Limb-darkening

TransitFit was built with two primary motivations. First, to facilitate transmission spectroscopy surveys using observations from heterogeneous telescopes, and second, to allow the user to fit light curves while accounting for the effects that filter profiles and host parameters have on the LDCs, which we refer to as ‘coupling’ the LDCs. We will discuss the latter here.

The full justification for and impacts of this approach can be found in the accompanying paper but, in short, not including host parameters and filter profiles in likelihood calculations of LDCs can lead to biases in your measurements of \(R_p/R_\star\) of tens of percent. By including this, TransitFit has made it easy to conduct robust transmission spectroscopy studies using observations from heterogeneous sources.

Calculating LDC likelihoods

TransitFit uses the Limb Darkening Toolkit (LDTk) to calculate the likelihood values of sets of LDCs given the host characteristics and filter profiles. These values are then included in the likelihood calculations of transit models during retrieval.

In order to use this feature, the user must provide the following:

  • Host temperature

  • Host \(z\)

  • Host \(\log(g)\)

  • A filter profile for each filter used in the observations.

The first three are provided as arguments to run_retrieval(), and the filter profiles are specified using the filter profiles input file.

Available limb-darkening models

Typically, stellar intensity profiles are described by analytical functions \(I_\lambda\left(\mu\right)\), where \(\mu\) is the cosine of the angle between the line of sight and the emergent intensity. \(\mu\) can also be expressed as \(\mu = \sqrt{1-r^2}\) where \(r\) is the unit-normalised radial coordinate on the stellar disk, and as such, all limb-darkening models must be valid for \(0 \le \mu < 1\).

There are 5 limb darkening models provided by TransitFit, which can be selected using the limb_darkening_model argument in run_retrieval(). These are:

  • 'linear' - the linear law given by
    \[\frac{I\left(\mu\right)}{I\left(1\right)} = 1 - u_{0,l} \left(1 - \mu\right)\]
  • 'quadratic' - the quadratic law given by
    \[\frac{I\left(\mu\right)}{I\left(1\right)} = 1 - u_{0,q} \left(1 - \mu\right) - u_{1,q} \left(1-\mu\right)^2\]
  • 'squareroot' - the square-root law given by
    \[\frac{I\left(\mu\right)}{I\left(1\right)} = 1 - u_{0,\textrm{sqrt}} \left(1 - \mu\right) - u_{1,\textrm{sqrt}} \left(1-\sqrt{\mu}\right)\]
  • 'power2' - the power-2 law given by
    \[\frac{I\left(\mu\right)}{I\left(1\right)} = 1 - u_{0,\textrm{p2}}\left(1 - \mu^{u_{1,\textrm{p2}}}\right)\]
  • 'nonlinear' - the non-linear law given by
    \[\begin{split}\begin{split} \frac{I\left(\mu\right)}{I\left(1\right)} = 1 & - u_{0,\textrm{nl}} \left(1 - \mu^{1/2}\right) - u_{1,\textrm{nl}} \left(1-\mu\right) \\ &- u_{2,\textrm{nl}} \left(1-\mu^{3/2}\right) - u_{3,\textrm{nl}} \left(1-\mu^{2}\right). \end{split}\end{split}\]

where each of \(u_0\), \(u_1\), \(u_2\), and \(u_3\) are the limb-darkening coefficients to be fitted. With the exception of the non-linear law, all of these models are constrained to physically-allowed values by the method in Kipping (2013), which we have extended to include the power-2 law.

LDC Fitting modes

TransitFit offers three modes for LDC fitting, which can be selected using the ld_fit_method argument in run_retrieval().:

  • 'independent'

    This is the traditional approach of fitting LDCs for each filter separately. TransitFit still uses the Kipping parameterisations, but LDTk is not used to couple LDCs across filters.

  • 'coupled'

    Using the Kipping parameterisations, each LDC is fitted as a free parameter, with LDTk being used to estimate the likelihood of sets of LDCs, using information on the host star and the observation filters. To use the coupled mode, the filter response file should have atleast 4 individual datapoints. TransitFit also provides the functionality to use the uncertainty multiplier from LDTk. This can be specified by adding ‘ldtk_uncertainty_multiplier’ while calling run_retrieval().

  • 'single'

    When fitting with multiple wavebands, the number of parameters required to be fitted can increase dramatically. The 'single' LDC fitting mode freely fitting LDC for only one filter, and uses LDTk to extrapolate LDC values for the remaining filters. The \(i\)-th coefficient of a filter \(f\), is calculated as

    \[c_{i, f} = u_i \times \frac{\langle c_{i, f}\rangle}{\langle u_{i}\rangle}\]

    where \(u_i\) is the sampled value of the \(i\)-th LDC in the actively fitted filter, and \(\langle c_{i, f}\rangle\) and \(\langle u_{i}\rangle\) are the maximum likelihood values initially suggested by LDTk.