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Short answer: Use the quadratic law as the usual practical choice for an exoplanet-transit model when the light curve cannot independently constrain many coefficients. Use the Claret four-parameter law when atmosphere-model fidelity matters and its coefficients can be fixed or strongly constrained externally. Treat the linear law mainly as a simple approximation or pedagogical baseline: its one coefficient cannot reproduce the curvature seen in realistic stellar-atmosphere profiles.
The correct choice depends on the observing passband, stellar-atmosphere grid, transit geometry, and whether the coefficients are fitted from the data or supplied by models. A law with more coefficients is not automatically better if the light curve does not contain enough information to identify them.
What limb darkening describes
A star usually appears brighter at the center of its visible disk than near its apparent edge, or limb. This center-to-limb decline is called limb darkening. In transit modeling, it determines how much stellar intensity the planet blocks as it crosses different parts of the disk, so it affects the transit shape and the inferred planet-to-star radius ratio, impact parameter, and related quantities.
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A limb-darkening law is therefore a compact mathematical approximation to a tabulated atmosphere-model intensity profile. The coefficients are not universal properties of a star: they change with wavelength or filter, effective temperature, surface gravity, metallicity, atmospheric geometry, intensity normalization, and the fitting procedure used to derive them.
The three commonly compared laws
Linear law: one coefficient
The linear law is
I(μ)/I(1) = 1 − u(1 − μ)
Here u is the sole coefficient. Increasing u steepens the decline from disk center to the limb. The law is fast to evaluate and easy to interpret, which makes it useful for demonstrations, rough checks, and models where computational simplicity is more important than profile fidelity.
Its limitation is structural: a straight dependence on μ has only one degree of freedom. Realistic atmosphere profiles generally show curvature and more complicated behavior near μ = 0, so one coefficient cannot match both the central part of the disk and the edge accurately. The comparative literature describes the linear form as a rather poor representation of detailed numerical atmosphere intensities.
Quadratic law: two coefficients
The quadratic law adds a term in (1 − μ)2:
I(μ)/I(1) = 1 − a(1 − μ) − b(1 − μ)2
The same parameters are often called u1 and u2. The first coefficient controls the broad slope and the second captures curvature, giving the model substantially more shape freedom than the linear law while retaining only two fitted quantities.
Quadratic limb darkening is popular in analytic transit calculations because it is inexpensive, supported by many light-curve codes, and small enough to constrain or regularize in typical data sets. It is still an approximation. If a passband or stellar atmosphere has strong center-to-limb structure, forcing the true profile into a quadratic shape can leave systematic residuals and bias transit parameters.
Claret four-parameter law
The four-parameter law, commonly associated with Claret, is
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I(μ)/I(1) = 1 − Σj=14 aj[1 − μj/2]
Its four half-power terms provide much more flexibility, especially near the limb. The coefficients a1 through a4 are normally obtained by fitting an atmosphere-grid intensity profile or by adopting tabulated values calculated for the target star and passband.
Among the common laws compared in the MPS-ATLAS review, this form gives the highest fidelity for detailed profiles. Its extra flexibility is valuable when the profile itself is the scientific object or when transit parameters must be protected from model mismatch. The trade-off is identifiability: ordinary transit photometry rarely contains enough independent information to estimate four free limb-darkening coefficients reliably at the same time as the planet and orbital parameters.
How the laws compare in practice
| Law | Free coefficients | Profile fidelity | Typical transit use | Main limitation |
|---|---|---|---|---|
| Linear | 1 | Lowest; misses realistic curvature and detailed edge behavior | Simple demonstrations, rough checks, or deliberately minimal models | One coefficient cannot represent the shape of numerical atmosphere profiles |
| Quadratic | 2 | Intermediate; captures broad slope and curvature | Practical default for constrained transit fits and fast analytic calculations | Can still produce systematic errors when treated as an exact profile |
| Claret four-parameter | 4 | Highest of these three for detailed atmosphere profiles | Use with externally supplied or tightly constrained coefficients when fidelity is important | Four coefficients are usually poorly identifiable from ordinary transit photometry alone |
Parameter count is therefore not a simple quality ranking. Four coefficients can reduce approximation error while increasing statistical uncertainty and degeneracy. A quadratic fit with well-chosen external information can produce more stable planetary parameters than a four-parameter fit whose coefficients are allowed to wander without informative constraints.
What accuracy improvements have been reported
A 2011 review by John Southworth states that the four-coefficient form reproduces model-atmosphere intensities to approximately one part in 1,000 over a wide range of parameter space. That figure describes the agreement with the cited model profiles, not a guaranteed error for every star, filter, atmosphere family, or transit analysis.
Ian D. Howarth’s 2011 model-grid comparison found that the four-coefficient law improved precision by up to an order of magnitude over standard two-coefficient options. The exact improvement depends on how intensities are normalized and which goodness-of-fit metric is used, so the result should be read as an upper-end comparison rather than a universal factor.
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Why transit geometry changes the fitted coefficients
Coefficients fitted directly to a transit light curve are not automatically the same quantities as coefficients fitted to every point in a stellar-atmosphere intensity grid. A planet samples a chord, not the entire stellar disk. The impact parameter determines whether that chord passes near the bright center, through intermediate radii, or close to the limb.
Because different chords weight different portions of the profile, two transits of the same star observed with different impact parameters can prefer different effective photometric coefficients. Cadence, noise, unocculted spots, finite integration time, and correlations with the radius ratio and orbital geometry add further sensitivity. This is why a coefficient pair inferred from one light curve should not be treated as a universal stellar constant.
Morello and collaborators reported that at least 26 percent of planet candidates and almost half of confirmed exoplanets in their studied analyses had significant systematic errors associated with the treatment examined. Those percentages are specific to that study and sample; they are not a universal error rate for all current surveys. They do illustrate why limb-darkening assumptions deserve explicit testing rather than being hidden as an innocuous implementation detail.
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Choosing a law for an exoplanet transit
Choose the linear law only for a deliberately minimal model
A linear law is defensible when the goal is a quick sanity check, a teaching example, or a deliberately low-dimensional model whose limitations are accepted. It is a poor choice when precision radius measurements, grazing transits, or detailed residual analysis are central, because the missing curvature can be absorbed by other transit parameters.
Choose the quadratic law for most constrained practical fits
Use the quadratic form when the data quality and science case favor a small number of parameters. You can fix both coefficients to atmosphere-model values, impose informative priors, or float them with suitable physical constraints when the light curve has enough information. The two-parameter structure is often the best compromise between computational convenience, interpretability, and resistance to overfitting.
Choose four parameters when fidelity outweighs parsimony
Use the four-parameter form when the intensity profile itself must be represented closely, when multiple high-quality transits provide strong leverage, or when coefficients are supplied by an atmosphere calculation and are fixed or tightly constrained. Floating all four coefficients without external information is usually impractical for ordinary ground- or space-based transit photometry.
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Consider the scientific risk of mismatch
If the target is a grazing system, the transit chord lies near the limb, or the desired radius precision is high, profile mismatch can have a larger effect than in a central, high signal-to-noise transit. In those cases, compare fits with alternative coefficient treatments and inspect whether the inferred radius ratio, impact parameter, or residual structure changes materially.
A reproducible coefficient-selection workflow
- Define the observing band. Select the exact filter or passband used for the light curve. Limb darkening is wavelength-dependent, so coefficients from a visually similar but different band are not interchangeable.
- Choose an atmosphere grid. Match the star’s effective temperature, surface gravity, metallicity, and any relevant geometry assumptions. Decide whether a plane-parallel or spherical model is appropriate for the target and analysis.
- Obtain or fit coefficients. Use tabulated coefficients or fit the selected law to the model intensities with a documented procedure. The resulting values depend on the fitting method, the sampled
μpoints, weighting, intensity normalization, and whether the limb points are included. - Match the model dimension to the data. Adopt a quadratic law when the transit cannot identify four coefficients. Use the four-parameter form when profile fidelity is more important and external coefficients or informative priors are available.
- Propagate coefficient uncertainty where possible. If atmosphere parameters are uncertain, vary or marginalize over plausible coefficient values instead of treating a single tabulation as exact. This is especially important when the transit geometry makes the light curve sensitive to the limb.
- Check geometry-dependent behavior. Fit or simulate the actual impact parameter and exposure integration. Confirm that changes in limb-darkening treatment do not produce an unexplained shift in the radius ratio or other reported quantities.
- Document every assumption. Record the law, coefficient values or priors, coefficient source, passband, atmosphere parameters, geometry choice, fitting method, and whether coefficients were fixed or floated.
Coefficient fitting and identifiability
Fitting coefficients to an atmosphere grid is a different operation from fitting them to a transit. The grid provides intensities at many viewing angles, while a transit light curve is a time series generated by a moving occultor and convolved with the observing cadence. A least-squares fit to grid intensities can therefore yield coefficients that do not reproduce the coefficients preferred by a particular transit geometry.
Four parameters also create covariance with the radius ratio, impact parameter, and sometimes the baseline or stellar variability model. If the posterior allows many combinations of coefficients and transit parameters to produce nearly identical light curves, the extra flexibility increases uncertainty rather than revealing four independently measured physical quantities.
Useful safeguards include informative priors from an atmosphere grid, reparameterizations that reduce coefficient covariance, fitting multiple passbands jointly with separate coefficient sets, and sensitivity tests in which coefficients are fixed, constrained, and partially or fully floated. The selected treatment should follow the information content of the data, not a preference for the largest formula.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Passband and atmosphere assumptions that must travel with every coefficient
- Passband: State the filter response or instrument band. A coefficient without its wavelength context is incomplete.
- Stellar parameters: Report the effective temperature, surface gravity, metallicity, and any other atmosphere inputs used to generate the profile.
- Geometry: Identify whether the atmosphere calculation is plane-parallel or spherical and whether that choice is appropriate for the star.
- Normalization: Explain how intensities were normalized and whether the disk-center value or another convention was used.
- Fitting method: State the weighting, angular sampling, limb-point treatment, and goodness-of-fit definition used to derive the coefficients.
- Statistical treatment: Say whether coefficients were fixed, assigned priors, jointly fitted, or independently fitted for each light curve.
Common failure modes and fixes
Using coefficients from the wrong filter
Symptom: The model appears smooth, but residuals are structured or the inferred radius changes when the passband is changed.
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Fix: Recalculate or retrieve coefficients for the exact bandpass and document the instrument response used.
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Floating four coefficients in a low-information light curve
Symptom: Broad, correlated posteriors, unstable coefficients between visits, or large changes in planetary parameters with little improvement in residuals.
Fix: Fix or constrain the coefficients with an atmosphere model, reduce the model to quadratic, or combine additional high-quality transits before increasing dimensionality.
Assuming atmosphere-grid coefficients are geometry-free
Symptom: Coefficients fitted to one transit do not transfer to another transit of the same star.
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Reporting a law without its provenance
Symptom: Another analyst cannot reproduce the light curve or determine whether a discrepancy comes from the atmosphere model or the transit implementation.
Fix: Publish the equation, coefficient values or priors, passband, atmosphere assumptions, normalization, and fitting procedure together.
What to report in a paper or data release
- The exact limb-darkening equation and coefficient notation.
- Whether the coefficients were fixed, constrained with priors, or fitted freely.
- The coefficient source or atmosphere-grid family and its stellar parameters.
- The passband and instrument response.
- The intensity normalization and coefficient-fitting method.
- The transit geometry assumptions and exposure-time integration treatment.
- A sensitivity comparison showing whether the chosen law changes the reported radius ratio or other key parameters.
Following this checklist makes a two-coefficient model auditable rather than arbitrary and makes a four-coefficient model defensible rather than merely elaborate.
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Bottom line for model selection
The linear law is the fastest but least faithful approximation. The quadratic law is the practical middle ground for most constrained transit analyses. The Claret four-parameter law offers the closest representation of detailed model-atmosphere profiles among these choices—about 1 part in 1,000 in the comparison summarized by Southworth (2011), and up to an order-of-magnitude precision improvement over standard two-coefficient options in Howarth’s 2011 comparison—but its four coefficients generally require external constraints. Select the law that matches both the required profile fidelity and the information content of the data, then report enough passband, atmosphere, geometry, and fitting detail for the result to be reproduced.
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