Numerous literature and major Tc#, Tc, and TTV tabulation references can be found on my "Summary" webpage following the table.
• Fabrycky et al., arXiv-1201.5415; P_ttv = 1/|(nmmri_a/P_a - nmmri_b/P_b)|.
• Bovaird & Lineweaver, 2013, arXiv:1304.3341.

1 July 2015

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Exoplanet-Science.com

        Kepler KOI-148 (Kepler-48, KIC-5735762) 4-(or more?)-Planet System

Discussion:
A Lomb-Scargle Periodogram (LSP) of the [(O-C) vs. Time] data for KOI-148.01 (Kepler-48b) showed no credible periodicity (none with FAP < 0.597) even after removal of 14 transits that overlapped with transits for the other two known planets and three (> 3 sigma from mean) outliers.  These data are plotted in Figure 1.

A Lomb-Scargle Periodogram (LSP) of the [(O-C) vs. Time] data for KOI-148.02 (Kepler-48c) showed a highly credible periodicity at 388.86 days; a sinusoidal curve-fit of the same data (Figure 2.) showed a periodicity (P_ttv) of 389.30 ± 10.52 days.

A Lomb-Scargle Periodogram (LSP) of the [(O-C) vs. Time] data for KOI-148.03 (Kepler-48d) showed a medium-credibility periodicity at 232.58 days; a sinusoidal curve-fit of the same data showed (Figure 3.) a periodicity (P_ttv) of 232.95 ± 6.43 days.

These results could be accommodated by the presence of a fourth planet between 148.02 and 148.03; see the Lineweaver-Bovaird-modified Titus-Bode Rule discussion below.

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-148.01, P = 4.78 days [Plot avg. error bars = ± 3.39 min.]
Linear ephemeris (this work): Tc = [4.77800312 ± 0.00000481](Tc#) + [57.06206446 ± 0.00083412]


Figure: KOI-148.02, P = 9.67 days [Plot avg. error bars = ± 2.12 min.]
TTV_maximum: 74.51 ± 24.13 days, Amp_ttv_maximum: 3.25 ± 0.69 min.
TTV_minimum: 269.16 ± 25.12 days, Amp_ttv_minimum: -3.30 ± 0.69 min.
TTV_maximum: 463.80 ± 27.12 days, Amp_ttv_maximum: 3.25 ± 0.69 min.
TTV_minimum: 658.45 ± 29.92 days, Amp_ttv_minimum: -3.30 ± 0.69 min.
TTV_maximum: 853.10 ± 33.32 days, Amp_ttv_maximum: 3.25 ± 0.69 min.
TTV_minimum: 1047.75 ± 37.15 days, Amp_ttv_minimum: -3.30 ± 0.69 min.
TTV_maximum: 1242.40 ± 41.30 days, Amp_ttv_maximum: 3.25 ± 0.69 min.
TTV_minimum: 1437.05 ± 45.68 days, Amp_ttv_minimum: -3.30 ± 0.69 min.
P_ttv: 389.30 ± 10.52 days.
Amp_ttv: 6.55 ± 0.98 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 388.86 days; Power: 13.88; FAP: 0.00117.
Linear ephemeris (this work): Tc = [9.67395148 ± 0.00000714](Tc#) + [58.34037827 ± 0.00061520]


Figure: KOI-148.03, P = 42.90 days [Plot avg. error bars = ± 4.32 min.]

TTV_maximum: 90.32 ± 22.94 days, Amp_ttv_maximum: 6.96 ± 2.33 min.
TTV_minimum: 206.80 ± 23.51 days, Amp_ttv_minimum: -7.48 ± 2.33 min.
TTV_maximum: 323.27 ± 24.49 days, Amp_ttv_maximum: 6.96 ± 2.33 min.
TTV_minimum: 439.75 ± 25.83 days, Amp_ttv_minimum: -7.48 ± 2.33 min.
TTV_maximum: 556.22 ± 27.49 days, Amp_ttv_maximum: 6.96 ± 2.33 min.
TTV_minimum: 672.70 ± 29.41 days, Amp_ttv_minimum: -7.48 ± 2.33 min.
TTV_maximum: 789.17 ± 31.53 days, Amp_ttv_maximum: 6.96 ± 2.33 min.
TTV_minimum: 905.65 ± 33.83 days, Amp_ttv_minimum: -7.48 ± 2.33 min.
TTV_maximum: 1022.12 ± 36.27 days, Amp_ttv_maximum: 6.96 ± 2.33 min.
TTV_minimum: 1138.59 ± 38.83 days, Amp_ttv_minimum: -7.48 ± 2.33 min.
TTV_maximum: 1255.07 ± 41.47 days, Amp_ttv_maximum: 6.96 ± 2.33 min.
TTV_minimum: 1371.54 ± 44.19 days, Amp_ttv_minimum: -7.48 ± 2.33 min.
TTV_maximum: 1488.02 ± 46.97 days, Amp_ttv_maximum: 6.96 ± 2.33 min.
P_ttv: 232.95 ± 6.43 days.
Amp_ttv: 14.44 ± 3.29 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 232.58 days; Power: 5.10; FAP: 0.0497.
Linear ephemeris (this work): Tc = [42.89623352 ± 0.00012436](Tc#) + [79.06743850 ± 0.00223549]

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The Lineweaver-Bovaird Rule ("L-B Rule", ref. below) was used to estimate a semi-major axis for the potential unseen (non-transiting) planet, designated as “148.04”.  Using their equation:
a = 0.382 + 0.334(1.925^n)
where "a" is the semi-major axes, and "n" = -∞, 0, 1, 2, 3, …,
the sequence of the first 4 relative semi-major axes are estimated as: 0.3820, 0.7160, 1.0250, and 1.6197.
Plotting (see Figure 4.) the 1st, 2nd, and 4th of these values against the known semi-major axes of 148.01 (0.0547 AU), 148.02 (0.0876 AU), and 148.03 (0.2365 AU), respectively, gave an excellent (R^2 = 0.9913) straight line and the equation: y = 0.1506x - 0.0102.  From this equation, the semi-major axis (and, hence, period) could be estimated for the potential unseen planet (the 3rd from the host star) that could be the object producing the observed TTVs of 148.02 and 148.03:
"148.04": Semi-major axis: ~ 0.1441 AU; Period: ~ 20.4260 days.
Summarizing the overall sequence of periods and potential periods:
1st, 148.01:  4.7780 days,
2nd, 148.02:  9.6740 days,
3rd, "148.04":  ~ 20.4260 days
4th, 148.03:  42.8961 days.
MMRs between the 3rd and 2nd can be estimated as 2.1114, or ~ 2:1, and between the 4th and 3th as 2.1001, or ~ 2:1.  Using these MMRs and the equation for P_ttv of Fabrycky, et al. (shown below), it is found that
(1) a period of 19.84 days for “148.04" is consistent with a P_ttv for 148.02 of 389.30 days (found above: 389.30 ± 10.52 days), and
(2) a period of 19.64 days for "148.04” is consistent with a P_ttv for 148.03 of 232.95 days (found above: 232.95 ± 6.43 days).
The agreement between the three determined periods (20.43, 19.84, and 19.64 days) for the potential unseen planet (“148.04”) lends additional credence to the Lineweaver-Bovaird (L-B) Rule.