11institutetext: National Centre for Nuclear Research, ul. Pasteura 7, 02-093
Warsaw, Poland 11email<EMAIL_ADDRESS>22institutetext: INAF -
Osservatorio astronomico d’Abruzzo, Via Maggini SNC, 64100, Teramo, Italy
33institutetext: INFN - Sezione di Perugia, Via A. Pascoli SNC, 06123,
Perugia, Italy 44institutetext: SISSA, Via Bonomea 265, Trieste, Italy
55institutetext: IFPU – Institute for fundamental physics of the Universe, Via
Beirut 2, 34014 Trieste, Italy 66institutetext: Astronomical Observatory
Institute, Faculty of Physics, Adam Mickiewicz University, ul. Słoneczna 36,
60-286 Poznań, Poland 77institutetext: INAF – Osservatorio Astronomico di
Padova, Vicolo dell’Osservatorio 5, 35122 Padova, Italy

# Dust survival in harsh environments

Is photo-evaporation an important destruction mechanism?

A. Nanni 1122 S. Cristallo 2233 D. Donevski 114455 M. J. Michałowski 66 M.
Romano 1177 P. Sawant 11

(Received …; accepted…)

###### Abstract

Aims. We investigate the role of photo-evaporation of dust exposed to the
radiation field from hot young stars and planetary nebulae (PNe) as a possible
destruction mechanism of dust grains in the interstellar medium (ISM).

Methods. We estimate photo-evaporation induced by the feedback of individual
or clustered young stars, of PNe and in the presence of a variable radiation
field scaled with the interstellar radiation field. For PNe we investigate
dust photo-evaporation of both dust grains already present in the ISM as well
as those formed in the last phases of the evolution of thermally pulsing
asymptotic giant branch (TP-AGB) stars. We include dust photo-evaporation rate
in models of dust evolution in galaxies for different assumptions of the dust
growth scenario, dust-to-gas ratios, star formation histories, and initial
mass functions of the stars.

Results. For all the cases considered, we find that both photo-evaporation
from young stars and from PNe are negligible with respect to other dust
removal processes such as destruction from supernovae shocks, astration and
possibly outflow. Grains are stable against photo-evaporation if they are
exposed to a radiation field which is up to 107 times the interstellar
radiation field.

Conclusions. Dust grains of size $\geq 0.01$ $\mu m$ are not efficiently
destroyed by photo-evaporation also in the presence of a strong radiation
field.

###### Key Words.:

galaxies: evolution – galaxies: ISM - evolution - dust, extinction

##  1 Introduction

Formation and survival of dust grains in galaxies have paramount implications
for many astrophysical processes. Dust grains are responsible for the build-up
of different molecules in space, and for gas cooling, promoting the formation
of stars (e.g., Cuppen et al. 2017). Dust absorbs light mostly in the
ultraviolet (UV) and visible wavelength and re-emits this energy in the
infrared. Therefore, the physical quantities derived from the spectral energy
distribution (SED) fitting, e.g. the star formation rate (SFR), must take the
effect of dust into account. Despite the significance of dust formation,
destruction, and survival in the interstellar medium (ISM), such mechanisms
remain controversial.

UV photons impinging on dust grains remove electrons from their surface and
heat the gas. Through such a process grains become positively charged
(Weingartner & Draine, 2001b). Photo-evaporation of dust grains is suggested
to be relevant for the life-cycle of polycyclic aromatic hydrocarbons (PAHs,
Demyk, 2011). Indeed, the ISM may be enriched of small molecules from the
photo-evaporation of PAHs, while in photo-dissociation regions PAHs may be
created when UV photons and/or shock waves break down carbon grains (Cesarsky
et al., 2000; Berné et al., 2007). Nevertheless, the role of dust photo-
evaporation as a possible efficient destruction mechanism of dust grains in
the ISM has not been equally well explored so far. Dust grains undergo photo-
evaporation due to the intense radiation field in galaxies with high SFRs. In
spite of that, in simulations of dust evolution in galaxies, the role of such
a mechanism in relation to other processes, such as astration and supernovae
(SN) shocks and large-scale galactic outflows has not been thoroughly assessed
yet. In low-metallicity galaxies and the early Universe, where stars are
typically hotter than at solar-like metallicity, and where the formation of
massive stars with intense radiation fields may be favoured (Marks et al.,
2012), photo-evaporation may be potentially relevant.

Recent observational studies show that the specific mass of dust ($sM_{\rm
dust}$=$M_{\rm dust}/M_{\star}$, where $M_{\rm dust}$ and $M_{\star}$ are the
masses of dust and of stars, respectively) rises quickly at young ages and
then decreases with age. Different classes of galaxies show evidence of this
trend: massive and dwarf star-forming galaxies in the local Universe (De Vis
et al., 2019; Nanni et al., 2020; Galliano et al., 2021; Casasola et al.,
2022), dusty galaxies and Lyman Break Galaxies identified at very high
redshifts ($2<z<6$, Burgarella et al. 2020; Donevski et al. 2020; Kokorev et
al. 2021; Burgarella et al. 2022), as well as quiescent but dusty galaxies at
low and intermediate redshifts (Gobat et al. (2018); Michałowski et al.
(2019); Donevski et al. (2023); Leśniewska et al. (2023)). The core of the
correlation between the $sM_{\rm dust}$ and the specific star formation rate
($sSFR=SFR/M_{\star}$) is believed to be an age-evolutionary sequence.

From a theoretical viewpoint, various models tracking the dust, gas and metal
content of galaxies that include recipes for dust formation, grain growth and
destruction, as well as inflows and outflows, are able to reproduce the
observed decline of $sM_{\rm dust}$ with $sSFR$ (Li et al., 2019; Pantoni et
al., 2019; De Vis et al., 2019; Nanni et al., 2020; Galliano et al., 2021;
Burgarella et al., 2022). The majority of models find strong outflows driven
by stellar feedback and SN explosions to explain the decline of $sM_{\rm
dust}$ with age. Predicted outflow efficiencies ($ML=\dot{M}/SFR$, where
$\dot{M}$ represents the mass outflow rate) are up to 80. In contrast to what
simulations predict, recent observational works on both local and high-z
sources (e.g. Ginolfi et al. (2020); Romano et al. (2023); Salak et al.
(2023)) found a typical value of $ML\approx 1$. Using a novel spectral
selection, Donevski et al. (2023) examined quiescent galaxies that showed no
signs of energetic feedback from embedded active galactic nuclei. A
significant scatter follows an anti-correlation between $sM_{\rm dust}$ and
age, suggesting distinct dust removal pathways over a range of timescales.
Dust destruction from feedback from Type Ia SNe (Li et al., 2020), planetary
nebulae (PNe) (Leśniewska et al., 2023) or heating from winds of thermally
pulsing asymptotic giant branch (TP-AGB) stars have also been proposed to
explain the observed decline of $sM_{\rm dust}$ with age in quiescent galaxies
(Conroy et al., 2015).

This all motivates us to re-examine under what conditions dust can survive in
galaxies and what processes are dominant in producing the observed anti-
correlation of $sM_{\rm dust}$ and stellar age. Along with usually considered
dust removal processes in the ISM such as SN shocks, astration, and outflow we
additionally probe whether photo-evaporation due to the intense radiation
input from massive stars and PNe can be an efficient destruction process for
dust grains.

##  2 Method: dust evolution model

In this work, we investigate the efficiency of dust photo-evaporation by
including this process in calculations that follow dust evolution in the ISM
of galaxies. We here do not consider all the details of dust condensation and
destruction during the process of star formation or PNe, but only the effect
of radiation feedback from stars already in their main sequence, i.e. in HII
regions, or in the PN phase.

###  2.1 General framework

In order to model gas and dust evolution in the ISM of galaxies, we first
compute the chemical gas enrichment from stars by adopting the one-zone
chemical evolution code omega (Côté et al., 2017; Ritter et al., 2018, One-
zone Model for the Evolution of GAlaxies) which includes population III stars
(Heger & Woosley, 2010), Type II (Limongi & Chieffi, 2018; Prantzos et al.,
2018) and Type Ia supernovae (SNe, Iwamoto et al., 1999) and TP-AGB stars
(Cristallo et al., 2015). We therefore compute the evolution of the gas mass
$M_{\rm gas}$ as well as the the mass of each metal species, $M_{\rm gas,i}$,
and the mass of the dust species, j, either silicates (olivine and pyroxene)
or carbon, $M_{\rm dust,j}$, similarly to other works in the literature (e.g.
Dwek, 1998; Calura et al., 2008; Asano et al., 2013; Ginolfi et al., 2018; De
Vis et al., 2019; Nanni et al., 2020; Galliano et al., 2021).

If only inflow is neglected we obtain:

$\frac{dM_{\rm gas}}{dt}=\frac{dM^{\rm SP}_{\rm gas,ej}}{dt}-SFR-ML\times
SFR-\sum_{\rm j}\frac{dM_{\rm dust,j}}{dt},$ (1)

where the last term takes into account the amount of metals locked into dust
grains.

$\begin{multlined}\frac{dM_{\rm gas,i}}{dt}=\frac{dM^{\rm SP}_{\rm
gas,ej,i}}{dt}-SFR\frac{M_{\rm gas,i}}{M_{\rm gas}}-ML\times SFR\frac{M_{\rm
gas,i}}{M_{\rm gas}}-\\\ -\sum_{\rm j}n_{\rm i}\frac{m_{\rm i}}{m_{\rm
dust,j}}\frac{dM_{\rm dust,j}}{dt},\end{multlined}\frac{dM_{\rm
gas,i}}{dt}=\frac{dM^{\rm SP}_{\rm gas,ej,i}}{dt}-SFR\frac{M_{\rm
gas,i}}{M_{\rm gas}}-ML\times SFR\frac{M_{\rm gas,i}}{M_{\rm gas}}-\\\
-\sum_{\rm j}n_{\rm i}\frac{m_{\rm i}}{m_{\rm dust,j}}\frac{dM_{\rm
dust,j}}{dt},$ (2)

where the first two terms of each equation are computed by omega to which we
refer for all the details. The first term represents the gas return from
stellar population (SP). The initial mass function (IMF) of stars is assumed
to be constant with time. The integral is performed between the minimum
($M_{\rm L}=0.8$ M⊙) and the maximum ($M_{\rm U}=120$ M⊙) mass of stars. The
IMF is normalised in such a way that:

$\int^{M_{\rm U}}_{\rm M_{\rm L}}mIMF(m)dm=1\leavevmode\nobreak\ M_{\odot}$
(3)

We here test two different IMFs: the Chabrier IMF (Chabrier, 2003) and a top-
heavy IMF with the form:

$IMF(m)\propto m^{-\alpha},$ (4)

with $\alpha=1.35$. Such an IMF favours the formation of massive stars and can
impact the efficiency of the destruction processes involved.

The second term of the Eqs. 1 and 2 is the astration of gas and metals due to
formation of stars while the last term is gas removal from the ISM operated by
outflows. The last term of Eq. 2 represents metal depletion from the gas phase
into dust grains, where $n_{\rm i}$ is the number of atoms of the element i in
the monomer of the dust species j, and $m_{\rm i}$ and $m_{\rm dust,j}$ are
the atomic mass of the element i and the dust monomer j, respectively.

The evolution of dust grains of species $j$ is computed as:

$\begin{multlined}\frac{dM_{\rm dust,j}}{dt}=\frac{dM^{\rm SP}_{\rm
dust,ej,j}}{dt}-SFR\frac{M_{\rm dust,j}}{M_{\rm gas}}-\frac{dM^{\rm SN}_{\rm
destr,j}}{dt}-\frac{dM^{\rm YSs}_{\rm destr,j}}{dt}-\\\ -\frac{dM^{\rm
PN}_{\rm destr,j}}{dt}-ML\times SFR\frac{M_{\rm dust,j}}{M_{\rm
gas}}+\frac{dM_{\rm growth,j}}{dt}.\end{multlined}\frac{dM_{\rm
dust,j}}{dt}=\frac{dM^{\rm SP}_{\rm dust,ej,j}}{dt}-SFR\frac{M_{\rm
dust,j}}{M_{\rm gas}}-\frac{dM^{\rm SN}_{\rm destr,j}}{dt}-\frac{dM^{\rm
YSs}_{\rm destr,j}}{dt}-\\\ -\frac{dM^{\rm PN}_{\rm destr,j}}{dt}-ML\times
SFR\frac{M_{\rm dust,j}}{M_{\rm gas}}+\frac{dM_{\rm growth,j}}{dt}.$ (5)

The first term of the equation represents the dust enrichment where the dust
yields are approximated by making use of the metal yields:

$\frac{dM^{\rm SP}_{\rm dust,ej,j}}{dt}=\frac{f_{\rm key,j}}{n_{\rm
key,j}}\frac{dM^{\rm SP}_{\rm gas,ej,key,j}}{dt}\frac{m_{\rm dust,j}}{m_{\rm
key,j}},$ (6)

where $f_{\rm key,j}$ is the fraction of key element111The key element is
defined as the least abundant among the elements that form a certain dust
species divided by its number of atoms in the compound. locked into dust
grains, $n_{\rm key,j}$ is the number of atoms of the key element in one
monomer of dust, and $dM^{\rm SP}_{\rm gas,ej,key,j}/dt$ is the gas mass
injection rate of the key element from the SP. We assume $f_{\rm
key,olivine}=0.3$, $f_{\rm key,pyroxene}=0.3$, $f_{\rm key,carbon}=0.5$,
$f_{\rm key,iron}=0.01$ for TP-AGB stars, and $f_{\rm key,olivine}=0$, $f_{\rm
key,pyroxene}=0.5$, $f_{\rm key,carbon}=0.5$, $f_{\rm key,iron}=0.5$, for SNe
(Nanni et al., 2020). The quantity $m_{\rm key,j}$ is the atomic mass of the
key element. The second term in Eq. 5 represents dust astration due to star
formation. The terms $dM^{\rm SN}_{\rm destr,j}/dt$, is the dust destruction
from SN shocks. We add the terms $dM^{\rm YSs}_{\rm destr,j}/dt$ and $dM^{\rm
PN}_{\rm destr,j}/dt$, corresponding to photo-evaporation due to young stars
and PNe, respectively. The destruction term in PNe includes both the dust
destroyed in the ISM ($dM^{\rm PN,ISM}_{\rm destr,j}/dt$) and in-situ
($dM^{\rm PN,in-situ}_{\rm destr,j}/dt$):

$\frac{dM^{\rm PN}_{\rm destr,j}}{dt}=\frac{dM^{\rm PN,ISM}_{\rm
destr,j}}{dt}+\frac{dM^{\rm PN,in-situ}_{\rm destr,j}}{dt}$ (7)

Similarly to Eqs. 1 and 2 dust removal from the outflow is parameterised
through the mass-loading factor. The last term is dust growth occurring in the
ISM.

We adopt the commonly-used delayed star-formation history (SFH) for the
galaxy:

$SFR\propto\frac{1}{\tau^{2}}e^{-t/\tau},$ (8)

where $\tau=10,1000$ Myrs are representative of a rapid burst of star
formation and of a more continuous star formation, respectively. The SFH is
normalised in such a way that $M_{\star}=1$ M⊙ after 13 Gyrs. We however
checked that the same kind of normalisation at different final ages of the
galaxy does not change the results.

###  2.2 Dust destruction from SN shocks

The dust destruction operated by SN shocks is modelled as in many works in the
literature, e.g. Dwek (1998). The destruction time-scale is given by:

$\tau_{\rm destr,SN}=\frac{M_{\rm gas}}{R_{\rm SN}(t)M_{\rm swept}},$ (9)

where $M_{\rm gas}$ evolves according to Eq. 1, $R_{\rm SN}$ is the SN rate
and $M_{\rm swept}$ is the mass of gas swept-up for each SN event. We here
assume $M_{\rm swept}=1200$ M⊙ (Dwek et al., 2007). The destruction rate of
dust is therefore:

$\frac{dM^{\rm SN}_{\rm destr,j}}{dt}=\frac{M_{\rm dust,j}}{\tau_{\rm
destr,SN}}.$ (10)

###  2.3 Dust photo-evaporation

The stellar parameters of individual stars have been calculated with the FUNS
code (Straniero et al., 2006; Cristallo et al., 2009, 2011, 2015). Surface
luminosities and temperatures have been extracted when 10% of the central
hydrogen has been burnt. We consider a metallicity of Z=0.0001 plus enrichment
of $\alpha$-elements (0.7 dex, for oxygen, and 0.4 dex for the other
$\alpha$-elements) and stellar masses between $0.8$ and $120$ M⊙. However, we
verify that by increasing the upper limit of stellar mass up to $300$ M⊙ does
not significantly change the results. The low metallicity and stellar
evolutionary phase allow for the maximum possible effective temperatures, and
therefore the largest possible dust photo-evaporation. At each distance from
the star we compute the dust equilibrium temperature $T_{\rm j}$ for each dust
species $\rm j$, either carbon or silicate:

$\int_{\nu}\kappa_{\rm abs,j}(\nu)B(\nu)(T_{\rm
eff})W(r)d\nu=\int_{\nu}\kappa_{\rm abs,j}(\nu)B(\nu)(T_{\rm j}),d\nu,$ (11)

where $\kappa_{\rm abs,j}(\nu)$ is the mass absorption coefficients of the
dust species $j$, in cm2 g-1 as a function of the frequency, $B(\nu)(T_{\rm
eff})$ and $B(\nu)(T_{\rm j})$ are the black body emission at the effective
temperature of the star and of the dust temperature, respectively, and $W(r)$
is the dilution factor of radiation with the distance from the star, $r$,
which for stars of radius $R_{*}$ reads:

$W(r)=\frac{1}{2}\Bigg{[}1-\sqrt{1-\Bigg{(}\frac{R_{*}}{r}\Bigg{)}^{2}}\Bigg{]}.$
(12)

Stars are seldomly born isolated. Therefore, we consider the more realistic
case of stars situated in stellar clusters. We assume for the radiation field
the one obtained from a simple stellar population (SSP) computed for a single
burst of star formation with $Z=0.0001$ and both the Chabrier and top-heavy
IMF by means of the code PÉGASE.3 (Fioc & Rocca-Volmerange, 2019) with a lower
and upper limit of 0.8 and 120 $M_{\odot}$, respectively. We adopt the stellar
population based on “Padova” tracks (Bressan et al., 1993; Fagotto et al.,
1994a, b, c; Girardi et al., 1996). We consider a zero-age SSP that provides
the maximum of the photo-evaporation efficiency, and an upper limit for the
star cluster mass of $10^{5}$ M⊙ (Portegies Zwart et al., 2010). In this case,
Eq. 11 becomes:

$\int_{\nu}\kappa_{\rm abs,j}(\nu)\frac{L(\nu)_{\rm SSP}}{4\pi
r^{2}}d\nu=\int_{\nu}\kappa_{\rm abs,j}(\nu)B(\nu)(T_{\rm j}),d\nu,$ (13)

where $L(\nu)_{\rm SSP}$ is the spectrum of the SSP.

The mass absorption coefficients in Eqs 11 and 13 for the main populations of
silicate and carbon are computed by starting from their optical properties of
Laor & Draine (1993) for astronomical silicate and graphite in the wavelength
range from 0.001–1000 $\mu$m which covers the entire emission spectrum of
young stars and PNe222Absorption and scattering coefficients are available at
https://www.astro.princeton.edu/~draine/dust/dust.diel.html. We adopt the same
size distribution of Weingartner & Draine (2001a); Draine & Li (2007). We do
not consider PAHs evolution in our treatment since they usually represent a
minor component in terms of the total dust mass in the ISM. For this reason,
we exclude from the grain size distribution of carbon the contribution of
small grains of size given by Eq. 2 of Weingartner & Draine (2001a).

We compute the variation of the dust size with time due to evaporation of dust
heated by starlight $da/dt^{\rm sub}_{\rm i}$ in analogy to Eq. 3 of Kobayashi
et al. (2011) and adopting the data in their Table 1:

$\frac{da}{dt}^{\rm sub}_{\rm j}=-\frac{1}{\rho_{\rm j}}\sqrt{\frac{A_{\rm
j}m_{H}}{2\pi kT_{\rm j}}}P_{0}\exp\Bigg{(}-\frac{A_{\rm j}m_{H}L_{\rm
j}}{kT_{\rm j}}\Bigg{)},$ (14)

where $\rho_{\rm j}$ and $\rm L_{\rm j}$ are the mass density and the latent
heat of sublimation of the j-th dust species, $A_{\rm j}$ is its mean
molecular weight, $P_{0}$ is the saturated vapour pressure, $k$ is the
Boltzmann constant, and $m_{\rm H}$ is the hydrogen mass. We conservatively
adopt the same data of olivine also to compute the evaporation of pyroxene.
Indeed, olivine is predicted to evaporate at temperature lower than the one of
pyroxene. As a consequence, our choice maximises the evaporation efficiency of
silicates.

Small grains evaporate quicker than large ones, therefore, in order to find an
upper limit of dust photo-evaporation, we assume that carbon and silicate
grains formed have a typical size of $a_{\rm j}=0.01$ $\mu$m which is about
one order of magnitude smaller than the peak of distribution found by
Weingartner & Draine (2001a) for the Milky Way and adopted in Draine & Li
(2007).

We define the dust temperature for which the grains are stable against photo-
evaporation as the temperature at which the time needed for evaporation for a
grain of size $a_{\rm j}=0.01$ $\mu$m is greater or equal to a critical age,
$\rm age_{\rm crit}$, which we set equal to the age at which the galaxy has
built its entire stellar mass, $\rm age_{\rm crit}=13$ Gyrs:

$\frac{a_{\rm j}}{da/dt^{\rm sub}_{\rm j}}={\rm age}_{\rm crit}$ (15)

We obtain $T_{\rm carbon}\approx 1330$ K for carbon dust and $\rm T_{\rm
silicate}\approx 877$ K for silicates.

The distance from the source at which the equilibrium temperature equals this
sublimation temperature (Eqs. 11 and 13) is the dust sublimation radius,
$R_{\rm subl,j}$.

In case of individual stars (ISs) or PNe, at each time-step the mass of dust
destroyed in the volume of a shell of thickness equal to $R^{\rm IS/PN}_{\rm
subl,j}-R_{*}$ is:

$M^{\rm IS/PN}_{\rm destr,j}=\frac{M_{\rm dust,j}}{M_{\rm
gas}}\frac{4}{3}\pi\Big{(}R^{3}_{\rm subl,j}-R_{*}^{3}\Big{)}\rho_{\rm gas}.$
(16)

For stars in the ISM, $\rho_{\rm gas}$ is either the gas density in the region
of star formation or where the PN resides. For young stars, we set as density
that of a molecular cloud $\rho_{\rm gas}=10^{5}m_{\rm H}\mu$ where $\mu$ is
the mean molecular weight $\mu=1.37$. Such a high value of the density is
typical of star-forming clumps and cores (Draine, 2011). The contribution of
the destruction by each isolated star is weighted over the IMF (which is
normalised as in Eq. 3):

$M^{\rm IS,W}_{\rm destr,j}=\int^{M_{\rm U}}_{M_{\rm L}}IMF(m)\frac{M_{\rm
dust,j}}{M_{\rm gas}}\frac{4}{3}\pi\Big{(}R^{3}_{\rm
subl,j}-R_{*}^{3}\Big{)}\rho_{\rm gas}dm.$ (17)

For the calculation of the dust destroyed by a single PN in the ISM we use Eq.
16 and assume $\rho_{\rm gas}=50m_{\rm H}\mu$, a value typical of the cold
atomic medium (Ferrière, 2001) which is an upper limit of the density of the
medium in which PNe are expected to reside. The quantity $M_{\rm
dust,j}/M_{\rm gas}$ is the dust-to-gas ratio in the ISM calculated from Eqs.
1 and 5.

Planetary Nebulae however are complex objects in which the central hot white
dwarf is surrounded by dust and gas at higher density than the one of the cold
atomic medium. Dust is produced during the TP-AGB phase when the star loses
mass at high rates (up to a few $10^{-5}$ $M_{\odot}yr^{-1}$). For computing
dust destruction in-situ for PNe we assume $\rho_{\rm gas}=10^{4}m_{\rm H}\mu$
(e.g. Stanghellini et al., 2012) and an upper limit of the total dust-to-gas
ratio of 0.01 (Stasińska & Szczerba, 1999). We assume that each of the dust
species considered (olivine, pyroxene and carbon) has $M_{\rm dust,j}/M_{\rm
gas}$ equal to one third of the total value.

In case the stars are born in clusters we assume that the radiation comes from
a point source. In this case, the mass of dust destroyed is the one in a
sphere of radius $R^{\rm SP}_{\rm subl,j}$:

$M^{\rm SP}_{\rm destr,j}=\frac{M_{\rm dust,j}}{M_{\rm gas}}\frac{4}{3}\pi
R^{3}_{\rm subl,j}\rho_{\rm gas},$ (18)

where $\rho_{\rm gas}$ is assumed to be the same as in the case of individual
stars ($\rho_{\rm gas}=10^{5}m_{\rm H}\mu$).

The final dust destruction operated by star formation is therefore computed
given the SFR as a function of time:

$\frac{dM^{\rm YSs}_{\rm destr,j}}{dt}=\frac{M^{\rm IS,W/SP}_{\rm
destr,j}}{M_{\rm gas}}SFR,$ (19)

Photo-evaporation rate induced by PNe is instead calculated as:

$\frac{dM^{\rm PN,ISM/in-situ}_{\rm destr,j}}{dt}=M^{\rm PN,ISM/in-situ}_{\rm
destr,j}\frac{dn_{\rm AGB}}{dt},$ (20)

where $dn_{\rm AGB}/dt$ is the AGB “birth” rate that end their evolution as
PNe which is extracted from the code omega. We assume an effective temperature
for all the PNe of $T_{\rm eff}=2\times 10^{5}$ K and luminosity $L=4\times
10^{4}$ L⊙ which are upper limits values for PNe with initial mass of 6 M⊙
(see FRUITY database333fruity.oa-abruzzo.inaf.it, Cristallo et al., 2015).

###  2.4 Dust growth in the ISM

The dust destruction processes included in our calculations depends on the
dust-to-gas ratio as a function of time. Since such a quantity varies if dust
growth in the ISM is included, we analyse two extreme cases: 1) no dust growth
in the ISM; 2) fully efficient dust growth in the ISM. The variation of mass
of dust due to growth is expressed as:

$\frac{dM_{\rm growth,j}}{dt}=4\pi\frac{da_{\rm j}}{dt}a_{\rm j}^{2}\rho_{\rm
j}n_{\rm s,j},$ (21)

where $\rm da_{\rm j}/dt$ is the variation of the dust size due to the
addition of atoms and molecules on the grain surface and is computed following
Nanni et al. (2020), $n_{\rm s,j}$ is the number of seed nuclei computed by
dividing the mass of each dust species by the mass of one dust grain (Asano et
al., 2013). For this calculation we implicitly assume that all the grains in
the ISM can potentially act as seed nuclei.

##  3 Results

###  3.1 A simple case

In order to give a sense of how much dust is destroyed for each solar mass of
stars formed and for each PN, we here perform some simple calculations by
considering a single value for the dust-to-gas ratio of $0.01$ in the ISM
(Bohlin et al., 1978). As sublimation radius of dust in Eqs. 16, 17 and 18 we
adopt the one of silicates, $R_{\rm subl,silicate}$, since this is larger than
the one of carbonaceous grains.

####  3.1.1 Dust photo-evaporation by isolated young stars

Based on Eq. 17 we first estimate how much dust is destroyed per $M_{\odot}$
of stars formed with the IMF distributed according to the Chabrier or top-
heavy functions and normalised as in Eq. 3. For this test, we assumed stars to
be formed in isolation. We obtain the fraction of $\approx 1.2\times 10^{-8}$
and $\approx 6.7\times 10^{-8}$ of dust destroyed, where the stars are
distributed according to the Chabrier and top-heavy IMF, respectively.

####  3.1.2 Dust photo-evaporation by stars in clusters

From Eq. 18 we find that the dust mass fraction destroyed per solar mass of
stars formed in a cluster is $\approx 1.0\times 10^{-8}$ and $\approx
1.3\times 10^{-7}$ where stars are distributed according to the Chabrier and
top-heavy IMF, respectively. In the case of the top-heavy IMF, such an
estimate is larger than the one found for isolated stars, while the two values
are comparable for the Chabrier IMF. The difference with the isolated stars
case may be due to the diverse estimates of flux impinging on the grain
surface in Eqs. 11 and 13, and of the total volume cleared-up from dust in
Eqs. 16 and 18.

The sublimation radius for the different dust species are $R_{\rm
subl,silicate}\approx 9\times 10^{-3}$ pc, and $R_{\rm subl,carbon}\approx
5\times 10^{-3}$ pc. These values suggest that dust is destroyed by photo-
evaporation in the immediate vicinity of the cluster, given that a typical
radius of an HII region is of a few pc (Draine, 2011).

####  3.1.3 Dust photo-evaporation by PNe

We estimate from Eq. 16 that the amount of dust destroyed in the ISM for each
PN is $\approx 2.4\times 10^{-11}$ M⊙.

We obtain that the mass of dust in-situ destroyed by photo-evaporation is
$\approx 4.9\times 10^{-9}$ $M_{\odot}$. Such a value is negligible compared
to the dust yields produced along the entire duration of the TP-AGB phase
(e.g. Zhukovska et al., 2008; Ventura et al., 2012; Nanni et al., 2013), and
to the dust mass inferred from observations which is of the order of $\approx
10^{-4}$ $M_{\odot}$ (Stasińska & Szczerba, 1999; Dell’Agli et al., 2023).
Recently, Dell’Agli et al. (2023) have found that the amount of dust possibly
destroyed by PN feedback is up to $\approx 60\%$. Therefore, an alternative
mechanism different from photo-evaporation may be at work. As for the case of
clusters, photo-evaporation is efficient in the vicinity of the source. We
find $R_{\rm subl,silicate}\approx 7\times 10^{-4}$ pc, and $R_{\rm
subl,carbon}\approx 9\times 10^{-5}$ pc.

###  3.2 Dust exposed to an interstellar radiation field- (ISRF-) like
radiation

Dust in the ISM as well as in the outer part of the cirmcumstellar envelopes
of TP-AGB stars, is exposed to a background radiation field. We here assess if
photo-evaporation may be relevant when dust is exposed to an external
radiation field as often assumed in the literature (e.g. Draine & Li, 2007).
In such an approach, the radiation field is parameterised as $U\times ISRF$
where $U$ is a scaling which can be up to $U_{\rm max}=10^{7}$ and the ISRF is
from Mathis et al. (1983). For such a maximum value of the scaling factor we
obtain from Eq. 11 (by substituting $B(\nu)(T_{\rm eff})$ with the ISRF flux
by Mathis et al., 1983) an equilibrium temperature for silicate dust of
$T_{\rm silicate}\approx 154$ K and of $T_{\rm carbon}\approx 245$ K for
carbon. Such values are well below the sublimation temperatures derived in
Section 2.3 and therefore both silicate and carbon dust grains are stable
against evaporation.

###  3.3 Including dust destruction in dust evolutionary models

We include dust photo-evaporation by considering the case of young stars born
in clusters and PNe, following the description provided in Section 2. In Fig.
1 we show the overall evolution of $sM_{\rm dust}$ in the ISM (upper panel)
and we compare the efficiency of different dust depleting mechanisms by
plotting the integrated amount of the specific mass of dust destroyed or
removed as a function of time, $sM_{\rm destr}=M_{\rm destr}/M_{*}$ (middle
and lower panels). In other words, we integrate in time each of the
destruction mechanisms corresponding to the terms from the 2nd to the 6th in
Eq. 5 and we divide by the stellar mass444The contribution of photo-
evaporation from PNe has been splitted in the two terms of Eq. 7.. We run a
reference model by assuming $M_{\rm gas,ini}=4$ $M_{\rm*,fin}$ (where $M_{\rm
gas,ini}$ and M∗,fin are the initial mass of gas and the final stellar mass,
respectively), $\tau=1000$ Myrs in Eq. 8, $M_{\rm swept}=1200$ M⊙, a Chabrier
IMF, and a typical value of the mass-loading factor $ML=1$. Note that with
such an assumption for $ML$ astration and outflow provide the same
contribution to dust removal. We consider the cases with and without dust
growth in the ISM, and we vary $M_{\rm gas,ini}$, $\tau$ and the IMF.

Fig. 1 demonstrates that destruction from photo-evaporation is orders of
magnitudes lower than SN shocks destruction, than dust astration, and than
outflow removal. This is true both with and without dust growth in the ISM.
The difference in the destruction efficiency for the two cases is minor. Only
the model in which dust growth is not included shows a decline in $sM_{\rm
dust}$ as a function of time mainly due to dust destruction from Type II SNe.
Such a trend is in qualitative good agreement with the one observed in the
literature, however, a number of observable parameters like metallicity and
gas fraction, beside specific mass of dust should be reproduced consistently
by models before drawing some firm conclusion.

In Fig. 2 we show the same model as in Fig. 1 (magenta lines) together with a
model representative of gas rich galaxies with $M_{\rm gas,ini}=20\times
M_{\rm*,fin}$. In the right panel we can appreciate that in this latter case
the efficiency of the various removal processes is reduced due to the effect
of lower values of $M_{\rm dust,j}/M_{\rm gas}$ (see Eqs. 9, 10 and 16).
Photo-evaporation remains negligible with respect to the other processes. The
plot shows that for gas rich systems with large $M_{\rm gas}/M_{\star}$ the
predicted $sM_{\rm dust}$ does not show a decline as instead in the case with
a smaller $M_{\rm gas}/M_{\star}$.

In Fig. 3 we show the same model as in the previous plots (magenta lines)
together with a model computed by assuming $\tau=10$ Myrs in Eq. 8
representing the case of a very short burst of star formation. In this latter
case, the decline in the $sM_{\rm dust}$ with time occurs at earlier times
with respect to the case with $\tau=1000$ Myrs due to the fast production of
stars which explodes as Type II SN, and it is followed by an increase in the
dust content due to the remaining contribution of Type II SNe, AGB stars and
Type Ia SNe. Such a trend showing a decline followed by an increase in the
dust mass seems to be at odd with the observed trend, but is obtained for the
specific choice of parameters (e.g $M_{\rm gas,ini}/M_{\rm\star,fin}$) beside
the SFH.

In Fig. 4 we show the effect of assuming different IMF (either Chabrier (2003)
or top-heavy). A top-heavy IMF may be typical of low-metallicity environments
(e.g. Marks et al., 2012). In the upper panel, the model with the top-heavy
IMF shows a more rapid decline with time with respect to the Chabrier case due
to the efficient destruction by Type II SNe which are more numerous than for
the Chabrier IMF. The dust produced by the top-heavy model, encompasses the
one in which the Chabrier IMF is employed. Because of the larger $sM_{\rm
dust}$ the destruction mechanisms are more efficient than in the Chabrier
case. Photo-evaporation remains negligible in both cases. For both the models,
the trend found is in qualitative good agreement with the observations.

Figure 1: Upper panel: overall evolution of dust computed as the enrichment
from stars minus the dust removal from different mechanisms. The cases with
and without dust growth in the ISM are shown. The initial mass of the gas is
equal to $4$ times the final mass of stars, $M_{\rm gas,ini}=4\times
M_{\rm\star,fin}$ and $\tau=1000$ Myrs in Eq. 8. Middle and lower panels: dust
removal due to different mechanisms included in the models in the upper panel
as mentioned in the legend, where “PE” stands for “photo-evaporation”.

Figure 2: Upper panel: overall evolution of dust computed as the enrichment
from stars minus the dust removal from different mechanisms. Two cases with
different initial gas masses, $M_{\rm gas,ini}=4\times M_{\rm\star,fin}$ and
$M_{\rm gas,ini}=20\times M_{\rm\star,fin}$, are compared. We select
$\tau=1000$ Myrs in Eq. 8 and we do not include dust growth in the ISM. Middle
and lower panels: dust removal due to different mechanisms included in the
models in the upper panel, similar to Fig. 1.

Figure 3: Upper panel: overall evolution of dust computed as the enrichment
from stars minus the dust removal from different mechanisms. Two cases with
different values of $\tau=10,1000$ Myrs in Eq. 8 are shown. $M_{\rm
gas,ini}=4\times M_{\rm\star,fin}$. Middle and lower panels: dust removal due
to different mechanisms corresponding to the models in the upper panel,
similar to Fig. 1.

Figure 4: Upper panel: overall evolution of dust computed as the enrichment
from stars minus the dust removal from different mechanisms. Two cases with
different choices of the IMF are shown. We select $M_{\rm gas,ini}=4\times
M_{\rm\star,fin}$, $\tau=1000$ Myrs in Eq. 8 and we do not include dust growth
in the ISM. Middle and lower panels: dust removal due to different mechanisms
corresponding to the models in the upper panel, similar to Fig. 1.

##  4 Conclusions

We find that photo-evaporation due to young stars and PNe has only a minor
role in dust destruction, independently of the assumed efficiency of dust
growth in the ISM, the initial mass of gas, the SFH, and the IMF. We do not
exclude however the possibility of dust being destroyed by shocks generated by
fast winds in HII regions and PNe. The investigation of such processes is
beyond the scope of this work. The dust exposed to an interstellar radiation
field $U\times ISRF$ is stable against sublimation up to the largest value
assumed in the literature $U_{\rm max}=10^{7}$ (Draine & Li, 2007). These
findings also imply that the total yields from TP-AGB stars are not normally
destroyed by photo-evaporation induced by the ambient radiation field or when
the PN phase is reached.

The observed trend with increasing $sM_{\rm dust}$ at early times followed by
a decrease is qualitatively well reproduced for low values of $M_{\rm
gas}/M_{\star}$ found in passive galaxies, while for larger values of $sM_{\rm
dust}$ representative of systems with a large gas fraction (e.g. DGS) the
observed trend is difficult to be reproduced if outflow is not extremely
efficient (see also Nanni et al. (2020)).

It is worth noticing that observationally we are able to probe specific dust
mass above $10^{-4}$ at each redshift. Therefore, at such values we don’t
expect photo-evaporation to be a relevant dust destruction process even with
low metallicity and intense radiation field.

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###### Acknowledgements.

A.N, M.R., P.S. acknowledge support from the Narodowe Centrum Nauki (NCN),
Poland, through the SONATA BIS grant UMO-2020/38/E/ST9/00077. D.D.
acknowledges support from the NCN through the SONATA grant
UMO-2020/39/D/ST9/00720. M.J.M. acknowledges the support of the NCN through
the SONATA BIS grant UMO-2018/30/E/ST9/00208 and the Polish National Agency
for Academic Exchange Bekker program grant BPN/BEK/2022/1/00110. M.R.
acknowledges support from the Foundation for Polish Science (FNP) under the
program START 063.2023. We thank the anonymous referee for the careful reading
of the manuscript and for her/his thoughtful comments.