TTVs are derived from Q1-Q16 Kepler data.  x-axes: “Observed Tc” (Mid-Transit Time): EXOFAST’s best-fits from Kepler light flux vs. time data.  y-axes: “(O – C)”: difference between Observed Tc and the Calculated Tc from the graphically obtained linear ephemeris.  The plots are pictured in the order of orbital periods.

Discussion:
Near Mean Motion Resonance integers in this system:
P_377.02/P_377.01: 2.0186/1.0000 = ~ 2/1
Exquisitely anti-correlated TTVs with very large amplitudes are observed for 377.01 and 377.02:
377.01: Amp_maximum - Amp_minimum = ~ 999 minutes (or ~ 17 hours over the course of ~ 1290 days).
377.02: Amp_maximum - Amp_minimum = ~ 2234 minutes (or ~ 37 hours over the course of ~ 1261 days).


The precision of the anti-correlation is particularly evident in the final plot above where the TTV arrays are co-plotted (via scaling and inverting 377.02); note that the overlay is nearly perfect throughout the entire observation time: approximately 1426 days (or 3.9 Earth years).

The Q1-Q16 sinusoidal best-fits for both 377.01 and 377.02 were reasonably good for the middle of each data array but showed some deviation at the extremes.  This would probably vanish with longer observation times for this system.

Exofast was used (as usual) in the Offline mode for 377.03.  However, because of the very high TTVs, it was used in the Online mode for 377.01 and 377.02.  Most of the transits, because of TTVs, were so far removed from ephemeris-predicted (non-TTV) values that the data input ranges had to be appropriately slid up or down to insure that they actually contained the transit and that it was reasonably centered within the data range.

1st: KOI-337.03, P = 1.59 days [Plot error bars (smaller than symbol) = ± 7.39 min.]
No observable TTV.
Lomb-Scargle periodogram: no credible periodicities.
Linear ephemeris (this work): Tc = 1.59296301(Tc#) + 115.09121782

2nd: KOI-377.01, P = 19.27 days [Plot error bars (smaller than symbol) = ± 2.23 min.]
TTV_minimum: 167.82 ± 11.27 days, Amp_ttv_minimum: -461.39 ± 10.48 min.
TTV_maximum: 812.84 ± 15.51 days, Amp_ttv_maximum: 537.94 ± 10.48 min.
TTV_minimum: 1457.86 ± 22.43 days, Amp_ttv_minimum: -461.39 ± 10.48 min.
P_ttv: ~ 1290.05 ± 17.27 days.
Amp_ttv: 999.33 ± 14.82 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 1382.74 days; Power: 32.93; P-value: 3.42 x 10^-12.
Linear ephemeris (this work): Tc = 19.24717180(Tc#) + 115.97422268
Estimated P_ttv = 1/|(nmmri_a/P_a - nmmri_b/P_b)| =1664.90 days

3rd: KOI-377.02, P = 38.91 days [Plot error bars (smaller than symbol) = ± 2.61 min.]
TTV_maximum: 181.58 ± 18.02 days, Amp_ttv_maximum: 963.47 ± 40.67 min.
TTV_minimum: 812.14 ± 24.64 days, Amp_ttv_minimum: -1271.13 ± 40.67 min.
TTV_maximum: 1442.70 ± 35.34 days, Amp_ttv_maximum: 963.47 ± 40.67 min.
P_ttv: ~ 1261.12 ± 26.78 days.
Amp_ttv: 2234.60 ± 57.52 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 1424.10 days; Power: 16.84; P-value: 0.0000
Linear ephemeris (this work): Tc = 38.94456558(Tc#) + 107.84003271

4th plot (above): Overlay or TTVs for 377.01 and 377.02.

KOI-377.01 & KOI-377.02: TTV-periodicity Variation (P_ttvV):
As was discussed previously for KOI-1573, KOI-277, and KOI-152 (see "P_ttvV" web page), the current system as also particularly interesting and characteristic in the same way.

For KOI-377.01, Table 1. and Figure 1. show the "P_ttvV" effect vs. numbers of Kepler quarters for P_ttv's obtained from Lomb-Scargle periodograms and from sinusoidal best-fits and for Amp_ttv's from sinusoidal best-fits.  The figure highlights the reasonably good agreement between the two methods.

For KOI-377.02, Table 2. and Figure 2. demonstrate the same.

Figure 3. is a plot of both sets of sinusoidal best-fit P_ttv's (note the close agreement) along with the values calculated using the Fabrycky equation (ref. below) and periods determined for each set of Q1-Qx data (in this work).  Interestingly, it appears that the series of calculated values is approaching the same asymptote as the sinusoidal best-fit values; that asymptote should be somewhere in the Kepler Q17-Q19 time frame. (Unfortunately, at the present time, it looks like high quality Kepler data will not be available to more precisely constrain this asymptote.)
KOI-377 (Kepler-9, KIC-3323887) 3-Planet System

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References:
• P_ttv equation: P_ttv = 1/|(nmmri_a/P_a - nmmri_b/P_b)|…Reference: Fabrycky, et al., 2012, arXiv-1201.5415;where nmmri_a and nmmri_b are the near mean motion resonance integers a and b, and P_a and P_b are the respective periods.
• For Q1-Q6 TTV data of 377.01 and 377.02 (thru ~ 563 (BJD-2454900) days), see: Ford2012arXiv-1201.1892.
• For Q1-Q12 TTV data, see: Mazeh, Fabrycky, Ford, Agol, et al., 2013, arXiv:1301.5499v1, 23 Jan 2013.

25 Aug 2013