Broadening the Canonical Picture of EUV-Driven Photoevaporation of Accretion Disks

0000-0002-1803-0203]Riouhei Nakatani
NASA Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Dr, Pasadena, CA 91109, USA
RIKEN Cluster for Pioneering Research, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
0000-0001-8292-1943]Neal J. Turner
NASA Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Dr, Pasadena, CA 91109, USA
0000-0003-3882-3945]Shinsuke Takasao
Department of Earth and Space Science, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan

Photoevaporation driven by hydrogen-ionizing radiation, also known as extreme-ultraviolet (EUV), profoundly shapes the lives of diverse astrophysical objects.
Focusing here mainly on the dispersal of protoplanetary disks, we construct an analytical model accounting for the finite timescales of photoheating and photoionization.
The model offers improved estimates for the ionization, temperature, and velocity structures versus distance from the central source, for a given EUV emission rate and spectral hardness.
Compared to the classical picture of fully-ionized and isothermal winds with temperatures $\approx 10^4\Kelvin$ and speeds $\approx 10\kms$, our model unveils broader hydrodynamical and thermochemical states of photoevaporative winds.
In contrast to the classical picture, T Tauri stars with EUV luminosities around $10^{30}\erg\sec^{-1}$ have non-isothermal ionized winds at lower temperatures than the classical value if the spectrum is soft, with an average deposited energy per photoionization less than about 3.7 eV.
Conversely, if the spectrum is hard, the winds tend to be atomic and isothermal at most radii in the disk.
For lower EUV intensities, even with soft spectra, atomic winds can emerge beyond $\sim 10\au$ through advection.

We demonstrate that the analytical model's predictions are in general agreement with detailed radiation-hydrodynamics calculations.
The model furthermore illustrates how the energy efficiency of photoevaporation varies with the intensity and spectral hardness of the EUV illumination, as well as addressing discrepancies in the literature around the effectiveness of X-ray photoevaporation.
These findings highlight the importance of considering the finite timescales of photoheating and photoionization, both in modeling and in interpreting observational data.

§ INTRODUCTION

Hydrogen-ionizing radiation, also called extreme-ultraviolet (EUV;$13.6\lesssim h\nu \lesssim 100 \eV$), is widely recognized for its crucial role in various astrophysical contexts, including planetary atmospheres <cit.>, protoplanetary disks <cit.>, molecular clouds <cit.>, and galactic minihalos <cit.>.
EUV radiation induces not only photoionization of hydrogen-dominated gas but also associated heating, driving photoevaporative winds that can disperse structures within a finite time.
Essentially, this means that ionizing radiation has exerted a significant influence on the evolution and formation processes of planets, stars, and galaxies since the dawn of the cosmic reionization epoch.

Classically, it is commonly assumed that EUV-heated gas is fully ionized and exhibits temperatures of $\approx 10^4\Kelvin$, regulated by a balance between heating and cooling processes, resulting in supersonic photoevaporative winds.
However, this scenario is valid only when the timescales of heating and ionization are much shorter than the dynamical timescales, and thus it could vary widely depending on factors such as the magnitude of gravity exerted by the object involved, its configuration, and EUV radiation environments, e.g., EUV flux and spectrum.
Consequently, a diverse range of modes is possible for EUV-driven photoevaporation.
Given that EUV photoevaporation occurs in diverse environments, there is a pressing need for a systematic and comprehensive understanding of this process. However, such an understanding is not yet fully developed.

In the context of protoplanetary disk dispersal, which constitutes the primary focus of this study, EUV-driven photoevaporation is considered as one of the mechanisms responsible for the final-stage dispersal of the gas disks [Alexander et al., 2014, Pascucci et al., 2023],

consequently determining the time limit available for (gas giant) planet formation.
Therefore, a comprehensive understanding of EUV photoevaporation is of great importance for advancing the planet formation theory.

The current canonical picture of EUV-driven photoevaporation in protoplanetary disks can be outlined as follows:
EUV energy is injected into the H1 layer of the disk just ahead of the ionization front, where the EUV optical depth $\tau_\mathrm{EUV}$ is approximately unity (fig:schematic),
Subsequently, the gas undergoes immediate photoionization to form ionized thermal winds emanating from the disk surface.
These winds typically attain around $\approx 10^4\Kelvin$, a value determined by the thermal balance between photoionization heating and various radiative cooling within the H2 region, including metal forbidden lines, Ly$\alpha$ emissions, and radiative recombination.
The resulting sound speed corresponds to $\approx 10\kms$, providing a typical speed of the transonic flows.

This conceptual framework was initially proposed in the context of disk photoevaporation around massive stars [Hollenbach et al., 1993, Hollenbach et al., 1994] and has been applied to the EUV photoevaporation of protoplanetary disks in general, including around young low-mass stars <cit.>.

However, the validity of this generalization is not straightforward, given that the typical EUV emission rate $\Phi_\mathrm{EUV}$ of young low-mass stars <cit.>
is likely several orders of magnitude less than that of massive stars <cit.>.
Qualitatively, the lower $\Phi_\mathrm{EUV}$ implies a longer time required for heating and ionizing the gas.
Consequently, the canonical picture, which implicitly assumes an infinitely short photoheating and photoionization timescale, is expected to be applicable only in cases where $\Phi_\mathrm{EUV}$ is sufficiently high (see also discussions in Appendix A of Hollenbach et al., 1994).

Moreover, the EUV spectrum of massive stars notably includes a significant contribution from photospheric emission with an effective temperature of $ \approx 30000$–$50000\Kelvin$.
This spectral attribute is softer in comparison to the EUV spectrum of young low-mass stars,
where metal line emissions from $\sim 10^5\Kelvin$ plasma are predominant [Shoda & Takasao, 2021].
Since the absorption cross-section of the gas for EUV is generally lower for higher photon energies, the difference in spectrum hardness can profoundly influence even the qualitative chemo-dynamical aspects of EUV-driven photoevaporation.

These considerations regarding application limits raise pertinent questions: Under what parameter space (EUV emission rate, spectral hardness, and distance from the source) does the canonical picture hold true? And what chemical and hydrodynamical states might an EUV-driven wind exhibit in the other parameter spaces?
How might these insights be relevant to photoevaporation induced by alternative photoheating processes, such as far-ultraviolet (FUV) grain photoelectric heating and X-ray photoionization heating?
To tackle these questions, we discuss how the picture of EUV photoevaporation evolves contingent upon the stellar mass $M_*$, EUV emission rate $\Phi_\mathrm{EUV}$, and EUV spectrum while incorporating the finite timescales of photoheating and photoionization.
We develop an analytical model based on phenomenology that adequately treats these timescales as well as the dynamical timescale of winds and recombination timescale.
Our phenomenological model draws inspiration from the analytical model for the Compton-heated winds of active galactic nuclei presented in Begelman et al., 1983 (hereafter ).

This paper is organized as follows.
We begin by outlining the basic setup and introducing fundamental quantities for our model in sec:analytic_theory.
Following this, in sec:classification, we provide a qualitative overview of the key findings from our model, reserving detailed quantitative derivations for sec:cs.
These sections delve into discussing the typical temperature, wind speed, and ionized state EUV-driven winds would have across various regions of the parameter space.
In sec:comparison_HD_simulations, we compare model predictions with hydrodynamics simulations incorporating self-consistent radiative transfer and nonequilibrium thermochemistry to assess the predictability.
To demonstrate the practical utility of the model in understanding the physical properties of photoevaporating disks, we explore the radial extension of the H2 region in sec:limiting_radius and analyze the energy efficiency of EUV heating in driving winds in sec:work_efficiency.
In sec:discussion, we extend the application of our model to photoevaporation driven by far-ultraviolet (FUV) and X-ray radiation.
Based on this application, we address a possible cause for a long-standing issue in the field — the divergent conclusions regarding the effectiveness of X-ray photoevaporation.
Additionally, caveats and model limitations are presented in this section.
Finally, we provide a summary of our major findings and outline potential avenues for future prospects in sec:summary.

§ FUNDAMENTAL QUANTITIES

The purpose of this study is to construct an updated picture of EUV-driven thermal winds.
To that end, this section focuses on introducing the fundamental setup of the model and the key quantities for the energetics and thermochemical structures of the system, such as the gravitational timescale and the rates of photoheating and photoionization.
A list of the symbols used in our model is available in Appendix <ref>.

The basic setup of our model is schematically depicted in fig:schematic:
a geometrically thin disk predominantly composed of hydrogen is irradiated by the central EUV source.
Hydrogen exists in atomic form in the upper layers of the disk due to the photodissociation of molecular hydrogen, which is the dominant species in the deeper layers.

The energy of EUV photons is deposited to the gas at the atomic layer by heating associated with photoionization, H + EUV -> H+ + e-,
and the gas can escape from the gravitational binding of the central object if it has reached a sufficiently high temperature.
This escaping gas gives rise to steady thermal winds emanating from the so-called base, corresponding to the optically thick surface of the disk where $\tau_\mathrm{EUV}\sim 1$.

Schematic picture of a disk under EUV-driven photoevaporation.
The radiation source with a mass of $M_*$ and an EUV emission rate of $\Phi_\mathrm{EUV}$ is illuminating the disk.
The EUV photon hits the disk surface, where the optical depth to EUV is approximately unity, and photoionizes atomic hydrogen there, depositing excessive energy remaining after photoionization to the gas.
As a result, photoevaporative winds arise from the disk surface.
Vigorous winds are possible only beyond the critical radius $R_\mathrm{c}$ owing to weak gravity at a distance.
In a classical picture, the winds are ionized and have an equilibrium temperature of $\approx 10^4\Kelvin$.
The corresponding isothermal sound speed is $c_\mathrm{s} \approx 10\kms$, which sets a typical flow speed of the winds due to the thermally-driven nature of photoevaporation.

§.§ Critical Radius

The winds can have a temperature of $\approx 10^4\Kelvin$ (thermal equilibrium temperature) at most because cooling due to forbidden lines and Ly$\alpha$ emission quickly becomes effective as the temperature approaches the equilibrium value.
The corresponding sound speed is $\ceq  \approx 10\kms$, and this sets the order of a maximum achievable speed for EUV-driven winds.

We can define a threshold radius below which EUV-heated gas cannot escape from the gravitational binding even when it achieves the maximum flow speed $\ceq $.
Hollenbach et al., 1994 introduce such a threshold radius (termed “gravitational radius”) as

\[
R_{\rm g} \equiv \frac{GM_*}{\ceq ^2}.
\]

through dimensional analysis comparing the thermal energy and gravitational potential energy.

The gravitational radius exactly corresponds to the Compton radius $R_{\rm IC}$ in .
predicted that the actual threshold radius separating the wind-excited and wind-inhibited regimes might be present at the cylindrical distance from the source of $R \approx 0.1$–$1R_{\rm IC}$ instead of $R = R_\mathrm{IC}$.
This is directly examined by hydrodynamics simulations in Woods et al., 1996, and they found the actual threshold radius to be $\approx 0.2R_{\rm IC}$.

Liffman, 2003 suggests that the threshold radius of $0.2R_{\rm IC}$ corresponds to the radius at which the sum of photoheated gas's enthalpy and mechanical energy equals zero.
Although the physical origin of the threshold radius is still worth exploring, in this study, we follow Liffman, 2003 to define the “critical radius” as

\begin{equation}
R_{\rm c}
% = \frac{\gamma -1}{2\gamma} R_{\rm g}
\equiv \frac{R_{\rm g}}{2c_p},
\label{eq:critical_radius}
\end{equation}

$c_p$ is a normalized specific heat at constant pressure

\[
c_p \equiv \frac{\gamma}{\gamma - 1 }
\]

with $\gamma$ being the specific heat ratio of the gas.
eq:critical_radius is a modified version of $R_\mathrm{g}$, including correcting factors from enthalpy and centrifugal force due to gas's angular momentum.

We define the gravitational velocity $\vg $ as

\begin{equation}
\vg
\equiv \sqrt{\frac{GM_*}{2c_p R}}
= \ceq    x ^{-1/2},
\label{eq:vg_ceq}
\end{equation}

so that $\ceq  = \vg $ at $R = R_\mathrm{c}$.
Here, $ x $ is a normalized distance

\[
x  \equiv \frac{R}{R_\mathrm{c}}.
\]

The corresponding gravitational timescale is

\begin{equation}
t_{\rm g} \equiv
% \frac{\sqrt{2c_p}R_0}{v_{\rm K}} =
\frac{R}{\vg }
\sqrt{\frac{2c_p R^3}{GM_*}},
\label{eq:gravitational_timescale}
\end{equation}

The normalized radius $x$ corresponds to the normalized radius “$\xi$” in  except for the factor of $1/2c_p$ and the equilibrium temperature.
We adopt denoting “$x$” for the normalized distance to highlight these differences and to avoid confusion with the ionization parameter, which is often denoted by $\xi$.

In this paper, we focus on the region where vigorous winds possibly form,
and thus we only consider the region with $ x  \geq 1$, which is a necessary condition for EUV-heated gas to escape freely without being inhibited by gravity.
We leave modeling the inner $x < 1$ region for future studies, since its nearly hydrostatic structure requires a different approach from the one presented in this study, as discussed in sec:discussion:model_limitation.

§.§ EUV Heating Rate

We calculate the frequency-dependent (specific) photon number flux from the point radiation source with radial ray tracing,

\[
F_\nu (r, N_{\rm HI})= \frac{\Phi_\nu }{4\pi r^2} e^{-\sigma_\nu N_{\rm HI}},
\]

where $\nu$ is EUV frequency,
$\Phi_\nu$ is the specific EUV emission rate of the radiation source,
$r$ is the spherical distance from the source,
$\sigma_\nu $ is the absorption cross-section of atomic hydrogen for EUV,
and $N_{\rm HI}$ is the radial column density of atomic hydrogen along an EUV ray defined as

\[
N_\mathrm{HI} \equiv \int \dd r \, y_\mathrm{HI} \nh
\]

with $y_\mathrm{HI}$ and $\nh$ being atomic hydrogen abundance and hydrogen nucleus number density, respectively. The atomic hydrogen number density is computed by $n_\mathrm{HI} = y_\mathrm{HI}\nh$.

Energetic EUV photons can also be absorbed by He1 and He2 in general, but we ignore their contributions.
This simplification has a minimal impact on the results of our order-of-magnitude phenomenological discussions in this study.
We will delve into the effect of this simplification in more detail in sec:discussion:model_limitation.

The expression of $F_\nu$ implicitly assumes the direct EUV component dominating over the diffuse component.
This assumption is based on the findings of [Tanaka et al., 2013], where the authors performed full 2D radiative transfer calculations and observed that the direct EUV dominates over the diffuse EUV.
This is in contrast to the classical findings of Hollenbach et al., 1994, where they employed the 1+1D radiative transfer and concluded that the inner ionized atmosphere completely cuts off the direct EUV within the gravitational radius.
This discrepancy likely results from the difference in the treatment of the diffuse component transfer.
Similarly to Tanaka et al., 2013, the direct EUV has been observed to reach the outer radii in more recent self-consistent radiation hydrodynamics simulations [Nakatani et al., 2018, Nakatani et al., 2018, Komaki et al., 2021, Nakatani & Takasao, 2022], even when the critical radius is well resolved.

A nearly hydrostatic, ionized atmosphere is formed at inner radii, agreeing with the prediction of the previous hydrostatic models, but it does not terminate the radial EUV rays within the gravitational radius.
These updated results justify our assumption on $F_\nu$ above.

As for photoheating, the excess energy of EUV photons over the ionization potential energy ($h\nu_1 \approx 13.6\eV$) is responsible for EUV heating, and thus the specific heating rate is

\begin{equation}
\begin{split}
\Gamma_{\rm EUV} \equiv
&    \frac{y_\mathrm{HI}}{m} \int_{\nu_1}^{\nu_\mathrm{max}} \sigma_\nu (h\nu - h\nu_1) \frac{\Phi_\nu }{4\pi r^2}
e^{-\sigma_\nu N_{\rm HI}}\, \dd \nu \\
&   =   \frac{y_\mathrm{HI} }{m}  F_0
\delta \langle \sigma \rangle
\langle \Delta E \rangle_i
% \braket{\equiv \frac{y_\mathrm{HI}}{m} \frac{R_*^2}{r^2} \gamma_\mathrm{EUV}}
\end{split}
\label{eq:gamma_EUV}
\end{equation}

where $m$ is the total gas mass per hydrogen nucleus, which can include the mass of other minor elemental species, like Helium, and is related to the gas density $\rho$ through $m \equiv \rho/\nh$;
$\nu_\mathrm{max}$ is the upper limit of the EUV frequency, typically set to $\sim 100\eV$;
and $F_0 $ is the unattenuated total EUV number flux at the spherical distance $r$,

\[
F_0 \equiv  \int_\mathrm{\nu_1}^\mathrm{\nu_\mathrm{max}} \dd \nu \, {F_\nu (r, 0)}  = \frac{\phieuv }{4\pi r^2}.
\]

Here, $\phieuv $ is the EUV emission rate of the radiation source.
The dimensionless quantity $\delta$ is the ratio of the attenuated flux to the unattenuated flux $F_0$ at $N_{\rm HI}$,

\[
\delta (N_{\rm  HI})\equiv
\dfrac{\int \dd \nu \,  F_\nu (r, N_\mathrm{HI}) }{F_0}
\dfrac{\int \dd \nu \, \Phi_\nu \exp \braket{- \sigma_\nu N_{\rm HI}}}{\phieuv }.
\]

This parameter describes the fraction of the EUV photons remaining at $N_{\rm HI}$ without being attenuated.
The average cross-section of the remained EUV, $\langle \sigma \rangle $, is defined by

\[
\langle \sigma \rangle (N_{\rm  HI}) \equiv
\frac{\int \dd \nu \, \sigma_\nu F_\nu (r, N_\mathrm{HI})}{\int \dd \nu \, F_\nu (r, N_\mathrm{HI}) },
\]

and $\langle \Delta E \rangle_i$ is an average deposited energy per ionization at $N_{\rm HI}$

\[
\langle \Delta E \rangle_i (N_{\rm  HI}) \equiv
\frac{\int \dd \nu \, \sigma_\nu (h\nu - h\nu_1) F_\nu (r, N_\mathrm{HI})}{\int \dd \nu \, \sigma_\nu F_\nu (r, N_\mathrm{HI})}.
\]

All of $\delta$, $\langle \sigma \rangle$, and $\langle \Delta E \rangle_i$ are dependent only on $N_{\rm HI}$ and a priori calculable, once $\Phi_\nu$ is given.
The former two are monotonically decreasing functions because the remaining EUV photons decrease as they go through the gas, and the average energy of the remaining photons increases to have a smaller cross-section on average. On the other hand, $\langle \Delta E \rangle_i$ is a monotonically increasing function since the average energy of remaining EUVs increases as they go through the gas.
The product $ \delta \langle \sigma\rangle \langle \Delta E\rangle_i$ is a monotonically decreasing function with respect to $N_{\rm HI}$,
which simply indicates that the less attenuated, the more energy is available for heating.

For convenience, we rewrite the product $ \delta \langle \sigma\rangle \langle \Delta E\rangle_i$ in the dimensionless form of

\[
\begin{split}
\chi_\mathrm{e} & \equiv \delta
\braket{\frac{\langle\sigma\rangle}{\avesigma }}
\braket{\frac{\langle \Delta E \rangle_i }{\aveenergy }}
\\ &
=\int \dd \nu \frac{\Phi_\nu}{\phieuv} \frac{\sigma_\nu}{\avesigma} \frac{h\nu - h\nu_1}{\aveenergy} e^{-\sigma_\nu N_\mathrm{HI}}.
\end{split}
\]

\[
\begin{gathered}
\avesigma  \equiv \left.\langle \sigma \rangle\right|_{N_{\rm HI} = 0}
= \phieuv^{-1} \int \dd \nu \, \sigma_\nu \Phi_\nu
% \quad
% \tilde{\sigma} \equiv \langle\sigma\rangle/\avesigma ,\\
\\
\begin{aligned}
\aveenergy  &
\equiv \left. \langle \Delta E \rangle_i\right|_{N_{\rm HI} = 0}
\\
& = \braket{\int \dd \nu \, \sigma_\nu \Phi_\nu  }^{-1} \int \dd \nu \, \sigma_\nu \Phi_\nu \braket{h\nu - h\nu_1}
\end{aligned}
% \quad
% \tilde{E} \equiv \frac{\langle \Delta E \rangle_i }{\aveenergy } .
\\
\end{gathered}
\]

With these notations, eq:gamma_EUV can be expressed in a simpler form

\begin{equation}
\Gamma_\mathrm{EUV}
= \frac{y_\mathrm{HI}}{m}
\frac{\phieuv }{4\pi r^2}\avesigma  \aveenergy  \chi_\mathrm{e}.
\label{eq:gamma_EUV:2}
\end{equation}

The attenuation factor $\chi_\mathrm{e}$ depends only on $N_\mathrm{H}$ and is a monotonically decreasing function ranging in $0 < \chi_\mathrm{e} \leq 1$; the upper and lower limits are achieved when $N_\mathrm{HI} = 0$ and $N_\mathrm{HI}\rightarrow \infty$, respectively.

In this paper, we perform our analysis under the idealization that a large portion of the wind region is optically thin ($\chi_\mathrm{e} \sim 1$), following the approach of .
While this idealization has its limitations and somewhat restricts the applicability of our model, it allows us to streamline our analysis.
Importantly, despite this limitation, constructing a model based on this simplification is a crucial initial step towards developing a more comprehensive model available for a broader range of spectra.
Furthermore, the insights obtained from our simplified model lay the groundwork for addressing more general cases and prove valuable in systematically interpreting the outcomes of such broader contexts.
The idealization would serve as a reasonable zeroth-order approximation in cases where spectra have minimal dispersion in photon energy, so that most photons are intensively absorbed at a certain column density. For instance, this includes relatively soft spectra, where most photons have energies close to the Lyman limit, or harder spectra with delta-function-like shapes.
(See sec:discussion:model_limitation for the potential impacts of this adopted approximation.)

§.§ Photoionization and Recombination Timescales

The EUV-driven winds undergo photoionization, H + EUV -> H+ + e-, and radiative recombination, H+ + e- -> H + photon, after launched.
The photoionization rate coefficient is the number of photons absorbed per unit time per hydrogen atom

\begin{equation}
\begin{split}
& \equiv \int \dd \nu \,  \sigma_\nu  \frac{\Phi_\nu}{4\pi r^2}
e^{- \sigma_\nu N_{\rm HI}}
% \\
% & = \frac{\Phi_{\rm EUV}}{4\pi r^2}
% \frac{\int \dd \nu \,  \frac{\Phi_\nu }{4\pi r^2} \exp \braket{- \sigma_\nu N_{\rm HI}}}{\int \dd \nu \,   \frac{\Phi_\nu }{4\pi r^2} }
% \frac{\int \dd \nu \,  \sigma_\nu \frac{\Phi_\nu }{4\pi r^2} \exp \braket{- \sigma_\nu N_{\rm HI}}}{\int \dd \nu \,   \frac{\Phi_\nu }{4\pi r^2} \exp \braket{- \sigma_\nu N_{\rm HI}}} \\
% & = F_0 \avesigma  \delta  \tilde{\sigma}
= \frac{\Phi_{\rm EUV}}{4\pi r^2} \avesigma  \chi_\mathrm{i}
% & =  0.000178 \sec^{-1}
% \braket{\frac{\Phi_{\rm EUV}}{10^{41}\sec^{-1}}}
% \braket{\frac{r}{1\au}}^{-2}
% \braket{\frac{\avesigma }{5\e{-18}\cm^2}}
% \delta \tilde{\sigma}
\end{split}
% In [3]: 1.e41/ (np.pi * 4. * ms.const_au**2) * 5.e-18
% Out[3]: 0.00017779076251283333
\label{eq:kioni}
\end{equation}

where $\chi_\mathrm{i} \equiv \delta  \langle\sigma\rangle/\bar{\sigma}_0 $ is another $N_\mathrm{HI}$-dependent, a priori calculable EUV attenuation factor, and $0 < \chi_\mathrm{i}\leq 1$.
Again, similarly to $\chi_\mathrm{e}$, we use the assumption of $\chi_\mathrm{i} \sim 1$ in this study, as mentioned below eq:gamma_EUV:2.

The photoionization timescale is the inverse of the photochemical reaction rate coefficient

\begin{equation}
t_{\rm ioni}
\equiv k_\mathrm{ioni}^{-1}
\braket{\frac{\Phi_{\rm EUV}}{4\pi r^2} \avesigma }^{-1} .
% \chi_\mathrm{i}}^{-1}.
\label{eq:tioni}
\end{equation}

This quantity sets the average timescale on which a single hydrogen atom takes to be photoionized.

Next, we introduce the recombination timescale

\[
t_\mathrm{rec} \equiv \braket{\nh \alpha} ^{-1}
\]

with $\alpha$ being the case B recombination coefficient.
We give $\alpha$ in the form of

\begin{equation}
\alpha %= \alpha_\mathrm{eq} (T/10^4\Kelvin)^{-\beta}.
=   \alpha_\mathrm{eq} \braket{\frac{\cs }{\ceq }}^{-2 \beta},
%     In [1]: 3.5e-12 * (1.e4/300)**-0.75 * 2/3.
% Out[1]: 1.681967989894328e-13
\label{eq:recombination_coeff}
\end{equation}

where $\alpha_\mathrm{eq}$ is the case B recombination coefficient at $T\approx 10^4\Kelvin$ and, typically, $\alpha_\mathrm{eq} \sim 2\e{-13}\cm^3\sec^{-1}$; $\beta$ is the power-law exponent for temperature and typically $0.5 \lesssim \beta \lesssim 0.75 $ <cit.>.

In contrast to $t_\mathrm{ioni}$, the recombination timescale is dependent on the gas density (and temperature) of the winds.
Hollenbach et al., 1994 and Tanaka et al., 2013 characterize $\nh$ at the base of EUV-driven winds by Strömgren condition, i.e., ionization-recombination equilibrium along line of sights from the radiation source, as

\begin{equation}
\begin{split}
n_\mathrm{base} & = C \sqrt{\frac{3\phieuv }{4\pi r^3 \alpha }}    \\
& \approx 7.6\e{5} \cm^{-3}
\braket{\frac{\phieuv}{10^{40}\sec^{-1}}}^{1/2}
\braket{\frac{r}{1\au}}^{-3/2}
\\
\times
\braket{\frac{\alpha}{2\e{-13}\cm^3\sec^{-1}}}^{-1/2}
\braket{\frac{C}{0.4}}
\end{split}
% \quad   \braket{\equiv n_\mathrm{S}}
% In [13]: 0.4 * (3 * 1e40 / (4 * np.pi * ms.const_au**3 * 2e-13))**0.5/1e5
% Out[13]: 7.552878144930797
\label{eq:base_density}
\end{equation}

Here, $C$ is an artificial dimensionless factor to match the numerical results, originating from the geometry of the ionized region <cit.>.

Substituting this density into $t_\mathrm{rec}$,
the recombination timescale is expressed as

\begin{equation}
=  \braket{\frac{3C^2\phieuv  \alpha_\mathrm{eq} }{4\pi r^3}}^{-1/2} \braket{\frac{\cs }{\ceq }}^\beta.
\label{eq:trec_base_density}
\end{equation}

Strictly, this equation specifically provides the recombination timescale near the launch point in the case where recombination dominates the opacity for EUV in the wind region.
Typically, as the evaporating gas flows away from the disk, its density decreases due to expansion, leading to a longer recombination timescale and consequently a higher ionization degree.
Moreover, the base density could vary depending on the EUV spectral hardness in practice.
Eqs.(<ref>) and (<ref>) do not account for such variations and are probably applicable only under conditions where recombination dominates the opacity over absorption by advected atomic hydrogen.
This condition typically holds for softer spectra.
Regardless of these uncertainties, in this study, we uniformly use eq:trec_base_density as a rough estimate for the recombination timescale to the height of $\sim R$ and for any spectra.
We discuss the potential impacts of this simplification in sec:discussion:model_limitation.

The ratio of the ionization timescale to the recombination timescale is a key metric for characterizing the ionization degree of the gas.
By using Eqs.(<ref>) and (<ref>), this can be expressed as

\begin{equation}
\begin{split}
\frac{t_\mathrm{ioni}}{t_\mathrm{rec}}
=  q^{1/2} % \braket{\frac{3C^2\alpha_\mathrm{eq}}{\ceq \avesigma}}^{1/2}
\braket{\frac{t_\mathrm{ioni} }{R/\ceq}}^{1/2}
\braket{\frac{\cs }{\ceq }}^{-\beta}
% \chi_\mathrm{i}^{-1}
\quad \braket{ = \frac{3C^2 \chi_\mathrm{i}^{-1}}{ n_\mathrm{base}\avesigma R}}
\end{split}
\label{eq:tioni_trec_ratio}
\end{equation}

with $q$ being a dimensionless constant

\begin{equation}
q \equiv \frac{3C^2\alpha_\mathrm{eq}}{\ceq \avesigma}.
\label{eq:q}
\end{equation}

This constant is uniquely set once an EUV spectrum is given; a harder spectrum results in a smaller average cross-section $\avesigma$, leading to a larger $q$.

We will explain this spectral hardness parameter is one of the parameters used for the classification of EUV photoevaporation in sec:sub:euv_classes.

To interpret the physical meaning of the $q$ parameter in a more direct manner, it is convenient to evaluate the ratio of $t_\mathrm{ioni}$ to $t_\mathrm{rec}$ with $\cs = \ceq$:

\[
\frac{t_\mathrm{ioni}}{t_\mathrm{rec, eq}}
% \braket{\frac{\avesigma^2 \phieuv }{12\pi C^2 \alpha_\mathrm{eq}  r} }^{-1/2}
% \approx
% \braket{\frac{3 C^2 \alpha_\mathrm{eq}}{\avesigma \ceq  } }^{-1/2}
\braket{\frac{t_\mathrm{ioni} }{R/\ceq}}^{1/2},
\]

where $t_\mathrm{rec,eq} \equiv t_\mathrm{rec}(\cs = \ceq)$.

The above equation gives

\[
q =  \braket{\frac{R/\ceq}{t_\mathrm{rec,eq}}}  \braket{\frac{t_\mathrm{ioni} }{t_\mathrm{rec,eq}}}
=  \braket{\alpha_\mathrm{eq} n_\mathrm{base} \frac{R}{\ceq} }\braket{k_\mathrm{ioni} t_\mathrm{rec, eq}}^{-1}.
% \sim y_\mathrm{HI} \frac{R/\ceq}{t_\mathrm{rec}}.
\]

Hence, $q$ is the number of recombination while the gas traveling to the height of $R$, with respect to the number of photoionization during one recombination event.
Thus, $q $ assesses the efficiency of reproducing atomic hydrogen through recombination against photoionization.

An increase in spectral hardness leads to a decrease in $\avesigma$, thereby relatively weakening photoionization compared to recombination by extending the photoionization timescale.
This effectively enhances the relative importance of atomic hydrogen reproduction, meaning a large $q$.

§ CLASSIFICATION OF PHOTOEVAPORATING DISKS

In this section, we summarize a qualitative overview of the major findings derived from our phenomenological model.
Our model is designed to characterize the typical temperature, namely the isothermal sound speed $\cs $, typical flow speed, and the ionization state of winds launched at a distance $R$ under the influence of a radiation source emitting EUV with a rate of $\phieuv $.
Since photoevaporative winds are thermally driven, the sound speed can also dictate the typical flow velocity.
We identify three distinct classes in EUV photoevaporation, each contingent upon the hardness of the EUV spectrum.

Here, we provide the general pictures for each class, outlining the key parameters employed in their classification.
We begin by discussing the possible ionization states of winds, which form the foundational basis of our model (sec:sub:wind_types).
Subsequently, we delve into the heating timescale, which stands as the most important quantity to define the hydrodynamical states of winds (sec:sub:t_h).
Expanding on this foundation, the typical temperature and velocity of winds are estimated (sec:typical_flow_velocities), followed by the introduction of the EUV photoevaporation classes (sec:sub:euv_classes).
Finally, we illustrate the distinctive characteristics of the three classes (one each in sec:sub:H-class–<ref>).
For detailed derivations of the quantities, parameter spaces, and classes reviewed in this section, the readers are referred to sec:cs.

As mentioned in Sections <ref> and <ref> and similarly to the approach of , this study focuses specifically on the cases where a significant portion of the wind region is optically thin (i.e., $\chi_\mathrm{e}\sim  1$ and $\chi_\mathrm{i} \sim 1$ in Eqs.(<ref>) and (<ref>)).
While this simplifying approach does place limitations on the applicability, the constructed model provides a foundation and valuable insights for addressing more general cases.
We discuss the potential impacts of this simplification in sec:discussion:model_limitation.
Throughout this section, we adopt a specific value of $\beta = 0.75$ for the power-law index of the recombination rate coefficient (eq:recombination_coeff) for simplicity.

§.§ Wind Types: Ionized VS Atomic

Two possible configurations of EUV-driven winds.
(left) The wind is essentially atomic while traveling to the height of $\sim R$.
This would occur when $\phieuv $ is small so that the ionization timescale is longer than either the wind crossing timescale or recombination timescale within the wind.
We term this type of atomic wind as Type I.
The variable $s$ here is coordinates along streamlines.
(right) In contrast to Type I, the photoionization timescale is shorter than the recombination timescale and wind crossing timescale.
In this case, the energy-injected gas can be ionized at some point $s_1$ between the launching point and the height $\sim R$.
Hydrogen atoms within the launched gas can survive for a photoionization timescale $t_\mathrm{ioni}$ on average, and thus the transition height can be given as $s_1\sim \mach \cs  t_\mathrm{ioni}$.
We refer to this ionized flow as Type II.
Note that the neutral region ($s \lesssim s_1$) can be geometrically very thin, but here, we draw it vertically extended for visualization purposes.

To discuss the ionization state of winds, we introduce the coordinate $s$ along a streamline of steady winds, with $s = 0$ denoting the launch point (fig:streamline_types).
Initial energy injection takes place in the H1 layer near the launch point.
If photoionization proceeds sufficiently rapidly, the launched gas can undergo complete photoionization, transitioning into the ionized wind before reaching a height of $s \sim R$ above the disk.
In essence, for an ionized wind to form, the photoionization timescale must be shorter than both the recombination timescale and wind crossing timescale.

This consideration leads us to categorize EUV-driven thermal winds into two types:

(I) Winds characterized by atomic composition, i.e., chemically neutral, within $s\leq R$ (left panel in fig:streamline_types).
This can arise when either the photoionization timescale is longer than the recombination timescale or the crossing timescale of the launched gas

\begin{equation}
t_\mathrm{ioni} > \mathrm{min} \braket{t_\mathrm{rec}, \frac{R}{\mach \cs }},
\label{eq:typeI_condition}
\end{equation}

where $\mach$ is a representative Mach number in the wind within $s \leq R$.

Qualitatively, this wind type is expected when $\phieuv $ is relatively low yet capable of gradually depositing energy into the gas to drive winds or when the spectrum is relatively hard, so $y_\mathrm{HI}$ in the ionization-recombination equilibrium is nearly unity.

(II) Winds transitioning into an ionized state at a certain height, $s = s_1 (< R)$, and being atomic below that (right panel in fig:streamline_types).
This scenario aligns with the canonical picture of EUV photoevaporation.
It occurs when $\phieuv $ is sufficiently high, causing the ionization timescale ($t_\mathrm{ioni}$) to be shorter than both the recombination and crossing timescales.

\begin{equation}
t_\mathrm{ioni} < \mathrm{min} \braket{t_\mathrm{rec}, \frac{R}{\mach \cs }}.
\label{eq:typeII_condition}
\end{equation}

Henceforth, we refer to the first and second types as Type I and Type II, respectively.
While the picture of Type II is more conventional for EUV photoevaporation,
the numbering convention is chosen to facilitate immediate association with the chemical states of the winds. Specifically, Type I corresponds to H1 winds, and Type II corresponds to H2 winds.

§.§ Characteristic Sound Speed and Heating Timescale

The magnitude of photoheating determines the typical temperature and speed of winds.
Photoheating can be gauged by the attainable sound speed as a result of heating.

In general, the wind's sound speed does not necessarily reach the equilibrium value $\ceq$ when the heating rate is small due to a finite photoheating rate.

Following , we evaluate the attainable sound speed by the characteristic sound speed $c_\mathrm{ch}$ defined as

\begin{equation}
c_p c_\mathrm{ch}^2 =  \bar{\Gamma}_\mathrm{EUV} \frac{R}{c_\mathrm{ch}}
%\int \Gamma_\mathrm{EUV} \frac{\dd s}{c_\mathrm{ch}} ,
\label{eq:c_ch_def}
\end{equation}

where $\bar{\Gamma}_\mathrm{EUV}$ is a time-average specific heating rate

\[
\bar{\Gamma}_\mathrm{EUV} \equiv \braket{\frac{R}{c_\mathrm{ch}}}^{-1} \int \Gamma_\mathrm{EUV} \frac{\dd s}{c_\mathrm{ch}}.
\]

The integral is taken from the base to the height $s = R$, treating $c_\mathrm{ch}$ as a constant while taking into account the variability of $y_\mathrm{HI}$ in $\Gamma_\mathrm{EUV}$ (cf. eq:gamma_EUV); thus, the form of $c_\mathrm{ch}$ differs by the wind types as we will see below.
The RHS of Eq.(<ref>) indicates the total specific energy deposited until the gas reaches the height of $R$.
Defining $c_\mathrm{ch}$ using the volume-integrated photoheating rate divided by the area of a flow bundle on the right-hand side of eq:c_ch_def can give a more accurate estimation of wind's sound speed.
Nevertheless, using the path integral instead, as in eq:c_ch_def, suffices for the order-of-magnitude estimate in the present study.

The characteristic sound speed corresponds to the isothermal sound speed that the launched gas would attain if all the specific energy integrated during its ascent to the height $\sim R$ were converted into the specific enthalpy.
It is important to note that the characteristic sound speed serves as a metric of the heating magnitude and does not necessarily reflect the actual sound speed of the gas $\cs $, as cooling and gravity play roles in determining the hydrodynamics of winds.
We will elaborate on this aspect later in sec:typical_flow_velocities.

In , $c_p $ does not appear on the left-hand side (LHS) of eq:c_ch_def.

This coefficient would not be uniquely determined, and one could potentially use another constant as long as it is $\mathcal{O}(c_p)$; this introduces some arbitrariness into the model.
such an arbitrariness does not significantly alter the results and conclusions presented in this paper.

Correspondingly to eq:c_ch_def, the heating timescale is defined as

\[
t_\mathrm{h} \equiv %  .
\frac{c_p c_\mathrm{ch}^2}{\bar{\Gamma}_\mathrm{EUV}}
= \frac{R}{c_\mathrm{ch}}.
\]

If the heating timescale is longer than the crossing timescale of an isothermal wind, $R/\ceq$, the wind does not have long enough time to be heated to the equilibrium temperature by reaching $\sim R$, within which the bulk of energy deposition occurs.
Moreover, if $t_\mathrm{h}$ is further longer than the gravitational timescale, the wind is significantly inhibited by gravity.

Since the specific heating rate $\Gamma_\mathrm{EUV}$ (eq:gamma_EUV) depends on the H1 abundance $y_\mathrm{HI}$, the expression for $c_\mathrm{ch}$ varies depending on whether the wind encompasses an ionized region or not.
For atomic winds (Type I; left panel in fig:streamline_types), the hydrogen abundance can be approximated to $y_\mathrm{HI}\approx 1$ in $\Gamma_\mathrm{EUV}$ of eq:c_ch_def, and $c_\mathrm{ch}$ reduces to

\begin{equation}
c_\mathrm{ch} =
\equiv  \braket{\frac{\Gamma_\mathrm{EUV}R}{c_p}}^{1/3}
= \ceq  \braket{\frac{\varphi}{x}}^{1/3},
\label{eq:cchI_def}
\end{equation}

where $\varphi$ is the dimensionless EUV emission rate, which is $\phieuv $ normalized by the critical EUV emission rate,

\begin{equation}
\Phi_\mathrm{c }
\equiv
\frac{4\pi m \ceq GM_*}{2\avesigma  \aveenergy },
\label{eq:critical_luminosity}
\end{equation}

Again, $x$ is the distance from the source in units of the critical radius $R_\mathrm{c}$ (eq:critical_radius):

\[
x \equiv \frac{R}{R_\mathrm{c}}.
\]

In this study, we focus on the region with $x > 1$, where strong winds can be anticipated.
We defer investigation of the inner region with $x < 1$ for future studies (see sec:discussion:model_limitation).

Note that the critical EUV emission rate is defined in such a way that it depends solely on spatially invariant constants determined by the properties of the radiation source and the disk gas.
It is physically interpreted as the EUV emission rate when the power required to evaporate the gas against the effective gravity at the equilibrium sound speed, $m \ceq  GM_*/2 R^2$, equilibrates with the heating rate yielded by the local flux, $\aveenergy  \avesigma  \Phi_\mathrm{c}/4\pi R^2$ (see also sec:work_efficiency).

It is noteworthy that the specific internal energy $\sim \braket{c_\mathrm{ch}^\mathrm{I}}^2$ is not proportional to $\Gamma_\mathrm{EUV}$ but exhibits a slightly slower increase, scaling as $\Gamma_\mathrm{EUV}^{2/3}$.

This originates from the drawback of increasing the gas temperature: while higher $\phieuv$ can increase the internal energy, the resulting short crossing time ($R/c_\mathrm{ch}$) leads to a shorter integration time of $\Gamma_\mathrm{EUV}$ and, thereby, a reduced total deposited energy during the gas traveling to the height of $\sim R$, where the bulk of photoheating and photoionization takes place.

This also implies that the fraction of the deposited energy going into the kinetic energy of the gas (and being drained by cooling) increases with $\phieuv$ (sec:work_efficiency).

For Type II winds (right panel of fig:streamline_types),

the total energy gain is the sum of the initially injected energy until the launched atomic hydrogen gets ionized, plus the additional energy resulting from the reionizations of recombined hydrogen within the H2 region.
The initial deposited energy is estimated as the product of $t_\mathrm{ioni}$ and $\Gamma_\mathrm{EUV}$ with $y_\mathrm{HI}\approx 1$, which is equivalent to $\aveenergy$.
To quantify this, we can introduce a spectral hardness parameter by normalizing the initially injected energy with the enthalpy of the gas at the equilibrium temperature:

\begin{equation}
\begin{split}
\varepsilon  & \equiv \frac{\aveenergy }{m c_p \ceq ^2}
\\ &
\approx 1
\braket{\frac{\aveenergy}{3.7 \eV}}
\braket{\frac{m}{1.4m_\mathrm{H}}}^{-1}
\braket{\frac{c_p}{5/2}}^{-1}
\braket{\frac{\ceq}{10\kms}}^{-2}
\end{split}
\label{eq:tildes}
\end{equation}

When $\varepsilon  > 1$, it signifies a hard spectrum, indicating that the gas could potentially be heated to a temperature higher than the equilibrium temperature by a single photoionization event.

The additionally supplied energy through reionizations is obtained by calculating the integration (eq:c_ch_def) from $s = s_1$ to $s = R$ with approximating $y_\mathrm{HI}$ in $\Gamma_\mathrm{EUV}$ to the ionization-recombination equilibrium abundance $y_\mathrm{HI}\sim t_\mathrm{ioni}/t_\mathrm{rec}$ (eq:tioni_trec_ratio).
Hence, the expression for $c_\mathrm{ch}$ for Type II winds are derived by solving this implicit equation with respect to $c_\mathrm{ch}^\mathrm{II}$:

\[
% \braket{\frac{c_\mathrm{ch}^\mathrm{II}}{\ceq }}^2
%   \approx \varepsilon   \left[ 1
%   + \braket{\frac{\varphi q}{ x  \varepsilon   }}^{1/2}
%   \braket{\frac{c_\mathrm{ch}^\mathrm{II}}{\ceq }}^{-1.75}
%   \right].
\braket{\frac{c_\mathrm{ch}^\mathrm{II}}{\ceq }}^2
= \varepsilon
+ \braket{\frac{\varphi \varepsilon   q}{ x }}^{1/2}
\braket{\frac{c_\mathrm{ch}^\mathrm{II}}{\ceq }}^{-1.75}
\braket{1 - \varphi^{-1}\varepsilon   x \frac{c_\mathrm{ch}^\mathrm{II}}{\ceq }}
\]

The first and second terms of the RHS correspond to the initially injected energy and additional energy through reionizations, respectively.
We refer the readers to sec:typeII (eq:chII_original_def:dimensionless) for the derivation of this equation.

The root of the above implicit equation can be described by simple expressions in some special cases.
When $t_\mathrm{rec}$ is sufficiently longer than $R/\cs $ so that energy addition due to reionizations can be ignored, $c_\mathrm{ch}^\mathrm{II}$ is approximated to

\[
c_\mathrm{ch}^\mathrm{II} \approx c_1 \equiv \varepsilon ^{1/2} \ceq ,
\]

by dropping the second term on the RHS of the above equation.

On the other hand, when recombination is sufficiently rapid, $t_\mathrm{rec} \ll R/\cs $, the reionization term dominates, and $c_\mathrm{ch}^\mathrm{II} $ is approximated to

\begin{equation}
c_\mathrm{ch}^\mathrm{II} \approx c_2  \equiv \ceq  \braket{\frac{\varphi_\mathrm{II}}{x}}^{1/7.5},
\label{eq:cchII_def}
\end{equation}

where $\varphi_\mathrm{II}$ is the EUV emission rate normalized by the secondary critical EUV emission rate for recombination-dominated Type II winds,

\[
\Phi_\mathrm{c,rec}^\mathrm{II} \equiv \frac{\Phi_\mathrm{c}}{\varepsilon  q}.
\]

Again, both $c_\mathrm{ch}^\mathrm{I}$ and $c_\mathrm{ch}^\mathrm{II}$ represent different forms of $c_\mathrm{ch}$ and thus serve as indicators of the heating timescale, not necessarily providing the actual sound speed of the wind, $\cs $, which is presented in the following section.

§.§ Typical Flow Velocities

When photoheating operates on a timescale shorter than gravitational deceleration, such that $t_\mathrm{h} < t_\mathrm{g}$, winds can be rapidly heated to have $\cs  $ exceeding the escape velocity $\vg $ without feeling the gravity, implying that the wind does not experience a large pressure drop.
In such instances, the typical flow velocity of the freely photoevaporating winds can approximate the isothermal sound speed, i.e., $\mach \sim 1$.

In a case of strong photoheating where $c_\mathrm{ch} > \ceq $, the gas undergoes immediate heating to a high temperature where cooling is activated.
This regulates the gas temperature to maintain $\cs \approx \ceq $.
For $x > 1$, this renders vigorous isothermal winds at the equilibrium temperature with a typical velocity of $\ceq $.
The temperature equilibration occurs within a quite short distance, $s \approx \ceq  t_\mathrm{h} = R (\ceq/c_\mathrm{ch})$.

In cases where $\vg  < c_\mathrm{ch} < \ceq $, gravitational deceleration is negligible, but heating is not rapid enough to heat the gas to the equilibrium temperature within $s \lesssim R$.
Here, neither cooling nor gravity significantly influences the wind, resulting in forming free winds with $\cs  \approx c_\mathrm{ch}$ and $\mach \sim 1$.

Conversely, when the heating timescale is longer than the gravitational timescale, or equivalently $c_\mathrm{ch} < \vg $, gravity can inhibit winds significantly.
This takes place typically at small distances with either low EUV emission rates or hard spectra.
Regardless of the emphasized gravitational effect, photoheated gas will not be completely hydrostatic, as the gas does not have an alternative option other than forming winds to process the continuously deposited energy due to its too low temperature to activate cooling.
In these escaping winds, a steady state is achieved such that the sound speed at a height of $\sim R$ is maintained at $\sim \vg $.
The typical flow speed of such gravity-inhibited winds can be estimated using the self-consistent condition (cf. eq:c_ch_def):

\[
c_p \vg ^2 \approx \int \Gamma_\mathrm{EUV} \frac{\dd s}{\mach \vg }.
\]

This reflects that when the heating timescale is long, the flow speed is self-regulated so that the gas can take a long time to reach $\sim R$ to obtain $\sim c_p \vg ^2$ worth of energy with a low heating rate.
Essentially, the above definition results in an estimated Mach number of

\begin{equation}
\mach_\mathrm{g} \equiv \mach \approx \braket{\frac{t_\mathrm{g}}{t_\mathrm{h}}}^3 ,
\label{eq:mach_g}
\end{equation}

indicating that winds are subsonic at $\sim R$.
The location of the sonic point is considerably farther from the base, $\sim (t_\mathrm{g}/t_\mathrm{h})^{-6} R $ [Begelman et al., 1983].

The specific form of the Mach number (eq:mach_g) for gravity-inhibited Type I winds is derived as

\[
\mach   \sim \mach_\mathrm{g}^\mathrm{I}
\equiv  \varphi x^{1/2},
\]

and that for gravity-inhibited Type II winds is

\[
\mach   \sim
\mach_\mathrm{g}^\mathrm{II}
\equiv
\braket{\frac{\varphi q x^\beta}{\varepsilon  }}^{1/2}
\left[
x^{-1} \varepsilon ^{-1} - 1
+ \braket{\frac{\varepsilon  q  x ^{\beta + 1} }{\varphi}}^{1/2}
\right]^{-1}.
\]

In either wind type, the Mach number increases with the EUV emission rate and reaches unity at $t_\mathrm{h} = t_\mathrm{g}$.

Regime     Description     Condition     Typical Sound Speed:      Typical Mach Number:
A     Isothermal wind     c_ch >     ∼    ∼1

B     Free wind      < c_ch <     ∼c_ch     ∼1

C     Gravity-inhibited wind     c_ch <     ∼    ∼(t_g/t_h)^3

Summary of wind regimes.

The hydrodynamical characteristics of the isothermal wind, free wind, and gravity-inhibited wind regimes can be summarized as tab:regime_summary.
In , the corresponding parameter spaces are labeled as Regions A, B, and C, respectively.
We follow this notation, but in our model, each regime bifurcates depending on the wind's chemical state, as we will see in the following sections (see also tab:sound_spees).
The three regimes cover different areas on the $x$–$\varphi$ phase diagram, depending on the EUV photoevaporation class.

§.§ EUV Photoevaporation Classes

Varying spectral hardness results in differing relative magnitudes of photoheating and photoionization.
For instance, a soft spectrum yields a relatively low amount of energy deposited into the gas per photoionization.
In such cases, while winds are readily photoionized, efficiently heating them proves challenging.
Hence, differences in spectral hardness suggest distinct combinations of the thermal and ionization states for winds at given $\phieuv$ and $R$.

Illustration of the spectrum-hardness diagram for the EUV photoevaporation classes (cf. tab:sq_plane). When the spectrum parameter $\varepsilon $ (eq:tildes) and ratio of recombination to photoionization $q$ (eq:q) for a given EUV spectrum is positioned in $\varepsilon  > 1$ or $\varepsilon  q > 1$, it is categorized into the hard-spectrum class (H-class).
The other cases are classified into the soft-spectrum classes.
For $q < \varepsilon ^{1.25}$, recombination is slow, and advection is more effective in maintaining atomic hydrogen within the launched gas. We call this class Sa-class.
The opposite class $q > \varepsilon ^{1.25}$, where recombination is rapid, is termed as Sr-class.
Here, $\beta = 0.75$ is used.

By evaluating Eqs.(<ref>) and (<ref>),
we find the distinct classes of EUV photoevaporation that are uniquely determined by the two spectrum-hardness parameters, $\varepsilon$ and $q$ (Eqs.(<ref>) and (<ref>)),

\[
\begin{gathered}
\varepsilon  \equiv \frac{\aveenergy }{m c_p \ceq ^2}
\approx 1
\braket{\frac{\aveenergy}{3.7 \eV}}
\braket{\frac{m}{1.4m_\mathrm{H}}}^{-1}
\braket{\frac{c_p}{5/2}}^{-1}
\braket{\frac{\ceq}{10\kms}}^{-2}
% \approx 0.27
% \braket{\frac{\aveenergy}{1\eV}}
% \braket{\frac{m}{1.4m_\mathrm{H}}}
% In [2]: ms.const_eV / 2.5 / 1.4/ms.const_mH / 10.e5**2
% Out[2]: 0.27357001360152305
% In [2]: 1.4 * ms.const_mH * 10e5**2 * 2.5 / ms.const_eV
% Out[2]: 3.6553713867799162
\\
\begin{split}
q  \equiv \frac{3C^2 \alpha_\mathrm{eq} }{\ceq \avesigma }
& \approx 0.1 \braket{\frac{C}{0.4}}^2
\braket{\frac{\alpha_\mathrm{eq}}{2\e{-13}\cm^3\sec^{-1}}}
\\ & \quad \times
\braket{\frac{\ceq }{10\kms}}^{-1}
\braket{\frac{\avesigma }{10^{-18}\cm^2}}^{-1}
\end{split}
% In [22]: 3 * 0.4**2 * 2.e-13 / 10.e5 / 1.e-18
% Out[22]: 0.09600000000000002
% \label{eq:q}
% \end{equation}
\end{gathered}
\]

Class name     Parameter space     Relevant source

H-class      {(ε, q) | ε> 1   ∪  εq > 1 }
T Tauri star

Sa-class     {(ε, q) |ε< 1,   q < ε^1.25 }
MS IM star

Sr-class       {(ε, q) | εq < 1,   q > ε^1.25 }

EUV photoevaporation classes and corresponding parameter spaces. $\beta = 0.75$ is used here. (See also fig:sqplane.)
The right column shows potentially relevant radiation sources to each class, where “MS IM star” denotes main-sequence intermediate-mass star.

The class of EUV photoevaporation is identified by the location of the given spectrum's $\varepsilon $ and $q$ on the $\varepsilon $–$q$ phase diagram (fig:sqplane).
The corresponding parameter spaces are summarized in tab:sq_plane.
The hard spectrum class is referred to as “H-class,” while the soft spectrum classes are denoted as “Sa-class” or “Sr-class.”
The lower-case letters of the soft spectrum classes stand for “advection” and “recombination,” respectively.
We will delve into the characteristics of each class in Sections <ref>–<ref>.

For young low-mass stars, in particular, EUV spectra might be classified into the H-class since $\varepsilon $ easily exceeds unity if there is a contribution of energetic EUV originating from the stellar magnetic activity.
However, the possibility remains that the spectra of low-mass stars are classified into the soft-spectrum classes if those low-mass stars have dominating accretion-originated soft EUV.
A blackbody with the effective temperatures of $\lesssim 5\e{4}\Kelvin$ falls into the soft spectrum class.

A necessary condition to be in the soft-spectrum classes is that the average deposited energy $\aveenergy$ is less than $m c_p \ceq^2 \approx 3.7\eV$.
This indicates most EUV photons have energies close to the Lyman limit.
Such extremely soft spectra would be possible for stars without convective zones, like main-sequence stars with intermediate masses, where EUV emission is mostly covered by moderately hot photospheres.

§.§ H-Class

Phase diagram for the H-class.
The horizontal axis is normalized radius $x = R/R_\mathrm{c}$, and the vertical axis is normalized EUV emission rate $\varphi = \phieuv /\Phi_\mathrm{c}$.

The red- and blue-shadings are the isothermal-wind (Regime A) and gravity-inhibited (Regime C) regimes, respectively.
The parameter space of steadily-heated, free winds (Regime B, or equivalently Regime B-I) is colored in green.
Different colors indicate the typical sound speed $\cs $
(see also tab:regime_summary for the chemical and hydrodynamical characteristics of each regime.)
The gray vertical stripes represent the Type I regime, where photoevaporative winds are atomic.
The remaining area is Type II.
Regime A has both Type I and II regimes, while the others do not.
This is a unique feature for the H-class.

This scenario corresponds to cases where the EUV spectrum is relatively hard (see tab:sq_plane).
In such instances, EUV can deposit a relatively large amount of energy into the gas per photoionization, enabling efficient heating of the gas.
In other words, this class has a relatively short heating timescale for a photoionization timescale.
Stars with active surface magnetic activities are anticipated to exhibit EUV spectra falling under this class.

The fast-ionization condition (eq:typeII_condition) is rewritten to

\begin{equation}
\varphi > \mathrm{max}\braket{1, q} \times \varepsilon  x.
\label{eq:typeII_condition_H_class}
\end{equation}

In this parameter space, the wind becomes ionized (Type II).
The remaining parameter space corresponds to Type I winds, depicted by the gray vertical lines in fig:H-class.

Photoheating is notably intense when \(c_\mathrm{ch}^\mathrm{I} > \ceq \), or equivalently \(\varphi > x\), instantly raising the gas temperature to the equilibrium value.
Consequently, the wind becomes isothermal with $\cs  \approx \ceq $ in this parameter space.
The corresponding area is visually highlighted by the red-shading in fig:H-class and is labeled as Regime A, following Region A of  (tab:sound_spees).
Regime A occupies the upper left corner of the $x$–$\varphi$ plane, reflecting the tendency for the heating timescale to be generally shorter at smaller $R$ and with higher $\phieuv $, i.e., where EUV flux is intense.

Dynamical state

Chemical state

Isothermal Wind

Atomic (Type I)

Ionized (Type II)

red  (w/ gray vertical stripes)

red  (w/o gray vertical stripes)

Steadily Heated, Free Wind

Free Wind

Atomic (Type I)

Ionized (Type II)

Gravity-inhibited wind

Atomic (Type I)

Ionized (Type II)

blue (w/ gray vertical stripes)

blue (w/o gray vertical stripes)

Qualitative characteristics of the chemical and hydrodynamical states for winds in each regime.
Each of the fundamental three regimes (Regimes A, B, and C; see tab:regime_summary) has two subregimes according to the wind types.
The listed “Color” in the last column is used in Figures <ref>, <ref>, <ref>, and <ref> to represent the corresponding parameter space on the phase diagrams, differentiating the typical sound speeds of winds $\cs$.
Note that Regime A-I is only possible in the H-class, while Regimes B-II and C-II are present only in the Sa- and Sr-classes.

Ionized winds (Type II) can exist only in Regime A in this class.
It is also possible that the winds manifest as an isothermal atomic flow for a lower $\phieuv $.
With the short heating timescale in this regime, the effect of gravity is minimal, and winds can freely escape at a typical flow speed of $\sim \ceq $.

If $\phieuv $ is small, or the distance $R$ is large, the heating timescale gets long, and in the ultimate case, $t_\mathrm{h}$ becomes longer than the gravitational timescale.
This condition is equivalent with $c_\mathrm{ch}^\mathrm{I} < \vg $, and in this case, the wind is significantly inhibited by gravity.
The energy deposited by photoheating charges to the gas
keeps the isothermal sound speed at $\cs  \approx \vg $.
This regime is represented by $\varphi < x^{-1/2}$, or equivalently $c_\mathrm{ch}^\mathrm{I} < \vg $, shown by the blue-shading in fig:H-class (Regime C).
The fast-ionization condition is not met there, meaning that gravity-inhibited winds are atomic in this class.
Correspondingly, the typical Mach number of gravity-inhibited winds in the H-class is $\sim \mach_\mathrm{g}^\mathrm{I}$.

In the other regime, $x^{-1/2} < \varphi < x$, neither gravity nor cooling significantly affects the system.
This regime corresponds to Region B of , but we label it as Regime B-I in this study to highlight the fact that the winds are atomic, and the sound speed is given by the Type I characteristic sound speed $c_\mathrm{ch}^\mathrm{I}$.
Similarly to Regime A, the heating timescale is sufficiently short compared to the gravitational timescale, and thus winds can freely escape at $\mach\sim 1$ without being influenced by gravity.
However, the heating timescale is not short enough so that gas temperature can reach the equilibrium temperature while traveling to the height of $\sim R$,
and hence $\cs  \approx c_\mathrm{ch}^\mathrm{I}$.

§.§ Sa-class

Phase diagram for the Sa-class, which is overall similar to fig:H-class.
However, Regime B is separated into two subcategories: Regime B-I (green) and Regime B-II (orange).
Both are free wind regimes, but the former is for atomic winds (Type I), and the latter is for ionized winds (Type II).
The blue-dashed line indicates where the ionization timescale is equal to the crossing timescale $t_\mathrm{ioni} = R/\mach \cs $; the ionization timescale is shorter above this line.
The green-dashed line is where the ionization and recombination timescales are equal;
again, the ionization timescale is shorter above this line.
The boundary where the recombination and wind crossing timescale is equal is represented by the red-dashed line, below which the recombination timescale is longer.
This class is visually distinguished from the other classes by the parameter space having the presence of the orange area attaching to the blue-dashed line and the red-dashed line positioning in the Type II regime.

This class is the case when the spectrum is soft, and recombination occurs relatively slowly in the wind.
The deposited energy per photoionization is small, and one single photoionization is not sufficient to supply the energy worth of the gas's internal energy at the equilibrium temperature, as opposed to the H-class.
A relatively high $\phieuv $ is required to heat the gas at a certain level than is needed in the H-class.
Due to the slow recombination, winds in this class do not necessarily undergo secondary photoionization following the initial photoionization at the launching point until the escaping gas travels to the height of $\sim R$.
Stars lacking surface magnetic activities, such as main-sequence intermediate-mass stars, would have EUV spectra classified into this class.

The fast-ionization condition is rewritten to

\begin{equation}
\left\{
\begin{array}{ll}
% \varphi > \varepsilon    x ^{1/2}
\varphi > \varepsilon  q x^{1 + \beta}
& \quad \mathrm{for}~ 1\leq  x  < \varepsilon  ^{-1} \\
\varphi >  \varepsilon  ^{3/2}  x & \quad \mathrm{for}~  x  > \varepsilon  ^{-1} \\
\end{array}
\right.
\label{eq:Sa-class:TypeIIcondition}
\end{equation}

which is equivalent to $t_\mathrm{ioni} < R/\cs $ for $x < \varepsilon ^{-1}$ (green dashed line) and $t_\mathrm{ioni} < R/\cs $ for $x > \varepsilon ^{-1}$ (blue dashed line).
In this parameter space, the wind becomes ionized (Type II), while the remaining parameter space (the area covered by gray vertical lines in fig:Sa-class) represents the atomic-wind regime (Type I).
For $x>\varepsilon ^{-1}$, atoms in the winds are maintained primarily through advection from the upstream region rather than reproduced by recombination, and vice versa for $x < \varepsilon ^{-1}$.
The blue dashed line, where $t_\mathrm{ioni} = R/\cs $, is vertical at $x = \varepsilon ^{-1} $ in fig:Sa-class.
This implies that steady winds cannot develop without additional energy deposition through reionizations in $x < \varepsilon $, since the initially injected energy is never sufficient for heating the gas to yield an escaping speed.

Contrary to the H-class, isothermal winds of this class (Regime A; the red-shade in fig:Sa-class) are always in an ionized state.
The corresponding parameter space is where the heating timescale is sufficiently short, i.e., $c_\mathrm{ch}^\mathrm{II} > \ceq $, or equivalently

\[
\varphi  > f_\mathrm{eq} ( \varepsilon , q) x,
\]

\[
f_\mathrm{eq} (\varepsilon , q)
\equiv \frac{4 \varepsilon  q }
{\braket{\sqrt{(\varepsilon ^{-1}-1)^2 + 4q} - (\varepsilon ^{-1}-1)}^2}.
\]

The factor $f_\mathrm{eq}$ is larger than unity for soft spectrum classes ($\varepsilon  < 1, ~\varepsilon  q< 1$), meaning that on the $x$–$\varphi$ plane, the $c_\mathrm{ch}^\mathrm{II} = \ceq $ line is located above $c_\mathrm{ch}^\mathrm{I} = \ceq $ line, which separated Regimes A and B in the H-class.
This results from more efficient heating in an atomic gas than in an ionized gas since the absorber per gas mass is higher.

The reduced heating rate necessitates a higher $\phieuv $ to heat the gas to the equilibrium value ($\cs  = \ceq $), compared to what would be required if the gas were atomic ($\varphi > x$, or equivalently $c_\mathrm{ch}^\mathrm{I} > \ceq $).

In the limit of $(\varepsilon^{-1}-1)^2 \gg 4 q$, which is often the case for Sa-class spectra (tab:sound_spees), $f_\mathrm{eq}$ is approximated to

\[
f_\mathrm{eq} \approx \frac{(1 - \varepsilon)^2}{\varepsilon q} \quad    \braket{\text{for } (\varepsilon^{-1}-1)^2 \gg 4 q}.
\]

A distinct characteristic of the soft-spectrum classes is that ionized winds can be inhibited by gravity.
This occurs when the heating timescale is shorter than the gravitational timescale.
For Type II winds, this condition is expressed by $c_\mathrm{ch}^\mathrm{II}< \vg $, which is rewritten to

\[
\begin{split}
\varphi&  < \varphi_\mathrm{g} (x; \varepsilon , q, \beta)
\\
& \equiv
\braket{
\frac{2\sqrt{q\varepsilon }x^{(\beta+1)/2}}
{\sqrt{(x^{-1}\varepsilon ^{-1}-1)^2 + 4 q x^{(2\beta + 1)/2}} - (x^{-1}\varepsilon ^{-1}-1)}
\end{split}
\]

Hence, the parameter space for the gravity-inhibited, Type II winds is expressed by

\[
\left\{
(x, \varphi)~|~
x < \varepsilon ^{-1},
\varphi > \varepsilon  q x^{\beta + 1},
% \varphi_\mathrm{II} < x^{-(\beta + 2)}
\varphi < \varphi_\mathrm{g}
\right\}.
\]

In this parameter space, typical sound speed and flow speed are $\sim \vg $ and $\sim \mach_\mathrm{g}^\mathrm{II}$, respectively.
This space has a shape that looks like extruding the blue-shaded area upward on the $x$–$\varphi$ plane within $x < \varepsilon ^{-1}$, compared to Regime C of the H-class.
Again, this distinction highlights the lower heating efficiency in an ionized wind than in an atomic wind.
Therefore, in order to deposit sufficient energy for the gas to form steadily escaping winds, shortening the recombination timescale by increasing flow density, i.e., increasing $\phieuv  $, is needed.
This facilitates the (re)production of atomic hydrogen, which is subsequently reionized in the wind, resulting in secondary energy deposition.

Another unique characteristic of the soft-spectrum classes is having the parameter space of free winds that are ionized (orange-shaded area in fig:Sa-class).
In this regime, the heating timescale is sufficiently shorter than the gravitational timescale while not being short enough to heat the gas to reach the equilibrium temperature.
The photoionization timescale is also sufficiently short to make the photoheated gas ionized in this regime.
As a result, the typical sound speed and Mach number of the ionized winds are given by $\sim c_\mathrm{ch}^\mathrm{II}$ and $\sim 1$, meaning lower-temperature ionized winds in contrast to the canonical picture.

The soft-spectrum classes are distinguished by the rapidity of recombination within free-ionized winds.
It differentiates the magnitude of the additionally deposited energy through reionizations with respect to the initially injected energy.
The Sa-class has relatively slow recombination and thus always has a parameter space where the contribution of additional energy deposition is minimal.
This is especially the case in the vicinity of the Type I and Type II boundary in Regime B (blue-dashed line in fig:Sa-class).
The sound speed and flow speed can be well approximated to $\sim c_1$ there.
In the zeroth order, this approximation is valid as long as the recombination timescale is longer than the wind crossing timescale.
This parameter space corresponds to the area bounded by the blue- and red-dashed lines in fig:Sa-class.

§.§ Sr-class

Same as fig:Sa-class, but we omit the blue-dashed line ($t_\mathrm{ioni} = R/\cs $) and red-dashed line ($t_\mathrm{rec} = R/\cs $), which position below the green-dashed line (cf. the bottom left panel in fig:classmaps).
Having the orange area attached to the green-dashed line is a visual uniqueness of the parameter space for this class.

This is the other soft-spectrum class in addition to the Sa-class.
Overall characteristics are the same as those of the Sa-class, but this class has a shorter recombination timescale.
In this class, the boundary between the Type I and Type II regimes is set by $t_\mathrm{rec} = t_\mathrm{ioni}$ (green dashed line in fig:Sr-class) instead of $t_\mathrm{ioni} = R/\cs $.
Thus, the slow-recombination regime (the area bounded by the blue and red dashed lines in fig:Sa-class) is absent.
This absence means that ionized winds are invariably associated with (multiple) reionizations of recombined hydrogen to heat the gas.
However, this class may not manifest with realistic EUV spectra, at least in the context of protoplanetary disk photoevaporation.

This class is distinguished by the $x$–$\varphi$ plane having the $t_\mathrm{ioni} = t_\mathrm{rec}$ line (the green dashed line), positioning above both the $t_\mathrm{ioni} = R/\cs $ (blue dashed) and $t_\mathrm{rec} = R/\cs $ (red dashed) lines (cf. fig:classmaps in sec:typeI:regime).
Note that we omit the blue and red dashed lines in fig:Sa-class for clarity.

§ DERIVATIONS OF PARAMETERS AND CLASSES

In this section, we provide the detailed derivations of the parameters and EUV photoevaporation classes reviewed in sec:classification.

We start with the estimation of the sound speed and flow speed for Type I winds (sec:c_ch), introducing photoionization-relevant parameters (sec:sub:photoionization_parameters), and identifying the parameter space where these winds are anticipated (sec:typeI:regime).
Following this, we extend our analysis to Type II winds in Sections <ref> and <ref>.

§.§ Characteristic Sound Speed: Type I

Following , the characteristic sound speed (eq:c_ch_def) is defined as

\[
c_p c_\mathrm{ch}^2 = \int \Gamma_\mathrm{EUV} \frac{\dd s}{c_\mathrm{ch}}.
\]

For Type I winds, this can be approximated to

\[
c_p c_\mathrm{ch}^2 \approx \frac{R}{c_\mathrm{ch}} \Gamma_\mathrm{EUV},
\]

assuming that $y_\mathrm{HI}$ in $\Gamma_\mathrm{EUV}$ (eq:gamma_EUV) is nearly unity for the Type I winds uniformly.

The above equation can be solved with respect to $c_\mathrm{ch}$,

\[
c_{\rm ch} = \braket{\frac{\Gamma_\mathrm{EUV} R }%{\mu m_{\rm H} \rho / m}
\quad \braket{y_\mathrm{HI} \approx 1},
% = \braket{\frac{H R_0 }{\rho }\sum_i y_i } ^{1/3},
%             \label{eq:c_ch_def}
% \end{equation}
\]

retrieving eq:cchI_def.

Then, the heating timescale is given by

\[
t_\mathrm{h } \equiv \frac{c_p c_\mathrm{ch}^2}{\Gamma_\mathrm{EUV}}
= \frac{R}{c_\mathrm{ch}},
\]

and it characterizes the strength of the photoheating.
We stress again that the characteristic sound speed $c_\mathrm{ch}$ is a virtual sound speed that a gas parcel would achieve if the photoheating energy is purely deposited to the internal energy of freely flowing gas.
In other words, $c_\mathrm{ch}$ does not necessarily give the actual isothermal sound speed of the gas $\cs $, which is also influenced by the effects of cooling and gravity, as mentioned in sec:classification.

We assume that $r$ in $\Gamma_\mathrm{EUV}$ can be approximated to $r \approx R$ within $s \leq R$ for our order-of-magnitude estimate,

and then eq:c_ch_def is rewritten to

\begin{equation}
\begin{split}
c_{\rm ch}
% & =  \ceq   x ^{-1/3} \braket{\frac{2 y_\mathrm{HI} R_*^2 F_0 \avesigma  \aveenergy  \chi_\mathrm{e} }{m \ceq GM_*}} ^{1/3}
% \\
% & = \ceq   x ^{-1/3}\braket{\frac{2 y_\mathrm{HI} \phieuv  \avesigma  \aveenergy  \chi_\mathrm{e} }{4 \pi m \ceq GM_*}} ^{1/3}
% \\
& = \ceq  \braket{\frac{\phieuv y_\mathrm{HI}\chi_\mathrm{e}}{\Phi_\mathrm{c}  x }}^{1/3}
% \chi_\mathrm{e}^{1/3}
% y_\mathrm{HI}^{1/3}
\quad \braket{y_\mathrm{HI} \approx 1}
\end{split}
\label{eq:cch_VS_ceq},
\end{equation}

\[
\begin{split}
\Phi_\mathrm{c}& \equiv \frac{ 4\pi m  \ceq GM_*} {2  \avesigma  \aveenergy }
% =  \Phi_\mathrm{c,i} \frac{ m  c_p \ceq ^2} {\aveenergy }
\\
&   \approx
\braket{\frac{M_*}{1\Msun}}
\braket{\frac{m}{1.4m_{\rm H}}}
\braket{\frac{\ceq }{10\kms}}
\\      &   \times
\braket{\frac{\avesigma  \aveenergy }{7\e{-18}\cm^2 \eV}}^{-1}
\end{split}
% In [10]:  4. * np.pi * 1.4 * ms.const_mH * ms.const_GMsun * 10.e5 / 2. / 7.e-18 / ms.const_eV
% Out[10]: 1.741746588375182e+38
% \label{eq:critical_luminosity}
% \end{equation}
\]

is the critical EUV emission rate (eq:critical_luminosity).
The critical EUV emission rate is spatially independent but depends only on the stellar property, namely the spectral shape of EUV and the stellar mass, and the disk gas property, namely gas mass per hydrogen nucleus and the equilibrium sound speed.

Using the EUV emission rate normalized by the critical EUV emission rate

\[
\varphi \equiv \frac{\phieuv }{\Phi_\mathrm{c}} ,
\]

we denote the characteristic sound speed of eq:cch_VS_ceq as

\[
\equiv \ceq
\braket{\frac{\varphi}{ x }}^{1/3}
%     \label{eq:cchI_def}
% \end{equation}
\]

to emphasize this being $c_\mathrm{ch}$ for Type I winds (cf. eq:cchI_def).

In this paper, we use $c_\mathrm{ch}^\mathrm{I}$ as the characteristic sound speed of the Type I wind instead of the more strict expression eq:cch_VS_ceq for simplicity.

This means that we analyze under the idealization that a large portion of the wind region is optically thin ($\chi_\mathrm{e} \sim 1$), following Begelman et al., 1983.
While this idealization restricts our model's applicability, it simplifies our analysis.
This idealization is a reasonable approximation for spectra with minimal photon energy dispersion, where most photons are intensively absorbed at a certain column density.
Examples include relatively soft spectra dominated by photons near the Lyman limit or harder spectra with delta-function-like shapes.
Regardless of these limitations, the adopted approach is an essential first step towards developing a more comprehensive model available for a wider range of spectra.
Besides, the insights gained from our simplified model provide a foundation for tackling more general cases.
In sec:discussion, We delve deeper into the potential impacts of this adopted approximation.

The Type I characteristic sound speed $c_\mathrm{ch}^\mathrm{I}$ increases with $\phieuv $ and eventually exceeds $\ceq $ when $\phieuv  > \Phi_\mathrm{c} x $.
This means the photoheating is so strong that the gas temperature could potentially exceed the equilibrium value $\ceq $.
However, as explained in sec:typical_flow_velocities, it does not happen for real; instead, cooling gets activated to force the actual sound speed $\cs $ to stay at $\ceq $.
In other words, coolants act as a resistance force for gas temperature so as not to exceed the terminal value $\ceq $.

On the contrary, when heating is weak, the characteristic sound speed can get lower than $\vg $, or equivalently $t_\mathrm{h} > t_\mathrm{g}$.

This appears as if the gas would be bounded so that it forms a hydrostatic atmosphere; however, this is not the case.
Even if the gas is indeed in hydrostatic equilibrium at first,
the energy will be continuously charged to the gas to increase the temperature gradually, as the deposited energy cannot be processed by radiative cooling at this low temperature.
At some point, $\cs $ will eventually reach the escaping velocity $\vg $, forming slow winds.
The system would achieve a steady state such that the sound speed is maintained of the order of $\vg  (= \sqrt{GM/2c_p R})$ with a small flow velocity of

\[
v \sim \braket{\frac{t_\mathrm{g}}{t_\mathrm{h}}}^3 \vg  = \braket{\frac{c_\mathrm{ch}^\mathrm{I}}{\vg }}^3 \vg
= \varphi x^{1/2} \vg ,
\]

which indicates

\[
\mach_\mathrm{g}^\mathrm{I}  = \varphi x^{1/2} \quad\braket{\text{for Type~I}}
\]
(cf. \eqnref{eq:mach_g}).
This gravity-inhibited-wind regime occurs when $c_\mathrm{ch}^\mathrm{I} < \vg $, i.e., when the ratio of
\begin{equation}
\frac{c_\mathrm{ch}^\mathrm{I}}{\vg }
= \braket{\varphi  x ^{1/2}}^{1/3}
% = \braket{\varphi }^{1/3}  x ^{1/6}
% y_\mathrm{HI}^{1/3} \chi_\mathrm{e}^{1/3}.
\label{eq:cchI_vg}
\end{equation}
gets lower than unity.
% checked right by mathematica (also for the factors involving y_HI and epsilon)
% \mycomment{Plot $\cs $ as a function of $\phieuv $ here?}
% Now we complete comparing all combinations of $\ceq $, $c_\mathrm{ch}$, and $\vg \equiv \sqrt{GM_*/2c_p R}$.
% For $R > R_\mathrm{c}$, there are three flow velocity regimes: $\vg  < c_\mathrm{ch} < \ceq $.
% For $R < R_\mathrm{c}$, the gas is assumed to be bound ($v\approx 0$) for simplicity (strictly, there is a very slow wind [Begelman} {et~al.}, 1983]).

\subsection{Photoionization-Related Parameters}
\label{sec:sub:photoionization_parameters}
% When Type~I streamline occurs? In other word, in which region $c_\mathrm{ch}^\mathrm{I}$ is available on the $ x $--$\varphi$ plane? or in which region assuming $y_\mathrm{HI}\approx1$ uniformly within $s\leq R$ is justified? Timescale comparison. Quantify the recombination timescale and ionization timescale.
As explained in \secref{sec:sub:wind_types},
the picture of the Type~I wind is valid only when the photoionization timescale is longer than the recombination or crossing timescale of the wind (left panel of \fref{fig:streamline_types}).
This condition is expressed by the slow photoionization condition, \eqnref{eq:typeI_condition},
% \begin{equation}
\[
t_\mathrm{ioni} > \mathrm{min}\braket{t_\mathrm{rec}, R/\mach \cs }.
%     \label{eq:typeI_condition}
% \end{equation}
\]
The Type~I characteristic sound speed $c_\mathrm{ch}^\mathrm{I}$ (\eqnref{eq:cchI_def}) is available only in this parameter space, for which we will identify the expressions in the following sections.
To that end, we introduce photoionization-relevant quantities in this section.

% \begin{figure}
%     \centering
%     \includegraphics[clip, width = \linewidth]{figs/HIIregionconfig.pdf}
%     \caption{Caption}
%     \label{fig:Hiiregion_config}
% \end{figure}
After the photo-energy is injected into a new \ion{H}{1} layer close to the $\tau_\mathrm{EUV}\sim 1$ surface, a launched hydrogen atom takes $t_\mathrm{ioni}$ to be photoionized in general.
The flowing hydrogen atom can survive to a distance of
\[
\begin{split}
\Delta s_\mathrm{suv} & \equiv \mach \cs  t_\mathrm{ioni} \\
& \approx 1.9 \au \braket{\frac{v}{10\km \sec^{-1}}}
\braket{\frac{\phieuv }{10^{40} \sec^{-1}}}\\
& \quad \times
\braket{\frac{r}{10\au}}^{-2}
\braket{\frac{\avesigma }{10^{-18} \cm^2}}
\braket{\frac{\mach}{1}}
\end{split}
% In [8]: (1.e-18 * 1.e40 / 4 / np.pi / ms.const_au**2/10**2 )**-1 * 10.e5 / ms.const_au
% Out[8]: 1.8799025539988772
\]
at the shortest.
We note that $\Delta s_\mathrm{suv}$ is a {\it potential} thickness of the photoheated \ion{H}{1} layer and
does not necessarily represent the actual thickness of it.
When recombination is strong, i,e., $t_\mathrm{rec} < R/\mach \cs $,  %$\cs  t_\mathrm{rec}$
the recombination timescale determines the thickness of the photoheated \ion{H}{1} layer,
as implied by the slow photoionization condition, \eqnref{eq:typeI_condition}.

% $R/vt_\mathrm{ioni}> 1$. If this is met, multiple photoionizations will occur during one crossing time, i.e., recombination is required in the steady state; otherwise, the ionization front advances into the neutral gas unsteadily.
% The criterion corresponds to
The condition, $\Delta s_\mathrm{suv} > R$, is a sufficient condition for winds to be Type~I.
This is rewritten to
\begin{equation}
\frac{R}{\mach \cs t_\mathrm{ioni}}
\mach^{-1}
\braket{\frac{\cs }{\ceq }}^{-1}
\frac{\phieuv }{\Phi_\mathrm{c,i}  x }\chi_\mathrm{i}  < 1,
\label{eq:tioni_tcross}
\end{equation}
and $\Phi_\mathrm{c,i}$ is the photoionization critical EUV emission rate
\[
\begin{split}
\Phi_\mathrm{c,i} & \equiv \frac{4 \pi GM_* }{ 2c_p \ceq  \avesigma }
% = \Phi_\mathrm{c} \varepsilon
= \Phi_\mathrm{c} \frac{\aveenergy }{c_p m \ceq ^2}\\
& \approx 3.3\e{38} \sec^{-1}
\braket{\frac{M_*}{1\Msun}}
\braket{\frac{c_p}{5/2}}^{-1}
\\& \times
\braket{\frac{\ceq }{10\kms}}^{-1}
\braket{\frac{\avesigma }{1\e{-18}\cm^2}}^{-1}.
\end{split}
% In [12]: 4. * np.pi * ms.const_GMsun / 2. / 2.5 / 10.e5 / 1.e-18
% Out[12]: 3.335427465105434e+38
\]  % checked right by mathematica
Again, the critical EUV emission rate of photoionization depends only on the stellar parameters and disk gas properties.
Similarly to $\chi_\mathrm{e}$, $\chi_\mathrm{i}$ is expected to be of the order of unity in the photoheated layer.
Therefore, we ignore $\chi_\mathrm{i}$ in the following sections to remove complexity from our order-of-magnitude estimation (again, see \secref{sec:discussion:model_limitation} for the potential impact of this simplification).

% For later convenience, we define
Here, we retrieve one of the spectral hardness parameters, $\varepsilon $ (\eqnref{eq:tildes}),
% \begin{equation}
\[
\begin{split}
\varepsilon   &
\equiv \frac{ \Phi_\mathrm{c,i} }{ \Phi_\mathrm{c} }
= \frac{\aveenergy }{c_p m \ceq ^2}
% In [24]: ms.const_eV / 2.5 / 1.4/ms.const_mH / 10.e5**2
% Out[24]: 0.27357001360152305
\\
& 1
\approx \braket{\frac{\aveenergy }{3.7 \eV}}
\braket{\frac{c_p}{5/2}}^{-1}
\braket{\frac{m}{1.4 m_\mathrm{H}}}^{-1}
\braket{\frac{\ceq }{10\kms}}^{-2}.
%     In [15]: 2.5 * ms.const_mH * 1.4 * 10.e5**2 / ms.const_eV
% Out[15]: 3.6553713867799162
\end{split}
% \label{eq:tildes}
% \end{equation}
\]
This parameter characterizes the spectral hardness for the advection of atomic hydrogen.
The inverse $\varepsilon  ^{-1}$ is physically interpreted as the number of photoionization necessary to heat the gas to the equilibrium temperature, or equivalently to have an energy of $c_p m \ceq ^2$.
If $\varepsilon  >1$, the spectrum is heating-oriented;
the gas can get the heat exceeding $c_p m \ceq ^2$ with only single photoionization of atomic hydrogen.
% Heating is so efficient that the gas can be heated to $\ceq $ at a lower $\phieuv $ than is needed to fully ionize the gas.
Conversely, for $\varepsilon   < 1$, the spectrum is ionization-oriented;
multiple photoionization is needed to heat the gas to $c_p m \ceq ^2$.
% The gas can be fully photoionized at a lower luminosity than is needed to heat the gas to $\ceq $.
A relatively large amount of energy is consumed through photoionization rather than heating in this case.
We stress that $\varepsilon  $ is determined by the spectral shape of EUV and is independent of the absolute value of the incident flux or $\phieuv $.

Using the dimensionless quantities,
the recombination timescale \eqnref{eq:trec_base_density} is rewritten to
\begin{equation}
= \frac{R_\mathrm{c}}{\ceq }
\braket{
\frac{q\varepsilon  ^{-1} \varphi}{ x ^3}
\braket{
\frac{\cs }{\ceq }
\label{eq:trec_reduced}
\end{equation}
Here, we find the other spectral hardness parameter, $q$ (\eqnref{eq:q}),
% \begin{equation}
\[
\begin{split}
q & \equiv \frac{3C^2 \alpha_\mathrm{eq} }{\ceq \avesigma } \\
& \approx 0.096 \braket{\frac{C}{0.4}}^2
\braket{\frac{\alpha_\mathrm{eq}}{2\e{-13}}}
\\ & \quad \times
\braket{\frac{\ceq }{10\kms}}^{-1}
\braket{\frac{\avesigma }{10^{-18}\cm^2}}^{-1}
\end{split}
% In [22]: 3 * 0.4**2 * 2.e-13 / 10.e5 / 1.e-18
% Out[22]: 0.09600000000000002
% \label{eq:q}
% \end{equation}
\]
The $q$ parameter also characterizes the spectral hardness for hydrogen recombination, similarly to $\varepsilon  $. (See \secref{sec:sub:photoionization_recombination_timescales} for a physical interpretation.)
% We will discuss this point in detail later.

We can evaluate the right-hand side (RHS) of the slow ionization condition (\eqnref{eq:typeI_condition}) by considering the ratio between $t_\mathrm{rec}$ and $R/\mach \cs $,
\begin{equation}
\frac{R}{\mach \cs  t_\mathrm{rec}}
\braket{
\frac{\varphi q}{\varepsilon  x }
\braket{\frac{\cs }{\ceq }}^{-(1+\beta)}
\mach^{-1}.
\label{eq:trec_tcross}
\end{equation}
Also, we can compare the ratio between $t_\mathrm{ioni}$ and $t_\mathrm{rec}$ as
\begin{equation}
\begin{split}
\frac{t_\mathrm{ioni}}{t_\mathrm{rec}}
% &    =
% \frac{4\pi r^2 \nh \alpha}{\phieuv  \avesigma }
% \approx A \frac{ (12\pi r \alpha)^{1/2}}{\phieuv ^{1/2} \avesigma }
% \\
% &
=  \braket{\frac{\varepsilon  q  x  }{\varphi}}^{1/2}
\braket{\frac{\cs }{\ceq }}^{-\beta},
% In [21]: 4 * np.pi * ms.const_au**2 * 1.e5 * 2e-13 / 1e40 / 1.e-18
% Out[21]: 0.005624589184872964
\end{split}
\label{eq:tioni_trec_ratio:dimensionless}
\end{equation}
For $t_\mathrm{ioni} \ll t_\mathrm{rec}$, this ratio gives an approximate \ion{H}{1} abundance in the photoinoization-recombination equilibrium
\begin{equation}
% \approx A \frac{ (12\pi r \alpha)^{1/2}}{\phieuv ^{1/2} \avesigma }
\approx  \braket{\frac{\varepsilon  q  x  }{\varphi}}^{1/2}
\braket{\frac{\cs }{\ceq }}^{-\beta}
% In [19]: 0.4 * (12. * ms.const_au * 2.e-13)**0.5 / (1.e40)**0.5 / 1.e-18
% Out[19]: 0.023967809772275817
\quad \braket{\mathrm{for} ~ t_\mathrm{ioni} < t_\mathrm{rec}}.
\label{eq:y_HI_equi_reduced}
\end{equation}
% (See \appref{sec:pi_param} for derivation.)

% If photoionization is sufficiently fast, it should be given by the Stromgren analysis; if not, it can be approximated to $y_\mathrm{HI} \sim 1$.

\subsection{Type~I Regime}
\label{sec:typeI:regime}
Now we will find the expressions for the parameter space that satisfies the Type~I wind condition (\eqnref{eq:typeI_condition}).
Recalling that
%$\cs $ can be $\ceq $ at the highest and $\vg $ at the lowest
typical $\cs $ and flow velocity follow the three pattern summarized in \tref{tab:regime_summary} (regardless of the wind types),
the slow photoionization condition \eqnref{eq:typeI_condition} is equivalent with the sum of these three conditions:
\begin{description}
\item [Condition~I-1] $c_\mathrm{ch}^\mathrm{I} > \ceq  ~ \cap ~ t_\mathrm{ioni} > \mathrm{min}\left[t_\mathrm{rec} (\cs  = \ceq ), R/\ceq \right]$.
\item [Condition~I-2] $c_\mathrm{ch}^\mathrm{I} < \vg  ~ \cap ~ t_\mathrm{ioni} > \mathrm{min}\left[t_\mathrm{rec} (\cs  = \vg ), R/\mach_\mathrm{g}^\mathrm{I} \vg  \right]$.
\item [Condition~I-3] $ v_\mathrm{g } < c_\mathrm{ch}^\mathrm{I} < \ceq  ~ \cap ~ t_\mathrm{ioni} > \mathrm{min}\left[t_\mathrm{rec} (\cs  = c_\mathrm{ch}^\mathrm{I}), R/c_\mathrm{ch}^\mathrm{I} \right]$.
\end{description}

First, we consider Condition~I-1.
If $\varphi > \varepsilon  q^{-1}  x $ (cf. \eqnref{eq:trec_tcross}),
the recombination timescale is faster than the replenishment of atomic hydrogen through advection,
and Condition~I-1 reduces to
\[
\left\{
( x , \varphi)~|~
\varphi >  x ,
\varepsilon  q x  > \varphi,
\varphi > \varepsilon  q^{-1}  x
\right\}
\]
using Eqs.\eqref{eq:tioni_trec_ratio:dimensionless} and \eqref{eq:cchI_def}.
In the opposite case $\varphi < \varepsilon  q^{-1}  x $,
the Condition~I-1 reduces to
\[
\left\{
( x , \varphi)~|~
\varphi >  x ,
t_\mathrm{ioni} > R/\ceq ,
\varphi < \varepsilon  q^{-1}  x
\right\}
\]
\begin{gather}
\frac{R}{t_\mathrm{ioni} \ceq }
% = \frac{R_0}{\ceq } \frac{\Phi_{\rm EUV}}{4\pi r^2} \delta \langle \sigma\rangle
% =  \frac{R_0 R_{\rm g}}{r^2} \frac{2 \Phi_{\rm EUV} c_p \ceq }{4 \pi GM_* } \delta \langle \sigma\rangle
=  x  ^{-1} %\chi
\frac{\Phi_{\rm EUV}}{\Phi_\mathrm{c,i}}
= \varphi
% \chi
\varepsilon  ^{-1}  x ^{-1}
\label{eq:t_ioni_VS_c_eq}
\end{gather}  % checked right by mathematica
Hence, Condition~I-1 is equivalent to the sum of
\begin{equation}
\left\{
( x , \varphi)~|~
\varphi >  x ,
\varphi < \varepsilon  q  x ,
\varphi > \varepsilon  q^{-1}  x
\right\}
\end{equation}
\begin{equation}
\left\{
( x , \varphi)~|~
x > 1,
x < \varepsilon  ,
x < \varepsilon  q^{-1}
\right\}.
\end{equation}
After some math,
we learn that
\begin{equation}
\varepsilon   > 1 \quad \mathrm{or} \quad \varepsilon  q > 1
\label{eq:hard_spectrum_def}
\end{equation}
is necessary for $(x,\varphi)$ to exist, and in this case, Condition~I-1 reduces to
\[
x  < \varphi < \mathrm{max}\braket{\varepsilon  , \varepsilon  q} \times  x .
\]
% This is exactly the same as that I derived from qualitative consideration.
For $\{(\varepsilon , q)|~\varepsilon   < 1 , ~ \varepsilon  q < 1\}$, the mathematical set of Condition~I-1 becomes null, meaning that the Type~I winds will never be a high temperature flow at $\ceq $.
the Type~I winds with the equilibrium temperature ($\ceq $) form only when the spectrum is hard enough so that one photoionization deposits larger energy than $c_p m \ceq ^2$ ($\varepsilon   > 1$), or the gas can be heated to $\cs  = \ceq $ at a lower $\phieuv $ than is needed to get $t_\mathrm{ioni}/t_\mathrm{rec} < 1$.
When the spectrum is soft,
high-temperature flows ($\ceq $) cannot form unless the wind is Type~II.
We will see this point later in \secref{sec:typeII}.

Next, we consider Condition~I-2.
The condition $c_\mathrm{ch}^\mathrm{I} < \vg $ reduces to $\varphi <  x ^{-1/2}$ (see \eqnref{eq:cchI_vg}),
and thus the emission rate relevant here is those having $\varphi < 1$, as we are interested in the region outside the critical radius, $ x  > 1$.
In this case, it is physically trivial that when the spectrum satisfies the hard spectrum condition \eqnref{eq:hard_spectrum_def},
$t_\mathrm{ioni} > \mathrm{min}\left[t_\mathrm{rec}(\cs  = c_\mathrm{ch}^\mathrm{I}), R/\mach_\mathrm{g}^\mathrm{I} \vg \right]$ is always met.
This is also understood as follows:
$\ceq  > \vg $ always holds in the radius of interest, $ x  > 1$, by definition, and
\eqnref{eq:tioni_trec_ratio:dimensionless} indicates $t_\mathrm{ioni} > t_\mathrm{rec}$ for $\varepsilon  q > 1$.
Similarly, when $\varepsilon   > 1$,
% \begin{gather}
%     \frac{R}{t_\mathrm{ioni}v_g}
% = \frac{2c_p\ceq \phieuv \avesigma  \chi}{4\pi  x ^{1/2}GM_*}
% = \frac{\phieuv }{\Phi_\mathrm{c,i}} x ^{-1/2}
% = \varphi
% \chi
% \varepsilon  ^{-1}  x ^{-1/2}
%     \label{eq:t_ioni_VS_v_g}
% \end{gather}  % checked right by mathematica
\begin{equation}
\frac{R}{t_\mathrm{ioni} \mach_\mathrm{g}^\mathrm{I} v_g}
=   \varepsilon  ^{-1}  x ^{-1} < 1
\label{eq:t_ioni_VS_v_g}
\end{equation}
and thus $t_\mathrm{ioni} > R/\vg $, with $ x $ of interest.
Hence, Condition~I-2 simply reduces to $\varphi <  x ^{-1/2}$ for $\varepsilon  >1$ or $\varepsilon  q > 1$.

The non-trivial case of Condition~I-2 is when the spectrum satisfies $\varepsilon   < 1$ and $\varepsilon  q < 1$, i.e., when the spectrum is soft.
Using Eqs.(\ref{eq:vg_ceq}), (\ref{eq:trec_tcross}), (\ref{eq:tioni_trec_ratio:dimensionless}), and (\ref{eq:t_ioni_VS_v_g}), Condition~I-2 reduces to the sum of
\[
\left\{( x , \varphi)~|~
\varphi  <   x ^{-1/2},
\varphi  < \varepsilon   q  x ^{1+ \beta},
% \varphi  >  \varepsilon   q^{-1}  x ^{-\beta}
\varphi < q \varepsilon ^{-1} x^{- (1 - \beta )}
\right\}
\]
\[
\left\{( x , \varphi)~|~
\varphi  <   x ^{-1/2},
% \varphi  <  \varepsilon     x ^{1/2},
x  > \varepsilon   ^{-1},
% \varphi  <  \varepsilon   q^{-1}  x ^{-\beta}
\varphi > q \varepsilon ^{-1} x^{- (1 - \beta)}
\right\}.
\]
After tedious mathematics,
% this condition reduces to the following three cases depending on $\varepsilon  $ and $q$:
% \[
%     \left\{
%     \begin{array}{ll}
%          \varphi <  \varepsilon    x ^{1/2}& \quad \mathrm{for}~ 1\leq  x  < \varepsilon  ^{-1} \\
%          \varphi  <    x ^{-1/2}& \quad \mathrm{for}~  x  > \varepsilon  ^{-1} \\
%     \end{array}
%     \right.
% \]
% for $\left\{(\varepsilon , q)|~\varepsilon   < 1,~\varepsilon  q < 1, ~ q < \varepsilon  ^{(2\beta + 1)/2}\right\}$,
% \[
%     \left\{
%     \begin{array}{ll}
%          \varphi  <   \varepsilon    x ^{1/2}& \quad\mathrm{for}~ 1\leq  x  < q^{-2/(2\beta+1)} \\
%          \varphi  <   \varepsilon   q  x ^{1+\beta}& \quad\mathrm{for}~ q^{-2/(2\beta+1)} <  x  < (\varepsilon  q)^{-2/(3+2\beta)} \\
%          \varphi  <    x ^{-1/2}& \quad\mathrm{for}~  x  > (\varepsilon  q)^{-2/(3+2\beta)}
%     \end{array}
%     \right.
% \]
% for $\left\{ (\varepsilon , q)|~\varepsilon   < 1,~ \varepsilon  q < 1,~ \varepsilon  ^{(2\beta + 1)/2} < q < 1\right\}$,
% or
% \[
%     \left\{
%     \begin{array}{ll}
%          \varphi  <   \varepsilon   q  x ^{1+\beta}& \quad\mathrm{for}~ 1 \leq   x  < (\varepsilon  q)^{-2/(3+2\beta)} \\
%          \varphi  <    x ^{-1/2}& \quad\mathrm{for}~  x  > (\varepsilon  q)^{-2/(3+2\beta)}
%     \end{array}
%     \right.
% \]
% for $\left\{(\varepsilon , q)|~\varepsilon   < 1,~ \varepsilon  q < 1,~ q > 1\right\}$.
this condition reduces to the following two cases depending on $\varepsilon  $ and $q$:
\[
\left\{
\begin{array}{ll}
\varphi <  \varepsilon   q  x ^{1 + \beta }& \quad \mathrm{for}~ 1\leq  x  < \varepsilon  ^{-1} \\
\varphi  <    x ^{-1/2}& \quad \mathrm{for}~  x  > \varepsilon  ^{-1} \\
\end{array}
\right.
\]
for $\left\{(\varepsilon , q)|~\varepsilon   < 1,~\varepsilon  q < 1, ~ q < \varepsilon  ^{(2\beta + 1)/2}\right\}$,
\[
\left\{
\begin{array}{ll}
\varphi  <   \varepsilon   q  x ^{1+\beta}& \quad\mathrm{for}~ 1 <  x  < (\varepsilon  q)^{-2/(3+2\beta)} \\
\varphi  <    x ^{-1/2}& \quad\mathrm{for}~  x  > (\varepsilon  q)^{-2/(3+2\beta)}
\end{array}
\right.
\]
for $\left\{ (\varepsilon , q)|~\varepsilon   < 1,~ \varepsilon  q < 1,~ \varepsilon  ^{(2\beta + 1)/2} < q \right\}$.

Finally, we evaluate Condition~I-3.
Again, it is trivial that when the spectrum satisfies \eqnref{eq:hard_spectrum_def},
% $t_\mathrm{ioni} > \mathrm{min}\left[t_\mathrm{rec} (\cs  = c_\mathrm{ch}^\mathrm{I}), R/c_\mathrm{ch}^\mathrm{I} \right]$
the Type~I condition is always met for $c_\mathrm{ch}^\mathrm{I} < \ceq $.
Therefore, here we discuss the non-trivial cases, $\left\{(\varepsilon , q)|~\varepsilon   < 1,~ \varepsilon  q < 1\right\}$.
To that end, we calculate Eqs.(\ref{eq:tioni_trec_ratio:dimensionless}) and (\ref{eq:trec_tcross}) with $\cs  = c_\mathrm{ch}^\mathrm{I}$ in advance,
% Substituting \eqnref{eq:cchI_def},
\begin{gather}
\left.\frac{t_\mathrm{ioni}}{t_\mathrm{rec}}\right|_{\cs  = c_\mathrm{ch}^\mathrm{I}}
= \braket{
\frac{\phieuv }{\Phi_\mathrm{c} x  (\varepsilon  q)^{3/(3+2\beta)}}
\\
\frac{R}{t_\mathrm{rec}c_\mathrm{ch}^\mathrm{I}}
= \braket{
\frac{\phieuv (\varepsilon  q^{-1})^{3/(2\beta-1)}}{\Phi_\mathrm{c} x  }
\end{gather}
Using these relations and Eqs.(\ref{eq:cchI_def}) and (\ref{eq:cchI_vg}),
Condition~I-3 reduces to the sum of
\[
\left\{
( x , \varphi)~|~
x ^{-1/2} < \varphi  <  x ,
\varphi < \braket{{q}\varepsilon  }^{\frac{3}{3+2\beta}}  x ,
\varphi  < \braket{\frac{q}{\varepsilon  }}^{\frac{3}{2\beta-1}}  x
\right\}
\]
\[
\left\{
( x , \varphi)~|~
x ^{-1/2} < \varphi  <  x ,
\varphi  < \varepsilon  ^{3/2}  x ,
\varphi  > \braket{\frac{q}{\varepsilon  }}^{\frac{3}{2\beta-1}}  x
\right\}.
\]
Consequently, Condition~I-3 reduces to the following two cases depending on $\varepsilon  $ and $q$:
\[
\left\{
( x , \varphi)~|~
\varphi >   x ^{-1/2},
\varphi <  \varepsilon  ^{3/2}  x
\right\}
\]
for $\left\{(\varepsilon , q)|~\varepsilon   < 1,~ \varepsilon  q < 1,~ q < \varepsilon  ^{(2\beta+1)/2}\right\}$,
\[
\left\{
( x , \varphi)~|~
\varphi >   x ^{-1/2},
\varphi <  (\varepsilon   q)^{3/(2\beta+3)}  x
\right\}
\]
for $\left\{(\varepsilon , q)|~\varepsilon   < 1,~ \varepsilon  q < 1,~ q > \varepsilon  ^{(2\beta+1)/2}\right\}$.

To summarize this section, winds become Type~I when satisfying the Type~I condition (\eqnref{eq:typeI_condition}),
\[
t_\mathrm{ioni} > \mathrm{min} \braket{t_\mathrm{rec}, \frac{R}{\mach \cs }}.
\]
This condition covers a certain area on the $ x $--$\varphi$ plane, depending on the spectral hardness parameters, $\varepsilon  $ and $q$, which are uniquely set once an EUV spectrum is given.
Based on $\varepsilon  $ and $q$, we can classify the $ x $--$\varphi$ map into the following three different classes:
\paragraph{\bf H-class}
Hard spectrum class.
This class is defined by $\{(\varepsilon  , q) | \varepsilon   > 1 \cup \varepsilon  q > 1\}$.
Hot ($\cs  = \ceq $), atomic flow is possible only in this case.
The Type~I regime covers
\[
\varphi < \mathrm{max}\braket{1, q} \varepsilon     x  .
\]
on the $ x $--$\varphi$ plane.
\paragraph{\bf Sa-class}
Soft spectrum with advection replenishment.
This class is defined by $\{(\varepsilon  , q) | \varepsilon   < 1, \varepsilon  q < 1, q < \varepsilon  ^{(2\beta+ 1)/2)}\}$.
The Type~I regime corresponds to
\begin{equation}
\left\{
\begin{array}{ll}
% \varphi < \varepsilon    x ^{1/2}
\varphi <  \varepsilon   q  x ^{1 + \beta }
& \quad \mathrm{for}~ 1\leq  x  < \varepsilon  ^{-1} \\
\varphi <  \varepsilon  ^{3/2}  x & \quad \mathrm{for}~  x  > \varepsilon  ^{-1} \\
\end{array}
\right.
\label{eq:typeI_regime_Saclass}
\end{equation}
Type~I flow is possible only with temperatures below the equilibrium value, i.e., $\cs  < \ceq $, and the atomic flow is maintained by advection in the steadily heated, free winds ($\cs \sim c_\mathrm{ch}^\mathrm{I}$).
\paragraph{\bf Sr-class}
Soft spectrum with recombination reproduction.
This class is defined by $\{(\varepsilon  , q)| \varepsilon   < 1, \varepsilon  q < 1, q > 1 \}$.
The Type~I regime is
\begin{equation}
\left\{
\begin{array}{ll}
\varphi <  \varepsilon   q  x ^{1+\beta}& \quad\mathrm{for}~ 1 \leq   x  < (\varepsilon  q)^{-2/(3+2\beta)} \\
\varphi <  \braket{\varepsilon  q}^{3/(3+2\beta)}  x  & \quad\mathrm{for}~  x  > (\varepsilon  q)^{-2/(3+2\beta)}
\end{array}
\right.
\label{eq:typeI_regime_Srclass}
\end{equation}
Atomic flow is maintained by recombination rather than replenishment through advection.
% owing to the shorter timescale than $t_\mathrm{ioni}$.
% \begin{itemize}
% \item   {\it Soft spectrum with the mixture of advection replenishment and recombination reproduction} ({\bf Sar-class}):
% This class is defined by $\{(\varepsilon  , q)| \varepsilon   < 1, \varepsilon  q < 1, \varepsilon  ^{(2\beta +1)/2)} < q < 1 \}$.
% The Type~I regime is described by
% \begin{equation}
% \left\{
% \begin{array}{ll}
%      \varphi <  \varepsilon    x ^{1/2}& \quad\mathrm{for}~ 1\leq  x  < q^{-2/(2\beta+1)} \\
%      \varphi  <  \varepsilon   q  x ^{1+\beta}& \quad\mathrm{for}~ q^{-2/(2\beta+1)} <  x  < (\varepsilon  q)^{-2/(3+2\beta)} \\
%      \varphi  <  \braket{\varepsilon  q}^{3/(3+2\beta)}  x & \quad\mathrm{for}~  x  > (\varepsilon  q)^{-2/(3+2\beta)}
% \end{array}
% \right.
% \label{eq:typeI_regime:Sarclass}
% \end{equation}
% Similarly to Sa-class, the atomic flow is maintained by advection in the free-wind regime where $\cs  = c_\mathrm{ch}^\mathrm{I}$. However, it is maintained by both advection and recombination in the gravity-inhibited regime ($\cs  = \vg $).
% \end{itemize}

The top left, top right, and bottom left panels in \fref{fig:classmaps} show the parameter spaces $(x, \varphi)$ for the H-class, Sa-class, and Sr-class, respectively.
In each panel, the Type~I regime (\eqnref{eq:typeI_condition}) is represented by gray vertical lines.
Generally, the Type~I regimes can be subdivided into three regimes: the isothermal wind regime where $\cs  \sim \ceq $ (red-shaded area); the steadily heated, free wind regime where $\cs  \sim c_\mathrm{ch}^\mathrm{I}$ (green-shaded area); and the gravity-inhibited wind regime where $\cs  \sim \vg $ (blue-shaded area).
These three regimes correspond to Regions~A, B, and C in \begelman{}, respectively (cf. \tref{tab:regime_summary}).
We follow these notations for the regime labels, but we designate the Type~I steadily heated, free wind regime as Regime~B-I to highlight its classification as a Type~I regime (cf. \tref{tab:sound_spees}).
\begin{figure*}
% python3.9 EUVX.py class
\centering
% \includegraphics[clip, width = \linewidth]{subsections/cchI_valid_region.pdf}
\includegraphics[clip, width = \linewidth]{EUVXClassMaps.pdf}
\caption{Same as Figures~\ref{fig:sqplane}, \ref{fig:H-class}, \ref{fig:Sa-class}, and \ref{fig:Sr-class} but with $\beta$ left unspecified. The expressions for the boundaries are also annotated.
The top left, top right, and bottom left panels are those for the H-class, Sa-class, and Sr-class, respectively.
The bottom right is the spectral-hardness diagram.
% The areas covered by the light-gray vertical lines correspond to the Type~I regime (atomic winds); the others are the Type~II regime (ionized winds).
% Among these soft-spectrum classes, the red dashed line is present above the blue dashed line only in the Sa-class, meaning that slow-recombination, ionized winds can form.
% The intersection of the three dashed lines is included in $x > 1$ for the Sar-class but not for the Sr-class.
\label{fig:classmaps}
\end{figure*}

Again, the class to which a given EUV spectrum is assigned is determined by the location of the spectrum's $\varepsilon  $ and $q$ on the $\varepsilon  $--$q$ plane (the bottom right panel in \fref{fig:classmaps}).

% \begin{figure}
%     \centering
%     \includegraphics[clip, width = \linewidth]{figs/sq_soft_hard_categorize.pdf}
%     \caption{Caption}
%     \label{fig:sq_soft_hard_categorize}
% \end{figure}

% \mycomment{(Where to put the descriptions below)}
% In this paper, we assume that photoionization dominates over collisional ionization.
% Comparison between $t_{\rm ioni}$ and $R_0/\ceq $ sets another critical EUV emission rate
% These high- and low-end sound speeds are constants and independent of $\phieuv $, and thus the condition of \eqnref{eq:tioni_tcross} are rewritten to simple forms

% The ratio of $R_0/t_\mathrm{ioni} \ceq $ should be interpreted as the ratio between $R$ and the thickness of the \ion{H}{1} layer the gas flowing at the speed of $\ceq $ forms.
% For instance, if $\phieuv $ is equal to $\Phi_\mathrm{c,i}  x $ at a radius, this means the \ion{H}{1} layer can extend to a height of $\sim R$.
% If it is ten times higher, then the \ion{H}{1} layer thickness would be $\sim 0.1 R$, i.e., advection cannot form a thick \ion{H}{1} layer.
% Note that $\ceq  t_\mathrm{ioni}$ is the maximum thickness a flowing gas can achieve through advection, as $\ceq $ is the maximum flow speed an EUV-heated gas can achieve.
% However, recombination is another factor to determine the \ion{H}{1} layer thickness, which we will discuss later.

\subsection{Characteristic Sound Speed: Type~II}
\label{sec:typeII}
The remaining parameter space uncovered by the Type~I regime is where photoionization is faster than recombination and the replenishment of the atoms through advection (\eqnref{eq:typeII_condition}),
\[
t_\mathrm{ioni} < \mathrm{min} \braket{t_\mathrm{rec}, \frac{R}{\mach \cs }}
\]
so that the flow includes the ionized region within $s < R$ (Type~II; right panel in \fref{fig:streamline_types}).
In this case, the characteristic sound speed for the Type~I winds, $c_\mathrm{ch}^\mathrm{I}$ (\eqnref{eq:cchI_def}), is unavailable to characterize the heating timescale of Type~II flows since it has been derived with the assumption of fully atomic winds within $s < R$.
Thus, we need to redefine an alternative characteristic sound speed for Type~II flows, which we address in this section.

% \begin{figure}
%     \centering
%     \includegraphics[clip, width = \linewidth]{figs/TypeII_model.pdf}
%     \caption{Caption}
%     \label{fig:type2_model}
% \end{figure}
% We give the flow speeds as $c_1$ and $c_\mathrm{ch}^\mathrm{II}$ in $s < s_1$ and $s_1 < s < R$, respectively.
% We derive the Type~II characteristic sound speed in a similar manner as \eqnref{eq:definition_of_Tch}.
For Type~II winds, the total energy gain during the gas flowing to the height $\sim R$ is decomposed into two parts.
The first component is the initially injected energy until a lunched atomic hydrogen undergoes photoionization once.
The launched atomic hydrogen can survive, on average, for a timescale of $t_\mathrm{ioni}$ and thus can travel a distance of $\sim \mach \cs  t_\mathrm{ioni}$.
The surviving distance corresponds to $s_1$, the length of the neutral part depicted in the right panel of \fref{fig:streamline_types}.
It is worth noting that, by definition, the timescale of recombination and advection replenishment is guaranteed longer than the photoionization timescale in the Type~II wind, and thus $t_\mathrm{ioni}$ sets the average timescale for which a newly launched atomic hydrogen can persist in the wind.

The second component is the additional supplied energy by reionizations of recombined hydrogen in the ionized region.
Recombination can take place in the \ion{H}{2} region if the timescale is sufficiently shorter than the wind crossing timescale.
The recombined hydrogen will undergo reionization due to the short ionization timescale, depositing energy to the gas.
The \ion{H}{1 } abundance remains small since $t_\mathrm{ioni} < t_\mathrm{rec}$.

Hence, to define the characteristic sound speed for Type~II winds, $c_\mathrm{ch}^\mathrm{II}$, we separate the integration range of \eqnref{eq:c_ch_def} for the initially injected energy and secondary deposited energy as
% integrating the total energy gain from the launching point to the height of $\sim R$,
\begin{equation}
c_p {c_\mathrm{ch}^\mathrm{II}}^2
=  \int_0^{s_\mathrm{II}}
\frac{\avesigma  \aveenergy   \phieuv }{4\pi m r^2}
\frac{\dd s}{c_\mathrm{ch}^\mathrm{II}}
\int_{s_\mathrm{II}}  ^R
\frac{t_\mathrm{ioni}}{t_\mathrm{rec}}
\frac{\avesigma  \aveenergy   \phieuv }{4\pi m r^2}
\frac{\dd s}{c_\mathrm{ch}^\mathrm{II}} ,
\label{eq:cch:typeII:integral_form}
\end{equation}
where $s_\mathrm{II}$ is the surviving length of launched hydrogen, $s_\mathrm{II} = c_\mathrm{ch}^\mathrm{II} t_\mathrm{ioni}$.
The first and second terms on the RHS represent the initially injected energy and additional energy supply through reionizations, respectively.
Here, we have approximated the \ion{H}{1} abundance to unity while the gas undergoes the first photoionization (the first term on the RHS) and to the value set by ionization-recombination equilibrium, $y_\mathrm{HI}\sim t_\mathrm{ioni}/t_\mathrm{rec}$ (\eqnref{eq:y_HI_equi_reduced}), in the ionized region (the second term on the RHS).
% since advection has a limited impact on the determination of $y_\mathrm{HI}$ for $t_\mathrm{ioni} < R/\mach \cs $.

% The energy gain of a newly launched gas while the initial photoionization (cf. \eqnref{eq:c_ch_def}) is
The first term on the RHS of \eqnref{eq:cch:typeII:integral_form} is explicitly computed as
\begin{equation}
c_p c_1^2 \approx \frac{\avesigma  \aveenergy   \phieuv }{4\pi m r^2}
t_\mathrm{ioni} = \frac{\aveenergy}{m}
\label{eq:c1_def}
\end{equation}
where $c_1$ is the sound speed corresponding to the initially injected energy.
% therefore, $s_1 = v_1 t_\mathrm{ioni}$.
The above equation is rewritten to
\[
\frac{c_1}{\ceq } = \varepsilon  ^{1/2}.
\]
This indicates that for $\varepsilon  > 1$, heating due to the initially injected energy is sufficiently rapid that the launched gas can reach the equilibrium temperature within a short distance $s < \mach \cs  t_\mathrm{ioni}$.
% and thus $s_1 = v_1 t_\mathrm{ioni}$.
On the other hand, for $c_1 < \vg $, or equivalently $\varepsilon  x < 1$, gravity is so strong that the initially injected energy is not enough to form an escaping wind.
This suggests that in this range, the additional energy supplied through reionizations is indispensable to form escaping winds.
% Similarly to $c_\mathrm{ch}^\mathrm{I}$,
% the flow speed $v_1$, which is typically the actual sound speed of the gas $\cs $, does not get higher (lower) than $\ceq $ ($\vg $) due to the effects of cooling (gravity).
% Hence, $v_1 \approx \cs  \approx \ceq $ for $\varepsilon   > 1$, and $v_1 \approx \cs  \approx \vg $ for $ x  < \varepsilon  ^{-1}$.

The second integration on the RHS of \eqnref{eq:cch:typeII:integral_form} can also simplify to
\[
\int_{s_\mathrm{II}}  ^R
\frac{t_\mathrm{ioni}}{t_\mathrm{rec}}
\frac{\avesigma  \aveenergy   \phieuv }{4\pi m r^2}
\frac{\dd s}{c_\mathrm{ch}^\mathrm{II}}
\approx
\frac{t_\mathrm{ioni}}{t_\mathrm{rec}}
\frac{\avesigma  \aveenergy   \phieuv }{4\pi m r^2}
\frac{R - c_\mathrm{ch}^\mathrm{II} t_\mathrm{ioni}}{c_\mathrm{ch}^\mathrm{II}}
\]
and then the definition of the characteristic sound speed for Type~II winds reduces to
\begin{equation}
c_p {c_\mathrm{ch}^\mathrm{II}}^2
= c_p c_1 ^2
+ \frac{\avesigma  \aveenergy   \phieuv }{4\pi m r^2} \frac{t_\mathrm{ioni}}{t_\mathrm{rec}}
\frac{R - c_\mathrm{ch}^\mathrm{II} t_\mathrm{ioni}}{c_\mathrm{ch}^\mathrm{II}} .
\label{eq:chII_original_def}
\end{equation}
This equation can be rewritten to
\[
\begin{split}
c_p {c_\mathrm{ch}^\mathrm{II}}^2
& =  \frac{\avesigma  \aveenergy   \phieuv }{4\pi m r^2}
\left[
t_\mathrm{ioni} +
\frac{t_\mathrm{ioni}}{t_\mathrm{rec}}
\braket{
\frac{R}{c_\mathrm{ch}^\mathrm{II}}
- t_\mathrm{ioni}
\right]
\\
& =
\frac{\avesigma  \aveenergy   \phieuv }{4\pi m r^2}
\left[
t_\mathrm{ioni} \braket{1 - \frac{t_\mathrm{ioni}}{t_\mathrm{rec}}}
\frac{t_\mathrm{ioni}}{t_\mathrm{rec}}
\frac{R}{c_\mathrm{ch}^\mathrm{II}}
\right].
\end{split}
\]
One can observe from this form that the Type~II characteristic sound speed $c_\mathrm{ch}^\mathrm{II}$ smoothly transitions to the Type~I characteristic sound speed $c_\mathrm{ch}^\mathrm{I}$ since the boundary between Type~I and Type~II regimes is expressed by either $t_\mathrm{ioni} = t_\mathrm{rec}$ or $t_\mathrm{ioni} = R/c_\mathrm{ch}$ for all classes.
This continuity arises because $c_\mathrm{ch}^\mathrm{I}$ can be interpreted as a special case of $c_\mathrm{ch}^\mathrm{II}$ when the surviving distance $s_1$ is larger than $R$.
It also emphasizes that $c_\mathrm{ch}^\mathrm{II}$ is defined only in the parameter space where
\begin{equation}
t_\mathrm{ioni} < t_\mathrm{rec}
\quad
\cap
\quad
t_\mathrm{ioni} < \frac{R}{c_\mathrm{ch}^\mathrm{I}}
\label{eq:chII_available_condition}
\end{equation}
is met.
The Type~II regime is always included in this parameter space,
and thus $c_\mathrm{ch}^\mathrm{II}$ is always defined in the Type~II regime (\eqnref{eq:typeII_condition}).

\eqnref{eq:chII_original_def} can also be rewritten in dimensionless form as
\begin{equation}
\braket{\frac{c_\mathrm{ch}^\mathrm{II}}{\ceq }}^2
= \varepsilon
+ \braket{\frac{\varphi \varepsilon   q}{ x }}^{1/2}
\braket{\frac{c_\mathrm{ch}^\mathrm{II}}{\ceq }}^{-(\beta+1)}
\braket{1 - \varphi^{-1}\varepsilon   x \frac{c_\mathrm{ch}^\mathrm{II}}{\ceq }}
\label{eq:chII_original_def:dimensionless}
\end{equation}
% The last factor of the second term on the RHS is guaranteed positive for the Type~II winds because this regime is defined by
% \[
%     t_\mathrm{ioni} < \mathrm{min}\braket{t_\mathrm{rec}, \frac{R}{\mach \cs }}.
% \]
% Note that $\cs $ here indicates the actual sound speed in the neutral layer; hence $\cs  = v_1$.
% The first term on the RHS indicates the initially injected energy into the atomic gas until photoionized, and the second term indicates the secondary deposited energy by (multiple) reionizations of recombined hydrogen in the \ion{H}{2} region.
The last factor of the second RHS term represents the length of the ionized region.
\eqnref{eq:chII_original_def:dimensionless} is also equivalently rewritten to
\begin{equation}
\braket{\frac{c_\mathrm{ch}^\mathrm{II}}{\ceq }}^2
\varepsilon   \left[1 - \braket{\frac{x \varepsilon   q}{ \varphi }}^{1/2}
\braket{\frac{c_\mathrm{ch}^\mathrm{II}}{\ceq }}^{-\beta}\right]
+ \braket{\frac{\varphi \varepsilon   q}{ x }}^{1/2}
\braket{\frac{c_\mathrm{ch}^\mathrm{II}}{\ceq }}^{-(\beta+1)}.
\label{eq:chII_original_def:dimensionless:another_form}
\end{equation}
The relative strength of the second term to the first term in Eqs.\eqref{eq:chII_original_def:dimensionless} and \eqref{eq:chII_original_def:dimensionless:another_form} is determined by the ratio between the crossing timescale and recombination timescale,
\begin{equation}
\frac{R}{\cs  t_\mathrm{rec}}
\braket{
\frac{\varphi q}{\varepsilon  x }
\braket{\frac{\cs }{\ceq }}^{-(1+\beta)}
\label{eq:trec_tcross_wo_mach}
\end{equation}
(cf. \eqnref{eq:trec_tcross}).
Again, the RHSs of Eqs.\eqref{eq:chII_original_def:dimensionless} and \eqref{eq:chII_original_def:dimensionless:another_form} are guaranteed positive in the parameter space represented by \eqnref{eq:chII_available_condition}, which is rewritten to a dimensionless form as
\begin{equation}
\left\{
(x, \varphi) ~\left| ~
\braket{\frac{\varepsilon  q  x  }{\varphi}}^{1/2}
\braket{\frac{\cs }{\ceq }}^{-\beta} < 1
% \frac{\varphi}{\varepsilon  x} > \braket{\frac{\varphi}{x}}^{1/3}
\varphi > \varepsilon ^{3/2} x
\right.
\right\}
\label{eq:chII_available_condition:dimensionless}
\end{equation}
This is consistently satisfied in the Type~II regime.

The characteristic sound speed $c_\mathrm{ch}^\mathrm{II}$ resulting from \eqnref{eq:chII_original_def:dimensionless} is a monotonically increasing function of $\varphi$ at any distance $x$ and is a monotonically decreasing function of $x$ at any $\varphi$.
This is understood by taking the partial differences of both sides of \eqnref{eq:chII_original_def:dimensionless} with respect to $\varphi$ and $x$,
\[
\begin{gathered}
\begin{aligned}
\braket{\frac{\partial u}{\partial \varphi}}_x
& = \frac{u}{\varphi} \left[\frac{1}{2} + \braket{\frac{\varphi}{\varepsilon x u} - 1}^{-1}\right]
\\
& \quad \times
\left[
\frac{2 u^2}{u^2 - \varepsilon}
+ \beta + 1
+ \braket{\frac{\varphi}{\varepsilon x u} - 1}^{-1}
\right]^{-1}
\quad \braket{> 0}
\end{aligned}
\\
\begin{aligned}
\braket{\frac{\partial u}{\partial x}}_\varphi
&= - \frac{u}{x} \left[\frac{1}{2} + \braket{\frac{\varphi}{\varepsilon x u} - 1}^{-1}\right]
\\
& \quad \times
\left[
\frac{2 u^2}{u^2 - \varepsilon}
+ \beta + 1
+ \braket{\frac{\varphi}{\varepsilon x u} - 1}^{-1}
\right]^{-1}
\quad \braket{< 0}
\end{aligned}
\end{gathered}
\]
where $u \equiv c_\mathrm{ch}^\mathrm{II}/\ceq$.
This simply reflects that the heating timescale of Type~II winds gets shorter as $\phieuv $ increases
% due to increasing reionization
and longer at large distances.

The Type~II characteristic sound speed is a metric for the heating timescale to be compared with $\ceq $ and $\vg $.
Therefore, it is convenient to derive the equations, $\varphi = \varphi(x)$, at which $c_\mathrm{ch}^\mathrm{II}$ equals to $\ceq $ and $\vg $.
Since $c_\mathrm{ch}^\mathrm{II}$ derived by \eqnref{eq:chII_original_def:dimensionless} is a monotonically increasing function of $\varphi$ at any distance, there is a unique $\varphi$ at which $c_\mathrm{ch}^\mathrm{II} = \ceq $ at any $x$.
Substituting $c_\mathrm{ch}^\mathrm{II} = \ceq $ into \eqnref{eq:chII_original_def:dimensionless}, such $\varphi$ is derived as
\[
\varphi  = f_\mathrm{eq} (\varepsilon , q) x,
\]
\[
f_\mathrm{eq} (\varepsilon , q)
\equiv \frac{4 \varepsilon  q }
{\braket{\sqrt{(\varepsilon ^{-1}-1)^2 + 4q} - (\varepsilon ^{-1}-1)}^2}
\]
Hence, the rapid heating condition, $c_\mathrm{ch}^\mathrm{II} > \ceq $, reduces to $\varphi > f_\mathrm{eq} x$.
The factor $f_\mathrm{eq}$ is equal to unity when $\varepsilon  = 1$ or $\varepsilon  q = 1$, and the range of $f_\mathrm{eq}$ is tabulated in \tref{tab:f_eq} for other $(\varepsilon , q)$.
\begin{table}[htbp]
\centering
\begin{tabular}{LcC}
\text{Parameter space} & Class & \text{Range of }~f_\mathrm{eq} \\
\hline
\left\{(\varepsilon , q)|~\varepsilon  q > 1, ~\varepsilon  > 1\right\}
1< f_\mathrm{eq} < \mathrm{min}(\varepsilon , \varepsilon  q)
\\
\left\{(\varepsilon , q)|~\varepsilon  q > 1, ~\varepsilon  < 1\right\}
\varepsilon  < f_\mathrm{eq} < 1
\\
\left\{(\varepsilon , q)|~\varepsilon  q < 1, ~\varepsilon  > 1\right\}
\varepsilon  q < f_\mathrm{eq} < 1
\\
\left\{(\varepsilon , q)|~\varepsilon  q < 1, ~\varepsilon  < 1\right\}
Sa, Sr
f_\mathrm{eq} > 1
\end{tabular}
\caption{Possible values of $f_\mathrm{eq}$ taken in the parameter spaces, and the corresponding spectrum classes.}
\label{tab:f_eq}
\end{table}
% For $\left\{(\varepsilon , q)|~\varepsilon  q > 1, ~\varepsilon  > 1\right\}$, $1< f_\mathrm{eq} < \mathrm{min}(\varepsilon , \varepsilon  q)$.
% When $\varepsilon  < 1$, $f_\mathrm{eq} > \varepsilon $; and when $\varepsilon  > 1$, $f_\mathrm{eq} < \varepsilon $.
% When $\varepsilon  q < 1$, $f_\mathrm{eq} > \varepsilon  q $; and when $\varepsilon  q > 1$, $f_\mathrm{eq} < \varepsilon  q$.
For the H-class, the Type~II regime $\varphi > \mathrm{max}(1, q)\varepsilon  x$ always satisfies $\varphi  > f_\mathrm{eq} x$, indicating that Type~II winds are always isothermal in the H-class.
For the Sa- and Sr-classes, Type~II winds are not necessarily isothermal and can even be gravity-inhibited.
This is because the soft spectrum has a relatively small energy deposition per photoionization, and a large $\phieuv $ is needed to increase photoionization to heat the gas to $\cs  = \ceq $.
We will discuss these points in more detail in \secref{sec:typeII:regime}.

Similarly, there is a unique $\varphi$ at which $c_\mathrm{ch}^\mathrm{II} = \vg $.
Substituting $c_\mathrm{ch}^\mathrm{II} = \vg $ into \eqnref{eq:chII_original_def:dimensionless} reduces to
\begin{equation}
% x^{-1} = \varepsilon    + \braket{{\varphi \varepsilon   q x^\beta}}^{1/2} \braket{1 - \varphi^{-1}\varepsilon   x ^{1/2}},
\braket{\frac{\varphi q x^\beta}{\varepsilon }}^{1/2}
x^{-1} \varepsilon ^{-1} - 1
+ \braket{\frac{\varepsilon  q x^{1+\beta}}{\varphi}}^{1/2}.
\label{eq:varphi_g_def}
\end{equation}
Note that the LHS is $R/\vg  t_\mathrm{rec}$ with $\cs  = \vg $.
Solving this equation with respect to $\varphi$ gives the functional form of
\begin{equation}
\begin{split}
\varphi&  = \varphi_\mathrm{g} (x; \varepsilon , q, \beta)
\\
& \equiv
\braket{
\frac{2\sqrt{q\varepsilon }x^{(\beta+1)/2}}
{\sqrt{(x^{-1}\varepsilon ^{-1}-1)^2 + 4 q x^{(2\beta + 1)/2}} - (x^{-1}\varepsilon ^{-1}-1)}
\end{split}
\label{eq:varphi_g}
\end{equation}
at which $c_\mathrm{ch}^\mathrm{II} $ equals to $\vg $.
This function has the intersections with $\varphi = x^{-1/2}$ at $x = \varepsilon ^{-1}$ and $x = (\varepsilon  q)^{-2/(3+2\beta)}$ and asymptotically approaches $\varphi = \varepsilon  x^{1/2}$ for $x \gg \varepsilon ^{-1}$.
It also connects to $f_\mathrm{eq} x$ at $x = 1$, which motivates us to define the secondary critical EUV emission rate as
\begin{equation}
\begin{split}
\Phi_\mathrm{c}^\mathrm{II}
& \equiv f_\mathrm{eq} \Phi_\mathrm{c} \\
& = \frac{4 \varepsilon  q }
{\braket{\sqrt{(\varepsilon ^{-1}-1)^2 + 4q} - (\varepsilon ^{-1}-1)}^2} \Phi_\mathrm{c},
\end{split}
\label{eq:secondary_critical_luminosity}
\end{equation}
Above this critical EUV emission rate, ionized winds can be isothermal at $x = 1$.
However, since Type II winds in the H-class are guaranteed isothermal, the secondary critical EUV emission rate is physically meaningful only for the Sa and Sr classes.

% To grab the essential behavior for the root of Eq.\eqref{eq:chII_original_def:dimensionless},
The root of the implicit equation, Eq.\eqref{eq:chII_original_def:dimensionless}, can be explicitly expressed by simple forms in some special cases.
% When the first and second terms on the RHS of the equation equal, the root is
% \[
%     c_\mathrm{ch}^\mathrm{II} = c_\mathrm{mid} \equiv \sqrt{2}\varepsilon ^{1/2} \ceq .
% \]
% This holds only at
% \[
%     \frac{\varphi}{x} = f_\mathrm{mid}(\varepsilon , q, \beta) \equiv   \frac{2q (2\varepsilon )^{-\beta + 1}}
%     {\braket{\sqrt{1 + 4 q (2\varepsilon )^{-(2\beta + 1)/2}} - 1}^2}
%     .
% \]
% Recalling that $c_\mathrm{ch}^\mathrm{II}$ is an increasing function of $\varphi$ at any $x$, $\varphi < f_\mathrm{mid} x$ indicates the initially injected energy larger than the additionally deposited energy through reionizations, and vice versa, for determining the characteristic sound speed.
When the additional energy injection by reionization is negligible due to slow recombination,
\begin{equation}
\braket{\frac{c_\mathrm{ch}^\mathrm{II}}{\ceq }}^{\beta+1}
\gg
\braket{\frac{\varphi q}{ x  \varepsilon   }}^{1/2},
\label{eq:slow_recombination_condition}
\end{equation}
% \begin{equation}
%     \varphi \ll f_\mathrm{mid} x,
%     \label{eq:c2_in_slow_recombination_valid_paramspace}
% \end{equation}
(cf. \eqnref{eq:trec_tcross}),
the total energy gain during the gas traveling to the height of $\sim R$ is primarily covered by the initially injected energy.
In this case, we can drop the second term on the RHS of \eqnref{eq:chII_original_def:dimensionless} to obtain a zeroth-order root as
\begin{equation}
c_\mathrm{ch}^\mathrm{II} \approx c_1 = \varepsilon  ^{1/2} \ceq .
\label{eq:c2_in_slow_recombination}
\end{equation}
Substituting this root back into \eqnref{eq:slow_recombination_condition}, we get
\begin{equation}
\varphi \ll \varepsilon ^{\beta + 2} q^{-1} x,
\label{eq:c2_in_slow_recombination_valid_paramspace}
\end{equation}
a sufficient condition where \eqnref{eq:c2_in_slow_recombination} is available.
% This parameter space corresponds to where the recombination timescale is longer than the crossing timescale $R/c_\mathrm{ch}^\mathrm{II}$ (cf. \eqnref{eq:trec_tcross}).

On the other hand, when recombination is fast enough so that the additionally deposited energy dominates over the initially deposited energy, i.e.,
\begin{equation}
\braket{\frac{c_\mathrm{ch}^\mathrm{II}}{\ceq }}^{\beta+1}
\ll
\braket{\frac{\varphi q}{ x  \varepsilon   }}^{1/2},
\label{eq:rapid_recombination_condition}
\end{equation}
% \[
%     \varphi \gg f_\mathrm{mid} x,
% \]
we can neglect the first term of the RHS in \eqnref{eq:chII_original_def:dimensionless:another_form}.
Then, the zeroth order $c_\mathrm{ch}^\mathrm{II}$ is derived as
\[
\braket{\frac{c_\mathrm{ch}^\mathrm{II}}{\ceq }}
\approx
\braket{\frac{\varphi_\mathrm{II}}{ x   }}^{1/2(\beta + 3)},
\]
\[
\begin{gathered}
\varphi_\mathrm{II} \equiv \frac{\phieuv }{\Phi_\mathrm{c,rec}^\mathrm{II}} \\
\Phi_\mathrm{c,rec}^\mathrm{II} \equiv
\frac{\Phi_\mathrm{c}}{\varepsilon  q} .
\end{gathered}
\]
The normalization factor $\Phi_\mathrm{c,rec}^\mathrm{II}$ is the secondary critical EUV emission rate for recombination-dominated Type~II winds.
Here we have retrieved the Type~II characteristic sound speed (\eqnref{eq:cchII_def})
\[
c_2  \equiv \ceq  \braket{\frac{\varphi_\mathrm{II}}{ x   }}^{1/2(\beta + 3)}.
\]
Substituting $c_2 $ into Eq.\eqref{eq:rapid_recombination_condition},
we obtain a sufficient condition under which the above approximate root is available:
\begin{equation}
\varphi \gg \varepsilon  ^{\beta + 2} q^{-1}  x ,
\label{eq:rapid_recombination_condition_paramspace}
\end{equation}
which is opposite to \eqnref{eq:c2_in_slow_recombination_valid_paramspace}.

In the following subsections, we will evaluate the sound speeds and corresponding parameter spaces in the Type~II regime for each class using $c_\mathrm{ch}^\mathrm{II}$.

\subsection{Type~II Winds}
\label{sec:typeII:regime}
As mentioned in \secref{sec:typeII}, the remaining parameter space uncovered by the Type~I regime (the area covered by gray vertical lines in \fref{fig:classmaps}) is the Type~II regime.
The condition of the Type~II regime is expressed by the fast ionization condition (\eqnref{eq:typeII_condition}),
\[
t_\mathrm{ioni} < \mathrm{min} \braket{t_\mathrm{rec}, \frac{R}{\mach \cs }}.
\]
Similarly to the Type~I regime, it can equivalently be broken down into the following three conditions:
\begin{description}
\item [Condition~II-1] $c_\mathrm{ch}^\mathrm{II} > \ceq  ~ \cap ~ t_\mathrm{ioni} < \mathrm{min}\left[t_\mathrm{rec} (\cs  = \ceq ), R/\ceq \right]$
\item [Condition~II-2] $c_\mathrm{ch}^\mathrm{II} < \vg  ~ \cap ~ t_\mathrm{ioni} < \mathrm{min}\left[t_\mathrm{rec} (\cs  = \vg ), R/\mach_\mathrm{g}^\mathrm{II} \vg  \right]$
\item [Condition~II-3]
$ v_\mathrm{g } < c_\mathrm{ch}^\mathrm{II} < \ceq $ $\cap$ $t_\mathrm{ioni} < \mathrm{min}\left[t_\mathrm{rec} (\cs  = c_\mathrm{ch}^\mathrm{II}), R/c_\mathrm{ch}^\mathrm{II} \right]$
\end{description}
Here, $\mach_\mathrm{g}^\mathrm{II}$ is the typical Mach number for gravity-inhibited Type~II winds, which we will derive the specific form later in this section.

In this section, we analyze the fast ionization condition for each class to derive the sound speed and typical flow speed for Type~II winds and corresponding parameter spaces on the $x$--$\varphi$ plane.
% We first consider Condition~II-1.
% The fast ionization condition is immediately derived as
% \[
%     \varphi > \mathrm{max}\braket{1, q} \times \varepsilon  x,
% \]
% from the discussions in \secref{sec:typeI:regime}.
% When recombination is slow in the wind
% (\eqnref{eq:c2_in_slow_recombination_valid_paramspace}), $c_\mathrm{ch}^\mathrm{II} > \ceq $ is rewritten to $\varepsilon  > 1$.

\subsubsection{Type~II winds in H-class}
\label{sec:typeII:regime:H}
For the H class, it is trivial that $c_\mathrm{ch}^\mathrm{II} $ always exceeds $ \ceq $ from a physical point of view as follows.
The Type~I condition $t_\mathrm{ioni} > \mathrm{min}(t_\mathrm{rec}, R/\mach \cs )$ indicates that on average, the launched gas reaches $s = R$ with a hydrogen atom being photoionized less than once.
Such gas can still have a characteristic sound speed $c_\mathrm{ch}^\mathrm{I}$ exceeding $\ceq $ in this class.
In the case where a launched hydrogen experiences photoionization at least once (Type~II), the gas acquires more energy than the Type~I case through recombination followed by reionization.
Thus, the corresponding Type~II characteristic sound speed must exceed $\ceq $.
Hence, the sound speed and the typical flow speed in the ionized region of Type~II flows are given by $\cs  = \ceq $.
One can also directly prove $c_\mathrm{ch}^\mathrm{II} > \ceq $ mathematically from \eqnref{eq:chII_original_def:dimensionless} for $\varepsilon  > 1$ or $\varepsilon  q > 1$ (see Appendix~\ref{sec:math_proof}).

% Therefore, the \ion{H}{2} region, if formed ($t_\mathrm{ioni} < t_\mathrm{rec}$), has $\ceq $ when $\varepsilon  q > 1$.
% This is intuitively understood as follows:
% first of all, $c_\mathrm{ch}$ is an increasing function with respect to $\phieuv $,
% although $y_\mathrm{HI}$ decreases from unity to a small value $t_\mathrm{ioni}/t_\mathrm{rec} < 1$ at $\phieuv  = \varepsilon  q \Phi_\mathrm{c}  x $.
% This is because the base density increases with $\phieuv $, and the increasing recombination rate amplifies the heating rate as $\aveenergy  t_\mathrm{ioni}^{-1}/c_p m = \aveenergy  n_\mathrm{H} \alpha / c_p m$.
% When $\varepsilon   q > 1$, $c_\mathrm{ch}$ already reaches $\ceq $ at $\phieuv  = \Phi_\mathrm{c} x $, where $y_\mathrm{HI}\approx 1$.
% Therefore, $c_\mathrm{ch}$ always exceeds $ \ceq $ even when the flowing gas is fully ionized ($y_\mathrm{HI} = t_\mathrm{ioni}/t_\mathrm{rec}$).

Since $c_\mathrm{ch}^\mathrm{II} > \ceq $, the parameter spaces satisfied by Conditions~II-2 and II-3 are null, and thus the fast ionization condition is equivalent with $\varphi  > \mathrm{max}(1, q)\varepsilon  x$ for the H-class.
It indicates that the Type~II regime is always in Regime~A (isothermal winds) for the H-class.

The fact of $c_\mathrm{ch}^\mathrm{II} > \ceq $ indicates that heating is always rapid enough to form isothermal winds in the Type~II regime.
As discussed in Sections~\ref{sec:typical_flow_velocities} and \ref{sec:c_ch}, this parameter range has a shorter heating timescale than the gravitational timescale, and thus winds can escape without being influenced by gravity.
The typical flow speed is thus given by $\sim \ceq $.

\subsubsection{Type~II winds in Sa-class}
The Type~II regime in the Sa-class is expressed by
\begin{equation}
\left\{
\begin{array}{ll}
% \varphi > \varepsilon    x ^{1/2}
\varphi >  \varepsilon   q  x ^{1 + \beta }
& \quad \mathrm{for}~ 1\leq  x  < \varepsilon  ^{-1} \\
\varphi >  \varepsilon  ^{3/2}  x & \quad \mathrm{for}~  x  > \varepsilon  ^{-1} \\
\end{array}
\right.
\label{eq:typeII_regime_Saclass}
\end{equation}
(cf. Eq.\eqref{eq:typeI_regime_Saclass}).

First, we consider Condition~II-1.
The fast ionization condition $t_\mathrm{ioni} < \mathrm{min}\left[t_\mathrm{rec} (\cs  = \ceq ), R/\ceq \right]$ is rewritten to $\varphi  > \mathrm{max}(1, q)\varepsilon  x = \varepsilon  x$ in this class.
One can confirm that the parameter space of $c_\mathrm{ch}^\mathrm{II}>\ceq $, or equivalently $\varphi > f_\mathrm{eq} x$, is covered by the Type~II regime, $\varphi > \varepsilon  x$, in this class ($\varepsilon  < 1$).
% This reflects the fact that soft spectra are ionization-oriented, meaning that heating per ionization is relatively inefficient.
% This parameter space also satisfies \eqnref{eq:chII_available_condition:dimensionless} in the soft-spectrum classes ($\varepsilon  < 1$), implying that $f_\mathrm{eq} x$ is available in the Type~II regime at any $x$.
Consequently, Condition~II-1 reduces to $\varphi > f_\mathrm{eq} x$.

Next, we consider Condition~II-2.
We first evaluate $\mach_\mathrm{g}^\mathrm{II}$.
To form escape winds in this gravity-inhibited regime, the gas must obtain energy to reach $\cs  \sim \vg  $ until traveling to a height of $\sim R$.
Thus, the self-consistent condition is
\begin{equation}
\begin{split}
c_p \vg ^2
& \approx c_p c_1^2
\frac{\avesigma  \aveenergy   \phieuv }{4\pi m r^2} \frac{t_\mathrm{ioni}}{t_\mathrm{rec}}
\frac{R - \mach_\mathrm{g}^\mathrm{II}\vg  t_\mathrm{ioni}}{\mach_\mathrm{g}^\mathrm{II}\vg }
\\
&       = c_p \varepsilon  \ceq ^2
\braket{1 - \frac{t_\mathrm{ioni}}{t_\mathrm{rec}}}
+ c_p \varepsilon  \ceq ^2
\frac{R}{\mach_\mathrm{g}^\mathrm{II}\vg t_\mathrm{rec}}
\end{split}
\label{eq:typeII:mach_g:self_consistent_condition}
\end{equation}
which reduces to
\[
\frac{R}{\mach_\mathrm{g}^\mathrm{II}\vg t_\mathrm{rec}}
\approx
x^{-1} \varepsilon ^{-1} - \braket{1 - \frac{t_\mathrm{ioni}}{t_\mathrm{rec}}}
x^{-1} \varepsilon ^{-1} - 1 +
\braket{\frac{\varepsilon  q x^{1+\beta}}{\varphi}}^{1/2}
\]

This ratio is larger than unity in the parameter space,
\[
\left\{
(x, \varphi)~|~
x < \frac{\varepsilon ^{-1}}{2}
\left(
\varphi < \frac{\varepsilon  q x^{1+\beta}}{(2-x^{-1}\varepsilon ^{-1})^2}
x > \frac{\varepsilon ^{-1}}{2}
\right)
\right\}
\]