# A Primordial Origin for the Gas-Rich Debris Disks Around Intermediate-Mass
Stars

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-shi, Saitama 351-0198, Japan
<EMAIL_ADDRESS>Neal J. Turner NASA Jet Propulsion Laboratory,
California Institute of Technology, 4800 Oak Grove Dr, Pasadena, CA 91109, USA
Yasuhiro Hasegawa NASA Jet Propulsion Laboratory, California Institute of
Technology, 4800 Oak Grove Dr, Pasadena, CA 91109, USA  Gianni Cataldi
National Astronomical Observatory of Japan, Osawa 2-21-1, Mitaka, Tokyo
181-8588, Japan  Yuri Aikawa Department of Astronomy, Graduate School of
Science, The University of Tokyo, Tokyo 113-0033, Japan  Sebastián Marino
Department of Physics and Astronomy, University of Exeter, Stocker Road,
Exeter, EX4 4QL, UK  Hiroshi Kobayashi Department of Physics, Nagoya
University, Furo-cho, Chikusa-ku, Nagoya, Aichi 464-8602, Japan

###### Abstract

While most debris disks consist of dust with little or no gas, a fraction has
significant amounts of gas detected via emission lines of CO, ionized carbon,
and/or atomic oxygen. Almost all such gaseous debris disks known are around
A-type stars with ages up to 50 Myr. We show, using semi-analytic disk
evolution modeling, that this can be understood if the gaseous debris disks
are remnant protoplanetary disks that have become depleted of small grains
compared to the interstellar medium. Photoelectric heating by the A-stars’ FUV
radiation is then inefficient, while the stars’ EUV and X-ray emissions are
weak owing to a lack of surface convective zones capable of driving magnetic
activity. In this picture, stars outside the range of spectral types from A
through early B are relatively hard to have such long-lived gas disks. Less-
massive stars have stronger magnetic activity in the chromosphere, transition
region, and corona with resulting EUV and X-ray emission, while more-massive
stars have photospheres hot enough to produce strong EUV radiation. In both
cases, primordial disk gas is likely to photoevaporate well before 50 Myr.
These results come from 0D disk evolution models where we incorporate internal
accretion stresses, MHD winds, and photoevaporation by EUV and X-ray photons
with luminosities that are functions of the stellar mass and age. A key issue
this work leaves open is how some disks become depleted in small dust so that
FUV photoevaporation slows. Candidates include grains’ growth, settling,
radial drift, radiation force, and incorporation into planetary systems.

††software: Numpy (Harris et al., 2020), Matplotlib (Hunter, 2007), Astropy
(Astropy Collaboration et al., 2013, 2018, 2022), SciPy (Virtanen et al.,
2020)

##  1 Introduction

A fundamental goal of planet formation research is to learn the evolutionary
pathways that protoplanetary disks (PPDs) follow to become planetary systems
and their associated debris disks. Classically the PPD lifetime is estimated
to be no longer than $10\,{\rm Myr}$, based on near- and mid-infrared (IR)
observations which trace the inner hot dust ($\lesssim 1\text{--}10\,{\rm
au}$; Haisch et al., 2001; Meyer et al., 2007; Hernández et al., 2007;
Mamajek, 2009; Ribas et al., 2014, 2015) and ultraviolet (UV) and H$\alpha$
observations tracing accreting gas (e.g., Kennedy & Kenyon, 2009; Fedele et
al., 2010; Sicilia-Aguilar et al., 2010). Ten Myr is thus considered a typical
timescale on which gas-rich PPDs evolve into gas-free debris disks.

An explanation is therefore needed for the more than 20 gaseous debris disks
observed around stars older than $10\,{\rm Myr}$ (e.g., Kóspál et al., 2013;
Marino et al., 2016; Lieman-Sifry et al., 2016; Moór et al., 2017; Hughes et
al., 2017; Higuchi et al., 2017, 2019b, 2019a; Kral et al., 2020). Two
hypotheses have been proposed for the origins of this gas. In the primordial-
origin scenario, the gas is a remnant of the PPD. In the secondary-origin
scenario, the gas was released more recently from volatiles in planetesimals
associated with the dust in the debris disks.

While the secondary-origin scenario has been extensively examined through
modeling (Kral et al., 2016, 2019; Moór et al., 2019; Matrà et al., 2019;
Marino et al., 2020; Cataldi et al., 2020; Marino et al., 2022), less
attention has been paid to the primordial-origin scenario (Nakatani et al.,
2021; Smirnov-Pinchukov et al., 2022; Iwasaki et al., 2023). A primordial
origin has been considered unlikely because of the short canonical PPD gas
disk lifetime. However, hydrodynamical modeling by Nakatani et al. (2021)
shows that if the PPD around an intermediate-mass star ($M_{*}\approx
2{\,M_{\odot}}$) is depleted in submicron grains, including polycyclic
aromatic hydrocarbons (PAHs), then the photoelectric heating produced when FUV
photons strike grains is weak enough to slow photoevaporation so the gas could
survive well past $10\,{\rm Myr}$ and into the debris disk stage — without
small-grain depletion, the gas lifetimes are most likely $\lesssim 10\,{\rm
Myr}$ regardless of stellar masses (Komaki et al., 2023). Primordial origins
thus have the potential to account for the occurrence of gaseous debris disks
around A stars (Lieman-Sifry et al., 2016; Moór et al., 2017; Hughes et al.,
2018).

The outstanding questions include what gas lifetimes to expect quantitatively
in the primordial origin scenario, how these depend on the stellar mass, and
whether the results are consistent with the observed occurrence rates of
gaseous debris disks. We here address these issues by constructing disk
evolution models using a semi-analytic, 0D approach.

##  2 Overall picture of gas disk evolution

Theoretical works indicate the gas disks are dispersed by accretion onto the
central star (Shakura & Sunyaev, 1973; Lynden-Bell & Pringle, 1974) and by
ejection in winds, launched either by magnetic forces (e.g., Suzuki &
Inutsuka, 2009; Bai & Stone, 2013a, b) or by photoevaporation (e.g., Shu et
al., 1993; Hollenbach et al., 1994; Richling & Yorke, 1998). These three
processes’ relative importance varies over the course of disk evolution:
accretion and/or magnetohydrodynamic (MHD) winds dominate the mass loss in the
early stage when the disk is relatively massive, while photoevaporation
dominates later when the disk mass decreases to $\lesssim
10^{-2}{\,M_{\odot}}$ at ages of 1–10$\,{\rm Myr}$ (e.g., Clarke et al., 2001;
Gorti et al., 2015; Carrera et al., 2017; Kunitomo et al., 2020, 2021; Weder
et al., 2023; Komaki et al., 2023).

This picture of time-varying disk dispersal processes is consistent with
several observational facts. For instance, class 0/I jets have much larger
mass-loss rates ($10^{-7}\text{--}10^{-6}{\,M_{\odot}}\,{\rm yr}^{-1}$) than
photoevaporation can provide, implying the ejection is driven by MHD effects
in these early stages. For another example, the luminosity of a
photoevaporative wind tracer, the [Ne II] 12.8${\rm\,\upmu m}$ low-velocity
component, is higher for more evolved disks, i.e., slow accretors ($\sim
10^{-8}{\,M_{\odot}}\,{\rm yr}^{-1}$) and inner-dust-depleted disks (Pascucci
et al., 2020).

Jet and wind mass-loss rates in the upper end of the range imply possibly
enough material crossing the lines of sight between star and disk to block
much of the stellar ultraviolet (UV) and X-ray radiation. This can slow or
stop the disk’s photoevaporation (for a recent review, see Pascucci et al.,
2022). In 3D MHD modeling by Takasao et al. (2022), the outflow from the disk
around a solar-mass star has a column density high enough to attenuate the UV
and X-ray photons at accretion rates $\gtrsim 10^{-8}{\,M_{\odot}}\,{\rm
yr}^{-1}$. Overall, a fairly advanced age and low rates of accretion and
outflow appear necessary before photoevaporation can dominate gas disk
dispersal.

##  3 Gas Lifetime Estimate

For the primordial-origin hypothesis to be a possible scenario, one of the
critical necessary conditions PPDs must meet is to retain a mass reservoir
beyond $10\,{\rm Myr}$, especially at outer radii ($\gtrsim 1\text{--}10\,{\rm
au}$) where gaseous debris disks are detected. This study’s primary objective
is to demonstrate its feasibility, which was previously considered
implausible. Our model will be tailored for this purpose, as will described
below.

We focus on the disks where PAH and small grains (${\lesssim}0.01{\rm\,\upmu
m}$) are depleted compared to the interstellar medium. These are the systems
that potentially meet the necessary condition; otherwise, the disks
dissipating through rapid photoelectric heating most likely have short
lifetimes $<10\,{\rm Myr}$ (Gorti et al., 2009, 2015; Komaki et al., 2021,
2023); we ignore this population. Thus, we here derive the lifetime for the
sub-population of PPDs where the gas is the longest-lived. The estimated
lifetime is distinct from the average gas lifetime, which depends on the
still-unknown probability of achieving a small-grain-depleted state.

We do not specify here the processes responsible for grain depletion. Instead,
we simply assume the disks reach a small-grain-depleted state before
photoevaporation begins to dominate disk dispersal, regardless of the stellar
mass. Several effects can deplete the dust, especially for intermediate-mass
stars: Grain growth and radial drift reduce the local dust-to-gas ratio (e.g.,
Sellek et al., 2020). The drift is likely especially rapid around young
intermediate-mass stars (Pinilla et al., 2022). The intense radiation force
from intermediate-mass stars can also blow out small dust (Owen & Kollmeier,
2019; Nakatani et al., 2021).

The specific level of small-grain depletion needed to inhibit FUV
photoevaporation remains uncertain. Nakatani et al. (2018a, b) explored FUV
photoevaporation varying disk metallicity and demonstrated that strong FUV
photoevaporation demands small grain abundance exceeding $\sim
0.1\text{--}1\%$ of the ISM level. However, these investigations focus on a
$0.5{\,M_{\odot}}$ star with extremely high FUV luminosity, lowering oxygen
and carbon abundances with metallicity, which limits the results’ general
applicability. The critical small-grain abundance depends on stellar mass and
FUV/X-ray luminosities, yet prior studies explored a limited parameter space.
Further investigations are warranted. Nevertheless, it is likely that critical
small-grain abundances exist, below which FUV photoevaporation becomes
negligible. We anticipate this threshold to be around $\sim 0.1\text{--}1\%$
of the ISM level, following Nakatani et al. (2018a, b).

Observations suggest the PAH abundance in PPDs is at least ten times less than
the interstellar value (e.g., Geers et al., 2007; Oliveira et al., 2010;
Vicente et al., 2013). The hybrid disk HD 141569A is depleted to a level where
photoelectric heating may drive only weak photoevaporation (Li & Lunine, 2003;
Thi et al., 2014). Non-detection of PAHs occur with a reasonable fraction in
Herbig disks, where the detection rate is $\sim 70\%$ (e.g., Acke et al.,
2010), or in T Tauri disks, where it is $\lesssim 10\%$; (e.g., Geers et al.,
2006). Such non-detection hints at abundances potentially over 100 times lower
than interstellar levels. This significant depletion in the disk atmosphere is
also indicated from mid-IR spectra (e.g, Furlan et al., 2006, 2011) and
implied from IR scattered-light imaging (e.g., Mulders et al., 2013). These
findings lead us to infer that these disks might be in the small-grain-
depleted state.

###  3.1 Semi-Analytic Model

Our model employs a 0D approach, following the disk’s mass decrease from
accretion and wind mass loss in the early stage and from photoevaporation in
the late stage. It is based on the analytic lifetime estimate from the UV-
switch model (Clarke et al., 2001, detailed in Appendix A). In their model,
the major dispersal process is viscous accretion at the early stage and
switches to photoevaporation beginning at the “switching time” $t_{0}$, which
defines the transition point into the late stage. Thereafter, it disperses the
remaining disk mass of $M_{\rm disk,0}^{\prime}$. This model returns $t_{0}$,
$M_{\mathrm{disk,0}}^{\prime}$, and the gas disk lifetime
$t_{\mathrm{life,g}}$ analytically (cf. Eqs.A5–A7) once the disk mass
$M_{\mathrm{disk}}(t)$ (cf. Eq.(A4)) and photoevaporation rate $\dot{M}_{\rm
ph}(t)$ are given as functions of time.

The original UV-switch model uses purely viscous disks, but here we consider
disks evolving under angular momentum and mass extraction through a
magnetically-driven wind alongside the internal angular momentum
redistribution by turbulent stresses. Observational evidence that PPDs appear
to evolve by driving magnetized winds is discussed by Rafikov (2017), Manara
et al. (2022), and Pascucci et al. (2022). We adopt the analytic disk
evolution model of Chambers (2019) for $M_{\mathrm{disk}}(t)$, as set out in
§B. We first derive $t_{0}$ and $M_{\mathrm{disk,0}}^{\prime}$ by finding the
time when the disk mass loss $\dot{M}_{\mathrm{disk}}(t)$ and photoevaporation
rates $\dot{M}_{\rm ph}(t)$ equal; we will define $\dot{M}_{\rm ph}(t)$ later.
$t_{\mathrm{life,g}}$ is then the time till the remaining mass is
photoevaporated, found by numerically integrating

$M_{\mathrm{disk}}(t=t_{0})=M_{\rm
disk,0}^{\prime}=\int_{t_{0}}^{t_{\mathrm{life,g}}}{\rm d}t\,\dot{M}_{\rm
ph}(t).$

The resulting $t_{\mathrm{life,g}}$ is insensitive to the choice of the
threshold mass to define “dispersal”, as $M_{\mathrm{disk}}$ falls rapidly
after $t=t_{0}$. The disk’s initial mass is taken as proportional to the
stellar mass with $M_{\mathrm{disk,0}}=0.1M_{*}$ (e.g., Williams & Cieza,
2011; Andrews et al., 2013, however a steeper scaling $\propto
M_{*}^{1.3\text{--}2.0}$ is found by Pascucci et al. (2016); Ansdell et al.
(2016, 2017); see Manara et al. (2022) for a recent review). We explore
$t_{\mathrm{life,g}}$’s dependence on $M_{\mathrm{disk,0}}$ and the scaling in
Appendix D.

Photoevaporation is generally driven by far-ultraviolet (FUV), extreme-
ultraviolet (EUV), and X-ray radiation. However, as stated above, we here
consider only cases where FUV grain photoelectric heating is negligible, and
the photoevaporative winds are driven solely by the EUV and X-ray heating
associated with photoionization. We also assume that the disks are isolated
from massive stars, so external photoevaporation is negligible. We use the EUV
photoevaporation rate estimate

$\dot{M}_{\mathrm{EUV}}=4.1\times 10^{-10}{\,M_{\odot}}\,{\rm
yr}^{-1}\left(\frac{\Phi_{\rm EUV}}{10^{41}{\,\rm
s}^{-1}}\right)^{1/2}\left(\frac{M_{*}}{1{\,M_{\odot}}}\right)^{1/2}$ (1)

(Hollenbach et al., 1994; Clarke et al., 2001) since it agrees well with the
mass-loss rates derived by radiation hydrodynamics simulations (Nakatani et
al., 2021). Based on 1+1-D modeling, EUV photoevaporation rates were thought
to increase about tenfold after the disk center is cleared out, letting the
stellar EUV field dominate over the diffuse EUV from recombining ions.
However, 2D axisymmetric radiative transfer showed that the direct stellar
component dominates even in disks without central cavities (Tanaka et al.,
2013). Additionally, radiation hydrodynamics modeling found no significant
difference in photoevaporation rates with and without cavities of various
sizes (Owen et al., 2010; Picogna et al., 2019; Nakatani et al., 2021). Hence,
we apply the Eq.(1) photoevaporation rates uniformly throughout disk
evolution.

The stellar EUV emission rate $\Phi_{\rm EUV}$ can be decomposed into
photospheric, chromospheric ($\sim 10^{4}{\rm\,K}$), transition regional
($\sim 10^{5}{\rm\,K}$), and coronal ($\sim 10^{6}{\rm\,K}$) components. We
shall refer to the last three jointly as magnetic-origin EUV, . For the
photospheric EUV component, we use Kunitomo et al. (2021) Table 1 results from
long-term stellar evolution calculations for $0.5{\,M_{\odot}}\leq M_{*}\leq
5{\,M_{\odot}}$. We estimate the magnetic-origin EUV emission rate from the
X-ray luminosity $L_{\mathrm{X}}$ using the $\Phi_{\rm
EUV}\text{--}L_{\mathrm{X}}$ relation of Shoda & Takasao (2021),

$\log\left(\frac{\Phi_{\mathrm{EUV,mag}}}{1{\,\rm
s}^{-1}}\right)=20.40+0.66\log\left(\frac{L_{\mathrm{X}}}{1{\rm\,erg}{\,\rm
s}^{-1}}\right).$ (2)

Strictly, this relation applies to solar-type stars, but we apply it to all
the stars considered here ($0.5{\,M_{\odot}}\leq M_{*}\leq 5{\,M_{\odot}}$).
Since the theoretical and observational uncertainties in
$\Phi_{\mathrm{EUV,mag}}$ are large, we believe this estimate is a reasonable
first-order approximation.

We follow Kunitomo et al. (2021) for $L_{\mathrm{X}}$ time evolution,
excluding the $L_{\mathrm{X}}/L_{*}=10^{-7}$ floor applied to intermediate-
mass stars in the original paper. We expect the floor not to apply
universally, given that most Herbig stars do not present X-ray emissions
detected at this level (e.g., Hamaguchi et al., 2005; Stelzer et al., 2006).
Instead, we employ $L_{\mathrm{X}}=\mathrm{min}(10^{-3.13},5.3\times
10^{-6}\mathrm{Ro}^{-2.7})L_{*}$, where $\mathrm{Ro}$ represents the Rossby
number (Mangeney & Praderie, 1984; Noyes et al., 1984) defined as
$\mathrm{Ro}=P_{\mathrm{rot}}/\tau_{\mathrm{conv}}$ with $P_{\mathrm{rot}}$
and $\tau_{\mathrm{conv}}$ being the rotational period and the convective
turnover timescale. Kunitomo et al. (2021) set $P_{\mathrm{rot}}$ to 3 days
and calculate $\tau_{\mathrm{conv}}$ from the stellar evolution model. The
rotational period can range between 1–10 days in general, but the qualitative
trend in the time evolution of $L_{\mathrm{X}}$ is independent of
$P_{\mathrm{rot}}$ (see Section 5.2 of Kunitomo et al. (2021)). Thus, the
variation of $P_{\mathrm{rot}}$ would have a limited impact on our results.
The total EUV emission rate is $\Phi_{\rm
EUV}=\Phi_{\mathrm{EUV,ph}}+\Phi_{\mathrm{EUV,mag}}$. We omit accretion-
generated EUV and X-ray here but address their influence on
$t_{\mathrm{life,g}}$ in Appendix C.

Stellar/interstellar Lyman-Werner (LW) radiation-driven photoevaporation might
be important for small-grain-depleted disks around stars with
$2{\,M_{\odot}}\lesssim M_{*}\lesssim 3{\,M_{\odot}}$ (hereafter late
intermediate-mass stars). However, our radiation hydrodynamics simulations
indicate that LW photoevaporation is negligible even for late intermediate-
mass stars (Nakatani et al. in prep.). Thus, we do not consider LW
photoevaporation in this study. The dependence of LW photoevaporation on
stellar mass and luminosity is currently unknown but warrants investigation in
future research.

For X-ray photoevaporation, the mass-loss rate is still under debate. Some
studies obtained large mass-loss rates (Ercolano et al., 2009; Owen et al.,
2010; Picogna et al., 2019), while others found weaker mass loss (Gorti et
al., 2009; Wang & Goodman, 2017; Nakatani et al., 2018b; Komaki et al., 2021).
The different conclusions likely originate from the adopted X-ray spectra —
mass-loss rates are high if the spectrum has a certain level of soft X-rays
($\sim 0.1{\,\rm keV}$, Ercolano et al., 2009; Gorti et al., 2009; Nakatani et
al., 2018b; Sellek et al., 2022), from the incorporated cooling processes —
the studies with large X-ray photoevaporation rates neglect important cooling
processes, and from the adopted numerical methods. To cover the uncertainty of
X-ray photoevaporation rates, we estimate the lifetimes with and without X-ray
photoevaporation.

For X-ray photoevaporation rates, we use the formula of Owen et al. (2012),

$\dot{M}_{\mathrm{X}}=4.8\times 10^{-9}{\,M_{\odot}}\,{\rm
yr}^{-1}\left(\frac{L_{\mathrm{X}}}{10^{30}{\rm\,erg}{\,\rm
s}^{-1}}\right)^{1.14}\left(\frac{M_{*}}{1{\,M_{\odot}}}\right)^{-0.148}.$ (3)

(Using the formula of Picogna et al. (2019) instead results in very similar
lifetimes.) Given the disagreement over X-ray photoevaporation rates discussed
above, Eq.(3) must be considered an upper limit, so the corresponding
$t_{\mathrm{life,g}}$ is a lower limit. The total photoevaporation rate is

$\dot{M}_{\mathrm{ph}}=\left\\{\begin{array}[]{cl}\dot{M}_{\mathrm{EUV}}&\text{(w/o
X-ray photoevaporation)}\\\
\mathrm{max}\left(\dot{M}_{\mathrm{EUV}},\dot{M}_{\mathrm{X}}\right)&\text{(w/
X-ray photoevaporation)}\end{array}\right..$

We calculate the lifetimes for two cases: (i) EUV only and (ii) EUV + X-ray
with Eqs.(1) and (3). In Appendix E, we additionally discuss how the observed
spread in $L_{\mathrm{X}}$ (e.g., Güdel et al., 2007) can affect
$t_{\mathrm{life,g}}$. Note that using the sum instead of
$\mathrm{max}\left(\dot{M}_{\mathrm{EUV}},\dot{M}_{\mathrm{X}}\right)$ does
not make a difference, as $\dot{M}_{\mathrm{X}}$ dominates
$\dot{M}_{\mathrm{EUV}}$ significantly.

Figure 1:  Estimated gas disk lifetime $t_{\mathrm{life,g}}$ as a function of
$M_{*}$. The blue line indicates $t_{\mathrm{life,g}}$ for the EUV-only case.
The gray line adds X-ray photoevaporation. The spectral type corresponding to
each $M_{*}$ is presented on the top axis. Note that these lifetimes apply
only for small-grain-depleted disks and not to disks where FUV photoelectric
heating drives photoevaporation.

###  3.2 Results

To provide an overview, we start with Figure 1, the estimated
$t_{\mathrm{life,g}}$ as a function of $M_{*}$ using the Chambers (2019) model
for disks dominated by slow winds. Here the accretion is predominantly driven
by the wind, which also extracts a significant fraction of the disk’s mass
(see Section 4.3 of the original paper for details). The figure shows the
corresponding spectral type at each $M_{*}$ determined by $T_{\mathrm{eff}}$
when the stars reach the zero-age-main-sequence (ZMAS) in the table of
Kunitomo et al. (2021). The relation between the spectral type and
$T_{\mathrm{eff}}$ is taken from Pecaut & Mamajek (2013). The table of
Kunitomo et al. (2021) ends before stars with $M_{*}\leq 1{\,M_{\odot}}$ reach
the ZAMS. Thus, for $M_{*}=1{\,M_{\odot}}$, we used the last
$T_{\mathrm{eff}}$ value in the table to determine the spectral type; the
spectral types of $M_{*}<1{\,M_{\odot}}$ stars are undetermined, so not
specified in Figure 1.

In the EUV-only case (blue), the lifetime peaks at $M_{*}=2{\,M_{\odot}}$, or
equivalently spectral types around A0. Lifetimes longer than $10\,{\rm Myr}$
are found only around the late intermediate-mass stars
($2{\,M_{\odot}}\lesssim M_{*}\lesssim 3{\,M_{\odot}}$), corresponding to
spectral types between F0 and B8. The surface convective zones of these stars
disappear at $\sim 1$–$10\,{\rm Myr}$, and the magnetic-origin EUV emission is
already weak by the time photoevaporation comes to dominate mass loss. The
results indicate that primordial gas disks relatively easily survive to the
ages of young debris disks around A- and late B-type stars.

Stars with $M_{*}\geq 4{\,M_{\odot}}$ (hereafter early intermediate-mass
stars) produce much stronger photospheric EUV emission owing to their high
$T_{\mathrm{eff}}$, leading to shorter $t_{\mathrm{life,g}}$. For stars of
Solar mass and below, the magnetic-origin EUV emission remains strong
throughout the disk dispersal.

X-ray photoevaporation (gray line in Figure 1) reduces the lifetime by 1–2
orders of magnitude for $\lesssim 3{\,M_{\odot}}$ compared to the EUV-only
case. This is because $L_{\mathrm{X}}$ is kept high at $\gtrsim
10^{31}{\rm\,erg}{\,\rm s}^{-1}$, corresponding to $\dot{M}_{\mathrm{X}}\sim
5\times 10^{-8}\text{--}10^{-7}{\,M_{\odot}}\,{\rm yr}^{-1}$, until the disk
completely disperses. However, as mentioned above, the adopted X-ray
photoevaporation rates might be overestimated. While X-ray photoevaporation
can explain observational wind diagnostics (e.g., Weber et al., 2020; Rab et
al., 2022), such high X-ray photoevaporation rates are also known to be
unfavorable for explaining the observed disk mass and accretion rates with
viscous disk models (Sellek et al., 2020; Alexander et al., 2023) and
magnetized disk models (Weder et al., 2023); smaller mass-loss rates as EUV
photoevaporation alone are preferred. However, we do not rule out the
effectiveness of X-rays in driving photoevaporation, as they indeed deposit
energy to the gas, but with somewhat much smaller mass-loss rates than Eq.(3).
If so, the lifetimes of the EUV models set upper limits for low-mass stars.
For the early intermediate-mass stars ($\geq 4{\,M_{\odot}}$),
$t_{\mathrm{life,g}}$ matches the EUV-only case since the convective zone
disappears before $1\,{\rm Myr}$.

We then explore the dependence on the parameters of the MHD wind-driven
accretion and mass loss across the fast-wind and laminar-disk regimes in
Chambers (2019). In the fast-wind case, the wind mass loss is negligible, and
most of the accretion is driven by the wind, while in the laminar case, wind
mass loss is substantial, and the wind drives almost all the accretion
(Sections 4.2 and 4.4 of the original paper).

Figure 2:  Same as Figure 1. Solid, dashed, and dot-dashed lines show
lifetimes of slow-wind, fast-wind, and laminar disks, respectively. The dotted
lines show the median lifetimes of the population synthesis in Appendix D.

The resulting $t_{\mathrm{life,g}}$–$M_{*}$ relation (Figure 2) is essentially
the same as Figure 1, consistently peaking at $2{\,M_{\odot}}$. Small
differences are that in the fast-wind model, $t_{\mathrm{life,g}}$ at
$1.5{\,M_{\odot}}$ becomes comparable to $2{\,M_{\odot}}$, and in the laminar
model, the peak lifetime at $2{\,M_{\odot}}$ is reduced to $\sim 20\,{\rm
Myr}$. However, the assumption of no wind mass loss in the fast-wind model
seems difficult to reconcile with the high detection rate of [O I] $\lambda
6300$ low-velocity component ($\sim 80\%$; Nisini et al., 2018), making it an
unfavored PPD evolution scenario. The stellar-mass-dependent trends are
consistent across a wide parameter space, as shown by the population synthesis
median in Figure 2 (cf. Appendix D). Overall, varying the disk and wind
parameters yields a wide range of possible lifetimes, especially near
$1.5{\,M_{\odot}}$. We conclude that the average lifetime peaks at early
A-type stars with $t_{\mathrm{life,g}}>10\,{\rm Myr}$ for small-grain-depleted
disks.

We emphasize that the lifetimes derived here apply only to small-grain-
depleted disks. Figures 1 and 2 neglect any short-lived disks that might be
dispersed by FUV photoevaporation. To quantify the general frequency of gas
disks, it would be necessary to estimate PPDs’ probability of becoming
depleted in small grains. This is an issue for future studies.

We attribute the stellar mass dependence of $t_{\mathrm{life,g}}$ mostly to
the variation with $M_{*}$ in the EUV and X-ray luminosities and thus the
photoevaporation rates. However, the variation of MHD effects with stellar
mass may also play a role. The hard X-rays from low-mass stars ($\lesssim
1{\,M_{\odot}}$) can ionize the disk atmosphere to increase the local
effective $\alpha$. This enhances MHD winds’ effects, shortening the disk
lifetime. The soft X-rays due to shocks originating from line-driven winds of
early B- and O-type stars could have a similar effect and can also directly
drive soft X-ray photoevaporation. Both these effects would strengthen the gas
lifetime difference between the late intermediate-mass stars and the rest.

Currently, no evidence supports the presence of companions in known gas-rich
debris disks, but the possibility still remains that they could harbor unseen
low-mass, close-in companions. In these scenarios, the EUV and X-ray emission
may be dominated by these companions, boosting photoevaporation rates. Thus,
to yield long-lived gas disks, late intermediate-mass stars should not have
such companions.

##  4 Comparison with detection rate of gaseous debris disks

We now compare the modeled lifetimes from §3 with the population of debris
disks detected in the CO millimeter, [C II] $157{\rm\,\upmu m}$, or [O I]
$63{\rm\,\upmu m}$ lines, focusing on the dependence on host star spectral
type. The detections come from Supplemental Table 1 of Hughes et al. (2018).
Only 3 out of 152 sources listed in the table have ages of $<10\,{\rm Myr}$,
with all being A0 stars. The age of HD 166191 (F star) is presented as
$4\,{\rm Myr}$, but we adopt an updated value of ${\sim}10\,{\rm Myr}$
(Potravnov et al., 2018; Su et al., 2022). As they discussed, this sample
combines studies with diverse sensitivity limits and selection criteria and
thus is not well-suited for determining quantitative occurrence rates.
However, the spectral-type dependence resembles that in a more-uniform
sampling by Moór et al. (2017).

Keeping this limitation in mind, we replot the detections compiled by Hughes
et al. (2018) versus stellar type in Figure 3. We show not only the detections
in CO but also those in C II and O I, since the disks may retain these species
mixed with hydrogen even in the absence of CO.

Figure 3: Detection rates of gaseous debris disks are highest among young
A-type stars. The tracers are CO millimeter emission (left), C II
157${\rm\,\upmu m}$ (middle), and O I 63${\rm\,\upmu m}$ (right). The light-
green histograms show the all-survey count and are superposed by dark-green
histograms indicating the detection count, for sources younger than $50\,{\rm
Myr}$. The light-yellow and dark-yellow histograms are sources older than
$50\,{\rm Myr}$. The horizontal axis gives the stellar type. Integrated counts
are shown for M, K, and G stars. All data are from Hughes et al. (2018) Table
1. These trends are consistent with the $M_{*}$-dependence of our predicted
lifetimes in Figure 1.

We note that nine of twelve young CO-detected sources have CO mass
$M_{\mathrm{CO}}>10^{-4}{\,M_{\oplus}}$ (cf. Table A1 of Marino et al., 2020;
Moór et al., 2017, 2019). The three sources below this threshold are likely
better explained by secondary origins, based on the short photodissociation
lifetimes for primordial CO at such low gas masses. For F stars, one of the
two young CO-detected sources is above this mass threshold and is early F
(F2/3V); the one below the threshold is late F (F5/6V). The decreasing trend
of the incidence from A to FGK stars is likely real but not due to a
sensitivity bias. A comparison of debris disks with similar fractional
luminosities (Tables A1 and A2 of Marino et al., 2020) indicates that only 2
out of 9 sources exhibit gas (at levels $\lesssim 10^{-4}{\,M_{\oplus}}$)
among FGK stars, while 11 out of 17 A-type stars display gas presence (often
exceeding $10^{-3}{\,M_{\oplus}}$).

Figure 3 shows several interesting consistencies with the modeled
$t_{\mathrm{life,g}}$–$M_{*}$ relation. The detection rate is highest for
early A-type stars (more than half the stars with gas detected are early
A-type, A0–A3). Almost all the gas-rich debris disks with
$M_{\mathrm{CO}}>10^{-3}{\,M_{\oplus}}$ are hosted by early A-type stars. The
exception is HD 121191 ($3\times 10^{-3}{\,M_{\oplus}}$, Moór et al., 2017),
which is type A5 but does not necessarily disagree with our model considering
the range in $t_{\mathrm{life,g}}$ at $M_{*}\sim 1.5{\,M_{\odot}}$. Our model
naturally explains younger ($<50\,{\rm Myr}$) gaseous debris disks’ higher
numbers around A-type stars than around both lower-mass and O- and early B
stars. The decreasing trend toward higher-mass stars could be due to both
photoevaporation and intense CO photodissociation. For A stars, the weak
photodissociating fluxes would also be helping the long survival of CO.
Overall, within the limitations of the sample, the consistencies with the
model are remarkable.

In contrast, the gas detected in debris disks around late F-stars appears more
compatible with secondary origins, given the short gas lifetime around those
low-mass stars in our model. Also, a few debris disks have multiple lines of
evidence supporting a secondary origin, among them the A5–A6 star $\beta$ Pic
(e.g., Dent et al., 2014; Greaves et al., 2016; Iwasaki et al., 2023; Cataldi
et al., 2023). We thus consider primordial and secondary origins not mutually
exclusive. We stress that our hypothesized disk evolution does not necessarily
rule out a secondary origin for some gaseous debris disks, even among A-type
stars younger than $50\,{\rm Myr}$. Determining the gas origins in individual
sources is a challenge for the future. However, the results we present here
suggest the high overall frequency of gas-rich debris disks among A-type stars
can come from the primordial origin.

##  5 Summary and Outlook

The substantial amounts of gas found in some debris disks could either be
remnants of the primordial protoplanetary disk, or originate in the secondary
release of volatiles from bodies resembling comets or asteroids. While the
primordial-origin scenario was classically considered unlikely, we here
demonstrated primordial gas can survive to the ages of the known gaseous
debris disks, using semi-analytic models to derive lifetimes for the gas in
disks that have been depleted in small grains ($\lesssim 0.01{\rm\,\upmu m}$)
so that FUV photoelectric heating is ineffective. The resulting lifetimes are
longest around stars of $2{\,M_{\odot}}\lesssim M_{*}\lesssim 3{\,M_{\odot}}$,
whose evolution switches off their surface convection at $1\text{--}10\,{\rm
Myr}$ before the disk loses the bulk of its gas. Without a surface convective
zone, these stars lack surface magnetic activity and thus have low
luminosities in the EUV and X-ray bands that drive the photoevaporation of the
gas.

The gas lifetime exceeds $10\,{\rm Myr}$ for the late intermediate-mass stars
only. The shorter lifetimes of the gas disks around lower-mass stars
($\lesssim 1{\,M_{\odot}}$) are due to strong EUV and X-rays from the
chromosphere, transition region, and corona, while those around higher-mass
stars ($\gtrsim 4{\,M_{\odot}}$) are due to the high photospheric temperatures
and resulting EUV luminosities.

Our model predicts a higher frequency of gaseous debris disks around A- and
late B-type stars compared to types both earlier and later. This is
qualitatively consistent with the observed population, motivating further
investigation of the primordial-origin scenario. Longer gas disk lifetimes for
A-stars could be a factor in the giant planet occurrence rate peaking at
1.7–$1.9{\,M_{\odot}}$ (Reffert et al., 2015; Wolthoff et al., 2022), though
more detailed modeling of the dissipating disks would be needed to assess the
possibility of gas giant formation. Likewise, our results could relate to the
increasing occurrence of transition disks with stellar mass (van der Marel &
Mulders, 2021). Further investigations are needed to evaluate whether these
transition disks might serve as precursors to gaseous debris disks.

The primordial-origin picture allows the existence around late intermediate-
mass stars of long-lived gas-rich disks with very weak dust emission, which we
term “phantom” disks. Finding such phantom disks would indicate that the
abundant CO gas in at least some young debris disks is primordial. Searches
for gas-rich, dust-poor disks especially around A- and late B-stars aged
$10$–$50\,{\rm Myr}$ could thus help determine the range of evolutionary paths
followed by planet-forming disks. Note that the phantoms are different from
the relic disks, purely viscous, dusty disks with central cavities larger than
$100\,{\rm au}$ surviving $>10\,{\rm Myr}$ around Solar-type stars owing to
very weak X-ray emission (Owen et al., 2011).

The fraction of PPDs achieving the small-grain-depleted state and the critical
level of depletion required to inhibit FUV photoevaporation are open
questions. Further hydrodynamics investigations are needed considering a range
of stellar masses and FUV/X-ray luminosities. We anticipate that PPDs with
PAH/small-grain abundances below $\sim 0.1\text{--}1\%$ of the interstellar
level (see §3) may have reached this state and could be potential precursors
of gas-rich debris disks around intermediate-mass stars. PPDs with PAH/small-
grain abundances above the critical level would have FUV-driven H2 winds
extend to $\sim 10\text{--}100\,{\rm au}$ with temperatures exceeding a few
hundred Kelvin and mass-loss rates of $\sim
10^{-9}\text{--}10^{-8}{\,M_{\odot}}\,{\rm yr}^{-1}$ (Nakatani et al., 2018a,
b; Komaki et al., 2021). Searching for these molecular winds could also help
examine the depletion level in PPDs and very young ($<10\,{\rm Myr}$) hybrid
disks (e.g., HD 141569), including hybrid disk candidates (Iglesias et al.,
2023), around weak X-ray emitters.

The point of this work is that gas disks can persist beyond $10\,{\rm Myr}$, a
necessary condition for the primordial-origin scenario. A caveat is that our
0D approach cannot unambiguously predict the detailed surface density
evolution and the alignment with other observational characteristics of inner
disks, such as accretion and wind mass-loss rates (Fang et al., 2018, 2023).
Future studies involve 1D simulations to accurately follow the radial profile
evolution and make comparisons with those observables to evaluate the proposed
evolutionary pathways. Nevertheless, we expect the conclusion on the long
lifetimes around intermediate-mass stars will remain consistent even in more
advanced models, as the key determinant of outer gas disk lifetimes, which
potentially manifest as gaseous debris disks in observations, is the
photoevaporation rates set by the stellar emission rates. The speed of final-
stage disk dispersal would not be significantly influenced by how the inner
disks clear. Thermochemical modeling would also help understand how the CO and
C masses evolve in small-grain depleted disks and examine if the final values
reproduce observations (Iwasaki et al., 2023; Cataldi et al., 2023). For a
broader perspective, it is worth investigating what fraction of PPDs in
general can be long-lived considering environmental factors like external
photoevaporation, tidal truncation, and late infall.

We are grateful to the anonymous referees for their useful comments. We thank
Masanobu Kunitomo and Akimasa Kataoka for their practical and insightful
comments on this study. RN is supported by the Japan Society for the Promotion
of Science (JSPS), Overseas Research Fellowship. SM is supported by a Royal
Society University Research Fellowship (URF-R1-221669). This research was
performed in part at the Jet Propulsion Laboratory, California Institute of
Technology, under contract 80NM0018D0004 with the National Aeronautics and
Space Administration and with the support of the NASA Exoplanets Research
Program through grant 17-XRP17_2-0081. This work was strongly benefited from
the Core2disk-III residential program of Institut Pascal at Université Paris-
Saclay, with the support of the program “Investissements d’avenir”
ANR-11-IDEX-0003-01.

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##  Appendix A The UV-switch Model

The evolution of an accretion disk redistributing its angular momentum via
viscous friction can be described by

$\frac{\partial\Sigma}{\partial t}=\frac{3}{R}\frac{\partial}{\partial
R}R^{1/2}\frac{\partial}{\partial R}\nu\Sigma R^{1/2},$ (A1)

(Lynden-Bell & Pringle, 1974). If the disk evolves purely viscously with the
viscosity profile of $\nu(R)\propto R$, the surface density profile
$\Sigma(R,t)$ is analytically computed as

$\Sigma(R,t)=\frac{M_{\mathrm{disk,0}}}{2\pi
R_{1}^{2}}\left(\frac{R}{R_{1}}\right)^{-1}\left(1+\frac{t}{t_{\nu}}\right)^{-3/2}\exp\left(-\frac{R}{R_{1}}\left(1+\frac{t}{t_{\nu}}\right)^{-1}\right)$
(A2)

where $M_{\mathrm{disk,0}}$ is the initial disk mass, $R_{1}$ is the outer
cut-off radius, and $t_{\nu}\equiv R_{1}^{2}/3\nu(R_{1})$ is the viscous
timescale at $R_{1}$. The nominal value of $t_{\nu}$ is

$\begin{split}t_{\nu}&\sim 0.1\,{\rm
Myr}\left(\frac{\alpha}{10^{-3}}\right)^{-1}\left(\frac{M_{*}}{1{\,M_{\odot}}}\right)^{-1/2}\\\
&\quad\times\left(\frac{H(R_{1};M_{*},L_{*})/R_{1}}{0.06}\right)^{-2}\left(\frac{R_{1}}{10\,{\rm
au}}\right)^{2}\end{split}$

with the so-called turbulent $\alpha$ viscosity, $\nu=\alpha\Omega H^{2}$
(Shakura & Sunyaev, 1973). We note that the scale height $H=c_{\rm s}/\Omega$
depends on $M_{*}$ and $L_{*}$; e.g., a passive disk has a temperature profile
of

$\begin{split}T(R,L_{*})&=T_{\mathrm{eff}}\left(\frac{R}{R_{*}}\right)^{-1/2}\theta_{\mathrm{inc}}^{1/4}=\left(\frac{L_{*}\theta_{\mathrm{inc}}}{4\pi\sigma_{\mathrm{SB}}R^{2}}\right)^{1/4},\end{split}$

where $\theta_{\mathrm{inc}}$ is the incident angle of irradiation, and
$\sigma_{\mathrm{SB}}$ is the Stefan-Boltzmann constant (Kenyon & Hartmann,
1987; Hartmann et al., 1998; Pinilla et al., 2022). The viscous $\alpha$ may
also depend on $M_{*}$ as $\alpha\propto M_{*}$ so that the accretion rate
reproduces the observational scaling profile $\dot{M}_{\mathrm{acc}}\propto
M_{*}^{2}$ (Muzerolle et al., 2003, 2005; Calvet et al., 2004; Mohanty et al.,
2005; Natta et al., 2006; Alcalá et al., 2014, 2017; Manara et al., 2016,
2022, also see discussions in Gorti et al. (2009) and Kunitomo et al. (2021)).

We can calculate the disk mass and mass accretion rate from Eq.(A2) as

$\displaystyle M_{\mathrm{disk}}(t)=\int_{0}^{\infty}{\rm d}R\,2\pi
R\Sigma=M_{\mathrm{disk,0}}\left(1+\frac{t}{t_{\nu}}\right)^{-1/2}$ (A3)
$\displaystyle\dot{M}_{\mathrm{acc}}=-\dot{M}_{\mathrm{disk}}=\frac{M_{\mathrm{disk,0}}}{2t_{\nu}}\left(1+\frac{t}{t_{\nu}}\right)^{-3/2}.$
(A4)

When photoevaporation is taken into account in Eq.(A1), the effect appears on
the right-hand-side as a sink term

$\frac{\partial\Sigma}{\partial t}=\frac{3}{R}\frac{\partial}{\partial
R}R^{1/2}\frac{\partial}{\partial R}\nu\Sigma
R^{1/2}-\dot{\Sigma}_{\mathrm{ph}}.$

The solution is the so-called UV-switch model (Clarke et al., 2001), and its
behavior is described as follows: (1) the disk evolves effectively purely
viscously at the early stage. (2) After the accretion rate equals the
photoevaporation rate, material supply from the outer part halts. (3) It
quickly opens a cavity in the inner region. (4) Photoevaporation dominates
mass loss afterward and determines the dispersal time of the remaining disk.

The first transition point at (2) is when $\dot{M}_{\mathrm{acc}}$ is reduced
to $\dot{M}_{\rm ph}$. This transition occurs at (using Eq.(A4))

$\begin{split}t_{w}&=t_{\nu}\left[\left(\frac{M_{\mathrm{disk,0}}}{2t_{\nu}\dot{M}_{\rm
ph}}\right)^{2/3}-1\right]\approx
t_{\nu}\left(\frac{\dot{M}_{\mathrm{acc,0}}}{\dot{M}_{\rm ph}}\right)^{2/3}\\\
&\approx 2.2\,{\rm Myr}\left(\frac{\dot{M}_{\mathrm{acc,0}}}{1\times
10^{-6}{\,M_{\odot}}\,{\rm yr}^{-1}}\right)^{2/3}\left(\frac{\dot{M}_{\rm
ph}}{1\times 10^{-8}{\,M_{\odot}}\,{\rm yr}^{-1}}\right)^{-2/3}\end{split}$
(A5)

where $\dot{M}_{\mathrm{acc,0}}\equiv M_{\mathrm{disk,0}}/2t_{\nu}$ is the
accretion rate at $t=0$. By this point, the disk mass has reduced to

$\begin{split}&M_{\rm disk,0}^{\prime}\equiv
M_{\mathrm{disk}}(t_{w})=M_{\mathrm{disk,0}}\left(1+\frac{t_{w}}{t_{\nu}}\right)^{-1/2}=M_{\mathrm{disk,0}}\left(\frac{\dot{M}_{\mathrm{acc,0}}}{\dot{M}_{\rm
ph}}\right)^{-1/3}\\\ &\approx
0.043{\,M_{\odot}}\left(\frac{M_{\mathrm{disk,0}}}{0.2{\,M_{\odot}}}\right)\left(\frac{\dot{M}_{\mathrm{acc,0}}}{10^{-6}{\,M_{\odot}}\,{\rm
yr}^{-1}}\right)^{-1/3}\left(\frac{\dot{M}_{\rm
ph}}{10^{-8}{\,M_{\odot}}\,{\rm yr}^{-1}}\right)^{1/3}.\end{split}$ (A6)

Note that $M_{\mathrm{disk,0}}$, $t_{\nu}$, and $\dot{M}_{\mathrm{acc,0}}$ are
not independent but are constrained by
$M_{\mathrm{disk,0}}=2t_{\nu}\dot{M}_{\mathrm{acc,0}}$.

Material supply from the outer disk stops at $t=t_{w}$, resulting in a fast
draining of the inner region ($R\lesssim 1\,{\rm au}$) on the timescale of the
viscous time at the critical radius $R\approx 1\,{\rm au}$, where a gap opens.
Photoevaporation governs disk evolution afterward. Note that for disks without
MHD winds’ mass loss, $t_{w}$ and the switching time of our model $t_{0}$ are
identical. The remaining outer disk disperses by photoevaporation on a
timescale of the “initial” outer disk mass $M_{\rm disk,0}^{\prime}$ over the
photoevaporation rate,

$\begin{split}&\frac{M_{\rm disk,0}^{\prime}}{\dot{M}_{\rm
ph}}=2t_{\nu}\left(\frac{\dot{M}_{\mathrm{acc,0}}}{\dot{M}_{\rm
ph}}\right)^{2/3}=2t_{w}\\\ &\approx 4.3\,{\rm
Myr}\left(\frac{M_{\mathrm{disk,0}}}{0.2{\,M_{\odot}}}\right)\left(\frac{\dot{M}_{\mathrm{acc,0}}}{10^{-6}{\,M_{\odot}}\,{\rm
yr}^{-1}}\right)^{-1/3}\left(\frac{\dot{M}_{\rm
ph}}{10^{-8}{\,M_{\odot}}\,{\rm yr}^{-1}}\right)^{-2/3}.\end{split}$ (A7)

Hence, the gas disk lifetime is $t_{\mathrm{life,g}}=t_{w}+2t_{w}$ in cases of
a constant $\dot{M}_{\rm ph}$ before and after $t_{w}$.

Photoevaporation rates vary according to stellar evolution, where the
luminosities and spectra change by orders of magnitude. Also, if $\dot{M}_{\rm
ph}$ gets smaller at some point by, e.g., reduced photoelectric heating due to
small-grain depletion, the lifetime is extended. This is the basic idea of our
model (and Nakatani et al., 2021) to explain a long-lived gas disk in the
context of the primordial origin scenario.

##  Appendix B An analytic disk evolution model with MHD effects

The effects of magnetohydrodynamics (MHD) winds are essential in disk
evolution. They can remove not only the mass but also the angular momentum of
the disk. Chambers (2019) derived an approximate analytic solution for disk
evolution with MHD wind effects. Eq.(A1) is modified to

$\frac{\partial\Sigma}{\partial t}=\frac{3}{R}\frac{\partial}{\partial
R}R^{1/2}\frac{\partial}{\partial R}\nu\Sigma
R^{1/2}+\frac{1}{R}\frac{\partial}{\partial R}Rv_{w}\Sigma-\dot{\Sigma}_{w},$
(B1)

where $v_{w}$ is the inward velocity induced by the disk wind, and
$\dot{\Sigma}_{w}$ is the surface mass-loss rate due to MHD winds. Using a
reference radius $R_{0}$, the surface mass-loss rate is characterized by a
constant parameter $K$ as

$\begin{gathered}\dot{\Sigma}_{w}=-\frac{Kf_{w}v_{0}\Sigma}{R_{0}}\left(\frac{R}{R_{0}}\right)^{-3/2}\\\
K=\frac{1}{2}\left(\frac{R}{R_{0}}\right)^{-1/4}\left(\frac{\Omega
R}{v_{\mathrm{esc}}-\Omega R}\right),\end{gathered}$

where $v_{0}$ is the inward velocity at $R_{0}$, $f_{w}$ is the fraction of
$v_{0}$ induced by MHD winds, $v_{\mathrm{esc}}$ is the tangential velocity of
the winds, and $\Omega$ is the Keplerian orbital frequency. The second
equation is derived from the conservation of angular momentum.

Chambers (2019) modified Eq.(B1) to make it analytically solvable (Eq.(12) of
the original paper) and found analytic solutions for the modified equation
(Eq.(36)–(39) of the original paper). We omit to write the solutions
explicitly here to avoid complexity and refer the readers to the original
paper. The point of this analytic solution is that the surface density and
temperature evolutions are uniquely determined for a given set of model
parameters $v_{0}$, $f_{w}$, $K$, the temperature at $R_{0}$: $T_{0}$, and the
surface density at $R_{0}$: $\Sigma_{0}$.

For example, $f_{w}=0$ means purely viscous disk, and choosing a high
$f_{w}(\leq 1)$ indicates a disk that accretes mainly magnetically. The
parameter $K$ can be set to zero with a nonzero $f_{w}$. The underlying
assumption, in this case, is negligible mass-loss rates due to MHD winds.
Small $K$ cases are termed fast wind. On the other hand, using a high value
for $K$ ($\leq 1$) means that a significant fraction of the accreting gas
flows out of the disk before reaching the host star. This case is called a
slow wind.

The analytic formula of Chambers (2019) can overall well agree with the true
solutions of Eq.(B1) regardless of the approximation taken to make the
equation analytically tractable. The exception is when the wind mass-loss rate
is large; the analytic formula underestimates the surface density and outer
spreading at the later stage. It also fails to predict the positive slope of
$\Sigma$ created by the wind in the inner $\sim 1\,{\rm au}$ region.
Nevertheless, the errors between the analytic formula and the true solution
are small compared to the large uncertainties in the long-term evolution of
disk evolution with the MHD effects. We, therefore, use the formula as a valid
estimate for time-dependent disk mass evolution.

##  Appendix C Accretion-generated emissions

We have omitted accretion-generated UV and X-ray emissions from our model.
This extra radiation component can dominate the magnetic and photospheric
components, leading to accelerated photoevaporation and reduced
$t_{\mathrm{life,g}}$. We now examine if this omission affects the study’s
conclusions—specifically, the peak lifetime beyond $10\,{\rm Myr}$ at
$M_{*}\approx 2{\,M_{\odot}}$ (cf. Figures 1 and 2). Since the EUV models
potentially account for the observed higher occurrence of gas-rich debris
disks around A stars, our focus remains exclusively on the EUV models.

The exact portion of accretion luminosity ($L_{\mathrm{acc}}\approx
GM_{*}\dot{M}_{\mathrm{acc}}/R_{*}$) contributing to EUV radiation is
uncertain. For simplicity, we assume $\sim 4\%$ is processed into EUV
emission, aligning with analogous estimation for FUV (Gorti & Hollenbach,
2008) and X-ray (Hartmann et al., 2016). This can be represented as:

$\Phi_{\mathrm{EUV,acc}}\approx 2\times 10^{42}{\,\rm
s}^{-1}\left(\frac{M_{*}}{1{\,M_{\odot}}}\right)\left(\frac{\dot{M}_{\mathrm{acc}}}{10^{-8}{\,M_{\odot}}\,{\rm
yr}^{-1}}\right)\left(\frac{R_{*}}{1\,R_{\odot}}\right)^{-1},$

Here, the average EUV photon energy is approximated to match that of
$9000{\rm\,K}$ blackbody radiation, $14.5\,{\rm eV}$. The total EUV emission
rate is now
$\Phi_{\mathrm{EUV}}=\Phi_{\mathrm{EUV,ph}}+\Phi_{\mathrm{EUV,mag}}+\Phi_{\mathrm{EUV,acc}}$.
In practice, accretion-generated EUV radiation may terminate once the disk
develops an inner cavity, when photoevaporation and accretion rates balance at
$\sim 1\,{\rm au}$ (the critical radius for photoevaporation). However, as the
cavity-opening time is not determined unambiguously within the framework of
our model, we let the accretion-generated EUV emission last until the disk
dispersal; this scenario sets the lower limit for $t_{\mathrm{life,g}}$.

Figure 4: Derived $t_{\mathrm{life,g}}$ with accretion-generated EUV (blue).
The lifetimes of the fiducial runs are also plotted for comparison (light
blue).

Figure 4 shows the resulting lifetimes. The accretion-generated EUV dominates
$\Phi_{\mathrm{EUV,mag}}$ for the weak- and fast-wind disks especially at the
early time ($\lesssim 1\,{\rm Myr}$), while laminar disks show minimal total
emission rate enhancement. Nevertheless, the photoevaporation rate remains
orders of magnitude lower than the accretion and/or MHD wind mass-loss rates
at the early stages, with no substantial impact on disk evolution. Except for
fast-wind disks, the accretion-generated component becomes negligible as the
accretion rate drops in $1\,{\rm Myr}\lesssim t\lesssim 10\,{\rm Myr}$. This
occurs prior to the onset of the photoevaporation-dominant epoch at $t=t_{0}$,
leading to minimal shortening of $t_{\mathrm{life,g}}$ for the slow-wind and
laminar disks.

In contrast, fast-wind disks can maintain relatively high accretion rates
($\sim 10^{-10}$–$10^{-9}{\,M_{\odot}}\,{\rm yr}^{-1}$) even at $t=10\,{\rm
Myr}$, with accretion-generated EUV dominating over (or comparable to) the
other components throughout dispersal. This impacts $t_{\mathrm{life,g}}$ at
$M_{*}=1.5$–$2.5{\,M_{\odot}}$, as $\Phi_{\mathrm{EUV,acc}}$ is significantly
exceeds $\Phi_{\mathrm{EUV,ph}}+\Phi_{\mathrm{EUV,mag}}$. However, the
prevalence of jets and winds observed widely (e.g., Pascucci et al., 2022)
suggests that the fast-wind scenario is unlikely to be common among PPDs. It
is also difficult to align with the high detection rate of [O I] $\lambda
6300$ low-velocity component (Nisini et al., 2018). If termination of
accretion-generated emissions due to cavity opening is considered, the peak at
$M_{*}\sim 2{\,M_{\odot}}$ may remain even for fast-wind disks.

Overall, incorporating the accretion-generated component can moderately reduce
$t_{\mathrm{life,g}}$ and lessen the stellar-mass dependence for disks with no
wind mass loss. Nonetheless, not all disks would evolve as fast-wind disks,
supporting the highest plausibility of long-living ($>10\,{\rm Myr}$) gas
disks around late intermediate-mass stars. Further investigation is needed to
accurately determine $t_{\mathrm{life,g}}$, $\Phi_{\mathrm{EUV,acc}}$, and its
termination time.

##  Appendix D Disk Parameter Dependences

Figure 5: Resulting population. The blue and gray histograms are the
probability densities for the EUV and EUV+X-ray models, respectively.
Corresponding normalized cumulative probabilities are shown by the solid lines
with the same colors.

We explore how $t_{\mathrm{life,g}}$ depends on various disk parameters,
including the initial disk mass ($M_{\mathrm{disk,0}}$), disk mass scaling law
($M_{\mathrm{disk,0}}\propto M_{*}^{\beta}$), viscous $\alpha$, and initial
disk cut-off radius ($r_{\mathrm{exp}}$). These parameters are treated as
Monte Carlo variables in population synthesis to span a wide parameter space.

The initial disk mass is defined as
$M_{\mathrm{disk,0}}/M_{*}=C_{\mathrm{disk,0}}(M_{*}/1{\,M_{\odot}})^{\beta-1}$,
with our fiducial values being $C_{\mathrm{disk,0}}=0.1$ and $\beta=1$.
Exploration covers a range of $0.03\leq C_{\mathrm{disk,0}}\leq 0.3$ for
$C_{\mathrm{disk,0}}$ and $1\leq\beta\leq 2$, following observations (e.g.,
Pascucci et al., 2016; Ansdell et al., 2016, 2017). As for $\alpha$, recent
finding favor a low value at $10^{-4}\lesssim\alpha\lesssim 10^{-3}$ (Manara
et al., 2022), and therefore we vary it in $10^{-4}\leq\alpha\leq 2\times
10^{-3}$, with the upper limit encompassing the slow-wind and fast-wind disks.

The fiducial $v_{0}$ values are $30{\,\rm cm}{\,\rm s}^{-1}$ for slow-wind and
fast-wind disks and $10{\,\rm cm}{\,\rm s}^{-1}$ for the laminar disk. We
explore $10{\,\rm cm}{\,\rm s}^{-1}\leq v_{0}\leq 30{\,\rm cm}{\,\rm s}^{-1}$,
as values beyond this range lead to unrealistic accretion rate variations. The
wind mass-loss parameter $K$ is varied in $0\leq K\leq 1$. The initial cut-off
radius is tested in $10\,{\rm au}\leq r_{\mathrm{exp}}\leq 40\,{\rm au}$; the
fiducial is $r_{\mathrm{exp}}=15\,{\rm au}$. For reference, Clarke et al.
(2001) and Kunitomo et al. (2020) adopt $r_{\mathrm{exp}}=10\,{\rm au}$ and
$30\,{\rm au}$, respectively; the time evolution of disk radius in the model
by Trapman et al. (2022) aligns with the characteristic radii of observed CO
disks in Lupus and Upper Sco when $r_{\mathrm{exp}}\approx 20\,{\rm au}$ and
$M_{\mathrm{disk,0}}=0.1{\,M_{\odot}}$.

Monte Carlo variables assume uniform densities in linear space, except for
$C_{\mathrm{disk,0}}$ and $\alpha$, which follow log-uniform sampling. We
perform in total 90000 runs: 5000 parameter sets times 9 different stellar
masses for each of the EUV and EUV+X-ray models. Figure 5 illustrates the
resulting probability densities and cumulative probabilities for
$t_{\mathrm{life,g}}$, reflecting the trends seen in Figures 1 and 2.

The lifetime is mostly unaffected by $\beta$ and $K$. For the EUV models and
EUV+X-ray models with $M_{*}\geq 3{\,M_{\odot}}$, $M_{\mathrm{disk,0}}$ has
limited influence. However, for EUV+X-ray models with $M_{*}\leq
2.5{\,M_{\odot}}$, smaller initial disk masses ($C_{\mathrm{disk,0}}\leq 0.1$)
result in $t_{\mathrm{life,g}}\lesssim 5\,{\rm Myr}$.

Increasing $r_{\mathrm{exp}}$ moderately extends $t_{\mathrm{life,g}}$. In the
EUV models, $t_{\mathrm{life,g}}$ tends to exceed $10\,{\rm Myr}$ for the runs
with $M_{*}\leq 1.5{\,M_{\odot}}$ and $r_{\mathrm{exp}}(>25\,{\rm au})$.
Conversely, the most runs with $M_{*}=2$–$3{\,M_{\odot}}$ and
$t_{\mathrm{life,g}}<10\,{\rm Myr}$ have $r_{\mathrm{exp}}\lesssim 20\,{\rm
au}$. When $r_{\mathrm{exp}}$ is expanded while keeping $M_{\mathrm{disk,0}}$
constant, it decreases the initial surface density in the inner part, leading
to greater mass distribution in the outer disk. This leads to slower mass loss
via accretion and MHD winds.

Viscous $\alpha$ and $v_{0}$ also have a moderate impact on the EUV models,
with smaller $\alpha$ and higher $v_{0}$ causing shorter lifetimes due to
enhanced MHD-driven mass loss and accretion. The longest-$t_{\mathrm{life,g}}$
population ($>10^{2}\,{\rm Myr}$) at $M_{*}=1.5$–$3{\,M_{\odot}}$ has
$\alpha>10^{-3}$ and $v_{0}<15{\,\rm cm}{\,\rm s}^{-1}$, indicating a slow
decay of the accretion rate. Unrealistically long lifetimes ($\sim
10^{3}\,{\rm yr}$) can occur when $\alpha$ is increased to $\sim 10^{-2}$.
Hence, when the disk exhibits a characteristic of a viscous disk relatively
strongly, $t_{\mathrm{life,g}}$ increases significantly.

In summary, higher $\alpha$ and larger $r_{\mathrm{exp}}$ can extend
$t_{\mathrm{life,g}}$ for specific cases. The other parameters exert minor
impacts. Regardless of the parameter dependences, overall trends remain
consistent: Late intermediate-mass stars in the EUV models consistently
exhibit $t_{\mathrm{life,g}}>10\,{\rm Myr}$, peaking at $M_{*}=2{\,M_{\odot}}$
and declining for lower and higher masses. While the fraction of long-living
($t_{\mathrm{life,g}}>10\,{\rm Myr}$) disks for $M_{*}\leq 1{\,M_{\odot}}$,
most of which have $r_{\mathrm{exp}}>25\,{\rm au}$, is apparently higher than
observations (Figure 3), the actual CO disk lifetime is shorter than
$t_{\mathrm{life,g}}$. This is because complete CO photodissociation occurs
before disk dissipation, when $M_{\mathrm{disk}}$ reduces to a certain
threshold mass. In addition, X-ray photoevaporation likely causes mass loss at
somewhat much smaller rates than Eq.(3), implying that the the lifetimes of
the EUV models should be interpreted as upper limits for low-mass stars.

Thermochemistry modeling is needed to interpret these results fully, along
with the potentially crucial influence of atypically weak stellar
photodissociating fluxes around A stars. It is important to highlight that our
findings primarily demonstrate the plausibility of gas disk survival beyond
$>10\,{\rm Myr}$, a necessary condition for supporting the primordial-origin
scenario. Additionally, our model qualitatively explains the observed
incidence of gaseous debris disks versus stellar mass.

##  Appendix E X-ray Luminosity Spread

Observed X-ray luminosities are known to have a large spread (e.g., Güdel et
al., 2007). Accounting for the variability in X-ray luminosity would introduce
scattering in the derived lifetimes. In the EUV+X-ray models, amplifying X-ray
luminosity results in shorter lifetimes while reducing it aligns the lifetimes
with those of EUV models. Within the EUV models, the spread in luminosity
could influence lifetimes for $M_{*}<1{\,M_{\odot}}$, as their lifetimes are
linked to $\Phi_{\mathrm{EUV,mag}}$. However, combining Eqs. (1) and (2), the
scaling of EUV photoevaporation rate with $L_{X}^{0.33}$ places a relatively
restrained impact on lifetimes. Furthermore, the adopted X-ray luminosity
encompasses representative observational values (Fig.6 of Kunitomo et al.
(2021)). It suggests that derived lifetimes remain relatively stable even when
incorporating luminosity spread with a plausible distribution of
$L_{\mathrm{X}}$. Consequently, the inclusion of luminosity dispersion is
unlikely to significantly alter the core finding of this study.

Taking into account the spread of $L_{\mathrm{X}}$ along with its time
evolution would require solving stellar evolution across a wider parameter
space than covered by Kunitomo et al. (2021).