Abbreviations: P_ttv: Periodicity of Transit Timing Variation. Amp_ttv: Total amplitude of periodic sinusoidal TTV curve. LSP: Lomb-Scargle Periodogram (or Periodicity) FAP: False Alarm Probability P-value: False Alarm Probability (Used within the NASA Periodogram Service; URL: http://exoplanetarchive.ipac.caltech.edu/cgi-bin/Periodogram/nph-simpleupload ) P_tdv: Periodicity of Transit Duration Variation. P_t(t_fwhm)v: Periodicity of Transit (Time for Full Width @ Half Maximum) Variation. P_opv: Periodicity of Orbital Period Variation. P_tpv: Periodicity of Transit Depth Variation.
Conversion of EXOFAST parameters to TTV graphical displays: All output parameters (whether from Offline or Online use of EXOFAST) were obtained as lines of text, e.g. "T_c Time of transit (\bjdtdb) 574.111917". The entire output was copied to an Excel spreadsheet and values of interest (e.g., transit times (Tc) and sigma_Tc) were separated into columns for subsequent plotting through a series of [if-then-else] statements, [text-to-columns] selections, and time sorting using appropriate Macros. A transit time was removed from the output list (1) if the Exofast best-fit plot showed it to be a partial (due to a data gap) or distorted transit, (2) if the transit overlapped with the transit of another planet in the system, (3) if the SNR was sufficiently low to actually cause the code to produce a negative tau (i.e.: [ingress + egress]) value, or (4) if it's [O - C] plotted point (below) was separated as a clear "outlier" from the mean by more than ~ 2.5 standard deviations.
Plotting the Tc# (x-axis) vs. the observed Tc (units: BJD-2454900; y-axis) gave a straight line (R^2 usually greater than 0.999999) and an equation that corresponded to the linear ephemeris of the confirmed or candidate exoplanet in question; calculated Tc's were obtained from the equation of that straight line. TTV plots were then produced from Observed Tc (x-axis) vs. [Observed Tc - Calculated Tc; O - C] (y-axis). When error bars in the plots are not visible, it is simply because they are smaller than the data-point symbols being used.
Lomb-Scargle Periodograms were prepared from the TTV x-y data to independently pinpoint periodicities withing the time series (i. e.: candidate periodic transit variations (P_ttv)) using NASA's Exoplanet Archive Periodogram Service: http://exoplanetarchive.ipac.caltech.edu/cgi-bin/Periodogram/nph-simpleupload . In general, a significant candidate TTV period ("P_ttv") within a time-series is characterized by high "Power" (magnitude of a coefficient in a frequency domain associated with that periodic variation) and a small "P-Value" (FAP, False Alarm Probability, i. e.: probability that the Power was a number obtained by chance and not due to a real sinusoidally recurring effect). Bottom line, small P-Value (FAP) (i. e.: near 0) indicates a significant candidate P_ttv which was then possibly selected for highlighted graphical display on this website. More specifically, through personal communications with Eric Agol, Eric Ford, Dan Fabrycky, and Carolina von Essen, the following were adopted as working ranges for periodicity credibility:
High credibility: FAP < 0.01;
Medium credibility: 0.01 < FAP < 0.10;
Low credibility: FAP > 0.10.
Best-fit sinusoidal curves and equations were generated using the Kaleidagraph® program of Synergy Software; technical support by Steve Wilson is sincerely appreciated. Adam Power, UMass Honors, is gratefully acknowledged for providing intuition on the facile differentiation of the complex trigonometric function required to obtain xy-coordinates of minima and maxima (dy/dx = 0) from best-fit sinusoidal equations. In the case of fairly low SNR where more than one "best-fit" sinusoidal curve was obtained, agreement with the Lomb-Scargle periodogram was determinate.
Two forms of sinusoidal equations were used throughout this website; in each: y = [O - C] (in minutes), i.e.: the difference between observed (Exofast-determined) transit times and those calculated using the linear ephemeris obtained by plotting those observed transit time (Tc) against the number of the transit (Tc#); and x = Time (days)(BJD-2454900). Also, in both cases, times of maxima and minima were calculated by setting dy/dx = 0 and solving for x.
y = a + b*sin(cx + d); designated as "Fit 1": In the best-fit-curve search with the Kaleidagraph® program, priors for the coefficients [a = y offset, b = amplitude, c = frequency, and d = phase shift] could each be set independently but it was usually most efficient to adjust the amplitude "b" and the frequency "c" and then allow the best-fit search to proceed.
y = a*cos(bx) + c*sin(bx) + d*x; designated as "Fit 2": With this one, priors for the coefficients were: a = (mostly associated with) amplitude, b = (mostly associated with) frequency, c = (mostly associated with) phase shift, and d = y-offset. Once again, each could be set independently but it appeared most efficient to adjust the frequency "b" and the amplitude "a" and then allow the best-fit search to proceed.
The Lineweaver-Bovaird Rule ("L-B Rule", ref. below) predicts relative semi-major axes for multiple planetary systems: a = 0.382 + 0.334(1.925^n), where "a" is the semi-major axes, and "n" = -∞, 0, 1, 2, 3, …, (see ref. 9 below).
References: 1a. Ballard, Fabrycky, et al., 2011, arXiv-1109.1561 (KOI-84, Kepler-19, KIC-2571238). 1b. Cochran, Fabrycky, et al., 2011, arXiv-1110.0820 (KOI-137, Kepler-18, KIC-8644288). 1c. Carter & Agol, 2012, arXiv-1206.4718 (KOI-277, Kepler-36, KIC-11401755). 1d. Ofir & Dreizler, 2012, arXiv-1206.5347 (KOI-277). 1e. Dawson, Johnson, et al., 2012, arXiv-1206.5579 (KOI-1474, KIC-12365184). 1f. Nesvorny, Kipping, et al., 2012, arXiv-1208.0942 (KOI-872, KIC-7109675). 1g. Nesvorny, Kipping, et al., 2013, arXiv-1304.4283 (KOI-142, KIC-5446285). 1h. Numerous other literature references involving TTVs can be found on E. Ford's website: http://www.astro.ufl.edu/~eford/data/kepler/ttv_papers.html. 2a. E. Ford, 2012 website: lists Q0-Q6 TTV data for confirmed and candidate exoplanets; http://www.astro.ufl.edu/~eford/data/kepler/TT_Q0-6/tab_transit_times_electronic.txt. 2b. Mazeh, Nachmani, et al., 2013, arXiv-1301.5499; in this paper, it is reported that a table of Q0-Q12 TTVs is in preparation. 3. The NASA Exoplanet Archives (http://exoplanetarchive.ipac.caltech.edu) is a compilation of all Kepler exoplanet candidates, confirmed exoplanets, false positives, and numerous "not dispositioned" objects (all Kepler Objects of Interest: KOIs) including all those reported by others in 2011 (ref. 4), and 2012 (ref. 5). 4. Borucki, Koch, et al., 2011, arXiv-1102.0541. 5. Batalha, Rowe, et al., 2012, arXiv-1202.5852. 6. For another comprehensive source of physical properties, see the Kepler Community Follow-up Observing Program (CFOP) website: https://cfop.ipac.caltech.edu. 7. "EXOFAST: A FAST EXOPLANETARY FITTING SUITE IN IDL", Eastman, Gaudi, & Agol, 2013, arXiv-1206.5798v3; PASP, 125, 923, 83 (2013). 8a. NASA Exoplanet Archives ("NEA"): http://archive.stsci.edu/kepler/data_search/search.php?action=Search&ktc_kepler_id=[KIC# here]. 8b. See also: http://exoplanetarchive.ipac.caltech.edu/applications/ETSS/Kepler_index.html. 9. Bovaird & Lineweaver, 2013, arXiv:1304.3341. NUMEROUS OTHER REFERENCES ARE LISTED ON THE "Summary" PAGE AND A FEW APPEAR ON OTHER PAGES DEALING WITH INDIVIDUAL KOIs.
6 Mar 2014; updated 20 Sep 2014; updated 18 May 2015.