values of) tau (vide infra).

Figure 5-LC, T(T_fwhm)V
[error bars = ± 2.74 min.]
T(T_fwhm)V_minimum: 274.51 ± 42.31 days, Amp_t(t_fwhm)v_minimum: -4.51 ± 1.06 min.
T(T_fwhm)V_maximum: 623.93 ± 49.26 days, Amp_ t(t_fwhm)v_maximum: 4.48 ± 1.06 min.
T(T_fwhm)V_minimum: 973.35 ± 59.65 days, Amp_ t(t_fwhm)v_minimum: -4.51 ± 1.06 min.
T(T_fwhm)V_maximum: 1322.76 ± 72.01 days, Amp_ t(t_fwhm)v_maximum: 4.48 ± 1.06 min.
P_t(t_fwhm)v 698.84 ± 31.47 days.
Amp_t(t_fwhm)v: 9.00 ± 1.50 minutes.
Lomb-Scargle periodogram, P_t(t_fwhm)v: 704.02 days; Power: 11.08; FAP: 0.0185.
(Note how low the FAP is compared to that for the corresponding TDV; see text for discussion.)

Figure 5-SC, T(T_fwhm)V
[error bars = ± 0.99 min.]
T(T_fwhm)V_minimum: 317.57 ± 55.97 days, Amp_t(t_fwhm)v_minimum: -4.02 ± 0.80 min.
T(T_fwhm)V_maximum: 663.90 ± 64.25 days, Amp_ t(t_fwhm)v_maximum: 3.53 ± 0.80 min.
T(T_fwhm)V_minimum: 1010.23 ± 76.32 days, Amp_ t(t_fwhm)v_minimum: -4.02 ± 0.80 min.
T(T_fwhm)V_maximum: 1356.56 ± 90.70 days, Amp_ t(t_fwhm)v_maximum: 3.53 ± 0.80 min.
P_t(t_fwhm)v 692.65 ± 37.48 days.
Amp_t(t_fwhm)v: 7.55 ± 1.13 minutes.
Lomb-Scargle periodogram, P_t(t_fwhm)v: 680.78 days; Power: 13.14; FAP: 0.00203.
(Note how low the FAP is compared to that for the corresponding TDV; see text for discussion.)

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Transit Depth Variation (TPV).
The apparent temporary drop in depth centered near 1255 days was observed through the use of Short Cadence data and is consistent with Kepler-142b crossing a large star spot during several (possibly as many as 3 to 7) consecutive orbits.  The same was not observed using 1/30th the number of data points from the Long Cadence data.

Figure 8-SC, TPV
[error bars = ± 42 ppm]

Additional References:
• For Q0-Q6 TTV data of 142.01 (thru ~ 563 (BJD-2454900) days), see: Ford et al., 2012, arXiv-1201.1892.
• For Q0-12 TTV data of 142.01 (thru ~ 1116 (BJD-2454900) days), see: Mazeh et al., 2013, arXiv-1301.5499.

6 Mar 2014
Transit Duration Variation (TDV):
This follows a very similar periodicity to that of the TTVs.  As expected, scatter is somewhat greater for LC data case where each transit is defined by ~ 7 data points; for the SC cases, that number is closer to ~ 210.

Figure 4-LC, TDV
[error bars = ± 5.48 min.]
TDV_minimum: 290.42 ± 59.48 days, Amp_tdv_minimum: -5.41 ± 1.81 min.
TDV_maximum: 634.69 ± 69.31 days, Amp_tdv_maximum: 5.32 ± 1.81 min.
TDV_minimum: 978.96 ± 83.74 days, Amp_tdv_minimum: -5.41 ± 1.81 min.
TDV_maximum: 1323.22 ± 100.81 days, Amp_tdv_maximum: 5.32 ± 1.81 min.
P_tdv: 688.53 ± 43.41 days.
Amp_tdv: 10.73 ± 2.57 minutes.
Lomb-Scargle periodogram, P_tdv: 686.17 days; Power: 5.92; FAP: 0.962.
(This is definitely a very high false alarm probability; it is only included because the sinusoidal best-fit-curve-derived P_tdv is within such close agreement.)

Figure 4-SC, TDV
[error bars = ± 3.97 min.]
TDV_minimum: 289.61 ± 88.79 days, Amp_tdv_minimum: -3.37 ± 0.94 min.
TDV_maximum: 668.96 ± 104.28 days, Amp_tdv_maximum: 2.86 ± 0.94 min.
TDV_minimum: 1048.32 ± 127.41 days, Amp_tdv_minimum: -3.37 ± 0.94 min.
TDV_maximum: 1427.67 ± 154.79 days, Amp_tdv_maximum: 2.86 ± 0.94 min.
P_tdv: 758.71 ± 68.81 days.
Amp_tdv: 6.23 ± 1.33 minutes.
Lomb-Scargle periodogram, P_tdv: 772.47 days; Power: 7.37; FAP: 0.476.
(This is definitely a very high false alarm probability; it is only included because the sinusoidal best-fit-curve-derived P_tdv is within reasonable agreement.)