## Appendix D: Optical depths

<figure><img src="image_27.png" /><figcaption>Fig. D.1. 3D-PDR calculations using a density distribution representing the $A_{V,eff} - n_H$ relationship (Bisbas et al. 2019). The figure shows the optical depth of $^{13}$CO(4–3) and C$^{18}$O(4–3) lines as a function of the H$_2$ volume density. We find that both lines remain optically thin for $n_{H_2} \le 10^6$ cm$^{-3}$, while the $^{13}$CO(4–3) starts to have moderate optical depth with increasing H$_2$ densities.</figcaption></figure>

We performed a PDR calculation to explore how the optical depth of $^{13}$CO(4–3) and C$^{18}$O(4–3) builds along with the cloud depth. We constructed a density distribution resulting from the $A_{V,eff} - n_H$ relation presented in Bisbas et al. (2019) and Bisbas et al. (2023). This empirical relation connects the local visual extinction with the most probable local number density. The calculation was performed using an FUV intensity of $\chi/\chi_0 = 10$ (normalized to the spectral shape of Draine 1978), a CR ionization rate of $\zeta_{CR} = 10^{-13}$ s$^{-1}$ (as derived from Sect. 7), and a metallicity of $1Z_{\odot}$. The microturbulent velocity controlling the turbulent heating was taken to be $v_{turb} = 10$ km s$^{-1}$, and the velocity dispersion was taken to be $v_{disp} = 180$ km s$^{-1}$ to be in accordance to the observed linewidth. Furthermore, we assumed a $^{12}$C/$^{13}$C isotope ratio of 30 and $^{12}$CO/$^{18}$O of 200 (e.g., Martín et al. 2019b; Tang et al. 2019).

Figure D.1 shows the optical depths of the $^{13}$CO(4–3) and C$^{18}$O(4–3) lines as a function of the H$_2$ density, $n_{H_2}$. For values of $n_{H_2} \le 10^6$ cm$^{-3}$, both lines remain optically thin. However, when $n_{H_2}$ reach beyond $10^6$ cm$^{-3}$, the $^{13}$CO(4–3) line starts to have a moderate optical depth while C$^{18}$O(4–3) remains optical thin until the density getting close to $10^7$ cm$^{-3}$. Both lines start to get optically thick if $n_{H_2} \ge 10^7$ cm$^{-3}$. In the conditions of the $n_{H_2}$ between $10^6$ and $10^7$ cm$^{-3}$, the optical depth of the $^{13}$CO(4–3) will reach about 10. Such an optical depth will lead to a low

flux ratio of $^{13}$CO/$^{18}$O close to unity and a very low flux ratio of $^{12}$CO/$^{13}$CO, as pointed out by Zhang et al. (2018). However, the observed $^{12}$CO/$^{13}$CO ratios in both our sources (Table 9) are about 20–30, inconsistent with a optically thick scenario of the $^{13}$CO.

## Appendix E: Detailed chemical modeling of HCO$^+$

The left panel of Fig. E.1 displays the HCO$^+$ abundance versus the local number density, $n_H$, for five different CR ionization rates ($\zeta_{CR} = 10^{-16} - 10^{-12}$ s$^{-1}$). The FUV radiation is neglected in these models, as mentioned in the main text, and the microturbulent optical depth expression (Eq. 4) is adopted. Overall, the abundance of HCO$^+$ peaks at densities in the range of $n_H \approx 10^4$ to a few $10^5$ cm$^{-3}$, with the effect to be better seen when $\zeta_{CR} \ge 10^{-13}$ s$^{-1}$. In the low end of the above $\zeta_{CR}$ values, HCO$^+$ is formed mainly via the reactions

$$H_2 + HOC^+ \rightarrow HCO^+ + H_2 \quad (R1)$$

$$H_2 + CO^+ \rightarrow HCO^+ + H \quad (R2)$$

R1 and R2 together form approximately 80% of the total HCO$^+$ abundance, with R2 becoming more effective for densities $n_H \le 10^2$ cm$^{-3}$. For $n_H \ge 10^3$ cm$^{-3}$, the reaction

$$H_3^+ + CO \rightarrow HCO^+ + H_2 \quad (R3)$$

dominates the formation route of HCO$^+$. For very high $\zeta_{CR}$ values, HCO$^+$ abundance builds rapidly when $n_H \ge a$ few $10^3$ cm$^{-3}$, below which is in practice negligible (Fig. E.1). Contrary to the aforementioned case, R2 remains, here, the main reaction that is able to form more than 50% of the total HCO$^+$ abundance. In all cases, HCO$^+$ is destroyed with an efficiency of almost 100% due to its reaction with free electrons, leading to the formation of CO and H.

The right panel of Fig. E.1 shows the temperature profile of HCO$^+$. The temperature profile varies as a function of the CR ionization rate since it inserts heating. We performed various runs with different CR ionization rates, and we have overplotted iso-($\zeta/n$) lines, the range of which corresponds to the solutions we found for the two galaxies. Assuming the typical density for the dense gas, we find that their gas at high column densities is relatively warm (~50–100 K), consistent with the LVG-derived gas temperatures.

For low densities, an interesting behavior occurs in which the gas temperature decreases as the CR ionization ray increases. This is counter-intuitive, given that CRs heat up the gas. The decrease in gas temperature occurs because the production of C$^+$ abundance increases, making the emission line of [C II] increase. The latter is a strong coolant, which in turn decreases the overall gas temperature at low densities (Bisbas et al. 2017).
<figure><img src="image_28.png" /><figcaption>**Fig. E.1.** Chemical models for HCO⁺. Left panel: HCO⁺ abundance versus the local total H-nucleus number density, $n_H$, for different CR ionization rates. In all cases, the abundance of HCO⁺ peaks for densities $8 \times 10^3 \le n_H \le 3 \times 10^5 \text{ cm}^{-3}$. Right panel: Temperature profile, with the iso-(ζ/n) lines overplotted in orange, for different CR ionization rates ($\zeta_{CR} = 10^{-16} - 10^{-12} \text{ s}^{-1}$, with the same color code as the left panel).</figcaption></figure>