Quadratic Limb Darkening LawEdit
The quadratic limb darkening law is a practical, widely used description of how a star’s brightness diminishes from its center toward its edge. It is a two-parameter, analytic expression that captures the way light from different parts of a stellar disk contributes to observed flux during events such as transits of exoplanets or eclipses in binary systems. The law is written in terms of the angular dependence of intensity, often formulated with μ = cos θ, where θ is the angle between the line of sight and the local surface normal. In its standard form, the relative intensity is given by I(μ)/I(1) = 1 − a(1 − μ) − b(1 − μ)², where a and b are the limb-darkening coefficients. This compact formula sits within the broader topic of limb darkening, a consequence of the temperature and opacity structure of stellar atmospheres that makes the disk appear brighter at the center than at the edge. limb darkening stellar atmosphere
As a phenomenological description, the quadratic law serves as a middle ground between the simplicity of linear models and the more complex, higher-order parameterizations. It reduces to the linear limb-darkening law when b is set to zero, and it provides a better fit to many observed light curves than a purely linear term alone. The two coefficients encode how rapidly brightness falls off toward the limb and how the curvature of that falloff behaves. Because the coefficients depend on wavelength, they often vary across photometric bands and with the star’s effective temperature, surface gravity, and metallicity. In practice, the law is used in models of transit photometry and eclipsing binary light curves, where the goal is to infer stellar properties and, in the case of exoplanets, the planet’s radius relative to the star. Coefficients are sometimes drawn from grids computed from stellar atmosphere models such as ATLAS or PHOENIX, and other times are treated as free parameters to be fitted to the data with appropriate priors. exoplanet light curve
Quadratic limb darkening law
Formula and interpretation
The central quantity is the specific intensity distribution across the stellar disk, I(μ). The quadratic law provides a simple functional form to approximate that distribution with two adjustable parameters, a and b. The term (1 − μ) describes how far a given line of sight is from the center of the disk, and the coefficients determine how strongly brightness changes with μ. The coefficients are typically constrained to keep the intensity nonnegative for all μ in [0, 1], though in practice this condition is enforced through priors or direct bounds during fitting. The simplicity of the two-parameter form makes it convenient for analytic light-curve calculations and for exploring how changes in limb darkening influence inferred parameters such as the planet-to-star radius ratio. limb darkening transit photometry light curve
Coefficients and wavelength dependence
a and b are not universal constants; they vary with wavelength and with stellar properties. In optical bands for Sun-like stars, typical ranges reflect the broader behavior of limb darkening across spectral types, with coefficients that capture both the depth and curvature of the brightness profile. When coefficients are taken from model grids, they inherit dependencies on Teff, log g, and metallicity, and they must be mapped to the observational bandpass used in a given study. Because real instruments integrate over a filter transmission, the effective limb-darkening coefficients arise from band-averaged intensities, which can differ from monochromatic values. stellar atmosphere ATLAS PHOENIX
Practical use in transit modeling
In fitting transit or eclipse light curves, the quadratic law offers a balance between physical realism and parameter identifiability. The planet-to-star radius ratio, a key quantity, is partially degenerate with the limb-darkening coefficients; data quality, sampling, and wavelength coverage determine whether a and b can be well constrained or must be fixed from theory with priors. In multi-band observations, adopting band-dependent coefficients helps reduce systematic biases and improves the consistency of radius determinations across wavelengths. The quadratic form remains popular precisely because it is computationally light and sufficiently flexible for many practical datasets. exoplanet transit photometry light curve
Alternatives and debates
The choice of limb-darkening law is a point of ongoing methodological discussion. Alternatives include the linear limb-darkening law (one coefficient) and the non-linear, four-parameter law (often associated with Claret), which can more faithfully reproduce detailed intensity profiles from stellar atmosphere models. Some researchers argue for fixed coefficients drawn from model grids, while others advocate fitting coefficients directly from high-quality data to capture star-specific physics and unmodeled instrumental effects. The quadratic law is widely used because it typically provides a good compromise between model fidelity and the risk of overfitting, especially when data are not or cannot be perfectly precise. In cases of very high signal-to-noise or very precise spectrophotometric data, more complex laws may be favored to minimize biases in derived parameters. linear limb-darkening law non-linear limb-darkening law Claret coefficients transit photometry
Historical context
The two-parameter, quadratic form gained prominence in the exoplanet era as transit photometry matured into a precision discipline. It offered analytic tractability for light-curve models while capturing essential curvature in the stellar intensity profile. The approach has been standard in many analytic and numerical treatments of transit light curves, often implemented alongside grids of limb-darkening coefficients computed from stellar atmosphere models. The broader family of limb-darkening laws, including the linear and four-parameter forms, provides a spectrum of complexity that scientists choose from based on the data quality and scientific goals. Mandel & Agol model transit light curve