###### 4th: The **Lineweaver-Bovaird Rule** ("L-B Rule", ref. below) was used to estimate semi-major axes for the two potential unseen (non-transiting) planets. 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 5 relative semi-major axes are estimated as: 0.382, 0.716, 1.025, 1.6197, and 2.7645.

###### Plotting (see Figure 4.) the 1st, 2nd, and 5th of these values against the known semi-major axes of 1781.02 (0.0380 AU), 1781.01 (0.0720 AU), and 1781.03 (0.2854 AU), respectively, gave an excellent (R^2 = 1.0000) straight line and the equation: y = 0.1040x - 0.0020. From this equation, semi-major axes (and, hence, periods) could be estimated for the two potential unseen planets (the 3rd and 4th from their host star) that are the could be the planetary objects producing the observed TTVs of 1781.01 and 1781.03, respectively, i.e.:

###### "1781.04": Semi-major axis: ~ 0.1046 AU; Period: ~ 13.73 days.

###### "1781.05": Semi-major axis: ~ 0.1664 AU; Period: ~ 27.53 days.

###### Summarizing the overall sequence of periods and potential periods:

###### 1st, 1781.02: 3.01 days,

###### 2nd, 1781.01: 7.83 days,

###### 3rd, "1781.04": 13.73 days,

###### 4th, "1781.05": 27.55 days,

###### 5th, 1781.03: 58.02 days.

###### MMRs between the 3rd and 2nd can be estimated as 1.7513, or ~ 7/4, and between the 5th and 4th as 2.1066, 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 13.7309 days for "1781.04" predicts a P_ttv for 1781.01 of 1304.25 days (found: 1304.25 ± 161.83 days), and

###### (2) a period of 28.1765 days for "1781.05" predicts a P_ttv for 1781.03 of 980.95 days (found: 980.95 ± 65.28 days).The consistency between the P_ttv's and the periods for the two potential unseen planets lends additional credence to the Lineweaver-Bovaird (L-B) Rule.

###### References:

###### • Fabrycky et al., arXiv:1201.5415; P_ttv = 1/|(nmmri_a/P_a - nmmri_b/P_b)|.

###### • For Q1-Q6 TTV data of 1781.01 and 1781.02 (thru ~ 563 (BJD-2454900) days), see: Ford et al., 2012, arXiv:1201.1892.

###### • For Q1-12 TTV data of 1781.01 and 1781.02 (thru ~ 1116 (BJD-2454900) days), see: Mazeh et al., 2013, arXiv:1301.5499.

###### • For Q1-14 TTV data, see Ji-Wei Xie's website: http://www.astro.utoronto.ca/~jwxie/TTV/TTV_Home.html

###### • Bovaird & Lineweaver, 2013, arXiv:1304.3341.

###### 18 Dec 2013, updated 19-Dec-2013

###### 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__:

**The observed sinusoidal TTVs of transiting 1781.01 and 1781.03 not only show substantially different P_ttv's but they are clearly not anti-correlated. One explanation could be that two unseen (non-transiting) planets are present in between them that individually cause the observed TTV effects, respectively. This is discussed in more detail below.**

###### 1st: KOI-1781.02, P = 3.01 days [Plot error bars = ± 2.27 min.]

**Linear ephemeris (this work): Tc = 3.00515621(Tc#) + 70.10343649**

###### 2nd: KOI-1781.01, P = 7.83 days [Plot error bars = ± 1.11 min.]

###### TTV_maximum: 687.41 ± 123.55 days, Amp_ttv_maximum: 2.64 ± 0.44 min.

###### TTV_minimum: 1339.54 ± 188.72 days, Amp_ttv_minimum: -2.28 ± 0.44 min.

**P_ttv: 1304.25 ± 161.83 days.**

**Amp_ttv: 4.92 ± 0.62 minutes.**

**Lomb-Scargle periodogram, candidate P_ttv: 1400.56 days; Power: 24.7; P-value: 2.196 x 10^-8.**

**Linear ephemeris (this work): Tc = 7.83442767(Tc#) + 68.22314882**

###### 3rd: KOI-1781.03, P = 58.02 days [Plot error bars = ± 2.32 min.]

###### TTV_minimum: 340.91 ± 54.83 days, Amp_ttv_minimum: -31.05 ± 12.02 min.

###### TTV_maximum: 831.39 ± 74.51 days, Amp_ttv_maximum: 57.37 ± 12.02 min.

###### TTV_minimum: 1321.86 ± 101.14 days, Amp_ttv_minimum: -31.05 ± 12.02 min.

**P_ttv: 980.95 ± 65.28 days.**

**Amp_ttv: 88.41 ± 17.00 minutes.**

**Lomb-Scargle periodogram, candidate P_ttv: 1006.28 days; Power: 3.94; P-value: 0.0548.**

**Linear ephemeris (this work): Tc = 58.01952760(Tc#) + 84.85465578**

**KOI-1781 (Kepler-411, KIC-11551692): Pot'l of 5-Planets**