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Short answer: Generate limb-darkening coefficients for the star’s measured parameters and the exact instrument passband, then use those values as informative, non-zero-width priors or fit them jointly with the transit. Do not copy a coefficient pair from another filter or hard-fix uncertain values. For the quadratic law, sample Kipping’s q1 and q2 variables, integrate the model over each exposure, and propagate coefficient uncertainty into Rp/R* and transit depth.
Limb darkening is the decrease in a star’s observed specific intensity from disk center to its edge. Because a transiting planet blocks different stellar intensities during ingress, mid-transit and egress, the effect changes the light-curve shape and therefore the inferred radius ratio, impact parameter and sometimes other orbital parameters.
What limb darkening changes in a transit
Define μ = cos(γ), where γ is the angle between the line of sight and the direction of emergent stellar intensity. At disk center, μ = 1; near the projected stellar limb, μ approaches zero. A transit model uses the star’s intensity profile, not a uniformly bright disk, when calculating the flux blocked by the planet.
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Limb-darkening coefficients are not universal constants. They depend on effective temperature, surface gravity, metallicity, the adopted stellar-atmosphere grid, the instrument’s wavelength throughput, the sampling of μ points and the method used to fit the analytic law to the atmosphere profile. A pair that is appropriate for one TESS sector, Kepler band or HST filter is not automatically valid for another passband, let alone for a JWST spectroscopic channel.
Choose an intensity law that your data can identify
Use the simplest law that describes the observed profile without leaving structured residuals. More coefficients can reproduce an atmosphere calculation more closely, but they also introduce degeneracies with transit geometry and systematics, especially in sparse or noisy data.
| Law | Definition | Free coefficients | When it is useful | Main limitation |
|---|---|---|---|---|
| Quadratic | I(μ)/I(1) = 1 − u1(1−μ) − u2(1−μ)2 |
2 | Efficient starting point for most broadband transit fits; compact enough for informative priors or joint fitting. | Can leave wavelength-dependent or infrared profile structure in the residuals when the atmosphere is not well represented by a quadratic shape. |
| Square-root | I(μ)/I(1) = 1 − s1(1−μ) − s2(1−√μ) |
2 | Alternative two-parameter description when the center-to-limb curvature is not well captured by the quadratic form. | Still an approximation; its coefficients are not interchangeable with quadratic coefficients. |
| Power-2 | Use the power-2 form implemented by the chosen transit package and document its exponent and coefficient convention. | 2 | Can be effective for profiles with a different curvature pattern, including some infrared data. | Parameter names and physical constraints vary among software packages; do not mix conventions. |
| Nonlinear four-parameter | I(μ)/I(1) = 1 − c1(1−μ0.5) − c2(1−μ) − c3(1−μ1.5) − c4(1−μ2) |
4 | Wavelength-resolved or high-signal-to-noise observations where atmosphere profiles show systematic departures from a two-coefficient law. | Four coefficients are often weakly identified by a single broadband light curve and can broaden or destabilize the posterior. |
In the conditions reported for one atmosphere-profile study, a four-coefficient fit reproduced model intensities to approximately one part in 1,000. That profile fidelity does not guarantee that a transit observation can constrain all four coefficients; identifiability depends on cadence, photometric precision, coverage of ingress and egress and correlations with the transit geometry.
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Generate coefficients for the exact star and passband
1. Adopt a documented stellar solution
Start with the effective temperature (Teff), surface gravity (log g), metallicity and their uncertainties from the stellar analysis used for the planet system. If those values change, the coefficients change as well. NASA Exoplanet Archive guidance states: “If new stellar parameters are adopted, the most-correct approach is to re-fit the transit with new limb-darkening coefficients in order to re-measure planet size.”
2. Use the actual throughput curve
Obtain the measured or versioned response function for the detector, filter and observing mode. Integrate the model-atmosphere intensity at each wavelength against that throughput before fitting a limb-darkening law. A nominal filter name is not enough when detector response, optics, order-sorting filters or time-dependent sensitivity alter the effective bandpass.
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For TESS, Kepler, HST and JWST data, use the response appropriate to the specific instrument mode and data release. A JWST spectroscopic bin, for example, has a different effective passband from a neighboring bin even when both are described informally as “near infrared.” Record the throughput file or version in the analysis metadata.
3. Fit the analytic law with a stated convention
Atmosphere grids provide intensities at discrete μ values. Fit the selected law to those intensities using a documented sampling and weighting rule. Weighting every μ point equally, weighting by projected area, excluding the extreme limb, or using another convention can produce different coefficients from the same atmosphere model. A coefficient table without its atmosphere grid, passband, μ sampling and fitting method is incomplete.
Keep the coefficient convention consistent with the transit code. In particular, verify the sign and ordering of the terms, whether intensities are normalized to I(1), and how the software defines a power-2 or nonlinear law. Never paste values labeled u1 and u2 into a package that expects a different ordering.
Fix, constrain or fit the coefficients?
The decision is a bias-versus-variance trade-off. Hard-fixing an inaccurate profile can bias the radius; allowing several poorly measured coefficients to float can inflate uncertainties or let systematics mimic stellar intensity structure.
| Strategy | How to apply it | Best use | Risk |
|---|---|---|---|
| Hard fixed | Set coefficients to atmosphere predictions with zero uncertainty. | Only when the stellar solution, atmosphere model and passband are exceptionally secure and the data cannot constrain the coefficients. | Any model or passband mismatch is forced into Rp/R*, impact parameter or baseline terms. |
| Informative prior | Center a prior on the atmosphere prediction and give it a width that includes stellar-parameter, model-grid and passband uncertainty. | Default for most broadband or moderate signal-to-noise transits. | An unrealistically narrow prior reproduces the bias of a fixed coefficient; an excessively broad prior may be nearly unconstrained. |
| Joint fit with weak or broad priors | Let coefficients adjust while retaining physical bounds and, where appropriate, atmosphere-based guidance. | High signal-to-noise, well-sampled light curves with clear ingress and egress information or multiple transits. | Strong covariance with impact parameter, radius ratio and correlated noise; posterior may become multimodal or very broad. |
| Free coefficients without informative structure | Fit all coefficients with only physical bounds. | Rarely justified for a single noisy broadband transit. | Weak identifiability and unphysical-looking profiles can produce unstable planet parameters. |
Southworth and colleagues summarized the practical objective this way: “the best strategy to minimize the bias introduced in the transit parameters is to let them float in the transit fitting procedure.” In practice, floating does not require abandoning atmosphere information: informative, non-zero-width priors usually provide a better compromise when the light curve is not powerful enough to determine the profile independently.
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Sample quadratic coefficients with Kipping’s q1 and q2
Directly sampling u1 and u2 often produces strong correlations and proposals that describe a nonphysical intensity profile. Kipping’s transformed variables provide a convenient unit-square parameterization for the quadratic law:
q1 = (u1 + u2)2q2 = u1 / [2(u1 + u2)]
Sample q1 and q2 in the interval (0,1), then recover the quadratic coefficients with:
u1 = 2√q1 q2u2 = √q1 (1 − 2q2)
This transform is the implementation pattern demonstrated in exoplanet documentation. It reduces the coefficient correlation encountered in direct sampling and excludes many combinations that would yield an unphysical quadratic profile. The transform does not remove uncertainty from stellar parameters or make a wrong passband correct; those uncertainties still belong in the priors on q1 and q2 or in a hierarchical model.
When converting atmosphere predictions into priors, transform the predicted u1,u2 distribution into q1,q2 consistently rather than transforming only the central values and guessing independent errors. If the software samples in q1,q2, store both the transformed prior and the recovered physical coefficients in the results.
End-to-end workflow
- Characterize the star and observation. Gather
Teff,log g, metallicity and uncertainties, identify the exact detector/filter or spectroscopic bin, and obtain the corresponding throughput curve. - Select a law. Begin with the quadratic law for compact broadband modeling. Test square-root, power-2 or nonlinear four-parameter laws when wavelength-resolved or infrared data, or residuals from an atmosphere profile, justify the extra flexibility.
- Compute passband intensities. Interpolate the atmosphere grid at the adopted stellar parameters, integrate intensities over the throughput, and document the atmosphere-grid version and interpolation method.
- Fit the selected law. State the
μsampling, limb-point treatment and weighting rule. Export coefficient uncertainties, not only best-fit values. - Choose the sampling coordinates. For a quadratic model, sample
q1,q2in(0,1)and recoveru1,u2with the equations above. Use the equivalent physically constrained parameterization provided by your package for other laws. - Set priors. Propagate stellar and atmosphere uncertainty into finite-width priors. Do not use a delta-function prior unless the assumptions have been demonstrated to be more precise than the transit can test.
- Fit the transit and nuisance terms together. Include baseline or detector-systematics terms, integrate the model over the exposure time, and use a correlated-noise treatment appropriate to the instrument. Examine posterior covariance among limb darkening, impact parameter and radius ratio.
- Validate the result. Inspect ingress and egress residuals, generate posterior-predictive light curves, fit an alternate law, and verify that radius ratio and depth remain stable within the quoted uncertainties.
- Recompute after updates. If stellar parameters, atmosphere grids, throughput files or fitting conventions change, regenerate the coefficients and refit before reporting a new planet size.
Implement the transit model without hidden approximations
Integrate over the exposure, not only at the timestamp
For finite-cadence photometry, evaluate the instantaneous transit model at sub-times within each exposure and average the flux. Long exposures smear ingress and egress; evaluating only at the midpoint can make the apparent shape too sharp and increase the pressure on limb-darkening and geometry parameters to compensate.
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Fit baseline and correlated noise jointly
Detrending terms, stellar variability, pointing changes and detector systematics can have time scales similar to ingress or egress. Fit the baseline or systematics model at the same time as the transit, and use a noise model that matches the instrument. Otherwise, the limb-darkening coefficients may absorb trends that are not stellar-atmosphere information.
Preserve analysis metadata
- Law name and exact coefficient convention.
- Atmosphere grid, version, interpolation and stellar parameters.
- Throughput file, instrument mode and version or calibration date.
μsampling, weighting and any excluded limb points.- Prior centers and widths, including how stellar uncertainty was propagated.
- Sampler coordinates, such as
q1,q2, and the software implementation. - Exposure-time integration and correlated-noise treatment.
How much can limb darkening bias the radius?
The size of the effect depends on signal-to-noise, geometry, passband and how wrong the adopted coefficients are. It is not safe to attach a single universal correction to every transit.
| Study or test | Reported result | Interpretation |
|---|---|---|
| Espinoza & Jordán (2015) | Up to approximately 3% bias in Rp/R* when quadratic coefficients were fixed; approximately 1% in some cases when coefficients were fitted. |
Allowing coefficients to adjust can reduce, but does not guarantee elimination of, radius bias. |
| Yang et al. (2021) | Up to 0.82% (2.5σ) Rp/R* bias for the quadratic law and 0.32% (1.0σ) for the nonlinear law under biased priors. |
A more flexible law reduced the tested bias, but an incorrect prior can still shift the posterior. |
| Atmosphere-profile fit study | Four-coefficient fits reproduced model intensities to about 1 part in 1,000 under its stated conditions. | Profile fidelity is not the same as observational identifiability; four coefficients may remain weakly constrained by a transit. |
Because transit depth is approximately proportional to (Rp/R*)2, a shift in radius ratio also changes the inferred depth. Report the posterior uncertainty from limb darkening together with photometric, stellar-radius and other relevant error terms; do not quote a radius precision that treats the coefficients as exact when they are not.
Diagnose common failure modes
Coefficients copied from another filter
Symptom: A seemingly precise fit shows ingress or egress residuals, or the radius changes substantially between filters in a way not supported by the data. Fix: regenerate coefficients using the actual throughput and refit; do not simply widen the error bar on the copied pair.
Unphysical or highly correlated quadratic samples
Symptom: The sampler spends time at negative intensities, encounters divergent transitions or returns a long, curved u1,u2 posterior. Fix: sample q1,q2 in the unit square, verify the inverse transform and inspect the resulting intensity profile over 0 ≤ μ ≤ 1.
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Free coefficients run to prior boundaries
Symptom: One or more coefficients are effectively unconstrained and the radius estimate changes with arbitrary prior limits. Fix: use atmosphere-informed finite-width priors, combine multiple transits or bands when justified, and report the resulting sensitivity to the prior.
Ingress and egress remain systematically wrong
Symptom: Residuals have a repeatable sign pattern at both contacts after detrending. Fix: check exposure-time integration, transit timing, impact parameter, passband definition and the selected law before adding coefficients. A wrong cadence treatment can mimic a limb-darkening problem.
Radius changes after a stellar reanalysis
Symptom: Updating Teff, log g or metallicity changes the reported planet size even though the photometry is unchanged. Fix: this is expected when coefficients were tied to the old stellar solution; regenerate them and refit the transit rather than applying a post-hoc scale factor.
Guidance by observing mode
Broadband space photometry
For a single broadband transit, the quadratic law with informative priors is usually the practical starting point. Use the mission’s exact response and cadence, and avoid interpreting a weakly constrained free-coefficient fit as a direct atmosphere measurement.
HST or JWST spectroscopic data
Wavelength-resolved observations can reveal profile differences that a quadratic law hides. Fit each channel or justified wavelength bin with coefficients generated for that bin’s throughput. Compare a quadratic model with a square-root, power-2 or nonlinear law using posterior predictive residuals and parameter stability, not coefficient count alone.
Multiple instruments or filters
Give each passband its own coefficient set. Shared stellar parameters can connect the priors, but sharing the numerical coefficient pair across filters imposes an assumption that the atmosphere and throughput do not support.
What to report with a published transit fit
- The adopted limb-darkening law and mathematical definition.
- Whether coefficients were fixed, assigned informative priors or fitted jointly.
- Atmosphere model, stellar parameters and uncertainties used to generate the priors.
- Instrument throughput and calibration version.
- Coefficient fitting grid,
μsampling and weighting rule. - Sampling parameterization, including the
q1,q2transform when used. - Exposure-time integration, baseline terms and correlated-noise model.
- Tests with an alternate law and the resulting change in
Rp/R*and depth. - How coefficient and stellar-input uncertainty was propagated into the final planet parameters.
Bottom line
Model limb darkening as a passband-specific, uncertain part of the transit—not as a universal coefficient lookup. Generate coefficients from the exact stellar atmosphere and throughput, use a physically valid parameterization such as Kipping’s q1,q2 for the quadratic law, let the coefficients move when the data can support it, and validate the planet radius against alternate laws and updated stellar inputs.
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