11institutetext:  Department of Physics, University of Warwick, Gibbet Hill
Road, Coventry CV4 7AL, UK
11email<EMAIL_ADDRESS>Centre for Exoplanets and
Habitability, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, UK
33institutetext: Institute of Theoretical Physics and Astrophysics, University
of Kiel, Leibnizstraße 15, 24118 Kiel, Germany

# Impact of discontinuous grain size distributions
on the spectral energy distribution of debris disks

M. Kim Impact of discontinuous grain size distributions on the spectral energy
distribution of debris disksImpact of discontinuous grain size distributions
on the spectral energy distribution of debris disksImpact of discontinuous
grain size distributions on the spectral energy distribution of debris
disksImpact of discontinuous grain size distributions on the spectral energy
distribution of debris disks S. Wolf Impact of discontinuous grain size
distributions on the spectral energy distribution of debris disksImpact of
discontinuous grain size distributions on the spectral energy distribution of
debris disks

(Received 13 June, 2023; accepted XXXX XX, XXXX)

###### Abstract

Context. The collisional evolution of debris disks is expected to result in a
characteristic wavy pattern of the grain size distributions, i.e., an
under/overabundance of particles of specific sizes. This perturbed grain size
distribution potentially leaves characteristic patterns in the spectral energy
distribution (SED) of the disk system.

Aims. We aim to quantify and understand the specific influence of
discontinuous particle size distributions on the appearance of debris disks.
For this purpose, we consider dust emission models based on two different
grain size distributions, i.e., once with a single and once with a broken
power law. In particular, the impact of an overabundance of small grains and
an underabundance of medium-sized grains on the far-infrared/(sub-)millimeter
regime on the dust reemission radiation and the potential to constrain
discontinuities in the grain size distribution from (sub-)millimeter
photometric measurements of debris disks are in the focus of our study.

Methods. We compare the spectral index $\alpha$
($F_{\nu}\,\propto\nu^{\rm{\,\alpha}}$) in the case of a continuous grain size
distribution with that of a discontinuous grain size distribution. We perform
this comparison for central stars with different spectral types and two
different disk structures (e.g., slim and broad debris dust rings).

Results. Within the considered parameter space, we find a characteristic
difference between the spectral slopes of the SED in the different scenarios.
In particular, the overabundance of small grains resulting from collisional
events and thus parameters defining the outcome of disk events in debris disks
is potentially observable by comparison with the SED corresponding to a grain
size distribution resulting from an ideal collisional cascade. More
specifically, the overabundance of small grains leads to a steeper slope in
the far-infrared/submillimeter regime, while the spectral index in the mm
regime is hardly affected. On the other hand, the underabundance of medium-
sized grains results in a slight steepening of the far-infrared slope of SED,
while its primary effect is on the mm slope of SED, causing it to become
shallower. We also find that the impact of an overabundance of small dust
particles is more pronounced than that of an underabundance of medium-sized
dust particles. Furthermore, we find that the difference between the spectral
indices for the two different grain size distributions is largest for debris
disks around brighter central stars and broader disks. However, the impact of
the considered spatial distributions described by the fractional width of the
disk system is weak. Our results also show that the dust composition is not
the sole physical mechanism responsible for the spectral inversion observed in
the far-IR to mm part of the SED of debris disk systems. Furthermore, the
location of the spectral break is placed at different wavelength regimes if
the grain size distribution is considered as a function of blowout size and
stellar type.

###### Key Words.:

circumstellar matter – planetary systems – methods : numerical

##  1 Introduction

The lifetime of small dust grains in debris disks is much shorter than the
stellar age (Artymowicz & Clampin, 1997), indicating that dust cannot be
primordial in these systems. Thus, the dust in debris disks must be
continuously replenished through disruptive collisions. This process leads to
the creation of fragments with a broad size distribution, which can be
approximately described by a power law. For example, for a steady-state
collisional cascade (Dohnanyi, 1969), the power law with $\gamma$ = -3.5 is
valid for the differential size distribution, namely d$n(a)$ $\propto$
$a^{\gamma}$ d$a$, where $a$ is the size of the dust grain. However, due to
the finite size distribution and size-dependent collisional strength of the
colliding bodies/particles, the real slope of the size distribution may
strongly deviate from this value and thus may not accurately reflect the true
grain size distribution. For example, the collisional cascade does not extend
down to arbitrarily small particles because tiny particles with a high $\beta$
ratio between radiation pressure and stellar gravity are quickly blown away by
the stellar radiation pressure from the disks (Krivov et al., 2006).
Furthermore, nongravitational forces (e.g., Poynting-Robertson drag; Backman &
Paresce, 1993) acting on grains of particular sizes in the range of tens to
hundreds of micrometers may further modify the grain size distribution (Löhne
et al., 2017).

In previous studies based on numerical simulation, a characteristic wavy
pattern in the grain size distribution resulting from collisional evolution
within debris disks was found (e.g., Thébault & Augereau, 2007; Kim et al.,
2018). These patterns, not solely described by a simple power law, but, e.g.,
by broken power laws, indicate the overabundance or underabundance of
particles of specific sizes. In particular, the wavy pattern of the
discontinuous grain size distribution becomes more pronounced (deeper) and
broader in extreme scenarios involving lower material strength of dust grains
(Kim et al., 2018).

Furthermore, several observational findings have pointed toward the need for a
more complex model for grain size distribution beyond a simple single power
law. For instance, observational findings based on the comparison between flux
and polarization measurements of protoplanetary disks at sub-millimeter (sub-
mm) wavelengths have also raised questions regarding the corresponding grain
size distribution (e.g., Brunngräber & Wolf, 2021). Bürger et al. (2023) also
recently demonstrated an apparent overabundance of grains with radii between 3
and 6 mm in the derived grain size distribution of comet 67P/Churyumov-
Gerasimenko. Additionally, Rodigas et al. (2015) found that the grain size
distributions derived from the observation of thermal reemission radiation are
not consistent with those from scattered light colors and/or scattering
functions. Finally, Olofsson et al. (2023) recently demonstrated that the
overabundance of small grains in debris disks could also explain an intriguing
arc-like structure observed in total intensity. These findings suggest the
presence of a discontinuous or more complicated grain size distribution,
challenging the traditional single-power law model.

The grain size distribution directly influences the slope of the spectral
energy distribution (SED) in the far-infrared (far-IR) to
submillimeter/millimeter (mm) range of the thermal reemission by the dust
(Wyatt & Dent, 2002). Consequently, the presence of overabundant or
underabundant particle sizes at these wavelengths potentially leaves an
observable imprint in the SED. For example, the recent NIKA2 observation of HD
107146 indicates a break at the far-IR and sub-mm/mm wavelength regimes
(Lestrade et al., 2020), which is further supported by supplementary
observations, showing possibly a change in the slope of the SED (e.g., Holland
et al., 2017; Ricci et al., 2015b). In this particular case, the spatial
distribution of dust in this debris disk system cannot be a reason for this
break at long wavelength because HD 107146 is observed as a narrow belt system
(Marino et al., 2018). Moreover, the influence of the dust temperature on the
Planck function is weak in the Rayleigh-Jeans regime, diminishing its
importance at long wavelengths. However, given the small range of dust
temperatures in a narrow belt system, the impact of the underlying grain size
distribution becomes more pronounced. Consequently, a change in the SED slope
due to an overabundance/underabundance of grains is expected in the wavelength
region where grains of the corresponding size emit most efficiently.

The goal of the current study is to evaluate the impact of the discontinuous
size distribution of dust in debris disks on the SED of these systems. In
turn, the results of this study are expected to serve as a tool to also
address the opposite question, namely whether constraints on the parameters
describing the collisions in debris disks can be derived from the analysis of
the slope of their thermal reemission SED. The present study is structured as
follows: In Section 2, we describe our model for the simulations of debris
disk observables. In Section 3, we quantify and discuss the impact of the
grain size distributions on the observational appearance of debris disks, in
particular, their SEDs (e.g., the slope of SEDs). In Section 4, we discuss an
application of our results to observations and possible degeneracies and
connections to chemical and physical properties. We summarize our findings in
Section 5.

##  2 Modeling of debris disks

###  2.1 Model parameters of debris disk system

The parameters and characteristics of stars, disks, and dust in our models are
based on observations of currently known debris disk systems. Table 1 and 2
summarize the detailed characteristics of disk and dust parameters.

Table 1: Disk and dust model parameters for the simulations and corresponding
references.

Parameter | Value
---|---
Narrow belt’s fractional width |  $\Delta$$R$/$R$ $\sim$ 0.17 \rm{\,a}\rm{\,a}footnotemark: $\rm{\,a}$(Lestrade & Thilliez, 2015)
Broad belt’s fractional width |  $\Delta$$R$/$R$ $\sim$ 1.20 \rm{\,b}\rm{\,b}footnotemark: $\rm{\,b}$ (Marino et al., 2020)
Distance $d$ | 7.7 pc (Mamajek, 2012)
Radial density distribution $n\,(r)$ |  $n\,(r)$ $\propto$ $r^{\rm-1.5}$ (Krivov et al., 2006)
Dust composition & bulk density | Astrosil & 3.5 g cm-3 (Draine, 2003)
Minimum grain radius $a_{\rm{\,min}}$ |  $a_{\rm\,bo}$ \rm{\,c}\rm{\,c}footnotemark: $\rm{\,c}$(Kirchschlager & Wolf, 2013) or 0.264 $\mu\rm{m}$ (Kim et al., 2018)
Maximum grain radius $a_{\rm{\,max}}$ | 10 cm (Lestrade et al., 2020)

111$\rm{a,b,c}$$\rm{a,b,c}$footnotetext: See Table 2

Table 2: Stellar, disk, and dust model parameters for the simulations and
their reference. The chosen stellar parameters are selected as one of the
representative stars for each spectral type.

SpT | Temperature | Radius | Narrow belt | Broad belt | Blow-out grain size
---|---|---|---|---|---
| [K] | [R⊙] |  $R_{\rm\,in}$, $R_{\rm\,out}$ [au] |  $R_{\rm\,in}$, $R_{\rm\,out}$ [au] |  $a_{\rm\,bo}$ [$\mu$m]
A star ($\beta$ Pictoris) | 8052\rm{\,a}\rm{\,a}footnotemark: $\rm{\,a}$ | 1.8\rm{\,b}\rm{\,b}footnotemark: $\rm{\,b}$ | 92, 109 | 37, 163 | 1.99
F star (HD 139664) | 6681\rm{\,c}\rm{\,c}footnotemark: $\rm{\,c}$ | 1.26\rm{\,d}\rm{\,d}footnotemark: $\rm{\,d}$ | 76, 90 | 31 , 136 | 1.05
G star (HD 107146) | 5820\rm{\,e}\rm{\,e}footnotemark: $\rm{\,e}$ | 0.993\rm{\,e}\rm{\,e}footnotemark: $\rm{\,e}$ | 62, 73 | 25, 110 | 0.43
K star (HD 53143) | 5224\rm{\,f}\rm{\,f}footnotemark: $\rm{\,f}$ | 0.85\rm{\,g}\rm{\,g}footnotemark: $\rm{\,g}$ | 57, 67 | 23, 101 | -
M star (AU mic) | 3500\rm{\,h}\rm{\,h}footnotemark: $\rm{\,h}$ | 0.5\rm{\,h}\rm{\,h}footnotemark: $\rm{\,h}$ | 38, 45 | 15, 68 | -

222$\rm{a}$$\rm{a}$footnotetext: Gray et al., 2006
$\rm{b}$$\rm{b}$footnotetext: Kervella et al., 2004
$\rm{c}$$\rm{c}$footnotetext: Reiners, 2006 $\rm{d}$$\rm{d}$footnotetext: Rhee
et al., 2007 $\rm{e}$$\rm{e}$footnotetext: Marino et al., 2018
$\rm{f}$$\rm{f}$footnotetext: Kalas et al., 2006 $\rm{g}$$\rm{g}$footnotetext:
Watson et al., 2011 $\rm{h}$$\rm{h}$footnotetext: Donati et al., 2023

Disk Geometry: We consider the inner and outer radii of disks based on the
empirical relation between planetesimal belt radius and stellar luminosity
(Matrà et al., 2018) with two different belt fractional widths: a narrower
belt (e.g., $\Delta$$R$/$R$ $\sim$ 0.17 of debris disks around $\varepsilon$
Eridani; Lestrade & Thilliez, 2015) comparable to the classical Kuiper belt,
and a broader belt (e.g., $\Delta$$R$/$R$ $\sim$ 1.25 of debris disks around
HD 206893; Marino et al., 2020). We note that broader belt systems show a
broad range of dust temperatures. The mass of inner belts in a two-belt system
is typically only a few percent compared to outer belts (e.g., Ertel et al.,
2011; Marino et al., 2018), implying that the contribution of the inner belt
to the slope of the SED is negligible. Thus, we consider a single belt system
to reduce the complexity of the model.

Minimum/maximum grain size and dust composition: For the smallest grain size,
we apply a blow-out grain size $a_{\rm\,bo}$ (Backman & Paresce, 1993)
depending on the spectral type. For cases without a blowout size (i.e., for
central stars’ effective temperatures below 5250 K in our model setup;
Kirchschlager & Wolf, 2013), we consider a minimum grain size of 0.264 $\mu$m
that has been used in previous studies (e.g., Löhne et al., 2017; Kim et al.,
2018), which shows the similarity to the previous findings of debris disks
around M-type stars (e.g., Matthews et al., 2015) based on the strong stellar
wind (Plavchan et al., 2005), causing an effect similar to that of the
radiation pressure force (Augereau & Beust, 2006). Furthermore, using Mie
theory, silicate or ice grains comparable to this size are hardly affected by
the stellar radiation pressure in debris disks around a solar-type star
(Cataldi, 2016). We adopt a maximum grain radius of 10 cm (Lestrade et al.,
2020; Pawellek et al., 2021) to analyze the dust continuum up to a wavelength
of a few mm such as ATCA Survey (Ricci et al., 2015b). We note that the
contribution of larger grains to the net flux is negligible when considering a
steep grain size distribution. For the grain composition, we assume the
simplest dust grains to be compact and homogenous spheres composed of
astronomical silicate (referred to as ”Astrosil”) with a bulk density of 3.5 g
cm-3 and optical constants taken from Draine (2003).

Disk mass: Based on previous surveys at sub-mm wavelengths, we consider a dust
mass in debris disks of $\sim$ 10-7 ${\rm M}_{\,\odot}$ for the single power
law grain size distribution model (e.g., dust mass of most debris disks
typically ranges from $\sim$ 10-9 to several 10-7 ${\rm M}_{\,\odot}$; Greaves
et al., 2005). In the case of the broken power law grain size distribution
model, the dust mass is varied depending on the scale factor but remains
within a similar range (i.e., 1.0 to 1.26 $\times$ 10-7 ${\rm M}_{\,\odot}$;
see Table 3). This is because the contribution of small dust grains mainly
affects the dust reemission rather than significantly influencing the overall
mass budget, which is dominated by larger grains. We consider the same mass of
dust $M_{\rm{dust}}$ for both types of disk structures (i.e., narrow and broad
belt-like disks). We note that within this model the overall mass of a disk
does not significantly affect spectral index as long as the disk remains
optically thin; however, a specific range of dust masses with the under/over-
abundance of medium and small grains is indeed an important factor. The
detailed characteristics of the modeled classical single power law and wavy
pattern (i.e., broken power-law) grain size distribution are compiled in Fig.
1 and Table 3.

Central star: We consider central stars of the following five spectral types
(in brackets exemplary systems are listed): A star ($\beta$ Pictoris), F star
(HD 139664), G star (HD 107146), K star (HD 53143), and M star (AU mic). Table
2 summarizes the detailed characteristics of stellar parameters and
corresponding disk and dust parameters.

Simulation of SEDs: For the calculation of model SEDs, we use the software
tool DMS (Kim et al., 2018), which is optimized for optically thin emission of
debris disks. Based on wavelength-dependent refractive indices, we calculate
required the optical properties with the tool miex on the basis of Mie
scattering theory (Wolf & Voshchinnikov, 2004).

###  2.2 Modeling of grain size distributions

To understand how the grain size distribution affects the SED of debris disks,
we consider dust emission models that are based on two different grain size
distributions: first a single power law grain size distribution (Sect. 2.2.1)
and second a broken power law grain size distribution (Sect. 2.2.2). Based on
these distributions, we study the shape of the thermal reemission SED in the
far-IR to sub-mm/mm wavelength range.

####  2.2.1 Single power law grain size distribution

As a reference case, we consider the classical single power law grain size
distribution d$n\,(a)\propto a^{\,-3.5}$ d$a$ (see Fig. 1 and Table 3 top row
for details), resulting from a collisional cascade in debris disks at steady-
state equilibrium with size-independent collisional strength (Dohnanyi, 1969).
Since the absorption cross-section $C_{\rm abs}$ of dust grains gradually
decreases towards longer wavelengths irrespective of the dust composition (Kim
et al., 2019), the dust emission model employing the single power law grain
size distribution is expected to result in a continuous decrease in the SED at
long wavelengths, i.e., the far-IR to mm range.

Table 3: Considered grain size distributions, dust mass, and corresponding model ID ($a_{\rm{bo}}$: blow-out grain size, see Sect. 2.1 for discussion). Grain size distribution | Description | Factor | Mass of dust [M${}_{\,\odot}$] | Model ID
---|---|---|---|---
Single power law | Reference model standard abundance |  | 1.00 $\times$ 10-7 | OS1 + UM1 + SL1
d$n\,(a)$ $\propto$ $a^{\rm-3.5}$d$a$ for [$a_{\rm\,bo}$ or 0.264 $\mu\rm{m}$, 10 cm] |
res: Broken power law | Overabundance of small grains | 1 | 2.62 $\times$ 10-10 | OS1
d$n\,(a)$ $\propto$ $a^{\rm-3.5}$d$a$ for [$a_{\rm\,bo}$ or 0.264 $\mu\rm{m}$, 100 $\mu\rm{m}$] | 10 | 2.62 $\times$ 10-9 | OS10
50 | 1.31 $\times$ 10-8 | OS50
100 | 2.62 $\times$ 10-8 | OS100
Underabundance of medium grains | 1 | 6.65 $\times$ $10^{-9}$ | UM1
d$n\,(a)$ $\propto$ $a^{\rm-3.5}$d$a$ for [100 $\mu\rm{m}$, 1000 $\mu\rm{m}$] | 10 | 6.65 $\times$ 10-10 | UM10
50 | 3.33 $\times$ 10-11 | UM50
100 | 6.65 $\times$ 10-11 | UM100
Standard abundance of large grains |  | 1.00 $\times$ 10-7 | SL1
| d$n\,(a)$ $\propto$ $a^{\rm-3.5}$d$a$ for [1000 $\mu\rm{m}$, 10 cm] |

Figure 1: Simulated total cross-section of the dust in our debris disk around
A star (e.g., $a_{\rm\,bo}$ $\sim$ 1.99 $\mu$m) based on the considered grain
size distributions. The red solid line represents the classical single power
law (reference case; see Sect. 2.2.1). The yellow, green, and blue dashed
lines represent the perturbed size distribution with an overabundance of
grains with radii ranging from the blow-out size (see Table 2) to 100
$\mu{\rm{m}}$. The purple, grey, and navy dashed lines represent the
distribution with the underabundance in the size range of 100 $\mu{\rm{m}}$ to
1000 $\mu{\rm{m}}$, which are mainly motivated by the collisional model
presented in Thébault & Augereau (2007) and Kim et al. (2018).

####  2.2.2 Broken power law grain size distribution

As indicated by previous studies (Thébault & Augereau, 2007; Löhne et al.,
2017; Kim et al., 2018), the collisional evolution leaves an imprint on the
grain size distribution. In particular, the depletion of the smallest grains
to the blow-out size results in a surplus of slightly larger grains due to
reduced collisional erosion. This leads to the propagation of an overabundance
around small dust grains with radii ranging from several to $\sim$ hundred
micrometers, and an underabundance (i.e., local minimum) towards larger grains
with radii of several hundred to thousand micrometers. For example, an
observed peak in the wavy pattern of the grain size distribution occurs for
grains that are slightly larger than the blow-out size, specifically at about
$\sim$ 1.5 times this size. Additionally, the first minimum in this
distribution is noted at about 100 times the blow-out size (Thébault &
Augereau, 2007).

Furthermore, for the particular case of debris disks around HD107146,
referenced in Sect. 1, a local decrease in the long-wavelength regime of the
SED ranging from 800 $\mu$m to 6 mm has been observed (Lestrade et al., 2020).
A possible explanation for this finding is to link it to the peculiarities of
the grain size distribution, where the deep depletion of the abundance of
grains in the hundreds of micrometers size range coincides with the
characteristic grain size for strongly absorbing materials ($\sim$
approximately ranging from 100 to 1000 $\mu\rm{m}$; Backman & Paresce, 1993).

Figure 2: Simulated total cross-section of the dust in our debris disk around A, F, G, K, and M star (e.g., $a_{\rm\,bo}$ $\sim$ 1.99, 1.05, 0.43, 0.264, and 0.264, $\mu$m; red, orange, yellow, green, and blue color solid lines) based on the considered grain size distributions with exemplary cases (i.e., 100 times overabundance of small grains and underabundance of medium grains). The dashed red, orange, yellow, green, and blue color lines represent the classical single power law (reference case) around A, F, G, K, and M stars. The perturbed size distribution with an overabundance of small grains with radii ranging from the blow-out size to 100 times each blow-out size depending on spectral types (see Table 2). The one with an underabundance of medium grains with radii ranging from 100 times to 1000 times each blow-out size depending on spectral types (see Table 2). These size distributions are mainly motivated by the collisional model presented in Thébault & Augereau (2007) and Kim et al. (2018). Note that K star and M star do not have blow-out sizes, suggesting the minimum grain sizes are used often in previous studies (Löhne et al., 2017; Kim et al., 2018). Thus, both grain size distributions look the same. See Table 4 for details. Table 4: The locations of the first and second maximum/minimum in grain size distribution depending on the central star for the simulations. The chosen parameters are based on the findings from previous studies (e.g., Thébault & Augereau, 2007; Kim et al., 2018). SpT | Blow-out grain size | first maximum | first minimum | second maximum
---|---|---|---|---
| [$\mu$m] | [$\mu$m] | [$\mu$m] | [$\mu$m]
A star ($\beta$ Pictoris) | 1.99 | 2.985 | 199.9 | 1999
F star (HD 139664) | 1.05 | 1.575 | 105 | 1050
G star (HD 107146) | 0.43 | 0.645 | 43 | 430
K star (HD 53143) | (0.264) | 0.396 | 26.4 | 264
M star (AU mic) | (0.264) | 0.396 | 26.4 | 264

Based on the discussion in this section, we consider general but rather
simplistic grain size distribution that shows the key characteristics
potentially allowing us to explain discontinuities in the shape of the thermal
reemission SED of debris disks. Our discontinuous grain size distribution
models follow a broken power law pattern, comprising an overabundance of small
grains ranging from the blow-out grain size to 100 $\mu\rm{m}$ and an
underabundance of medium-sized grains (in short ”medium grain” in the
following) ranging from 100 to 1000 $\mu\rm{m}$ (Backman & Paresce, 1993). We
incorporate scaling factors of 10, 50, and 100 to represent the relative
abundance of these grain populations (see Fig. 1 and Table 3 for details). We
do not consider the depletion of the smallest dust grains to the blow-out
grain size that results in the excess of slightly larger grains (e.g., Kim et
al., 2018), as the effect of their absence on the spectral slope of the SED is
negligible in the spectral range explored in the present study. To avoid the
introduction of additional parameters, we set all slopes described by the
exponent of the grain size distribution $\gamma$ in the broken power law grain
size distribution to -3.5 (Dohnanyi, 1969) for all size ranges. We note that
the location of the subsequent minima/maxima and their overabundance and
underabundance may depend on not only the collisional circumstances but also
further disk parameters (e.g., Thébault & Augereau, 2007; Kim et al., 2018).
Thus, the location and abundance of the local minima/maxima of the size
distribution may not be directly indicated by the physical situations (e.g.,
the collisional state of the disk).

We note that a uniform overabundance and/or underabundance of grains of
certain sizes across all spectral types may not accurately represent the
actual conditions within these systems. Thus, we particularly consider grain
size distributions with an overabundance and/or underabundance of certain
grain sizes as a function of stellar spectral type (i.e., blow-out size). For
this purpose, we apply a grain size distribution model following the findings
of Thébault & Augereau, 2007; Kim et al., 2018, in which the first local
maximum of the grain size distribution typically occurs at about $\sim$ 1.5
times the blow-out size. Furthermore, the first minimum and second maximum are
observed at around 100 and 1000 times the blow-out size, respectively (see
also Fig. 2 and Table 4). In the case of disks around stars that are not
luminous enough to reach $\beta$ = 0.5 (e.g., K and M stars in our model;
Kirchschlager & Wolf, 2013), we consider 0.264 $\mu$m as the minimum grain
size (see also Sect. 2.1). The results based on the specifically proposed
grain size distribution, particularly concerning its impact on the
corresponding SEDs, can be found in Sect. 4.4.

##  3 Results

In the following, we systematically compare and evaluate the influence of the
different dust emission models based on the two different grain size
distributions on the SED for the various considered debris disk systems.
Specifically, we focus on quantifying the slope of the SED at long wavelengths
to assess how the presence of over- and underabundances of the specific size
of grains in the distribution impacts the emission characteristics of debris
disks. For this purpose, we compute the spectral index ${\alpha}$ as,

${\alpha}=\left|{\dfrac{{\rm{log}}\,({F_{\nu 1}}/{F_{\nu
2}})}{{\rm{log}}\,({\nu_{1}}/{\nu_{2}})}}\right|,$ (1)

where $F_{\nu}$ represents the radiative flux density and $\nu$ denotes the
frequency in the dust continuum emission.

Table 5: Comparison of the spectral index of the considered discontinuous grain size distributions depending on the overabundance of small grains, stellar parameters (e.g., stellar temperature and radius), and disk parameters (e.g., fractional width $\Delta$$R$/$R$). For the definition of model IDs (OS, UM, and SL) see Table 3. SpT | $\Delta$$R$/$R$ | Single power law | overabundance of small grains
---|---|---|---
OS1+UM1+SL1 | OS10+UM1+SL1 | OS50+UM1+SL1 | OS100+UM1+SL1
${\alpha}^{\,450\,\mu{\rm{m}}}_{\,850\,\mu{\rm{m}}}$ | ${\alpha}^{\,1000\,\mu{\rm{m}}}_{\,2000\,\mu{\rm{m}}}$ | $\Delta\,\alpha$ | ${\alpha}^{\,450\,\mu{\rm{m}}}_{\,850\,\mu{\rm{m}}}$ | ${\alpha}^{\,1000\,\mu{\rm{m}}}_{\,2000\,\mu{\rm{m}}}$ | $\Delta\,\alpha$ | ${\alpha}^{\,450\,\mu{\rm{m}}}_{\,850\,\mu{\rm{m}}}$ | ${\alpha}^{\,1000\,\mu{\rm{m}}}_{\,2000\,\mu{\rm{m}}}$ | $\Delta\,\alpha$ | ${\alpha}^{\,450\,\mu{\rm{m}}}_{\,850\,\mu{\rm{m}}}$ | ${\alpha}^{\,1000\,\mu{\rm{m}}}_{\,2000\,\mu{\rm{m}}}$ | $\Delta\,\alpha$
A | 0.17 | 2.07 | 2.75 | -0.68 | 2.59 | 2.77 | -0.18 | 3.67 | 2.86 | 0.81 | 4.24 | 2.95 | 1.28
1.25 | 2.08 | 2.75 | -0.67 | 2.6 | 2.78 | -0.18 | 3.68 | 2.87 | 0.81 | 4.25 | 2.96 | 1.29
F | 0.17 | 1.99 | 2.76 | -0.77 | 2.51 | 2.79 | -0.28 | 3.58 | 2.88 | 0.7 | 4.13 | 2.98 | 1.15
1.25 | 2.00 | 2.77 | -0.77 | 2.52 | 2.79 | -0.27 | 3.59 | 2.88 | 0.71 | 4.14 | 2.98 | 1.16
G | 0.17 | 1.93 | 2.76 | -0.83 | 2.44 | 2.79 | -0.35 | 3.49 | 2.88 | 0.61 | 4.03 | 2.98 | 1.05
1.25 | 1.94 | 2.77 | -0.83 | 2.45 | 2.79 | -0.34 | 3.51 | 2.89 | 0.62 | 4.05 | 2.98 | 1.06
K | 0.17 | 1.86 | 2.75 | -0.89 | 2.36 | 2.77 | -0.41 | 3.41 | 2.87 | 0.54 | 3.94 | 2.97 | 0.97
1.25 | 1.88 | 2.75 | -0.88 | 2.38 | 2.78 | -0.4 | 3.43 | 2.87 | 0.56 | 3.96 | 2.97 | 0.98
M | 0.17 | 1.53 | 2.64 | -1.11 | 2.06 | 2.67 | -0.61 | 3.12 | 2.78 | 0.34 | 3.63 | 2.89 | 0.74
1.25 | 1.57 | 2.65 | -1.08 | 2.1 | 2.68 | -0.58 | 3.15 | 2.79 | 0.36 | 3.67 | 2.89 | 0.77

Figure 3: SEDs of the case of the overabundance of small grains. Solid and
dashed lines indicate the modeled SED of the dust reemission for the two
different disk structures (thin belts and broad disks). The dotted lines
represent the direct stellar radiation. The top left plot shows the result
based on the single power law grain size distribution (OS1+UM1+SL1), while the
other three plots exhibit the results based on the broken power law grain size
distribution, specifically the overabundance of small grains with the
different scaling factors, 10, 50, and 100 (i.e., OS10, OS50, and OS100). See
Sect. 3.2 and Table 5 for details.

Considering our specific grain size distribution with overabundant small
grains and a deep depletion of grains in the hundreds of micrometers size
range, we expect spectral transitions to occur at sub-mm to mm wavelengths. To
characterize the SED in this wavelength range, we calculate the spectral index
within two distinct wavelength ranges: 450/850 $\mu{\rm{m}}$
(${\alpha}^{\,450\,\mu{\rm{m}}}_{\,850\,\mu{\rm{m}}}$) and 1000/2000
$\mu{\rm{m}}$ (${\alpha}^{\,1000\,\mu{\rm{m}}}_{\,2000\,\mu{\rm{m}}}$). These
spectral indices enable us to quantify the differences between the two
wavelength ranges, represented by $\Delta\,\alpha$ =
${\alpha}^{\,450\,\mu{\rm{m}}}_{\,850\,\mu{\rm{m}}}$ \-
${\alpha}^{\,1000\,\mu{\rm{m}}}_{\,2000\,\mu{\rm{m}}}$. Positive and negative
values of $\Delta\,\alpha$ indicate a lower and higher value of
${\alpha}^{\,1000\,\mu{\rm{m}}}_{\,2000\,\mu{\rm{m}}}$, representing a
shallower and steeper slope in 1000 to 2000 $\mu{\rm{m}}$ range compared to
450 to 850 $\mu{\rm{m}}$ range, respectively.

###  3.1 Single power law grain size distribution

We begin by examining the spectral index of the SED for the continuous grain
size distribution described by the single power law (i.e., $\gamma$ = -3.5).
This reference case will allow us to identify the influence of the
over/underabundance of grains of specific sizes. The top left plot of Figs. 3
and 5 shows the resulting SED for the two different disk structures (i.e.,
thin belt and broad disk) depending on the spectral type of the central stars
(see also Tables 5 and 6, column ”OS1+UM1+SL1”). For this reference case, the
SEDs show the ”typical” smooth and steady decrease in the FIR/mm wavelength
range, resulting from the steady decrease of the absorption cross-section
$C_{\rm{abs}}$ towards longer wavelengths and the steady decrease of the dust
grain temperature with increasing radial distance from the central star.

The slope of the SED slightly steepens with increasing the wavelength, i.e.,
the value of ${\alpha}^{\,1000\,\mu{\rm{m}}}_{\,2000\,\mu{\rm{m}}}$ is
slightly larger than ${\alpha}^{\,450\,\mu{\rm{m}}}_{\,850\,\mu{\rm{m}}}$. The
absolute amount of this difference between the spectral indices increases
towards stars of late spectral type. This can be explained by the shift of the
wavelength of the dust emission maximum towards longer wavelengths with
decreasing effective temperature of the central star. Consequently, the
spectral index ${\alpha}^{\,450\,\mu{\rm{m}}}_{\,850\,\mu{\rm{m}}}$ decreases,
leading to an increase of the absolute value of $\Delta\,\alpha$.

###  3.2 Broken power law grain size distribution

We now analyze the spectral index of the SED for models with a discontinuous
grain size distribution described by a broken power law. Our grain size
distribution model is defined by three distinct groups of grains: an
overabundance of small grains, a depletion of medium-sized grains, and a
standard abundance of larger grains (Sect. 2.2.2). We assess the individual
impact of each group on the resulting SEDs separately.

Overabundance of small grains Fig. 3 shows the modeled SED with a broken power
law grain size distribution depending on the overabundance of small grains,
disk structures (thin belts and broad disks), and varying stellar sources. The
corresponding spectral indices
${\alpha}^{\,450\,\mu{\rm{m}}}_{\,850\,\mu{\rm{m}}}$,
${\alpha}^{\,1000\,\mu{\rm{m}}}_{\,2000\,\mu{\rm{m}}}$, and their differences
$\Delta\,\alpha$ are listed in Table 5.

Figure 4: Modeled SED of debris disks with a narrow belt ($\Delta$$R$/$R$
$\sim$ 0.17) around an A-type star depending on the overabundance of small
grains (color dot lines). Total emissions (i.e., net SED including the
emission from the star and dust) are indicated with solid lines of the same
color for the overabundant of small grains (dashed line). See Table 5 for
details.

We find that an overabundance of small grains leads to a modest impact on the
mm slope of the SED, causing it to become slightly steeper (i.e., slightly
higher value of ${\alpha}^{\,1000\,\mu{\rm{m}}}_{\,2000\,\mu{\rm{m}}}$), but
predominantly affects and steepens the far-IR slope of the SED (i.e., higher
value of ${\alpha}^{\,450\,\mu{\rm{m}}}_{\,850\,\mu{\rm{m}}}$). This behavior
becomes more pronounced as the overabundance of small grains increases. In
particular, for models with an overabundance of small grains of a factor of 50
(OS50) – compared to the ideal collisional cascade model – we find a
significant kink in the slope of the SED in the 800 to 900 $\mu$m wavelength
range (see the bottom left plot of Fig. 3).

Furthermore, we find that the spectral index transition is more pronounced in
debris disks around brighter (hotter) central stars and larger radial extent.
The systems with broader debris rings (i.e., disks with an inner radius that
is close significantly close to the central star and an outer radius that is
located farther out, compared to their thin-ring counterpart) exhibit a larger
range of temperatures, with higher dust temperatures closer to the star,
resulting in stronger emissions with the location of their maximum shifted
towards shorter wavelengths. This leads to a steeper slope of the SED in the
450 - 850 $\mu\rm{m}$ wavelength range, e.g., larger absolute values of the
index ${\alpha}^{\,450\,\mu{\rm{m}}}_{\,850\,\mu{\rm{m}}}$. However, compared
to the impact of different abundance in the different grain size distribution,
the impact of the fractional width $\Delta\,R$/$R$ on the spectral index is
small.

Next, we investigate the contribution of grains in different size regimes to
the observed spectral index. For this purpose, we examine their individual
contributions to the total flux and thus their impact on the spectral indices
(see Fig. 4). Dust particles with sizes comparable to the observing wavelength
emit most efficiently in this wavelength regime (Backman & Paresce, 1993). We
find that small grains, which constitute a significant portion of the emitting
surface area in the grain size distribution, primarily affect the spectral
index ${\alpha}^{\,450\,\mu{\rm{m}}}_{\,850\,\mu{\rm{m}}}$ as they efficiently
re-emit in the far-IR, while their contribution to the long wavelength regime,
and thus their impact on
${\alpha}^{\,1000\,\mu{\rm{m}}}_{\,2000\,\mu{\rm{m}}}$, is negligible. On the
other hand, grains larger than the medium size (e.g., SL1), whose abundance is
held constant in this model, have a noticeable impact on the SED and the
spectral index in the mm range. In particular, grains larger than $\sim$ 1 mm,
which radiate like black bodies at longer wavelengths, cause a shallower slope
of the mm SED.

Underabundance of medium grains Fig. 5 shows the SED for the case of the
broken power law grain size distribution depending on the underabundance of
medium grains, disk structures (thin belts and broad disks), and spectral type
of the central stars. The corresponding spectral indices
${\alpha}^{\,450\,\mu{\rm{m}}}_{\,850\,\mu{\rm{m}}}$,
${\alpha}^{\,1000\,\mu{\rm{m}}}_{\,2000\,\mu{\rm{m}}}$, and their differences
$\Delta\,\alpha$ are listed in Table 6.

Table 6: Comparison of the spectral index of the considered discontinuous grain size distributions depending on the underabundance of medium grains, stellar parameters (e.g., stellar temperature and radius), and disk parameters (e.g., fractional width $\Delta$$R$/$R$). For the definition of model IDs (OS, UM, and SL) see Table 3. SpT | $\Delta$$R$/$R$ | Single power law | underabundance of medium grains
---|---|---|---
OS1+UM1+SL1 | OS1+UM10+SL1 | OS1+UM50+SL1 | OS1+UM100+SL1
${\alpha}^{\,450\,\mu{\rm{m}}}_{\,850\,\mu{\rm{m}}}$ | ${\alpha}^{\,1000\,\mu{\rm{m}}}_{\,2000\,\mu{\rm{m}}}$ | $\Delta\,\alpha$ | ${\alpha}^{\,450\,\mu{\rm{m}}}_{\,850\,\mu{\rm{m}}}$ | ${\alpha}^{\,1000\,\mu{\rm{m}}}_{\,2000\,\mu{\rm{m}}}$ | $\Delta\,\alpha$ | ${\alpha}^{\,450\,\mu{\rm{m}}}_{\,850\,\mu{\rm{m}}}$ | ${\alpha}^{\,1000\,\mu{\rm{m}}}_{\,2000\,\mu{\rm{m}}}$ | $\Delta\,\alpha$ | ${\alpha}^{\,450\,\mu{\rm{m}}}_{\,850\,\mu{\rm{m}}}$ | ${\alpha}^{\,1000\,\mu{\rm{m}}}_{\,2000\,\mu{\rm{m}}}$ | $\Delta\,\alpha$
A | 0.17 | 2.07 | 2.75 | -0.68 | 2.15 | 2.22 | -0.07 | 2.17 | 2.07 | 0.1 | 2.18 | 2.05 | 0.13
1.25 | 2.08 | 2.75 | -0.67 | 2.15 | 2.22 | -0.07 | 2.18 | 2.08 | 0.1 | 2.19 | 2.06 | 0.13
F | 0.17 | 1.99 | 2.76 | -0.77 | 2.09 | 2.23 | -0.14 | 2.13 | 2.07 | 0.06 | 2.14 | 2.04 | 0.1
1.25 | 2.00 | 2.77 | -0.77 | 2.1 | 2.24 | -0.14 | 2.14 | 2.07 | 0.07 | 2.15 | 2.05 | 0.1
G | 0.17 | 1.93 | 2.76 | -0.83 | 2.04 | 2.23 | -0.2 | 2.08 | 2.06 | 0.03 | 2.09 | 2.03 | 0.06
1.25 | 1.94 | 2.77 | -0.83 | 2.05 | 2.24 | -0.19 | 2.1 | 2.06 | 0.03 | 2.1 | 2.03 | 0.07
K | 0.17 | 1.86 | 2.75 | -0.89 | 1.97 | 2.22 | -0.25 | 2.03 | 2.04 | -0.01 | 2.04 | 2.01 | 0.02
1.25 | 1.88 | 2.75 | -0.88 | 1.99 | 2.23 | -0.24 | 2.04 | 2.04 | 0 | 2.05 | 2.02 | 0.03
M | 0.17 | 1.53 | 2.64 | -1.11 | 1.68 | 2.13 | -0.44 | 1.75 | 1.94 | -0.19 | 1.76 | 1.91 | -0.14
1.25 | 1.57 | 2.65 | -1.08 | 1.71 | 2.13 | -0.42 | 1.78 | 1.94 | -0.16 | 1.8 | 1.91 | -0.12

Figure 5: SEDs of the case of the underabundance of medium grains. Solid and
dashed lines indicate the modeled SED of the dust reemission for the two
different disk structures (thin belts and broad disks). The dotted lines
represent the direct stellar radiation. The top left plot shows the result
based on the single power law grain size distribution (OS1+UM1+SL1), while the
other three plots exhibit the results based on the broken power law grain size
distribution, specifically the underabundance of medium grains with the
different scaling factors, 10, 50, and 100 (i.e., UM10, UM50, and UM100). See
Sect. 3.2 and Table 6 for details.

For an exemplary case, the contribution of medium grains for different levels
of their underabundance to the SED is shown in Fig. 6. We find that an
underabundance of medium grains leads to a slight steepening of the far-IR
slope of the SED (i.e., the slightly higher value of
${\alpha}^{\,450\,\mu{\rm{m}}}_{\,850\,\mu{\rm{m}}}$). However, it primarily
affects the mm slope of the SED, which becomes shallower, corresponding to a
lower spectral index ${\alpha}^{\,1000\,\mu{\rm{m}}}_{\,2000\,\mu{\rm{m}}}$.
Therefore, the presence of both an overabundance of small grains and an
underabundance of medium grains increases the absolute difference between
${\alpha}^{\,450\,\mu{\rm{m}}}_{\,850\,\mu{\rm{m}}}$ and
${\alpha}^{\,1000\,\mu{\rm{m}}}_{\,2000\,\mu{\rm{m}}}$, leaving the 800 - 900
$\mu$m kink as a characteristic impact on the reemission SED.

Figure 6: Modeled SED of debris disks with a narrow belt ($\Delta$$R$/$R$
$\sim$ 0.17) around an A-type star depending on the underabundance of medium
grains (color dot lines). Total emissions (i.e., net SED including the
emission from the star and dust) are indicated with solid lines of the same
color for the underabundance of medium grains. See Table 5 for details.

However, we do not observe a significant spectral inversion solely due to the
underabundance of medium grains. Thus, we conclude that the impact of the
underabundance of medium grains on the spectral transition is weaker compared
to the effect of the overabundance of small grains. Additionally, we find that
the spectral index transition is more pronounced in debris disks around
brighter (hotter) central stars and broader disk systems, which is consistent
with the findings regarding the overabundance of small grains.

We note that the impact of the underabundance of medium grains could lead to
the higher value of ${\alpha}^{\,450\,\mu{\rm{m}}}_{\,850\,\mu{\rm{m}}}$ if
the disk is massive enough (e.g., more mass in the medium to large-sized
particles). However, this model could also lead to a steeper slope of the SED
at a longer wavelength (i.e., the higher value of
${\alpha}^{\,1000\,\mu{\rm{m}}}_{\,2000\,\mu{\rm{m}}}$). Thus, the grain size
model with the underabundance of medium grains hardly shows a noticeable
spectral inversion.

##  4 Discussion

###  4.1 Application to observations

Lestrade et al. (2020) highlighted that the spectral indices
${\alpha}^{\,1153\,\mu{\rm{m}}}_{\,2000\,\mu{\rm{m}}}$ derived from NIKA2
observations of the debris disks around Vega and HD 107146 are in agreement
with the expected values in the Rayleigh-Jeans regime, consistently around 2.0
$\pm$ 0.18 to 2.0 $\pm$ 0.32. Furthermore, MacGregor et al. 2016 found that mm
spectral indices ${\alpha}_{\rm mm}$ (i.e., the flux density for the complete
sample at a given mm wavelength combined with the VLA 9 mm or ATCA 7 mm
fluxes; Ricci et al., 2012, 2015a, 2015b) have the value of 1.63 to 3.08 with
mean values $\sim$ 2.55, inferring for the slope of the size distribution a
range from -2.84 (e.g., HD 141569) to -3.64 (e.g., HD 104860) with the
weighted mean of these value of -3.36 $\pm$ 0.02. These results indicate a
shallower slope of size distribution compared to standard collisional models.
Matthews et al. (2015) also indicated a spectral index of $\sim$ 2 desired at
sub-mm/mm wavelengths for AU Mic, which is lower than expected. These findings
are in contrast to the steeper spectral indices
${\alpha}^{\,450\,\mu{\rm{m}}}_{\,850\,\mu{\rm{m}}}$ measured at shorter
wavelengths, showing 2.8 to 2.9 (Holland et al., 2017). In general, the SONS
JCMT/SCUBA2 Legacy survey (Holland et al., 2017) found that the spectral
indices ${\alpha}^{\,450\,\mu{\rm{m}}}_{\,850\,\mu{\rm{m}}}$ in 48 debris
disks range from 2.7 to 4.7, indicating some size distributions steeper than
one based on the collisional cascade.

Further spectral index inversions were observed in the debris disks around q1
Eri (i.e., from 2.90 for ${\alpha}^{\,350\,\mu{\rm{m}}}_{\,860\,\mu{\rm{m}}}$
to 2.39 for ${\alpha}^{\,1260\,\mu{\rm{m}}}_{\,6800\,\mu{\rm{m}}}$; Holland et
al., 2017, Ricci et al., 2015b), HD 98800 (i.e., from 3.21 for
${\alpha}^{\,350\,\mu{\rm{m}}}_{\,870\,\mu{\rm{m}}}$ to 1.00 for
${\alpha}^{\,1350\,\mu{\rm{m}}}_{\,2000\,\mu{\rm{m}}}$; Nilsson et al., 2010,
Sylvester et al., 2001), and HD 95086 (i.e., from 3.18 for
${\alpha}^{\,500\,\mu{\rm{m}}}_{\,1300\,\mu{\rm{m}}}$ to 2.34 for
${\alpha}^{\,1300\,\mu{\rm{m}}}_{\,6800\,\mu{\rm{m}}}$; Su et al. 2017, Ricci
et al., 2015b).

###  4.2 Possible degeneracies with the grain composition

Table 7: Comparison of the spectral index of the considered discontinuous grain size distributions depending on the overabundance of small grains, disk parameters (e.g., fractional width $\Delta$$R$/$R$), and ice volume fraction $\mathcal{F_{\rm{ice}}}$. For the definition of model IDs (OS, UM, and SL) see Table 3. Composition | $\Delta$$R$/$R$ | Single power law |  overabundance of small grains (OS) or underabundance of medium grains (UM)
---|---|---|---
OS1+UM1+SL1 | OS10+UM1+SL1 | OS50+UM1+SL1 | OS100+UM1+SL1
${\alpha}^{\,450\,\mu{\rm{m}}}_{\,850\,\mu{\rm{m}}}$ | ${\alpha}^{\,1000\,\mu{\rm{m}}}_{\,2000\,\mu{\rm{m}}}$ | $\Delta\,\alpha$ | ${\alpha}^{\,450\,\mu{\rm{m}}}_{\,850\,\mu{\rm{m}}}$ | ${\alpha}^{\,1000\,\mu{\rm{m}}}_{\,2000\,\mu{\rm{m}}}$ | $\Delta\,\alpha$ | ${\alpha}^{\,450\,\mu{\rm{m}}}_{\,850\,\mu{\rm{m}}}$ | ${\alpha}^{\,1000\,\mu{\rm{m}}}_{\,2000\,\mu{\rm{m}}}$ | $\Delta\,\alpha$ | ${\alpha}^{\,450\,\mu{\rm{m}}}_{\,850\,\mu{\rm{m}}}$ | ${\alpha}^{\,1000\,\mu{\rm{m}}}_{\,2000\,\mu{\rm{m}}}$ | $\Delta\,\alpha$
Astrosil | 0.17 | 2.07 | 2.75 | -0.68 | 2.59 | 2.77 | -0.18 | 3.67 | 2.86 | 0.81 | 4.24 | 2.95 | 1.28
1.25 | 2.08 | 2.75 | -0.67 | 2.6 | 2.78 | -0.18 | 3.68 | 2.87 | 0.81 | 4.25 | 2.96 | 1.29
$\mathcal{F_{\rm{ice}}}$ = 0.5 | 0.17 | 2.55 | 2.82 | -0.26 | 2.92 | 2.85 | 0.07 | 3.72 | 2.99 | 0.73 | 4.12 | 3.11 | 1.01
1.25 | 2.56 | 2.82 | -0.26 | 2.93 | 2.86 | 0.07 | 3.73 | 3.00 | 0.73 | 4.13 | 3.12 | 1.02
$\mathcal{P}$ = 0.5 | 0.17 | 2.65 | 2.87 | -0.23 | 2.91 | 2.93 | -0.01 | 3.47 | 3.09 | 0.38 | 3.74 | 3.22 | 0.52
1.25 | 2.65 | 2.88 | -0.23 | 2.92 | 2.93 | -0.01 | 3.48 | 3.10 | 0.38 | 3.75 | 3.22 | 0.53
|  | OS1+UM1+SL1 | OS1+UM10+SL1 | OS1+UM50+SL1 | OS1+UM100+SL1
|  | ${\alpha}^{\,450\,\mu{\rm{m}}}_{\,850\,\mu{\rm{m}}}$ | ${\alpha}^{\,1000\,\mu{\rm{m}}}_{\,2000\,\mu{\rm{m}}}$ | $\Delta\,\alpha$ | ${\alpha}^{\,450\,\mu{\rm{m}}}_{\,850\,\mu{\rm{m}}}$ | ${\alpha}^{\,1000\,\mu{\rm{m}}}_{\,2000\,\mu{\rm{m}}}$ | $\Delta\,\alpha$ | ${\alpha}^{\,450\,\mu{\rm{m}}}_{\,850\,\mu{\rm{m}}}$ | ${\alpha}^{\,1000\,\mu{\rm{m}}}_{\,2000\,\mu{\rm{m}}}$ | $\Delta\,\alpha$ | ${\alpha}^{\,450\,\mu{\rm{m}}}_{\,850\,\mu{\rm{m}}}$ | ${\alpha}^{\,1000\,\mu{\rm{m}}}_{\,2000\,\mu{\rm{m}}}$ | $\Delta\,\alpha$
Astrosil | 0.17 | 2.07 | 2.75 | -0.68 | 2.15 | 2.22 | -0.07 | 2.17 | 2.07 | 0.1 | 2.18 | 2.05 | 0.13
1.25 | 2.08 | 2.75 | -0.67 | 2.15 | 2.22 | -0.07 | 2.18 | 2.08 | 0.1 | 2.19 | 2.06 | 0.13
$\mathcal{F_{\rm{ice}}}$ = 0.5 | 0.17 | 2.55 | 2.82 | -0.26 | 2.26 | 2.29 | -0.03 | 2.17 | 2.19 | -0.02 | 2.15 | 2.17 | -0.02
1.25 | 2.56 | 2.82 | -0.26 | 2.26 | 2.29 | -0.03 | 2.17 | 2.19 | -0.02 | 2.16 | 2.17 | -0.01
$\mathcal{P}$ = 0.5 | 0.17 | 2.65 | 2.87 | -0.23 | 2.24 | 2.33 | -0.08 | 2.13 | 2.23 | -0.10 | 2.11 | 2.22 | -0.11
1.25 | 2.65 | 2.88 | -0.23 | 2.25 | 2.33 | -0.08 | 2.13 | 2.23 | -0.10 | 2.13 | 2.23 | -0.10

Figure 7: SEDs of the case of the overabundance of small grains depending on
the dust composition (ice inclusion $\mathcal{F}_{\rm ice}$ = 0.5 and porosity
$\mathcal{P}$ = 0.5. Solid and dashed lines indicate the modeled SED of the
dust reemission for the two different disk structures (thin belts and broad
disks). The dotted lines represent the direct stellar radiation from A star.
The top left plot shows the result based on the single power law grain size
distribution (OS1+UM1+SL1), while the other three plots exhibit the results
based on the broken power law grain size distribution, specifically the
overabundance of small grains with the different scaling factors, 10, 50, and
100 (i.e., OS10, OS50, and OS100). See Sect. 3.2 and Table 7 for details.

Figure 8: SEDs of the case of the overabundance of small grains depending on
the dust composition (ice inclusion $\mathcal{F}_{\rm ice}$ = 0.5 and porosity
$\mathcal{P}$ = 0.5. Solid and dashed lines indicate the modeled SED of the
dust reemission for the two different disk structures (thin belts and broad
disks). The dotted lines represent the direct stellar radiation from A star.
The top left plot shows the result based on the single power law grain size
distribution (OS1+UM1+SL1), while the other three plots exhibit the results
based on the broken power law grain size distribution, specifically the
underabundance of medium grains with the different scaling factors, 10, 50,
and 100 (i.e., UM10, UM50, and UM100). See Sect. 3.2 and Table 7 for details.

Variations in dust grain compositions, such as the inclusion of ice or
consideration of porosity, affect the absorption and scattering efficiencies
(Kim et al., 2019), thereby influencing the overall shape of SED (Lebreton et
al., 2012; Schüppler et al., 2015). To understand the effect of dust
compositions on the spectral inversion and potential degeneracy, we examine
the implications of altering chemical compositions, specifically by
integrating crystalline water ice into icy-astrosilicate mixtures (e.g., 50%
ice volume fraction ${\mathcal{F}}_{\rm ice}$ = 0.5) and the microstructure of
the dust grains with introducing moderate porosity (e.g., 50% porosity
$\mathcal{P}$ = 0.5) through effective medium approximations (e.g., Maxwell-
Garnett rule; Maxwell Garnett, 1904) based on the grain size distribution
described by the single power law and broken power law.

In cases of single power law grain size distribution and a lower
over/underabundance of small/medium grains of the broken power law grain size
distribution (e.g., OS10+UM1+SL1 and OS1+UM10+SL1; see the first and second
columns of Table 7, and Figs. 7 and 8), resulting in a decrease in the
difference in their slopes (i.e., smaller $\Delta\alpha$) and might help the
situation. However, these modifications not only steepen the slope in the far-
IR range but also affect the mm range, resulting in relatively small spectral
inversions. Furthermore, both models lead to even smaller deviations in the
spectral slopes, particularly with an increase in over/underabundance of
small/medium grains based on the broken power law grain size distribution
model (e.g., OS50+UM1+SL1 and OS1+UM50+SL1; see the third and fourth columns
of Table 7, and Figs. 7 and 8). Consequently, these modifications to dust
compositions alone are insufficient to reproduce the observed spectral
inversion. This indicates that the specific composition is not the sole factor
responsible for the spectral inversions seen in the far-infrared to mm ranges
debris disk SEDs.

Overall, the effect of ice inclusion seems stronger than that of porosity in
cases of an increase in the over/underabundance of small/medium grains. On the
other hand, for the case of underabundance of medium grains, all of our models
with the modification of dust composition fail to reproduce spectral
inversion, highlighting the more significant effect of a higher abundance of
small grains. Moreover, the fractional width does not seem to play a
significant role in these outcomes, suggesting that the spectral
characteristics are largely independent of this aspect of grain size
distribution. This finding is consistent with our previous results in Sect. 3.

###  4.3 Possible connection between the inferred size distribution and the
collisional state of the disk

As MacGregor et al. (2016) suggested, the slope of the grain size distribution
$\gamma$ can be derived from the measurement of the mm spectral index
$\alpha_{\rm mm}$ based on the dust opacity power law index of small particles
(e.g., compared to observing wavelengths) $\beta_{\rm s}$ and an inferred
power-law exponent from the dust temperature for each disk (i.e., spectral
index of the Planck function $\alpha_{\rm PL}$) given wavelength ranges. Since
most astronomical dust compositions have very similar values of $\beta_{\rm
s}$ (e.g., 1.5 - 2) and the dust temperature only affects the Planck function
weakly at long wavelengths (e.g., $\alpha_{\rm PL}$ $\sim$ 1.9 to 1.95 with
the value of 2 as the Rayleigh-Jeans limit), $\alpha_{\rm mm}$ directly
corresponds to the slope of the grain size distribution $\gamma$.

The inferred range of the parameters $\gamma$ based on $\alpha_{\rm mm}$ for
the overabundance of small grains from our models (e.g., $\sim$ -3.5 to -3.6
regardless of stellar properties and dust compositions) is in good agreement
with the predictions of steady-state collisional models, both from the
classical model and modern models (e.g., Dohnanyi, 1969; Marshall et al.,
2017), while the one from underabundance of medium grains showing a bit
shallower slopes of grain size distributions (e.g., $\gamma$ $\sim$ -3.0),
which can be explained by the size dependence of the forces removing the
smaller dust grains from the system (e.g., PR drag and stellar radiation).

These observational results have yielded valuable insights, yet they also
highlight the intrinsic challenges faced in such measurements, primarily due
to the typical uncertainties (e.g., around 5 to 15%) present in observational
data, leading to spectral inversions that differ by approximately $\sim$ 10%.
These differences may stem from the varying sensitivities of telescopes, along
with the diverse calibration and reduction techniques employed to mitigate
atmospheric conditions, optical flaws, and environmental sensitivity, all of
which impact the accuracy and precision of observations. Additionally, our
studies show variability in spectral indices, which appears to depend on the
specific properties of the considered debris disks (see Sect. 3). The
complexity of these factors underscores advanced solutions, which are
anticipated to be addressed by the ngVLA (Next Generation Very Large Array; Di
Francesco et al. 2019).

###  4.4 Effect of the blow-out size dependent grain size distribution on the
SEDs

We also consider the grain size distribution with the overabundance and/or
underabundance of different grain sizes as a function of stellar properties
(i.e., blow-out size; see also Sect. 2.2.2). We find that the variations in
the blowout size and spectral type that are responsible for the location of
the first and second maximum/minimum in the grain size distribution (see also
Fig. 2 and Table 4), in turn, change the spectral slope of the SED at varying
wavelength regimes (Figs. 9 and 10). For example, a grain size distribution
with an overabundance of small grains in the case of an A-type star displays a
change of the slope at significantly longer wavelengths (e.g., $\sim$ a few
mm). On the other hand, K and M-type stars show this impact at even shorter
wavelengths (e.g., $\sim$ a few hundred micrometers; see Fig. 9). In the case
of an F-type star, the results are comparable to those discussed in previous
Sect. 2.2.2 because of similar grain size distribution. These results
illustrate that the location of the gap in the grain size distribution has a
direct impact on the wavelength of which the slope of SED is affected. This
finding highlights the impact of the blow-out size on the resulting -even sub-
mm/mm- slope of the SED. Furthermore, we find that the underabundance of
medium-sized grains has a rather weak impact on the changing slope of the SED
(see Figs. 9 and 10; Tables 8 and 9), which is consistent with our previous
findings discussed in Sect. 3.2.

The results discussed in this section suggest the need to constrain the slope
of the SED in the different wavelength regimes to disentangle the impact of
the spectral type from the impact of potentially, other physical mechanisms on
the grain size distribution and thus the slope of the SED. However, we note
that previous observational studies (e.g., Lestrade et al., 2020; MacGregor et
al., 2016; see also Sect. 4.1) have not identified a clear dependence of the
impact of stellar spectral type on the slope of the SED, showing consistent
observed spectral breaks occurring at similar wavelength regions, irrespective
of the central star types.

Figure 9: SEDs of the case of the overabundance of small grains depending on the stellar properties (i.e., blow-out sizes). Solid and dashed lines indicate the modeled SED of the dust reemission for the two different disk structures (thin belts and broad disks). The dotted lines represent the direct stellar radiations. The top left plot shows the result based on the single power law grain size distribution (OS1+UM1+SL1), while the other three plots exhibit the results based on the broken power law grain size distribution, specifically the overabundance of small grains with the different scaling factors, 10, 50, and 100 (i.e., OS10, OS50, and OS100). See Table 4 and Fig. 2 for the proposed grain size distributions depending on the central star and blow-out sizes (Thébault & Augereau, 2007; Kim et al., 2018). Table 8: Comparison of the spectral index of the proposed discontinuous grain size distributions (see Table 4 and Fig. 2) depending on the overabundance of small grains, stellar parameters (e.g., stellar temperature and radius), and disk parameters (e.g., fractional width $\Delta$$R$/$R$). For the locations of the first and second maximum/minimum in grain size distribution depending on the central star for the simulations see Table 4. SpT | $\Delta$$R$/$R$ | Single power law | overabundance of small grains
---|---|---|---
OS1+UM1+SL1 | OS10+UM1+SL1 | OS50+UM1+SL1 | OS100+UM1+SL1
${\alpha}^{\,450\,\mu{\rm{m}}}_{\,850\,\mu{\rm{m}}}$ | ${\alpha}^{\,1000\,\mu{\rm{m}}}_{\,2000\,\mu{\rm{m}}}$ | $\Delta\,\alpha$ | ${\alpha}^{\,450\,\mu{\rm{m}}}_{\,850\,\mu{\rm{m}}}$ | ${\alpha}^{\,1000\,\mu{\rm{m}}}_{\,2000\,\mu{\rm{m}}}$ | $\Delta\,\alpha$ | ${\alpha}^{\,450\,\mu{\rm{m}}}_{\,850\,\mu{\rm{m}}}$ | ${\alpha}^{\,1000\,\mu{\rm{m}}}_{\,2000\,\mu{\rm{m}}}$ | $\Delta\,\alpha$ | ${\alpha}^{\,450\,\mu{\rm{m}}}_{\,850\,\mu{\rm{m}}}$ | ${\alpha}^{\,1000\,\mu{\rm{m}}}_{\,2000\,\mu{\rm{m}}}$ | $\Delta\,\alpha$
A | 0.17 | 2.33 | 2.58 | -0.25 | 3.03 | 3.67 | -0.63 | 3.25 | 4.97 | -1.73 | 3.28 | 5.43 | -2.15
1.25 | 2.34 | 2.58 | -0.25 | 3.04 | 3.67 | -0.63 | 3.26 | 4.98 | -1.72 | 3.29 | 5.44 | -2.15
F | 0.17 | 2.27 | 2.76 | -0.50 | 3.61 | 2.89 | 0.72 | 4.60 | 3.24 | 1.36 | 4.84 | 3.45 | 1.39
1.25 | 2.28 | 2.77 | -0.49 | 3.62 | 2.90 | 0.72 | 4.62 | 3.24 | 1.37 | 4.85 | 3.46 | 1.40
G | 0.17 | 2.56 | 2.93 | -0.37 | 2.74 | 2.98 | -0.24 | 3.14 | 3.13 | 0.01 | 3.33 | 3.23 | 0.10
1.25 | 2.58 | 2.94 | -0.36 | 2.76 | 2.99 | -0.23 | 3.15 | 3.13 | 0.01 | 3.34 | 3.23 | 0.11
K | 0.17 | 2.59 | 3.21 | -0.62 | 2.73 | 3.24 | -0.51 | 3.03 | 3.32 | -0.29 | 3.18 | 3.36 | -0.19
1.25 | 2.61 | 3.22 | -0.61 | 2.74 | 3.25 | -0.51 | 3.04 | 3.32 | -0.29 | 3.18 | 3.37 | -0.19
M | 0.17 | 2.29 | 3.14 | -0.85 | 2.50 | 3.18 | -0.67 | 2.89 | 3.27 | -0.38 | 3.06 | 3.32 | -0.26
1.25 | 2.33 | 3.15 | -0.82 | 2.53 | 3.18 | -0.66 | 2.90 | 3.27 | -0.37 | 3.06 | 3.32 | -0.26

Figure 10: SEDs of the case of the underabundance of medium grains depending on the stellar properties (i.e., blow-out sizes). Solid and dashed lines indicate the modeled SED of the dust reemission for the two different disk structures (thin belts and broad disks). The dotted lines represent the direct stellar radiations. The top left plot shows the result based on the single power law grain size distribution (OS1+UM1+SL1), while the other three plots exhibit the results based on the broken power law grain size distribution, specifically the underabundance of medium grains with the different scaling factors, 10, 50, and 100 (i.e., UM10, UM50, and UM100). See Table 4 and Fig. 2 for the proposed grain size distributions depending on the central star and blow-out sizes (Thébault & Augereau, 2007; Kim et al., 2018). Table 9: Comparison of the spectral index of the proposed discontinuous grain size distributions (see Table 4 and Fig. 2) depending on the underabundance of medium grains, stellar parameters (e.g., stellar temperature and radius), and disk parameters (e.g., fractional width $\Delta$$R$/$R$). For the locations of the first and second maximum/minimum in grain size distribution depending on the central star for the simulations see Table 4. SpT | $\Delta$$R$/$R$ | Single power law | underabundance of medium grains
---|---|---|---
OS1+UM1+SL1 | OS1+UM10+SL1 | OS1+UM50+SL1 | OS1+UM100+SL1
${\alpha}^{\,450\,\mu{\rm{m}}}_{\,850\,\mu{\rm{m}}}$ | ${\alpha}^{\,1000\,\mu{\rm{m}}}_{\,2000\,\mu{\rm{m}}}$ | $\Delta\,\alpha$ | ${\alpha}^{\,450\,\mu{\rm{m}}}_{\,850\,\mu{\rm{m}}}$ | ${\alpha}^{\,1000\,\mu{\rm{m}}}_{\,2000\,\mu{\rm{m}}}$ | $\Delta\,\alpha$ | ${\alpha}^{\,450\,\mu{\rm{m}}}_{\,850\,\mu{\rm{m}}}$ | ${\alpha}^{\,1000\,\mu{\rm{m}}}_{\,2000\,\mu{\rm{m}}}$ | $\Delta\,\alpha$ | ${\alpha}^{\,450\,\mu{\rm{m}}}_{\,850\,\mu{\rm{m}}}$ | ${\alpha}^{\,1000\,\mu{\rm{m}}}_{\,2000\,\mu{\rm{m}}}$ | $\Delta\,\alpha$
A | 0.17 | 2.33 | 2.58 | -0.25 | 2.68 | 2.66 | 0.02 | 2.74 | 2.68 | 0.07 | 2.75 | 2.68 | 0.07
1.25 | 2.34 | 2.58 | -0.25 | 2.69 | 2.66 | 0.02 | 2.75 | 2.68 | 0.07 | 2.76 | 2.68 | 0.08
F | 0.17 | 2.27 | 2.76 | -0.50 | 2.87 | 2.28 | 0.59 | 3.05 | 2.13 | 0.92 | 3.08 | 2.11 | 0.97
1.25 | 2.28 | 2.77 | -0.49 | 2.88 | 2.28 | 0.60 | 3.07 | 2.13 | 0.94 | 3.10 | 2.11 | 0.99
G | 0.17 | 2.56 | 2.93 | -0.37 | 2.13 | 2.31 | -0.17 | 1.97 | 2.17 | -0.20 | 1.94 | 2.15 | -0.21
1.25 | 2.58 | 2.94 | -0.36 | 2.14 | 2.31 | -0.17 | 1.97 | 2.17 | -0.20 | 1.95 | 2.15 | -0.21
K | 0.17 | 2.59 | 3.21 | -0.62 | 2.06 | 2.29 | -0.24 | 1.89 | 2.15 | -0.27 | 1.86 | 2.13 | -0.27
1.25 | 2.61 | 3.22 | -0.61 | 2.07 | 2.30 | -0.23 | 1.90 | 2.16 | -0.26 | 1.87 | 2.14 | -0.27
M | 0.17 | 2.29 | 3.14 | -0.85 | 1.80 | 2.20 | -0.40 | 1.64 | 2.05 | -0.41 | 1.62 | 2.04 | -0.42
1.25 | 2.33 | 3.15 | -0.82 | 1.83 | 2.21 | -0.38 | 1.67 | 2.06 | -0.40 | 1.64 | 2.04 | -0.40

##  5 Summary

Motivated by the expected deviation of the grain size distribution from the
often assumed continuous distribution described by a single power law, we
investigated the impact of the over- and underabundance of grains in specific
size regimes on the SED of debris disks. Based on an exemplary, simple model,
we quantified and discussed the impact of different values of
over/underabundances of submicrometer to mm-sized grains in a debris dust ring
with two different fractional widths $\Delta\,R/R$ around central stars with
spectral types A, F, G, K, and M on the slope of the FIR to mm SED of these
systems. We described this shape and its the spectral indices
${\alpha}^{450\,\mu{\rm{m}}}_{850\,\mu{\rm{m}}}$ and
${\alpha}^{1000\,\mu{\rm{m}}}_{2000\,\mu{\rm{m}}}$, and discussed their
differences $\Delta\,\alpha$. Our key findings are:

  * –

The deviation of a grain size distribution from that described by a single
power law can be potentially derived from the analysis of the SED in the FIR
to mm wavelength range. More specifically, the characteristics of the slope of
the SED via spectral indices and wavelengths at which photometric data can be
obtained even from ground-based telescopes appear promising. In the considered
model, characterized by a chosen grain size distribution, i.e., the specific
size range of grains for which an over/underabundance was assumed, we find a
characteristic kink in the SED around 800 to 900 $\mu$m.

  * –

As the change of the slope and the wavelength range of its occurrence is
directly linked to the deviation of the grain size distribution from a single
power law distribution, these observations potentially provide the key for
constraining the underlying physical processes leading to the specific grain
size distribution. In particular, these observations provide the potential to
constrain the collisional physics (e.g., material strength) of the otherwise
hardly accessible properties of the larger grains ($\gg$ cm).

  * –

The presence of the overabundance of small grains and the underabundance of
medium grains both contribute to a steeper slope in the SED at the far-IR
range. However, their effects differ at mm wavelengths. Specifically, the
overabundance of small grains primarily results in a much higher value
${\alpha}^{\,450\,\mu{\rm{m}}}_{\,850\,\mu{\rm{m}}}$ (i.e., steeper slope),
while the ${\alpha}^{\,1000\,\mu{\rm{m}}}_{\,2000\,\mu{\rm{m}}}$ becomes
slightly higher (i.e, slightly steeper slope). On the other hand, the
underabundance of medium grains leads to a higher value of
${\alpha}^{\,450\,\mu{\rm{m}}}_{\,850\,\mu{\rm{m}}}$ (i.e, slightly steeper
slope), but primarily results in a lower value of
${\alpha}^{\,1000\,\mu{\rm{m}}}_{\,2000\,\mu{\rm{m}}}$ (i.e., shallower
slope). The impact of the overabundance of small dust particles is stronger
than that of the underabundance of medium-sized dust particles.

  * –

We found that the spectral index transition is more pronounced for debris
disks around brighter, i.e., early-type central stars and broader disks.
However, the disk structure such as the fractional width of the disk system
leaves only little effect on the FIR to mm slope of the SED if compared to the
impact of the grain size distributions.

  * –

We found that dust composition is not the sole physical mechanism responsible
for the spectral inversion observed in the far-IR to (sub-)mm part of the SEDs
of debris disk systems.

  * –

If we assume that the location of the first and second maximum/minimum of the
grain size distribution depends on the blow-out size, and thus on the stellar
spectral type, the corresponding impact on the sub-mm/mm on SED is wavelength-
dependent.

###### Acknowledgements.

The authors thank J.-F. Lestrade for fruitful discussions on the modeling of
specific grain size distribution in this study. MK gratefully acknowledges the
funding from the Royal Society. MK and SW gratefully acknowledge the financial
support by the Deutsche Forschungsgemeinschaft under contracts _WO 857/15-1_
and _WO 857/15-2_ (DFG).

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