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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 1.: KOI-448.02, P = 43.61 days [Plot avg. error bars = ± 8.37 min. (smaller than symbols)]
TTV_maximum: 743.79 ± 95.99 days, Amp_ttv_maximum: 190.17 ± 26.36 min.
TTV_minimum (extrapolated): 1537.52 ± 155.40 days, Amp_ttv_minimum (extrapolated): -195.98 ± 26.36 min.
[Caution: LESS THAN ONE P_ttv CYCLE AVAILABLE!]
P_ttv: 1587.44 ± 144.17 days.
Amp_ttv: 386.15 ± 37.28 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 1523.22 days; Power: 12.12; FAP: 8.176 x 10^-12.
Linear ephemeris (this work): Tc = 43.58241434(Tc#) + 127.62174828

Figure 2.: Residuals of Figure 1. [Plot avg. error bars = ± 26.61 min.]
TTV_maximum: 478.76 ± 47.94 days, Amp_ttv_maximum: 46.88 ± 9.55 min.
TTV_minimum: 914.22 ± 59.45 days, Amp_ttv_minimum: -53.73 ± 9.55 min.
TTV_maximum: 1349.67 ± 74.44 days, Amp_ttv_maximum: 46.88 ± 9.55 min.
P_ttv: 870.92 ± 39.31 days.
Amp_ttv: 100.61 ± 13.50 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 863.10 days; Power: 8.68; FAP: 5.964 x 10^-5.

Figure 3.: Residuals of Figure 2.; Residuals-of-the-Residuals of Figure 1. [Plot avg. error bars = ± 28.27 min.]
TTV_maximum: 371.20 ± 53.80 days, Amp_ttv_maximum: 21.73 ± 7.27 min.
TTV_minimum: 670.09 ± 61.70 days, Amp_ttv_minimum: -23.82 ± 7.27 min.
TTV_maximum: 968.97 ± 72.42 days, Amp_ttv_maximum: 21.73 ± 7.27 min.
TTV_minimum: 1267.86 ± 84.89 days, Amp_ttv_minimum: -23.82 ± 7.27 min.
P_ttv: 597.78 ± 32.38 days.
Amp_ttv: 45.55 ± 10.29 minutes.
Lomb-Scargle periodogram, candidate P_ttv: 602.19 days; Power: 5.14; FAP: 0.0513.

Figure 4.: Added combination of Figures 1., 2., and 3. [Plot error bars = ± 38.99 min.]
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Figure 5.: KOI-448.01, P = 10.14 days [Plot avg. error bars = ± 4.70 min.]
Lomb-Scargle periodogram: no credible periodicities.
Linear ephemeris (this work): Tc = 10.13961937(Tc#) + 111.45070135

Literature: Tc#, Tc, and TTV Major Tabulations and Other References: see my "Summary" webpage.

21 July 2014
KOI-448 (Kepler-159, KIC-5640085) 5(?)-(or more?)-Planets

Discussion:
The versatility inherent in Online EXOFAST was successfully applied in the case of KOI-448.02.  Specifically, Online EXOFAST allows for preliminary graphical testing and then, if warranted, making slight adjustments to the approximate Tc used as a Prior to better center the transit on a preliminary fit; this is especially useful when large TTVs (as in this case) have caused that transit to wander significantly from its ephemeris-predicted time.  Thus, each transit was individually evaluated by:
(1) selecting an approximate T_(n) time by modifying the NASA Exoplanet Archive's (NEA's) ephemeris prediction with the time difference found between the NEA's ephemeris predicted T_(n-1) and the observed T_(n-1),
(2) selecting a data range to be examined that included points ± 0.3 day on either side of the approximate T_(n),
(3) running the EXOFAST preliminary "Test Input" module to better define and adjust the transit Prior, and then,
(4) running the complete Online EXOFAST evaluation of T_(n).

Also, in this example, after a sinusoidal curve-fit of the (O-C) vs. Time data showed (Figure 1. below) a periodicity (P_ttv) of 1587.44 ± 144.17 days (1523.22 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 870.92 ± 39.31 days (LSP: 863.10 days).  Similarly, the Residuals-of-the-Residuals also gave a good sinusoidal curve-fit (Figure 3.) with a periodicity of 597.78 ± 32.38 days (LSP: 602.19 days).  The added 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 3 other planets in non-transiting orbits are responsible for the complex TTV distribution observed for 448.02.

For completeness, Figure 5. shows the plot of [(O-C) vs. Tc] for the 448.01; this and the corresponding Lomb-Scargle periodogram shows no evidence for a TTV.

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