11institutetext: European Southern Observatory, Karl-Schwarzschild-Str. 2,
D-85748 Garching b. München, Germany; email<EMAIL_ADDRESS>
# Dark dust II: Properties in the general field of the diffuse ISM

R. Siebenmorgen

(Received April 25, 2022 / Accepted xxx, xxx)

Distance estimates derived from spectroscopy or parallax have been unified by
considering extinction by large grains. The addition of such a population of
what is called dark dust to models of the diffuse interstellar medium is
tested against a contemporary set of observational constraints. The dark dust
model explains, by respecting representative solid-phase element abundances,
simultaneously the typical wavelength-dependent reddening, extinction, and
emission of polarized and unpolarized light by interstellar dust particles
between far UV and millimetre wavelengths. The physical properties of dark
dust are derived. Dark dust consists of micrometre-sized particles, which have
been recently detected in$-$situ. It provides significant wavelength-
independent reddening from the far UV to the near infrared. Light absorbed by
dark dust is re-emitted in the submillimeter region by grains at dust
temperatures of $8-12$ K. Such very cold dust has been frequently observed in
external galaxies. Dark dust contributes to the polarisation at
$\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise 5.0pt\hbox{$\char
62\relax$}\hss}\mathchar 29208\relax$}}\,1$ mm to $\sim 35$ % and at shorter
wavelengths marginally. Optical constants for silicate dust analogous are
investigated. By mixing 3% in mass of Mg0.8Fe${}^{2+}_{0.2}$ SiO3 to MgO$-$0.5
SiO2 a good fit to the data is derived that still can accommodate up to 5 -
10% of mass in dark dust. The extra diming of light by dark dust is unexplored
when discussing SN Ia light curves and in other research. Previous models that
ignore dark dust do not account for the unification of the distance scales.

###### Key Words.:

(ISM:) dust, extinction, Polarization, (ISM:) Infrared: ISM, Stars: distances

##  1 Introduction

Physical properties of dust in the diffuse interstellar medium (ISM) are
derived by confronting models against observational constraints. Two classes
of models emerged over the years, either those that employ distinct grain
populations (Mathis et al., 1977; Draine & Lee, 1984; Desert et al., 1990;
Weingartner & Draine, 2001; Zubko et al., 2004; Compiègne et al., 2011;
Siebenmorgen et al., 2014) or models which are using a mixture of different
grain constituents forming composite particles (Hage & Greenberg, 1990; Mathis
& Whiffen, 1989; Ossenkopf, 1991; Krügel & Siebenmorgen, 1994; Greenberg et
al., 1995; Voshchinnikov, 2012; Jones et al., 2013; Siebenmorgen et al., 2014;
Köhler et al., 2015; Ysard et al., 2015; Jones et al., 2017; Guillet et al.,
2018; Draine & Hensley, 2021a). The present observational status of ISM dust
characteristics after the Planck mission (Planck Collaboration et al., 2014)
has been reviewed by Hensley & Draine (2021). Dust models shall agree with
several major observations:

a) The solid-phase element abundances in the medium out of which the dust is
made (Hensley & Draine, 2021).

b) The reddening curves in the Milky Way (Fitzpatrick & Massa, 2007; Gordon et
al., 2009; Fitzpatrick et al., 2019).

c) The diffuse galactic dust emission that has been observed in the wavelength
range from a few $\mu$m up to several mm by space missions ISO, AKARI,
Spitzer, WMAP, DIRBE (Dwek et al., 1997), and Planck (Planck Collaboration et
al., 2014).

d) The optical/near-infrared (NIR) starlight polarization (Serkowski et al.,
1975), which is due to the dichroic extinction of aligned non-spherical dust
particles (Hong & Greenberg, 1980; Krügel, 2008; Draine & Fraisse, 2009;
Siebenmorgen et al., 2014; Voshchinnikov, 2012).

e) The polarized emission spectrum of the same grains which is predominantly
observed by Planck Collaboration et al. (2020). Strikingly previous dust
models in the pre-Planck era systematically under-predict the observed
submillimeter (submm) and mm emission of unpolarized light from about $0.3-3$
mm, and did not explain the flatness of the polarized dust emission spectrum
in that wavelength range.

f) In addition, the unification of spectroscopically derived distances with
the parallax is not closed.

However, previous dust models failed to explain all observational constraints
that have been available in the post-Planck era simultaneously (Ysard, 2020),
unless the model by Draine & Hensley (2021a), in which the optical constants
of amorphous silicates were modified to fit the Planck data.

For a star the spectro-photometric distance $D_{\rm{SpL}}$ shall agree with
the distance $D_{\rm{GAIA}}$ derived from parallax. Geometric distances using
GAIA (Gaia Collaboration et al., 2016) as derived by Bailer-Jones et al.
(2018, 2021) show for OB stars with available reddening curves at distances
below 2 kpc a fractional error of $\sim 5$ %. The photometric distance of a
star is connected via the apparent $m_{\rm V}$ and absolute magnitude $M_{\rm
V}$, and the dust extinction $A_{V}$ along that sightline:
$\log{D_{\rm{SpL}}}=0.2\ (m_{V}-M_{V}-A_{V}+5)$ pc. Photometric distances
require accurate calibration of the spectral type and luminosity class (SpL)
of the star allowing derivation of $M_{V}$ and $A_{V}$ (Eq.4). The original
SpL estimates of OB stars (Walborn, 1971; Walborn & Fitzpatrick, 1990) have
gained significantly in precision using results of the galactic O star
spectroscopic survey (Maíz-Apellániz et al., 2004) with SpL updates by Sota et
al. (2011, 2014) and applying quantitative spectral classification schemes
utilizing high-resolution optical spectroscopy (Markova et al., 2011; Martins,
2018). Standard grids of absolute magnitudes for OB stars are provided (Bowen
et al., 2008; Pecaut & Mamajek, 2013)111http://www.pas.rochester.edu/ẽmamajek/
as well as high-quality extinction estimates (Maíz Apellániz & Barbá, 2018).
This progress enables computing spectro-photometric distances of OB stars at
fractional precision of $\sim 15\%$ for the nearby (2 kpc) sample. Photometric
versus parallax distance estimates to galactic OB stars have been discussed by
Shull & Danforth (2019) and reveal larger uncertainties at larger distances.
In the nearby sample, the spectroscopic distances show a systematic
overestimate with a dispersion above these errors (Siebenmorgen et al., 2020).
Both distance estimates of the same source shall of course agree.

Already Trumpler (1930) included in the original form of the photometric
equation, in addition to the wavelength-dependent (selective) interstellar
extinction, a constant, wavelength-independent (non-selective) extinction
component in the optical. He proposed that light may be obscured by large
(’meteoritic’) particles. Large micrometre-sized dust particles are frequently
found in circumstellar shells (Strom et al., 1971; Jones, 1972; Lanz et al.,
1995; Steinacker et al., 2015; Kataoka et al., 2017), Herbig stars (Dunkin &
Crawford, 1998), $\eta$ Car (Andriesse et al., 1978) and other evolved stars
(Scicluna et al., 2015, 2022). Grains larger than $2\,\mu$m have been seen in
scattering light halos around X-ray sources (Witt et al., 2001) and grains as
large as $\sim 4\,\mu$m have been suggested for V404 Cygni by Heinz et al.
(2016). The emission from 100 $\mu$m large particles accounting for the
observed submm fluxes from evolved giants are derived by Jura et al. (2001).
Maercker et al. (2022) finds that for carbon stars preferentially
$\sim\,2\mu$m sized grains survive the AGB wind-ISM interaction regions and
act later as seeds for grain growth in the ISM.

Distance estimates of the Orion Trapezium star HD 37020 using spectral type-
luminosity distance $D_{\rm SpL}$ are a factor 2.5 larger than the Ca ii or
VLBI parallax distance estimates (Krełowski et al., 2016). For several nearby
($\la 400$ pc) OB stars a significant overestimate of the spectroscopic
distance over the HIPPARCOS parallax is reported by Skórzyński et al. (2003).
Assuming that the parallax has been correctly measured the derived absolute
magnitudes of these stars appear too faint. ”Super large” grains are suggested
for the extra weakening of the observed brightness of these stars. The column
density of this very large grain population is well correlated with the
strength of DIB 6367 Å and DIB 6425 Å and therefore appears distributed along
the sightline through the diffuse ISM. The spectral type-luminosity distances
of 132 OB stars have been compared to those derived by GAIA up to $D_{\rm
GAIA}<2$ kpc (Siebenmorgen et al., 2020) and for $\sim 10$% of the sample
$D_{\rm{SpL}}/D_{\rm{GAIA}}\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise
5.0pt\hbox{$\char 62\relax$}\hss}\mathchar 29208\relax$}}2$. Both distance
estimates have been unified introducing $0.3-0.7$ mag reddening by what they
called dark dust. The terminology follows the unveiling of dark gas by Grenier
et al. (2005) because gas and dust are intimately mixed in the diffuse ISM. It
describes a mysterious, dimmed and hidden nebulosity. Extensive clouds of dark
gas have been weighted using Fermi-LAT $\gamma$-ray data by Widmark et al.
(2022) and emerge in the cold ISM. The hidden dust component appears in
sightlines that are connected to the cold ISM as well and have line intensity
ratio of CH 4300 Å to CH+ 4233 Å greater than one and CN 3875 Å detected.
Possibly a sticking of smaller grains into larger units is favoured in cold
ISM environments. A circumstellar nature of dark dust towards these stars is
excluded from inspecting WISE imaging between $3-22\,\mu$m. Micrometre-sized
particles from the diffuse ISM have been measured in$-$situ from the Ulysses,
Galileo, and Stardust space probes to the outer solar system (Landgraf et al.,
2000; Westphal et al., 2014; Krüger et al., 2015).

The emission of dark dust as a new component of the diffuse ISM should be
observable in the submm/mm wavelength range. It will absorb a fraction of the
interstellar radiation field, ISRF (Mathis et al., 1983). As such grains are
big they are cold and will emit at long wavelengths. Originally very cold
($\la 15$ K) dust emission has been detected in the general field of the ISM
of non-active galaxies (Krügel et al., 1998; Siebenmorgen et al., 1999) and in
our Galaxy towards high-density regions (Chini et al., 1993). More recently,
the Herschel KINGFISH (Kennicutt et al., 2011) and Dwarf Galaxy (Madden et
al., 2013) surveys find that 35 out of 78 galaxies have excess emission at 0.5
mm that cannot be explained by a single modified black-body temperature
component with dust emissivity spectral index of two (Rémy-Ruyer et al.,
2013a). Other galaxies, e.g. Haro 11 (Galliano et al., 2005; Galametz et al.,
2009), show the excess emission at even longer wavelengths in the mm range and
were missed by Herschel surveys. The excess emission is due to very cold dust
at temperatures as low as 10 K. A solution for explaining the submm excess
emission is often favoured by ad-hoc adjustments of the emissivity law of the
grains (Rémy-Ruyer et al., 2013b; Guillet et al., 2018). This can be derived
by changes in the porosity, shape, or adjustments of the optical constants
(Draine & Hensley, 2021a). Models changing the spectral index of the dust
emissivity do not touch on the issue of distance unification.

In this paper, a model for the general field in the diffuse ISM is described
that accounts simultaneously for observations of solid-phase elemental
abundances, average Milky Way reddening, IR to mm emission at high galactic
latitudes, average Milky Way polarized extinction and polarized emission by
dust. A particular feature is the inclusion of dark dust as an additional dust
population that accounts for the very cold emission that is detected in the
submm/mm range. Dark dust may solve the puzzle between discrepant
spectroscopic and parallax-derived distance estimates. First, the
observational basis is specified, then the dust model is described, and a
vectorized fitting procedure of the 11-parameter dust model to the
observational constraints is detailed. Results will be presented that show the
overall success of the model in fitting the data and deriving the physical
properties of dark dust. Particular attention is given to the use of optical
constants of amorphous silicates (Demyk et al., 2022). The main findings are
summarized in the conclusions.

##  2 Observations

###  2.1 Soild-phase abundances

Depletion of elements from the gas into dust is estimated by Voshchinnikov &
Henning (2010) as the difference between abundances in the Sun (Asplund et
al., 2009) and that of the gas (Jenkins, 2009). Elemental depletion is used to
infer dust compositions and some form of grains with silicate and carbon is
widely accepted. Abundances of element [X] to that of hydrogen [H] are
summarized by Hensley & Draine (2021) for the gas and the dust phase,
respectively. The most abundant elements in dust are: O ($249\pm 94$), C
($126\pm 56$), Mg ($46\pm 5$), Fe ($43\pm 4$), Si ($38\pm 3$), S (7.6), Al
(3.4), Ca (3.2), and Ni (2.0), where numbers in parenthesis give [X]/[H]
(ppm). An over-abundance of O in the dust is noted and upper limits of the
$16\,\mu$m and $22\,\mu$m bands of iron-oxides indicate that most of the Fe
remain unexplained as well (Draine & Hensley, 2021a). However, the Fe
abundance is insufficient to form large grains (Zhukovska et al., 2018). Fe
particles heated by the ISRF contribute to the emission at $\sim 60\,\mu$m
(Fischera, 2004). Draine & Hensley (2021a) showed that the absorption cross-
section remains in the IR - mm unaltered when including the available Fe in
form of impurities in large particles.

Dust abundances are uncertain and estimates of the C and Si abundance in dust
scatter by 50%. Mulas et al. (2013) find an average of [C]/[H] $=145$ ppm. The
Si abundance in the dust is estimated by Sofia & Meyer (2001) to be $18\pm 9$
ppm, while Nieva & Przybilla (2012) derive $32\pm 3$ ppm. Voshchinnikov &
Henning (2010) find a sharp difference in the dust abundances for sightlines
located at low and high galactic latitudes and give an average abundance of
[Si]/[H] $=23\pm 5$ ppm.

Absorption and emission signatures of the dust provide important constraints
on the composition. A striking feature is the 2175 Å extinction bump, where
graphite and polycyclic aromatic hydrocarbons (PAHs; Allamandola et al.
(1989)) have strong electronic transitions. In the IR, there are conspicuous
emission bands at 3.3, 6.2, 7.7, 8.6, 11.3, and 12.7 $\mu$m, as well as a
wealth of weaker bands in the $12-24\,\mu$m region. These bands are ascribed
to vibrational transitions in PAH molecules, which are planar structures that
consist of benzol rings with hydrogen attached. PAH feature strengths depend
on the hardness of the exciting radiation field and the ionisation or
hydrogenation coverage of the molecules.

Absorption features in the diffuse ISM between $3-8\,\mu$m are interpreted as
being due to either carbonaceous material (Mennella et al., 2003) or, guided
by laboratory spectra, by ices mixed with silicates (Potapov et al., 2021).
Dust models including interstellar ice mixtures have been presented by
Siebenmorgen & Gredel (1997). Solid-state water mixed with silicates would
explain a good fraction of the unaccounted oxygen depletion in the diffuse ISM
(Potapov et al., 2021). A comparison of future JWST observations against
laboratory spectra is needed to confirm the existence of solid water in the
diffuse ISM. Ices are not further considered in this study.

The 9.7 $\mu$m and 18 $\mu$m broadband features are assigned to Si-O
stretching and O-Si-O bending modes of silicate grains, respectively. A
comparison of the observed band profiles and laboratory spectra favours
amorphous rather than crystalline silicates. Intriguing is the detection of
the $11.1\,\mu$m absorption band that is attributed in the atlas of ground-
based mid-infrared (MIR) spectra by Do-Duy et al. (2020) to forsterite,
providing a mass abundance of crystalline silicates of $\sim 2$% in the
diffuse ISM. Olivine (Mg2x Fe2-2x SiO4 has $x\sim 0.8$), have been detected in
AGB and T Tauri stars, and comet Hale-Bopp (Henning, 2010). The silicate
stoichiometry and grain geometry can be revealed by MIR spectro-polarimetry
(Sec. 2.4). As nominal composition Draine & Hensley (2021a) adopt
Mg1.3(Fe,Ni)0.3SiO3.6 having a molecular weight of $\mu=134.5$. In that
structure, the silicate grain abundance is limited by Mg with a dust abundance
ratio $\rm{[C]/[Si]}\sim\rm{[C]\ 1.3/[Mg]}\sim 3.5$. To accommodate the
reported large scatter of $\sim 50$ % in the solid-phase element abundances
the dust models shall respect a limit of the dust abundance ratio of

$\rm{[C]/[Si]}\la 5.25\ .$ (1)

###  2.2 Reddening and extinction

The interstellar reddening is derived by measuring the flux ratio of a
reddened and unreddened star with the same SpL luminosity class, e.g. the
standard pair method (Stecher, 1965). The flux of a star is derived from the
spectral luminosity $L(\lambda)$, the distance $D$, and the extinction optical
depth $\tau(\lambda)$, which is due to the absorption and scattering of
photons along the sightline excluding emission. The observed flux of the
reddened star is given by

$F(\lambda)=\frac{L({\lambda})}{4\pi\ D^{2}}\ e^{-\tau({\lambda})}\ .$ (2)

In photometry it is customary to express the flux of an object by the apparent
magnitude, which is related to the flux through
$m(\lambda)=2.5\log_{10}\Big{(}F(\lambda)/w_{\lambda}\Big{)}$ and
$w_{\lambda}$ as the zero point of the photometric system. The difference in
magnitudes between two stars is $\Delta
m(\lambda)=1.086\times\Big{(}\tau(\lambda)+2\log_{10}(D/D_{0})\Big{)}$ .
Unfortunately, distances to hot, early-type stars, commonly used to measure
interstellar extinction, are often subject to large errors (Siebenmorgen et
al., 2020). Hence one relies on relative measurements of two wavelengths and
defines the colour excess $E(\lambda-\lambda^{\prime})=\Delta
m(\lambda)-\Delta m(\lambda^{\prime})$. The reddening curve $E(\lambda)$ is
traditionally represented by a colour excess and normalisation to avoid the
uncertainties between both distances,

$E(\lambda)=\frac{E(\lambda-\rm{V})}{E(\rm{B}-\rm{V})}=\frac{A_{\lambda}-A_{\rm{V}}}{A_{\rm{B}}-A_{\rm{V}}}=\frac{\tau_{\lambda}-\tau_{\rm{V}}}{\tau_{\rm{B}}-\tau_{\rm{V}}}\
.$ (3)

Naturally, for the V and B band, $E(\rm{V})=0$ and $E(\rm{B})=1$. The
extinction in magnitudes at wavelength $\lambda$ is denoted by $A(\lambda)$.
One obtains some extrapolated estimate of the visual extinction $A_{\rm{V}}$
from photometry. This requires measuring $E(\rm{B}-\rm{V})$ and extrapolating
the reddening curve to infinite wavelengths $E(\infty)$. In practice, one
derives $E(\lambda-\rm{V})$ at the longest wavelength, which is not
contaminated by any kind of emission. From this wavelength, e.g. the H-band,
one extrapolates to infinite wavelengths assuming some prior shape of
$E(\lambda)$ and hence estimates $E(\infty,V)$. By introducing the ratio of
the total-to-selective extinction

$R_{\rm V}=\frac{A_{\rm V}}{E(\rm{B}-\rm{V})}\ ,$ (4)

where obviously $E(\infty)=-{R_{\rm V}}$. The total-to-selective extinction
can also be written using Eq. 3 or applying a dust model as

${R_{\rm
V}}=\frac{\tau_{\rm{V}}}{\tau_{\rm{B}}-\tau_{\rm{V}}}=\frac{K_{\rm{V}}}{K_{\rm{B}}-K_{\rm{V}}}\
,$ (5)

where $K=K_{\rm{abs}}+K_{\rm{sca}}$ is the extinction cross-section which is
the sum of the total absorption and scattering cross-section of the dust. The
relation between reddening and extinction is

$\frac{\tau(\lambda)}{\tau_{\rm V}}=\frac{K(\lambda)}{K_{\rm
V}}=\frac{E(\lambda)}{R_{\rm V}}+1\ ,$ (6)

Figure 1: Reddening (left) and extinction (right) of the diffuse ISM
(Fitzpatrick & Massa, 2007; Fitzpatrick et al., 2019; Indebetouw et al., 2005;
Chen et al., 2018) and of HD 046202 (Gordon et al., 2009). Model (black) with
$r_{\rm{Dark}}^{+}=1\,\mu$m (Table 1) and individual dust components as
labelled. The area in grey shows the contribution of dark dust.

The International Ultraviolet Explorer (IUE) operated between
$0.185-0.33\,\mu$m and the Far Ultraviolet Spectroscopic Explorer (FUSE)
observed between $0.119-0.905\,\mu$m. IUE and FUSE provide the legacy of (far)
UV spectra with a total of 895 reddening curves towards more than 568 early-
type (OB) stars (Valencic et al., 2004; Fitzpatrick & Massa, 2007; Gordon et
al., 2009). The database of available reddening curves was scrutinized against
systematic errors by the following means: 1) IUE/FUSE spectra were verified to
include only a single star in the observing aperture and not beside the
program star and other almost equally bright objects. 2) Although most early-
type stars are binaries (Chini et al., 2012) their reddening curves are
generally derived assuming a single star. Whenever the companion contributes
significantly to the total flux of the system, the derived extinction includes
large systematic errors. 3) A large fraction of the OB stars show emission
components above the photosphere in the IR (Siebenmorgen et al., 2018a; Deng
et al., 2022), which prohibits deriving meaningful reddening in the IR. 4)
Reddening curves are derived assuming steady stellar systems. Stellar
variability will thus systematically impact the derived reddening. Stars with
detected variability between ground-based and Hipparcos V-band photometry
($\ga 0.1$ mag), significant B - V colour changes, or GAIA G-band photometry
of more than ($\ga 0.033$ mag) shall be rejected. 5) some stars show
inconsistencies in the GAIA parallaxes between data release two (Gaia
Collaboration et al., 2018) and data release three (Gaia Collaboration et al.,
2020) that are not instrumental so that their reddening curves shall be
declared as spurious. 6) Finally, the quality of the derived reddening curve
depends critically on the fidelity of the SpL estimate. The SpL determination
of fast rotators is highly uncertain and UV spectral diagnostics indicate
considerably earlier SpL (hotter) classifications than optically assigned SpL.
Reddening curves with largely deviating SpL assignments shall be ignored. All
these systematic effects impact the derivation more dramatic at
$\lambda\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise 5.0pt\hbox{$\char
62\relax$}\hss}\mathchar 29208\relax$}}1\,\mu$m and the extinction more than
the reddening. In total, 48 stars with one or more reddening curves passing
the rejection criteria are available with 21 classified as multi-component
sightlines and 27 as single-cloud sightlines (Siebenmorgen et al., 2020). The
reddening curves in the ISM show significant variations from sightline-to-
sightline and the derived total-to-selective extinction are between $2\la
R_{\rm V}\la 6.4$. For the same star the published $R_{\rm V}$ that are mostly
extrapolated from JHK photometry, scatter typically by 10%. Even in the high-
quality sample, the peak-to-peak scatter in $R_{\rm V}$ estimates of the same
star are up to $\pm 0.6$. The scatter might be reduced by detailed physical
modelling of the dust applying Eq. (5). Whenever possible the reddening
instead of the extinction curve shall be discussed.

For the diffuse ISM a mean reddening of ${E(\rm{B}-\rm{V})}=0.45$ mag and a
median and mean value of $R_{\rm V}=3.22$ and $2.99\pm 0.27$ is derived by
Fitzpatrick & Massa (2007), respectively; Wang & Chen (2019) find $R_{\rm
V}=3.16$, and Voshchinnikov (2004); Fitzpatrick et al. (2019) give ${A_{\rm
V}}\sim 3.1\ E({B-V)}$. Here $R_{\rm V}=3.1$ is adopted as by Hensley & Draine
(2021). The mean reddening curve of the Milky Way is derived for $1/\lambda\la
1./8\,\mu$m by Fitzpatrick & Massa (2007) and Fitzpatrick et al. (2019) and
both are shown together with the reddening curve of HD 046202 (Gordon et al.,
2009) in Fig. 1. The reddening curve of that star perfectly matches the mean
curve derived from IUE and FUSE. Reddening in spectral regions close to wind
lines at 6.5 and 7.1 $\mu$m-1 and Ly-$\alpha$ at $8\,\mu$m${}^{-1}\leq x\leq
8.45\,\mu$m-1, or with apparent instrumental noise at $x\mathbin{\lower
3.0pt\hbox{$\hbox to0.0pt{\raise 5.0pt\hbox{$\char 60\relax$}\hss}\mathchar
29208\relax$}}3.6\,\mu$m-1 shall be ignored. Overall a typical error in the
derived reddening is $\sim 10$ %. Various mean extinction curves of the
diffuse ISM between $0.5-7\,\mu$m are also shown in Fig. 1. Noticeable is the
large scatter which is increasing towards longer wavelengths.

Figure 2: Dust emission of the diffuse ISM observed at high galactic latitudes
and normalised per H atom by Dwek et al. (1997) and Planck Collaboration et
al. (2015) and as tabulated by Hensley & Draine (2021). Model (black) with
$r_{\rm{Dark}}^{+}=1\,\mu$m (Table 1) and individual dust components as
labelled. The area in grey shows the contribution of dark dust.

###  2.3 Diffuse galactic dust emission

Diffuse emission of the ISM from dust grains heated by the ambient ISRF has
been observed from the NIR through the microwave region. The dust emission of
the diffuse ISM is not uniform across the sky and there is evidence that the
properties of the dust as well as the spectral distribution and strength of
the ISRF that heats the dust vary (Fanciullo et al., 2015). Observations of
this component for high galactic latitude ($|b|\ga 25\degr$) from the Cosmic
Background Explorer using DIRBE data are given by Dwek et al. (1997) and from
the Planck Collaboration et al. (2015). A colour-corrected composite spectrum
with error estimates is tabulated by Hensley & Draine (2021). The emission
spectrum is shown in Fig. 2.

###  2.4 Starlight and FIR polarization

Stellar light that passes a cloud of moderate extinction by aligned grains
becomes linearly polarized. From the far UV to the NIR, between
$0.12\,\mu\rm{m}-1.6\,\mu$m, the polarization curves are fit by an empirical
formula given by Serkowski et al. (1975)

${p(\lambda)}={p_{\max}}\ \exp\left[-k_{\rm p}\
\ln^{2}\left(\frac{\lambda_{\max}}{\lambda}\right)\right]\,.$ (7)

Figure 3: Polarized extinction of the diffuse ISM (circles) with error bars
indicating the observed scatter (Voshchinnikov et al., 2012; Bagnulo et al.,
2017). The model (black) with $r_{\rm{Dark}}^{+}=1\,\mu$m (Table 1) and
contributions (dashed) from large amorphous carbon (brown), large silicate
grains (green), and dark dust area in grey, when treated as prolate particles,
is shown.

where $\lambda_{\max}$ is the wavelength at maximum polarization $p_{\max}$
and $k_{\rm p}$ the width of the spectrum. Significant variations in the width
$0.5\la k_{\rm p}\la 1.5$ towards different sightlines are observed, using
HPOL222http://www.sal.wisc.edu, the Wisconsin UV Photo-Polarimeter WUPPE
satellites, and ground-based instruments as compiled by Efimov (2009) and by
the Large Interstellar Polarization Survey (Bagnulo et al., 2017). The latter
authors could not confirm the linear trend of $k_{\rm p}\propto\lambda_{\rm
max}$ claimed in earlier work (Wilking et al., 1980; Whittet et al., 1992).
Observations of 160 sightlines of mildly reddened stars show that
$p_{\rm{V}}/E_{B-V}\la 9$ %/mag and $p_{\rm{max}}\la 10$ % (Voshchinnikov et
al., 2016). Starlight polarization reaches a maximum in the V band at
$\lambda_{\rm max}=0.545\,\mu$m and $k_{\rm p}=1.15$ is selected to represent
the mean Serkowski curve that shows a typical scatter of 5% near
$p_{\rm{max}}$, 10% at $p/p_{\rm{max}}\la 0.4$, and in the NIR $\sim 15$ %. At
$\lambda>1.5\,\mu$m the polarization spectrum smoothly matches onto a power-
law with an exponent of 1.6 (Martin et al., 1992). The fit naturally breaks in
the MIR near the silicate band. The typical wavelength dependence of the
observed polarized extinction is shown in the optical/NIR in Fig. 3 and in the
MIR in Fig. 4.

Figure 4: Polarized extinction in the MIR. A composite spectrum (circles) of
two Wolf-Rayet stars WR 48A and AFGL 2104 by Wright et al. (2002), the model
(full line) with $r_{\rm{Dark}}^{+}=1\,\mu$m (black) and contribution (dashed)
from large amorphous carbon (brown), large silicate grains (green), and dark
dust (grey) when treated as prolate particles, is shown, as well as the model
with $r_{\rm{Dark}}^{+}=10\,\mu$m (blue). Parameters as in Table 1.

A mean MIR polarized extinction spectrum, which covers the silicate band, is
constructed by Wright et al. (2002) by averaging observations of two Wolf-
Rayet stars, WR 48A and AFGL 2104. The analysis by Wright shows that the
polarized extinction of silicates with stoichiometry and optical constants
$(n,k)$ of Mg0.4Fe0.6SiO3, MgFeSiO3, as well as MgFeSiO4 provide a pure fit
whereas Mg0.8Fe1.2SiO4 give an almost perfect fit to the polarization spectrum
(Wright et al., 2002). The molecular weight of these materials varies between
$\mu=97-141$, an uncertainty directly impacting estimates of the gas-to-dust
mass ratio. The silicate composition observed towards the Wolf-Rayet stars is
distinct from Mg1.3(Fe,Ni)0.3SiO3.6 that is adopted for the diffuse ISM
(Draine & Hensley, 2021a). There are marked differences between the
environments of the dust-producing circumstellar shells of the bright Wolf-
Rayet stars and the diffuse ISM. Towards both sightlines, the stars have a
factor $\sim 3$ higher reddening than what is typical for the diffuse ISM and
indications of ice absorption at $6.1\,\mu$m are found (Marchenko & Moffat,
2017). Therefore, the silicate composition of the diffuse ISM is retained.
Nevertheless, the analysis exemplifies that characterization of the silicate
stoichiometry and particle shape by MIR spectro-polarization is viable.
Unfortunately, such observations were recorded in the past century (Smith et
al., 2000) and novel high-sensitive instrumentation is needed for detecting
polarized MIR extinction in the diffuse ISM.

The polarized dust emission spectrum in the submm is observed by Planck
Collaboration et al. (2014, 2020). If the polarized emission in the submm and
the polarized extinction in the optical is due to the same grains then this
ratio in principle includes information on the grain elongation (Krügel,
2003). For the ratio of 850 $\mu$m to V band polarization a characteristic
value of

$\tau_{\rm{V}}\ \frac{p_{850\,\mu{\rm m}}}{p_{\rm{V}}}=4.31$ (8)

is adopted by Hensley & Draine (2021). In the diffuse ISM variations for
different sightlines are noted in the hydrogen column density $N_{\rm{H}}$,
$E(\rm{B}-\rm{V})$, $R_{\rm V}$, and hence $\tau_{\rm{V}}$ (Eq.6).

##  3 Model

###  3.1 Dust populations

A dust model for the diffuse ISM is presented that shall be in line with
present observational constraints: the [C]/[Si] abundance ratio (Eq. 1), the
spectral variation of the reddening, the starlight polarization, the diffuse
galactic emission, and the polarized dust emission (Sec. 1). Two major grain
materials are considered: amorphous silicates and carbon. Dust particles need
to be of different sizes to fit the reddening and polarization curves. A
power-law size distribution $\mathrm{d}n(r)/\mathrm{d}r\propto r^{-q}$ (Mathis
et al., 1977) that ranges from the molecular domain ($r_{-}\sim 5$ Å ) to a
rather unconstrained upper size limit of several microns ($r_{+}\la 10\,\mu$m)
is applied. Three different dust populations are distinguished:

1) Nanoparticles with sizes below 6 nm, which are in form of very small
silicates (vSi), very small graphite (VSG), and PAHs. Two kinds of PAH are
treated, small molecules having 30 C and 12 H atoms and clusters with 200 C
and 40 H atoms, respectively. The cross-sections of the nanoparticles and PAHs
are taken from Siebenmorgen et al. (2014).

2) Large grains with radii between $6-300$ nm, which are partly aligned and of
prolate shape with an axial ratio $a/b=2$. Large grains are made of silicates
(Si) and amorphous carbon (aC). Scattering, absorption, and polarization
cross-sections of spheroids are computed with the procedure outlined in Sec.
3.3.

Figure 5: Optical constants $n-1$ (top) and $k$ (bottom) for various grain
materials.

3) Dark dust, is included as a new additional grain component of micro-meter-
sized particles. Dark dust grains are taken to be some kind of fluffy
aggregates that are made up of porous composites of large silicate and large
amorphous carbon grains that are sticking loosely together. The frosting of
molecules on dark dust can be envisioned as an attractive reservoir of the
reported O depletion but is not considered because of the missing detection of
ice absorption bands in the diffuse ISM. Dark dust particles are treated as
spheres unless otherwise stated. For fluffy grains, there are only approximate
methods for calculating the cross-sections. The numerical simple tool by
Krügel & Siebenmorgen (1994) is used utilizing the Bruggemann mixing rule. An
averaged complex dielectric function, $\epsilon=m^{2}$ is computed by solving

$\sum_{\rm i}f_{\rm i}\ {\frac{\epsilon_{\rm
i}-\epsilon_{\rm{av}}}{\epsilon_{\rm i}+2\epsilon_{\rm{av}}}}=0\ \quad\
{\rm{with}}\quad\ \sum_{\rm i}f_{\rm i}=1\ ,$ (9)

where the volume fraction of each component $i$ is denoted by $f_{\rm i}$. The
considered inclusions in a dark dust particle are: silicates with $f_{\rm
Si}=0.5$, amorphous carbon $f_{\rm aC}=0.3$, and vacuum with $f_{\rm vac}=0.2$
to represent porosity, while ices are ignored.

###  3.2 Porosity

Optical constants $m=n-{\rm i}\ k$ with $k\ga 0$ of various grain materials
are displayed in Fig. 5. Optical constants are shown for the aC-H2 mixture of
amorphous carbon by Zubko et al. (1996), silicate and graphite by Draine
(2003), and a 97:3 mix in mass of amorphous silicate grains with 2% vaccum
inclusion that are composed of MgO$-$0.5 SiO2 and Mg0.8Fe${}^{2+}_{0.2}$ SiO3
using optical constants by Demyk et al. (2022) and as discussed in Sect. 6.
The Si and aC grains differ by large in $n$ and $k$, respectively.

By considering fluffy particles the optical constants are shifted to lower
values and follow a similar spectral shape as for pure materials that do not
have vacuum inclusions. This is seen in Fig. 5 by comparing aC-H2 grains by
Zubko et al. (1996) with fluffy aC (Krügel & Siebenmorgen, 1994) with 20%
vaccum inclusion and using Eq. (9). The spectral shape of the agglomerates by
Draine & Hensley (2021a) is shown in Fig. 5 in orange without (full line) and
with 20% porosity (dashed), which can be compared with fluffy composites in
black. The latter is a representation of dark dust. Again in the latter two
sets similarities in the spectral shape of $n$ and $k$ are visible as well as
a larger shift in the FIR/submm where differences in the slope become more
pronounced. Also important is that by changing the vacuum content of materials
the spectral slope remains unaltered. Thus the slope of the dust emissivity of
a material, e.g. in the submm, does not change with porosity. For the same
mass the strength of the dust emissivity increases by increasing the porosity
because fluffy grains are larger than particles of the same material that have
less or no vacuum inclusions.

###  3.3 Aligned spheroidal grains

The interstellar polarization phenomenon in the optical and IR/mm cannot be
explained by spherical dust particles. Interstellar polarization of starlight
and polarized dust emission is explained by partially aligned non-spherical
dust grains that wobble and rotate about the axis of the greatest moment of
inertia. Most simple representations of finite-sized non-spherical grains are
spheroids. They are characterised by the ratio $a/b$ between major and minor
semi-axes. Spheroids can be either in form of oblates such as pancakes or
prolates such as needles. The volume of a prolate is the same as a sphere with
a radius given by $r^{3}=a\cdot b^{2}$.

The dust optical depth at frequency $\nu$ is given by the product of the
column density $N_{\rm d}$ of the dust along the sightline and the dust
extinction cross-section $C_{\rm ext}=C_{\rm abs}+C_{\rm sca}$, which is the
sum of absorption and scattering

$\tau({\nu})=N_{\rm d}\ C_{\rm ext}({\nu})\ .$ (10)

Linear polarization is

$p(\nu)=N_{\rm d}\ C_{\rm p}({\nu})\ ,$ (11)

where $C_{\rm{p}}$ is the linear polarization cross-section.

Physical processes that led to grain alignment are under debate and various
mechanisms have been proposed, see reviewed by Voshchinnikov (2012) and
Andersson et al. (2015). In picket fence (Dyck & Beichman, 1974) a fraction of
the grains is in perfect alignment while the other particles are randomly
oriented. A modified version of it is used by Draine & Hensley (2021a), where
it is assumed that the cross-sections along the ${\bf E}$ and ${\bf B}$ field
depends on $\cos^{2}{\theta}$ of the scattering angle (the angle between the
wave vector ${\bf k}$ of the incoming light and the symmetry axis of the
grain). There exist also non-magnetic alignment processes such as in gas
streams of stellar winds or by an-isotropic illumination where photons are
predominantly absorbed from one side of the particle. Radiative torque
alignment has also become popular. In the Milky Way the direction of the dust-
induced polarization in the optical is well correlated with the orientation of
the magnetic field, which is derived from synchrotron emission in the radio.
Hence magnetic fields are often taken to be responsible for the grain
alignment in the ISM. In that picture, questions arise when estimating the
relaxation times for the magnetic alignment of the grain in comparison to the
time when the disorder is again established through collisions with gas atoms.
For solving that puzzle supra-thermal grain rotation (at a frequency of
$10^{5}$ Hz) were proposed as well as (super-)paramagnetic or ferromagnetic
relaxation. In the following the imperfect Davies-Greenstein mechanism (IDG)
is applied in which Fe atoms in the dust interact with the weak ${\bf B}$
field of the ISM. One advantage of IDG is that the orientation of a spinning
and wobbling spheroid can be described mathematically in closed form. For
angles $\psi$, $\varphi$, precision angle $\beta$, and magnetic field
direction $\Omega$, applied as in the notation by Hong & Greenberg (1980) and
suffix TM for the transverse magnetic and TE for transverse electric
polarization direction as in the notation by Bohren & Huffman (1983), the
alignment function becomes

${f}(\beta)=\frac{\frac{r+0.1\ \delta_{0}\
}{r+\delta_{0}}\,\sin\beta}{\left(\left(\frac{r+0.1\ \delta_{0}\
}{r+\delta_{0}}\right)^{2}\cos^{2}\beta+\sin^{2}\beta\right)^{3/2}}\ .$ (12)

The alignment efficiency $\delta_{0}\sim 0.1-10\,\mu$m (Das et al., 2010;
Siebenmorgen et al., 2014) is related to the physical picture. It impacts the
maximum of the polarization $p_{\rm max}$ but not the shape $p(\lambda)$ of
the polarization (Voshchinnikov & Das 2008). The cross-sections are computed
for given efficiency factors $Q$ (Sec. 4.1) as average over the orientation of
the wobbling particles

${C}_{{\rm ext}}(\nu)=\frac{2}{\pi}\int(Q_{{\rm ext}}^{\rm{TM}}+Q_{{\rm
ext}}^{\rm{TE}})\,r^{2}\,f(\beta)\,\mathrm{d}{\varphi}\,\mathrm{d}{\Omega}\,\mathrm{d}{\beta}\
,$ (13)

${C}_{{\rm p}}(\nu)=\frac{1}{\pi}\int(Q_{{\rm ext}}^{\rm{TM}}-Q_{{\rm
ext}}^{\rm{TE}})\,r^{2}\,f(\beta)\,\cos(2{\psi})\,\mathrm{d}{\varphi}\,\mathrm{d}{\Omega}\,\mathrm{d}{\beta}\
.$ (14)

###  3.4 Reddening

The mass extinction cross-sections $K_{{\rm{ext}},i}(r)$ (cm2/g-dust) of a
dust particle of population $i\in\\{{\rm{Si,aC,vSi,VSG,Dark}}\\}$, radius $r$,
and density $\rho_{i}$ is

$K_{{\rm ext},i}(r)=\frac{m_{i}}{{\displaystyle{\frac{4\pi}{3}}\ \rho_{i}}}\
\frac{r^{-q}}{\displaystyle\int_{r_{-,i}}^{r_{+,i}}r^{3-q}\ \mathrm{d}r}\
C_{{\rm ext},i}(r)\ .$ (15)

The cross-sections $C$ are derived using efficiency factors $Q$, which are
computed for spheres by Mie theory and spheroids as in Sec. 3.3. The relative
weight, also called specific mass, in 1 g of dust in component $i$, is

$m_{\rm{i}}={\frac{\frac{[\rm{X}]}{[\rm{H}]}_{\rm{i}}\ \mu_{\rm
X}}{\left(\frac{[\rm{C}]}{[\rm{H}]}_{\rm{aC}}+\frac{[\rm{C}]}{[\rm{H}]}_{\rm{gr}}+\frac{[\rm{C}]}{[\rm{H}]}_{\rm{PAH}}\right)\
\mu_{\rm
C}+\left({\frac{[\rm{Si}]}{[\rm{H}]}_{\rm{Si}}}+\frac{[\rm{Si}]}{[\rm{H}]}_{\rm{sSi}}\right)\
\mu_{\rm Si}}}\ ,$ (16)

where relative dust abundances of element X, which is either C or Si, with
respect to H are denoted by $[\rm{X}]/[\rm{H}]_{\rm i}$ together with a
subscript $i$ for each of the dust population, $\mu_{\rm C}=12$ is the
molecular weight of carbon, $\mu_{\rm Si}=134.46$ that of silicate grains with
bulk densities $\rho_{C}\sim 1.6$ g/cm3 and $\rho_{\rm Si}\sim 3.41$ g/cm3
(Draine & Hensley, 2021a), respectively. The specific mass for PAH is computed
in similar terms (Siebenmorgen et al., 2014). Specifying the mass extinction
cross-section per gram of dust has the advantage that only relative element
abundances $[\rm{X}]/[\rm{H}]_{\rm i}$ need to be specified instead of the
more uncertain absolute solid-phase element abundances (in ppm), which
however, may be used as a guideline. The total mass extinction cross-section
averaged over the dust size distribution in cm2/g-dust is

$K_{\rm ext}=\sum_{i}\ \int_{r_{-}}^{r_{+}}K_{{\rm ext},i}(r)\ \mathrm{d}r\ .$
(17)

The wavelength dependence in the above expressions (Eq. 13 \- 17) is dropped
for clarity. The reddening curve of the dust model is derived using Eqs. (5,
6) and noting $K$ as the mass extinction cross-section

$E(\lambda)=\frac{K_{\rm{V}}}{K_{\rm{B}}-K_{\rm{V}}}\left(\frac{K(\lambda)}{K_{\rm{V}}}-1\right)\
.$ (18)

Note by using $R_{\rm V}$ (Eq. 5) that the reddening $E(\lambda)$ is derived
self-consistently from the dust model without extrapolation to infinite
wavelengths.

###  3.5 Emission

The emission $\epsilon_{\nu,i}(r)$ of a dust particle of radius $r$, grain
material $i$ at frequency $\nu$ is

$\epsilon_{\nu,i}(r)=K_{\nu,i}^{abs}(r)\
{\int}{P_{i}(r,T)\,B_{\nu}(T)\,\mathrm{d}T}\,,$ (19)

where $K_{\nu,i}^{abs}$ is the mass absorption cross-section (Eq. 15),
$B_{\nu}(T)$ is the Planck function, and $P_{i}(r,T)$ is the temperature
distribution function that gives the probability of finding a particle of
material $i$ and radius $r$ at temperature $T$. This function is determined
from the energy balance between the emission and absorption of photons from
the mean intensity $J_{\nu}$, for which here the ISRF by Mathis et al. (1983)
is applied

${\int}{K_{\nu,i}^{abs}(r)\,J_{\nu}(T)\,\mathrm{d}\nu}={\int}{K_{\nu,i}^{abs}(r)\,P_{i}(r,T)\,B_{\nu}(T)\
\mathrm{d}T\,\mathrm{d}\nu}\ .$ (20)

It is evaluated using an iterative scheme that is described by Krügel (2008).
The $P(T)$ function only needs to be evaluated for small grains as it
approaches a $\delta$-function for large particles where the temperature
fluctuates very little around the equilibrium temperature. The total emission
$\epsilon_{\nu}$ of the dust at frequency $\nu$ is given as the sum of the
emission $\epsilon_{\nu,i}(r)$ of all dust components.

###  3.6 Gas-to-dust mass ratio

Observations of the diffuse emission of the galaxy are given per hydrogen
column density, $\lambda\ I_{\lambda}/N_{\rm{H}}$ (erg/s/sr/ H-atom), whereas
the dust emission of the model is computed (Eq. (19) per dust mass (erg/s/sr/
g-dust). For the necessary conversion of the dust to the gas column densities
often a procedure is applied assuming that for sightlines in the diffuse ISM
the extinction is proportional to the hydrogen column density $A_{\rm
V}=1.086\times\tau_{\rm V}\propto N_{\rm H}$. In addition, one presumes that
$A_{\rm V}$ can be reasonably estimated by applying a mean value of the total-
to-selective extinction and the derived reddening using $A_{\rm V}={R_{\rm
V}}\times{E(\rm{B}-\rm{V})}$. Further, one assumes that the ratio
$\zeta=N_{\rm H}/{E(\rm{B}-\rm{V})}$ stays roughly constant in the ISM.
Traditionally $\zeta=5.8$ ($10^{21}\,\rm{H}\,\rm{cm}^{-2}\,\rm{mag}^{-1}$) is
applied, which is derived from 75 sightlines observed within 3.4 kpc by
Copernicus (Bohlin et al., 1978). For translucent clouds at $\tau_{\rm V}\ga
0.5$ observed by FUSE $\zeta=5.94\pm 0.37$, from here onwards in units as
before (Rachford et al., 2009), and from X-ray observations $\zeta\sim
6.3-6.5$ (Zhu et al., 2017) using ${R_{\rm V}}=3.1$. However, striking
differences are derived from radio observations of HI with lower values in
earlier work of $\zeta=4.6$ (Mirabel & Gergely, 1979) and 5.1 (Knapp & Kerr,
1974) or higher values in recent studies of $\zeta=8.3$ (Liszt, 2014), 8.8
(Lenz et al., 2017) and 9.4 (Nguyen et al., 2018). Dust emission of the
diffuse ISM is observed towards high galactic latitudes where there is less
dust per H atom as in the Galactic plane so that the latter value of $\zeta$
has been favoured by Hensley & Draine (2021).

In the procedure applied here these uncertainties in $\zeta$, $R_{\rm V}$, and
${E(\rm{B}-\rm{V})}$ are avoided. A gas-to-dust mass ratio
${M_{\rm{gas}}}/{M_{\rm{dust}}}$ is introduced that is given by scaling the
dust emission spectrum of Eq. (19) to the Planck data at 350 $\mu$m. The
observed emission is fit by

$\frac{\lambda\,I_{\lambda}}{N_{\rm{H}}}=\frac{M_{\rm{dust}}}{M_{\rm{gas}}}\
\frac{\lambda\ \epsilon_{\lambda}}{m_{\rm p}}\ ,$ (21)

with H-atom mass $m_{\rm p}$. The procedure has the advantage that
${M_{\rm{gas}}}/{M_{\rm{dust}}}$ is derived in the same direction as to where
the data were taken. Of course, uncertainties that arise due to possible
variations in the strength of the ISRF towards the observed fields remain and
are assumed to be small.

###  3.7 Polarized extinction

Starlight polarization is derived by dichroic extinction of the dust (Eqs. 10
\- 11, 14)

$p(\nu)/\tau_{\rm V}=K_{\rm{p}}(\nu)/K_{\rm{ext,V}}\ ,$ (22)

$K_{\rm p}$ denotes the total linear mass polarization cross-section
(cm2/g-dust) and is computed similar to $K_{\rm ext}$ by replacing in Eq. (15)
the cross-section $C_{{\rm ext},i}$ (Eq. 13) with $C_{{\rm p},i}$ (Eq. 14).

###  3.8 Polarized emission

Polarized dust emission of component $i$ is computed by integrating over the
minimum ${r_{pol,i}^{-}}$ to maximum ${r_{pol,i}^{+}}$ alignment radii

$\epsilon_{pol,i}{(\nu)}={\int}_{r_{pol,i}^{-}}^{r_{pol,i}^{+}}K_{pol,i}(\nu,r)\,B_{\nu}(T)\,\mathrm{d}r\
.$ (23)

Note that $C_{\rm p}=K_{\rm p}=0$ for spherical or non-aligned grains. The
total polarized dust emission $\epsilon_{\rm{pol}}$ is given as the sum of the
polarized emission $\epsilon_{pol,i}$ of all components contributing to the
polarization. These are large aC and Si prolate grains and dark dust when also
considered of prolate shape. Polarization by nanoparticles is not considered.

##  4 Method

The dust model is confronted with the observational constraints presented in
Sec. 2. As described by Zubko et al. (2004) the fitting procedure leads to a
typical ill-posed inversion problem where the solution is extremely sensitive
to small changes in input data and with several priors such as the size
distribution or grain composition as unknowns. Therefore, a least
$\chi^{2}$-technique is applied for which first, a method for finding the best
fit of the reddening curve is presented that respects dust abundance
constraints (Eq. 1). For that model, the parameters which impact the shape of
the starlight polarization are varied to find the best fit to the mean
Serkowski curve. The latter model is then compared to the diffuse galactic
emission, which requires applying a gas-to-dust mass ratio. The flatness in
the submm/mm polarization spectrum as well as the ratio of the starlight
polarization to submm polarization is tested. The parameter space of the model
is explored and the goodness of the fit is quantified as the sum of
differences between observations and the model, each squared and divided by
the observed data. For the reddening curve the goodness parameter is denoted
by $\chi^{2}_{\rm r}$ and for the polarized extinction $\chi^{2}_{\rm p}$,
respectively.

###  4.1 Cross-sections

The various dust cross-sections for extinction, scattering, and polarization
are computed in the spectral range between $90\,\rm{nm}\la\lambda\la 1$ cm and
for grain radii range between 5 Å $\la r\la 10\,\mu$m. The challenge is
computing the efficiency factors $Q=C/\pi r^{2}$ for the two polarization
directions of elongated particles (Eqs. 13, 14). For small size parameter
$x=2\pi r/\lambda<<1$ the Rayleigh approximation might be used (Krügel, 2008).
For typical ISM grains with sizes $r\leq 0.3\,\mu$m scattering becomes small
at $\lambda\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise 5.0pt\hbox{$\char
62\relax$}\hss}\mathchar 29208\relax$}}8\,\mu$m and the Rayleigh limit $2\pi
r<<\lambda$ is held. The Rayleigh approximation breaks at $x\mathbin{\lower
3.0pt\hbox{$\hbox to0.0pt{\raise 5.0pt\hbox{$\char 62\relax$}\hss}\mathchar
29208\relax$}}0.25$ (Voshchinnikov, 2004; Draine & Hensley, 2021b) and cannot
be used in most of the interesting cases. Electromagnetic absorption and
scattering by spheroids can be treated using different methods. A numerical
solution of separation of variables in the Maxwell equation is presented by
Voshchinnikov & Farafonov (1993). The discrete dipole approximation by Draine
& Flatau (1994) offers a possibility for the treatment of various grain
structures. In addition, the extended boundary condition method by Mishchenko
(2000) also known as T-matrix approximation is available. The methods have
been tested extensively by the authors and excellent agreement at $x\sim 10$
is exemplified (Draine & Hensley, 2021b). The program333Fortran code SPVV8.5
kindly provided by N. Voshchinnikov†. by Voshchinnikov & Farafonov (1993) is
used. It returns for a given complex optical constant $m$ and the size
parameter $x$, the efficiency factors $Q_{{\rm ext}}^{\rm{TM}}(x,m)$ and
$Q_{{\rm ext}}^{\rm{TE}}(x,m)$ (Eqs. 13 \- 14). The code converges up to
$\lvert m-1\rvert\ x\ \la 25$, which is found to be at larger $x$ than in most
of the other applications.

###  4.2 Fitting procedure

The radial grid in grain size is ascending using $r_{i+1}=1.05\ r_{i}$.
Intermediate radii between two consecutive grid points are linearly
interpolated when necessary. The upper radius of the dark dust is varied for
six values of $r_{\rm{Dark}}^{+}=0,0.5,0.8,1.0,5.0$, and $10\,\mu$m, where
$r_{\rm{Dark}}^{+}=0\,\mu$m refers to a model without dark dust. Models
assuming even larger grains were also inspected without finding striking
features. By assuming a continuous distribution in the particle size the
minimum radius of the dark dust follows that of the maximum radius of the
large aC and Si grains, hence
$r_{\rm{Dark}}^{-}=\rm{max}(r_{\rm{ac}}^{+},r_{\rm{Si}}^{+})$. Models fitting
the observational constraints show $r_{\rm{ac}}^{+}<r_{\rm{Si}}^{+}$. The
exponent of the size distribution $q$ is kept as a free parameter, however,
the same $q$ is used for all three dust populations. After some experiments
the mass in dark dust is adopted to be 10 % of the total dust mass; this
parameter will be fine-tuned when fitting the dust emission in the mm range.
Dust abundances are given in the model in relative terms and one parameter can
be kept fixed, and $[\rm{Si}]/[\rm{H}]_{\rm{Si}}=15$ ppm is set.

The polarization data are solely fit by large grains unless otherwise
specified. A detailed fit to the Serkowski curve is obtained when the
alignment of both large grain materials aC and Si differ. Models fitting the
absolute values of the polarization shall be applied towards individual
sightlines that are dominated by a single absorbing cloud (Siebenmorgen et
al., 2018b). Hence parameters that change $p_{\rm{max}}$ but do not affect the
spectral shape of the polarization are held constant (Voshchinnikov, 2012).
This concerns the magnetic field direction that is set to
$\Omega=60^{\rm{o}}$, the ratio of the major axis of prolates $a/b=2$, and the
alignment efficiency of aC grains $\delta_{0}\,(\rm{aC})=10\,\mu$m, while
variations of Si grains of $\delta_{0}\,(\rm{Si})=0.5,1.0$, and $10\,\mu$m are
explored. The maximum alignment radius $r_{\rm{pol}}^{+}$ shows no strong
variations of the starlight polarization spectrum whereas the minimum
alignment radius $r_{\rm{pol}}^{+}$ has a stong impact (Siebenmorgen et al.,
2014). As maximum alignment radius the upper radius derived from the reddening
fit is applied, $r_{\rm{pol,Si}}^{+}=r_{\rm{Si}}^{+}$ and
$r_{\rm{pol,aC}}^{+}=r_{\rm{aC}}^{+}$, respectively. The minimum alignment
radius is taken as a free parameter for aC and Si grains. The Si grains are
dominating the polarization curve at shorter wavelengths and generally
$r_{\rm{pol,Si}}^{-}\la r_{\rm{pol,aC}}^{-}$ is hold.

The reddening curve is fit using the $3\times 6$ values as above for
$\delta_{0}$ and $r_{\rm{Dark}}^{+}$, which have minor impact on
$E_{\rm{mod}}(\lambda)$ and in addition the seven parameters of the dust
model: $[\rm{C}]/[\rm{H}]_{\rm{aC}}$, $[\rm{C}]/[\rm{H}]_{\rm{VSG}}$,
$[\rm{C}]/[\rm{H}]_{\rm{PAH}}$, $[\rm{Si}]/[\rm{H}]_{\rm{vSi}}$, $q$,
$r_{\rm{ac}}^{+}$, and $r_{\rm{Si}}^{+}$. A set of best fit parameters are
computed by a least $\chi^{2}$-technique utilizing the Levenberg–Marquardt
algorithm as implemented in MPFIT444http://purl.com/net/mpfit (Markwardt,
2009). The algorithm can find local minima. The challenge is to identify the
global $\chi^{2}_{\rm r}(r)$ minimum of the reddening curve fit. This minimum
is derived by starting the algorithm using many different initial parameter
values. As an initial guess the mean dust parameters of ISM sightlines fit by
Siebenmorgen et al. (2018b) are applied: $[\rm{C}]/[\rm{H}]_{\rm{aC}}=67$,
$[\rm{Si}]/[\rm{H}]_{\rm{vSi}}=5$, $[\rm{C}]/[\rm{H}]_{\rm{VSG}}=17$,
$[\rm{C}]/[\rm{H}]_{\rm{PAH}}=10$ (ppm), and $q=3$. In addition one initial
radius $r_{\rm i}$ is selected out of 12 different upper sizes of large aC and
Si grains of the radial grid between $180\la r_{\rm
i}=r_{\rm{ac}}^{+}=r_{\rm{Si}}^{+}\la 300$ (nm). This limit is based on an
exploration of the model space which has shown that start values of $r_{\rm
i}>300$ nm did not result in acceptable solutions. An initial MPFIT run is
started keeping $r_{\rm{ac}}^{+}=r_{\rm{Si}}^{+}$ fixed and after converging
the resulting 5 parameters $[\rm{C}]/[\rm{H}]_{\rm{aC}}$,
$[\rm{C}]/[\rm{H}]_{\rm{VSG}}$, $[\rm{C}]/[\rm{H}]_{\rm{PAH}}$,
$[\rm{C}]/[\rm{H}]_{\rm{vSi}}$, and $q$ are held constant, while MPFIT is re-
started a second time with $r_{\rm{ac}}^{+}$ and $r_{\rm{Si}}^{+}$ set as
free.

Table 1: Parameters of the dark dust model for the diffuse ISM using Draine
(2003) optical constants for amorphous silicate grains.

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
Model | Mass ratio (%) | Sizes | Derived quantities
|  | Abundances (ppm) |  |  |  |  |  |  |  |  |  |
$r_{\rm{Dark}}^{+}$ | $\chi^{2}$ | $m_{\rm{Dark}}$ | $m_{\rm{Si}}$ | $m_{\rm{vSi}}$ | $m_{\rm{aC}}$ | $m_{\rm{VSG}}$ | $m_{\rm{PAH}}$ | $q$ | $r_{\rm{Si}}^{+}$ | $r_{\rm{ac}}^{+}$ | $r_{\rm{pol,Si}}^{-}$ | $r_{\rm{pol,aC}}^{-}$ | $\delta_{0}(\rm{Si})$ | $\frac{[\rm{C}]}{[\rm{Si}]}$ | $\frac{M_{\rm{gas}}}{M_{\rm{dust}}}$ | $R_{\rm{V}}$ | $\tau_{\rm{V}}$
($\mu$m) | $N\ (\chi^{2}<1)$ |  | $\frac{[\rm{Si}]}{[\rm{H}]}_{\rm{Si}}$ | $\frac{[\rm{Si}]}{[\rm{H}]}_{\rm{vSi}}$ | $\frac{[\rm{C}]}{[\rm{H}]}_{\rm{aC}}$ | $\frac{[\rm{C}]}{[\rm{H}]}_{\rm{VSG}}$ | $\frac{[\rm{C}]}{[\rm{H}]}_{\rm{PAH}}$ |  | (nm) | (nm) | (nm) | (nm) |  ($\mu$m) |  |  |  |
0‡ | 0.5 | - | 46 | 23 | 25 | 4 | 3 | 3.0 | 243 | 203 | 80 | 166 | 0.5 | 5.2 | 125 | 3.09 | 1.01
| 0 |  | 15 | 7 | 92 | 15 | 10 |  |  |  |  |  |  |  |  |  |
0.75† | 0.7 | $7$ | 43 | 20 | 24 | 4 | 2 | 3.0 | 228 | 201 | 84 | 166 | 0.5 | 5.5 | 138 | 3.16 | 1.42
| 0 |  | 15 | 7 | 96 | 15 | 8 |  |  |  |  |  |  |  |  |  |
1 | 0.6 | $6$ | 44 | 21 | 23 | 4 | 2 | 3.0 | 217 | 207 | 76 | 143 | 1.0 | 5.0 | 131 | 3.09 | 1.37
| 5 |  | 15 | 7 | 88 | 15 | 8 |  |  |  |  |  |  |  |  |  |
5 | 0.6 | $7$ | 41 | 22 | 23 | 3 | 3 | 2.9 | 220 | 208 | 80 | 166 | 0.5 | 5.2 | 124 | 3.11 | 1.30
| 3 |  | 15 | 8 | 95 | 13 | 11 |  |  |  |  |  |  |  |  |  |
10 | 0.7 | $6$ | 49 | 17 | 22 | 3 | 2 | 3.1 | 239 | 222 | 88 | 183 | 0.5 | 4.6 | 124 | 3.18 | 1.36
| 3 |  | 15 | 6 | 91 | 14 | 8 |  |  |  |  |  |  |  |  |  |

Notes: In Col. (1) the upper radius of the dark dust agglomerates
$r_{\rm{Dark}}^{+}$ is specified, in Col. (2) the goodness $\chi^{2}$ of the
best fit model and below the number $N$ of models that are consistent with the
observational constraints and fit the reddening and Serkowski curve at
$\chi^{2}<1$. Corresponding parameters of the specific mass in % per gram dust
(Col. 3-8) of dark dust as lower limit $m_{\rm{Dark}}$, and for large
silicates $m_{\rm{Si}}$, nano-sized silicates $m_{\rm{vSi}}$, large amorphous
carbon $m_{\rm{aC}}$, very small graphite $m_{\rm{VSG}}$, and PAHs
$m_{\rm{PAH}}$. Below, in the second row of Col. 3-8, exemplified dust
abundances [X]/[H] (ppm) are given by adopting [Si]/[H] = 15 ppm in large
silicates. Col. (9) gives the exponent of the dust size distribution $q$. The
upper radius of large silicates $r_{\rm{Si}}^{+}$ (Col. 10) and amorphous
carbon $r_{\rm{ac}}^{+}$ (Col. 11), their lower alignment radii
$r_{\rm{pol,Si}}^{-}$ and $r_{\rm{pol,aC}}^{-}$ (Col. 12, 13), respectively,
and the alignment efficiency (Col. 14) of large silicates
$\delta_{0}(\rm{Si})$ is given. Derived quantities of the dust models are
given for the dust abundance ratio [C]/[Si] (Col. 15), the gas-to-dust mass
ratio $M_{\rm{gas}}/M_{\rm{dust}}$ (Col. 16), the total-to-selective
extinction $R_{\rm{V}}$ (Eq. 5, Col. 17), and the optical depth (Col. 18) that
matches the optical-to-submm polarization ratio of ${p_{850\,\mu{\rm
m}}}/{(p_{\rm{V}}/\tau_{\rm{V}})}=4.31$ (Eq. 8) by Planck Collaboration et al.
(2020).
† Model violates abundance constrain (Eq. 1).
‡ Models without dark dust do not fit the Planck Collaboration et al. (2015,
2020) data at $\ga 0.8$ mm.

This procedure is iterated, generally twice, until $\chi^{2}_{\rm r}$ is not
reduced further. Then the starlight polarization curve is fit minimizing
$\chi_{\rm
p}^{2}\left(\delta_{0}\,(\rm{Si})\,,r_{\rm{pol}}^{-}\,(\rm{Si})\,,r_{\rm{pol}}^{-}\,(\rm{aC})\right)$
between the observed Serkowski curve (Eq. 7) and the dust model (Eq. 22). In
that method the 7 best-fit dust parameters derived from the reddening curve
procedure are held constant and $\chi_{\rm p}^{2}$ is computed for the three
$\delta_{0}\,(\rm{Si})$ and all combinations of the minimum alignment radii of
aC and Si grains between
$6\,{\rm{nm}}<r_{\rm{pol}}^{-}<\rm{min}(r^{+}_{\rm{aC,Si}})$, respectively.

For keeping the computational time within reasonable limits the fitting
procedure is vectorized by running calls to MPFIT with the many different
start values parallel.

##  5 Results

A $r_{\rm{Dark}}^{+}\times\delta_{0}(\rm{Si})\times r_{\rm{ac}}^{+}$ tuple
with $3\times 6\times 12$ models is computed applying the procedure of Sec.
4.2. The model grid includes for each combination of $\delta_{0}(\rm{Si})$ and
$r_{\rm{Dark}}^{+}$, the seven derived best-fit parameters to the reddening
$[\rm{C}]/[\rm{H}]_{\rm{aC}}$, $[\rm{C}]/[\rm{H}]_{\rm{VSG}}$,
$[\rm{C}]/[\rm{H}]_{\rm{PAH}}$, $[\rm{C}]/[\rm{H}]_{\rm{vSi}}$, $q$,
$r_{\rm{ac}}^{+}$, $r_{\rm{Si}}^{+}$, the two best fit parameters to the
starlight polarization $r_{\rm{ac}}^{+}$, $r_{\rm{Si}}^{+}$, and the
corresponding goodness parameters $\chi^{2}_{\rm r}$ and $\chi^{2}_{\rm p}$,
respectively. The majority of models provide a reasonable fit to the reddening
and polarization curves but do not respect the abundance constraint of
$\rm{[C]/[Si]}\la 5.25$ (Eq. 1) and are rejected. Out of the 216 models, 27
are consistent with the dust abundance ratio (Eq. 1) and 22 remain after
applying a $3\ \sigma$ outlier rejection in $\chi^{2}_{\rm r}$ and
$\chi^{2}_{\rm p}$, respectively. Their $\chi^{2}_{\rm r}$ and $\chi^{2}_{\rm
p}$ are each normalised to a mean of one so that they can be combined. The
model that simultaneously fits both curves best is selected from a minimum
$\chi^{2}$ condition where each parameter is given the same weight using

$\chi^{2}(r_{\rm{Dark}}^{+})=\frac{7}{10}\ \chi^{2}_{\rm r}\ +\ \frac{3}{10}\
\chi^{2}_{\rm p}\ .$ (24)

Model parameters with the minimum in $\chi^{2}<1$ for given
$r_{\rm{Dark}}^{+}$ are listed in Table 1. These models agree with the
observed reddening and polarization. Models that ignore dark dust
($r_{\rm{Dark}}^{+}=0\,\mu$m) underpredict the observed emission in the Planck
bands at wavelength $\ga 0.8$ mm (Fig. 6). There is no model with
$r_{\rm{Dark}}^{+}=0.5\,\mu$m that fits reddening and polarization at
$\chi^{2}\la 1$.

Figure 6: Ratio $F/F_{{\rm{no}}\ {\rm{dark}}}$ of the dark dust models and the
observed fluxes (circles, using data as in Fig. 2) divided by the model
neglecting dark dust ($r_{\rm{Dark}}^{+}=0\,\mu$m). Best fits (lines) to the
1.4 mm and 2.1 mm data derived by varying the amount of dark dust
$m_{\rm{Dark}}$ for models with $r_{\rm{Dark}}^{+}\mathbin{\lower
3.0pt\hbox{$\hbox to0.0pt{\raise 5.0pt\hbox{$\char 62\relax$}\hss}\mathchar
29208\relax$}}0.75\,\mu$m as labelled. Parameters as in Table 1.

For $r_{\rm{Dark}}^{+}=0.75\,\mu$m there is only one such model, it has
$\chi^{2}=0.7$ but $\rm{[C]/[Si]}=5.5$ (Table 1, Col. 15). Hence it violates
the abundance constraint (Eq. 1). Therefore, at $r_{\rm{Dark}}^{+}<1\,\mu$m no
model is consistent with the observations at $\chi^{2}<1$ and the number of
such models is $N\ (\chi^{2}<1)=0$ (Table 1, Col. 2).

Models that agree with the observing constraints consider micrometre-sized
dark dust particles at $r_{\rm{Dark}}^{+}\ga 1\,\mu$m. The global minimum over
all models is found for $r_{\rm{Dark}}^{+}=1\,\mu$m at $\chi^{2}=0.6$. For
this radius, there are 5 models at $\chi^{2}<1$. The best fitting models of
$r_{\rm{Dark}}^{+}=5$ and $10\,\mu$m have similar $\chi^{2}\ $ ($\la 0.7$),
respectively. Their fits to the observed reddening, polarization, and dust
emission are of similar quality as for the best fit
$r_{\rm{Dark}}^{+}=1\,\mu$m model. For each of these radii there are $N\
(\chi^{2}<1)=3$ models that fit the reddening and polarization at $\chi^{2}<1$
and fulfill the abundance constraint [C]/[Si] ¡ 5.25 (Eq. 1). The scatter in
the individual parameters between the selected models is small (Table 1). The
peak-to-peak variations in the abundances for a given dust population (Table
1, Col. 3-8) and the particular size parameters stay well below 10 % (Table 1,
Col. 10-13).

The fit of the $r_{\rm{Dark}}^{+}=1\,\mu$m model with parameters of Table 1 to
the observed reddening and extinction is shown in Fig. 1. The individual dust
components show the known behaviour of the reddening: nanoparticles are
responsible for the rise in the far UV, PAH and VSG fit the 2175Å extinction
bump, and large aC and Si grains give a rather flat contribution from the U
band to shorter wavelengths. Remarkable is the constant reddening of dark dust
from the far UV to the NIR, up to wavelengths, which are similar to the upper
grain radius $r_{\rm{Dark}}^{+}$, which is for that model at $\sim 1\,\mu$m
(Fig. 1, right). The non-dispersed reddening provided by dark dust to the
UBVRI bands is significant. Dark dust causes extra dimming of light not
accounted for in dust models omitting this component.

The model fit to the observed starlight polarization curve is shown in Fig. 3.
For deriving such a detailed fit it is necessary that both large Si and aC
grains contribute to the polarization and that both dust materials have
distinct characteristics in the alignment radii and the polarization
efficiency (cmp. Col 10-14 of Table 1). Polarized extinction is dominated in
the UV by Si grains and in the R band by aC grains.

Dark dust treated as prolates with axial ratio $a/b=2$, $\delta_{o}=10\,\mu$m,
and $r_{\rm{Dark}}^{+}=1\,\mu$m contribute to $p/p_{\rm{max}}$ most near $\sim
2\,\mu$m at less than a few percent.

Dark dust polarized extinction in the MIR is shown for models with
$r_{\rm{Dark}}^{+}=1\,\mu$m and $r_{\rm{Dark}}^{+}=10\,\mu$m in Fig. 4,
respectively. Large aC grains show a flat polarization at $\sim 20$ % of the
maximum MIR polarization $p_{\rm{max}}$, while Si grains dominate the
spectrum. Dark dust, even when elongated and aligned particles are assumed,
contributes to the normalised MIR polarization by less than
$p/p_{\rm{max}}<10$ %. The composite polarization spectrum derived from
observations of two Wolf-Rayet stars refers to greatly distinct environments
from the diffuse ISM. Nevertheless, the data are shown in Fig. 4 demonstrates
the capabilities of using MIR polarization as a method to estimate the
stoichiometry of silicate grains.

The dust emission of the models is compared to the diffuse emission of the ISM
that is observed at high galactic latitudes normalised per H atom and is shown
for the best fit $r_{\rm{Dark}}^{+}=1\,\mu$m model in Fig. 2. The models are
scaled to the 0.3 mm data, which allows for an estimate of the gas-to-mass
ratio (Eq. 21). A mean of ${M_{\rm{gas}}}/{M_{\rm{dust}}}=126\pm 4$ is derived
for the best fitting models (Table 1, Col 16). The emission of the individual
dust components shows the known behaviour: nanoparticles are responsible for
the MIR emission, PAH to the IR emission bands and large grains dominate the
FIR/submm. The lowest temperature of aC grains is 16.8 K, that of Si grains
14.8 K, and dark dust is as cold as 12.3 K for the $r_{\rm{Dark}}^{+}=1\,\mu$m
model and 8.1 K for the $r_{\rm{Dark}}^{+}=10\,\mu$m, respectively. Dark dust
emission peaks at $\sim 0.2$ mm and becomes a more important contributor in
the mm range where it even overshines the emission by large Si grains. This is
as expected by comparing the extinction cross-sections of the different dust
ingredients. Noticeable in Fig. 7 are the various PAH features, the 9.7 $\mu$m
and 18 $\mu$m bands of the silicates, and that aC grains dominate the cross-
sections over the entire spectrum at $\lambda\ga 0.1\,\mu$m, except near the
$9.7\,\mu$m band. Furthermore, in the submm, the cross-section of dark dust
has similar strength as that of Si grains. Dark dust shows in the mm range a
slightly shallower decline in the emissivity than the other grains, which is
as expected inspecting the imaginary part $k$ of the optical constants (Sec.
3.2).

Figure 7: Total extinction cross-section normalized to V for the
$r_{\rm{Dark}}^{+}=1\,\mu$m model (Table 1) with individual components as
labelled.

Models that exclude dark dust or models that do not alter the FIR/mm spectral
index of the grain emissivity do not account for the dust emission observed by
Planck Collaboration et al. (2015, 2020). This is exemplified in Fig. 6. The
observed flux $F$ is divided by the flux $F_{{\rm{no}}\ {\rm{dark}}}$ of the
models that do not consider dark dust ($r_{\rm{Dark}}^{+}=0\,\mu$m, Table 1).
The latter is a kind of Siebenmorgen et al. (2014) model; they include non-
porous prolate grains and fit the DIRBE and Planck data up to 0.35 mm. These
models are at $F/F_{{\rm{no}}\ {\rm{dark}}}\sim 1$ in Fig. 6. They
systematically underpredict the emission in the Planck bands at $0.8-3$ mm by
$15-30$ %, which is significant considering the unprecedented precision of the
Planck data. Fluffy and spheroidal grains show greater dust emissivity in the
FIR/mm range than do non-porous and spherical grains of the same mass. The
FIR/mm spectral slope of the dust emissivity stays invariable by increasing
the porosity or the axial ratio of the spheroidal particles (cmp. Fig. 5 and
Fig. 3 in Siebenmorgen et al. (2014)). Hence for fitting the emission in the
mm range one must either change the spectral index of the dust emissivity by a
suitable set of optical constants (Draine & Hensley, 2021a) or include dark
dust.

Figure 8: Polarized emission of the diffuse ISM between $0.2-3$ mm normalized
to the polarization at $850\,\mu$m. Observations (circles) with $1\sigma$
error bars in blue by Planck Collaboration et al. (2015, 2020), in magenta by
Gandilo et al. (2016), in brown by Shariff et al. (2019), in orange by Ashton
et al. (2018); as tabulated by Hensley & Draine (2021). Model with
$r_{\rm{Dark}}^{+}=1\,\mu$m (Table 1) and $m_{\rm{Dark}}=6$ % of mass in dark
dust (full line), the contributions (dashed) from large amorphous carbon
(brown) and large silicate grains (green) as well as the total polarization
fraction adopting $m_{\rm{Dark}}=12$ % (dashed) is shown. The contribution
from dark dust when treated as prolate particles are indicated by the area in
grey.

The Planck bandpasses at 0.85 mm and 3 mm includes dust and other emission
components such as the CO $(3-2)$ or CO $(1-2)$ line transitions, and at
longer wavelengths free–free or synchrotron radiation (Galametz et al., 2014).
The two least contaminated bandpasses near 1.4 mm and 2.1 mm are used to
estimate a lower limit of the amount of dark dust. The mass of dark dust is
varied in the $r_{\rm{Dark}}^{+}\ga 0.75\,\mu$m models until a best fit is
found. Typically the mass in dark dust is at least $m_{\rm{Dark}}\ga 6$ % of
the total dust mass (Table 1, Col. 3) to fit the Planck observed emission.

The polarized dust emission at $\lambda\mathbin{\lower 3.0pt\hbox{$\hbox
to0.0pt{\raise 5.0pt\hbox{$\char 62\relax$}\hss}\mathchar
29208\relax$}}0.85\,\mu$m is observed by Planck Collaboration et al. (2015,
2020) and between $0.25\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise
5.0pt\hbox{$\char 62\relax$}\hss}\mathchar
29208\relax$}}\lambda\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise
5.0pt\hbox{$\char 62\relax$}\hss}\mathchar 29208\relax$}}0.85\,\mu$m for
selected areas on the sky by BLASTPol (Gandilo et al., 2016; Ashton et al.,
2018; Shariff et al., 2019). The polarization spectrum normalised to the
fractional polarization at 0.85 mm is tabulated by Hensley & Draine (2021) and
is shown in Fig. 8. It features an astonishingly flat spectrum that is within
1$\sigma$ of 11% constant and challenges several dust models (Draine &
Hensley, 2021a). The best fit of the $r_{\rm{Dark}}^{+}=1\,\mu$m dark dust
model to the polarized emission spectrum is shown in Fig. 8 using parameters
of Table 1. The total fractional FIR/mm polarization spectrum of the model is
within $\sim 5$ % constant and aC grains are the dominating contributor over
the Si grains.

Dark dust when treated as prolate particles and alignment efficiency as aC
grains give in the FIR a contribution of $\la 10$ % and in the mm range $\sim
35$ % to the total polarization. The flatness of the submm/mm polarization
spectrum provides an upper limit of the total amount of dark dust. This is
shown in Fig. 2 where the polarization in the FIR is overestimated when
increasing the dark dust mass to $m_{\rm{Dark}}\ga 12$ % of the total dust
mass.

The consistency of the models between optical and submm polarization is
verified. Polarized extinction and polarized emission are tightly connected
when produced by the same grains. For the diffuse ISM, a characteristic value
is provided by Planck Collaboration et al. (2020) and is given in Eq. 8. The
dark dust models cope with this optical-to-submm polarization ratio by
adopting an optical depth of $1.3\la\tau_{\rm V}\la 1.37$ (Col. 18, Table 1).
This is somewhat smaller than derived for sightlines with translucent clouds
by Guillet et al. (2018), which have a slightly larger reddening than the
diffuse ISM. The optical depth is converted to a dust reddening using the
model-derived total-to-selective extinction $R_{\rm V}$ (Eq. 5). The reddening
of the models is ${E(\rm{B}-\rm{V})}=\tau_{\rm V}/1.086/R_{\rm V}$ (Col. 17-18
of Table 1) and range between $0.40-0.42$ mag, which is in good agreement with
the observationally derived mean of 0.45 mag (Fitzpatrick & Massa, 2007).

Table 2: Parameters of the dust model for the diffuse ISM using Demyk et al.
(2022) optical constants for amorphous silicate grains.

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
Sample |  Abundances${}^{\dagger}\frac{[\rm{X}]}{[\rm{H}]}$ | Sizes |  |  Derived quantities‡
ID | Composition | $\mu$ | $\rho$ | vSi | aC | VSG | PAH | $q$ | $r_{\rm{Si}}^{+}$ | $r_{\rm{ac}}^{+}$ | $r_{\rm{pol,Si}}^{-}$ | $r_{\rm{pol,aC}}^{-}$ | $\chi^{2}$ | $\frac{[\rm{C}]}{[\rm{Si}]}$ | $\frac{M_{\rm{gas}}}{M_{\rm{dust}}}$ | $R_{\rm{V}}$ | $\tau_{\rm{V}}$
|  |  ($u$) |  (g/cm${}^{3})$ |  | (ppm) |  | (nm) |  |  |  |  |
X35 |  0.65 MgO$-$0.35 SiO2 | 141 | 2.7 | 3 | 72 | 11 | 9 | 3.2 | 261 | 262 | 92 | 222 | 1.7 | 5.0 | $<115$ | $-$ | $-$
X40 |  0.60 MgO$-$0.40 SiO2 | 121 | 2.7 | 4 | 29 | 7 | 8 | 2.9 | 286 | 261 | 72 | 88 | 0.5 | 2.4 | $<168$ | 2.94 | $-$
X50A |  0.50 MgO$-$0.50 SiO2 | 100 | 2.7 | 3 | 57 | 10 | 6 | 3.0 | 267 | 248 | 54 | 72 | 0.8 | 4.0 | $<129$ | 3.10 | 1.04
X50B |  0.50 MgO$-$0.50 SiO2 | 100 | 2.7 | 4 | 63 | 11 | 7 | 3.0 | 257 | 251 | 102 | 233 | 0.8 | 4.3 | $<162$ | 3.14 | $-$
E10 |  Mg0.9Fe${}^{3+}_{0.1}$ SiO3 | 99.9 | 2.8 | 5 | 77 | 12 | 8 | 2.9 | 246 | 230 | 59 | 84 | 0.7 | 4.8 | $<148$ | 3.08 | $-$
E20 |  Mg0.8Fe${}^{3+}_{0.2}$ SiO3 | 99.3 | 2.9 | 3 | 41 | 10 | 6 | 3.0 | 266 | 253 | 72 | 97 | 0.7 | 3.1 | $<141$ | 3.09 | $-$
E30 |  Mg0.7Fe${}^{3+}_{0.3}$ SiO3 | 99.8 | 3.0 | 5 | 76 | 13 | 9 | 3.1 | 259 | 227 | 84 | 183 | 0.5 | 4.9 | $<165$ | 2.99 | $-$
E40 |  Mg0.6Fe${}^{3+}_{0.4}$ SiO3 | 99.3 | 3.1 | 5 | 75 | 12 | 9 | 3.1 | 235 | 252 | 57 | 92 | 0.6 | 4.9 | $<161$ | 3.05 | 1.15
E10R |  Mg0.9Fe${}^{2+}_{0.1}$ SiO3 | 99.9 | 2.8 | 4 | 64 | 10 | 8 | 3.0 | 257 | 238 | 66 | 92 | 0.7 | 4.2 | $<159$ | 3.05 | $-$
E20R |  Mg0.8Fe${}^{2+}_{0.2}$ SiO3 | 99.3 | 2.9 | 3 | 46 | 9 | 6 | 3.0 | 267 | 260 | 69 | 92 | 0.7 | 3.3 | $<160$ | 3.11 | $-$
E30R |  Mg0.7Fe${}^{2+}_{0.3}$ SiO3 | 99.8 | 3.0 | 6 | 85 | 13 | 10 | 3.0 | 235 | 234 | 88 | 192 | 0.5 | 5.2 | $<159$ | 3.00 | 1.18
E40R |  Mg0.6Fe${}^{2+}_{0.4}$ SiO3 | 99.3 | 3.1 | 5 | 73 | 13 | 8 | 3.2 | 237 | 244 | 59 | 102 | 0.6 | 4.8 | $<162$ | 2.98 | 1.14

Notes. Notation as in Table 1. † Abundance of large amorphous silicate grains
are [Si]/[H] = 15 ppm. ‡ Models that do not fit the reddening or polarised
emission (Planck Collaboration et al., 2020) are indicated by symbol ’$-$’ and
models that do not fit the dust emission (Dwek et al., 1997; Planck
Collaboration et al., 2015) are indicated by symbol ’$<$’.

Figure 9: Best fits (full line) to the reddening (left) and starlight
polarisation (right) using Demyk et al. (2022) optical constants as labelled.
Model parameters as in Table 2 and data as in Fig. 1 and Fig. 3, respectively.
Figure 10: Dust emission of the diffuse ISM for models using Demyk et al.
(2022) optical constants (as labelled) that fit simultaneously the reddening
and starlight polarization (Fig. 9). Model parameters as in Table 2 and data
as in Fig. 2.

##  6 Laboratory studies of cosmic dust analogous

Figure 11: Flux ratio to the observed photometry (left) and the polarised
emission spectrum (right) with data as in Fig. 2 and Fig. 8, respectively.
Models using Demyk et al. (2022) optical constants (as labelled) that do not
fit the polarised emission spectrum are shown by dashed lines otherwise by a
full line.

Figure 12: As Fig. 11 for a 97:3 mix in mass of the MgO$-$0.5 SiO2 (X50A) and
Mg0.8Fe${}^{2+}_{0.2}$ SiO3 (E20R) Demyk et al. (2022) samples. This model
with a contribution to the total dust mass of 0 (green), 5% (brown), and 10%
(magenta) of dark dust is shown. The best fit using Draine (2003) optical
constants and parameters as in Table 1 for the $r_{\rm{Dark}}^{+}=1\,\mu$m
model (black) is shown for comparison.

Laboratory studies of cosmic dust analogous foster our knowledge in several
research fields (Jäger et al., 2020): The experiments enlighten our
understanding of processes leading to the formation or destruction of dust
particles (Jones et al., 1994), the growth of grains to pebbles and
planetesimals, and how dust eventually evolves into planets (Wurm & Blum,
1998; Blum & Wurm, 2008; Wurm & Teiser, 2021). In the laboratory, chemical
reaction paths of interstellar gas with charged or uncharged dust and
nanoparticles and their interaction on the grain surfaces can be simulated
(Salama et al., 1996; Herbst, 2001; Jones, 2021). Scattering matrices and
phase functions of several irregular shaped particles have been measured
(Muñoz et al., 2020).

Experimental results of optical constants for a range of dust materials have
been obtained. Although, they are limited in the wavelength coverage
(Dorschner et al. (1995); Mennella et al. (1998); Jäger et al. (2003), see the
Heidelberg - Jena - St. Petersburg database 555https://www2.mpia-
hd.mpg.de/HJPDOC). Models aiming to explain simultaneously the dust absorption
and emission of polarized and unpolarised light require a consistent set of
optical constants from the Lyman limit to about 1 cm, at least. In that
wavelength range, optical constants derived from laboratory experiments are
available for various carbon materials. These have been incorporated in dust
models; see Zubko et al. (2004) for a comprehensive study. For amorphous
silicate grains, a complete set of laboratory-derived optical constants suited
for dust modelling was not available until the work by Demyk et al. (2022).
Commonly, and as used in Sect. 4.2, the semi-empirical set by Draine (2003) is
applied. These optical constants are based on laboratory measurements in the
UV/optical by Huffman & Stapp (1973), on observations in the NIR and MIR, and
some extrapolation by a power-law to longer wavelengths (Draine & Lee, 1984).
The same extrapolation is used by Jones et al. (2017), who considered in the
UV/optical range optical constants of amorphous silicate dust by Scott & Duley
(1996) and added metallic Fe and FeS inclusions to reproduce NIR observations.
The exponent of the power-law was modified by Draine & Hensley (2021a) to
accommodate the deficit submm/mm emission that was present in all of the
previous dust models when confronted with the observations by the Planck
mission (Ysard, 2020).

Recently, Demyk et al. (2022) calculated optical constants between
$10^{-2}-10^{5}\,\mu$m of four Mg-rich glassy silicate dust particles with
stoichiometry from about enstatite to olivine and eight samples of Mg- and Fe-
rich silicates with stoichiometry close to pyroxene. For the samples, the mass
absorption coefficients were measured at temperatures between $10-300$ K and
between 5 to $\sim 1000\,\mu$m (Demyk et al., 2017a, b). The optical constants
of the samples show at temperatures above 30 K and wavelengths
$\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise 5.0pt\hbox{$\char
62\relax$}\hss}\mathchar 29208\relax$}}\,80\,\mu$m a significant temperature
dependence. The dust absorption cross section at 300 K is increased by about
one order of magnitude when compared to the sample measured at 30 K. This
temperature dependence of the cross-section has so far been neglected in dust
models. It results in an overestimate of the derived dust masses that is
important for environments where dust is heated to such high temperatures
(Fanciullo et al., 2020). The optical constants remain constant between 10 K
and 30 K. In the diffuse ISM of the Milky Way the temperatures of large grains
stay below $\sim 20$ K. Demyk et al. (2022) adopted in the wavelength range
between 0.5 to $1\,\mu$m, the refractive indices as measured for similar kinds
of amorphous silicate grains by Dorschner et al. (1995) and Jäger et al.
(2003). The extrapolation of the measurements to the entire wavelength range
is a delicate issue. Besides the numerical challenges, detailed knowledge of
the sample is required such as the bulk density of the materials, the grain
shapes and structures, as well as the particle size distribution of the
agglomerates. These parameters were derived by investigating images of the
samples obtained by using Transmission Electron Microscopy (Demyk et al.,
2022). Samples labelled ’E’ consist of submicron-sized particles with
irregular shapes that are characterised by prolates with axial ratios of
$a/b\sim 2$ and glassy silicate with $a/b\sim 1.5$. These axial ratios are
used in the following.

The fitting procedure of Sect. 4.2 is applied to the dust models by replacing
the optical constants of the large silicate grains with those by Demyk et al.
(2022) measured at 10 K. The alignment efficiency of silicates is set to
$\delta_{0}=1\ \mu$m. First, the contribution of dark dust is neglected. The
models that are consistent with the abundance constraints (Eq. 1) and that
best fit the reddening and starlight polarisation and the emission of
polarised and unpolarised light simultaneously are shown in Fig. 9 \- Fig. 11.
Sample characteristics, dust parameters, and derived quantities of these
models are summarised in Table 2. In that table, columns 1 - 4 provide the
identifier, composition, molecular weight $\mu$ of the mean composition, and
bulk density $\rho$ of the sample as given by Demyk et al. (2022). Parameters
for the abundances of the dust populations (columns 4 - 8) and their size
parameters (columns 9 - 13) are specified. The derived quantities such as the
dust abundance ratio (col. 15), the dust-to-gas mass ratio (col. 16), the
total-to-selective extinction (col. 17), and the optical depth (col. 18) that
match the optical-to-submm polarization ratio of ${p_{850\,\mu{\rm
m}}}/{(p_{\rm{V}}/\tau_{\rm{V}})}=4.31$ (Eq. 8) by Planck Collaboration et al.
(2020). The goodness parameter of the fit is given in col. 14.

The mean reddening curve is fit by all samples reasonably well except for
sample X35, which overestimates the reddening between $5-7$ ($\mu$m-1) and
underestimates the far UV rise (Fig. 9, left). The starlight polarisation
spectrum is fit within the errors by all models (Fig. 9, right), although at
somewhat larger dispersion than the dark dust model shown in Fig. 3.

However, none of these models fit the dust emission (Fig. 10). All models
except sample X35 underestimate the FIR and all models except samples E20R and
X50B underestimate the emission $>0.5$ mm (Fig. 10). Thus estimates of the
dust-to-gas mass ratios for these models shall be taken with caution. The
model features of the dust emission become more apparent in the left panel of
Fig. 11, where the ratio of the flux to the observed photometry is shown.
Samples that do not fit the observed polarised emission spectrum are indicated
by a dashed line and those that do are shown by full lines, those are samples
E30R, E40R, E40 and X40. The potpourri of the fits to the polarised emission
spectrum is shown in the right panel of Fig. 11. Samples that do not fit the
$0.25-0.5$ mm polarisation spectrum by Gandilo et al. (2016), Shariff et al.
(2019), and Ashton et al. (2018) within 15% or deviates by more than $5\sigma$
from the Planck Collaboration et al. (2020) polarisation spectrum in the
millimetre are indicated by dashed lines.

Mixing the curves that are shown in the panels of Fig. 11 allows finding a
flattened curve that better fits the data. The sample X50A appears particular
attractive as it fits the polarization data and the FIR/submm emission up to
0.8 mm already. That model only falls short in the mm-region and this deficit
emission may be filled-up using sample E20R that gravely overestimates the
submm/mm emission. Indeed a 97:3 mix in mass of the MgO$-$0.5 SiO2 (X50A) and
Mg0.8Fe${}^{2+}_{0.2}$ SiO3 (E20R) provides a simultaneous fit to all the
observing constraints. This dust mix without a contribution by dark dust is
shown in Fig. 12 together will models that include dark dust contributing to
5% and 10% to the total dust mass, respectively. The best fit using Draine
(2003) optical constants and parameters as in Table 1 for the
$r_{\rm{Dark}}^{+}=1\,\mu$m model is shown for comparison. All four models
provide a similarly good fit to the data.

##  7 Conclusion

Dark dust was suggested for the unification of spectroscopic and parallax-
derived distances. Such grains have a wavelength-independent reddening and
non-selective extinction in the optical were already considered by Trumpler
(1930). The extinction properties of these particles require that they are
large. Their absorbed energy will be re-emitted in the FIR-mm.

In this paper dark dust has been included as an additional grain population.
The model copes with the current state of observations and provides
constraints on the physical properties of dust in the diffuse ISM. Major
observations that have been verified are representative solid-phase element
abundances, far UV-NIR reddening, optical-NIR polarized extinction, FIR-mm
dust emission of polarized and unpolarized light, and the ratio of the
optical-to-submm polarization. A dust model with 11 parameters for the various
grain abundances, sizes, and alignment properties accounts for these data
simultaneously. The parameter space is explored by a vectorised iterative
fitting procedure to derive the physical properties of dark dust. Different
sets of optical constants for amorphous silicate grains are analysed. The
principal findings of that study are:

  1. 1.

A large set of models that account for the mean reddening, extinction and
starlight polarization curves of the Milky Way fail to respect the adopted
abundance constraints.

  2. 2.

A detailed fit to the Serkowski curve is derived assuming different alignment
characteristics of large aC and Si grains.

  3. 3.

Dark dust consists of micrometre-sized particles of at least
$r_{\rm{Dark}}^{+}\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise
5.0pt\hbox{$\char 62\relax$}\hss}\mathchar 29208\relax$}}1\,\mu$m.

  4. 4.

Dark dust provides a significant wavelength-independent reddening in the
UV/optical and up to wavelengths $\lambda\la r_{\rm{Dark}}^{+}$.

  5. 5.

In the FIR/mm the extinction cross-section of dark dust is of similar
strengths than Si grains and shows a slightly less steep slope than large
grains.

  6. 6.

The detected Planck excess emission at $0.8-3$ mm that previous models cannot
explain without ad-hoc changes in the slope of the dust emissivity or
adjustments of the optical constants is provided by the emission of very cold
($8-12$ K) dark dust. Very cold dust is frequently observed in other non-
active galaxies.

  7. 7.

The observed flatness of the submm/mm polarization spectrum is held by the
model unless the mass in dark dust is $\la 12$ % of the total dust mass.

  8. 8.

Dark dust, when treated as aligned prolate particles with
$r_{\rm{Dark}}^{+}=1\,\mu$m, does not provide a significant contribution to
the observed polarized extinction, polarization in the $10\,\mu$m silicate
band, and polarized emission at $\lambda\la 1$ mm. At longer wavelengths
($1-3$ mm) dark dust may contribute to $\sim 1/3$ of the maximum polarization.

  9. 9.

Dark dust models fit the characteristic value of the optical-to-submm
polarization ratio assuming an optical depth of $\tau_{\rm V}=1.25\pm 0.13$.
The total-to-selective extinction varies between $2.9\la R_{\rm V}\la 3.2$,
which translates to a mean reddennig of ${E(\rm{B}-\rm{V})}\sim 0.4$ mag.

  10. 10.

The gas-to-dust mass ratio of ${M_{\rm{gas}}}/{M_{\rm{dust}}}=126\pm 4$ in the
models using optical constants by Draine (2003). The estimate is consistent
when using a 97:3 mix in mass of silicate dust analogous to the samples by
Demyk et al. (2022) of MgO$-$0.5 SiO2 and Mg0.8Fe${}^{2+}_{0.2}$. The later
mixture provides a good fit to the observing constraints and still
accommodates up to 5 - 10% of mass in dark dust.

Present spectro-polarization in the MIR demonstrates the ability to constrain
the stoichiometry of silicate grains. More sensitive MIR polarization spectra
are required along sightlines of the diffuse ISM to constrain the models
presented here. Dark dust with icy grain mantles would foster grain growth and
provide a reservoir for the yet unaccounted O depletion. For the manifestation
of such a model ice absorption features in the diffuse ISM need to be
detected. Observations shall be possible utilizing the high sensitivity of the
JWST spectrometers.

The dark dust model will be applied to individual sightlines to explain the
absolute reddening and polarization towards selected stars. This requires a
unification of distance estimates to the star either when derived by
spectroscopy or from parallax. Dust models that ignore dark dust do not
account for the unification of both distance estimates. Dark dust causes
significant dimming of light that is not treated in some contemporary research
for example by estimating spectroscopic distances and the calibration of SN Ia
light curves (Phillips, 1993; Riess et al., 1996). This deserves further
investigation.

###### Acknowledgements.

I am grateful to Karen Demyk for granting me early access to optical constants
of the laboratory samples of amorphous silicate grains. I am grateful to
Endrik Krügel and Rolf Chini for helpful discussions.

## References

  * Allamandola et al. (1989) Allamandola, L. J., Tielens, A. G. G. M., & Barker, J. R. 1989, ApJS, 71, 733
  * Andersson et al. (2015) Andersson, B. G., Lazarian, A., & Vaillancourt, J. E. 2015, ARA&A, 53, 501
  * Andriesse et al. (1978) Andriesse, C. D., Donn, B. D., & Viotti, R. 1978, MNRAS, 185, 771
  * Ashton et al. (2018) Ashton, P. C., Ade, P. A. R., Angilè, F. E., et al. 2018, ApJ, 857, 10
  * Asplund et al. (2009) Asplund, M., Grevesse, N., Sauval, A. J., & Scott, P. 2009, ARA&A, 47, 481
  * Bagnulo et al. (2017) Bagnulo, S., Cox, N. L. J., Cikota, A., et al. 2017, A&A, 608, A146
  * Bailer-Jones et al. (2021) Bailer-Jones, C. A. L., Rybizki, J., Fouesneau, M., Demleitner, M., & Andrae, R. 2021, AJ, 161, 147
  * Bailer-Jones et al. (2018) Bailer-Jones, C. A. L., Rybizki, J., Fouesneau, M., Mantelet, G., & Andrae, R. 2018, AJ, 156, 58
  * Blum & Wurm (2008) Blum, J. & Wurm, G. 2008, ARA&A, 46, 21
  * Bohlin et al. (1978) Bohlin, R. C., Savage, B. D., & Drake, J. F. 1978, ApJ, 224, 132
  * Bohren & Huffman (1983) Bohren, C. F. & Huffman, D. R. 1983, Absorption and scattering of light by small particles (ISBN)
  * Bowen et al. (2008) Bowen, D. V., Jenkins, E. B., Tripp, T. M., et al. 2008, ApJS, 176, 59
  * Chen et al. (2018) Chen, X., Wang, S., Deng, L., & de Grijs, R. 2018, ApJ, 859, 137
  * Chini et al. (2012) Chini, R., Hoffmeister, V. H., Nasseri, A., Stahl, O., & Zinnecker, H. 2012, MNRAS, 424, 1925
  * Chini et al. (1993) Chini, R., Krügel, E., Haslam, C. G. T., et al. 1993, A&A, 272, L5
  * Compiègne et al. (2011) Compiègne, M., Verstraete, L., Jones, A., et al. 2011, A&A, 525, A103
  * Das et al. (2010) Das, H. K., Voshchinnikov, N. V., & Il’in, V. B. 2010, MNRAS, 404, 265
  * Demyk et al. (2022) Demyk, K., Gromov, V., Meny, C., et al. 2022, arXiv e-prints, arXiv:2209.06513
  * Demyk et al. (2017a) Demyk, K., Meny, C., Leroux, H., et al. 2017a, A&A, 606, A50
  * Demyk et al. (2017b) Demyk, K., Meny, C., Lu, X. H., et al. 2017b, A&A, 600, A123
  * Deng et al. (2022) Deng, D., Sun, Y., Wang, T., Wang, Y., & Jiang, B. 2022, ApJ, 935, 175
  * Desert et al. (1990) Desert, F. X., Boulanger, F., & Puget, J. L. 1990, A&A, 500, 313
  * Do-Duy et al. (2020) Do-Duy, T., Wright, C. M., Fujiyoshi, T., et al. 2020, MNRAS, 493, 4463
  * Dorschner et al. (1995) Dorschner, J., Begemann, B., Henning, T., Jaeger, C., & Mutschke, H. 1995, A&A, 300, 503
  * Draine (2003) Draine, B. T. 2003, ApJ, 598, 1017
  * Draine & Flatau (1994) Draine, B. T. & Flatau, P. J. 1994, Journal of the Optical Society of America A, 11, 1491
  * Draine & Fraisse (2009) Draine, B. T. & Fraisse, A. A. 2009, ApJ, 696, 1
  * Draine & Hensley (2021a) Draine, B. T. & Hensley, B. S. 2021a, ApJ, 909, 94
  * Draine & Hensley (2021b) Draine, B. T. & Hensley, B. S. 2021b, ApJ, 919, 65
  * Draine & Lee (1984) Draine, B. T. & Lee, H. M. 1984, ApJ, 285, 89
  * Dunkin & Crawford (1998) Dunkin, S. K. & Crawford, I. A. 1998, MNRAS, 298, 275
  * Dwek et al. (1997) Dwek, E., Arendt, R. G., Fixsen, D. J., et al. 1997, ApJ, 475, 565
  * Dyck & Beichman (1974) Dyck, H. M. & Beichman, C. A. 1974, ApJ, 194, 57
  * Efimov (2009) Efimov, Y. S. 2009, Bulletin Crimean Astrophysical Observatory, 105, 82
  * Fanciullo et al. (2015) Fanciullo, L., Guillet, V., Aniano, G., et al. 2015, A&A, 580, A136
  * Fanciullo et al. (2020) Fanciullo, L., Kemper, F., Scicluna, P., Dharmawardena, T. E., & Srinivasan, S. 2020, MNRAS, 499, 4666
  * Fischera (2004) Fischera, J. 2004, A&A, 428, 99
  * Fitzpatrick & Massa (2007) Fitzpatrick, E. L. & Massa, D. 2007, ApJ, 663, 320
  * Fitzpatrick et al. (2019) Fitzpatrick, E. L., Massa, D., Gordon, K. D., Bohlin, R., & Clayton, G. C. 2019, ApJ, 886, 108
  * Gaia Collaboration et al. (2018) Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2018, A&A, 616, A1
  * Gaia Collaboration et al. (2020) Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2020, arXiv e-prints, arXiv:2012.01533
  * Gaia Collaboration et al. (2016) Gaia Collaboration, Prusti, T., de Bruijne, J. H. J., et al. 2016, A&A, 595, A1
  * Galametz et al. (2014) Galametz, M., Albrecht, M., Kennicutt, R., et al. 2014, MNRAS, 439, 2542
  * Galametz et al. (2009) Galametz, M., Madden, S., Galliano, F., et al. 2009, A&A, 508, 645
  * Galliano et al. (2005) Galliano, F., Madden, S. C., Jones, A. P., Wilson, C. D., & Bernard, J. P. 2005, A&A, 434, 867
  * Gandilo et al. (2016) Gandilo, N. N., Ade, P. A. R., Angilè, F. E., et al. 2016, ApJ, 824, 84
  * Gordon et al. (2009) Gordon, K. D., Cartledge, S., & Clayton, G. C. 2009, ApJ, 705, 1320
  * Greenberg et al. (1995) Greenberg, J. M., Li, A., Mendoza-Gomez, C. X., et al. 1995, ApJ, 455, L177
  * Grenier et al. (2005) Grenier, I. A., Casandjian, J.-M., & Terrier, R. 2005, Science, 307, 1292
  * Guillet et al. (2018) Guillet, V., Fanciullo, L., Verstraete, L., et al. 2018, A&A, 610, A16
  * Hage & Greenberg (1990) Hage, J. I. & Greenberg, J. M. 1990, ApJ, 361, 251
  * Heinz et al. (2016) Heinz, S., Corrales, L., Smith, R., et al. 2016, ApJ, 825, 15
  * Henning (2010) Henning, T. 2010, Annual Review of Astronomy and Astrophysics, 48, 21
  * Hensley & Draine (2021) Hensley, B. S. & Draine, B. T. 2021, ApJ, 906, 73
  * Herbst (2001) Herbst, E. 2001, in Spectroscopy from Space, ed. J. Demaison, K. Sarka, & E. A. Cohen, 1
  * Hong & Greenberg (1980) Hong, S. S. & Greenberg, J. M. 1980, A&A, 88, 194
  * Huffman & Stapp (1973) Huffman, D. R. & Stapp, J. L. 1973, in Interstellar Dust and Related Topics, ed. J. M. Greenberg & H. C. van de Hulst, Vol. 52, 297
  * Indebetouw et al. (2005) Indebetouw, R., Mathis, J. S., Babler, B. L., et al. 2005, ApJ, 619, 931
  * Jäger et al. (2003) Jäger, C., Dorschner, J., Mutschke, H., Posch, T., & Henning, T. 2003, A&A, 408, 193
  * Jäger et al. (2020) Jäger, C., Potapov, A., Rouillé, G., & Henning, T. 2020, in Laboratory Astrophysics: From Observations to Interpretation, ed. F. Salama & H. Linnartz, Vol. 350, 27–34
  * Jenkins (2009) Jenkins, E. B. 2009, ApJ, 700, 1299
  * Jones (2021) Jones, A. P. 2021, arXiv e-prints, arXiv:2111.04516
  * Jones et al. (2013) Jones, A. P., Fanciullo, L., Köhler, M., et al. 2013, A&A, 558, A62
  * Jones et al. (2017) Jones, A. P., Köhler, M., Ysard, N., Bocchio, M., & Verstraete, L. 2017, A&A, 602, A46
  * Jones et al. (1994) Jones, A. P., Tielens, A. G. G. M., Hollenbach, D. J., & McKee, C. F. 1994, ApJ, 433, 797
  * Jones (1972) Jones, B. F. 1972, ApJ, 171, L57
  * Jura et al. (2001) Jura, M., Webb, R. A., & Kahane, C. 2001, ApJ, 550, L71
  * Kataoka et al. (2017) Kataoka, A., Tsukagoshi, T., Pohl, A., et al. 2017, ApJ, 844, L5
  * Kennicutt et al. (2011) Kennicutt, R. C., Calzetti, D., Aniano, G., et al. 2011, PASP, 123, 1347
  * Knapp & Kerr (1974) Knapp, G. R. & Kerr, F. J. 1974, A&A, 35, 361
  * Köhler et al. (2015) Köhler, M., Ysard, N., & Jones, A. P. 2015, A&A, 579, A15
  * Krełowski et al. (2016) Krełowski, J., Galazutdinov, G. A., Strobel, A., & Mulas, G. 2016, Acta Astron., 66, 469
  * Krügel (2003) Krügel, E. 2003, The physics of interstellar dust (IOP)
  * Krügel (2008) Krügel, E. 2008, An introduction to the physics of interstellar dust (IOP)
  * Krügel & Siebenmorgen (1994) Krügel, E. & Siebenmorgen, R. 1994, A&A, 288, 929
  * Krügel et al. (1998) Krügel, E., Siebenmorgen, R., Zota, V., & Chini, R. 1998, A&A, 331, L9
  * Krüger et al. (2015) Krüger, H., Strub, P., Grün, E., & Sterken, V. J. 2015, ApJ, 812, 139
  * Landgraf et al. (2000) Landgraf, M., Baggaley, W. J., Grün, E., Krüger, H., & Linkert, G. 2000, J. Geophys. Res., 105, 10343
  * Lanz et al. (1995) Lanz, T., Heap, S. R., & Hubeny, I. 1995, ApJ, 447, L41
  * Lenz et al. (2017) Lenz, D., Hensley, B. S., & Doré, O. 2017, ApJ, 846, 38
  * Liszt (2014) Liszt, H. 2014, ApJ, 780, 10
  * Madden et al. (2013) Madden, S. C., Rémy-Ruyer, A., Galametz, M., et al. 2013, PASP, 125, 600
  * Maercker et al. (2022) Maercker, M., Khouri, T., Mecina, M., & De Beck, E. 2022, A&A, 663, A64
  * Maíz Apellániz & Barbá (2018) Maíz Apellániz, J. & Barbá, R. H. 2018, A&A, 613, A9
  * Maíz-Apellániz et al. (2004) Maíz-Apellániz, J., Walborn, N. R., Galué, H. Á., & Wei, L. H. 2004, ApJS, 151, 103
  * Marchenko & Moffat (2017) Marchenko, S. V. & Moffat, A. F. J. 2017, MNRAS, 468, 2416
  * Markova et al. (2011) Markova, N., Puls, J., Scuderi, S., Simón-Díaz, S., & Herrero, A. 2011, A&A, 530, A11
  * Markwardt (2009) Markwardt, C. B. 2009, in Astronomical Society of the Pacific Conference Series, Vol. 411, Astronomical Data Analysis Software and Systems XVIII, ed. D. A. Bohlender, D. Durand, & P. Dowler, 251
  * Martin et al. (1992) Martin, P. G., Adamson, A. J., Whittet, D. C. B., et al. 1992, ApJ, 392, 691
  * Martins (2018) Martins, F. 2018, A&A, 616, A135
  * Mathis et al. (1983) Mathis, J. S., Mezger, P. G., & Panagia, N. 1983, A&A, 500, 259
  * Mathis et al. (1977) Mathis, J. S., Rumpl, W., & Nordsieck, K. H. 1977, ApJ, 217, 425
  * Mathis & Whiffen (1989) Mathis, J. S. & Whiffen, G. 1989, ApJ, 341, 808
  * Mennella et al. (2003) Mennella, V., Baratta, G. A., Esposito, A., Ferini, G., & Pendleton, Y. J. 2003, ApJ, 587, 727
  * Mennella et al. (1998) Mennella, V., Brucato, J. R., Colangeli, L., et al. 1998, ApJ, 496, 1058
  * Mirabel & Gergely (1979) Mirabel, I. F. & Gergely, T. E. 1979, A&A, 77, 110
  * Mishchenko (2000) Mishchenko, M. I. 2000, Appl. Opt., 39, 1026
  * Muñoz et al. (2020) Muñoz, O., Moreno, F., Gómez-Martín, J. C., et al. 2020, ApJS, 247, 19
  * Mulas et al. (2013) Mulas, G., Zonca, A., Casu, S., & Cecchi-Pestellini, C. 2013, ApJS, 207, 7
  * Nguyen et al. (2018) Nguyen, H., Dawson, J. R., Miville-Deschênes, M. A., et al. 2018, ApJ, 862, 49
  * Nieva & Przybilla (2012) Nieva, M. F. & Przybilla, N. 2012, A&A, 539, A143
  * Ossenkopf (1991) Ossenkopf, V. 1991, A&A, 251, 210
  * Pecaut & Mamajek (2013) Pecaut, M. J. & Mamajek, E. E. 2013, ApJS, 208, 9
  * Phillips (1993) Phillips, M. M. 1993, ApJ, 413, L105
  * Planck Collaboration et al. (2014) Planck Collaboration, Abergel, A., Ade, P. A. R., et al. 2014, A&A, 566, A55
  * Planck Collaboration et al. (2015) Planck Collaboration, Ade, P. A. R., Aghanim, N., et al. 2015, A&A, 576, A106
  * Planck Collaboration et al. (2020) Planck Collaboration, Aghanim, N., Akrami, Y., et al. 2020, A&A, 641, A12
  * Potapov et al. (2021) Potapov, A., Bouwman, J., Jäger, C., & Henning, T. 2021, Nature Astronomy, 5, 78
  * Rachford et al. (2009) Rachford, B. L., Snow, T. P., Destree, J. D., et al. 2009, ApJS, 180, 125
  * Rémy-Ruyer et al. (2013a) Rémy-Ruyer, A., Madden, S. C., Galliano, F., et al. 2013a, A&A, 557, A95
  * Rémy-Ruyer et al. (2013b) Rémy-Ruyer, A., Madden, S. C., Galliano, F., et al. 2013b, A&A, 557, A95
  * Riess et al. (1996) Riess, A. G., Press, W. H., & Kirshner, R. P. 1996, ApJ, 473, 88
  * Salama et al. (1996) Salama, F., Bakes, E. L. O., Allamandola, L. J., & Tielens, A. G. G. M. 1996, ApJ, 458, 621
  * Scicluna et al. (2022) Scicluna, P., Kemper, F., McDonald, I., et al. 2022, MNRAS, 512, 1091
  * Scicluna et al. (2015) Scicluna, P., Siebenmorgen, R., Wesson, R., et al. 2015, A&A, 584, L10
  * Scott & Duley (1996) Scott, A. & Duley, W. W. 1996, ApJS, 105, 401
  * Serkowski et al. (1975) Serkowski, K., Mathewson, D. S., & Ford, V. L. 1975, ApJ, 196, 261
  * Shariff et al. (2019) Shariff, J. A., Ade, P. A. R., Angilè, F. E., et al. 2019, ApJ, 872, 197
  * Shull & Danforth (2019) Shull, J. M. & Danforth, C. W. 2019, ApJ, 882, 180
  * Siebenmorgen & Gredel (1997) Siebenmorgen, R. & Gredel, R. 1997, ApJ, 485, 203
  * Siebenmorgen et al. (2020) Siebenmorgen, R., Krełowski, J., Smoker, J., Galazutdinov, G., & Bagnulo, S. 2020, A&A, 641, A35
  * Siebenmorgen et al. (1999) Siebenmorgen, R., Krügel, E., & Chini, R. 1999, A&A, 351, 495
  * Siebenmorgen et al. (2018a) Siebenmorgen, R., Scicluna, P., & Krełowski, J. 2018a, A&A, 620, A32
  * Siebenmorgen et al. (2014) Siebenmorgen, R., Voshchinnikov, N. V., & Bagnulo, S. 2014, A&A, 561, A82
  * Siebenmorgen et al. (2018b) Siebenmorgen, R., Voshchinnikov, N. V., Bagnulo, S., et al. 2018b, A&A, 611, A5
  * Skórzyński et al. (2003) Skórzyński, W., Strobel, A., & Galazutdinov, G. A. 2003, A&A, 408, 297
  * Smith et al. (2000) Smith, C. H., Wright, C. M., Aitken, D. K., Roche, P. F., & Hough, J. H. 2000, MNRAS, 312, 327
  * Sofia & Meyer (2001) Sofia, U. J. & Meyer, D. M. 2001, ApJ, 558, L147
  * Sota et al. (2014) Sota, A., Maíz Apellániz, J., Morrell, N. I., et al. 2014, ApJS, 211, 10
  * Sota et al. (2011) Sota, A., Maíz Apellániz, J., Walborn, N. R., et al. 2011, ApJS, 193, 24
  * Stecher (1965) Stecher, T. P. 1965, ApJ, 142, 1683
  * Steinacker et al. (2015) Steinacker, J., Andersen, M., Thi, W. F., et al. 2015, A&A, 582, A70
  * Strom et al. (1971) Strom, K. M., Strom, S. E., & Yost, J. 1971, ApJ, 165, 479
  * Trumpler (1930) Trumpler, R. J. 1930, PASP, 42, 214
  * Valencic et al. (2004) Valencic, L. A., Clayton, G. C., & Gordon, K. D. 2004, ApJ, 616, 912
  * Voshchinnikov (2004) Voshchinnikov, N. V. 2004, Astrophys. Space Phys. Res., 12, 1
  * Voshchinnikov (2012) Voshchinnikov, N. V. 2012, J. Quant. Spec. Radiat. Transf., 113, 2334
  * Voshchinnikov & Farafonov (1993) Voshchinnikov, N. V. & Farafonov, V. G. 1993, Ap&SS, 204, 19
  * Voshchinnikov & Henning (2010) Voshchinnikov, N. V. & Henning, T. 2010, A&A, 517, A45
  * Voshchinnikov et al. (2012) Voshchinnikov, N. V., Henning, T., Prokopjeva, M. S., & Das, H. K. 2012, A&A, 541, A52
  * Voshchinnikov et al. (2016) Voshchinnikov, N. V., Il’in, V. B., & Das, H. K. 2016, MNRAS, 462, 2343
  * Walborn (1971) Walborn, N. R. 1971, ApJS, 23, 257
  * Walborn & Fitzpatrick (1990) Walborn, N. R. & Fitzpatrick, E. L. 1990, PASP, 102, 379
  * Wang & Chen (2019) Wang, S. & Chen, X. 2019, ApJ, 877, 116
  * Weingartner & Draine (2001) Weingartner, J. C. & Draine, B. T. 2001, ApJ, 548, 296
  * Westphal et al. (2014) Westphal, A. J., Stroud, R. M., Bechtel, H. A., et al. 2014, Science, 345, 786
  * Whittet et al. (1992) Whittet, D. C. B., Martin, P. G., Hough, J. H., et al. 1992, ApJ, 386, 562
  * Widmark et al. (2022) Widmark, A., Korsmeier, M., & Linden, T. 2022, arXiv e-prints, arXiv:2208.11704
  * Wilking et al. (1980) Wilking, B. A., Lebofsky, M. J., Martin, P. G., Rieke, G. H., & Kemp, J. C. 1980, ApJ, 235, 905
  * Witt et al. (2001) Witt, A. N., Smith, R. K., & Dwek, E. 2001, ApJ, 550, L201
  * Wright et al. (2002) Wright, C. M., Aitken, D. K., Smith, C. H., Roche, P. F., & Laureijs, R. J. 2002, in The Origin of Stars and Planets: The VLT View, ed. J. F. Alves & M. J. McCaughrean, 85
  * Wurm & Blum (1998) Wurm, G. & Blum, J. 1998, Icarus, 132, 125
  * Wurm & Teiser (2021) Wurm, G. & Teiser, J. 2021, Nature Reviews Physics, 3, 405
  * Ysard (2020) Ysard, N. 2020, in Laboratory Astrophysics: From Observations to Interpretation, ed. F. Salama & H. Linnartz, Vol. 350, 53–60
  * Ysard et al. (2015) Ysard, N., Köhler, M., Jones, A., et al. 2015, A&A, 577, A110
  * Zhu et al. (2017) Zhu, H., Tian, W., Li, A., & Zhang, M. 2017, MNRAS, 471, 3494
  * Zhukovska et al. (2018) Zhukovska, S., Henning, T., & Dobbs, C. 2018, ApJ, 857, 94
  * Zubko et al. (2004) Zubko, V., Dwek, E., & Arendt, R. G. 2004, ApJS, 152, 211
  * Zubko et al. (1996) Zubko, V. G., Mennella, V., Colangeli, L., & Bussoletti, E. 1996, MNRAS, 282, 1321