[1] Mao-Yuan Wan

[1] University of Shanghai for Science and Technology, Shanghai, 200093, China

2] Shanghai Advanced Research Institute, Chinese Academy of Sciences,
Shanghai, 201210, China

# Absorption and scattering of massless scalar wave from Regular Black Holes

<EMAIL_ADDRESS>Chen Wu  *  [

###### Abstract

In this work, we numerically investigate the scattering and absorption cross
section of the massless scalar field from some well-known regular black holes
by using the partial wave approach. Our computational results indicate that
the larger the parameters, the lower the associated total absorption cross
section maxima. When compared to the Schwarzschild black hole, the scattering
cross section is enhanced in some regular black hole spacetimes, meanwhile the
scattering width is narrow in the forward orientation. Moreover, it is found
that the null geodesics of the critical impact parameter and the geometrical
optical value in the high frequency regime have similar changing behavior.

###### keywords:

regular black hole, scalar wave, black hole scattering theory, null geodesic

##  1 Introduction

The problem of understanding how to avoid singularities in black hole
spacetimes is important in general relativity. Bardeen bib1  generated the
first regular black hole which is non-singularity during 1968. Such a black
hole spacetime geometry without singularity that obeys the weak energy
condition, some exotic matter or adjustment in gravity theory must be
included. The spacetime geometry is achieved by adding a regular and bounded
energy-momentum tensor that vanishes at the infinity and matches the weak
energy requirement. Before Ayón-Beato et al. bib2  in 2000, this regular black
hole spacetime was deficient in a accordant physical explanation that
describing it as the gravitational field of nonlinear magnetic monopole with a
mass $M$ and a charge $q$. Bardeen’s solution has motivated deeper works about
singularity avoidance may be realized generally. Several other scholars paid
attention to theories of gravity coupled to nonlinear electrodynamics, and
proposed other solutions in different contexts bib3 ; bib4 ; bib5 ; bib6 ;
bib7 ; bib8 ; bib9 . The lack of singularity is associated with topological
change beyond the event horizon for regular black hole solutions. Borde bib10
claimed that verifying the general singularity theorem is unattainable unless
strong energy conditions or global hyperbolicity are presented.

In the 1970s and 1980s, considerable effort was devoted to the study of the
scattering and absorption of planar waves of frequency $\omega$ by black holes
bib11 . People can regard the scattering results as a black hole fingerprint
and possibly observed. Recently, these topics have been also arousing lots of
interest bib12 ; bib13 ; bib14 ; bib15 ; bib16 .

The reason that this subject attracts increasing attentions are as follows:
1)quasinormal modes of black holes are thought to representing the poles of
the corresponding black hole scattering matrix. 2)scattering and absorption
from black holes is related to many interesting phenomena, such as glory,
photon orbit and superradiant scattering bib13 ; bib17 ; bib18 ; bib19 ; bib20
; bib21 ; bib22 ; bib23 ; bib24 ; bib25 ; bib26 . In Ref. bib27 ; bib28 ;
bib29 , Sánchez noticed that the absorption cross section of the Schwarzschild
black hole for massless scalar wave oscillating near the geometrical optical
limit via numerical approaches.

Chen and his coworkers bib30  investigated the scattering and absorption cross
section of massless scalar wave from the black hole surrounded by magnetic
field. Crispino et al. bib31  extended study of the scattering theory of
scalar waves in the Reissner-Nordström black hole spacetime. Huang et al.
bib32  studied the scattering and absorption cross section of massless scalar
waves by the Bardeen black hole.

In this article, we mainly focus on the scalar scattering process of some
well-known regular black holes, and how different nonlinear interactions
effects on scalar absorption and scattering cross sections. The following is
the structure of this paper: Sect. 2 provide a quick overview of some regular
black holes and the null geodesics in these spacetimes. In Sect. 3, we perform
an analysis of the classical dynamics of massless scalar field. The
concentration of Sect. 4 is the absorption and scattering cross sections of
massless scalar fields for regular black holes. Finally, we come to a
conclusion in Sect. 5. Throughout this paper we use natural units
$c=\hbar=G=1$.

##  2 Classical analysis

###  2.1 The basic equations

In this subsection, we will first give a brief introduction to the regular
black holes. The nonlinear electrodynamic term bib1  is incorporated in the
classical action to obtain the solution

$S=\frac{1}{16\pi G}\int\mathrm{d}^{4}\sqrt{-g}\left[R-\mathcal{L}(F)\right]$
(1)

where $g$ is the determinant of the metric tensor $g^{\mu\nu}$, $G$ is the
gravitational constant, $R$ is the scalar curvature, and $\mathcal{L}(F)$ is
the nonlinear electrodynamics Lagrangian $F=(F_{\mu\nu}F^{\mu\nu})/4$, where
$F_{\mu\nu}=\bigtriangledown_{\mu}A_{\nu}-\bigtriangledown_{\nu}A_{\mu}$
denotes the intensity of the electromagnetic field.

The generic line element for spherically symmetric regular black hole solution
is as described in the following

$ds^{2}=-f(r)c^{2}dt^{2}+f^{-1}(r)dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})$ (2)

where $(t,r,\theta,\phi)$ are the four dimensional spacetime spherical
coordinates, and specific forms of the lapse function $f(r)$ distinguish
between the different spacetimes which relies on the energy momentum.

The lapse function $f(r)$ of the Bardeen black hole is determined by the
formula in Ref. bib1

$f(r)=1-\frac{2Mr^{2}}{(r^{2}+q^{2})^{\frac{3}{2}}}$ (3)

where $q$ and $M$ are the magnetic charge and the mass of the magnetic
monopole, respectively. Ayón-Beato et al. bib2  interpreted this black hole as
the gravitational field of a magnetic monopole arising from nonlinear
electrodynamics. The particular nonlinear electrodynamics Lagrangian is given
by $\mathcal{L}(F)=\frac{3M}{\lvert
q\rvert^{3}}[\sqrt{2q^{2}F}(1+\sqrt{2q^{2}F})]^{5/2}$. There has horizon only
if $\lvert q\rvert\leq\frac{4M}{3\sqrt{3}}$. However the horizon is degenerate
if $q=\frac{4M}{3\sqrt{3}}$ and there are no horizon if
$q>\frac{4M}{3\sqrt{3}}$.

There has another renowned regular black hole is presented by Ayón-Beato and
García in Ref. bib3 . they took the nonlinear electric field as a source of
charge for the solution of Maxwell field equations to obtaining this regular
black hole solution. Its lapse function $f(r)$ is given by

$f(r)=1-\frac{2Mr^{2}}{(r^{2}+q^{2})^{3/2}}+\frac{q^{2}r^{2}}{(r^{2}+q^{2})^{2}}$
(4)

where $M$ and $q$ are the total mass and charge respectively. This solution is
obtained from a nonlinear electrodynamics with Lagrangian density
$\mathcal{L}(F)=\frac{X^{2}}{-2q^{2}}\frac{1-8X-3X^{2}}{(1-X)^{4}}-\frac{3M}{2q^{3}}\frac{X^{(15/2-5X)}}{(1-X)^{7/2}}$,
where $X=\sqrt{-2q^{2}F}$.

Bronnikov bib4  and Berej et al. bib5  originally constructed the regular
black hole by introducing the Lagrangian for nonlinear electrodynamics to
first order. The lapse function for this black hole is given as

$f(r)=1-\frac{2M}{r}\left(1-\tanh\frac{r_{0}}{r}\right)$ (5)

where the parameter $r_{0}$ denotes a length scale defined as $r_{0}=\frac{\pi
q^{2}}{8M}$ related to the electric charge.

Dymnikova bib6  presented an exact, regular spherically symmetric, charged
black hole solution by using the idea proposed by Bronnikov bib4  and Berej et
al. bib5 . This solution is constructed from a nonlinear electrodynamic theory
with a Hamiltonian-like function. The lapse function for Dymnikova’s solution
is given as

$f(r)=1-\frac{2M}{r}\frac{2}{\pi}\left(\arctan\frac{r_{0}}{r}-\frac{r_{0}}{r^{2}+r^{2}_{0}}r\right)$
(6)

the parameter $r_{0}$ in the solution (6) is a length scale, where $M$ and $q$
is the total mass and the charge respectively.

In 2006, Hayward bib7  found a new regular black hole spacetime that resemble
the physical interpretation of the Bardeen one and has center flatness. This
simple regular black hole implies a specific matter energy-momentum tensor
that is de Sitter at the core and vanishes at large distances $r\to\infty$.
The lapse function $f(r)$ for the Hayward black hole takes a simple form

$f(r)=1-\frac{2Mr^{2}}{(r^{3}+2\alpha^{2})}$ (7)

where $\alpha$ is a constant. As like the Bardeen black hole, this black hole
can also have zero, one, or two horizons depending on the values of $M$ and
$\alpha$.

Balart and Vagenas bib8  constructed charged regular black holes in the
framework of Einstein-nonlinear electrodynamics theory. They built the general
lapse function for mass distribution functions that are inspired by continuous
probability distributions. In this paper, we focus on one instances of black
hole solutions that use their technique. The lapse function is of the form

$f(r)=1-\frac{2M}{r}\exp\left(-\frac{q^{2}}{2Mr}\right)$ (8)

in the metric function $M$ and $q$ are associated with total mass and charge
respectively.

All the regular black hole solutions used in this paper are summarized in
Table 1. The mass $M$ of black holes is normalized to 1. It is necessary to
emphasize that in all the mentioned black holes the lapse function $f(r)$ can
have zero, one, or two horizons depending on the value of charge parameters.
In Table 1, we list the extreme charge parameters for which the inner horizon
and outer horizon coincide.

Table 1:  Summary of some well-known regular black holes in the paper. For more details, one can refer to the Refs. bib1 ; bib2 ; bib3 ; bib4 ; bib5 ; bib6 ; bib7 ; bib8  listed in the subsection 2.1. When the parameter is equal to or greater than the value presented in the Extremal column, the event horizon degenerates or even disappears. In this work, the mass $M$ of black holes is normalized to 1. The letters A-F represent different black holes respectively.  Label | Lapse function | Eq. | Extremal | Originator | Ref.
---|---|---|---|---|---
A | $1-\frac{2Mr^{2}}{(r^{2}+q^{2})^{3/2}}$ | (3) | $q\approx 0.77$ | Bardeen | bib1 ; bib2
B | $1-\frac{2Mr^{2}}{(r^{2}+q^{2})^{3/2}}+\frac{q^{2}r^{2}}{(r^{2}+q^{2})^{2}}$ | (4) | $q\approx 0.63$ | Ayón-Beato&García | bib3
C | $1-\frac{2M}{r}\left(1-\tanh\frac{r_{0}}{r}\right)$ | (5) | $r_{0}\approx 0.55$ | Bronnikova | bib4 ; bib5
D | $1-\frac{2M}{r}\frac{2}{\pi}\left(\arctan\frac{r_{0}}{r}-\frac{r_{0}}{r^{2}+r^{2}_{0}}r\right)$ | (6) | $r_{0}\approx 0.45$ | Dymnikova | bib6
E | $1-\frac{2Mr^{2}}{(r^{3}+2\alpha^{2})}$ | (7) | $\alpha\approx 1.06$ | Hayward | bib7
F | $1-\frac{2M}{r}\exp\left(-\frac{q^{2}}{2Mr}\right)$ | (8) | $q\approx 1.21$ | Balart&Vagenas | bib8
|  |  |  |  |

###  2.2 Geodesics analyze

We discuss the null geodesics in regular black hole spacetimes in this
subsection. In general, the geodesics in regular black hole spacetimes are
described using the formula bib31

$\dot{s}^{2}=-f(r)\dot{t}^{2}+f^{-1}(r)\dot{r}^{2}+r^{2}(\dot{\theta}^{2}+\sin^{2}\theta\dot{\phi}^{2})=\lambda,$
(9)

where the over dot denotes the derivative with respect to an affine parameter
$\lambda$. For massless particles we have $\lambda=0$.

It is useful to introduce the function $h(r)^{2}$

$h(r)^{2}=\frac{r^{2}}{f(r)},$ (10)

In terms of $h(r)^{2}$, the radius of the outermost photon sphere
$r_{\rm{ph}}$ is the largest root of the equation bib33

$\frac{\mathrm{d}}{\mathrm{d}r}h(r)^{2}=0,$ (11)

Figure 1:  The null geodesics for different critical impact parameters
$b_{\rm{cr}}$ in several regular black hole spacetimes. We find that metric
parameters for different black hole spacetimes considerably affect the
magnitude of the photon sphere radii. The letters A-F represent different
black holes respectively in Table 1.

The critical impact parameter $b_{\rm{cr}}$ and the radius of the photon
sphere $r_{\rm{ph}}$ are related by

$b_{\rm{cr}}=h(r_{\rm{ph}})$ (12)

On the equatorial plane, the orbit equation of light rays can be written as

$\left(\frac{\mathrm{d}r}{\mathrm{d}\varphi}\right)^{2}=\frac{r^{2}}{f^{-1}(r)}\left(\frac{h(r)^{2}}{b_{{cr}}^{2}}-1\right)$
(13)

In Fig. 1, we plot the null geodesics for different critical impact parameters
$b_{\rm{cr}}$ in several regular black hole spacetimes. Then we find that
metric parameters for different black hole spacetimes considerably affect the
magnitude of the photon sphere radii.

Because of the existence of the innermost unstable photon sphere
$r=r_{\rm{ph}}$, the photon can deviate from the photon sphere at any angle.
This feature indicates that the glory phenomena will occur bib31 . Glory
phenomena, like those seen in optics, are bright spots or halos that occur in
the scattering intensity in the epipolar direction, the intensity and size of
which depend on the wavelength of the incident perturbation. The approximation
developed by Matzner et al. bib34  may be used to determine the magnitude and
size of bright spots.

##  3 Massless scalar field equation

###  3.1 Effective scattering potential

The dynamics of a massless test scalar field $\Psi$ is governed by the Klein-
Gordon equation, namely

$\frac{1}{\sqrt{-g}}\partial_{\mu}(\sqrt{-g}g^{\mu\nu}\partial_{\nu}\Psi)=0.$
(14)

Regular black holes are spherically symmetric spacetime with a global timelike
Killing vector field, $\partial_{t}$, for $r>r_{\rm{EH}}$ bib31 , where
$r_{\rm{EH}}$ is the outer event horizon. As a result, we can separate the
variables

$\Psi=\frac{\psi_{\omega l}(r)}{r}Y_{lm}(\theta,\phi)e^{-i\omega t}$ (15)

$Y_{lm}(\theta,\phi)$ denotes a scalar spherical harmonic function. The
radially Schrödinger-like equation for the massless scalar field may therefore
be obtained

$\left[f(r)\frac{\mathrm{d}}{\mathrm{d}r}f(r)\frac{\mathrm{d}}{\mathrm{d}r}+\omega^{2}-V_{\rm{eff}}(r)\right]\psi_{\omega
l}(r)=0$ (16)

$V_{\rm{eff}}(r)$ denotes the effective scattering potential:

$V_{\rm{eff}}(r)=f(r)\left[\frac{1}{r}\frac{\mathrm{d}f(r)}{\mathrm{d}r}+\frac{l(l+1)}{r^{2}}\right]$
(17)

We present Regge-Wheeler type coordinates $r_{*}$, also known as Tortoise
coordinates, for a better study of the asymptotic infinity limit of radial
equation (16). If we introduce the tortoise coordinate

$r_{*}=\int f(r)^{-1}\mathrm{d}r,$ (18)

the semi-infinite interval $r\in(2M,+\infty)$ is mapped to the infinite
interval $r_{*}\in(-\infty,+\infty)$, and the effective scattering potential
$V_{\rm{eff}}(r)$ changes into $V_{\rm{eff}}(r_{*})$. After introducing this
coordinate transition, the equation of motion of scalar field has a
Schrödinger-like form

$\left[\frac{\mathrm{d}^{2}}{\mathrm{d}r_{*}^{2}}+\omega^{2}-V_{\rm{eff}}(r_{*})\right]\psi_{\omega
l}(r_{*})=0$ (19)

In Fig. 2 , the effective potential $V_{\rm{eff}}(r)$ is plotted with
$l=0,1,2$. From this figure, we can see that the peak value of potential
barrier gets upper when the angular quantum number $l$ increases. When the
value of $l$ is fixed, the height of the scattering barrier increases as the
metric parameter increases. The results also reveal that as the radial
distance increases, the numerical difference between the curves diminishes.

Figure 2:  The effective scattering potentials $V_{\rm{eff}}$ as a function of
$r$ for different choices of $l$ in the background of different regular black
holes. The larger the value of $l$ is, the higher the scattering barrier is.
The mass $M$ is normalized to 1. The letters A-F represent different black
holes respectively in Table 1.

For our scattering problem, we consider plane scalar waves incoming from the
infinite null past. Therefore, we are interested in solutions of equation (19)
subjected to the following boundary conditions

$\psi_{\omega l}(r_{*})\sim\begin{cases}A_{\rm{in}}e^{-i\omega
r_{*}}+A_{\rm{out}}e^{i\omega r_{*}}&\rm{for\quad}r_{*}\to+\infty\\\
e^{-i\omega r_{*}}&\rm{for\quad}r_{*}\to-\infty\\\ \end{cases}$ (20)

with the relation $1+\lvert A_{\rm{out}}\rvert^{2}=\lvert
A_{\rm{in}}\rvert^{2}$ satisfied. The phase shifts of the scattered waves are
defined by

$e^{2i\delta_{l}}=(-1)^{l+1}\frac{A_{\rm{out}}}{A_{\rm{in}}}$ (21)

Based on the quantum mechanics, when an incident wave travels through the
potential barrier, part of the wave is reflected back and part of the wave
goes through the potential barrier. By using the the conservation of the flux,
the transmission amplitude $T_{l}$ and reflection amplitude $R_{l}$ satisfy

$\lvert T_{l}\rvert^{2}+\lvert R_{l}\rvert^{2}=1,$ (22)

where $T_{l}$ and $R_{l}$ are defined by $T_{l}=1/A_{in}$ and
$R_{l}=A_{out}/A_{in}$, respectively.

###  3.2 The Mashhoon method

To calculate the absorption cross section and scattering cross section of the
scalar field scattered by regular black holes in this study, we need to solve
the radial equation under proper boundary conditions to obtain the phase shift
by using the Numerov numerical method and the Pöschl-Teller potential
approximation bib35 .

We apply the Pöschl-Teller potential $V_{\rm{PT}}$ to approximate the
effective scattering potential in order to obtain analytic results. The
potential of Pöschl-Teller is formulated as

$V_{PT}(r_{*})=\frac{V_{0}}{\cosh^{2}[\kappa(r_{*}-r_{*,\rm{max}})]}$ (23)

where $V_{0}$ and $-2V_{0}\kappa^{2}$ denote the peak value of the effective
potential and the function curvature when the effective potential is at its
highest value $r_{*}=r_{*,\rm{max}}$, respectively. More specifically

$\kappa^{2}=-\frac{1}{2V_{0}}\left.\frac{\mathrm{d}^{2}V_{eff}}{\mathrm{d}r_{*}^{2}}\right\rvert_{r_{*}=r_{*,\text{max}}}=-\frac{1}{2V_{0}}f(r)\left.\frac{\mathrm{d}}{\mathrm{d}r}\left[f(r)\frac{\mathrm{d}V_{eff}}{\mathrm{d}r}\right]\right\rvert_{r=r_{\text{max}}}$
(24)

The potential $V_{\rm{PT}}$ can be adjusted to matching its maximum of the
effective potential with parameters.

One can analytically obtain the radial wave function by replacing
$V_{\rm{eff}}(r_{*})$ with the Pöschl-Teller potential $V_{PT}(r_{*})$. As a
consequence, the transmission coefficient $T_{l}$ and reflection coefficient
$R_{l}$ are given by bib36

$\displaystyle T_{l}$
$\displaystyle=\frac{\Gamma(\frac{-i\omega}{\kappa})\Gamma(1+\beta+\frac{i\omega}{\kappa})\Gamma(-\beta+\frac{i\omega}{\kappa})}{\Gamma(\frac{i\omega}{\kappa})\Gamma(1+\beta)\Gamma(-\beta)}$
(25) $\displaystyle R_{l}$
$\displaystyle=\frac{\Gamma(1+\beta+\frac{i\omega}{\kappa})\Gamma(-\beta+\frac{i\omega}{\kappa})}{\Gamma(1+\frac{i\omega}{\kappa})\Gamma(\frac{i\omega}{\kappa})}$

where

$\beta=-\frac{1}{2}+\left[\frac{1}{4}-\frac{V_{0}}{\kappa^{2}}\right]^{\frac{1}{2}}$
(26)

##  4 Numerical results

###  4.1 Absorption cross section

It is well known that the total absorption cross section can be calculated as

$\sigma_{\rm{abs}}=\sum\limits_{l=0}^{\infty}\sigma^{(l)}_{\rm{abs}}$ (27)

where $\sigma^{(l)}_{\rm{abs}}$ denotes the partial absorption cross section
and can be expressed by transmission coefficient:

$\sigma^{(l)}_{\text{abs}}=\frac{\pi}{\omega^{2}}(2l+1)(1-\lvert
e^{2i\delta_{l}}\rvert^{2})=\frac{\pi}{\omega^{2}}(2l+1)\lvert T_{\omega
l}\rvert^{2}$ (28)

Figure 3:  The partial absorption cross sections (solid lines)
$\sigma^{(l)}_{\rm{abs}}$, i.e. $l=0\sim 10$ and total absorption cross
sections (dashed lines) $\sigma_{\rm{abs}}$ of the regular black holes with
different metric parameters. The horizontal lines (short dash) denote the
geometrical optics limit value $\sigma^{\rm{hf}}_{\rm{abs}}$. The mass $M$ is
normalized to 1. The letters A-F represent different black holes respectively
in Table 1.

In Fig. 3, we show the partial absorption cross sections
$\sigma^{(l)}_{\rm{abs}}$, i.e. $l=0\sim 10$ and total absorption cross
sections $\sigma_{\rm{abs}}$ of the regular black holes with different metric
parameters. The mass $M$ is normalized to 1. It is obvious that the partial
absorption cross section $\sigma^{(l)}_{\rm{abs}}$ tends to vanish as $\omega
M$ increases. In the null energy limit, the partial wave with $l=0$ provides a
nonzero absorption cross section bib31 . Furthermore, for each value of $l>0$,
the corresponding partial absorption cross section $\sigma^{(l)}_{\rm{abs}}$
starts from zero, reaches a peak, and decreases asymptotically. The larger the
value of $l$ is, (i)the smaller the corresponding maximum of
$\sigma^{(l)}_{\rm{abs}}$ is and (ii) the larger the value of $\omega M$
associated with the peak of $\sigma^{(l)}_{\rm{abs}}$ is. This is compatible
with the fact that the larger the value of $l$ is, the higher the scattering
potential $V_{\rm{eff}}$ is.

Fig. 3 also shows that the total absorption cross section $\sigma_{\rm{abs}}$
is smaller for bigger value of the metric parameter. The total absorption
cross section $\sigma_{\rm{abs}}$ oscillates around the limit of geometrical
optics. When $\omega M\gg 1$ goes to the high frequency zone, the total
absorption cross section tends to the geometric optics limit value
$\sigma^{\rm{hf}}_{\rm{abs}}=\pi b^{2}_{\rm{cr}}$ bib31 , where $b_{\rm{cr}}$
is the critical impact parameter mentioned in subsection 2.2. From Fig. 1, we
can see that the larger the value of the metric parameter is, the smaller the
critical impact parameter is, which is numerically consistent with the results
we discussed above.

Figure 4:  The effective scattering potentials $V_{\rm{eff}}$ as a function of
$r_{*}$ for different choices of $l=0,1,2$ in the background of Bardeen case
(red, green and blue lines) and Schwarzschild case (black lines). The dashed
lines represent the Mashhoon method we utilized and the solid lines stand for
the results of numerical integration.

The absorption of planar massless scalar waves by Bardeen regular black holes
has been studied in bib37 . In the low-frequency regime, the total absorption
cross section is less than the Shwarzschild case in the Figure 6 of bib37  and
is similar as shown in the Fig. 3A of this paper. While in the high-frequency
regime, the trend of the total absorption cross section decreasing with
parameter increasing is consistent with Fig. 6 of bib37  and the oscillation
of the total absorption cross section tends to geometrical optics limit
$\sigma^{\rm{hf}}_{\rm{abs}}$ in Fig. 3A. Because of the Mashhoon method we
utilized. It can be observed that the accuracy of our estimate increases as
the shape of the effective potential becomes more consistent with the Pöschl-
Teller potential. We present the effective potential $V_{\rm{eff}}(r_{*})$
with $l=0,1,2$ in Fig. 4. Despite the fact that this approach is fairly simple
to compute, it is clear that not all effective potentials of perturbation
fields in black holes accord well with the Pöschl-Teller potential.

###  4.2 Scattering cross section

Figure 5:  Logarithmic value of the total scattering cross sections of
different regular black holes with spacetime parameters in the range of
$\theta=0^{\circ}$ to $\theta=180^{\circ}$. $\omega M$ is fixed to 1. The
letters A-F represent different black holes respectively in Table 1.

It is widely known that the scattering amplitude is defined as

$g(\theta)=\frac{1}{2i\omega}\sum\limits_{l=0}^{\infty}(2l+1)(e^{2i\delta_{l}}-1)P_{l}(\cos\theta)$
(29)

and the differential scattering cross section

$\frac{\mathrm{d}\sigma_{\text{sca}}}{\mathrm{d}\Omega}=\lvert
g(\theta)\rvert^{2}$ (30)

As a direct consequence, the scattering cross section is obtained

$\sigma_{\text{sca}}(\omega)=\int\frac{\mathrm{d}\sigma}{\mathrm{d}\Omega}\mathrm{d}\Omega=\frac{1}{2i\omega}\sum\limits_{l=0}^{\infty}(2l+1)\lvert
e^{2i\delta_{l}}-1\rvert^{2}$ (31)

Fig. 5 shows the total scattering cross section as a function of the
scattering angle. For comparison, we also plot the results of the
Schwarzschild black hole. For some black holes, we find the scattering flux is
strengthened and its width becomes narrower in the forward direction. The
scalar field scattering becomes more concentrated. With fixed frequency in the
low frequency range, the glory peak is higher and the glory width becomes
narrower. As a result, the glory phenomena at the forward and backward lends
itself advantageously to astronomical observation.

##  5 Conclusions

In summary, we have investigated the scattering and absorption cross section
of the massless scalar field from some well-known regular black holes by using
the Numerov approach and the Pöschl-Teller potential approximation. Our
computational results indicated that the larger the spacetime parameter is,
the lower the associated total absorption cross section maxima is. When
compared to the Schwarzschild black hole, the scattering cross section is
enhanced in some regular black hole spacetimes, meanwhile the scattering width
is narrow in the forward orientation. We also studied the null geodesics in
the regular black hole spacetimes and compared the critical impact parameter
with the geometrical optical limit. The total absorption cross section tends
to the geometrical optical limit in the high frequency region. We found that
in several cases the scattering flux is strengthened and scattering cross
section width becomes narrower in the forward direction in comparison to the
Schwarzschild case.

Acknowledgments

The authors thank Dr. Z.N. Yan for his positive help and useful discussion.
This work was supported by the National Key Research and Develop Program of
China under Contract No. 2018YFA0404404.

Data Availability Statement The data generated in this study are available on
request from the corresponding authors.

## References

  *   * (1) Bardeen, J.: Non-Singular General-Relativistic Gravitational Collapse (1968)
  * (2) Ayón-Beato, E., García, A.: The bardeen model as a nonlinear magnetic monopole. Physics Letters B 493(1), 149–152 (2000). https://doi.org/10.1016/S0370-2693(00)01125-4
  * (3) Ayón-Beato, E., García, A.: Regular black hole in general relativity coupled to nonlinear electrodynamics. Phys. Rev. Lett. 80(23), 5056–5059 (1998). https://doi.org/10.1103/PhysRevLett.80.5056
  * (4) Bronnikov, K.A.: Regular magnetic black holes and monopoles from nonlinear electrodynamics. Phys. Rev. D 63(4), 044005 (2001). https://doi.org/10.1103/PhysRevD.63.044005
  * (5) Berej, W., Matyjasek, J., Tryniecki, D., Woronowicz, M.: Regular black holes in quadratic gravity. General Relativity and Gravitation 38(5), 885–906 (2006). https://doi.org/10.1007/s10714-006-0270-9
  * (6) Dymnikova, I.: Regular electrically charged vacuum structures with de sitter centre in nonlinear electrodynamics coupled to general relativity. Classical and Quantum Gravity 21(18), 4417–4428 (2004). https://doi.org/10.1088/0264-9381/21/18/009
  * (7) Hayward, S.A.: Formation and evaporation of nonsingular black holes. Phys. Rev. Lett. 96(3), 031103 (2006). https://doi.org/10.1103/PhysRevLett.96.031103
  * (8) Balart, L., Vagenas, E.C.: Regular black holes with a nonlinear electrodynamics source. Phys. Rev. D 90(12), 124045 (2014). https://doi.org/10.1103/PhysRevD.90.124045
  * (9) Bronnikov, K.A., Fabris, J.C.: Regular phantom black holes. Phys. Rev. Lett. 96(25), 251101 (2006). https://doi.org/10.1103/PhysRevLett.96.251101
  * (10) Borde, A.: Regular black holes and topology change. Phys. Rev. D 55(12), 7615–7617 (1997). https://doi.org/10.1103/PhysRevD.55.7615
  * (11) Futterman, J., Handler, F., Matzner, R.A.: Scattering from Black Holes. Cambridge University Press, Cambridge; New York (1988)
  * (12) Glampedakis, K., Andersson, N.: Scattering of scalar waves by rotating black holes. Classical and Quantum Gravity 18(10), 1939–1966 (2001). https://doi.org/10.1088/0264-9381/18/10/309
  * (13) Doran, C., Lasenby, A., Dolan, S., Hinder, I.: Fermion absorption cross section of a schwarzschild black hole. Phys. Rev. D 71(12), 124020 (2005). https://doi.org/10.1103/PhysRevD.71.124020
  * (14) Dolan, S., Doran, C., Lasenby, A.: Fermion scattering by a schwarzschild black hole. Phys. Rev. D 74(6), 064005 (2006). https://doi.org/10.1103/PhysRevD.74.064005
  * (15) Crispino, L.C.B., Oliveira, E.S., Higuchi, A., Matsas, G.E.A.: Absorption cross section of electromagnetic waves for schwarzschild black holes. Phys. Rev. D 75(10), 104012 (2007). https://doi.org/10.1103/PhysRevD.75.104012
  * (16) Dolan, S.R.: Scattering and absorption of gravitational plane waves by rotating black holes. Classical and Quantum Gravity 25(23), 235002 (2008). https://doi.org/10.1088/0264-9381/25/23/235002
  * (17) Matzner, R.A.: Scattering of massless scalar waves by a schwarzschild “singularity”. Journal of Mathematical Physics 9(1), 163–170 (1968). https://doi.org/10.1063/1.1664470
  * (18) Mashhoon, B.: Scattering of electromagnetic radiation from a black hole. Phys. Rev. D 7(10), 2807–2814 (1973). https://doi.org/10.1103/PhysRevD.7.2807
  * (19) Starobinskiǐ, A.A.: Amplification of waves reflected from a rotating ”black hole”. Zh. Eksp. Teor. Fiz 7(1), 28–32 (1973)
  * (20) Starobinskiǐ, A.A., Churilov, S.M.: Amplification of electromagnetic and gravitational waves scattered by a rotating “black hole”. Zh. Eksp. Teor. Fiz 65(3), 3–11 (1973)
  * (21) Crispino, L.C.B., Higuchi, A., Matsas, G.E.A.: Quantization of the electromagnetic field outside static black holes and its application to low-energy phenomena. Phys. Rev. D 63(12), 124008 (2001). https://doi.org/10.1103/PhysRevD.63.124008
  * (22) Ju-Hua, C., Yong-Jiu, W.: Quasinormal modes of the scalar field in five-dimensional lovelock black hole spacetime. Chinese Physics B 19(6), 060401 (2010). https://doi.org/10.1088/1674-1056/19/6/060401
  * (23) Kobayashi, T., Tomimatsu, A.: Superradiant scattering of electromagnetic waves emitted from disk around kerr black holes. Phys. Rev. D 82(8), 084026 (2010). https://doi.org/10.1103/PhysRevD.82.084026
  * (24) Chen, J., Liao, H., Wang, Y.: Absorption of massless scalar wave by high-dimensional lovelock black hole. Physics Letters B 705(1), 124–128 (2011). https://doi.org/10.1016/j.physletb.2011.09.091
  * (25) LIAO, H., CHEN, J., WANG, Y.: Scattering of scalar wave from black hole in hoŘava–lifshitz gravity. International Journal of Modern Physics D 21(05), 1250045 (2012). https://doi.org/10.1142/S0218271812500459
  * (26) Frolov, V.P., Shoom, A.A.: Scattering of circularly polarized light by a rotating black hole. Phys. Rev. D 86(2), 024010 (2012). https://doi.org/10.1103/PhysRevD.86.024010
  * (27) Sánchez, N.: Wave scattering theory and the absorption problem for a black hole. Phys. Rev. D 16(4), 937–945 (1977). https://doi.org/%****␣RegularBHwithScaler1020.bbl␣Line␣425␣****10.1103/PhysRevD.16.937
  * (28) Sánchez, N.: Absorption and emission spectra of a schwarzschild black hole. Phys. Rev. D 18(4), 1030–1036 (1978). https://doi.org/10.1103/PhysRevD.18.1030
  * (29) Sánchez, N.: Elastic scattering of waves by a black hole. Phys. Rev. D 18(6), 1798–1804 (1978). https://doi.org/10.1103/PhysRevD.18.1798
  * (30) Chen, J., Liao, H., Wang, Y., Chen, T.: Absorption and scattering of scalar wave from schwarzschild black hole surrounded by magnetic field. The European Physical Journal C 73(4) (2013). https://doi.org/10.1140/epjc/s10052-013-2395-9
  * (31) Crispino, L.C.B., Dolan, S.R., Oliveira, E.S.: Scattering of massless scalar waves by reissner-nordström black holes. Phys. Rev. D 79(6), 064022 (2009). https://doi.org/10.1103/PhysRevD.79.064022
  * (32) Huang, H., Liao, P., Chen, J., Wang, Y.: Absorption and scattering cross section of regular black holes. Journal of Gravity 2014 (2014). https://doi.org/10.1155/2014/231727
  * (33) Perlick, V., Tsupko, O.Y.: Calculating black hole shadows: Review of analytical studies. Physics Reports 947, 1–39 (2022). https://doi.org/10.1016/j.physrep.2021.10.004
  * (34) Thorne, K.S., Misner, C.W., Wheeler, J.A.: Gravitation. Freeman, San Francisco (2000)
  * (35) Pöschl, G., Teller, E.: Bemerkungen zur quantenmechanik des anharmonischen oszillators. Zeitschrift für Physik 83(3), 143–151 (1933). https://doi.org/10.1007/BF01331132
  * (36) Ferrari, V., Mashhoon, B.: New approach to the quasinormal modes of a black hole. Phys. Rev. D 30(2), 295–304 (1984). https://doi.org/10.1103/PhysRevD.30.295
  * (37) Macedo, C.F.B., Crispino, L.C.B.: Absorption of planar massless scalar waves by bardeen regular black holes. Phys. Rev. D 90(6), 064001 (2014). https://doi.org/10.1103/PhysRevD.90.064001


11institutetext: Faculty of Aerospace Engineering, Delft University of
Technology, Delft, The Netherlands 11email<EMAIL_ADDRESS>
22institutetext: Leiden Observatory, Leiden University, P.O. Box 9513, NL 2300
RA Leiden, The Netherlands
33institutetext: Centro de Astrobiología (CSIC-INTA), Ctra. de Ajalvir, km 4,
Torrejón de Ardoz, 28850 Madrid, Spain
44institutetext: Max Planck Institute for Extraterrestrial Physics, Postfach
1312, 85741 Garching, Germany
55institutetext: Observatorio Astronómico Nacional (OAN, IGN), Apdo 112, 28803
Alcalá de Henares, Spain

# Photoprocessing of H2S on dust grains: building S chains in translucent
clouds and comets

S. Cazaux 1122 H. Carrascosa 33 G. M. Muñoz Caro 33 P. Caselli 44 A. Fuente 55
D. Navarro-Almaida 55 P. Riviére-Marichalar 55

(Received September 15, 1996; accepted March 16, 1997)

###### Abstract

Context. Sulfur is a biogenic element used as a tracer of the evolution from
interstellar clouds to stellar systems. However, most of the expected sulfur
in molecular clouds remains undetected. Sulfur disappears from the gas phase
in two steps. One first depletion occurs during the translucent phase,
reducing the gas phase sulfur by 7-40 times, while the following freeze-out
step occurs in molecular clouds, reducing it by another order of magnitude.
This long-standing dilemma awaits an explanation.

Aims. The aim of this study is to understand under which form the missing
sulfur is hiding in molecular clouds. The possibility that sulfur is depleted
onto dust grains is considered.

Methods. Experimental simulations mimicking H2S ice UV-photoprocessing in
molecular clouds were conducted at 8 K under ultra-high vacuum. The ice was
subsequently warmed up to room temperature. The ice was monitored using
infrared spectroscopy and the desorbing molecules were measured by quadrupole
mass spectrometry in the gas phase. Theoretical Monte Carlo simulations were
performed for interpretation of the experimental results and extrapolation to
the astrophysical and planetary conditions.

Results. H2S2 formation was observed during irradiation at 8 K. Molecules H2Sx
with $x$ ¿ 2 were also identified and found to desorb during warm-up, along
with S2 to S4 species. Larger Sx molecules up to S8 are refractory at room
temperature and remained on the substrate forming a residue. Monte Carlo
simulations were able to reproduce the molecules desorbing during warming up,
and found that residues are chains or sulfur consisting of 6-7 atoms.

Conclusions. Based on the interpretation of the experimental results using our
theoretical model, it is proposed that S+ in translucent clouds contributes
notoriously to S depletion in denser regions by forming long S-chains on dust
grains in few times 104 years. We suggest that the S2 to S4 molecules observed
in comets are not produced by fragmentation of these large chains. Instead,
they probably come either from UV-photoprocessing of H2S-bearing ice produced
in molecular clouds or from short S chains formed during the translucent cloud
phase.

###### Key Words.:

Astrochemistry – ISM: molecules – Solid state: volatile – Solid state:
refractory – Methods: laboratory: solid state – Methods: numerical

##  1 Introduction

Sulfur is an important element for life as we know it and its chemistry is
particularly relevant for linking interstellar clouds to star forming regions,
to protoplanetary disks and stellar systems such as our own. S-bearing
molecules are frequently used to trace the kinematics and the chemical
evolution of star- (e.g. Zhou et al., 1993; Wakelam et al., 2004) and planet-
forming regions (e.g. Dutrey et al., 1997; Le Gal et al., 2019). Several
S-bearing species have also been recently detected in the coma of comet 67P
(Calmonte et al., 2016). There is however a big problem related to sulfur
chemistry, which was first recognized in the early 70s: it is not yet clear in
which form most of the sulfur resides in molecular clouds. In fact, while
cosmic abundances of (ionized) sulfur are found in diffuse clouds (Jenkins,
2009), the abundance of the most abundant S-bearing molecule in molecular
clouds, CS, cannot be reproduced by astrochemical models, unless the S
abundance in the gas phase is reduced by several orders of magnitude (e.g.
Penzias et al., 1971; Oppenheimer & Dalgarno, 1974; Hasegawa et al., 1992;
Bulut et al., 2021).

Several attempts have been done in the past to shed light on the ”missing
sulfur” problem in molecular clouds. Caselli et al. (1994) considered the
possibility that most of the S in molecular clouds is residing on the surface
of dust grains; if this is the case, the rapid formation of H2S, and the
production of H2 from the H2S+H reaction in the ice, create a sink of H atoms,
which could also explain the very slow conversion of CO ice in CH3OH, which is
indeed only observed in dense cores of molecular clouds and star-forming
regions (e.g. Boogert et al., 2015; Goto et al., 2020). However, only upper
limits have been measured for H2S in ice (e.g. Smith, 1991; van der Tak et
al., 2003; Jiménez-Escobar & Muñoz Caro, 2011), thus excluding that the
majority of solid S is in H2S form. Ruffle et al. (1999) proposed that most of
the S depletion happens in translucent clouds, where S atoms are still mainly
in atomic ionized form (see also Sternberg & Dalgarno, 1995) while the dust
grains are mainly negatively charged; in this scenario, the Coulomb-enhanced
freeze-out rate of S+ will rapidly deplete sulfur from the gas phase (see also
Umebayashi & Nakano, 1980, for generic cations). Vidal et al. (2017) compared
their astrochemical modeling results to observations toward the TMC-1(CP) dark
cloud and concluded that, depending on the cloud age, most of the sulfur could
be either in (the not observable) atomic form or on HS and H2S in solid form.
Laas & Caselli (2019) expanded the chemistry of sulfur by including S-bearing
organic molecules and the Ruffle et al. (1999) freeze-out mechanism; they
found a good agreement with observations in molecular clouds and concluded
that the majority of S is trapped in organo-sulfur species in icy mantles of
dust grains.

In laboratory experiments where H2S ice is irradiated either with UV photons
or energetic particles (proxy for the interstellar cosmic rays), it has been
found that a significant fraction of H2S can be transformed in allotropic
forms of S, including the most stable form S8 (e.g. Jiménez-Escobar & Muñoz
Caro, 2011; Jiménez-Escobar et al., 2014). This has also been confirmed
theoretically by Shingledecker et al. (2020) using astrochemical models
inclusive of radiation chemistry; interestingly, these models predict that the
impact of energetic particles in H2S-rich ices reduces the abundance of H2S,
while enhancing SO2, OCS and S8. The first interstellar molecule containing
more than one S atom (S2H) was detected recently by Fuente et al. (2017)
toward the Horsehead photodissociation region, suggesting an interesting
interplay between gas-phase and surface chemistry. The GEMS large project
(Rodríguez-Baras et al., 2021) at the IRAM 30m telescope is also providing
important observational constraints on sulfur chemistry in molecular clouds;
for example Navarro-Almaida et al. (2020) measured gas-phase H2S across
several sources, finding good agreement with chemical models assuming non
depleted S abundances, which however highly overestimate the CS abundance.
With the ROSINA instrument onboard Rosetta, Calmonte et al. (2016) measured a
variety of S-bearing molecules in comet 67P, which appears to preserve pre-
stellar ice (Altwegg et al., 2019; Drozdovskaya et al., 2019, 2021): the
majority of S is in H2S ice (about 57%), followed by atomic S, SO2, SO, OCS,
H2CS, CH3SH, CS2, S2 and other S-bearing organics.

Despite the many observations and chemical models available, there are still
many open questions about sulfur chemistry in the interstellar medium: what
happens to S atoms when adsorbed onto dust grains in translucent and molecular
clouds? What is the chemistry of irradiated H2S ice? Which fraction of sulfur
atoms will form chains? Which S-bearing molecules should be observed to test
the models? In this paper we focus our attention on detailed experimental work
and its simulation with a Monte Carlo chemical model, with the aim of putting
quantitative constraints on the S depletion and chemical processing in
translucent and molecular clouds.

The paper is structured as follows: Section 2 describes the laboratory
experiments and Section 3 presents the results of the experimental work; Monte
Carlo simulations are described in Section 4. Section 5 discusses photo-
processes of H2S ice in simulations and experimental work, while applications
to interstellar molecular clouds and to Solar System conditions can be found
in Sections 6 and 7, respectively. Our conclusions are in Section 8.

##  2 Experiments: H2S irradiation with UV photons

The experimental results presented here have been obtained using the
InterStellar Astrochemistry Chamber (ISAC) at the Centro de Astrobiología
(Muñoz Caro et al., 2010). ISAC is an ultra-high vacuum (UHV) chamber with a
base pressure dominated by background H2 in the range $P$ = 2.5-4.0 x 10-11
mbar, which corresponds to dense cloud interiors. H2S ice samples were grown
by accretion of gas molecules onto the tip of a cold finger at 8 K, where a
MgF2 infrared (IR) and ultraviolet (UV) transparent window serves as the
substrate for deposition. This temperature was achieved by means of a closed-
cycle helium cryostat. The samples were then warmed up until a complete
sublimation was attained. A silicon diode temperature sensor, and a LakeShore
Model 331 temperature controller are used, reaching an accuracy of about 0.1
K.

The evolution of the solid sample was monitored by in situ transmittance
Fourier Transform Infrared spectroscopy (FTIR) with a spectral resolution of 2
cm-1. Column densities of each species in the ice were calculated from the IR
spectra using the formula

$N=\frac{1}{A}\int_{band}{\tau_{\nu}d\nu},$ (1)

where $N$ is the column density in cm-2, ${\tau}_{\nu}$ the optical depth of
the band, $d\nu$ the wavenumber differential in cm-1, and $A$ the band
strength in cm molecule-1. The integrated absorbance is equal to 0.43 $\times$
$\tau$, where $\tau$ is the integrated optical depth of the band. The adopted
band strengths, $A$, derived from laboratory experiments are provided in Table
1.

Species | Feature | Position | Position |  $A$$\times$ 10-17
---|---|---|---|---
|  |  $\mu$m | cm-1 | cm molec-1
H2S | S-H stretch | 3.940 | 2538 | 2.0 a
H2S2 | S-H stretch | 4.016 | 2490 | 2.4$\pm$0.5 b

a Jiménez-Escobar & Muñoz Caro (2011)
b This work. Band strength of H2S2 was calculated between 100 K and 120 K,
showing a constant value in this range, see Sect. 3.2.

Table 1: Photo-dissociation reactions with associated rates considered in the
present study. These rates are taken from Shingledecker et al. (2020).

The accretion rate measured by IR spectroscopy of the deposited ice was 1 ML
s-1. Samples were UV-irradiated using a microwave-stimulated hydrogen flow
discharge lamp (MDHL) that provides a flux of 2.5 $\times$ 1014 photons cm-2
s-1 at the sample position with an average photon energy of 8.6 eV. The MDHL
spectrum is reported in Cruz-Diaz et al. (2014). UV spectra were measured
routinely in situ during the experiments with the use of a McPherson 0.2 meter
focal length VUV monochromator (Model 234/302) with a photomultiplier tube
(PMT) detector equipped with a sodium salicylate window, optimized to operate
from 100-500 nm (11.27-2.47 eV), with a spectral resolution of 0.4 nm. UV
absorption spectra of H2S ice samples served to monitor the detection of
photoproducts generated upon UV-irradiation at 8 K or later during warm-up.
For more details on the experimental protocol employed for VUV spectroscopy,
see Cruz-Diaz et al. (2014).

##  3 Experimental results

As reported by Jiménez-Escobar & Muñoz Caro (2011), thermal processing of H2S
ice leads to crystallization at temperatures above 40 K. This process reduces
the number of randomly oriented molecules and the vibration modes become
narrower, see inlet of Fig. 1, which agrees with Fig. 1 in Jiménez-Escobar &
Muñoz Caro (2011). The antisymmetric and symmetric vibrations appear at 2553
and 2530 cm-1, respectively. The remaining amorphous domains are responsible
for the IR band centered at 2538 cm-1.

###  3.1 Photoproduct formation in UV-irradiated H2S ice

Figure 1: Evolution of the H2S band strength as a function of temperature.
Above 70 K, thermal desorption becomes important, reducing H2S column density,
preventing us from a quantification of the band strength.

UV-irradiation of H2S ice samples produces HS$\cdot$ radicals very readily,
according to Reaction 2. HS$\cdot$ radicals can also react with hydrogen atoms
in the backwards reaction to reform the original H2S molecule. IR spectroscopy
shows that H2S is dissociated along the first steps of UV irradiation, when
recombination to form H2S is inhibited, as a consequence of the low abundance
of HS$\cdot$ radicals. For larger irradiation periods, however, H2S
recombination takes place, reducing the overall H2S dissociation.

$\displaystyle\rm{H}_{2}\rm{S}\thinspace\rightleftharpoons\thinspace\rm{HS\cdot}\thinspace+\thinspace\rm{H\cdot}$
(2)

The rapid formation of HS$\cdot$ radicals favors Reaction 3 to produce H2S2
molecules as well as a second hydrogen elimination (Reaction 4) forming S
atoms. H2S2 molecules were detected at 8 K by IR spectroscopy through its H-S
stretching mode centered at 2485 cm-1 (Isoniemi et al., 1999). On the other
hand, S atoms can only be detected during the warm-up phase (see Sect. 3.2).

$\displaystyle
2\thinspace\rm{HS\cdot}\thinspace\xrightarrow{}\thinspace\rm{H_{2}S_{2}}$ (3)
$\displaystyle\rm{HS\cdot}\thinspace\xrightarrow{}\thinspace\rm{S}\thinspace+\thinspace\rm{H\cdot}$
(4)

The rapid formation of H2S2 enables further reactions. H2S2 molecules can be
dissociated by UV-photons according to Reaction 5. H2S2 molecules can also
recombine with HS$\cdot$ radicals to produce H2S3 molecules (Reaction 6).

$\displaystyle\rm{H_{2}S_{2}}\thinspace\xrightarrow{}\thinspace\rm{H_{2}S\cdot}\thinspace+\thinspace\rm{H\cdot}$
(5)
$\displaystyle\rm{H_{2}S_{2}}\thinspace+\thinspace\rm{HS\cdot}\thinspace\xrightarrow{}\thinspace\rm{H_{2}S_{3}}$
(6)

Finally, H2S4 was also detected in our experiments. The presence of
HS${}_{3}\cdot$ radicals will produce H2S4 by reaction with the abundant
HS$\cdot$ radical (Reaction 7). Secondly, 2 HS${}_{2}\cdot$ radicals can react
producing H2S4 molecules (Reaction 8). The relatively low abundance of
HS${}_{3}\cdot$ radicals, and the low probability of 2 HS${}_{2}\cdot$
radical-radical reactions in an environment dominated by HS$\cdot$ radicals
determine the low formation rate of H2S4 molecules. Nevertheless, H2S4 was
detected during warm-up of the irradiated H2S ice by quadrupole mass
spectrometry (QMS), as explained in Sect. 3.2.

$\displaystyle\rm{HS\cdot}\thinspace+\thinspace\rm{HS_{3}\cdot}\thinspace\xrightarrow{}\thinspace\rm{H_{2}S_{4}}$
(7) $\displaystyle
2\thinspace\rm{HS_{2}\cdot}\thinspace\xrightarrow{}\thinspace\rm{H_{2}S_{4}}\thinspace$
(8)

Non-hydrogenated molecules can also be formed. Reaction between S atoms
produces S2 molecules. Further S atom additions, as well as dehydrogenation of
H2Sx species can lead to larger Sx molecules. Some Sx species do not absorb in
the IR and others display only very weak absorption bands. These products
could not be detected by IR spectroscopy in our experiments, but they were
measured by QMS during warm-up.

###  3.2 Warm-up of UV-irradiated H2S ice

Some of the radical species made during UV irradiation of H2S ice react at low
temperatures, such as HS$\cdot$ radicals, while larger radical species may
remain in the ice matrix until thermal energy during warm-up enables
subsequent reactions. IR spectroscopy confirmed the production of H2S2
molecules at 8 K (Isoniemi et al., 1999). Overlapping of H-S IR stretching
modes of H2S2 with those of larger H2Sx species, expected to present lower
formation rates, hindered the estimation of their formation temperatures.

Figure 2: IR spectra collected during warm-up of a previously UV-irradiated
H2S ice sample. After sublimation near 90 K of most H2S molecules, H2S2
molecules are allowed to interact forming a crystalline structure.
Crystallization leads to narrowing of the antisymmetric and symmetric
vibrations of H2S2 that appear at 2495 and 2482 cm-1, respectively.

After the sublimation of amorphous and crystalline H2S molecules at 96 K, IR
spectra show the crystallization of H2S2 (Fig. 2). From the IR band centered
around 2490 cm-1, two absorption features can be clearly distinguished at
temperatures between 110 K and 150 K. These bands belong to the antisymmetric
(2495 cm-1) and the symmetric (2482 cm-1) vibration modes of H2S2. Zengin &
Giguère (1959) measured the IR spectrum of crystalline H2S2, reporting values
in close agreement with our experiments and the work reported by Moore et al.
(2007).

The band strength of the H2S2 IR band at 2490 cm-1 was estimated as follows.
Martín-Doménech et al. (2015) provided a method to calculate the column
density of a given species from the integrated QMS signal, after calibrating
the QMS by using CO ice, by applying Eq. 9, where $N(mol)$ is the column
density of a given species in molec $\cdot$ cm-2, $A\left(\frac{m}{z}\right)$
is the integrated area taken from the QMS, $k_{CO}$ is the proportionality
constant from the calibration of the QMS in a CO ice irradiation experiment
leading to photodesorption of the CO molecules, $\sigma^{+}$(mol) is the
ionization cross-section of each species ionized at a voltage of 70 eV in the
QMS (data adopted from the National Institute of Standards and Technology),
$IF(z)$ is the ionization factor, which has been considered unity for all
molecules, $FF$ is the fragmentation factor, derived from the QMS spectrum of
each species, and $S\left(\frac{m}{z}\right)$ is the sensitivity of the QMS
(Martín-Doménech et al., 2015).

$N(mol)=\frac{A\left(\frac{m}{z}\right)}{k_{CO}}\cdot\frac{\sigma^{+}(CO)}{\sigma^{+}(mol)}\cdot\frac{IF(CO^{+})}{IF(z)}\cdot\frac{FF(28)}{FF(m)}\cdot\frac{S(28)}{S(\frac{m}{z})}\\\
$ (9)

Then, the ratio between column density of H2S and H2S2
(R${}_{\frac{H_{2}S}{H_{2}S_{2}}}$) can be obtained from the QMS during
thermal desorption of H2S and H2S2, using Eq. 10, and taking the area below
the thermal desorption peak in the QMS:

$R_{\frac{H_{2}S}{H_{2}S_{2}}}=\frac{N(H_{2}S)}{N(H_{2}S_{2})}=\frac{A(34)}{A(66)}\cdot\frac{\sigma^{+}(H_{2}S_{2})}{\sigma^{+}(H_{2}S)}\cdot\frac{IF(H_{2}S_{2}^{+})}{IF(H_{2}S^{+})}\cdot\frac{FF(66)}{FF(34)}\cdot\frac{S(66)}{S(34)}\\\
$ (10)

The band strength of the 2490 cm-1 IR feature of H2S2 can be finally obtained
assuming that $\sigma^{+}(H_{2}S_{2})$ = $\sigma^{+}(H_{2}S)$, as there is no
reported value for $\sigma^{+}(H_{2}S_{2})$, and calculating the column
density of H2S from IR data before its thermal desorption:

$N(H_{2}S_{2})=\frac{1}{R_{\frac{H_{2}S}{H_{2}S_{2}}}}\cdot N(H_{2}S)$ (11)

Because the factor $k_{CO}$ cancels out and does not appear in Eq. 10, this
method reduces the error due to i) variation in the QMS sensitivity between
the CO photodesorption experiment and the H2S irradiation experiment due to
degradation of the QMS filament with time and ii) different environmental
conditions between those experiments, since the measurement of the reference
species, H2S, and that of the target molecule, H2S2, were made during the same
experiment. Entering the estimated values of $N$(H2S2) and its corresponding
integrated IR absorption in Eq. 1, an IR band strength value of $A$(H2S2) =
2.4$\pm$0.5 $\times$ 10-17 cm molecule-1 was obtained.

Figure 3: IR spectra acquired during warm-up of previously irradiated H2S ice.
This band is evidence for the presence of H2Sx molecules with $x$ ¿ 2 that
were detected during desorption in the gas phase by QMS, H2S3 and H2S4
desorbing at temperatures below 225 K, see Fig. 4. Therefore, the small IR
absorption that remains above 240 K is likely due to H2Sx species with $x$ ¿ 4
that are refractory at this temperature and could thus not be detected by the
QMS.

After thermal desorption of H2S2, a new IR feature becomes visible at 2481
cm-1 (Fig. 3). This IR band disappears gradually above 150 K during the TPD,
suggesting that H-S stretching modes from species less volatile than H2S2
contribute to this vibration mode.

Fig. 4 shows the thermal desorption of H2Sx molecules. Two peaks are clearly
observed during H2S thermal desorption. A first desorption peak appearing at
88 K coincides with the disappearance of the 2538 cm-1 IR feature between 80 K
and 90 K in Fig. 1, that is, thermal desorption of amorphous H2S molecules.
More stable crystalline H2S molecules sublimate at slightly larger
temperatures, producing a second peak at 96 K, see Fig. 4.

Furthermore, QMS data in Fig. 4 shows evidence for H2S2 desorption,
$\frac{m}{z}=66$, with a maximum at 144 K, which coincides with the
disappearance of the IR features at 2495 and 2482 cm-1 attributed to this
species.

The better sensitivity of QMS compared to IR spectroscopy allowed the
identification of larger species during ice warm-up. Fig. 4 shows a maximum of
$\frac{m}{z}=98$ at 184 K related to desorbing H2S3 molecules, and the peak of
$\frac{m}{z}=130$ at 204 K corresponds to thermally desorbed H2S4. The latter
identification of H2S4 was confirmed in several irradiation and warm-up
experiments.

Figure 4: Thermal programmed desorption (TPD) of an irradiated H2S ice sample.
Up to 4 different H2Sx species were detected with decreasing abundances. Most
intense $\frac{m}{z}$ fragment for all the species is shown.

The fragmentation of desorbing H2Sx species when they impact the filament of
the QMS can lead to S${}_{x}^{+}$ fragments that might be erroneously
attributed to the desorption of Sx species formed in the ice. Nevertheless,
this was not a problem since the m/z values of Sx molecules were detected at
temperatures that did not overlap significantly with those of H2Sx species.
Fig. 5 shows the thermal desorption of Sx species. Thermal desorption of S
atoms reaches its maximum at 58 K, dragging some S2 molecules during this
process. The main desorption of S2 molecules takes place at 113 K. If we
assume that the molecular mass is the main factor determining the desorption
temperature, S3 molecules are expected to desorb around 180-200 K, but they
were not detected in our experiments. The non-detection of S3 and the
detection of S4, with a maximum of desorption at 283 K, suggests that a
preferential formation scheme encompass the dimerization of S2 molecules,
favoring the formation of Sx species with a pair number of S atoms.

Figure 5: TPD of an irradiated H2S ice sample. It should be noted that a given
m/z value can correspond to the molecular ion of a species or to a fragment of
a larger species. The desorption temperatures of Sx molecules correspond to
the solid vertical lines. Dashed lines indicate the desorption of H2Sx
species. Apart from the molecular ion of each species, $\frac{m}{z}$ values of
the main fragments used to confirm the presence of these species are
displayed.

##  4 Monte Carlo simulations

We used a step-by-step Monte Carlo simulation to follow the deposition of H2S
molecules on the surface as well as their photo-dissociation into other
products. Our model is described in Cazaux et al. (2015, 2017). H2S molecules
originating from the gas phase arrive at a random time and location on the
substrate, which is represented as a grid, and follow a random walk. The
arrival time depends on the rate at which gas species collide with the
surface. The molecules arriving on the surface can be bound to the substrate
and to other H2S, HS or S species via van der Waals interactions. In the
present study we use on-lattice KMC simulations. Other types of simulations,
such as off-lattice KMC method allow to determine the distance of the species
explicitly (Garrod, 2013). In the present study, we consider the distance
between two adsorption sites (and therefore the distance between two adjacent
adsorbed species) to be equal for the entire grid, and concentrate on defining
the reactions occurring within the chemical species that are deposited on the
surface (H2S) or are produced by photo-dissociation reactions. Our model is
meant to constrain the experimental results as accurately as possible, before
being exported to interstellar medium conditions in sections 6 and 7.
The different mechanisms used in this model are accretion, diffusion,
sublimation, chemical reactions and photo-dissociation. These different
mechanisms (accretion, diffusion and sublimation) occur at rates which have
been described in a previous work (Cazaux et al., 2018). The accretion rate in
number of molecules per second is

$R_{\rm{H_{2}S}}=n_{\rm{H_{2}S}}\rm{V}_{\rm{H_{2}S}}\ \sigma\ \rm{Stick},$
(12)

where V${}_{\rm{H_{2}S}}=\sqrt{\frac{8\ k\ T_{\rm{gas}}}{\pi\
m_{\rm{H_{2}S}}}}\sim 2.5\times 10^{4}\sqrt{\frac{T_{\rm{gas}}}{100}}$ cm s-1
is the thermal velocity of H2S, and $\sigma$, the cross-section of the
surface. Stick is the sticking coefficient that we consider to be unity in
this study. The distance between two sites is assumed to be 3 Å, which is
consistent with the density of the number of sites typically assumed on
surfaces NS = $({3\AA})^{-2}$ $\sim$ 1015 cm-2. The cross-section scales with
the size of the grid considered in our calculations (which is 40$\times$40
sites) as $\sigma\sim(3\ 10^{-8}\times 40)^{2}$ cm2= 1.4 10-12 cm2. The
accretion rate of H2S molecules in the experiment is 1ML/s. In our
calculations, the accretion rate in ML/s can be written as
R${}_{\rm{H_{2}S}}=\frac{n_{\rm{H_{2}S}}\rm{V}_{\rm{H_{2}S}}}{NS}$ ML/s.
Therefore, we chose the density of H2S in our simulations to be around
$n_{\rm{H_{2}S}}\sim 10^{10}$ cm-3 in order to reproduce experimental
conditions.

The diffusion from one site to another one is described in previous work
(Cazaux et al., 2017). In the present study, we consider an alpha coefficient,
e.g. the ratio between diffusion barriers and thermal desorption barriers, of
0.9. Such slow diffusion was determined in previous work (Cazaux et al., 2017)
and is considered in the present study.

In the present work, we also consider photo-dissociation reactions as well as
the reactions between the photo-products. These mechanisms are described in
the following subsections.

###  4.1 Photo-dissociation reactions

Once a photon is absorbed by a H2S molecule, it will dissociate it in HS + H.
The product of the reaction, HS, can also receive a photon and be dissociated
further in S + H. For our model, the dissociation rates are taken from
Shingledecker et al. (2020) and are reported in Table 2. As the photo-
dissociation of H2S can produce S atoms, these S atoms can find each other and
make chains, which increases their binding energies and lowers their photo-
dissociation rates as shown in Table 2. The rates are computed as follow:

$R_{diss}=G_{0}*A\exp(-B\times Av).$ (13)

In the experiments, each H2S molecule receives a photon every 10 seconds. The
parameter G0 is the Habing radiation field (Habing, 1968) which is the far
ultraviolet (FUV) radiation field (where 1 G0 equals a flux of 1.6 $\times$
10-3 erg cm-2 s-1, which is equivalent to 2$\times$ 107 cm-2 s-1). In order to
have 0.1 photon per second, to mimic experimental conditions, we use
G0=5$\times$ 106. The dissociation rate of H2S molecules is therefore Rdiss=
0.015 s-1 for H2S and is of 0.006 s-1 for HS.

React1 | React2 | Prod1 | Prod2 | A | B
---|---|---|---|---|---
HS | Photon | H | S | 0.12E-08 | 2.4
H2S | Photon | H | HS | 0.31E-08 | 2.6
H2S2 | Photon | H | HS | 0.55E-09 | 1.7
H2S2 | Photon | H | HS | 0.45E-09 | 1.7
S2 | Photon | S | S | 0.6E-09 | 1.9
S3 | Photon | S2 | S | 0.1E-09 | 0.0
S4 | Photon | S3 | S | 0.8E-10 | 0.0
S5 | Photon | S4 | S | 0.5E-10 | 0.0
S6 | Photon | S5 | S | 0.1E-10 | 0.0
Table 2: Photo-dissociation reactions with associated rates considered in the
present study. These rates are taken from Shingledecker et al. (2020).

###  4.2 Chemical reactions

In the present simulations, we consider that species present in the solid
phase (on the surface or in the ice) can react with each other. We consider a
small solid phase chemical network allowing 5 possible reactions, and report
the associated pre-exponential factors and barriers in table 3. These
reactions have been selected in the chemical network from Shingledecker et al.
(2019). In the present study we do not consider chemical desorption.

React1 | React2 | Prod1 | Prod2 |  $\nu$ (s-1) | Ea(K)
---|---|---|---|---|---
H | H | H2 |  | 1E12 | 0
H | S | HS |  | 1E12 | 0
H | HS | H2S |  | 1E12 | 0
H | H2S | H2 | HS | 1E12 | 1530
H | HS | H2 | S | 1E12 | 0
Table 3: Reactions considered in the present study. Reactions are taken from
Shingledecker et al. (2020).

###  4.3 Binding energies

The binding energies of S atoms, as well as H, H2, HS, H2S and S-chains are
reported in Table 4. The binding energies for the S-chains are extrapolated by
using the S-chains observed in the experiments. S4 desorbs at 283 K (which
implies a binding energy of $\sim$8490 K) while S3 desorbs around 200 K
(binding energy of $\sim$6000 K). These binding energies are reported in Table
4. This means that the difference in desorption peak in the TPD when a S is
added is around 80 K, which corresponds to a additional binding energy due to
an additional S atom to the chain around 2400 K. We take this value as an
addition to the binding energy of the S-chain when an extra S atom is added.
We therefore reach a binding energy of 18000 K for S8. It should be noted that
this is a simple assumption, and the change in binding energy could become
less with the number of S atoms added, as this is the case in clusters (Lin et
al., 2005).

Species | Binding | Reference
---|---|---
| energy |
| (K) |
H | 500 | Dulieu et al. 2005
H2 | 400 | Dulieu et al. 2005
H2S | 2640 | Jiménez-Escobar & Muñoz Caro 2011
HS | 1100 | Wakelam et al. 2017
S | 1800 | Wakelam et al. 2017
HS2 | 4264 | Jiménez-Escobar & Muñoz Caro 2011
| 5500 | Jiménez-Escobar & Muñoz Caro 2011
S2 | 3390 | this work
S3 | 6000 | this work
S4 | 8490 | this work
S5 | 10800 | extrapolation
S6 | 13200 |
S7 | 15600 |
S8 | 18000 |
Table 4: Species considered in this study with their associated binding
energies. Note that 2 values of HS2 correspond to the 2 desorption peaks in
the TPD implying different orientation of the molecules.

###  4.4 Results

####  4.4.1 TPD simulations of H2S

We performed Monte Carlo simulations for the deposition and irradiation of H2S
in order to reproduce the experimental measurements. We first performed
simulations without irradiation in order to reproduce a simple experimental
TPD curve. Our results are reported in Figure 6. In this figure, we are able
to reproduce the two peaks located around 88 K and 96 K as found in Fig. 4. To
reproduce these two peaks, we considered binding energies of species in Table
4. In our calculations, the strongest interaction between a species with a
neighbor is considered. However, the width of the peaks seen in the
experiments are of few tens of K. This implies that the binding energy is not
a single value but a distribution of values. To reproduce the width of the
peak, we consider the strongest interaction of the species (from Table 4) and
add the contribution of the neighbors around the considered species for 2%
(this value is taken as it reproduces the width the best). For example, an H2S
molecule bound with 2 neighbors would have a binding energy of 2640 + 0.02
$\times$ 2640 = 2693 K. This allows to take into account the fact that the
binding energy is increased if more neighbors are involved.

Figure 6: TPD of H2S with MC simulations.

####  4.4.2 TPD simulations of irradiated H2S: case without ice self shielding

In these simulations, we deposit H2S on the surface and irradiate the
molecules with UV photons in rates similar to experimental rates. The
vizualisation of our Monte Carlo simulations for different surface temperature
is presented in Figure 7. In this figure, we show the grid of 40 by 40 sites.
Each cube shows one species, the color allowing to identify the species. The
H2S molecules are presented in yellow, the HS in green and S atoms in blue.
The first panel represents the H2S after deposition, while the second, third,
fourth and fifth panels show the coverage of the surface during the TPD at 50,
100, 200 and 300 K respectively. Our results show that after deposition, the
H2S layers have not interacted much with the photons since most of the ice is
yellow (and traces of green and blue). The deposition in our simulation takes
around 8 seconds, and the molecules receive a photon every 10 seconds. This
means that the number of products due to photo-dissociation are less important
during the deposition that during the TPD (in our simulations). This is
reflected by the composition of the surface at 8 K, showing that most of H2S
molecules are still present on the surface with a minority of HS and S
species. The surface at 50 K, shown in the top middle panel of Figure 7,
illustrates a longer irradiation time, since the time spent after deposition
is 42 minutes. The photo-dissociation rate of H2S is Rdiss = 0.015 s-1 in our
simulations, which implies that one H2S molecule would not survive such a long
time since in 42 minutes it would have received around 40 photons, and the
first photon would have already dissociated the molecule in HS + S. The
surface coverage at 50 K is therefore mainly composed of S atoms but also HS
and few H2S. The photo-products HS and S are dominating compared to the
coverage at 8 K after deposition. In Figure 7, top right panel, the surface
temperature is of 100 K. Most of the H2S and HS have desorbed from the
surface, and mostly S atoms are present. The species are organized as chains,
which allow them to have larger binding energies compared to individual
species. These binding energies are reported in Table 4. For high
temperatures, only chains can still be present on the surface. The bottom left
panel of Figure 7 shows the surface at 150 K. Only chains with S atoms and
sometimes an inclusion of HS or H2S can be observed. At 300 K, on the bottom
middle panel, only few chains are remaining on the surface and all other
species have desorbed. These residues in our simulations under the form of
chains cannot be seen experimentally in the TPD because their binding energies
are too high. In our simulations, the longest chain has 7 S atoms, and few
chains have 6 S atoms. The bottom right panel shows the same simulation and
temperature than the bottom middle panel, but in this case the chains are
highlighted by different colors while in the previous figures, the colors were
showing which species were composing the chain (S, HS or H2S). We can clearly
see that some chains can reach a maximum of around 6-7 S atoms in the present
conditions.

In order to compare our results to the experimental measurements from Fig. 2,
we present in Figure 8 the simulated number of desorbed species during the
TPD. Our results show different peaks that are attributed to S atoms, S chains
as well as HS, H2S and H2S2. This TPD agrees very well with the experimental
results, showing chains with up to 4 members. In the present simulations, we
used a coverage of few 3–5 ML (see top left panel of Figure 7). This differs
from the experimental conditions where thousands of monolayers were deposited
on the surface. The ratios between the different peaks of the TPD would change
significantly if the simulations were run for high H2S ice thickness. Our
model however, cannot compute such thick ices, but in the present study we can
compare the temperature of the desorption peaks in the TPD and estimate the
residuals remaining in the surface at high temperatures.

Figure 7: H2S on the surface after deposition at 8 K (top left) and during TPD
at 50 K (top middle), at 100 K, (top right), at 200 K (bottom left) and 300 K
(bottom middle). The bottom right panel also shows the surface at 300 K but
with the chains being highlighted by using one color per chain. In these
simulations, the molecules in the ice do not shield each other and photons can
therefore dissociate molecules from the top layers to the bottom layers. The
deposition rate of H2S is 1ML/s, the heating ramp is 1K/min and G0=5 $\times$
106. Figure 8: TPD from Monte Carlo simulations of H2S. In these simulations,
shielding of molecules within the ice is not considered.The deposition rate of
H2S is 1ML/s, the heating ramp is 1K/min and G0=5$\times$ 106.

####  4.4.3 TPD simulations of irradiated H2S: case with self shielding

We perform similar simulations than in the previous section but this time,
considering that molecules shield themselves from the UV radiation. For
simplicity, we consider that if a species is under another one, then it is
shielded from radiation and will not receive photons. It is then expected that
the layers under the surface will remain H2S as long as they are covered by
other species, and will then be exposed only when the layers above sublimate.
The results are presented in Figure 9. The surface coverage is shown from
temperatures of 8 K (top left panel), 50 K (top middle panel), 100 K (top
right panel) and 150 K (bottom left panel). These figures clearly show that
there are much fewer chains on the surface than when shielded is not
considered, as in Figure 8, and that instead species such as H2S2 (2 green
boxes attached together) or H2S3 (1 blue box surrounded by 2 green boxes), are
formed on the surface. However, no chains longer than S3 are created, which is
shown in Figure 9 bottom left panel, when chains are highlighted in different
colors (for a surface temperature of 150 K). Therefore, there is no species on
the surface after $\sim$ 180 K as the binding energies of these chains are too
low. This is reflected by the TPD in Figure 10. This figure shows that
S-chains are present, but only containing 2 to 3 sulfur atoms. On the other
hand, many hydrogenated sulfur species are present, desorbing at temperatures
lower than 180 K.

Figure 9: H2S on the surface after deposition at 8 K (top left) and during TPD
at 50 K (top middle), at 100 K, (top right), at 150 K (bottom left).The bottom
right panel also shows the surface at 150 K but with the chains being
highlighted by using one color per chain.In these simulations, the molecules
in the ice do shield each other and photons cannot dissociate molecules if
they are under another molecule. The deposition rate of H2S is 1ML/s, the
heating ramp is 1K/min and G0=5$\times$ 106.  Figure 10: TPD from Monte Carlo
simulations of H2S. In these simulations, the molecules in the ice can shield
each other. The deposition rate of H2S is 1ML/s, the heating ramp is 1K/min
and G0=5$\times$ 106.

##  5 Photo-processes in H2S ice

In our simulations, we considered at first that icy molecules are not
shielding each other from photons. As a result, we obtained a TPD dominated by
atomic sulfur as well as S-chains, showing the presence of S-chains up to S4
and a small amount of hydrogenated sulfur species. Our results show that
residues are present on the surface at the end of the TPD, and that these
residues are under the form of chains containing up to 6-7 sulfur atoms. Such
TPD is reported in figure 7. We also considered in a second step that icy
species could shield each other from UV photons, and obtained a different TPD,
dominated by hydrogenated sulfur species, as shown in Figure 9. In this case,
the species on the surface desorb below 200 K and no residue is found after
the TPD.
The experimental results reported in Figure 5 show that the irradiation of H2S
ice allows for the formation of hydrogenated sulfur species which are
dominating the TPD. The S-chains are also present and are desorbing until very
high temperatures. These results are reproduced by our simulations if icy
species can shield each other (hydrogenated sulfur species dominate the TPD)
but also if icy species do not shield each other (S-chains up to 4 S atoms in
the TPD, residue on the surface after the TPD). The experimental results shows
that residues are still present on the surface after the TPD. Therefore, we
can assume that H2S species have a small shielding properties so that
hydrogenated species can survive under the present experimental conditions,
and S-chains can be formed.
Note that we do not consider in this study the effect of photodesorption or
chemical desorption. Photodesorption will reduce the amount of solid H2S,
meaning that less H2S is available for photo-dissociation to form HS, HS2 and
H2S2 as well as chains. This implies that the total number of ice species
could differ. However, the fraction of H2S versus other molecules should
remain similar. Chemical desorption, on the other hand, would remove some
formed species and send them into the gas phase (especially via H+HS
$\rightarrow$ H2S Oba et al. (2018); Shingledecker et al. (2020)). This will
reduce the amount of H2S species compared to other chemical species in the
solid phase. Our present study aims at reproducing general experimental trends
in the TPDs, so neglecting these processes will only slightly affect the
abundances of the species in the ices which will not affect our conclusions.

##  6 Application to molecular clouds: sulfur depletion

Only $\sim$10 % of the currently detected molecules contain sulfur atoms. This
apparent lack of chemical diversity in sulfur-bearing species is somewhat
reflective of a great problem in astrochemistry: while the observed gaseous
sulfur accounts for its total cosmic abundance (S/H$\sim$1.5$\times$10-5) in
diffuse clouds, there is an unexpected paucity of sulfur bearing molecules
within molecular clouds. Indeed, the sum of the observed gas phase abundances
of S-bearing molecules (mainly SO, SO2, H2S, and CS) constitute only ¡1 % of
the cosmic abundance in cold and dense cores (n(H2) $>$ 104 cm-3) (see, e.g.,
Agúndez & Wakelam, 2013). One could think that most of the sulfur is locked on
the icy grain mantles in these regions but, surprisingly, the abundances of
S-bearing species in the ice as measured by near-IR observations could only
account for $<$ 5% of the total sulfur (Geballe et al., 1985; Palumbo et al.,
1995; Boogert et al., 1997; Jiménez-Escobar & Muñoz Caro, 2011). Fuente et al.
(2019) investigated the sulfur content in the translucent part of TMC 1 where
the freeze-out of molecules on the grain surfaces is expected to be
negligible. These observations revealed that gas-phase sulfur is already one
order of magnitude lower than the cosmic value in translucent gas, i.e., 90%
of S atoms incorporate to the solid grains in the transition between the
diffuse medium and the translucent cloud, corresponding to an abundance of S
in the gas $\sim$8 10-7 (Fuente et al., 2019). sulfur atoms impinging in
interstellar ice mantles are expected to form H2S preferentially. The
understanding of H2S chemistry is therefore linked to the sulfur depletion
problem. Chemical desorption of H2S upon formation from HS + H is significant
(Oba et al., 2018). Navarro-Almaida et al. (2020) investigated the H2S
abundance at the cloud edges and concluded that chemical desorption of H2S
from bare grains is needed to explain the high H2S abundance observed in the
translucent gas (n(H2) $<$ 104 cm-3) towards TMC 1 and Barnard 1b.
In the present study we concentrate on the fraction of S atoms locked onto
dust particles in the translucent phase, and therefore follow the fate of S
atoms impinging the dust under these conditions. We can then address whether
the high depletion of sulfur could be explained by our models and more
specifically, could be under the form of sulfur chains or HS or H2S. In
translucent clouds, sulfur is mainly under the form of S+ (Laas & Caselli,
2019) and in dense PDRs S+ can be present until extinction of $\sim$4 (for
PDRs with nH=106 cm-3 and $\chi$=2 105, see Sternberg & Dalgarno 1995). The
fact that sulfur is under cationic form influences its accretion rates, as
discussed in Ruffle et al. (1999). We therefore adopt a correction factor of
1+$\frac{167}{T}$ to account for the fact that the accretion rates are
increased due to the charges of the sulfur as mentioned in Ruffle et al.
(1999). We performed simulations for translucent cloud conditions taking into
account the fact that the accretion rate of S+ is higher due to the effect of
the charge. The conditions we used in our simulations are reported in Table 5
and are taken from Laas & Caselli (2019) and Snow & McCall (2006). In these
simulations, reported in the left panel of Figure 11, we show the coverage of
S compared to HS and H2S as a function of time in translucent cloud
conditions. The figure shows the increase of atomic sulfur under the form of S
chains compared to H2S and HS. The fact that S forms chains preferentially
compared to HS or H2S comes from the fact that in the gas phase, the ratio
between atomic hydrogen and sulfur is 20 (see conditions in table 5). However,
due to the fact that sulfur is under cationic form, its accretion rate is
increased by 1+167/T which implies an accretion rate 7.5 times faster. This
means that for each S+ accreting on the dust, 3 H atoms accrete. At
temperatures of 17K, the residence time of H atoms is very short, and
therefore S atoms can make S chains. These chains are not due to diffusion but
to the fact that S+ arrive on an adsorbed S atom (or a S-chain) and makes a
(larger) chain. In order to address whether the sulfur adsorbed on the dust
surface could be responsible for the depletion of sulfur in the gas phase, we
estimate how much gas phase sulfur is needed to obtain the coverage obtained
in our simulations. The surface area represented by carbon dust is nc$\times$
$\sigma_{c}$ = 7.2 10-22 cm-1, by silicate dust is ns$\times$ $\sigma_{s}$ =
4.4 10-21 cm-1, and by PAHs is $\sigma_{c}$ = 3.4 10-21 cm-1 (Weingartner &
Draine (2001); we consider the possibility for S-chains to form on PAHs). This
corresponds to a total area of 8.5 10-21 cm-1. This total area is multiplied
by the number of sites per area (1015 cm-2) to obtain the number density of
sites per cm3 which is 8.5 10-6 cm-3. This implies that if the grains would be
covered by one monolayer of S (or HS or H2S), then 8.5 10-6 S atoms would be
missing from the gas phase and the gas phase density of sulfur would drop to
1.4 10-6. The abundances of sulfur in translucent clouds have been reported to
be $\sim$ 8 10-7 (this is the most likely value taking into account the
uncertainties 0.4-2.2 10-6; Fuente et al. 2019) . Looking at our results, we
show that such depletion of sulfur can occur in few times $\sim$104 years. We
therefore suggest that the sulfur under the form of S+ in translucent clouds
plays a significant role in setting the S depletion in denser regions, as the
timescales to reach such depletion are quite short. The fact that sulfur atoms
are creating chains on the surface allows for layers to be built that can
represent more than 100$\%$ surface coverage. The increase in S on the surface
is due to the formation of S-chains. The right panel of Figure 11 shows the
surface coverage at the end of the simulations, the blue boxes showing the
sulfur atoms and how they are organized on the surface. Our simulations
suggest the creation of long S-chains under translucent cloud conditions which
can explain the important depletion observed in the gas phase. As translucent
clouds evolved further to molecular clouds and dense cores phases, and
extinction increases, these S-rich grains will then be covered with ices. Some
of these ices is composed of H2S which will adsorb on the icy grains during
these phases as a fraction of sulfur is still in the gas phase. The sulfur
chains from the translucent cloud phase, buried underneath the ice, cannot
desorb in the hot core or hot corinos phases since the temperature of the dust
in these environments is too small to allow desorption (we observe up to S4
until 300 K). These buried chains could be ejected again in the gas phase upon
shocks that would vaporize the ice, dust and these chains. These chains could
also be observed in our solar system within Interplanetary dust particles
(IDPs), meteorites and comets under a refractory form.

nH | nHI/nH a | nS+/nH | Tgas (K) | Tdust (K) | AV
---|---|---|---|---|---
5 103 | 2 10-4 | 10-5 | 25 | 17 | 1.6
Table 5: Parameters used in our simulations for translucent clouds conditions.
Taken from Laas & Caselli (2019); a taken from Cazaux & Spaans (2009).

Figure 11: Sulfur coverage of S, HS or H2 S as a function of time for
translucent cloud models with increase of accretion rate due to the charge
(left panel). The composition of the surface is visualised (right panel). In
these simulations, the temperature of the dust is 17K, and G0=1.

##  7 Application to solar system conditions: origin of sulfur found in
cometary material

In the present study, we show that UV irradiation of H2S ice results in the
formation of S chains. These chains are increasingly resistant to UV as their
size increase. We show that H2S species in the ice can shield other sulfur
bearing species to some extent. Our study therefore shows that photo-processes
of H2S ices can lead to the formation of S chains that either desorb (below
250 K for chains up to S4) or stay on the surface as residue for chains with
more than 4 sulfur atoms.
Comets are known to be rich in H2S ice representing $\sim$56% of the total
sulfur budget, with some minor contribution from SO2, SO and S2 (0.14%,
Calmonte et al., 2016). The abundance of H2S in comets is up to 1.5$\%$
relative to H2O as inferred from millimeter and submillimeter observations
(Bockelée-Morvan et al., 2000; Boissier et al., 2007). In comet 67P, data
analysis from Rosina (Rosetta Orbiter Spectrometer for Ion and Neutral
Analysis) showed the presence of S3 and S4. Calmonte et al. (2016) discussed
whether these are real parent molecules or the products of even heavier Sn,
fragmenting into S3 and S4. In the present study, we show that large Sn do not
desorb at temperatures below 300 K (experimental temperature range in the
TPD), but stay as residue in the ice. We therefore suggest that the S3 and S4
chains measured in cometary material do not come from fragmentation from large
Sn, but from UV processing of H2S bearing-ices leading to small sulfur chains.

The detection of S2, S3 and S4 makes radiolysis of H2S a very likely formation
process, as already discussed by other authors (Grim & Greenberg 1987;
Jiménez-Escobar et al. 2012; Woods et al. 2015) which could well connect the
cometary ice to the ISM. In particular, the S2/H2S abundance ratio in comet
67P varied from less than 1% to about 3.4% (Calmonte et al., 2016), the latter
value requires a substantial UV dose to convert H2S into S2 in the ice
(Jiménez-Escobar et al., 2012). Other species, such as H2S2 and S2H, that are
efficiently formed in H2S ice irradiation experiments (Jiménez-Escobar et al.,
2012, this work), could not be detected in the coma of 67P; this was probably
due to overlap with other species of similar mass-to-charge ratio (Calmonte et
al., 2016). The present study suggests that small sulfur chains come either
from UV photo-processes of H2S or from S depletion in the cloud prior to the
formation of the protosun. The different environments at which the formation
of these S-chains occur are illustrated in figure 12. In the translucent
stage, S-chains are formed on bare grains due to the accretion of cationic S
on grain surfaces, while in dense clouds, S-chains are formed via UV photo-
processes of H2S ice. In this case, these chains are within the ices
Therefore, observation of small S chains alone cannot allow distinguishing
between the two possible formation scenarios. Large chains, produced in the
second scenario (translucent cloud phase), are present in the residues within
and/or beneath ice mantles that do not desorb at temperatures typical of
cometary conditions, and are therefore impossible to observe in the gas phase.
Infrared bands are expected for the Sn species that would be very difficult to
detect since they will be masked by the intense absorption features of ice
species in the same spectral region. In particular, S8 displays a band near
465 cm-1 and other bands in the far-IR (Trofimov et al., 2009). These species
could have been detected by the COSAC-Rosetta instrument on board the Philae
lander, but the non-nominal landing on 67P comet nucleus did not allow to
extract a sample for analysis (Goesmann et al. (2015). Looking at the present
experiments and simulations, UV photolysis of H2S ices produces a fraction of
H2S/Sn (with n $\leq$3) of the order of 102 (from Fig. 5), while the Sn built
in the translucent cloud conditions has mainly n $>$ 4\. The observations of
S-chains in comet 67P seem to be related with UV processes of H2S ice, rather
than S-chains originating from the translucent cloud phase. This is in
agreement with Calmonte et al. (2016) suggesting an ice origin of the
S-chains. Comparison of the different periods in the evolution of 67P show
that at least in 2 periods the S follows more or less the H2S abundances,
suggesting a common origin in the ice matrix.

Figure 12: Sketch summarizing the results for the formation of S-chains via
accretion in the translucent cloud phase, and via UV photo-processes of H2S
ices in molecular cloud phase. Adapted from Fuente et al. (2019).

##  8 Conclusions

  1. 1.

The 2538 cm-1 band profile (H-S stretching) and integrated band strength of
H2S ice change gradually up to about 60 K. Above this temperature the
structure is dominated by the crystalline phase, as evidenced by the
distinctive narrow features of antisymmetric and symmetric S-H modes, see Fig.
1.

  2. 2.

Upon H2S irradiation, the efficient formation of H2S2 molecules occurs already
at 8 K, the lowest temperature in our experiments. Above 100 K, H2S ice
sublimates and H2S2 molecules are allowed to form a crystalline ice structure,
see Fig. 2. H2S2 desorbs around 144 K.

  3. 3.

Molecules H2Sx with $x$ ¿ 2 are also observed in the IR, Fig. 3. This is
confirmed by the desorption of H2S3 during warm-up near 184 K, while H2S4
desorbs near 204 K, see Fig. 4.

  4. 4.

During warm-up of the previously irradiated H2S ice, the more volatile Sx
species are detected. S desorbs near 58 K, S2 desorbs around 113 K, and S4
desorbs at 283 K, see Fig. 5. Larger Sx species are more refractory and remain
on the substrate at room temperature, being S8 by far the most abundant
species in these residues (Muñoz Caro, 2003).

  5. 5.

Monte Carlo simulations of the photo-dissociation of H2S ice shows that
hydrogenated sulfur species observed in the experiments could be reproduced if
species in the ice can shield each other. However, this shielding should be
not too efficient to be able to reproduce the S-chains observed in the
experiments. We predict that the residues remaining after the TPD in the
experiments are S-chains with a length up to 6-7 S atoms.

  6. 6.

Applying our Monte Carlo simulations to translucent clouds conditions, we show
that long S-chains are created in a rather short timescales of few 104 years.
The important depletion observed in these environments can therefore be
explained by grains covered by Sn. These grains will be buried under ices in
later stages as the cloud evolves.

  7. 7.

Small S-chains observed in comets could in theory originate from processed H2S
ices (from molecular cloud phases) or from S-chains from the translucent cloud
phase. Since small chains are being mainly produced by photo-processes of H2S
ice, we suggest that observations of S3 and S4 in 67P have an icy origin, and
are not due to processes of long chains from the translucent cloud phase.

###### Acknowledgements.

Part of this work was supported by the German _Deutsche
Forschungsgemeinschaft, DFG_ project number Ts 17/2–1. H. C. was supported by
PhD fellowship FPU-17/03172. AF, PRM and DNA thank the Spanish MICINN for
support under grant PID2019-106235GB-I100.

## References

  * Agúndez & Wakelam (2013) Agúndez, M. & Wakelam, V. 2013, Chemical Reviews, 113, 8710
  * Altwegg et al. (2019) Altwegg, K., Balsiger, H., & Fuselier, S. A. 2019, ARA&A, 57, 113
  * Bockelée-Morvan et al. (2000) Bockelée-Morvan, D., Lis, D. C., Wink, J. E., et al. 2000, A&A, 353, 1101
  * Boissier et al. (2007) Boissier, J., Bockelée-Morvan, D., Biver, N., et al. 2007, A&A, 475, 1131
  * Boogert et al. (2015) Boogert, A. C. A., Gerakines, P. A., & Whittet, D. C. B. 2015, ARA&A, 53, 541
  * Boogert et al. (1997) Boogert, A. C. A., Schutte, W. A., Helmich, F. P., Tielens, A. G. G. M., & Wooden, D. H. 1997, A&A, 317, 929
  * Bulut et al. (2021) Bulut, N., Roncero, O., Aguado, A., et al. 2021, A&A, 646, A5
  * Calmonte et al. (2016) Calmonte, U., Altwegg, K., Balsiger, H., et al. 2016, MNRAS, 462, S253
  * Caselli et al. (1994) Caselli, P., Hasegawa, T. I., & Herbst, E. 1994, ApJ, 421, 206
  * Cazaux et al. (2015) Cazaux, S., Bossa, J. B., Linnartz, H., & Tielens, A. G. G. M. 2015, A&A, 573, A16
  * Cazaux et al. (2018) Cazaux, S., Bossa, J. B., Martin-Doménech, R., et al. 2018, Astrophysics and Space Science Library, Vol. 451, Monte Carlo Simulations of the Formation and Morphology of Interstellar Ices, ed. G. M. Muñoz Caro & R. Escribano, 95
  * Cazaux et al. (2017) Cazaux, S., Martín-Doménech, R., Chen, Y. J., Muñoz Caro, G. M., & González Díaz, C. 2017, The Astrophysical Journal, 849, 80
  * Cazaux & Spaans (2009) Cazaux, S. & Spaans, M. 2009, A&A, 496, 365
  * Drozdovskaya et al. (2021) Drozdovskaya, M. N., Schroeder I, I. R. H. G., Rubin, M., et al. 2021, MNRAS, 500, 4901
  * Drozdovskaya et al. (2019) Drozdovskaya, M. N., van Dishoeck, E. F., Rubin, M., Jørgensen, J. K., & Altwegg, K. 2019, MNRAS, 490, 50
  * Dulieu et al. (2005) Dulieu, F., Amiaud, L., Baouche, S., et al. 2005, Chemical Physics Letters, 404, 187
  * Dutrey et al. (1997) Dutrey, A., Guilloteau, S., & Guelin, M. 1997, A&A, 317, L55
  * Fuente et al. (2017) Fuente, A., Goicoechea, J. R., Pety, J., et al. 2017, ApJ, 851, L49
  * Fuente et al. (2019) Fuente, A., Navarro, D. G., Caselli, P., et al. 2019, A&A, 624, A105
  * Garrod (2013) Garrod, R. T. 2013, ApJ, 778, 158
  * Geballe et al. (1985) Geballe, T. R., Baas, F., Greenberg, J. M., & Schutte, W. 1985, A&A, 146, L6
  * Goesmann et al. (2015) Goesmann, F., Hendrik Bredehöft, J., Cabane, M., et al. 2015, in EGU General Assembly Conference Abstracts, EGU General Assembly Conference Abstracts, 8615
  * Goto et al. (2020) Goto, M., Vasyunin, A. I., Giuliano, B. M., et al. 2020, arXiv e-prints, arXiv:2012.10883
  * Grim & Greenberg (1987) Grim, R. J. A. & Greenberg, J. M. 1987, A&A, 181, 155
  * Habing (1968) Habing, H. J. 1968, Bull. Astron. Inst. Netherlands, 19, 421
  * Hasegawa et al. (1992) Hasegawa, T. I., Herbst, E., & Leung, C. M. 1992, ApJS, 82, 167
  * Isoniemi et al. (1999) Isoniemi, E., Khriachtchev, L., Pettersson, M., & Räsänen, M. 1999, Chemical Physics Letters, 311, 47
  * Jenkins (2009) Jenkins, E. B. 2009, ApJ, 700, 1299
  * Jiménez-Escobar & Muñoz Caro (2011) Jiménez-Escobar, A. & Muñoz Caro, G. M. 2011, A&A, 536, A91
  * Jiménez-Escobar et al. (2014) Jiménez-Escobar, A., Muñoz Caro, G. M., & Chen, Y. J. 2014, MNRAS, 443, 343
  * Jiménez-Escobar et al. (2012) Jiménez-Escobar, A., Muñoz Caro, G. M., Ciaravella, A., et al. 2012, ApJ, 751, L40
  * Laas & Caselli (2019) Laas, J. C. & Caselli, P. 2019, A&A, 624, A108
  * Le Gal et al. (2019) Le Gal, R., Öberg, K. I., Loomis, R. A., Pegues, J., & Bergner, J. B. 2019, ApJ, 876, 72
  * Lin et al. (2005) Lin, C. S., Zhang, R. Q., Lee, S. T., et al. 2005, J. Chem. Phys. B, 109, 14183
  * Martín-Doménech et al. (2015) Martín-Doménech, R., Manzano-Santamaría, J., Muñoz Caro, G. M., et al. 2015, A&A, 584, A14
  * Moore et al. (2007) Moore, M. H., Hudson, R. L., & Carlson, R. W. 2007, Icarus, 189, 409
  * Muñoz Caro (2003) Muñoz Caro, G. M. 2003, PhD thesis, Leiden University, P.O. Box 9513, 2300 RA Leiden, The Netherlands
  * Muñoz Caro et al. (2010) Muñoz Caro, G. M., Jiménez-Escobar, A., Martín-Gago, J. Á., et al. 2010, A&A, 522, A108
  * Navarro-Almaida et al. (2020) Navarro-Almaida, D., Le Gal, R., Fuente, A., et al. 2020, A&A, 637, A39
  * Oba et al. (2018) Oba, Y., Tomaru, T., Lamberts, T., Kouchi, A., & Watanabe, N. 2018, Nature Astronomy, 2, 228–232
  * Oppenheimer & Dalgarno (1974) Oppenheimer, M. & Dalgarno, A. 1974, ApJ, 187, 231
  * Palumbo et al. (1995) Palumbo, M. E., Tielens, A. G. G. M., & Tokunaga, A. T. 1995, ApJ, 449, 674
  * Penzias et al. (1971) Penzias, A. A., Solomon, P. M., Wilson, R. W., & Jefferts, K. B. 1971, ApJ, 168, L53
  * Rodríguez-Baras et al. (2021) Rodríguez-Baras, M., Fuente, A., Riviére-Marichalar, P., et al. 2021, A&A, 648, A120
  * Ruffle et al. (1999) Ruffle, D. P., Hartquist, T. W., Caselli, P., & Williams, D. A. 1999, MNRAS, 306, 691
  * Shingledecker et al. (2020) Shingledecker, C. N., Lamberts, T., Laas, J. C., et al. 2020, ApJ, 888, 52
  * Shingledecker et al. (2019) Shingledecker, C. N., Vasyunin, A., Herbst, E., & Caselli, P. 2019, ApJ, 876, 140
  * Smith (1991) Smith, R. G. 1991, MNRAS, 249, 172
  * Snow & McCall (2006) Snow, T. P. & McCall, B. J. 2006, Annual Review of Astronomy and Astrophysics, 44, 367
  * Sternberg & Dalgarno (1995) Sternberg, A. & Dalgarno, A. 1995, ApJS, 99, 565
  * Trofimov et al. (2009) Trofimov, B., Sinegovskaya, L., & Gusarova, N. 2009, Journal of Sulfur Chemistry, 30, 518
  * Umebayashi & Nakano (1980) Umebayashi, T. & Nakano, T. 1980, PASJ, 32, 405
  * van der Tak et al. (2003) van der Tak, F. F. S., Boonman, A. M. S., Braakman, R., & van Dishoeck, E. F. 2003, A&A, 412, 133
  * Vidal et al. (2017) Vidal, T. H. G., Loison, J.-C., Jaziri, A. Y., et al. 2017, MNRAS, 469, 435
  * Wakelam et al. (2004) Wakelam, V., Caselli, P., Ceccarelli, C., Herbst, E., & Castets, A. 2004, A&A, 422, 159
  * Wakelam et al. (2017) Wakelam, V., Loison, J.-C., Mereau, R., & Ruaud, M. 2017, Molecular Astrophysics, 6, 22–35
  * Weingartner & Draine (2001) Weingartner, J. C. & Draine, B. T. 2001, ApJ, 548, 296
  * Woods et al. (2015) Woods, P. M., Occhiogrosso, A., Viti, S., et al. 2015, MNRAS, 450, 1256
  * Zengin & Giguère (1959) Zengin, N. & Giguère, P. A. 1959, Canadian journal of chemistry, 37
  * Zhou et al. (1993) Zhou, S., Evans, Neal J., I., Koempe, C., & Walmsley, C. M. 1993, ApJ, 404, 232