Read atmospheric asymmetry and resolved-line evidence from the original papers

Pin the paper and the data product

The first paper is Espinoza and collaborators, Inhomogeneous terminators on WASP-39 b, published in Nature 632, 1017–1020, DOI 10.1038/s41586-024-07768-4. The fixed preprint is arXiv 2407.10294v1. Use the published full text for Figures 1–3 and the robustness Methods, keeping the preprint version separate in notes.

Its measurement uses NIRSpec light curves across roughly 2–5 micrometres. It extracts separate morning and evening limb spectra from ingress, egress and the transit shape. The actual NIRISS tutorial instead uses a previously fitted whole-planet spectrum below 2 micrometres. The instrument, wavelength interval and inferred observable all differ.

The second paper is Brogi and Line, Retrieving Temperatures and Abundances of Exoplanet Atmospheres with High-resolution Cross-correlation Spectroscopy, published AJ 157, 114, DOI 10.3847/1538-3881/aaffd3. The fixed accepted manuscript is arXiv 1811.01681v2. Read Section II.2.2 and the tests accompanying Figures 2, 5, 7 and 10.

For the teaching model, the checked solver and inference-grid differences against a declared independent 1280-layer reference are below two ppm only within the complete frozen sampled opacity and specified temperature–pressure interpolation. Halving its native wavelength sampling changes the five presets by about 40–145 ppm; omitting a known temperature row changes them by about 3.4–31.8 ppm. These source sensitivities are comparable to measurement errors, and they neither define Gaussian error bars nor bound exact high-resolution spectra. The tutorial posteriors remain conditional demonstrations, with no water or starspot detection claim.

This older methods paper remains useful because it makes the probability calculation explicit. Its resolved-line spectra and processing cannot be reproduced by treating our sampled resolving-power-15,000 water cross section as a high-resolution line template. The comparison below is conceptual and mathematical, with the original figures as the evidence.

Reading: Espinoza et al. (2024) · Inhomogeneous terminators on WASP-39 b · Feinstein et al. (2023) · Early Release Science of WASP-39 b with JWST NIRISS · Brogi and Line (2019) · Retrieval with high-resolution cross-correlation spectroscopy

Figure 1 asks where the asymmetry enters the light curve

Start with the transit model in the limb paper. Its two semicircles represent different effective morning and evening radii. They change which area is covered during ingress and egress. The method infers that geometric difference from a time series; it does not resolve a photograph of two atmospheric hemispheres.

In Figure 1, trace the observed light curve, the asymmetric fit and the corresponding residuals. Locate the times when the two limb prescriptions matter most. Then read the model and error assumptions alongside the figure. The two fitted limb radii share a transit geometry and need not have independent uncertainties.

Write down the null model before looking at the claimed difference: equal limb radii at the fitted wavelength. Ask whether changing the orbital parameters or stellar limb-darkening law could create a similar ingress–egress pattern. The limb paper tests these alternatives with the time-dependent geometry.

Reading: Espinoza et al. (2024) · Inhomogeneous terminators on WASP-39 b · Rackham, Apai and Giampapa (2019) · The Transit Light Source Effect II

Figures 2 and 3 separate an observed contrast from its explanation

Figure 2 reports that the evening limb is deeper than the morning limb, with a quoted average contrast of $405\pm88\,\mathrm{ppm}$ in the paper’s analysis. The authors infer a warmer evening limb, reporting a temperature difference of about 177 kelvin with asymmetric uncertainties. Keep the measured spectral contrast and the model-derived temperature contrast in separate lines of your notes.

Compare wavelengths within the figure. A common offset, a changed molecular contrast and a cloud cutoff can affect the spectrum differently. Use the individual limb errors and their fitted covariance; subtracting central values and treating every error bar as independent would discard part of the inference.

Figure 3 compares different circulation and atmospheric prescriptions. Read which parts of each spectrum agree and which miss. The authors discuss enhanced morning cloudiness as a possible explanation and describe incomplete agreement among the model treatments. A cloud interpretation depends on those atmospheric assumptions; a measured limb contrast alone does not uniquely specify cloud composition or particle sizes.

A useful figure-reading exercise is to choose a feature that the preferred explanation still misses. Record its wavelength and sign, then identify which missing process might plausibly change that sign. Tie each proposed process to a visible discrepancy that it could change.

Reading: Espinoza et al. (2024) · Inhomogeneous terminators on WASP-39 b · Heng and Kitzmann (2017) · Transmission spectra and the radius–pressure degeneracy

The robustness Methods are part of the result

Read the limb-darkening and orbital-parameter tests before repeating the headline. In the paper’s injected tests, an imposed limb-darkening error of order 0.01 can generate a false limb contrast of about 200–300 ppm under the tested conditions. Allowing the relevant coefficients to vary recovers the symmetric injected case.

The orbital tests explore eccentricity and timing choices. The reported worst-case eccentric configuration produces a contrast estimate of $582\pm188\,\mathrm{ppm}$ . That sensitivity calls for precise orbital information. It should not be substituted for the main result as though it were another independent measurement.

These tests answer concrete alternatives to atmospheric asymmetry. They also show why fitting only a published mean depth spectrum cannot reproduce the limb paper. Information about when light is lost has already been compressed away. A new timing-sensitive claim requires the appropriate light curves and a joint geometric model.

Reading: Espinoza et al. (2024) · Inhomogeneous terminators on WASP-39 b · Feinstein et al. (2023) · Early Release Science of WASP-39 b with JWST NIRISS

Resolved moving lines need a likelihood for the processed data

Turn to Section II.2.2 of Brogi and Line. High-resolution spectra contain many moving planetary lines among much stronger stellar and terrestrial signals. Processing removes large components, and the model must be treated consistently with the retained data. Figure 2 is a starting point for following that processing.

For the paper’s continuum-subtracted, homogeneous-noise derivation, let $f_n$ and $g_n$ be the processed data and model samples. Their mean-square values are $s_f^2$ and $s_g^2$ , and their cross product is $R=N^{-1}\sum_n f_ng_n$ . The squared residual depends on all three quantities, not just a normalized correlation peak.

Profiling the unknown common Gaussian noise scale gives the log-likelihood below, up to constants independent of the model under those assumptions. It is a specific mapping from processed residuals to probability. A cross-correlation score pasted into an exponential is not an interchangeable likelihood.

Read Figure 5 with its likelihood-calibration test. Then inspect Figure 7 for the effect of changing the line list and Figure 10 for missing absorbers. A strong apparent preference within an incomplete set of models can coexist with biased temperatures or abundances. The paper discusses how prior volume can even favor an incomplete model in an evidence comparison.

\ln\mathcal L=-\frac{N}{2}\ln\!\left(s_f^2-2R+s_g^2\right)+C

This is the paper’s profiled homogeneous-noise expression for consistently processed, continuum-subtracted samples. Its assumptions differ from the known, bin-dependent Gaussian errors in the NIRISS tutorial.

Reading: Brogi and Line (2019) · Retrieval with high-resolution cross-correlation spectroscopy · Polyansky et al. (2018) · The POKAZATEL water line list

Leave with three distinct tests

For broad-band molecular evidence, check whether several predicted bands occur together under a stated cloud and stellar model. For limb asymmetry, check whether a joint time-dependent fit survives geometric and limb-darkening alternatives. For resolved-line evidence, check the Doppler pattern, line data, processing and likelihood calibration together.

These measurements can inform one another, but the compressed data products are not interchangeable. The public tutorial supplies a whole-planet conditional retrieval and its failed residuals. The limb paper adds timing information. The high-resolution paper develops a route from processed line patterns to a probability model.

For each original figure, keep one short record: the quantity actually measured, the model operation converting it into an atmospheric statement, and a tested alternative that changes the conclusion. That record is a useful starting point for a future replication with the correct raw data and source resolution.

Reading: Espinoza et al. (2024) · Inhomogeneous terminators on WASP-39 b · Brogi and Line (2019) · Retrieval with high-resolution cross-correlation spectroscopy · Feinstein et al. (2023) · Early Release Science of WASP-39 b with JWST NIRISS

Locate the information the teaching spectrum keeps

Compare cloud and unocculted-star changes in the spectrum. Identify which depends on atmospheric paths and which changes the stellar denominator.
Explain why this spherical snapshot has no separate ingress and egress information. Use the original limb paper’s Figure 1 to identify the additional measurement needed.

Exercises with explanations

Can the $405\pm88\,\mathrm{ppm}$ limb contrast be checked by subtracting two curves from the 74-bin tutorial input?
Read the explanation
No. The input is a different instrument, wavelength range and whole-planet product. The limb contrast comes from a joint time-series fit with a shared geometry and covariance. Reproducing it requires the appropriate NIRSpec light curves and the paper’s limb model.
In the resolved-line likelihood, why does normalizing away the model variance change more than the plot’s appearance?
Read the explanation
The squared residual contains the model mean-square amplitude as well as the data–model cross product. Independently renormalizing the model changes the residual probability model unless an amplitude parameter is explicitly treated. The full likelihood retains the amplitude and cross-product terms together.
Which part of the limb paper would you read after noticing a small asymmetric signal in a new light curve?
Read the explanation
Read the limb-darkening and orbital robustness Methods, define a symmetric injected case and test whether the full fitting procedure recovers it. The published examples demonstrate a concrete route for checking false asymmetry; their numerical outcomes are tied to the stated instrument, geometry and error conditions.

Sources and calculation scope

Feinstein et al. (2023) · Early Release Science of WASP-39 b with JWST NIRISS
Published Nature 614, 670–675. Read Figures 1–3 and the extraction, light-curve and atmospheric-model Methods. The tutorial uses a fixed author-released nirHiss spectrum, with 74 selected order-one bins; it does not repeat the paper’s full chemical and cloud retrieval.
Heng and Kitzmann (2017) · Transmission spectra and the radius–pressure degeneracy
Fixed accepted version, arXiv 1702.02051v2; published MNRAS 470, 2972–2981, DOI 10.1093/mnras/stx1453. Section 2.2 derives the thin, constant-gravity limit; Sections 3.2–3.3 discuss the reference radius and pressure. The course derives its inverse-square-gravity extension separately.
Polyansky et al. (2018) · The POKAZATEL water line list
Primary line-list description for the dominant water isotopologue. The delivered opacity is the derived ExoMolOP product, not a new computation of the full line list. ExoMol data and the extracted opacity derivatives retain CC BY-SA 4.0.
Rackham, Apai and Giampapa (2019) · The Transit Light Source Effect II
Fixed arXiv 1812.06184v1; published AJ 157, 96, DOI 10.3847/1538-3881/aaf892. Section III.1, Equations 3–6 define the unocculted-region correction and its uniform-intensity approximation. Section IV.3 explains why molecular contamination is generally weak at typical FGK activity levels.
Espinoza et al. (2024) · Inhomogeneous terminators on WASP-39 b
Published Nature 632, 1017–1020; fixed preprint arXiv 2407.10294v1. Read Figures 1–3 and the orbital-parameter and limb-darkening robustness Methods. Its NIRSpec 2–5 micrometre limb spectra are a different product from the NIRISS tutorial spectrum. The guide performs figure-reading exercises, not a new limb retrieval.
Brogi and Line (2019) · Retrieval with high-resolution cross-correlation spectroscopy
Fixed accepted arXiv 1811.01681v2; published AJ 157, 114, DOI 10.3847/1538-3881/aaffd3. Section II.2.2 derives the likelihood mapping; Figures 2, 5, 7 and 10 test processing, calibration, line-list choice and missing absorbers. The low-resolution teaching opacity cannot reproduce the paper’s resolved-line analysis.