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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: KOI-720.04, P = 2.80 days [Plot avg. error bars = ± 5.29 min.]
The [(O-C) vs. Tc} data gave NO CREDIBLE Lomb-Scargle periodicity.
From its EXOFAST-determined mid-transit times, its linear ephemeris is:
Tc = 2.79590034(Tc#) + 65.72929193

Figure: KOI-720.01, P = 5.69 days [Plot avg. error bars = ± 2.53 min.]
The [(O-C) vs. Tc} data gave NO CREDIBLE Lomb-Scargle periodicity.
From its EXOFAST-determined mid-transit times, its linear ephemeris is:
Tc = 5.69058408(Tc#) + 107.04991231

Figure: KOI-720.02, P = 10.04 days [Plot avg. error bars = ± 3.33 min.]
The [(O-C) vs. Tc} data gave NO CREDIBLE Lomb-Scargle periodicity.
From its EXOFAST-determined mid-transit times, its linear ephemeris is:
Tc = 10.04156186(Tc#) + 70.08569647

Figure: KOI-720.03, P = 18.37 days [Plot avg. error bars = ± 3.15 min.]
SINUSOIDAL CURVE-FIT: TTV_minimum: 88.52 ± 166.73 days, Amp_ttv_minimum: -7.67 ± 2.44 min.
SINUSOIDAL CURVE-FIT: TTV_maximum: 812.87 ± 233.50 days, Amp_ttv_maximum: 8.63 ± 2.44 min.
ESTIMATE (just beyond last data point at 1497.70 days):
TTV_minimum: 1537.23 ± 352.41 days, Amp_ttv_minimum: -7.67 ± 2.44 min.
P_ttv: 1448.72 ± 293.09 days.
Amp_ttv: 16.30 ± 3.45 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 1558.85 days; Power: 10.97; FAP: 0.009493.
Linear ephemeris (this work): Tc = 18.36972877(Tc#) + 64.86607272
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The Lineweaver-Bovaird Rule ("L-B Rule", ref. below), a recent modification of the Titius-Bode Rule, was used to estimate the semi-major axis and, via Kepler's Third Law, the period for the unseen planetary perturber "720.05".  Bovaird and Lineweaver, using only their "Rule", also proposed the presence of a fifth planet in this system.  The above TTV finding now adds substantial credibility to that proposal.

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KOI's (Blue)  or
K2 Objects (Green)
KOI-720 (Kepler-221, KIC-9963524) 5-(or more?)-Planets

Discussion:
The four confirmed planets in this system are:
• KOI-720.04, Kepler-221c, period: 2.80 days;
• KOI-720.01, Kepler-221d, period: 5.69 days;
• KOI-720.02, Kepler-221e, period: 10.04 days;
• KOI-720.03, Kepler-221b, period: 18.37 days.
The first three show no obvious TTVs.  However, the fourth (720.03) does indeed show a strong orbital periodicity (TTV) through its [(O-C) vs. Tc] data in a Lomb-Scargle Periodogram and a best-fit sinusoidal array.  These results are consistent with a fifth unseen (non-transiting) planet, designated below as "720.05", farther from the star than 720.03; estimates of its orbital period and semi-major axis are discussed lower on this page.
Using the linear equation derived from the 4 known planets [y = 0.0744x + 0.0069; correlation coefficient: 0.9990; error bars much smaller than symbols used in plot.], the following are estimated for "KOI-720.05":
semi-major axis: 0.2125 ± 0.0013 AU,
period: 39.88 ± 0.14 days.
The B&L13* literature estimate was 35 ± 4 days.
P_"720.05" : P_720.03 = 2.1678 (or a near mean motion resonance (nmmr) of 2:1).
Using the sinusoidally derived P_ttv [1448.72 days], one can also estimate from F12's** equation:
P_ttv = 1/|(nmmri_a/P_a - nmmri_b/P_b)| a period of 37.21 days, in good agreement with the two estimates above.

References:
* B&L13: Bovaird & Lineweaver, 2013, arXiv:1304.3341.
** F12: Fabrycky et al., arXiv-1201.5415.
• Tc#, Tc, and TTV Major Tabulations and Other References: see my "Summary" webpage.

17 Sep 2014