Detrending and transit modelling

For the detrending and the transit modelling of the extracted KOINet observations, initial transit model parameters of good precision are necessary. The determination of them is described in the first part of this section.

Part of this thesis project was a contribution to the KOINet by providing initial transit parameters for the sixty planets and planet candidates. The parameters are derived from modelling all transits of the objects in theKepler long cadence data. To reduce the impact of the TTVs in the data every two consecutive transits are mod-elled simultaneously. From theKepler data the transits are extracted with one transit duration out-of-transit data before and after the transit. For detrending the observa-tions, the outside transit data belonging to one transit are modelled by a second-order polynomial and the extracted light curve is divided by this fitted function. The mod-elling of every two consecutive transits simultaneous is done with the transit Mandel

& Agol (2002) model assuming a quadratic limb darkening law. The limb darkening coefficients are extracted from the values calculated by Claret et al. (2013) for the fundamental stellar parameters of the objects host star (effective temperature, metal-licity, surface gravity) from theNASA Exoplanet Archive. To reduce the impact of the sampling rate on the derived transit model parameters (Kipping, 2010), the model is computed on a fine grid of thirty points per observation time step and rebinned to the data points afterwards. Assuming circular orbits, the transit modelling delivers values for the semi-major axis, orbital inclination, planet-to-star radius ratio, orbital period, and mid-transit time. The parameter space is explored by the Markov chain Monte Carlo (MCMC) algorithm PyMC (Patil et al., 2010) accessed from the PyAstronomy⁵ package and the mean and standard deviation of the MCMC posterior distribution gives the fitting parameters. The initial input transit parameters used to model the ground-based observations are derived from the mean and standard deviation of the model parameter distribution of analysing all two consecutive transits of an object.

For KOINet objects for which a dynamical modelling like a photodynamical analysis (see section 1.4) has already been performed, the transit parameter predictions of this analysis are used as initial parameters for the KOINet observation modelling.

For extracting the transit times of ground-based data by modelling the transit, a more refined detrending model than just a polynomial is necessary to carefully respect the influence of the Earth’s atmosphere and the individual instruments of the observations. The detrending model is also important for the photodynamical analysis (section 1.4) when including KOINet observations. For this reason, a more complicated model was developed in cooperation with C. von Essen as part of this thesis project. Due to the high time consumption of the photodynamical model by numerical integrations, the detrending model needed to be fast. Therefore, a linear combination of an extraction of components that induce trends in the photometric

5http://www.hs.uni-hamburg.de/DE/Ins/Per/Czesla/PyA/PyA/index.html

1.3. THE KOINET 19 light curve is calculated for a given transit model. The possible detrending components are the seeing ˆS, airmass ˆχ, and per measured star,i, theX_i,Y_i-centroid positions, sky background counts BG_i, and flat and dark fluxes,F C_i and DK_i. The full detrending model (DM) when considering all components has the form

DMptq “c₀`c<sub>1</sub>¨χˆ`c₂¨Sˆ`

N`1

ÿ

i“1

x_i¨X_i`y<sub>i</sub>¨Y<sub>i</sub>`bg_i¨BG_i`f c<sub>i</sub>¨F C<sub>i</sub> `dk_i¨DK_i withN`1 denoting the total number of stars, target and reference, and the coefficients of the model arec₀,c₁,c₂,x_i,y_i,bg_i,f c_i, anddk_i. The coefficients are calculated from this linear combination while considering a transit model for the photometric light curve. In order not to overfit the observations, not all possible detrending compo-nents are considered, but a sub-model is searched that sufficiently detrends the data with a minimum amount of detrending components. The individual star components can be considered for only the target star, for all stars or as a combination of all stars. To determine the best matching detrending model of an observation an array of trial transit mid-times from the predicted time plus/minus the transit duration is arranged. A transit Mandel & Agol (2002) model with the initial transit model pa-rameters of the object and quadratic limb darkening law together with a detrending model is calculated. The quadratic limb darkening coefficients are computed for the Johnson-Cousins R-filter transmission response and angle-resolved synthetic spectra from spherical atmosphere models using PHOENIX (Husser et al., 2013) with stellar parameters closely matching the ones of the star (see, e.g. von Essen et al., 2013).

With the combined model a minimisation statistic to the data is computed for the transit time array and the sub models with the different combinations of detrending components. The mean of four different minimisation statistics is used to ensure a good number of fitting parameters. These statistics are the reduced-χ², the Bayesian Information Criterion, BIC, the standard deviation of the residuals enlarged by the number of fitting parameters, and the Cash (1979) statistic. For a trial transit mid-time near the actual transit mid-time, all statistics should be minimised independent of the chosen detrending model. In this way, the detrending model which most minimises the mean of the statistics at the transit time is chosen, while taking the number of detrending components into account. In Figure 1.7 two different KOINet light curves and there detrending models are visualised as an example. The top plot shows an example where large trends and jumps due to observation breaks are corrected by the detrending procedure. A full transit observation and the correction of small effects with the detrending, resulting in a reduction in the scatter in the light curves is visible in the bottom plot.

The last part of DIP²OL contains a transit fitting routine for transit time de-termination only when this value is of interest and for accurate errorbars that en-sure a correct subsequent use of the ground-based light curve in, for instance, the photodynamical modelling. To calculate accurate errorbars, correlated noise in the observations needs to be considered (see e.g. Carter & Winn, 2009). The observed light curve is fitted with an MCMC algorithm by the transit Mandel & Agol (2002) model with quadratic limb darkening coefficients combined with the fitted detrending

Figure 1.7: Two different examples of KOINet light curves to show the detrending.

The plots show from top to bottom the raw light curve with the combined detrending and transit light curve model, the detrended light curve with the transit model and the residuals of the data from the combined models. The curves are artificially shifted for better visualisation. Both plots show KOINet light curves of Kepler-9 with a transit of planet c (top) observed by the 2 m Liverpool telescope (Steele et al., 2004) and a transit of planet b (bottom) observed by the 2.5 m Nordic Optical Telescope.

1.4. PHOTODYNAMICAL MODELLING 21