Reconstructive and simultaneous trend removals

Of course, we are not really interested in the de-trending of non-variable stars. Unless one wants to quantify the generic quality of a certain photometric pipeline, the importance of any trend removal algorithm are relevant only in the cases where the stars have intrinsic brightness variations. In the following, we suppose that the physical variations can be quantified by a small set of parameters {Ar}, namely the fiducial signal of a particular star

can be written as

m0_i = m0+ F (ti, A1, . . . , AR) (2.90)

where F is some sort of model function.

In principle, one can manage variable stars by four considerations. First, even stars with physical brightness variations are treated as non-variable stars. This naive method is likely to distort the signal shape by treating the intrinsic changes in the brightness to be unexpected. In the cases where the periodicity of these intrinsic variations are close to the periodicity of the generic trends24_{or when the period is comparable or longer with the observation window,}

either EPD or TFA tend to kill the real signal itself. Second, one can involve the method of signal reconstruction, as it was implemented by Kov´acs, Bakos & Noyes (2005). In this method, the signal model parameters {Ar} are derived using the noisy signal, and then the

fit residuals undergo either the EPD or TFA. The model signal F (ti, . . .) is added to the

de-trended residuals, yielding a complete signal reconstruction. The steps can be repeated until convergence is reached. Third, one can involve the simultaneous derivation of the Ar

model parameters and the Ek/Ft coefficients by minimizing the merit function

χ2 =X i wi " mi− m0− F (ti, {Ar}) − X k Ekp(k)i #2 . (2.91)

(This merit function shows the simultaneous trend removal for EPD. The TFA and the joint EPD+TFA can be applied similarly.) The fourth method derives the Ek and/or Tf

coefficients on sections of the light curve where the star itself shows no real variations. This is a definitely useful method in the analysis of planetary transit light curves, since the star itself can be assumed to have constant brightness within noise limitations25 _{and therefore}

24_{For instance, trends with a period of a day are generally very strong.}

25_{At least, in the most of the cases. A famous counter-example is the star CoRoT-Exo-2 of Alonso et al.}

the light curve should show no variations before and after the transit. If these out-of-transit sections of the light curves are sufficiently long, the trend removal coefficients Ek and/or Tf

can safely be obtained.

There are some considerations regarding to the F (ti, A1, . . . , AR) function and its param-

eters {Ar} that should be mentioned here. In principle, one can use a model function that is

related to the physics of the variations. For instance, a light curve of a transiting extrasolar planet host star can be well modelled by 5 parameters26_{: period (P ), epoch (E), depth of the}

transit (d), duration of the transit (τ14) and the duration of the ingresses/egresses (τ12) (see

e.g. Carter et al., 2008, about how these parameters are related to the physical parameters of the system, such as normalized semimajor axis, planetary radius and orbital inclination). Although the respective model function, Ftransit(ti, P, E, d, τ14, τ12) is highly non-linear in its

parameters, the simultaneous signal fit and trend removal of equation (2.91) can be per- formed, and the fit yields reliable results in general27_{. In the cases where we do not have}

any a priori knowledge of the source of the variations, but the signal can be assumed to be periodic, one can use a periodic model for F , that is, for instance, a linear combination of step functions. Although the number of free parameters (which must be involved in such a fit) are significantly larger, in the cases of HATNet light curves, the fit can be achieved properly. The signal reconstruction algorithm of Kov´acs, Bakos & Noyes (2005) use a step function (also known as “folded and binned light curve models”) for this purposes. Like so, F can also be written as a Fourier series with finite terms. If the period and epoch are kept fixed, both assumptions for the function F (i.e. step function or Fourier expansion) yield a linear fit for both the model parameters and the EPD/TFA coefficients.

It should be mentioned here that the signal reconstruction mode and the simultaneous trend removal yields roughly the same results. However, a prominent counter-example is the case of HAT-P-11(b) (Bakos, Torres, P´al et al., 2009), where the reconstruction mode yielded an unexpectedly high impact parameter for the system. In this case, only the method of simultaneous EPD and TFA was able to reveal a refined set of light curve parameters that are expected to be more accurate on an absolute scale. Further discussion of this problem can be found in Bakos, Torres, P´al et al. (2009).

26_{Other parameters might be present if we do not have a priori assumptions for the limb darkening and/or}

the planetary orbit is non-circular and the signal-to-noise of the light curve is sufficiently large to see the asymmetry.

27_{Only if the transit instances inter/extrapolated from the initial guess for the epoch E and period P}

sufficiently cover the observed transits. Otherwise, all of the parametric derivatives of F will be zero and only methods based on systematic grid search (e.g. BLS) yield reliable results.

2.11. MAJOR CONCEPTS OF THE SOFTWARE PACKAGE