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A Multi-Species Atmospheric Escape Model with Excited Hydrogen and Helium: Application to HD209458b

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper argues that the weak helium and hydrogen absorption seen in the archetypal hot Jupiter HD209458b does not require an atmosphere that is poor in helium.

desk verdict Serious escape model with new atomic data and a good sensitivity story, but the diffusive-separation claim clashes with the paper's own crossover-mass estimate and the Hα validation is partial. read the letter →

arxiv 2506.08232 v1 pith:DGJPDSCI submitted 2025-06-09 astro-ph.EP astro-ph.SR

Anna Ruth Taylor , Tommi T. Koskinen , Luca Argenti , Nicholas Lewis , Chenliang Huang , Anthony Arfaux , Panayotis Lavvas This is my paper
classification astro-ph.EPastro-ph.SR
keywords atmosphericescapeexoplanetatmosphereshotJupiterhelium10830ÅHalphadiffusiveseparationphotoelectronheatingHD209458b
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 argues that the weak helium and hydrogen absorption seen in the archetypal hot Jupiter HD209458b does not require an atmosphere that is poor in helium. Instead, a self-consistent hydrodynamic model of the whole atmosphere reproduces the observed He I 10830 Å transit depth and the Hα upper limit through strong diffusive separation, which depletes helium at high altitudes, and through reduced photoelectron heating. The model matches observations with a photoelectron heating efficiency of $20\text{--}40\%$, corresponding to a mass-loss rate of $1.9\text{--}3\times10^{10}$ g/s. If this is right, system-to-system scatter in helium transit depths can be explained by stellar activity, mixing, and heating structure rather than by ad hoc helium abundances. A sympathetic reader would care because it turns a single planet's anomalous signal into a diagnostic of atmospheric mixing and stellar variability.

What carries the argument

The machinery is a 1D thermosphere-ionosphere and hydrodynamic escape model coupled to a lower/middle atmosphere photochemical model, spanning the full atmosphere from 1000 bar to the exosphere. It solves continuity, momentum, and energy equations with multi-species transport, so diffusive separation emerges rather than being imposed. The observable is the density of metastable helium, the long-lived excited $2^3S$ state that produces the 10830 Å absorption, whose production is dominated by recombination of He$^+$ and whose loss at high altitude is dominated by photoionization; the model feeds new high-resolution photoionization cross-sections for this state, computed with the B-spline K-matrix method, into the chemistry. Hydrogen $n=2$ populations are set by a separate non-LTE calculation driven by Lyman-$\alpha$ resonant scattering, which is why H$\alpha$ responds differently to stellar activity and diffusion than He I.

What would settle it

Measure the He I 10830 Å transit depth of HD209458b in several epochs across the host star's activity cycle while monitoring a stellar activity indicator; the model predicts substantial variation (roughly a factor of two to three between solar minimum and maximum), and a stable depth would undercut the stellar-activity story. Also, an independent constraint putting $K_{zz}$ at or above $10^7$ m$^2$/s would largely erase the diffusive separation and force a different explanation.

Watch

Extended reading notes

Core claim

The central claim is that the observed He I 10830 Å transit depth of HD209458b, together with the upper limit on Hα, is reproduced by a multi-species hydrodynamic escape model that includes molecular and eddy diffusion and self-consistently computed non-LTE populations of excited hydrogen and helium. Starting from a solar-composition atmosphere, the model develops strong diffusive separation: the elemental He/H ratio falls from $8\%$ near the thermosphere base to about $2.5\%$ at high altitudes in the best-fit case. Matching the observations requires a photoelectron heating efficiency of $20\text{--}40\%$, lower than the $100\%$ often assumed, and yields mass-loss rates of $1.9\text{--}3\times10^{10}$ g/s depending on whether metals are included in the upper atmosphere. The paper also updates the metastable helium rate coefficients, most notably a high-resolution photoionization cross-section that includes EUV resonances and a temperature-dependent Penning ionization rate, which together lower the predicted He I transit depth by about a factor of $2.5$ relative to earlier rate sets.

Load-bearing premise

The claim depends on the model's diffusion treatment producing strong helium separation even though the standard crossover-mass criterion says helium should stay mixed with hydrogen at the inferred mass-loss rate, and on adopting an eddy diffusion coefficient of $K_{zz}=10^5$ m$^2$/s, at the low end of the literature range.

Editorial extensions

If this is right

  • He I 10830 Å transit depths can no longer be read as direct proxies for mass-loss rate; the same depth can arise from different combinations of heating efficiency, eddy diffusion, and stellar activity, so population-level scatter does not require diverse atmospheric helium abundances.
  • Simultaneous observations of He I, H$\alpha$, and stellar activity indicators can separate the effects of stellar XUV variability from planetary mixing and reveal the degree of diffusive separation.
  • Mass-loss rates inferred for HD209458b from isothermal Parker-wind fits are degenerate; the self-consistent model favors lower rates, $1.9\text{--}3\times10^{10}$ g/s, at the low end of the previously published $1\text{--}10\times10^{10}$ g/s range.
  • If the host star is as variable as the Sun, the He I transit depth of HD209458b should change over time, while H$\alpha$ should stay comparatively stable, offering a testable prediction.

Reading between the lines

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

  • Editorial inference: the crossover-mass discrepancy suggests the standard limiting-escape criterion is not valid for strongly ionized, non-isothermal outflows with eddy diffusion, so other planets modeled with the same machinery may also show more separation than simple criteria predict.
  • Editorial inference: because the new He($2^3S$) photoionization cross-section has strong resonances overlapping stellar coronal emission lines, the depth of the 10830 Å line may depend on the detailed shape of the stellar EUV spectrum, not just its integrated flux, which could explain some outliers in population studies.
  • Editorial inference: the metal run's enhanced electron density boosting recombination implies that He I 10830 Å absorption could serve as an indirect probe of electron abundance in the thermosphere, connecting helium observations to ionization balance and heating.
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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. The paper presents a 1D multi-species hydrodynamic escape model for the upper atmosphere of the hot Jupiter HD209458b, coupled to a lower/middle-atmosphere photochemical model. It updates several He I(2^3S) excitation and de-excitation rates, including a new high-resolution photoionization cross-section, a temperature-dependent Penning ionization rate, proton de-excitation, and photoelectron excitation. The model is used to compute He I 10830 Å and H-alpha transit depths, and a 'best-fit' model is obtained by adjusting the photoelectron heating efficiency and eddy diffusion coefficient. The central claims are that the model reproduces the observed He I transit depth and the H-alpha upper limit, that this requires a photoelectron heating efficiency of 20–40% with mass-loss rates of 1.9–3e10 g/s, and that strong diffusive separation of helium (He/H dropping from 8% to about 2.5% at high altitudes) explains the weak signals without invoking a helium-poor bulk composition. The paper also explores sensitivity to stellar activity, eddy diffusion, tidal forcing, and the inclusion of metals.

Significance. If the central claims hold, the paper provides a physically motivated explanation for the weak He I 10830 Å and H-alpha absorption of HD209458b, connecting it to reduced photoelectron heating and diffusive separation, and it offers new atomic data (high-resolution He I(2^3S) photoionization cross-sections) that are made publicly available in machine-readable form. The systematic comparison among an isothermal Parker model, a self-consistent upper-atmosphere model, and a full-atmosphere model is a useful demonstration of how lower-boundary assumptions and energy balance affect inferred mass-loss rates. The main results are, however, load-bearing on two points that are not yet established: the internal consistency of the diffusive-separation prediction with the paper's own crossover-mass analysis, and the strength of the H-alpha constraints relative to the most restrictive published upper limits. These issues require substantial additional analysis before the headline conclusions can be accepted.

major comments (3)
  1. [Section 4, Eq. (8)] The paper itself reports that the crossover-mass limit is Mdot_lim = 6.4e9 g/s at the pressure-averaged temperature of 5336 K, about one third of the best-fit mass-loss rate of 1.9e10 g/s; standard diffusion theory therefore predicts that helium should remain mixed with hydrogen. The rebuttal in the text is that the crossover equation assumes no eddy diffusion, constant temperature, and constant mixing ratio, but each of these corrections acts to increase mixing: eddy diffusion is an explicit mixing process, a temperature gradient does not suppress momentum transfer from H to He, and a strongly ionized upper atmosphere should further couple the species. The manuscript needs a quantitative reconciliation—for example, an altitude-resolved effective crossover mass or a direct comparison of diffusion and advection timescales in the region where separation occurs—before the central claim of strong diffusive separation can be considered supported.
  2. [Table 3 and Section 3.4] The photoelectron heating efficiency, the eddy diffusion coefficient Kzz, and the stellar activity level are treated as free parameters, and the best-fit model is selected by matching the same observed He I transit depth and H-alpha upper limit that are later used to validate the model. Because the heating efficiency and Kzz are degenerate (as the paper acknowledges in Section 3.5.2), the reported agreement does not by itself constrain the mass-loss rate or the degree of diffusive separation. Please provide an explicit parameter-space exploration or demonstrate that the conclusions are robust across the allowed range of Kzz and heating efficiency, ideally using an independent observable such as Ly-alpha or a simultaneous multi-line fit.
  3. [Section 3.4 and Figure 12] The claim that the model reproduces the observed H-alpha upper limit is based on the broad Jensen et al. (2012) limit, but the model's line-core absorption exceeds more restrictive published measurements: Astudillo-Defru and Rojo (2013) report -0.123% ± 0.012% in a 1.125 Å bandpass while the model predicts -0.4%, and Casasayas-Barris et al. (2021) imply an upper limit of about -0.3% in a 0.5 Å bandpass versus the model's -0.73%. The paper acknowledges these discrepancies but does not quantify whether the tidal or stellar-activity variations discussed in Section 3.5 are sufficient to resolve them; without such a demonstration, the H-alpha consistency claim in the abstract and conclusion is overstated.
minor comments (6)
  1. [Section 2.3] The star is referred to as 'HD2098458' in the paragraph on the stellar flux; this appears to be a typographical error for HD209458.
  2. [Section 4, Eq. (10)] The stellar wind density is quoted as 4000 g/cm^3, which is unphysical and must be a typographical error (presumably 4000 amu cm^-3 or a similar value); as written, the Chapman-Ferraro and hydropause estimates are not reproducible.
  3. [References] The reference Lavvas & Arfaux (2021) appears twice in the reference list; please remove the duplicate.
  4. [Section 2.4] The description of the secondary-electron scheme ('⌊E_p/Ecs⌋-1') would benefit from an explicit definition of the energy binning and the treatment of sub-threshold primaries; as written, the number of secondaries is ambiguous.
  5. [Table 2] In the He(2^3S) + e^- rate entries, the notation 'Υ31a 3' separates the oscillator-strength label from the factor of three in a way that is easy to misread; please restructure the rate column so that the oscillator strengths and numerical factors are unambiguous.
  6. [Section 3.5.2] The sentence stating that increasing the eddy diffusion coefficient 'reduces the mass loss rate by a factor of 1.1' is confusing; please report the actual mass-loss rates and the factor explicitly.

Circularity Check

1 steps flagged · score 6.0 of 10

He I match is a fit to the observed depth via the photoelectron heating efficiency, so the headline 'reproduces the observed He I transit depth' reduces by construction; diffusive separation and the Hα limit provide partial independent content.

  1. fitted input called prediction [Abstract; Section 3.4 'Best Fit Model']
    "Our model reproduces the observed He I transit depth and Hα upper limit, showing strong diffusive separation. We match the observations assuming a photoelectron efficiency of 20-40%, depending on the composition of the atmosphere, corresponding to mass-loss rates of 1.9-3×10^10 g/s. ... The free parameters of this model include the photoelectron heating efficiency, the eddy diffusion coefficient, and the activity level of the star, none of which are well constrained for this system."

    The observed He I 10830 Å depth is the target used to set the free photoelectron heating efficiency (and to select Kzz and activity level). The abstract then reports the resulting match as 'reproduces the observed He I transit depth', and the quoted mass-loss rate range is a consequence of that fitted efficiency rather than an independent model prediction. The Hα upper limit and the altitude-dependent He/H diffusive-separation profile are not fitted to the same observable and provide partial independent support, so the circularity is partial rather than total.

full rationale

The paper is largely a self-consistent forward model with updated atomic rates and full-atmosphere coupling, and much of its content is not circular: the T/P structure, the updated He I photoionization cross-sections and Penning rate, the sensitivity to stellar activity, the metal runs, the magnetosphere estimates, and the comparison with previous Parker-wind fits all stand on their own. The one clear reduction is the headline He I transit-depth 'reproduction': the photoelectron heating efficiency is a free parameter tuned to the observed He I depth (Section 3.4 explicitly lists it among free parameters), and the abstract then quotes the same matched depth as a success. The Hα upper limit is an independent, though weaker, constraint, and the diffusive-separation result is a genuine model output. However, its strength depends on the co-author-supplied low eddy diffusion coefficient Kzz = 1e5 m2/s (Arfaux & Lavvas 2023) and is internally in tension with the paper's own crossover-mass analysis (Section 4, Eq. 8), which predicts mixing at the fitted mass-loss rate. That tension is a physical-consistency concern, not a circular one. The updated atomic rates and cross-sections are ab initio or externally sourced and are not fitted to the target observables. Overall, the central He I agreement reduces by construction to a fitted parameter, while the Hα and diffusion content remain independent, giving partial circularity.

Assumptions & free parameters 5 free parameters · 8 assumptions · 0 invented entities

The central fit relies on several fitted/assumed parameters: photoelectron heating efficiency, eddy diffusion coefficient, stellar activity level, and continuum normalization choices. No new entities are introduced. The main physics assumptions are 1D geometry, diffusion approximation for multi-species transport, a simplified photoelectron secondary-electron treatment, and a solar-minimum stellar spectrum.

free parameters (5)
  • photoelectron heating efficiency = 0.40 best-fit H/He; 0.23 metal-rich; paper quotes 20-40%
    Fitted to match observed He I 10830 A depth and H alpha upper limit; not derived from first principles (Sections 3.4, 4).
  • eddy diffusion coefficient Kzz = 1e5 m2/s (best fit); 1e6-1e7 tested
    Chosen from Arfaux & Lavvas (2023); literature ranges 1e6-1e9 m2/s; directly controls diffusive separation and is degenerate with heating efficiency (Section 3.5.2).
  • stellar activity level (XUV spectrum) = solar minimum baseline; solar average and maximum tested
    The actual HD209458 XUV SED is not measured; this sets photoionization and heating rates, changing mass-loss and He I depth by large factors (Sections 2.3, 3.5.1).
  • photochemical haze production rate = 1e-14 g cm-2 s-1
    Adopted to improve the lower/middle atmosphere continuum fit; affects transit normalization around the lines (Section 2.1).
  • JWST vs HST continuum shift = Delta d = 0.000169 (Eureka), 0.000179 (Sparta)
    Fitted by BIC to align JWST and HST transit spectra; affects continuum level, not line-core physics (Section 3.3).
assumptions (8)
  • domain assumption The terminator transit geometry is represented by a 1D dayside-averaged illumination (solar zenith angle 60 degrees, incident flux divided by 2).
    Used in best-fit model; substellar geometry gives different transit depths, so transit geometry is load-bearing (Sections 2.3, 3.5.3).
  • domain assumption Multi-species transport is approximated with a diffusion approximation instead of separate momentum equations per species.
    The paper asserts this reproduces Xing et al. 2023 and Schulik & Owen 2024, but the discrepancy with the crossover mass equation makes the approximation important (Sections 1, 3.2, 4).
  • ad hoc to paper Best-fit model uses Kzz=1e5 m2/s with no independent constraint on eddy diffusivity.
    Controls the degree of diffusive separation; cited literature spans 1e6-1e9 m2/s (Section 3.5.2).
  • domain assumption The He(2 3S) recombination rate is obtained by assuming all higher triplet states feed 2 3S and ground-state recombination equals total minus triplet case B.
    Used for metastable helium production; affects transit depth (Section 2.4).
  • domain assumption Photoelectron excitation is bounded with a simplified secondary-electron scheme that ignores collisions of primaries with the background gas.
    Paper uses this only as an upper limit and finds negligible effect on HD209458b (Section 2.4, Appendix).
  • domain assumption H(n=2) population and H alpha depth rely on a plane-parallel Ly-alpha Monte Carlo model iterated with the escape model.
    H alpha is almost entirely controlled by Ly-alpha excitation, so radiative-transfer geometry matters (Sections 2.5, 3.5).
  • domain assumption The stellar XUV spectrum is a scaled solar minimum spectrum, with solar average/maximum for sensitivity tests.
    HD209458 is Sun-like and inactive, but the actual EUV spectrum is unmeasured; He I depth is sensitive to this input (Section 2.3).
  • standard math Reaction rates from measured cross-sections are averaged over a Maxwell-Boltzmann velocity distribution.
    Standard kinetic theory used for Penning ionization and proton de-excitation rates (Appendix).

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Cite this review

Pith. "Pith review of A Multi-Species Atmospheric Escape Model with Excited Hydrogen and Helium: Application to HD209458b." pith.science (2026). https://pith.science/paper/DGJPDSCI

@misc{pith2026250608232,
  author       = {Pith},
  title        = {Pith review of: A Multi-Species Atmospheric Escape Model with Excited Hydrogen and Helium: Application to HD209458b},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DGJPDSCI}},
  note         = {Machine review of arXiv:2506.08232}
}
abstract

Atmospheric escape shapes exoplanet evolution and star-planet interactions, with He I 10830 \AA\ absorption serving as a key tracer of mass loss in hot gas giants. However, transit depths vary significantly across observed systems for reasons that remain poorly understood. HD209458b, the archetypal hot-Jupiter, exhibits relatively weak He I 10830 \AA\ and H$\alpha$ absorption, which has been interpreted as evidence for a high H/He ratio (98/2), possibly due to diffusive separation. To investigate this possibility and other processes that control these transit depths, we reassess excitation and de-excitation rates for metastable helium and explore the impact of diffusion processes, stellar activity, and tidal forces on the upper atmosphere and transit depths using a model framework spanning the whole atmosphere. Our model reproduces the observed He I transit depth and H$\alpha$ upper limit, showing strong diffusive separation. We match the observations assuming a photoelectron efficiency of 20-40\%, depending on the composition of the atmosphere, corresponding to mass-loss rates of $1.9-3\times10^{10}$ g/s. We find that the He I 10830 \AA\ transit depth is sensitive to both stellar activity and diffusion processes, while H$\alpha$ is largely unaffected due to its strong dependence on Lyman-$\alpha$ excitation. These differences may help explain the system-to-system scatter seen in population-level studies of the He I line. While He I data alone may not tightly constrain mass-loss rates or temperatures, they do confirm atmospheric escape and help narrow the viable parameter space when interpreted with physically motivated models. Simultaneous observations of He I, H$\alpha$, and stellar activity indicators provide powerful constraints on upper atmosphere dynamics and composition, even in the absence of full transmission spectra.

Figures

Figures reproduced from arXiv: 2506.08232 by the authors.

Figure 1
Figure 1. Recombination rates to the metastable He I (23S) triplet state versus electron temperature from different sources. shows the calculated cross-section compared with the Norcross (1971) cross-section. In the right-hand panel of [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. The photoionization cross section for He I (23S) versus wavelength. The black line indicates what has been used in previous literature (Norcross 1971), while the pink lines show high-resolution cross-sections used in our model. Penning ionization with hydrogen is another important loss mechanism for He I (23S), especially in the lower thermosphere and below it (see Section 3.4). Previous studies have used a temperat… view at source ↗
Figure 3
Figure 3. Isothermal model (Model A) results for HD209458b. The left panel shows the temperature (pink) and velocity (orange) profiles as a function of planetary radii, with a mass loss rate of 1.86×1010 g/s. The right panel presents the number densities of key atmospheric species, including neutral hydrogen (H), ionized hydrogen (H+), neutral helium (He), ionized helium (He+), doubly ionized helium (He2+), metastable helium … view at source ↗
Figures from the paper (17 more)
Figure 4
Figure 4. Figure 4: Transit depth results from our baseline model (Model A), comparing the modeled He I 10830 ˚A (left) and H I Balmer α (right) transit depths from our baseline model (orange solid lines) with the best-fit model to the observed transit depth for the He I 10830 ˚A line fro…
Figure 5
Figure 5. Figure 5: Left: Temperature and bulk outflow velocity predicted by our full upper atmosphere model for HD209458b (Model B). Right: Composition predicted for HD209458b by Model B. Note that the H+ profile is practically identical to the electron (e−) density profile. Lamp´on et a…
Figure 6
Figure 6. Figure 6: The elemental H/He ratio (left panel) and transport and diffusion timescales for helium (middle panel) and the eddy versus molecular diffusion coefficients (right panel) for our upper atmosphere model (Model B) [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: Left: Excited helium number density comparison between our isothermal model (Model A) with no multi-species processes in orange and our upper atmosphere model (Model B) in teal. Middle: Predicted transit depth for He I (23S) for our upper atmosphere model (Model B) in …
Figure 8
Figure 8. Figure 8: Left: P-T profile of our full atmosphere model (Model C). Middle: Transmission spectrum around the He I triplet. In teal is the self-consistent upper atmosphere model (Model B), and in pink is the full atmosphere model (Model C). Right: Transmission spectrum around the…
Figure 9
Figure 9. Figure 9: Transit depth versus wavelength for our full atmosphere model of HD209458b (Model C). We zoom in past the extent of the Hα and He I 10830 ˚A line to show the comparison with HST observations (Sing et al. 2016) and JWST observations (with different data reduction pipeli…
Figure 10
Figure 10. Figure 10: Left: Temperature and bulk outflow velocity predicted by our best-fit atmosphere model for HD209458b. Middle: Composition predicted for HD209458b. Right: The elemental H/He ratio for our best-fit model. nbar is 5336 K, and above 7 nbar it is 7970 K. Somewhat by design…
Figure 11
Figure 11. Figure 11: Left: Production mechanisms relevant for the formation of He I (23S). Right: Loss mechanisms relevant for the formation of He I (23S). Hα transit depth is also consistent with the broad upper limit from Winn et al. (2004). Using the same observations as Winn et al. (2…
Figure 12
Figure 12. Figure 12: Transit depth results from our best-fit model, comparing the modeled He I 10830 ˚A (left) and H I Balmer α (right) transit depths from our best-fit model (green solid lines) with the best-fit to the observed transit depth for the He I 10830 ˚A line from Alonso-Florian…
Figure 13
Figure 13. Figure 13: Transmission spectrum around the He I triplet (left) and the H-α Balmer line (right). The transit depth produced using a maximum, average, and minimum solar spectrum is in pink, orange, and yellow respectively [PITH_FULL_IMAGE:figures/full_fig_p019_13.png]
Figure 14
Figure 14. Figure 14: In contrast, the effect of diffusion is much weaker for the Hα transit depth (right panel of [PITH_FULL_IMAGE:figures/full_fig_p019_14.png]
Figure 15
Figure 15. Figure 15: Transmission spectrum around the He I triplet (left) and the H-α Balmer line (right). The model in green is our best-fit model, in blue is the zenith angle of 60◦ with tidal forcing, and in purple is the model run at the substellar line with tidal forcing. balance. Th…
Figure 16
Figure 16. Figure 16: Left: Temperature and bulk outflow velocity predicted by our atmosphere model for HD209458b including heavy elements (blue) compared to our best-fit model (green). Middle and Right: Transmission spectrum around the He I triplet (middle) and the H-α Balmer line (right)…
Figure 17
Figure 17. Figure 17: Plasma beta (β) as a function of radius (r/Rp) at the magnetic equator (pink) and pole (teal). The high values of β indicate that thermal pressure dominates over magnetic pressure at all altitudes, leading to an extended and dynamic magnetosphere influenced by planeta…
Figure 18
Figure 18. Figure 18: illustrates a schematic overview of all the radiative and collisional transitions considered in our calculation. The radiative transition between the metastable state and the 23P state, responsible for the 10830 ˚A line, is not included in the modeled reaction network…
Figure 19
Figure 19. Figure 19: Cross sections versus energy for for proton de-excitation (top left, Augustoviˇcov´a et al. (2014)), and penning ionization (top right, Cohen & Lane (1971); Morgner & Niehaus (1979)). Reaction rates (cm3 s −1 ) versus temperature (K) for proton de-excitation (bottom l…
Figure 20
Figure 20. Figure 20: Photoelectron cross-section as a function of energy for the impact reaction between photoelectrons and helium. REFERENCES Allan, A. P., & Vidotto, A. A. 2025, arXiv e-prints, arXiv:2504.02578, doi: 10.48550/arXiv.2504.02578 Alonso-Floriano, F. J., Snellen, I. A. G., C…

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