The linear, quadratic, and Claret four-parameter laws are progressively more flexible formulas for describing how a star’s specific intensity falls from the disk center to its apparent edge. The linear law is simplest but usually least faithful; the quadratic law is the practical default for many transit fits; and the four-parameter law can match atmosphere-model profiles far more closely, provided its coefficients are supplied or constrained externally.
There is no universally best law. The right choice depends on passband, stellar-atmosphere assumptions, transit geometry, data quality, and whether the coefficients are fixed, constrained, or fitted freely.
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What limb darkening measures
A resolved stellar disk is not equally bright everywhere. The center is viewed along a line of sight that passes through a relatively deep, hotter layer of the atmosphere. Toward the apparent edge, or limb, the line of sight samples higher and generally cooler layers, so the measured specific intensity decreases.
Limb-darkening laws approximate that center-to-limb intensity profile with a small number of coefficients. Define μ = cos γ, where γ is the angle between the outward stellar-surface normal and the observer’s line of sight. At disk center, γ = 0 and μ = 1. At the idealized edge of the projected disk, γ approaches 90 degrees and μ approaches 0.
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Most published laws normalize intensity to disk-center intensity, I(1). The quantity being modeled is therefore I(μ)/I(1)), not an absolute flux. A coefficient set is meaningful only together with the passband, atmosphere model, geometry, normalization, and fitting method used to obtain it.
The three laws at a glance
| Law | Equation | Free coefficients | Typical role | Main limitation |
|---|---|---|---|---|
| Linear | I(μ)/I(1) = 1 − u(1 − μ) |
1 | Fast, compact exploratory models | Cannot reproduce realistic curvature well |
| Quadratic | I(μ)/I(1) = 1 − a(1 − μ) − b(1 − μ)² |
2 | Constrained transit analyses and analytic light curves | Still approximates detailed atmosphere profiles |
| Claret four-parameter | I(μ)/I(1) = 1 − Σ(j=1 to 4) aj[1 − μj/2] |
4 | High-fidelity profiles with external coefficient information | Four free coefficients are difficult to identify from ordinary transit photometry |
Quadratic coefficients are also commonly written as u1 and u2 instead of a and b. The notation changes, but the two-coefficient functional form is the same.
How each limb-darkening law behaves
Linear law
The linear expression, I(μ)/I(1) = 1 − u(1 − μ), has one degree of freedom. Increasing u makes the intensity decline more strongly toward the limb; reducing it produces a flatter profile.
Its appeal is computational and statistical simplicity. A light-curve calculation needs only one coefficient, and a fit has fewer opportunities for parameter covariance. That simplicity is also the problem: a straight dependence on μ has no independent term for curvature. Real atmosphere profiles generally bend across the disk and can change slope near μ ≈ 0, behavior the linear law cannot reproduce.
The literature therefore characterizes the linear law as a rather poor representation of detailed numerical atmosphere intensities. It can still be useful for quick sensitivity checks, low-information data, or pedagogical calculations, but a good fit to a light curve does not prove that the underlying intensity profile is physically accurate.
Quadratic law
The quadratic law adds a second shape parameter:
I(μ)/I(1) = 1 − a(1 − μ) − b(1 − μ)²
The first term controls the broad center-to-limb slope, while the squared term permits curvature. That extra flexibility usually follows model-atmosphere profiles much better than a straight line and remains inexpensive to evaluate in analytic transit models.
Quadratic is often the practical middle ground because two coefficients can sometimes be constrained, or at least usefully checked, with high-quality transit data. It is still not an exact description of a complex atmosphere. If the adopted coefficients are wrong for the star or passband, the resulting mismatch can be absorbed by the planet-to-star radius ratio, impact parameter, or other transit parameters.
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Claret four-parameter law
The Claret form uses four half-power terms:
I(μ)/I(1) = 1 − a1(1 − μ1/2) − a2(1 − μ) − a3(1 − μ3/2) − a4(1 − μ²)
Equivalently, it is written as I(μ)/I(1) = 1 − Σ(j=1 to 4) aj[1 − μj/2]. The half-power terms provide substantially more freedom to reproduce both smooth curvature and detailed behavior close to the limb.
That flexibility is valuable when coefficients come from a suitable atmosphere grid or are otherwise constrained. It is usually not sensible to float all four coefficients independently in an ordinary transit light curve: the data sample only the stellar chord crossed by the planet, and the coefficients can be strongly correlated with one another and with geometric parameters.
Accuracy: what the published comparisons actually show
Accuracy should be stated relative to a specified atmosphere profile, passband, normalization, and goodness-of-fit metric. A ranking measured in one grid or filter is not a universal ranking for every star.
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A free scan shows the junk files, broken settings and background clutter dragging Windows down - then fixes them in one click.Free scan · Windows 10 & 11- A 2011 review by Southworth reports that the four-parameter form reproduces model-atmosphere intensities to approximately 1 part in 1000 over a wide range of parameter space. This is a statement about comparison with the referenced model atmospheres, not a guaranteed error for every observed star.
- 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 normalization and the goodness-of-fit measure.
- The same comparative literature describes the linear law as a poor representation of detailed numerical results and the quadratic law as better but still approximate.
These results establish a fidelity-versus-complexity trade-off rather than a rule that four coefficients must always be used. A profile can be represented extremely well while the corresponding coefficients remain unidentifiable from the available photometry.
Why transit geometry changes the answer
A transit does not measure every point on the stellar disk equally. The planet blocks a moving chord whose location is set largely by the impact parameter. A near-central transit samples a different range of μ values from a grazing or high-impact-parameter transit.
Consequently, coefficients inferred directly from a transit light curve are not automatically the same quantities as coefficients obtained by fitting a complete stellar-atmosphere intensity grid. The transit fit is weighted by the chord, cadence, noise, stellar activity, and the correlations among radius ratio, impact parameter, inclination, and limb darkening.
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This distinction matters because a law can provide an acceptable local approximation along the observed chord while being poor elsewhere on the disk. Conversely, forcing a globally accurate four-parameter profile into a data set that samples only a narrow chord can create unconstrained coefficients and unstable posteriors.
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Which law should you use?
Choose linear only for deliberately simple models
Use the linear law when one coefficient is an explicit priority: rapid exploratory calculations, educational demonstrations, rough sensitivity tests, or data too weak to support even a two-parameter shape. Treat the result as a simplified approximation, especially if the inferred planet radius or impact parameter is scientifically important.
Choose quadratic for most constrained transit analyses
Quadratic is a sensible default when you need a compact analytic light curve and have realistic atmosphere-based priors or coefficient estimates. It captures curvature without introducing four highly correlated free parameters. For many data sets, the best practice is to place informative priors on the two coefficients, or to fit transformed combinations that have better-behaved physical bounds, rather than allowing an unconstrained pair to wander.
Choose four-parameter when profile fidelity is the priority
Use the Claret law when detailed center-to-limb structure matters, the passband and stellar parameters are well characterized, and coefficients can be fixed or tightly constrained from atmosphere calculations. It is especially attractive for precision work in which small profile mismatches could bias radius, geometry, or atmospheric inferences.
Do not interpret “four-parameter” as “four coefficients should be freely estimated from the transit.” In many cases the statistically safer implementation is to obtain the four coefficients from an appropriate grid, propagate uncertainty in the stellar parameters, and test sensitivity to plausible coefficient variations.
A reproducible coefficient workflow
- Specify the star and observation. Record effective temperature, surface gravity, metallicity, stellar geometry assumptions, and the exact observing passband. Coefficients are not portable between filters or arbitrary atmosphere models.
- Select an atmosphere grid. Use a grid whose parameter range and geometry match the target as closely as possible. Decide whether the model is plane-parallel or spherical and document that choice.
- Obtain or fit coefficients. Use tabulated coefficients when available, or fit the selected law to the tabulated intensity profile. State the sampling in μ, the weighting, the treatment of the limb, the normalization, and whether flux conservation or another constraint was imposed.
- Match the law to the data. For ordinary, moderate-quality transit photometry, start with a constrained quadratic model. For high-precision data or analyses sensitive to subtle profile structure, use a four-parameter profile with external constraints.
- Propagate uncertainty. Vary stellar parameters and coefficient priors within credible ranges. Check whether the posterior for radius ratio, impact parameter, and related quantities changes materially.
- Validate geometry. Compare the transit chord’s μ range with the region where the chosen law is most accurate. A globally excellent fit can still be weakly tested if the observed chord covers little of the disk.
- Run sensitivity checks. Refit with a defensible alternative law or coefficient prior. A large shift in the scientific parameter of interest signals that limb-darkening assumptions are influential and should be reported.
Fitting methods and identifiability
Atmosphere-grid fitting
In atmosphere-grid fitting, the law is evaluated against intensities calculated at several μ values. The resulting coefficients describe the adopted model atmosphere under the chosen fitting convention. Changing μ sampling, weights, normalization, or the excluded region near the extreme limb can change the coefficients even when the underlying atmosphere grid is unchanged.
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Transit-light-curve fitting
In photometric fitting, the coefficients are inferred indirectly through the shape of ingress, egress, and the in-transit flux profile. Noise, finite cadence, stellar variability, and geometric correlations reduce the information available. Four coefficients generally require informative external constraints; otherwise the posterior may be broad, multimodal, or apparently precise only because of imposed parameter boundaries.
Fixed versus fitted coefficients
Fixing coefficients reduces variance but can introduce bias if the atmosphere assumptions are wrong. Floating them allows the data to reveal mismatch but can inflate uncertainties and transfer profile errors into transit parameters. A middle approach is to impose priors centered on atmosphere predictions and include uncertainty broad enough to reflect stellar-model and fitting-method limitations.
Common failure modes and fixes
Using coefficients from the wrong passband
Symptom: residuals show structured ingress or egress deviations, or the inferred radius changes when the filter changes.
Fix: obtain coefficients for the actual transmission band, including the instrument response when appropriate. Do not reuse a visually similar filter as if it were equivalent.
Mixing coefficient conventions
Symptom: a published pair is inserted into a model that expects a different notation or normalization.
Fix: verify whether the source uses a,b, u1,u2, or another reparameterization, and confirm that intensities are normalized to I(1).
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Floating too many coefficients
Symptom: four coefficients have large correlations, unphysical profiles, or unstable transit parameters.
Fix: constrain them with atmosphere-based priors, reduce the model to quadratic, or use a data-informed parameterization. More coefficients do not create information that the light curve does not contain.
Trusting a low residual without checking parameters
Symptom: the light curve appears well fit, but radius ratio or impact parameter differs substantially between limb-darkening choices.
Fix: compare posterior shifts and physically relevant derived quantities, not only the aggregate residual or likelihood.
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Ignoring the extreme limb
Symptom: a fit looks acceptable over most μ values but fails near μ ≈ 0, affecting grazing transits most strongly.
Fix: inspect the profile as a function of μ, use a law validated for the relevant limb region, and account for the transit geometry when selecting the model.
What to report so another analyst can reproduce the result
- The exact limb-darkening law and equation used.
- Coefficient values, uncertainties, and whether they were fixed, fitted, or assigned priors.
- The coefficient source, including atmosphere-grid family and version when applicable.
- Passband definition and any instrument-throughput treatment.
- Stellar effective temperature, surface gravity, metallicity, and geometry assumptions.
- The μ sampling, weighting, normalization, and fitting method used to derive coefficients.
- Transit geometry and whether sensitivity tests with another law or coefficient set changed the reported parameters.
Without these details, two analyses can cite the same named law yet use materially different coefficient definitions.
A practical decision framework
| Situation | Preferred starting point | Reason | Watch for |
|---|---|---|---|
| Teaching, code checks, or rough estimates | Linear | One parameter and minimal computation | Systematic profile mismatch |
| Routine transit modeling with limited or moderate signal-to-noise | Constrained quadratic | Captures curvature while remaining identifiable | Bias from incorrect passband or stellar assumptions |
| High-precision transit with reliable stellar characterization | Four-parameter, externally constrained | Best available shape flexibility | Coefficient correlations and model-grid uncertainty |
| Grazing or high-impact-parameter transit | Quadratic or four-parameter after profile inspection | More sensitivity to the limb region | Extreme-limb mismatch affecting geometry |
The central decision is not simply “one, two, or four coefficients.” It is whether the information in the data supports the chosen number of free parameters and whether the coefficients describe the same star, passband, and fitting convention as the observation.
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