Limb Darkening Calculator Tool
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.
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(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.
Using the web interface
The web interface requires 4 types of input: stellar parameters, 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 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.
- 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.
If you specify parameter values outside these ranges and click the Calculate Coefficients button, the interface will print a red error message at the bottom of the stellar parameters section. If you specify a value between grid points, the tool linearly interpolates intensities for the 6 grid bounding points.
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. Currently, NIRISS SOSS (GR700XD) is the only JWST spectroscopic mode available in the menu. For other JWST spectroscopic modes, select the Top Hat option and manually update the minimum and maximum wavelength to appropriate values from JWST instrumentation documentation (e.g., MIRI, NIRCam, and NIRSpec). The Top Hat spectral response is adequate for proposals. Actual spectral response functions for JWST dispersers will be available before the start of cycle 1 observations. 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 bandpass is used, on which 100 spectroscopic channels are requested.
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.
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.
Husser, T.-O., et al. 2013, A&A, 553, 6, 9.
A new extensive library of PHOENIX stellar atmospheres and synthetic spectra
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batman: BAsic Transit Model cAlculatioN in Python