DETERMINACIÓN DE LA EXPRESIÓN DE LRP-1

to well represent the limb profiles of most stars, rather than have to utilise different limb darkening laws for different types of stars. A major disadvantage in using this law is that there are four coefficients, which can cause difficulties in fitting lightcurves if not fixed to values found from model stellar atmospheres.

Sing et al.(2009) introduced a variation on the four parameter law in Equation3.7, which dropped theµ12 _term:

I(µ)

I(1) =1−c1(1−µ)−c2(1−µ

3

2)−_c

3(1−µ2), (3.8)

where c₁–c₃ are the coefficients of the law. The justification for making this adjustment was primarily because the µ12 term affects mostly the shape of the limb profile at lowµ values,

and it is well known that the variation for this part of the limb profile is smooth and well approximated by a linear function, so the term is not required to provide a good representation of the CLIV. The second reason to remove the term is that it reduces the number of coefficients in the model, which is then easier to fit.

3.5

Tables of LDCs

Tabulated LDCs are the result of fitting the outputs of a grid of stellar atmosphere models for the coefficients of the chosen law for a chosen photometric passband. Several tables of LDCs have been calculated for different grids of stellar atmosphere models. Each of the grids of stellar atmosphere models mentioned here for fitting LDCs are introduced in Section3.3.

Claret & Bloemen(2011) made use of grids of the ATLAS9 plane-parallel models, and also the PHOENIX spherically symmetric models.Neilson & Lester(2013) made use of a version of the ATLAS9 model adapted to function with a spherically symmetric geometry rather than plane- parallel (Lester & Neilson,2008). Sing(2010) also made use of a grid of ATLAS9 models. The table of LDCs presented inMagic et al.(2015) were generated from the STAGGER grid of 3D models, although the number of models included in this grid is significantly smaller than for those based on grids of 1D atmosphere models.

The basic process for generating a table of LDCs is briefly described in the following steps. The atmosphere model outputs take the form of specific intensities for a range of wavelengths and µ angles for each model. The specific intensities at eachµ are then convolved with a photometric bandpass, and integrated over wavelength, resulting in an intensity for each angle

for each passband used. The grid of coefficients may be calculated using the least squares method, which optimises the fit of the parametric law for each atmosphere model across the calculated intensities.

The tabulated LDCs inClaret & Bloemen(2011) also include LDCs found using a second method to determine their values for each atmosphere model – the flux conservation method (FCM). Intuitively basing the determination of the LDCs on conserving flux appears the optimal method to utilise. The FCM does not however constrain the LDCs to well represent the form of the atmosphere model output for any non-linear limb darkening formulation, only that the integrated fluxes match. Another condition must be used in conjunction with flux conservation to allow a solution to be found for laws with more than one coefficient, and to force the fit to use some information about the form of the limb profile. Despite the desire for parameterised limb darkening to conserve the flux of the atmosphere model output, using the FCM leads to significantly worse fits of the LDCs than using a least squares fitting procedure (Claret,2000).

4

Limb darkening coefficient interpolation

4.1

Limb darkening and exoplanet transits

This chapter introduces a new approach for determining the limb darkening coefficients for a transit lightcurve, interpolating between tabulated values of limb darkening coefficients using a Gaussian Process (GP).

The forms of the parameterisations for limb darkening are discussed in Section3.4. In or- der to apply these parameterisations to an exoplanet transit fit, some approach must be chosen to implement the model into the fitting procedure. When choosing which limb darkening law to utilise, a balance needs to be struck between accurately mapping the stellar surface varia- tion with a higher order law, and having too many ill-constrained free parameters to fit. Once a limb darkening law has been selected, what fitting procedure to use must be determined – the coefficients can be freely fit, constrained only by the transit observation data; some ex- isting knowledge of the stellar parameters can be used to apply a prior to some or all of the coefficients, or some or all of the coefficients can be fixed to the values expected for that host

star.

If choosing not to apply an informative prior to the coefficient fit, the coefficients for certain laws can be reparameterised to allow uniform sampling of the coefficients. The adjustment reduces the correlation between the parameters, and means that the parameters used must sit within the region[0,1]where the probability of each value is equal (a flat prior is appropriate). This reduces the parameter space that any MCMC fit must explore1, and prevents unintended or non-physical non-uniformity in the prior. This form of reparameterisation has however only been found for the quadratic law (Kipping,2013), and for theSing et al.(2009) 3-parameter law (Kipping,2016).

Many studies of exoplanet transit lightcurves have used these uninformative priors on the limb darkening coefficients for the quadratic law (eg. Barragán et al.(2016);Foreman-Mackey et al.(2016)), or for the 3-parameter law (eg. Sandford & Kipping(2017)). This method is most effective when the lightcurve is of a good SNR, and the impact parameter is low enough, so that enough information about the limb darkening for the system is encoded into the tran- sit lightcurve. With a low SNR transit lightcurve, freely fitting the coefficients risks offering too much flexibility in the fit, and can result in over-fitting or biasing the transit parameters (Espinoza & Jordán,2016).

The LDCs can be fixed to the expected values for the host star (eg. Adams et al.(2017);

Wilkins et al.(2017)), which ensures that the modelled CLIV is physically justified for that star, and prevents the fit for the limb darkening over-fitting any other signal in the transit lightcurve. This approach does assume that the measurements of the stellar atmospheric parameters used to find the fixed coefficients are accurate and precise, and the atmosphere models and the form of the parametric law used to determine the coefficients reflect the physics of the atmosphere of the host star. The uncertainties in the determination of the stellar atmospheric parameters are also not propagated through this fitting procedure.

Instead of fitting the LDCs, prior information about the stellar atmosphere parameters can be used to inform the fitting of the LDCs. This approach provides some flexibility in the LDCs, as it allows the uncertainties in the stellar parameters to be propagated through the fit. Southworth(2008) proposes JKTLD2, which performs bi-linear interpolation between tabulated LDCs from a choice of tabulations. The method interpolates between the Teffand

1_{Markov Chain Monte Carlo (MCMC) methods are introduced in Section2.4}

4.1. Limb darkening and exoplanet transits

logggrid provided in the tables for a given set of input stellar parameters. This method can then be called during an MCMC transit lightcurve fit to sample the limb darkening coefficients with stellar parameters as jump parameters.

LDTK (Parviainen & Aigrain, 2015)3 generalises the interpolation between tabulated co- efficients to produce fits for LDCs for arbitrary photometric passbands from theHusser et al.

(2013) grid of stellar atmosphere models. A prior on the stellar parameters (Teff, logg, and

[Fe/H]) is then sampled to produce a multivariate prior on the LDCs to be applied to a transit fit.

4.1.1

Biases from incorrect or insufficient fitting of limb darkening

Insufficiently or incorrectly modelling the stellar limb darkening when fitting an exoplanet transit can introduce biases in the transit parameters determined. The scales and extents of the biases are explored by several authors.

Work bySouthworth(2008) found that in general fixing the limb darkening coefficients to those tabulated did not bias the transit parameters, but would result in an underestimate of the uncertainties.Howarth(2011) showed more significant differences between limb darken- ing determined from the lightcurve shape and from tabulated coefficients, although the scales of the differences showed some dependence on the stellar parameters of the host star.Müller et al.(2013) demonstrated that the predicted LDCs from theoretical atmosphere models sys- tematically over-predicts the value of the second quadratic coefficient, which would induce biases in the transit parameters if fixed to that value. The study also highlights that fixing the LDCs to the tabulated values (or tightly constraining the coefficients) is the only appropriate approach for systems with high impact parameters, as not enough of the stellar surface is oc- culted during the transit to provide enough information about the limb darkening to determine the LDCs solely from the transit lightcurve.

Hayek et al. (2012) were able to show that fixing LDCs to values determined from the outputs of 3D stellar atmosphere models produced far lower residuals in transit fits than LDCs determined from 1D stellar atmosphere models. Espinoza & Jordán (2015) quantify some of the differences seen between LDC modelling approaches for transit lightcurves, which are significant, particularly when fixing the coefficients to tabulated values for the quadratic law

– up to a 3% bias inRp/R⋆. They recommend freely fitting the LDCs as part of a transit fit to

prevent this bias.

Neilson et al.(2017) found significant differences (up to 300 ppm in the ingress/egress) between the transit shapes for limb darkening determined directly from stellar atmosphere models with plane-parallel geometry and those with spherically symmetric geometry. They did not however fully take into account the differences in definition of the stellar radius between the two geometries, so the scale of the true effect will be much smaller.

Ensuring that limb darkening is precisely and accurately modelled is particularly important when fitting for third order effects, which are potentially observable in the ingress and egress of an exoplanet transit lightcurve. The scale of these effects can be ∼ 100 ppm, which is comparable to the scales of biases introduced by ineffective modelling of limb darkening. A few examples of third order transit features, which will be more readily observable in transit lightcurves with a strong understanding of the stellar limb darkening – exo-rings (Brown et al.,

2001), stellar granulation (Chiavassa et al.,2017), and planetary thermal emission (Kipping & Tinetti,2010).