Source-linked AI summary
LDTk: Limb Darkening Toolkit
Hannu Parviainen, Suzanne Aigrain
TL;DR
Limb-darkening treatment can bias and over-narrow astrophysical inferences, while custom narrow passbands require profiles beyond standard tabulations. LDTK uses PHOENIX spectra to generate uncertainty-aware profiles and coefficients, and integrates them into model posteriors. The package supports soft, profile-based constraints across transit, eclipsing-binary, microlensing, and interferometric analyses.
Problem
Fixed or unconstrained limb darkening can bias estimates, underestimate uncertainties, or overwhelm narrow-band transmission-spectrum variations.
Method
LDTK generates custom passband profiles from PHOENIX spectra, propagates stellar-parameter uncertainties, estimates model coefficients, and integrates profile constraints into posterior calculations.
Results
LDTK provides uncertainty-aware profiles and coefficients for freely defined passbands, with direct integration into existing modelling codes.
Takeaways & Limitations
The package enables soft stellar-model constraints, coefficient-prior construction, direct profile likelihoods, and direct use of sampled profiles.
Takeaways & Limitations
Profile sampling and stellar-edge definitions affect coefficient estimates, limiting direct comparison with previous tabulations unless settings are matched.
Abstract
from arXiv · showhide
We present a Python package LDTk that automates the calculation of custom stellar limb darkening (LD) profiles and model-specific limb darkening coefficients (LDC) using the library of PHOENIX-generated specific intensity spectra by Husser et al. (2013). The aim of the package is to facilitate analyses requiring custom generated limb darkening profiles, such as the studies of exoplanet transits--especially transmission spectroscopy, where the transit modelling is carried out for custom narrow passbands--eclipsing binaries (EBs), interferometry, and microlensing events. First, LDTk can be used to compute custom limb darkening profiles with uncertainties propagated from the uncertainties in the stellar parameter estimates. Second, LDTk can be used to estimate the limb-darkening-model specific coefficients with uncertainties for the most common limb-darkening models. Third, LDTk can be directly integrated into the log posterior computation of any pre-existing modelling code with minimal modifications. The last approach can be used to constrain the LD model parameter space directly by the LD profile, allowing for the marginalization over the LD parameter space without the need to approximate the constraint from the LD profile using a prior.
1 INTRODUCTION
Fixed limb-darkening coefficients can bias parameter estimates and underestimate uncertainties, especially when stellar models or passbands do not match the analysis. LDTK addresses this by generating passband-specific profiles, propagating stellar-parameter uncertainties, and supporting soft or direct profile-based constraints.
- Stellar limb darkening is a major uncertainty source in exoplanet transits, interferometric radius estimates, and microlensing analyses.
- Fixed coefficients can bias parameter estimates when limb-darkening models fail to reproduce true stellar intensity profiles.
- Fixed coefficients also underestimate uncertainties because stellar-parameter uncertainties are not propagated into limb-darkening profile uncertainties.
- In transmission spectroscopy, unconstrained limb darkening can overwhelm radius variations, while narrow non-standard passbands prevent using pretabulated coefficients.
- LDTK generates custom passband profiles with propagated uncertainties and uses them for coefficient priors, direct model constraints, or arbitrary-profile analyses.
2 EXAMPLES
The examples show LDTK creating stellar limb-darkening profiles and estimating model coefficients for chosen filters, then integrating profile constraints directly into posterior evaluation. Its workflow supports both likelihood-based and prior-based treatments.
- 2 EXAMPLES: LDTK estimates maximum-likelihood values and full posterior distributions for coefficients of the listed limb-darkening models.
- 2.1 Calculation of model coefficients: The example defines boxcar filters and stellar parameters, downloads uncached PHOENIX spectra, creates profiles, and estimates quadratic coefficients with uncertainties.
- 2.1 Calculation of model coefficients: The example also calculates quadratic coefficients for 19 narrow 15 nm passbands spanning 500–800 nm.
- 2.2.1 Log likelihood evaluation: LDTK can evaluate the limb-darkening log likelihood during posterior sampling or construct a multivariate normal prior during initialization.
- 2.2.1 Log likelihood evaluation: Profiles are created during initialization and saved, after which the model adds their likelihood to the main log-posterior calculation.
3 IMPLEMENTATION
LDTK provides Python tools for generating filter-specific limb darkening profiles from PHOENIX spectra, propagating stellar-parameter uncertainties through Monte Carlo sampling. It also supports interpolation and integration into existing modelling workflows.
- Integration: The package is designed for straightforward integration into existing Python-based modelling code and includes usage and transit-analysis examples.It depends on standard Python modules, NumPy, SciPy, and PyFITS.
- Spectrum preparation: LDTK uses PHOENIX specific intensity spectra spanning 50–2600 nm at 0.1 nm resolution for 78 values of µ.Only the spectra required for a given star are downloaded, with cached files reused.
- Spectrum preparation: LDPSetCreator selects spectra inside a 2nσ-wide cube of the three stellar parameters, with n defaulting to three.Required files are calculated, checked against a local cache, and downloaded when absent.
- Profile construction: Filter-specific profiles combine detector quantum efficiency, filter transmission, and stellar spectra, then are renormalized and placed on a (T_eff, log g, z) interpolation grid.The filter transmission can therefore define custom passbands for profile construction.
- Uncertainty propagation: Monte Carlo sampling draws stellar parameters from a multivariate normal distribution and computes a profile for each sample.The final profile is the sample mean, while its uncertainty is the sample standard deviation.
3.3 Limb darkening coefficient estimation
LDTK estimates coefficients for common limb-darkening laws and can quantify their uncertainties using either default likelihood approximations or MCMC sampling. The resulting covariance information can support use of the estimates as modelling priors.
- Coefficient estimation: LDPSet provides model-specific methods to estimate limb-darkening coefficients and their uncertainties for the laws listed in Table 1.The method naming convention is coeffs_xx, where xx identifies the limb-darkening model.
- Default uncertainty estimates: By default, coefficient estimates are maximum-likelihood values with 68% central likelihood intervals assuming zero coefficient correlation.These defaults provide a multivariate-normal approximation to the coefficient likelihood.
- Default uncertainty estimates: The likelihood-based standard deviation is obtained from the curvature of the log likelihood at the maximum-likelihood estimate using a numerical derivative.The equation defines P as likelihood density and x_0 as the maximum-likelihood estimate.
- MCMC estimation: MCMC sampling provides more robust uncertainty estimates and allows covariance matrices to be estimated for use in modelling priors.The chain can be tuned through iteration count, thinning, and burn-in settings, then retained for convergence analysis.
3.4 Log likelihood evaluation
LDTK evaluates joint likelihoods for supplied limb-darkening coefficients by comparing model profiles with profile values and their uncertainties across datapoints and filters.
- Likelihood evaluation: The lnlike_xx methods evaluate model likelihoods for a supplied set of limb-darkening coefficients.For quadratic models, the joint likelihood can use one coefficient pair per filter.
- Likelihood assumptions: The likelihood assumes normally distributed profile values at each µ, yielding the usual Gaussian log-likelihood form.The calculation compares observed profile values P_i with model values M_i.
- Likelihood assumptions: The likelihood uses N datapoints, profile uncertainties σ_i, profile values P_i, model values M_i, and an uncertainty multiplier ϵ.The multiplier scales the uncertainties according to trust in the stellar spectrum models.
3.5 Uncertainty multiplier
The uncertainty multiplier ϵ controls how strongly the generated limb-darkening profile constrains the final analysis by representing trust in the stellar atmosphere models.
- Uncertainty multiplier: ϵ is a subjective factor that increases profile uncertainties when the stellar atmosphere models are trusted less.It is applied both to log-likelihood calculations and to model-coefficient uncertainty estimation.
3.6 Resampling
LDTk supports resampling limb darkening profiles because the original μ sampling emphasizes the stellar edge and sparsely samples the disk interior.
- The Husser et al. (2013) μ sampling resolves the stellar edge in detail but sparsely samples the inner stellar disk.This can cause likelihood evaluation and model fitting to be dominated by the stellar edge unless sample weighting is used.
- LDPSet can resample profiles onto arbitrary μ or z vectors, including linear sampling in μ or z.The package provides resample(mu=mu_array,z=None), resample_linear_mu(nsamples), and resample_linear_z(nsamples).
3.7 Filters
LDTk represents each filter as a passband transmission function and includes utilities for boxcar and tabulated transmission curves.
- A filter returns transmission values between zero and one for an array of wavelengths in nm.Filters may be implemented as functions or callable classes.
- LDTk includes BoxcarFilter and TabulatedFilter utilities for defining passbands.TabulatedFilter uses transmission values listed as a function of wavelength.
3.8 Quantum efficiency
LDTk allows instrumental quantum efficiency to be supplied as a wavelength-dependent function or tabulated curve.
- Any function evaluating quantum efficiency as a function of wavelength can define the instrumental quantum efficiency.The interface parallels the package's filter representation.
- LDPSetCreator accepts a detector quantum-efficiency curve, and TabulatedQE provides a tabulated implementation.TabulatedQE can be used in the same way as TabulatedFilter.
4 DISCUSSION
Stellar spectrum libraries enable custom-passband limb darkening profiles with propagated stellar-parameter uncertainties, supporting more advanced analyses. LDTK profiles can differ substantially from earlier tabulations because geometry and sampling choices affect the fitted coefficients, especially in blue passbands.
- 4 DISCUSSION: Current stellar spectrum libraries can calculate limb darkening profiles for custom passbands while propagating stellar-parameter uncertainties.The discussion argues that analyses should use this capability when integrating stellar-atmosphere information.
- 4 DISCUSSION: LDTK coefficients cannot be directly compared with earlier tabulations because geometry and sampling differ, including stellar-edge redefinition and uniform z-space sampling.Both choices can be changed by the user.
- 4 DISCUSSION: Claret et al. (2013) fits agree with LDTK near the stellar limb but deviate across much of the stellar disk in blue passbands.The comparison used g′, r′, i′, and z′ profiles for a star with Teff = 6500 ± 50 K, log g = 4.5 ± 0.1, and z = 0.0 ± 0.05.
5 CONCLUSIONS
LDTK incorporates stellar-atmosphere information to softly constrain limb darkening while supporting flexible passbands, model coefficients, uncertainties, and direct integration into modelling codes.
- LDTK uses researcher-defined trust in stellar atmosphere models to set the strength of the limb-darkening constraint.
- The package calculates limb darkening coefficients and uncertainties for freely defined passbands before modelling.
- LDTK can be integrated directly into modelling code or generate limb darkening profile samples for arbitrary-profile analyses.
- Arbitrary profiles are useful when stellar features poorly captured by simple models, especially near the star’s edge, affect the analysis.