# A comprehensive model of morphologically realistic Cosmic Dust particles: an
application to mimic the unusual Polarization properties of the interstellar
Comet 2I/Borisov

Prithish Halder Indian Institute of Astrophysics, Bangalore
Karnataka 560034, India  Sujan Sengupta Indian Institute of Astrophysics,
Bangalore
Karnataka 560034, India

###### Abstract

The cosmic dust particles found in space are mainly porous aggregates of
smaller grains. Theoretically, these aggregates are replicated using fractal
geometry, assuming a cluster of spheres. Although, the light scattering
response of cosmic dust aggregates has been thoroughly studied using clusters
of spherical grains in the past few decades, yet, the effect of irregularities
on the surface of each grain in an entire aggregate has mostly been neglected.
We, for the first time, introduce a visually realistic cosmic dust model which
incorporates a mixture of rough fractal aggregates (RFA) and agglomerated
debris (Solids) to replicate the unusual polarization-phase curve observed in
case of the interstellar comet 2I/Borisov at multiple wavelengths. The
authenticity of the RFA structures has been verified by replicating light
scattering results of circumstellar dust analogues from the Granada Amsterdam
Light Scattering Database. We demonstrate that the light scattering response
from the RFA structures has a very close resemblance with the experimental
values. Finally, we model the observed polarization-phase curve of the
interstellar comet 2I/Borisov using a mixture of RFA and solid particles. The
best-fit data indicates presence of higher percentage of porous RFA structures
(80%) owing to the fact that the comet carries higher percentage of small and
highly porous pristine cosmic dust particles. Further, the model indicates
that the unusually steeper polarimetric slope and the high dust-to-gas ratio
in relatively newer comets is mainly due to higher porous-to-compact ratio.

Cosmic Dust — Radiative transfer simulations — Polarimetry — Interstellar
objects — Comets — Coma Dust

##  1 Introduction

The subject of interstellar comets is a very recent development in the field
of astronomy, which started to emerge after the discovery of the Oort cloud.
It was Sen & Rana (1993) who for the first time predicted that one should
detect an interstellar comet once in 200 years. The discovery of the
interstellar comet 2I/Borisov in 2019, by Gennady Borisov after 180 years of
cometary research has proved the above prediction to be true. Similar to other
Solar System comets, 2I/Borisov exhibited a distinct coma allowing various
researchers to study the physics and chemistry of the material content using
spectroscopic and polarimetric observations. The spectroscopic studies of the
comet 2I/Borisov indicate a dust-to-gas ratio similar to those observed in
carbon depleted comets of the Solar System (Aravind et al., 2021; Yang et al.,
2021). On the other hand polarimetric observations indicate an unusually
steeper slope (Bagnulo et al., 2021). Generally, Solar System comets are
categorized in two polarimetric classes: low and high polarization comets
depending on the different dust-to-gas ratio observed in the coma (Chernova et
al., 1993; Levasseur-Regourd et al., 1996). Apart from these two classes there
exists a third class (Hadamcik & Levasseur-Regourd, 2003) having polarization
higher than that of high polarization comets, which was observed only in case
one Solar System comet, C/1995 O1 (Hale-Bopp). Such high polarization is
believed to be due to presence of extremely small Rayleigh size dust
particles. The presence of Rayleigh size dust particles, in Hale-Bopp can be
traced back to its origin in the outer regions of our Solar System, where the
physical environment is comparable to that of interstellar medium. Similarly,
2I/Borisov is believed to have originated from the outer regions of its host
stellar system. Although, Hale-Bopp might have visited the Sun once before its
last apparition, 2I/Borisov on the hand, has not encountered any star before
passing close to Sun, thereby the comet may hold a huge population of pristine
cosmic dust particles. The interpretation and analysis of astronomical
observations of dust in comets is mainly based on our knowledge of light
scattering by morphologically irregular particles. The significance of dust
particles having size comparable to the wavelength of incident light has been
widely acknowledged (A’Hearn et al., 1995, 2011; Kimura et al., 2006; Das et
al., 2008; Kolokolova et al., 2015; Zubko et al., 2006, 2020; Deb Roy et al.,
2017; Halder et al., 2018; Halder & Ganesh, 2021). The cosmic dust particles
found in space are mainly porous fractal aggregates of smaller grains formed
due to coagulation and ballistic agglomeration in the circumstellar or
interstellar environment. Theoretically, these aggregates are replicated using
fractal geometry, assuming a cluster of spherical grains. But the studies of
modelling of the third class of comets (Hale-Bopp) done by Lasue & Levasseur-
Regourd (2006); Lasue et al. (2009) and Markkanen et al. (2015) used
aggregates of non-spherical monomers/grains and kept the monomer size fixed
for multiple wavelengths. In order to verify whether 2I/Borisov holds
relatively high amount of pristine cosmic dust particles or to have an
estimate of the amount of pristine dust present in the coma of the comet, it
is necessary to conduct light scattering simulations over modelled pristine
cosmic dust aggregates and replicate the observed unusual polarization with
exact computer modelled replica of cosmic dust. The dust particles studied by
the Rosetta/MIDAS and Rosetta/COSIMA suggest presence of porous aggregated
dust particles that resemble the morphology of interplanetary dust (IDP)
(Bentley et al., 2016; Güttler et al., 2019; Mannel et al., 2019; Schulz et
al., 2015). The IDPs collected from the Earth’s stratosphere and Antarctic ice
having irregular geometry, fluffy aggregates and fractal nature represents the
physical morphology of Solar system cosmic dust (Brownlee, 2003; Noguchi et
al., 2015). Again, due to flash heating in the upper atmosphere, these IDP
samples may not purely represent pristine cosmic dust. The cosmic dust analog
aggregates prepared in the Granada Amsterdam Light Scattering facility using
Condensation Flow Apparatus shall represent the most pristine morphology of
cosmic dust which are devoid of flash heating. The microgravity and laboratory
experiments of dust-dust interactions conducted to replicate the conditions
prevailing in the early Solar System suggest formation of fractal assemblage
of dust via ballistic agglomeration (Blum & Wurm, 2000; Krause & Blum, 2004;
Wurm & Blum, 1998). In a similar way, small dust particles in the interstellar
medium may coagulate in the vicinity of dense molecular clouds. Hence,
astronomers around the world use fractal aggregates/cluster of spheres to
study the physical and/or optical properties of cosmic dust. Numerically
fractal aggregates are prepared using ballistic agglomeration techniques.
These agglomeration techniques hold the physics behind the dust coagulation in
circumstellar and protoplanetary disks, but the morphology of each grain in an
aggregate lacks surface roughness or irregularities. Roughness has been a
matter of concern for a longer period of time and hence, dust structures such
as gaussian random sphere, agglomerated debris and rough spheroids are
developed by various researchers (Kolokolova et al., 2015; Muinonen et al.,
1996; Zubko et al., 2006) to include the contribution of irregularities or
roughness. Although, these rough or irregular structures explain the
contribution of roughness in case of single particles and debris particles,
the contribution of roughness or irregularities on the surface of each grain
of a fractal aggregate remains unknown.

In the present investigation, for the first time, we use a visually realistic
cosmic dust model which is represented by a mixture of highly porous Rough
Fractal Aggregate (RFA) Halder (2022) and low porous Solids (Agglomerated
Debris) to model the unusual polarization properties of the interstellar comet
2I/Borisov. The highly porous RFA structures which are aggregates of
irregular/rough grains have a very close resemblance with the IDPs collected
from Earth’s stratosphere. Initially, the RFA modelled dust particles are
validated by replicating light scattering results from Granada Amsterdam Light
Scattering Database for the different aggregates samples (1-6)(Volten et al.,
2007) of circumstellar or cosmic dust analogs (Nuth et al., 2000; Rietmeijer
et al., 1999). Then, we model the observed polarization-phase curve and the
polarimetric spectral gradient of the interstellar comet 2I/Borisov using a
mixture of RFA model structures (high porous) and Solids (low porous) at the
three wavelengths, $\lambda$ = 0.557$\mu$m ($V_{f}$ filter), 0.655$\mu$m
($R_{f}$ filter) and 0.768$\mu$m ($I_{f}$ filter) respectively. Finally, we
compare the observed dust-to-gas ratio with the intrinsic dust parameter
porous-to-compact ratio for the extremely high-polarization comets (2I/Borisov
& Hale-Bopp) and low-polarization comets (67P/C-G & 1P/Halley) to understand
the dependence of dust-to-gas ratio on the intrinsic dust parameters.

##  2 Modelling Methodology

In this section, we describe the techniques employed to generate RFA (Rought
Fractal Aggregates) and Solid particles used to replicate the light scattering
results from the Granada Amsterdam Light Scattering Database and to model the
polarization properties of the of the interstellar comet 2I/Borisov. The light
scattering technique and the related light scattering parameters are also
discussed in this section.

###  2.1 Fractal Aggregates (FA)

Fractal aggregates (FA) having polydisperse spheres are created following the
BPCA and BCCA agglomeration techniques using the Java package FLAGE111FLAGE
https://scattering.eu/ (Skorupski et al., 2014). The structure of the FA is
loaded in the package REST (Halder, 2022) where the x, y, z coordinates and
radii of each sphere of an aggregate are scaled into an equi-volume sphere
made up of unit dipoles/lattice points. In REST the RFA algorithm remove those
dipoles/lattice points which do not fall within the radii of each sphere and
hence forming the resultant structure which is a fractal aggregate (FA) of
spheres but made up of dipoles/lattice points as shown in Figure-1(a). REST is
a structure tool that generates realistic cosmic dust particles from spheres,
superellipsoids and fractal aggregates (FA). FLAGE is a very useful java tool
to create aggregates of spherical grains. It takes the following physical
parameters as input to create a proper fractal aggregate:

  1. 1.

Number of spheres (N).

  2. 2.

Radius of each sphere/primary particle (rp).

  3. 3.

Radius of aggregate (Ra = $\sqrt{5/3}$Rg, where Ra is the characteristic
radius of the aggregate and Rg is the radius of gyration which is defined in
equation 1).

  4. 4.

Fractal dimension (Df, a dimensionality constant that characterises a fractal
structure and is defined by equation 1).

  5. 5.

Fractal prefactor (kf, a proportionality constant that is a prefactor of the
fractal scaling relation defined by equation 1).

  6. 6.

Porosity (The degree or percentage of space present within a fractal
aggregate).

  7. 7.

Aggregate Type:

    1. (a)

Ballistic Particle Cluster Agglomeration (BPCA).

    2. (b)

Ballistic Cluster-Cluster Agglomeration (BCCA).

    3. (c)

Diffusion Limited Agglomeration (DLA).

    4. (d)

Reaction Limited Agglomeration (RLA).

The structural arrangement of an aggregate having $N$ monodisperse (spheres
having same size) spherical grains (each having radius $r$) is defined by the
following equation (Sorensen et al., 1992),

$N=k_{f}\left(\frac{R_{g}}{r}\right)^{D_{f}}$ (1)

But the aggregates found in space are polydisperse (spheres having different
sizes) in nature. An aggregate having polydisperse spherical grains with
primary particle (PP) radius $r_{p}$, average PP mass $\overline{m_{p}}$ and
aggregate mass $m_{a}$ is defined by Eggersdorfer & Pratsinis (2011) as

$\frac{m_{a}}{\overline{m_{p}}}=k_{f}\left(\frac{R_{g}}{r_{p}}\right)^{D_{f}}$
(2)

where $m_{a}/\overline{m_{p}}=N$ is the number of PP in the aggregate.

The fractal dimension is directly related to the porosity of an aggregate. The
BCCA aggregates have porosity $\geq$ 95% and Df $<$ 2, while the BPCA
aggregates have porosity $\leq$ 90% and Df $>$ 2\. The porosity of different
FA structures is controlled by changing Df = 1.8 to 2.5. This is done to
incorporate minute variation in porosity following the explanations provided
for the different aggregate samples from the Granada Amsterdam Light
Scattering Database. The aggregate samples 1 and 2 (Volten et al., 2007) are
made up of the same material, with the same grain size and the same aggregate
size, yet the polarization maximum (Pmax) for the two samples are different.
It is already clear from previous studies that a slight increase in porosity
shall induce an increase in the Pmax value (Kimura et al., 2006; Halder et
al., 2018). Also, the authors of the Granada Amsterdam Light Scattering
Database have mentioned that this shift in Pmax for the aggregate samples 1
and 2 is possibly due to minute difference in porosity.

Figure 1: (a)Algorithm to generate Fractal Aggregates (FA) having Nd dipoles
and (b) algorithm to generate Rough Fractal Aggregates (RFA) using REST
(Halder, 2022).

###  2.2 Rough Fractal Aggregates (RFA)

The Rough Fractal Aggregate (RFA) structures used in this study are generated
using the java package REST222Rough Ellipsoid Structure Tools (REST)
https://rest-package.readthedocs.io/ (Rough Ellipsoid Structure Tools)
(Halder, 2022) from the loaded FA structure file. The RFA algorithm in REST
crafts surface roughness/irregularities on the surface of each spherical grain
of a FA structure [see Figure-1(b)]. The algorithm to generate the RFA
structures is discussed below:

  1. 1.

Browse and select the structure file of a fractal aggregate/cluster of spheres
having following format ($i$, $R$, $X$ $Y$ $Z$, _mtag_ , _mtag_), where $i$ is
the sphere number, $R$ is the $i^{th}$ sphere radius (in $\mu$m), $X$, $Y$,
$Z$ are the coordinates of each sphere (monomer) and the _mtag_ is the
composition tag.

  2. 2.

Multiply each coordinate and radii with an integer scale factor $n$. This done
to achieve a desired number of dipoles (Nd) for the entire RFA structure.

  3. 3.

Measure the distance ($d_{mono}$) of each monomer from the centre
($0$,$0$,$0$).

  4. 4.

Create the initial sphere having $N_{d}$ dipoles and radius $R_{d}$ =
maximum($d_{mono}$) (in number of dipoles) from the centre of the initial
sphere.

  5. 5.

Randomly choose $N_{ss}$ surface seed cells on the surface of the FA inside
the initial sphere.

  6. 6.

Randomly choose $N_{m}$ material seed cells inside the surface of the FA.

  7. 7.

Measure the distance $D_{m}^{i,j}$ between the $j$th material seed cell and
$i$th dipole of the base structure, where $j$ = 1 to $N_{m}$ and $i$ = 1 to
$N_{d}$.

  8. 8.

Measure the distance $D_{ss}^{i,k}$ between the $k$th surface space seed cell
and $i$th dipole of the base structure, where $k$ = 1 to $N_{ss}$ and $i$ = 1
to $N_{d}$.

  9. 9.

Print those dipoles for which, $D_{m}^{i}$ $<$ $D_{ss}^{i}$ in the final RFA
structure file.

The porosity $\mathrm{P}$ of a fractal aggregate is determined by the ratio of
total number of space seed cells in entire volume of sphere circumscribing the
RFA structure by the total volume of the circumscribing sphere (Halder, 2022),

$\texttt{Volume of initial structure}(V_{i})=N_{d}(\texttt{initial})\times
d^{3}$ $\texttt{Volume of final structure}(V_{f})=N_{d}(\texttt{final})\times
d^{3}$

The total volume of space seed cells is given by,

$V_{T}=[N_{d}(\texttt{final})-N_{d}(\texttt{initial})]\times d^{3}$

Therefore, the degree of porosity is,

$\begin{split}\mathrm{P}&=\frac{V_{T}}{V_{f}}=\frac{[N_{d}(\texttt{final})-N_{d}(\texttt{initial})]\times
d^{3}}{N_{d}(\texttt{final})\times d^{3}}\\\
&=\frac{[N_{d}(\texttt{final})-N_{d}(\texttt{initial})]}{N_{d}(\texttt{final})}\end{split}$
(3)

###  2.3 Generating Solid structures

The Solid structures used in this study are low porous Agglomerated Debris
particles generated using REST following the Poked Structure (PS)
option/algorithm. The steps to generate PS shape are as follows:

  1. 1.

Generate initial spherical structure file target.out having $N_{d}$ dipoles
and radius $R$ (in number of dipoles) using CALLTARGET module.

  2. 2.

Randomly choose $N_{m}$ number of material seed cells from the $N_{d}$ dipoles
present in the target.out file.

  3. 3.

Randomly choose $N_{is}$ number of internal space seed cells from the $N_{d}$
dipoles present in the target.out file.

  4. 4.

Randomly choose $N_{ss}$ number of surface space seed cells from the $N_{d}$
dipoles present in the target.out file. The surface thickness should be $t$
times $r$.

  5. 5.

Measure the distance $D_{m}^{i,j}$ between the $j$th material seed cell and
$i$th dipole of the base structure, where $j$ = 1 to $N_{m}$ and $i$ = 1 to
$N_{d}$.

  6. 6.

Measure the distance $D_{is}^{i,k}$ between the $k$th internal space seed cell
and $i$th dipole of the base structure, where $k$ = 1 to $N_{is}$ and $i$ = 1
to $N_{d}$.

  7. 7.

Measure the distance $D_{ss}^{i,l}$ between the $l$th surface space seed cell
and $i$th dipole of the base structure, where $l$ = 1 to $N_{ss}$ and $i$ = 1
to $N_{d}$.

  8. 8.

Print those dipoles for which, $D_{m}^{i}$ $<$ $D_{is}^{i}$ and $D_{m}^{i}$
$<$ $D_{ss}^{i}$ in the final structure file.

The steps to generate Solid structures (AD particles) using PS algorithm are:

  * •

$R_{d}$ (in number of dipoles) = 64

  * •

$N_{m}$ = 21

  * •

$N_{is}$ = 20

  * •

$N_{ss}$ = 100

  * •

$t$ = 1%

###  2.4 Light scattering simulations

The coma of a comet is optically thin having low volume concentration of dust
particles. Hence, in theoretical modelling of the light scattering by the dust
particles in the coma of comet, multiple scattering effects are neglected. The
scattering phenomenon for a mirror symmetric and macroscopically isotropic
particulate medium is defined by the scattering matrix which represents far
field transformation of the Stokes parameters of the incident light ($I_{i}$,
$Q_{i}$, $U_{i}$, $V_{i}$) to that of the scattered light ($I_{s}$, $Q_{s}$,
$U_{s}$, $V_{s}$). This scattering matrix is given by Bohren et al. (1998):

$\left(\begin{array}[]{cccc}I_{s}\\\ Q_{s}\\\ U_{s}\\\
V_{s}\end{array}\right)=\frac{1}{k^{2}d^{2}}\left(\begin{array}[]{cccc}S_{11}&S_{12}&0&0\\\
S_{12}&S_{22}&0&0\\\ 0&0&S_{33}&S_{34}\\\
0&0&-S_{34}&S_{44}\end{array}\right)\left(\begin{array}[]{cccc}I_{i}\\\
Q_{i}\\\ U_{i}\\\ V_{i}\end{array}\right)$ (4)

where $k$ is the wave-number and $d$ is the distance between the scatterer and
the observer and $S_{ij}$ represents the orientationally symmetric scattering
matrix elements. The angle $\alpha$ between Sun-Comet-Earth is called the
_Phase angle_. Angle $\theta$ = 180 - $\alpha$ is called the _Scattering
angle_ , $\theta$ = [0∘,180∘].

In this work, we study the following light scattering parameters defined by
the scattering matrix elements:

  1. 1.

_Phase function_ : $S_{11}$

  2. 2.

_Degree of linear polarization_ : $DP=-S_{12}/S_{11}$.

  3. 3.

$\frac{S_{22}}{S_{11}}$

The anisotropy condition for a non spherical scatterer is, $S_{11}\neq S_{22}$
and $S_{33}\neq S_{44}$.

We use the Discrete Dipole Approximation(DDA) scattering codes333Discrete
Dipole Approximation (DDA) DDSCAT version 7.3.3 http://ddscat.wikidot.com/
Draine & Flatau (1994) in parallel mode, for the numerical simulations of
light scattering.

Further, the results are averaged using the power-law size distribution
$r^{-n}$ where $n$ ranges between 2.0 to 3.0. The power-law size distribution
is modelled by considering aggregates of different sizes from smallest to
largest. The aggregate sizes are increased by increasing the number of
monomers/grains and keeping the monomer size fixed.

Table 1: ModelX RFA parameters: details of different parameters of RFA structures such as number of monomers (N), aggregate size parameter (Xa), fractal dimension (Df), monomer radius (rp) and aggregate radius (Ra) in the three different wavelengths for the minimum and maximum sizes respectively. N | Xa | Df |  rp($\lambda=0.557\micron$) |  Ra($\lambda=0.557\micron$) |  rp($\lambda=0.655\micron$) |  Ra($\lambda=0.655\micron$) |  rp($\lambda=0.768\micron$) |  Ra($\lambda=0.768\micron$)
---|---|---|---|---|---|---|---|---
45 | 5 | 1.8 |  0.062$\mu$m |  0.44$\mu$m |  0.073$\mu$m |  0.52$\mu$m |  0.085$\mu$m |  0.61$\mu$m
625 | 21 | 1.8 |  0.062$\mu$m |  1.91$\mu$m |  0.073$\mu$m |  2.25$\mu$m |  0.085$\mu$m |  2.62$\mu$m
Table 2: ModelX Solid parameters: details of different parameters of Solid structures such as size parameter (X), packing fraction ($\rho_{f}$) and radius (R) in the three different wavelengths for the minimum and maximum sizes respectively. X | $\rho_{f}$ |  R($\lambda=0.557\micron$) |  R($\lambda=0.655\micron$) |  R($\lambda=0.768\micron$)
---|---|---|---|---
0.65 | 0.26 |  0.057$\mu$m |  0.067$\mu$m |  0.079$\mu$m
20 | 0.27 |  1.77$\mu$m |  2.08$\mu$m |  2.44$\mu$m
Table 3: ModelR RFA parameters: details of different parameters of RFA structures such as number of monomers (N), fractal dimension (Df), monomer radius (rp), aggregate radius (Ra) and aggregate size parameter (Xa) in the three different wavelengths for the minimum and maximum sizes respectively. N | Df | rp | Ra | Nd |  Xa($\lambda=0.557\micron$) |  Xa($\lambda=0.655\micron$) |  Xa($\lambda=0.768\micron$)
---|---|---|---|---|---|---|---
10 | 1.8 |  0.073$\mu$m | 0.22 | 1,675 | 2.48 | 2.11 | 1.79
625 | 1.8 |  0.073$\mu$m | 2.00 | 90,780 | 22.5 | 19.18 | 16.36
Table 4: ModelR Solid parameters: details of different parameters of Solid structures such as radius (R), packing fraction ($\rho_{f}$) and size parameter (X) in the three different wavelengths for the minimum and maximum sizes respectively. Ra | $\rho_{f}$ |  X($\lambda=0.557\micron$) |  X($\lambda=0.655\micron$) |  X($\lambda=0.768\micron$)
---|---|---|---|---
0.07$\mu$m | 0.26 | 0.79 | 0.67 | 0.57
2.0$\mu$m | 0.27 | 22.5 | 19.2 | 16.4
Figure 2: Variation of complex refractive index n (red line) and k (blue line)
with increasing wavelength for (a) amorphous forsterite and (b) amorphous
carbon.

###  2.5 Dust Model

In this study, we have considered two kinds of modelling approaches to extract
the best possible results. The first modelling approach considers fixed values
of monomer size parameters (x) for all the three wavelengths and we term it as
ModelX. While the second modelling approach considers fixed values of monomer
size (r) for all the three wavelengths and we term it as ModelR. In both
models we use a mixture of highly porous RFA structures and Solid agglomerated
debris particles. As silicates and carbonaceous materials majorly constitute
the composition of dust found in comets (Bardyn et al., 2017), we have
considered the refractive indices of amorphous forsterite to represent
silicates and amorphous carbon to represent carbonaceous materials. We have
considered the refractive indices of amorphous forsterite to represent
silicates and amorphous carbon to represent carbonaceous materials. Figure-2
shows the wavelength dependence of refractive index for both amorphous
forsterite (Scott et al., 1996) and amorphous carbon (Jenniskens et al., 1993;
Li et al., 1997), while the specific values of refractive indices in the three
wavelengths for silicate and carbon are shown below.

$Silicate\left\\{\begin{array}[]{lr}1.68+0.0035i,&\text{for
}\lambda=0.557~{}\micron\\\ 1.67+0.0040i,&\text{for
}\lambda=0.655~{}\micron\\\ 1.66+0.0048i,&\text{for
}\lambda=0.768~{}\micron\end{array}\right\\}$

$Carbon\left\\{\begin{array}[]{lr}1.97+0.23i,&\text{for
}\lambda=0.557~{}\micron\\\ 1.99+0.22i,&\text{for }\lambda=0.655~{}\micron\\\
2.0+0.2i,&\text{for }\lambda=0.768~{}\micron\end{array}\right\\}$

The details of both the modelling approaches are explained in the following
sub-sub-sections.

Table 5: Physical properties of RFA Model structures (1-6) Sample444The different samples of circumstellar dust analog prepared in the Granada Amsterdam Light Scattering setup is described in Table-3 of Volten et al. (2007). |  RFA Structures555The different realizations of the RFA Model structures are shown in Figure-12 in Appendix-A. |  $n$666For simplicity we considered the imaginary part of the refractive index $k$ = 0.0001 for all the structures, as it was not detected in the experiments. & colour777The colour refers to the observed colours of different samples from the experiments. | porosity | grain radius |  aggregate radius (Ra) |  Average no. of dipoles (Nd)
---|---|---|---|---|---|---
1 | 1.1, 1.2, 1.3 | 1.7 dark brown | 91% - 95% |  0.05-0.12$\mu$m |  0.65$\mu$m | 25,896
2 | 2.1, 2.2 | 1.7 dark brown | 93% - 94% |  0.05-0.12$\mu$m |  0.65$\mu$m | 24,500
3 | 3.1, 3.2, 3.3 | 1.6 light brown | 94% - 95% |  0.03-0.05$\mu$m |  0.65$\mu$m | 25,423
4 | 4.1, 4.2, 4.3 | 1.6 light brown | 93% - 94% |  0.03-0.05$\mu$m |  0.6$\mu$m | 18,809
5 | 5.1, 5.2, 5.3 | 1.8 black | 97% - 99% |  0.015-0.06$\mu$m |  0.375$\mu$m | 3,305
6 | 6.1, 6.2, 6.3 | 1.7 black | 91% - 96% |  0.015-0.06$\mu$m |  0.6$\mu$m | 19,324
Figure 3: Variation of the degree of linear polarization -S12/S11 (a-e) and
phase function S11 (f-j) with scattering angle for RFA model structures (1-6)
(Solid lines) compared with the experimental results for Aggregate Samples
(1-6) (hollow squares and triangles) from the Granada Amsterdam Light
Scattering Database (Volten et al., 2007). Figure 4: Variation of S22/S11
(a-c) with scattering angle for RFA model structures (1-4) (Solid lines)
compared with the experimental results for Aggregate Samples (1-4) (hollow
squares and triangles) from the Granada Amsterdam Light Scattering Database
(Volten et al., 2007).

####  2.5.1 ModelX: constant size parameter

In this modelling approach we consider fixed primary particle (monomer) size
parameter (xp = 0.7) for the RFA model structures which corresponds to
following PP (monomer) radii ($r_{p}$),

$x_{p}=\frac{2\pi}{\lambda}r_{p}=0.7\left\\{\begin{array}[]{lr}r_{p}=0.062\micron,&\text{for
}\lambda=0.557~{}\micron\\\ r_{p}=0.073\micron,&\text{for
}\lambda=0.655~{}\micron\\\ r_{p}=0.085\micron,&\text{for
}\lambda=0.768~{}\micron\end{array}\right\\}$

The upper and lower size cuttoffs of monomer size under ModelX is defined
using lognormal size distribution having standard deviation of $\sigma_{p}$ =
$\pm$ 0.03$\micron$. Thus the smallest monomer sizes are 0.032$\micron$,
0.043$\micron$ and 0.055$\micron$ at $V_{f}$, $R_{f}$ and $I_{f}$ filters
respectively. While the largest monomer sizes are 0.092$\micron$,
0.103$\micron$ and 0.115$\micron$ at $V_{f}$, $R_{f}$ and $I_{f}$ filters
respectively. In total, 50 RFA aggregates are considered having aggregate size
parameters ($X_{a}$) in the range 5 to 20 and number of monomers ($N$) in the
range 45 to 625 (see Table-1 for more details).

On the other hand, for the low porosity solid particles we have considered
agglomerated debris particles generated ussing REST. A total of 50 solid
particles are considered having minimum size parameter 0.65 and maximum size
parameter 20. The respective minimum and maximum radii of the solids for the
three wavelengths are depicted below,

$X_{min}=\frac{2\pi}{\lambda}R=0.65\left\\{\begin{array}[]{lr}R=0.057\micron,&\text{for
}\lambda=0.557~{}\micron\\\ R=0.067\micron,&\text{for
}\lambda=0.655~{}\micron\\\ R=0.079\micron,&\text{for
}\lambda=0.768~{}\micron\end{array}\right\\}$

$X_{max}=\frac{2\pi}{\lambda}R=20\left\\{\begin{array}[]{lr}R=1.77\micron,&\text{for
}\lambda=0.557~{}\micron\\\ R=2.08\micron,&\text{for
}\lambda=0.655~{}\micron\\\ R=2.44\micron,&\text{for
}\lambda=0.768~{}\micron\end{array}\right\\}$

Table-1 & 2 shows all the details of different parameters used in ModelX.

####  2.5.2 ModelR: constant monomer radius

In this modelling approach we consider fixed value of primary particle
(monomer) radius or mean monomer radius rp = 0.073$\micron$ for all the three
wavelengths, while the aggregate radius (Ra) is considered to be in the range
0.22 to 2.0 $\micron$ for all the wavelengths (see Table-3 for more details).
On the other hand the radius/size (R) of solid particles are also fixed for
all the wavelengths. The minimum and maximum radii of PP (monomers) and the
related size parameters for the three respective wavelengths are shown below,

$r_{p}=\frac{\lambda}{2\pi}x_{p}=0.073\micron\left\\{\begin{array}[]{lr}x_{p}=0.82,&\text{for
}\lambda=0.557~{}\micron\\\ x_{p}=0.70,&\text{for }\lambda=0.655~{}\micron\\\
x_{p}=0.60,&\text{for }\lambda=0.768~{}\micron\end{array}\right\\}$

The upper and lower size cuttoffs of monomer size under ModelR is defined
using lognormal size distribution having standard deviation of $\sigma_{p}$ =
$\pm$ 0.03$\micron$. Thus the smallest monomer size is 0.043$\micron$ and
0.103$\micron$ for all the three wavelengths. For the low porous solid
particles, 50 structures are generated in the size range 0.07$\micron$ to
2.0$\micron$ for all the wavelengths. The minimum and maximum radii of Solids
and the related size parameters for the three respective wavelengths are shown
below,

$R_{min}=\frac{\lambda}{2\pi}X=0.07\micron\left\\{\begin{array}[]{lr}X=0.79,&\text{for
}\lambda=0.557~{}\micron\\\ X=0.67,&\text{for }\lambda=0.655~{}\micron\\\
X=0.57,&\text{for }\lambda=0.768~{}\micron\end{array}\right\\}$

$R_{max}=\frac{\lambda}{2\pi}X=2.0\micron\left\\{\begin{array}[]{lr}X=22.5,&\text{for
}\lambda=0.557~{}\micron\\\ X=19.2,&\text{for }\lambda=0.655~{}\micron\\\
X=16.4,&\text{for }\lambda=0.768~{}\micron\end{array}\right\\}$

Table-4 shows all the details of different parameters used in ModelR.

##  3 Results

In this section, we discuss the results from light scattering simulations of
RFA structures for the aggregate samples (1-6) from the Granada Amsterdam
Light Scattering Database. Finally, we explain the RFA+SOLID model results
used to model the observed polaeization from the comet 2I/Borisov.

###  3.1 Validating RFA model structures

To replicate the morphology of cosmic dust aggregates, we generate the
polydisperse RFA model structures using the package REST (Halder, 2022). REST
takes the structure file of a polydisperse fractal aggregate (FA) and crafts
roughness and/or irregularities on the surface of each spherical grain of a FA
structure and thereby creating the most realistic computer modelled cosmic
dust analog, as shown in Figure-1. To proceed further, it is necessary to
cross-check whether the RFA Model structures are reliable enough to be
considered as pristine cosmic dust candidate to model the observed
polarization of the comet 2I/Borisov. Thus, to validate the RFA Model
structures light scattering simulations are performed for each of the RFA
Model structures using the discrete dipole approximation (DDSCAT) to extract
the light scattering parameters S11 (phase function), -S12/S11 (degree of
linear polarization) and S22/S11 ratio for the respective model structures,
where S(ij) are the orientationally symmetric scattering matrix elements. The
simulations are performed considering the similar values of monomer size,
aggregate size, refractive index and porosity as provided in the Granada
Amsterdam Light Scattering Database for the aggregate samples (1-6) (see
Table-5). Figure-3 shows the variation of -S12/S11 and S11 [normalized by
S11(30∘) as in case of experiments] with scattering angle for RFA model
structures (1-6) compared with the experimental results for Aggregate Samples
(1-6) from the Granada Amsterdam Light Scattering Database shows the variation
of the degree of linear polarization with the scattering angle for the
different RFA model structures and those obtained from the experiments.
Figure-4 shows variation of S22/S11 with scattering angle for RFA model
structures (1-4) compared with those which are experimentally obtained for
Aggregate Samples (1-4). It is clear from the figures that when roughness is
induced on the surface of spherical monomers of a fractal aggregate, a better
agreement with the experimental data can be achieved. Hence, the RFA Model
structure not only look similar to original cosmic dust aggregates but can
also produce light scattering response similar to those obtained from light
scattering experiments over cosmic dust analogs.

Figure 5: 3D visualization of RFA + Solid mixed morphology used to model the
observed polarization-phase data of comet 2I/Borisov. Figure 6: Variation of
the degree of linear polarization with phase angle using ModelX for mixed
morphology RFA:Solid = 80:20 over the power-law index range _n_ = 2.0 - 3.0
having C:Si = 50:50 (a-c) and C:Si = 60:40 respectively for the wavelengths
$\lambda$ = 0.557$\mu$m ($V_{f}$ filter), 0.655$\mu$m ($R_{f}$ filter) and
0.768$\mu$m ($I_{f}$ filter).

###  3.2 Modelling the observed polarization of 2I/Borisov

After validating the RFA model structures, we proceed further to develop a
comet dust model considering both porous cosmic dust aggregates (RFA) and low
porous solid particles (Solids) (see Figure-5). In this model, the RFA
structures are considered to represent the porous cosmic dust aggregates,
while the agglomerated debris particles are considered to represent low porous
solid particles. The recent study on modelling of cometary polarization by
Halder & Ganesh (2021) shows a detailed modelling technique where relatively
larger particles are used to model both the short and long period comets using
hierarchical aggregates, fluffy solids and agglomerated debris particles.
Although, the model is able to explain the observed polarization of short
period comets 1P/Halley, 67P/C-G (Halder & Ganesh, 2021) and 156P/Russel-
Linear (Aravind et al., 2022), but it showed certain discrepancies in the
negative polarization in case of the comet Hale-Bopp. Thus, it is clear that
the third class of comets require a special treatment with much simpler
approach. Hence in the present study we consider relatively smaller size
particles $\leq$ 2.5µm with more pristine morphology.

We have used DDSCAT to compute the degree of linear polarization for the high
porous RFA model structures and the low porous Solid particles under the
parameterization schemes of ModelX and ModelR discussed in the Section-2.5. It
is clear from the Rosetta/MIDAS findings that both high porous aggregates and
low porous solids are present in a comet. Also, the Carbon to Silicate ratio
or high absorbing material to low absorbing material ratio was found to be
50:50. Initially we mixed the simulated polarization data of porous RFA
structures for amorphous silicate and amorphous carbon under the mixing ratio
C:Si = 60:40 and 50:50. Finally, we mixed the inhomogeneous RFA polarization
data with Solid silicate data under the mixing ratios RFA:Solid = 80:20. The
best-fit results obtained under both the schemes are explained below.

####  3.2.1 Best-fit results using ModeX

Figure-6 depicts the variation in the degree of linear polarization using
ModelX for RFA:Solid = 80:20 with varying power-law index n under the
different wavelength filters $\lambda$ = 0.557$\mu$m ($V_{f}$ filter),
0.655$\mu$m ($R_{f}$ filter) and 0.768$\mu$m ($I_{f}$ filter) respectively for
C:Si=50:50 (a-c) and C:Si = 60:40 (d-f). These figures portray a multi-
dimensional approach of the model where we compare the observations of
2I/Borisov and Hale-Bopp for the particular power-law index over all the three
wavelengths. One can easily notice from all the three figures that the
polarimetric observations of the comets 2I/Borisov and Hale-Bopp show good
agreement with model curves in the power-law index range of 2.4 to 2.8 in all
the three wavelengths. Although the power-law index must remain consistant
over all the three wavelengths. This discrepancy may arise due to the
consideration of fixed monomer size parameter which is the basis of ModelX.
This issue is resolved when fixed monomer size is considered over all the
three wavelengths as explained in the next section.

Figure 7: Visualisation of six out of 50 RFA structures for ModelR having
radii (Ra) (a) Ra = 0.22$\micron$ (b) Ra = 0.60$\micron$ (c) Ra =
1.06$\micron$ (d) Ra = 1.27$\micron$ (e) Ra = 1.76$\micron$ (f) Ra =
2.0$\micron$. Figure 8: Variation of the degree of linear polarization with
phase angle using ModelR for mixed morphology RFA:Solid = 80:20 over the
power-law index range _n_ = 2.0 - 3.0 having C:Si = 50:50 (a-c) for the
wavelengths $\lambda$ = 0.557$\mu$m ($V_{f}$ filter), 0.655$\mu$m ($R_{f}$
filter) and 0.768$\mu$m ($I_{f}$ filter) Figure 9: Best fit model results
using ModelR for the interstellar comet 2I/Borisov with the observations (red
filled circles) at wavelengths $\lambda$ = 0.557$\mu$m ($V_{f}$ filter),
0.655$\mu$m ($R_{f}$ filter) and 0.768$\mu$m ($I_{f}$ filter) and compared
with the polarimetric observations of the comet C/1995 O1 (Hale-Bopp) (blue
filled circles) at $\lambda$ = 0.4845$\mu$m (Ganesh et al., 1998), 0.620$\mu$m
(Ganesh et al., 1998) and 0.730$\mu$m (Kikuchi, 2006) for C:Si = 50:50 (_n_
=2.6) (a-c).

####  3.2.2 Best-fit results using ModeR

In case of ModelX, the monomer size parameter is fixed for all the three
wavelengths, the monomer radii is scaled according to the size parameter. But
in a realistic case the monomer radii shall remain constant for the smaller
and larger aggregates over all the wavelengths. This anomaly is corrected in
ModelR which considers fixed value of monomer radii for all the aggregates in
all the three wavelengths. In this case study the variation of the degree of
linear polarization and phase angle for changing aggregate sizes keeping the
monomer radii fixed in all the three wavelengths for silicate and carbon
respectively for six out of 50 RFA structures (see Figute-7) and 50 Solid
particles following the parameterization scheme of ModelR explained in
Section-2.5.2. Figure-8 depicts the variation in the degree of linear
polarization using ModelR for RFA:Solid = 80:20 with varying power-law index n
under the different wavelength filters $\lambda$ = 0.557$\mu$m ($V_{f}$
filter), 0.655$\mu$m ($R_{f}$ filter) and 0.768$\mu$m ($I_{f}$ filter)
respectivly for C:Si=50:50 (a-c). It is clear from the figure that the best-
fit size distribution index $n$ remains consistant at 2.6 which becomes more
clear from Figure-9 shows the best-fit curves at n=2.6 obtained using ModelR
for the comet 2I/Borisov at $\lambda$ = 0.557$\mu$m ($V_{f}$ filter),
0.655$\mu$m ($R_{f}$ filter) and 0.768$\mu$m ($I_{f}$ filter) respectively for
C:Si = 50:50.

Figure 10: Best-fit results of polarimetric spectral gradient for the two
pairs of wavelengths (a) PSG0.557,0.655 and (b) PSG0.655,0.768 (h) using
ModelX (dashed line) and ModelR (solid line) over the observed PSG values for
Hale-Bopp (blue hollow square) and 2I/Borisov (red solid circle) Figure 11:
Comparison of _dust-to-gas_ ratio (measured) (Jewitt et al., 1999; Yang et
al., 2021; McDonnell et al., 1991; Biver et al., 2019), _porous-to-compact_
ratio (modelled) and polarimetric slope (observed & modelled) for dynamically
newer and older comets.

###  3.3 Modelling the Polarimetric Spectral Gradient (PSG)

In this section we discuss the observed and modelled Polarimetric Spectral
Gradient (PSG) which is defined by the following equation,

$\frac{dP(\lambda)}{d\lambda}\simeq
PSG_{(\lambda_{1},\lambda_{2})}=\frac{P(\lambda_{2})-P(\lambda_{1})}{\lambda_{2}-\lambda_{1}}$
(5)

where P is the value of polarization (observed/modelled), while $\lambda_{1}$
and $\lambda_{2}$ are the two subsequent wavelengths. The polarimetric
observations of the comet 2I/Borisov indicate that in the positive branch,
polarization increases with increasing wavelength and hence the polarization
spectral gradient (PSG) remains positive. The model PSG curves shown in
Figure-10 under ModelX and ModelR indicates a similar trend which is a common
feature for all comets including the third class of comets having polarization
higher than the high polarization comets. But in the negative branch, the
observed PSG becomes negative at around 20∘ phase angles for both Hale-Bopp
and 2I/Borisov. Surprisingly, the modelled PSG also becomes negative around
20∘ phase angle. Although the model PSG($V_{f}$,$R_{f}$) curve obtained using
Modelx indicate a significant negative trend in low phase angles, but it does
not show a promising fit with the observed PSG points. Moreover, the model
PSG($R_{f}$,$I_{f}$) curve obtained using ModelX does not indicate any
significant negative trend which is observed in case of Hale-Bopp. On the
other hand, the model PSG($V_{f}$,$R_{f}$) and PSG($R_{f}$,$I_{f}$) obtained
using ModelR show significant negative trend in the low phase angles and also
the model curves almost fit with some of the observed points and remains
relatively close to rest of the observed values. Hence, ModelR which considers
fixed value of monomer radii in all the three wavelengths produces better
results, while ModelX which considers fixed monomer size parameter is
physically incorrect and it is clear from the PSG plot.

###  3.4 Exploring relation between _dust-to-gas_ ratio and intrinsic dust
parameters

The best-fit model data for the observed polarization and PSG of the third
class of comets 2I/Borisov & Hale-Bopp and those of short period comets
1P/Halley and 67P/C-G obtained from the light scattering simulations explained
in this study and Halder & Ganesh (2021) respectively, indicate that the
intrinsic properties of dust play a crucial role in defining the signature
polarimetric slope of different class of comets. For example, in this study
the porous-to-compact ratio (RFA:Solid) obtained for the comets 2I/Borisov &
Hale-Bopp (third-class of comets) is 4. While the porous-to-compact ratio
(HA:Solid) obtained for comets 1P/Halley & 67P/C-G (short period comets) is
0.25. Thus, the difference in polarization or the polarimetric slope of
different class of comets tends to be proportional to the porous-to-compact
ratio of dust in the coma of the comet. In a similar way, the dust-to-gas
ratio in the coma of a comet is proportional to the polarization or
polarimetric slope. Figure-11 depicts the respective dust-to-gas ratio and
porous-to-compact ratio for the aforesaid comets. It is clear from the figure
that high dust-to-gas ratio in the third class of comets is accompanied by
high porous-to-compact ratio of dust particles, while, in low polarization
comets the dust-to-gas ratio is accompanied by low porous-to-compact ratio.

##  4 Discussion

Under the framework of polarimetric observations of the interstellar comet
2I/Borisov (Bagnulo et al., 2021) and the light scattering experiments of
different aggregate samples of cosmic dust analogs from the Granada Amsterdam
Light Scattering Database (Volten et al., 2007) we develop a visually
realistic dust model to replicate the unusual polarization phase curve
observed in the comet 2I/Borisov. We obtained the best fit model for a mixture
of porous RFA particles 80% and Solids 20% having power-law size distribution
index $n$ = 2.7 using parameterization scheme of ModelX and $n$ = 2.6 using
parameterization scheme of ModelR respectively for C:Si = 50:50 under
wavelength filters $\lambda$ = 0.557$\mu$m ($V_{f}$ filter), 0.655$\mu$m
($R_{f}$ filter) and 0.768$\mu$m ($I_{f}$ filter). The higher percentage of
porous RFA structures and the higher power-law index indicate that the coma of
2I/Borisov is dominated by porous and relatively smaller dust particles. On
the other hand, best fit model results for short period comets 1P/Halley,
67P/Churyumov-Gerasimenko & 156P/Russel-LINEAR (Halder & Ganesh, 2021; Aravind
et al., 2022) indicate presence of lesser amount of porous aggregates. Also,
the study indicates that the porous-to-compact ratio of dust particles is
directly proportional to the dust-to-gas ratio observed in the coma of a
comet. In case of relatively newer comets, the coma is dominated by high
amount of small size porous dust aggregates of Rayleigh size grains indicating
high dust-to-gas ratio owing to high porous-to-compact ratio thereby producing
relatively higher polarization. On the other hand in case of short-period or
older comets, the coma is dominated by gas and large size compact dust
particles that are mainly concentrated in the inner coma and the near nucleus
regions indicating low dust-to-gas ratio owing to low porous-to-compact ratio
which in turn produces lower polarization. Thus, it is very much clear from
this study that dynamically new comets carry larger amount of porous pristine
cosmic dust particles, while in dynamically older comets or short period
comets a larger portion of the pristine dust particles are lost due to
frequent weathering by Solar wind. In this study we employ a multi-dimensional
approach by considering morphologically realistic dust particles having a
mixture of high and low porosity over a wide range of size distribution for
the three broadband filters. We all know 2I/Borisov as an interstellar comet,
yet it was once part of an extrasolar planetary system and hence it was an
exocomet before drifting away from its host star. Astronomers have recently
found signatures of exocomets using techniques such as photometric transits
and far IR/mm gas emission from within debris disks (Strøm et al., 2020;
Lecavelier des Etangs et al., 2022). On the other hand recent observations of
main-belt comets (active asteroids) reveal certain amount of dust and gas
production rates (Jewitt, 2012; Moreno et al., 2021). Hence, the correlation
between dust-to-gas ratio and porous-to-compact ratio indicated in this study
can be of great use to determine the intrinsic dust properties in main-belt
comets and exocomets. The model has been verified considering best-fit results
for the observed polarization and polarimetric spectral gradient of the
interstellar comet 2I/Borisov. This study ensures that the RFA model
structures represent the pristine morphology of cosmic dust particles and are
capable enough to reproduce experimental as well as observational data. Hence,
these structures will be highly useful for future studies related to cometary
dust polarization, polarimetric response from protoplanetary disks, atmosphere
of cloudy exoplanets and brown dwarfs (Chakrabarty et al., 2022; Marley &
Sengupta, 2011; Sengupta & Krishan, 2001), extinction of background starlight
in dense molecular clouds and polarimetric study of dust in the circumstellar
environments. Although, we tried to develop a realistic cosmic dust model
considering realistic dust particles having surface roughness/irregularities,
yet the model can be improved by considering large size hierarchical
aggregates and including compositions such as organics, FeS and different
kinds of ices. These limitations can be addressed in some future work to
develop a more realistic and generalised comet dust model.

##  5 Acknowledgements

The authors deeply thank the anonymous reviewers for their fruitful
suggestions, which has enriched the manuscript with greater details. The
authors acknowledge the high-performance computing facility (NOVA) of the
Indian Institute of Astrophysics, Bangalore, where all the intensive light
scattering simulations are conducted. The authors also acknowledge Prof.
Shashikiran Ganesh of PRL, Ahmedabad and Dr. Himadri Sekhar Das of Dept. of
Physics, Assam University, Silchar for important discussions.

##  Appendix A RFA Model Structures

The RFA model structures used in this study are the exact computer modelled
replica of circumstellar dust analogs prepared in the Amsterdam Granada Light
Scattering setup. Figure-12 depicts the RFA model realisations of
circumstellar dust analog samples (1-6) having similar refractive index (n),
porosity, grain size and aggregate size following Table-3 of Volten et al.
(2007). The different physical parameters of the RFA model structures are
shown in Table-5. For simplicity we considered the imaginary part of the
refractive index $k$ = 0.0001 for all the structures, as it was not detected
in the experiments.

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Figure 12: RFA Model structures (1-6) representing Aggregate Samples (1-6)
from the Granada Amsterdam Light Scattering Database.