###### Lomb-Scargle periodograms and sinusoidal curve-fits were derived for each set of Q-sequence TTV (x-y) data. The results, listed in Table 1. and plotted in Figure 1., demonstrate the importance of having a long-enough-time sequence of data to specify virtually-unchanging best-fit curvature and parameters (like P_ttv and periodicity amplitude (Amp_ttv)) derived therefrom.

###### Through Kepler Q16, none of the TTV plots for the four planet candidates in this system has yet shown more than 2 minima or 2 maxima (see also the "KOI-152" web page). Table 3 and Figure 3 summarize what is known for the P_ttv's which, in this case, were the periods obtained from Lomb-Scargle periodograms using TTV x-y data; in the Table, the high "by chance" probabilities are in bold font (and within gray areas). Note (a) except for the two high excursions in this case (probably arising from the paucity of points in its TTV x-y data), there is a general increase for 152.01, (b) there is an apparent continuous upward trend for 152.02 where all the values have credible (very low "by chance") periodic probabilities, and (c) there is almost a leveling off for 152.03 and 152.04. Overall, once again, this illustrates the danger of assuming that the P_ttv's obtained through sixteen Kepler quarters are unchanging. Presumably, more data (if it were coming) would give better and better sinusoidal fits and their P_ttv's should eventually reach asymptotic values.

**Since this is probably a fairly widespread problem due to the limited time, about 4 Earth-years, of Kepler's operation, especially with the growing number of people evaluating TTVs and their impact on exoplanetary science, all reports of P_ttv's, Amp_ttv's, or derived planetary properties should include a clear designation of the time intervals involved.**

Updated: 15 August 2013.

###### However, starting with the Q0-Q8 group data, the periodicity (Figure 2.) and amplitude (Table 2.) seem to nearly level off in the 450 day range. [While it's possible that some of the apparent irregularity in this "plateau" is due to issues in the original light curve, some could also be caused other as yet undefined natural phenomena like a third ("277.03") unseen planet, elliptical character of orbits, precessing of orbits, etc.]

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**• **__KOI-152.0x__: TTV-periodicity variation for all 4 planet candidates in this system.

Data: Table 3.; Plot: Figure 3.:

** P_ttvV ****TTV Periodicity Variation**** P_ttvV**

**Well before TTV data defines some characteristic periodicity, e.g. a sinusoid, its x-y plot may just show some simple binomial curvature. Doing evaluations like Lomb-Scargle Periodograms or Curve-Fits on such early data sequences will inevitably give misleading results. One must keep this in mind, for example, when comparing parameters generated from TTV data derived from different numbers of Kepler Quarters. The following three examples are instructive in this regard. (Also see the related web pages: KOI-1573, KOI-277, and KOI-152 (vide infra).)**

__• KOI-1573.01__: ultimate TTV periodicity (P_ttv) in the range of 800-1000 days.

TTV data from various Kepler-Quarter-sequences: Q1-Q6, Q1-Q7, …, through Q1-Q16 have been obtained from corresponding plots of Tc vs. Tc# and the corresponding Q-sequence-based linear ephemeris equations. These data show simple curvature for the Q1-Q6, Q1-Q7, and Q1-Q8 cases while the earliest hint of an eventual inflection point appears in the Q1-Q9 sequence. Thereafter, this becomes progressively more pronounced, eventually leading to a sinusoidal curve (see the series of some selected figures that follow).

###### In this case, while the P_ttv seemed to have reached a plateau in Q11, Q12, Q13, and Q14, in the neighborhood of 875 days, it then began to increase again through Q13 and Q14, possibly because the periodicity was approaching another minimum which had a more pronounced influence on an "eventual" unchanging best-fit. As in this case, it is likely that many of the Kepler exoplanetary systems that show TTVs will probably not be observed long enough to achieve unchanging best-fits.

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__• KOI-277.01__: ultimate TTV-periodicity and amplitude of about 450 days and 120 minutes, respectively. Once again, significant P_ttv and Amp_ttv variability is seen for this system in Table 2. and Figure 2. for early groups of quarters (Q0-Q4 through Q0-Q7) where still too little data is available.