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Unveiling the Atmosphere of HR 7672 B from the Near-Infrared High-Resolution Spectrum Using REACH/Subaru

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read High-resolution REACH spectra of the L4.5 brown dwarf HR 7672 B show water and iron hydride absorption and require an optically thick cloud deck near 14 bar.

desk verdict A credible first REACH science paper whose telluric model is the one soft spot; the detections and cloudy preference are likely robust, and it deserves a serious referee. read the letter →

arxiv 2508.01281 v1 pith:RMMXRHPS submitted 2025-08-02 astro-ph.EP

Yui Kasagi , Yui Kawashima , Hajime Kawahara , Takayuki Kotani , Kento Masuda , Kyohoon Ahn , Olivier Guyon , Teruyuki Hirano
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Nemanja Jovanovic Masayuki Kuzuhara Julien Lozi Motohide Tamura Taichi Uyama Sebastien Vievard Kenta Yoneta
This is my paper
classification astro-ph.EP
keywords browndwarfatmosphereshigh-resolutionspectroscopyatmosphericretrievalcloudopacityFeHwaterabsorptiondirectlyimagedcompanionsREACH
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper reports the first atmospheric retrieval applied to a faint companion observed with REACH, the R~100,000 near-infrared spectrograph on the Subaru Telescope. Using J- and H-band spectra of the L-type brown dwarf HR 7672 B, the authors identify water and iron hydride as the main absorbers and find that the data strongly prefer a model with an optically thick cloud whose top sits near 14 bar at about 2110 K. The cloud-top temperature falls between the condensation temperatures of titanium dioxide, aluminium oxide, and iron, so those species are plausible cloud materials. The same retrieval simultaneously corrects for light leaking from the host star and for Earth's telluric absorption, and it measures a projected rotation speed of about 41 km/s. If the claim holds, it shows that high-dispersion spectroscopy of faint directly imaged substellar companions can recover molecular abundances and cloud structure with a relatively simple model.

What carries the argument

The load-bearing machinery is a single autodifferentiable spectral model that simultaneously computes the brown dwarf spectrum, telluric transmission, and host-star leakage. The brown dwarf part uses a power-law temperature–pressure profile, constant volume-mixing-ratio opacities for water and iron hydride, collision-induced absorption, and a two-parameter cloud: a fixed, optically thick opacity ($\tau_\mathrm{cloud}=500$) inserted at pressures $P>P_\mathrm{top}$. Telluric transmission is modelled as the exponential of summed cross-sections for $\mathrm{H_2O}$, $\mathrm{CO_2}$, $\mathrm{CH_4}$, and $\mathrm{O_2}$ at fixed summit pressure and temperature, with a free Doppler shift. Host-star light is treated as a scaled version of the observed HR 7672 A spectrum. The model is fit jointly to the J-band photometric magnitude and the high-resolution spectra using Hamiltonian Monte Carlo with a jitter term, and the molecular detections are cross-checked with cross-correlation functions built from the best-fit model.

What would settle it

Re-analyse the two nights of data after dividing out the telluric absorption using a rapidly rotating standard star observed on the same night; if the retrieved H2O and FeH mixing ratios or the cloud-top pressure shift by more than the quoted uncertainties, the fixed-T-P telluric model is falsified.

Watch

Extended reading notes

Core claim

The central claim is that the J- and H-band spectrum of HR 7672 B is dominated by water ($\mathrm{H_2O}$) and iron hydride ($\mathrm{FeH}$) absorption, and that a wavelength-independent continuum source—an optically thick cloud at $P_\mathrm{top} = 10^{1.16}\,\mathrm{bar}\simeq14\,\mathrm{bar}$ with a cloud-top temperature near 2112 K—is required to reproduce the observed features. A cloud-free model can also fit the data, but only by letting collision-induced absorption supply the continuum, and the cloudy model is favoured by $\Delta\mathrm{BIC}\simeq22$. The retrieved temperature–pressure profile places the cloud top between the condensation curves of $\mathrm{TiO_2}$, $\mathrm{Al_2O_3}$, and Fe, making those the likely cloud materials. Cross-correlation analysis independently confirms the molecular detections, with S/N of 3.6 for $\mathrm{FeH}$ in the J band and 6.9 for $\mathrm{H_2O}$ in the H band. The paper also presents the first characterization of the periodic noise systematics that currently limit REACH's usable wavelength range.

Load-bearing premise

The analysis assumes the model of Earth's telluric absorption—computed at fixed temperature and pressure and fitted simultaneously with the brown dwarf spectrum—is accurate enough that its deviations of up to 0.1 in normalized flux around some telluric lines do not bias the retrieved molecular abundances or cloud-top pressure.

Editorial extensions

If this is right

  • REACH can deliver molecular detections and cloud-top constraints for faint companions at $R\sim100{,}000$, complementing K-band high-resolution instruments by probing deeper atmospheric layers near 4–15 bar.
  • The retrieved water abundance (log VMR $\simeq -3.8$) is consistent within $3\sigma$ with the earlier K-band retrieval, supporting a uniform composition across the altitudes probed by the two bands.
  • A cloud-top temperature near 2110 K provides a concrete benchmark for cloud-condensation models of L dwarfs, pointing to $\mathrm{TiO_2}$, $\mathrm{Al_2O_3}$, or Fe grains.
  • Characterizing the periodic noise systematics gives future REACH users a template for mitigating fringing-like signals in post-processing, which will extend the usable spectral range toward the longer H band.
  • In the mass-free cloudy retrieval, the mass upper limit is unconstrained, indicating that for cloudy atmospheres J/H-band spectra alone may not pin down surface gravity when cloud opacity hides the collision-induced continuum.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the retrieved cloud is as optically thick as assumed, the J-band continuum should be nearly grey; a test would be to compare the continuum slope across the full J band with the cloudy and clear-sky models fit to future full-band data.
  • The paper's 3.3σ shortfall of retrieved water relative to thermochemical equilibrium could reflect vertical mixing or a non-constant abundance profile that the constant-VMR assumption cannot capture; a per-order or per-night jackknife could reveal whether residual telluric error drives the deficit.
  • An empirical check of the telluric model—dividing the companion spectra by a same-night rapidly rotating standard star instead of forward-modelling telluric lines—would directly test the paper's weakest assumption; agreement would strengthen the cloud claim, disagreement would implicate telluric systematics.
  • The same joint modelling of companion, host-star leakage, and telluric transmission is a natural template for future REACH observations of directly imaged exoplanets, provided the targets rotate slowly enough to resolve molecular line structure.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. This paper presents the first atmospheric retrieval from REACH/Subaru high-resolution spectra (R ~ 100,000 in Y, J, H) of the benchmark L4.5 brown dwarf HR 7672 B. The forward model is built on ExoJAX and simultaneously treats brown-dwarf molecular opacity (H2O, FeH, CIA), a simplified optically thick cloud, telluric transmission, host-star light leakage, and a low-order continuum correction. The authors run four HMC retrievals (mass-constrained/free x cloudy/clear) over narrow J- and H-band windows and report detections of H2O and FeH, a cloud-top pressure log10(Ptop/bar) ~ 1.16, a cloud-top temperature near 2112 K, and a strong preference for the cloudy model (Delta BIC ~ 22). The molecular identifications are supported by a null-test cross-correlation procedure (Appendix B), and the paper also characterizes periodic noise in REACH faint-target spectra.

Significance. If the results hold, this paper demonstrates that high-contrast, high-resolution YJH spectroscopy with REACH can recover molecular abundances and cloud properties for a faint companion at 0.7 arcsec separation, a useful step for the field. The work benefits from a benchmark target with a dynamical mass, a publicly archived retrieval code and data (Zenodo/GitHub), and an explicit null test for CCF-based molecular detection. The comparison with the KPIC/Keck retrieval of Wang et al. (2022) is appropriate and places the new abundance and T-P constraints in context. The central atmospheric conclusions, however, rest on two assumptions that need quantitative stress-testing: the fixed-temperature-pressure telluric model and the post-hoc selection of very narrow wavelength windows for the cloud-vs-clear comparison.

major comments (3)
  1. [Section 3.2 / 3.2.1, Eq. (2)] The telluric transmission model fixes T = 273 K and P = 0.6005 bar for all species and fits only column-scale factors beta_i. Section 3.2.1 reports residuals up to 0.1 in normalized flux around telluric lines, which is roughly 100 times the mean per-pixel uncertainties (7.8e-4 and 8.1e-4) quoted in Figure 2 and Section 2.1. Because the telluric model is multiplied into and fitted simultaneously with the companion model (Eq. 5), a coherent line-shape error cannot be absorbed by the beta_i and can directly bias the retrieved molecular abundances and pseudo-continuum, particularly in the H band where telluric and brown-dwarf H2O lines overlap at 1.48-1.54 um. The statement that these residuals have no significant impact because they are 'comparable to the spectral noise' is not quantitatively supported: the jitter term sigma = 0.06 (Table 2) inflates the noise but does not correct a coherent systematic. I request a robustness test that varies the telluric T-P profile, or uses per-species effective temperatures, or repeats the retrieval after masking or down-weighting the strongest telluric lines, and reports the resulting shifts in logH2O, logFeH, and logPtop. Note also that the telluric-model validation was performed on a June 6 standard-star observation, while the companion was observed on June 24; the residual budget on the science night is assumed by the analysis, not demonstrated.
  2. [Section 4.1.3 / Section 5.1] The cloud-formation claim rests on the wavelength dependence of the continuum, but the retrieval uses only two narrow windows (J orders 43-45, 1.265-1.3 um; H orders 57-60, 1.48-1.54 um) that were selected after inspecting the data. The Y band and the 1.57-1.67 um H-band region are excluded (Sections 2.1 and 3). With such limited coverage, the Delta BIC = 21.9 preference for the cloudy model may reflect the particular choice of windows rather than a real, robust continuum shape. Please demonstrate that the cloud-vs-clear conclusion is stable to the inclusion or removal of wavelength ranges, or use injected cloud models to calibrate the sensitivity of this BIC comparison under the present spectral coverage.
  3. [Section 3.4.2] The J-band host-star leakage is fixed at 63% based on a retrieval around the single Paschen-beta line, with the stated expectation that deviations are absorbed by the linear trend parameter a in Eq. (6). A single-line measurement, combined with the assumption that the telluric transmission is identical on the two nights (TA = TB), is a fragile anchor for a parameter that directly sets the J-band continuum level and therefore affects the retrieved FeH abundance and line-to-continuum ratios. I recommend running a sensitivity test in which log(scale_star_y) is a free parameter and reporting the resulting change in logFeH and logPtop; if the degeneracy with the linear correction is strong, the abundance uncertainties should be enlarged accordingly.
minor comments (6)
  1. [Table 1 / Table 2] The notation T_1bar^0 and T^Ptop_0 is easy to confuse in the tables; please define the reference pressure explicitly in the table caption for each quantity.
  2. [Figure 2] The pink wavelength ranges used in the retrieval are difficult to distinguish in the printed figure; consider adding shaded vertical bands spanning both panels in each wavelength window.
  3. [Section 4.1.1 / Appendix B] The overall CCF peak S/N = 11.4 is quoted in the main text, but the molecular S/N values of 3.6 (FeH) and 6.9 (H2O) appear only in Appendix B; the main text should state these values explicitly because they are the actual detection statistics for the two species.
  4. [Section 3.6, Eq. (10)] The BIC formula uses the maximum of the likelihood, but the text does not specify how this maximum is obtained from the HMC posterior; please state whether it is the maximum over posterior samples or a separate optimization.
  5. [Section 3.2] The telluric absorber list includes O2 in the J band and H2O, CO2, and CH4 in the H band; please state explicitly whether O2 is also included in the H-band window or whether its opacity is negligible there.
  6. [Section 2.2 / Appendix A] The long-period noise characterization is thorough, but the paper does not translate the measured 15-22% noise amplitude into a quantitative uncertainty on the excluded 1.57-1.67 um region or on the continuum shape; a one-sentence quantitative summary would help readers judge the impact of the wavelength selection.

Circularity Check

1 steps flagged · score 1.0 of 10

No load-bearing circularity; the only mild issue is that a fitted photometric point is retrospectively labeled a prediction.

  1. fitted input called prediction [Section 4.1.1 (Results from a Cloudy Model), with the fitting setup in Section 3 and Equation (7).]
    "The prediction of the J-band magnitude is 14.48, with a 95 % credible interval of [13.86, 14.98]. This range includes the observed J-band magnitude of 14.39 ± 0.20."

    The observed J-band magnitude is not a withheld test point: Section 3 states that 'we computed the J-band photometric magnitude and fitted it to the magnitudes observed by Boccaletti et al. (2003)', and Equation (7) includes (D_J - M_J)^2 in the likelihood. The posterior predictive value for M_J is therefore conditioned on D_J, so its interval enclosing D_J is expected by construction rather than being an independent prediction. This is a minor rhetorical overstatement; the central H2O/FeH and cloud claims are not based on this 'prediction'.

full rationale

The paper's central derivations are self-contained inversions rather than circular constructions. The retrieval fits a high-resolution spectral model to the J/H-band data and then reports the fitted H2O and FeH abundances and cloud-top pressure; these are explicitly labeled retrieval results, not independent predictions. The CCF detections in Section 4.1.1 and Appendix B use templates from the best-fit model but include a genuine null test: the cross-correlation with the residual after subtracting the full model shows no peak, while the residual against the model without the target molecule shows a peak. This makes the molecular attribution an internal consistency check rather than a definitional tautology. The cloud preference is established by a BIC model comparison (Delta BIC ~ 21.9) between a cloudy model with a free Ptop and a clear-sky model, and the paper explicitly notes that the clear-sky model can also fit the spectra, so the cloudy conclusion is not forced by the parameterization. The self-citations (ExoJAX/Kawahara et al. 2022, 2025; Kawashima et al. 2025) are tool use, prior methodological precedent, or comparison, not load-bearing uniqueness arguments. The fixed-T/P telluric model with residuals up to 0.1 is a genuine systematic-error risk for the reported abundances and cloud-top pressure, but it is a correctness limitation, not a circular step, because no telluric parameter is defined in terms of the retrieved brown dwarf properties. The only mild circularity is the retrospective labeling of the fitted J-band magnitude as a 'prediction'.

Assumptions & free parameters 10 free parameters · 7 assumptions · 0 invented entities

The central retrieval depends on several simplified model choices: constant mixing ratios, power-law T-P profile, fixed optically thick cloud, fixed telluric temperature, and a scaling model for host-star contamination. All are common in atmospheric retrieval but are assumptions that could bias the results if not representative of the real atmosphere.

free parameters (10)
  • H2O volume mixing ratio (log10) = -3.83 (+0.09/-0.07)
    Fitted to H-band H2O lines; a central retrieval output.
  • FeH volume mixing ratio (log10) = -7.41 (+0.10/-0.09)
    Fitted to J-band FeH lines; a central retrieval output.
  • Temperature at cloud top (T_Ptop) = 2112 K (+46/-49)
    Fitted in cloudy model; sets cloud-top temperature.
  • Power-law index alpha = 0.12 (+0.02/-0.01)
    Fitted T-P profile slope.
  • Surface gravity log g = 5.46 (+0.09/-0.09)
    Retrieved; degenerates with mass and radius.
  • Projected rotation speed vsin(i) = 41.28 km/s (+0.92/-0.61)
    Fitted from line broadening.
  • Cloud top pressure log10(Ptop/bar) = 1.16 (+0.08/-0.08)
    Fitted; pressure at which optically thick cloud opacity sets in.
  • Host star leakage in H band (log scale_star_h) = -0.43 (+0.02/-0.02)
    Fitted fraction of host star light in H-band model.
  • Telluric column coefficients (log beta H2O, CO2, CH4, O2) = 21.48, 21.28, 17.49, 24.20
    Fitted scaling of telluric absorbers.
  • Jitter term sigma = 0.06
    Fitted noise inflation term to account for underestimated errors.
assumptions (7)
  • domain assumption Volume mixing ratios are constant with pressure for H2O and FeH.
    Section 3.1: 'The volume mixing ratios of these molecules are assumed to be constant across all pressure levels considered.'
  • domain assumption The T-P profile follows a power law T(P)=T0(P/P0)^alpha with 300 layers.
    Section 3.1: introduced as the adopted simple T-P profile.
  • domain assumption Cloud opacity is modeled as an optically thick layer with tau_cloud=500 fixed.
    Section 3.3: 'we fixed the value of taucloud to be 500, assuming the optically thick cloud is placed at the pressure Ptop.'
  • domain assumption Telluric transmission is modeled with fixed pressure and temperature (0.6005 bar, 273 K).
    Section 3.2: 'The pressure and temperature are fixed to the values at the summit of Mauna Kea, 0.6005 bar and 273 K, respectively.'
  • domain assumption Host star contamination can be represented by scaling the observed HR 7672 A spectrum with a single factor, and telluric transmission is identical for A and B observations.
    Section 3.4.2, Equation (5): assumes TA(ν)=TB(ν) and uses a scaled version of the observed star spectrum.
  • domain assumption Line lists (POKAZATEL, MoLLIST, etc.) and CIA coefficients from HITRAN are accurate in the modeled bands.
    Section 3.1 lists the adopted databases; the paper relies on their accuracy without independent validation.
  • standard math The HMC-NUTS sampling with 1000 post-warmup samples and R-hat below 1.1 indicates convergence.
    Section 3.6 uses NumPyro HMC-NUTS; convergence is assessed via R-hat.

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Cite this review

Pith. "Pith review of Unveiling the Atmosphere of HR 7672 B from the Near-Infrared High-Resolution Spectrum Using REACH/Subaru." pith.science (2026). https://pith.science/paper/RMMXRHPS

@misc{pith2026250801281,
  author       = {Pith},
  title        = {Pith review of: Unveiling the Atmosphere of HR 7672 B from the Near-Infrared High-Resolution Spectrum Using REACH/Subaru},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RMMXRHPS}},
  note         = {Machine review of arXiv:2508.01281}
}
abstract

Characterizing the atmospheres of exoplanets and brown dwarfs is crucial for understanding their atmospheric physics and chemistry, searching for biosignatures, and investigating their formation histories. Recent advances in observational techniques, combining adaptive optics with high-resolution spectrographs, have enabled detailed spectroscopic analysis for directly imaged faint companions. In this paper, we report an atmospheric retrieval on the L-type brown dwarf HR 7672 B using a near-infrared high-contrast high-resolution spectrograph, REACH (Y, J, H band, $R\sim100,000$), which combines SCExAO with IRD at the Subaru Telescope. Our model, developed based on the ExoJAX spectrum code, simultaneously accounts for several factors, including the presence of clouds in the L dwarf's atmosphere as well as contamination from the host star's light and telluric absorption lines in the observed spectra. Our analysis identified H2O and FeH as the primary absorbers in the observed J- and H-band spectra. Additionally, the observed features were reproduced with a model that includes cloud opacity, assuming an optically thick cloud at the pressure $P_\mathrm{top}$. The resulting temperature at the cloud top pressure suggests the potential formation of clouds composed of TiO2, Al2O3, or Fe. This study is the first science demonstration for faint spectra obtained by REACH, providing a foundation for future investigations into the atmospheres of exoplanets and brown dwarfs.

Figures

Figures reproduced from arXiv: 2508.01281 by the authors.

Figure 1
Figure 1. Image of HR 7672 observed by SCExAO internal fast IR detector (C-RED2) on June 7, 2021 (ID: S21A-062, PI: H. Kawa￾hara). HR 7672 B in a green circle is 0. ′′7 away from the host star. no significant differences, and the telluric lines in the spectra were also largely consistent. An astrometry prediction was not required for fiber placement on this target, as its posi￾tion was visually confirmed using the image obtai… view at source ↗
Figure 2
Figure 2. The observed J- and H-band spectra of HR 7672 B. The top panels of each figure show the telluric transmittance created by the LBLRTM (Clough et al. 2005) for reference. The bottom three panels show the spectra of HR 7672 A, HR 7672 B, and the speckle halo light, with each combining two frames. The last spectrum was obtained from the speckle halo fiber during the observation of HR 7672 B. The gray dots in the third p… view at source ↗
Figure 3
Figure 3. Heatmap of the H-band spectrum of κ And b, shown across spectral orders and pixels within each order. The color gradually changes in accordance with the normalized flux. The left panel is the spectrum obtained from the science fiber, while the right panel is the spectrum obtained from the speckle halo fiber. The periodic noise appears prominently especially from orders 61 to 66, and they look similar in both fibers … view at source ↗
Figures from the paper (15 more)
Figure 4
Figure 4. Figure 4: The example of spectra of order 64 (in the H band) ob￾served on the same date. The blue line is the spectrum of κ And b, the orange line is the FLAT spectrum scaled by a factor of 0.04 for comparison, the green line is the spectrum from speckle halo fiber of the κ And …
Figure 5
Figure 5. Figure 5: Comparison of the spectra of HR 7672 A and B was performed to assess light leakage from the host star in HR 7672 B’s spectrum. Panels (a)–(c) display the spectra around the strong absorption lines from the host star, as indicated by vertical colored lines, following th…
Figure 6
Figure 6. Figure 6: Retrieval results for the J-band spectrum with the model including cloud opacity. The top panel is the observed spectrum of HR 7672 B and the full model with 95 % credible intervals. The second panel shows the leakage of the host star’s light (dark gray), that is the s…
Figure 7
Figure 7. Figure 7: Same as [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]
Figure 8
Figure 8. Figure 8: Posterior distributions of the retrieval for the J- and H-band spectra when a model includes cloud opacity. Only the main parameters are shown here, excluding telluric-related ones. served spectrum. Since the prior range of cloud top pres￾sure includes values where clo…
Figure 11
Figure 11. Figure 11: Abundances assuming thermochemical equilibrium (solid lines). We set C/O = 0.56 and [Fe/H] = −0.04 for this calculation. The retrieved 1σ values of VMRs for H2O and FeH are shown by vertical shaded regions. According to the contri￾bution functions, we observe the atmo…
Figure 10
Figure 10. Figure 10: Comparison of the retrieved T-P profile (black line) with condensation curves (colored dashed lines). The gray lines repre￾sent 100 random samples from the posterior distributions, while the gray dashed line indicates the retrieved pressure at the cloud top. Reference…
Figure 12
Figure 12. Figure 12: Posterior distributions of the retrieval for the J- and H-band spectra with a clear sky model. Only the main parameters are shown here, excluding telluric-related ones. nation of surface gravity in this retrieval analysis, which uses both the high-resolution spectrum …
Figure 13
Figure 13. Figure 13: Spectrally averaged contributions overlaid with T-P profiles (left panel) and contribution functions (right panel). (a) Results for the mass-constrained cloudy model, showing the total opacity from the retrieval. (b) Results from (a), but isolating only the CIA contri…
Figure 14
Figure 14. Figure 14: Contribution functions with varying g and xi while maintaining g/xi = const.. The atmospheric parameters used to generate all figures are based on the results from the mass-constrained retrieval with a cloudy model. Contributions from both the J- and H-band wavelength…
Figure 15
Figure 15. Figure 15: Comparison of T-P profiles. The black and grey lines represent the T-P profiles derived from our retrievals using the cloudy and cloud-free models, respectively. The grey shaded re￾gion indicates the 1σ range from the retrieval in Wang et al. (2022), reproduced from …
Figure 16
Figure 16. Figure 16: Comparison of fluxes in the J, H, and Ks bands. The black dots represent the observed flux, while the red and purple dots correspond to the fluxes calculated from our model — red represents the cloudy model and purple represents the clear sky model. The J￾band values …
Figure 17
Figure 17. Figure 17: Mean combined spectra from the speckle halo fiber in order 64 observed on different dates. From top to bottom: June 6th, 7th, 22nd, 24th for HR 7672 B, and June 24th for κ And b. The colored points are selected by rejecting both edges of the order and clipping outlier…
Figure 18
Figure 18. Figure 18: Periodograms of spectra in order 64 of the speckle halo fiber observed on different dates, corresponding to the spectra in [PITH_FULL_IMAGE:figures/full_fig_p026_18.png]
Figure 19
Figure 19. Figure 19: Cross-correlation analysis demonstrating the detection of FeH in the J-band spectrum in panel (a) and the detection of H2O in the H-band spectrum in panel (b). The vertical black dashed line shows the rest frame of the brown dwarf. The optical depth of CIA is derived …

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