Joint modelling of transit and stellar temperature using an MCMC ...

CHAPTER 6. JOINT MODELLING OF TRANSIT AND STELLAR TEMPERATURE USING AN MCMC APPROACH 182
which might not be the global minimum.
The set-up stage of an MCMC can also be lengthy, as opposed to the setting up
of a grid search for instance. Factors to take into account in the setting-up stage on
a MCMC include: a) the format of the data (number of points) as it can lengthen
the calculation time, b) the choice of priors and their distributions, c) the initial values,
d) the typical step sizes, e) the length of the chain to ensure convergence. However,
once the MCMC is set-up, for the same resolution and statistical robustness of the solution,
running an MCMC can be faster than a grid search.
The transit modelling using the MCMC approach described in this chapter makes
use of a new photometric method to derive the stellar temperature. This method uses
stellar evolution models to translate the stellar densities (adjusted in the MCMC) into
stellar temperature. The precision of the derived T eff will depend on the finesse and
intrinsic accuracy of the grid of stellar evolution models.
Applied to the phase-folded IRF-filtered transit light curve of CoRoT-2b, the MCMC
approach derived is a more robust method to determine error bars on the planet parameters.
It also provides an independent measurement of the stellar temperature
from the transit shape. The addition of a prior in the stellar temperature was not found
to change the final values of the planet parameters, but did change the shape of the
distribution in stellar density.
6.3.2 Future work
Longer chains still need to be run to ensure the convergence for each parameter. This
will secure a robust solution for the parameters, for the given data and priors.
The uncertainties on the contaminant flux needs to be taken into account in the
uncertainties derived for the parameters. The suggested approach is to add to the
transit model, at each MCMC iteration, a constant flux drawn from a Gaussian distribution
(with a zero mean and a standard deviation equal to the uncertainty on the
contaminant flux), then re-normalise the model and use it in the calculation of the likelihood.
The posterior distribution of the parameter will thus include the uncertainty of
the contaminant flux.
The red noise in the light curve needs to be taken into account as a good merit
function needs to use a true estimate of the noise. The red noise can be taken into
account by replacing the σ white in the merit function by σ pink = σ white + N ∗ σ red (Pont
et al., 2006). σ red can be evaluated by binning the unfolded light curves with different
bin sizes and evaluating each time the standard deviation of the resulting signal. The
σ white will go as σ no bin /N , N being the number of points binned together, while σ red
should be constant, so the difference between the σ bin and σ no bin /N , e.g. for bin size
of 1 or 2h, is σ red .
The IRF filtered light curve was binned in order to reduce the number of data points
to speed up the MCMC. It will be interesting to see how this binning affects the ac-