The online Exoplanet Characterization Toolkit (ExoCTK) Limb Darkening Calculator, used for planning JWST exoplanet observations, returns limb darkening coefficients that describe center-to-limb variations of stellar intensity for a specified wavelength interval. 

See also: MIRI Filters and Dispersers, NIRISS GR700XD Grism, NIRCam Grisms, and NIRSpec Dispersers and Filters

The shape of simulated and observed transit light curves depends on how the stellar intensity varies along the exoplanet transit chord across the face of the host star.

At optical and infrared wavelengths, stars usually appear brightest at the disk center and get darker towards the limb. Physically, limb darkening occurs because stellar photospheres get hotter (and hence brighter) with increasing depth. Detecting photons from the star is possible up until the depth where opacity blocks their escape along the line of sight. Viewing geometry causes observations to see deeper (hotter, brighter) layers of the photosphere at disk center and shallower (cooler, fainter) layers towards the limb. Model atmosphere codes compute intensity as a function of wavelength for discrete points from center (µ = 1) to limb (µ = 0), where µ is cosine of the angle between a radial vector and the line of sight. To simplify subsequent calculations, an analytic function is often fitted to I(µ)/I(1) for each wavelength interval of interest. Coefficients of the analytic function are called limb darkening coefficients.

The online Limb Darkening Calculator, which is part of the online suite of ExoCTK tools, is described below. The limb darkening calculator is also available as a Python module in the ExoCTK python library.

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Using the web interface

The web interface requires 5 types of input: stellar parameters, planetary parameters (optional), choice of model grid, bandpass, and limb darkening "profile" (functional form).

In the Specify the Stellar Parameters section, you may explicitly enter the desired stellar effective temperature, logarithm (base 10) of the stellar surface gravity, and stellar metallicity. As an example, these parameters have nominal values of 5,770 K, 4.44 (cgs), and 0.0 for the Sun.

Alternatively, you may enter the name of a desired star (e.g., "HD 189733" or equivalently "Wolf 864") or planet (e.g., HD 189733 b") in the optional Target Name section and click the Resolve Target button. If exo.MAST is able to resolve the target name you entered, then the limb darkening tool will update stellar parameter values in the Specify the Stellar Parameters fields with nominal values for the specified star. If Exo.MAST is not able to resolve the target name, then the interface will print a red error message immediately below the Resolve Target button. An example for WASP-18 is shown in Figure 1.

For Choose a Model Grid, you can select between 2 available grids of model intensities; Currently, you may choose between those from Kurucz ATLAS9 (obtained from the Robert L. Kurucz website; Kurucz 1979) or Phoenix ACES (Husser 2013). 

• For the ATLAS9 grid, the stellar effective temperature range must be 3,500–8,750 K, logarithm of stellar surface gravity range must be 3–5, and stellar metallicity range must be from -0.5 to +0.5. Only wavelengths 0.1–30 µm are covered in these models.
• For the Phoenix grid, the stellar effective temperature range must be 2,300–7,800 K, logarithm of stellar surface gravity range must be 3–5, and stellar metallicity range must be from -0.5 to +0.5. Only wavelengths 0.1–2.6 µm are covered in these models, though a future release will extend this coverage out to 30 µm as well.

In addition to calculating the standard limb darkening coefficients, you can also calculate the quadratic SPAM coefficients by completing all the fields in the Specify the Planetary Parameters section. If this data is available via exo.MAST, it will also be automatically filled in if you use the Resolve Target button.

In the Choose a Bandpass section, first select a spectral response function (filter, disperser, or top hat) from the drop-down menu. Selecting a spectral response function sets the minimum and maximum wavelength to nominal values. NIRISS SOSS (GR700XD) orders 1 and 2 are avilable but order 3 is not yet supported. For dispersers, specify the desired number of spectral channels. The tool will return separate limb darkening coefficients for each spectral channel. Figure 2 shows an example in which the NIRISS SOSS order 1 bandpass is used, on which 100 spectroscopic channels are requested.In the Choose One or More Limb Darkening Profiles section, use the check boxes to indicate which functional forms to to fit the model intensities. The tool will return separate limb darkening coefficients for each functional form that you select. All functional forms defined in Kreidberg (2015) are available in the Limb Darkening Calculator, although the "nonlinear" case in that work is renamed 4-parameter in the calculator. The Limb Darkening Calculator also offers a 3-parameter functional form, which is equivalent to setting c₁ = 0 in the 4-parameter functional form (and renumbering the remaining coefficients). This section also includes an additional parameter, Minimum µ, that truncates the range of µ values considered when the tool fits model intensities with an analytic function. This parameter is useful for cases when model intensities near the stellar limb (i.e., intensities near µ = 0) deviate substantially from the specified functional form. You may enter an explicit value or simply accept the default value of 0.1. In Figure 3, an example is shown where only fits for the 4-parameter and quadratic laws are requested, and where the default value of 0.1 is selected for the minimum µ.After entering all input parameters described above, click the Calculate Coefficients button to start the calculation. 

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Understanding the outputs of the Limb Darkening Calculator Tool

Once the calculation finishes, you will be presented with output information about the calculation. Figure 4 shows an example output using the parameters defined above for WASP-18, using the ATLAS9 stellar atmospheric models.

The "Input" table contains a summary of the input values used for the calculation; in this case, the stellar parameters corresponding to WASP-18, the specified bandpass, limb darkening profiles calculated, the stellar model used for the calculation, and the planetary parameters, if applicable. The top right plot shows the throughput of each channel of the selected bandpass and the bottom plot shows the corresponding fit to each of the wavelength channels. In this latter panel, the top numbers indicate the central wavelength of each channel; by clicking them, one can visually inspect the fits (solid lines) made to the model intensity profiles (black points). 

Below these outputs, the values for the actual limb darkening coefficients are presented. An example is shown in Figure 5 for this same science case.

One big table is presented for each of the selected limb darkening parameterizations. The parametric form of the law is on the top of the table, and the coefficients—along with their errors—are presented in the table for the different wavelengths. The first column presents the effective ("central") wavelength, the second and third columns show the minimum and maximum wavelengths of the spectroscopic channel, and the rest of the columns present limb darkening coefficients with their associated errors. Although these can be directly copied from the webpage to be used, the Download Coefficient Tables button just above all the tables can be used to download all the results in text form. If the SPAM coefficients were also calculated, a separate table and download button to Download SPAM Coefficient Tables will also be available.

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References

Husser, T.-O., et al. 2013, A&A, 553, 6, 9.
A new extensive library of PHOENIX stellar atmospheres and synthetic spectra

Kurucz, R. L., 1979, ApJ, Suppl. Ser., 40, 1
Model atmospheres for G, F, A, B, and O stars. http://kurucz.harvard.edu/grids/

Kreidberg, L. 2015, PASP, 127, 957, 1161.
batman: BAsic Transit Model cAlculatioN in Python

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