Scroll Down  to

KOI's (Blue)  or
K2 Objects (Green)

Exoplanet-Science.com

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 1.: KOI-1573.01, P = 24.81 days [Plot avg. error bars = ± 2.63 min. (smaller than symbols)]
TTV_minimum: 424.31 ± 19.53 days, Amp_ttv_minimum: -51.67 ± 3.40 min.
TTV_maximum: 972.62 ± 26.45 days, Amp_ttv_maximum: 60.56 ± 3.40 min.
P_ttv: 1096.63 ± 22.36 days.
Amp_ttv: 112.23 ± 4.80 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 1076.52 days; Power: 21.73; FAP: 0.
Linear ephemeris (this work): Tc = 24.81072246(Tc#) + 89.51309020

Figure 2.: Residuals of Figure 1. [Plot avg. error bars = ± 4.29 min.]
TTV_minimum: 189.75 ± 42.90 days, Amp_ttv_minimum: -10.57 ± 2.64 min.
TTV_maximum: 517.33 ± 48.98 days, Amp_ttv_maximum: 12.18 ± 2.64 min.
TTV_minimum: 844.92 ± 58.96 days, Amp_ttv_minimum: -10.57 ± 2.64 min.
TTV_maximum: 1172.51 ± 71.21 days, Amp_ttv_maximum: 12.18 ± 2.64 min.
TTV_minimum: 1500.10 ± 84.76 days, Amp_ttv_minimum: -10.57 ± 2.64 min.
P_ttv: 655.18 ± 32.18 days.
Amp_ttv: 22.75 ± 3.73 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 660.59 days; Power: 9.29; FAP: 0.0003951.

Figure 3.: Residuals of Figure 2.; Residuals-of-the-Residuals of Figure 1. [Plot avg. error bars = ± 5.04 min.]
TTV_maximum: 250.79 ± 46.37 days, Amp_ttv_maximum: 7.21 ± 2.35 min.
TTV_minimum: 503.05 ± 51.49 days, Amp_ttv_minimum: -7.10 ± 2.35 min.
TTV_maximum: 755.30 ± 59.07 days, Amp_ttv_maximum: 7.21 ± 2.35 min.
TTV_minimum: 1007.55 ± 68.28 days, Amp_ttv_minimum: -7.10 ± 2.35 min.
TTV_maximum: 1259.80 ± 78.55 days, Amp_ttv_maximum: 7.21 ± 2.35 min.
P_ttv: 504.50 ± 25.91 days.
Amp_ttv: 14.31 ± 3.32 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 504.61 days; Power: 5.79; FAP: 0.0415.

Figure 4.: Added combination of Figures 1., 2., and 3. [Plot (purple) error bars = ± 7.12 min.]

Last Figure: KOI-1573.02, P = 7.14 day [Plot error bars (smaller than symbol) = ± 9.57 min.]
Only Q1-Q16 Kepler data.
Linear ephemeris (this work): Tc = 7.13697906(Tc#) + 64.75223997.

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

22 July 2013; updated 15 Aug 2013; updated 21 Oct 2014
Kepler KOI-1573 (KIC-5031857) 4-(or more?)-Planet System

Discussion:
When very few cycles of TTV periodicity (there are only ~ 1.3 in the case of 1573.01) have been observed by Kepler, the P_ttv values derived from best-fit sinusoidal curves and the corresponding Lomb-Scargle periodograms change with each Quarter of added data; this is shown quite clearly for 1573.01 on page "P_ttvV" of this website for Q1-Q16 data.  Even one more point (from Q17), now included on this 2014-update, led to another slight increase in P_ttv to 1096.63 days (from 1062.13 days).  Such continuing changes can only be terminated by longer-term observations (and more Quarters of data)…which may or may not be in the offing via the Kepler K2 project.

Further, in the case of 1573.01, after a sinusoidal curve-fit of the (O-C) vs. Time data showed (Figure 1. below) a periodicity (P_ttv) of 1096.63 ± 22.36 days (1076.52 days was observed in the Lomb-Scargle Periodogram (LSP) of the same data), a plot (Figure 2.) of the Residuals also gave a good sinusoidal curve-fit with a periodicity of 655.18 ± 32.18 days (LSP: 660.59 days).  Similarly, the Residuals-of-the-Residuals also gave a good sinusoidal curve-fit (Figure 3.) with a periodicity of 504.50 ± 25.91 days (LSP: 504.61 days).  The summed combination of all three sinusoidal curves is arrayed in Figure 4. and reasonably reproduces a complex overall curvature consistent with the initial data.  While it is certainly possible (see recent work of Lithwick and others) that some of this unusual curvature obtains from eccentric orbits of planetary objects in this system, it is also possible that at least 3 planets in near-circular orbits are interacting with 1573.01 to give the TTV distribution observed.  And, of course, it is eminently possible that one of those other 3 planets is 1573.02.

The ratio of periods for 1573.01 and 1573.02 is 3.476, which corresponds to a mean motion ratio (MMR) of ~ 7/2.  Using this MMR and the equation for P_ttv of Fabrycky, et al. (shown below), an approximate P_ttv of 524.80 days is predicted.  Interstingly, this corresponds closely to the P_ttv of the sinusoidal array of the TTV-Residuals-of-the-Residuals (Figure 3.): 504.50 ± 25.91 days.