# Vesicular dust particles – light scattering and radiation pressure –

Robert Botet, A Sen, A Zaman

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Robert Botet, A Sen, A Zaman. Vesicular dust particles – light scattering and radiation pressure – Monthly Notices of the Royal Astronomical Society, 2025, 10.1093/mnras/staf885. hal-05090873

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# Vesicular dust particles – light scattering and radiation pressure –

R. Botet,¹*A. K. Sen,² and A. Zaman²

¹Laboratoire de Physique des Solides, Université Paris-Saclay, CNRS UMR8502, Orsay-91405, France

²Department of Physics, Assam University, Silchar-788011, India

May 30, 2025

## Abstract

We present a new model for irregular vesicular particles, featuring both open and closed porosity. The model can simulate particles with large vesicles, comparable in size to the wavelength of light. Calculations of light polarization indicate that such particles might be suitable candidates to represent some cometary dust particles. For sub-micron sized dust, the radiation pressure exerted on these particles is found to be significantly higher than that exerted on porous particles of the same material, size, and overall porosity, but consisting only of small pores. It is demonstrated that the average vesicle size is relevant to characterize the motion of the porous particle in an environment with radiation pressure. In particular, pore size can be a key parameter in determining the location of the small porous particles in the comet's dust tail. Depending on the material, the numerical model of vesicular particles could also be relevant for other areas of physics, such as volcanic ash.

## 1 Introduction

Dust particle formation generally occurs in two stages. First, nucleation and growth of matter in a gas cloud produce small grains of relatively uniform size ([88, 46]), then quiescent aggregation of the grains gives rise to large irregular dust particles ([18, 60]). The resulting large size dust particles are fragile against mechanical, thermal and chemical stresses ([83]). Therefore, diverse processes subsequently cause the dust particles to evolve into aggregates of a wide variety of sizes and shapes. They can be: disintegration ([33, 36]), rotational disruption ([39]), sputtering ([91]), sintering ([75]), evaporation ([23]), or compaction ([14]).

It is difficult to characterize the various morphologies of the particles that compose a dust cloud. In the rare cases where particles are collected (interplanetary dust particles in the Earth's stratosphere; cometary dust particles from space probes), collection

<footnote>*E-mail: robert botet@universite-paris-saclay.fr</footnote>
destroys most structures except for small, compact particles ([55]). More often, dust particles are identified by their optical properties since electromagnetic interaction affects all particles, whether fragile or sturdy, large or small. However, it is known that complete reconstruction of the particle morphologies is impossible from the optical properties of the dust cloud alone. Thus, the characteristics of the dust particle structures – nonsphericity ([97]), inhomogeneities ([57, 26]), internal structures ([45, 47]), irregularities ([30, 102]) – are generally incomplete because they come from analysis of damaged particle images or partial optical properties.

In this paper, we present a new model of vesicular particle, that is porous particle with a broad pore size distribution (such particles will be named: ‘BPSD particles’, for ‘Broad Pore Size Distribution particles’). We discuss the properties and consequences of light scattered by them, viz. the polarization caused by the particle and the radiation pressure exerted on the particle, in relation to its pore size distribution.

Being fully aware that the last Rosetta Mission did not formally detect vesicular particles (for comet 67P/Churyumov–Gerasimenko), we went further with an open mind to explore whether optical polarization of comets can be made consistent with particles comprising wide pore size distribution. As mentioned above, the internal structure of highly porous particles is hard to identify in real situations because the particles are profoundly modified when collected. Similar vagueness appeared in the past, when comets exhibited very diverse properties in terms of gas and dust contents. For example, this has happened with the extremely different D/H ratios detected for comet Hartley2 and 67P, igniting a debate about the possible source of water on earth (see [35] for comet Hartley 2, and [2] for comet 67P). Another example is the detection of crystalline silicate dust in comet Wild 2 by STARDUST mission, which went against the common sense of most of the astronomers expecting comets to contain amorphous silicates only ([74]).

With this background, we make a case for cometary vesicular particles (Section 2), and introduce a new generic model for them (Section 3). Accordingly, we show in Section 4 that light polarization properties of various comets can be grossly explained even by these porous particles made of pyroxene or astronomical silicates. In Section 5, we compare between the radiation pressure on vesicular particles and that on porous particles of same material, size and porosity, but containing only small pores. We address the issue of characterizing porous particles using the *β* factor, that is the ratio of radiation pressure to gravitational force ([9]). Relationship between *β* and particle porosity is indeed not yet well understood.

The present study is intended for use in the field of cometary dust particles. In Section 2, we present a physical process which is known to transform naturally aggregated dust particles into porous particles with broad pore size distribution. We also explain why the definite shapes of such fragile aggregated particles are probably destroyed during the collection process. This explains why vesicular particles are actually absent as definite category from catalogues that attempt to organize the diversity of cometary particles.

Particles with wide distribution of inclusion sizes can be found in various contexts,
for example interplanetary dust particles often exhibit blobs of volatile elements ([24]).

Also, such porous particles are found among metallic micrometeorites ([31]). Incidentally, most of the porous pyroxene-rich chondritic micrometeorites found in Antarc-tica, are thought to have originated from comets ([72]).

On the Moon, vesicular metallic iron nanoparticles (diameter < 100 nm) are widely distributed among lunar samples ([15]). The cumulative volume of vesicles is expected to be an important helium reservoir of the Moon.

In a very different field, vesicular-shaped particles form volcanic ash because of gas bubbles trapped in molten material. Explosive eruptions cause rapid cooling of the molten substance, breaking it into small, solid vesicular fragments ([38]).

## 2 Porous particles with broad pore size distribution as possible components of comet dust

Every time a space probe is launched to capture dust particles from a comet's coma, our knowledge of cometary dust particles is refined. The most recent consensus is the concept of 'hierarchical aggregate' ([63]). It suggests a favoured structure for the cometary particles analysed in the MIDAS experiment, refining the classification scheme of [34] based on structure, porosity, and strength of fluffy and porous dust agglomerates, compact aggregates, and grains. In short, the classification into hierarchical aggregates is as follows (we introduce abbreviations for the three categories):

1) (particles ‘*Po*’) these large particles are agglomerates with subunits that might consist of smaller subunits ([62]). They are expected to not be packed extremely loosely, as the target surface is not visible behind them. However, they fragment easily into their micrometer-sized subunits. An example is particle F, analysed in [62]

2) (particles ‘*Fr*’) these large particles consist of similar subunits as for particles ‘*Po*’, and they are extremely loosely assembled. In [62], their structure is shown to follow a fractal ordering with a fractal dimension close and less than 2. It has been suggested that all dust particles found in comets originated from this fractal structure, and the majority of them subsequently densified to form the ‘*Po*’ particle group ([27])

3) (particles ‘*Sm*’) these small particles show bulb-shaped subunits with a moderate packing density of nanometric subunits.

The *Rosetta* mission showed that ‘*Fr*’ dust particles constitute a significant fraction of the refractory component of a comet nucleus ([62]). These particles can be quite large (MIDAS and GIADA instruments detected fractal particles a few microns and hundreds of microns respectively). They are probably pristine, both in terms of composition and morphology, due to gentle inclusion in the comet nucleus ([27]). These particles are known to have very low mechanical resistance ([42]), therefore they are soft and fragile (i.e. sensitive to any stress, whether it is mechanical, thermal, etc.).

At the very beginning of the release of a large aggregate from the nucleus, collisions of gas molecules or other dust particles lead to a series of fragmentations of the particle. Further out, the surrounding environment rapidly becomes much less dense, and heating from the Sun becomes the dominant stress process on cometary dust particles.
For CCA fractal aggregates of fractal dimension less than 2, heating is essentially homogeneous through the particle, due to the absence of shadowing effect. The bonds between the grains are weakened by the increasing temperature ([42]). We focus hereafter on the case of moderate thermal energy, $k_B T$, where only limited deformations of the aggregate of grains are allowed.

## 2.1 Model of aggregate restructuring (2D case)

A restructuring model of fractal aggregate is introduced to simulate transformation ‘Fr’ → ‘Po’ by thermal effect. The model is a variation of the restructuring process reported in [65]. We focus here on two-dimensional (2D) simulations because they allow easy visualization of the restructuring mechanisms. The three-dimensional case is discussed in the next Subsection.

Let us consider a fractal aggregate of fractal dimension $d_f$ and size (number of grains) $N$. The 2D RCCA (Reaction-limited Cluster-Cluster Aggregation model ([66])) is selected as a generic example of realistic fractal cluster. An example of RCCA aggregate with $N = 2048$ and $d_f = 1.54$ is shown in Fig. 1(a). Like any aggregate generated by regular CCA model, it is made up of random ramifications of self-avoiding filaments (no loops). There are exactly $2N - 2$ bonds in a loopless aggregate of $N$ grains ([29]).

Thermal energy is distributed equally among all bonds of a fractal aggregate. It is dissipated by random elastic bending of the bonds between neighbouring grains. In this situation, large blocks of grain can move around the weakest bonds ([95]). This is a collective rearrangement of the structure, different from the local compaction process involving single grain movements as modelled in [94]. The former reorganization process is expected to be valid for fractal aggregates, while the latter one is dominant for the densification of compact particles.

A step of ‘rotation of a block / irreversible sticking’ process is as follows:

• a grain (grain A) is randomly selected in the aggregate. Another grain (grain B) is randomly selected from the list of grains neighbouring A. If removing the A-B bond causes the aggregate to split into two pieces of grains, say cluster $C_1$ to which A belongs and cluster $C_2$ containing B, the piece $C_1$ rotates rigidly around the centre of the grain B until a new contact is formed, namely grain C of $C_1$ hits and sticks to grain D belonging to $C_2$. The bond C-D is created and the bond A-B is remade. The total number of bonds between neighbouring grains in the aggregate then increases by 1 because a new loop formed.

The entire restructuring process consists of a sequence of such random steps until no disassembly of the aggregate into two disconnected clusters is possible by removing a single bond. An example of restructured aggregate is presented in Fig. 1(b). The fractal aggregate in Fig. 1(a) became a porous particle with inner pores clearly identified. Fig. 1 (c) is identical to Fig. 1 (b) with grains in red colour and closed pores are coloured according to the pore size.
<figure><img src="image_2.png" /></figure>

Figure 1: (a): RCCA fractal aggregate (fractal dimension $d_f = 1.54$) made of $N = 2048$ spherical grains. (b): the RCCA fractal aggregate after complete restructuring (see text for details). (c): the same as in (b) with colours enhancing the porous structure of the restructured aggregate. Grains are red colour ; pore colour corresponds to pore size (yellow is the largest closed pore, blue the smallest closed pores). In this example, the area of the largest pore is 37% of the total particle area. Porosity of the compacted particle (b) is 0.66.

Understanding why pores appear naturally in the 2D restructuring process of a fractal aggregate is insightful. Loops of grains are mechanically stronger than filaments of grains (removing two bonds is required to cut a loop into two pieces, while a single bond break is enough to cut a filament). Then, the filaments of the aggregate gradually disappear in favour of loops. This process releases thermal energy. Finally, closed pores are the areas inside the loops of grains.

Amazingly, the compacted porous particle resulting from this process is still fractal (see Figure 2), even though it is not ramified. Fractality is not here the result of the tortuosity and branching of filaments of grains (as it is for CCA aggregates), but comes from the self-similarity of the pore size distribution. The distribution of the pore area, $s$, is indeed the power law: $n(s) \sim 1/s^{d'_f+1}$ with $d'_f$ the porous particle fractal dimension that is slightly larger than $d_f$ ([65, 67]) (see also [40, 3] for examples with circular pores and [98] for a general statistical approach). This kind of structure is named ‘trema random fractal’ in the terminology of fractals ([61]).
<figure><img src="image_3.png" /></figure>

Figure 2: Double logarithmic plot of the aggregate size, *N*, versus its gyration radius, $R_g$. Natural logarithms are used. Each symbol corresponds to average over $2^{16-a}$ aggregates of sizes $N = 2^a$, with $a = 1, \dots, 16$ (the largest aggregate is then made of 65 536 grains). Black circles correspond to the original 2D RCCA aggregates of fractal dimension = 1.54 (black line represents: $\ln N = 1.54 \ln R_g + 1.3$); red triangles correspond to the respective compacted aggregates (red line is: $\ln N = 1.6 \ln R_g + 1.6$ corresponding to the fractal dimension $d'_f = 1.60$ for these restructured aggregates).

## 2.2 3D model of aggregate restructuring

The spirit of the 3D restructuring process is similar to the 2D case, but the additional degree of freedom in the third dimension allows to perform three steps, viz. rotation, folding and twisting each time the aggregate can be split into two clusters by the thermal fluctuations. Full details about these steps and visual examples of 3D aggregates after restructuring can be found in [44].

Note however, that in [67, 44], restructuring occurs during the aggregation process. Indeed, the restructuring steps are performed on the two diffusing clusters immediately after their collision. In contrast, the restructuring steps in the version presented here are carried out on a large aggregate of grains already formed. Splitting the aggregate into two clusters results from simultaneous removal of a few random bonds by thermal fluctuations. Many more restructuring steps are then underway compared to the Meakin-Jullien model.

Due to the higher number of degrees of freedom, the 3D version leads to reorganized aggregates with much larger number of bonds per grain than the 2D version. In 3D, filaments and loops (in which each grain has about two neighbouring grains) soon disappear in favour of membranes of grains (in which grains end up with much more neighbours). Given the limited amount of thermal energy, it is unlikely that a membrane will separate into two pieces due to the high interconnectedness of the grains in it, hence
the creation of membranes organizing themselves in pores.

This kind of compaction process evolves the fractal, fluffy agglomerate to a fractal porous particle of fractal dimension slightly larger than the the fractal dimension of the initial particle ([67, 44]). As in the 2D case, the fractal structure of the particle is due to the broad power-law distribution of the pore sizes.

The fractal structure of the reorganized particles can be hard to detect on standard images since, though very porous, they are opaque (like the observed 'Po' particles). This kind of opaque, porous aggregates with self-similar pore size distribution can however be lumped together with data from the GIADA experiment onboard the Rosetta spacecraft that suggested most of the dust released from 67P consists of relatively compact (porosity ≈ 0.5 ([79, 6]) or ≈ 0.6 ([42]), fragile particles [63].

# 3 BPSD model for numerical simulation of jagged dust particles

We consider particles made of a solid material. The density of the material is $ρ$. The complex refractive index, $m$, characterizes the optical properties of the material. We restrict the present work to the case where vesicles are voids, i.e. the optical refractive index is $m_{pore} = 1$ inside a vesicle. This value can be modified when vesicles are filled with a particular material (e.g. vesicles = chondrules).

We name 'BPSD' (after the main characteristic of the particle: 'Broad Pore Size Distribution') the random irregular particle model as described below.

## 3.1 Input parameters

Three characteristic lengths are the input parameters of the BPSD model:

* $a_{cir}$ is the radius of the minimal sphere that circumscribes the particle ($a_{cir}$ is called: circumradius); this sphere represents the overall volume of the particle

* $a_{eff}$ is the effective radius of the particle; it is defined through the relation to the mass of the particle: $m = ρ 4πa_{eff}^3/3$

* $a_p$ is the mean radius of the pores; all pores are assumed to be spherical.

A BPSD particle is inherently porous and its porosity, $f$, is defined by the relation:

$$f = 1 - \left( \frac{a_{eff}}{a_{cir}} \right)^3, \qquad (1)$$

i.e. $f$ represents the volume fraction of voids inside the circumscribed sphere. The definition (1) of particle porosity was given in [103] for general agglomerates.
## 3.2 The basic BPSD algorithm

The present model is a generalization of the ‘roughened spheres model’ presented in [92], to the case of vesicular particles including closed porosity. It is also a variant with spherical debris of the model of ‘agglomerated debris particles’ introduced in [100] (where the role of matrix and vesicles is reversed).

As in previous studies ([100, 48, 59]), the irregular structure of the particles is simulated by randomly adding or subtracting dipoles according to certain rules. Hereafter are the details of the BPSD algorithm.

BPSD particles are built within the dipole-aggregate representation. That way, DDSCAT code ([22]) can be used readily to estimate the optical characteristics of the particles. The steps to build a BPSD particle are listed below:

* Defining dipoles

Space is divided into an array of small subunits, called dipoles, located at the vertices of a cubic lattice with the same lattice spacing $d$ for all BPSD particles to build. On such a lattice, each dipole has six neighbouring dipoles.

* Building the circumscribed sphere

The circumscribed sphere is defined as the sphere, $S_0$, with centre $O$ (the origin of the cubic lattice) and radius $a_{cir}$. There are $N_0$ dipoles inside this spherical volume. The number of dipoles targeted in the final particle is $N = (1 - f) N_0$

* Making the pores

The pores are spherical in shape and they may overlap. The values of the pore radii are randomly distributed according to the one-parameter exponential distribution with mean value $a_p$ (any other pore size distribution can be used in the same way).

The loop over the successive pores $j = 1, 2, 3, \dots$ is as follows:

0) start with $j = 1$

1) select a random radius $a_j$ for the $j$-th pore, according to the pore size distribution

2) select a random position, $O_j$ inside the sphere with centre $O$ and radius $a_{cir} + a_j$, where to place the centre of the next pore

3) if removing the dipoles inside the volume intersection of the particle under construction and the sphere (centre $O_j$ and radius $a_j$) leads to a set of disconnected dipoles (the spreading algorithm as in ([71]) is used to test the condition), go to step 1) ; if there is no fragmentation of the particle, these dipoles are removed

4) if the number, $N_j$, of dipoles constituting the actual particle is less than $N$, increment $j$ by 1 then go to step 1) ; otherwise, exit the loop 1)-4) and the total number of pores: $j_{max} = j$ is defined.
<figure><img src="image_4.png" /></figure>

Figure 3: Random BPSD particle with porosity $f = 0.5$ and circumradius $0.77~\mu\text{m}$. Top: 3D representation of the particle as an aggregate of 119 509 dipoles (each small sphere represents a dipole with diameter $0.02~\mu\text{m}$). A number of ‘craters’ (spherical pores connected to the outside) make the surface of the particle very irregular. They contribute to the open porosity of the particle. Bottom: central slice of the particle (matter is white, void is black). A section of a large isolated pore is visible inside the particle. Such internal pores form closed porosity because they are not connected to the outside of the particle. In contrast, a series of three pores (at the bottom of the particle) contribute to the open porosity.

* Defining the final particle

A connected particle of $N_{jmax}$ dipoles results from the algorithm above. However, the particle’s dipole number should be as close as possible to $N$, so the particle at step $j_{max} - 1$ is selected if $N - N_{jmax-1} < N_{jmax} - N$ (and the final number of dipoles making the particle is then $N_{BPSD} = N_{jmax-1}$), while the particle at step $j_{max}$ is selected in the opposite case (and the final number of dipoles of the particle is then $N_{BPSD} = N_{jmax}$).

* Final decision

The selected particle is finally accepted as a BPSD particle of porosity $f$ if $|f - (N_0 - N_{BPSD})/N_0| < 0.1$. Otherwise, the particle is considered not to meet the porosity condition and the algorithm starts again from (0).

## 3.3 An example: BPSD particles of porosity $f = 0.5$

A BPSD particle with porosity $f = 0.5$ is shown in Fig. 3. It consists of 119 509 dipoles. The input parameters are: $a_{cir}/d = 38.5$, $a_{eff}/d = 30.6$ and $a_p/d = 10$. Alternatively, if the interdipole distance is set to $d = 20$ nm, the parameters of this BPSD particle are: $a_{cir} = 0.77~\mu\text{m}$, $a_{eff} = 0.61~\mu\text{m}$, and the average pore size is: $a_p = 0.2~\mu\text{m}$. Such a random particle is irregular in shape and has both closed and open porosity.
Its appearance bears similarities to fine volcanic ash particles (see example in Fig. 4, left; see also examples in [70], [87]) and to some metallic micrometeorites (see example in Fig. 4, right).

In addition, examples of cometary particles show spherical metal grains embedded in a matrix of carbon-rich material ([58]).

Also, as explained in Section 2.2, such BPSD particle with power-law distribution of the pore size might be a model including the essential features of the category of cometary particles defined as: ‘large agglomerates that consist of micrometer-sized subunits in a fragile arrangement with moderate packing density of the subunits at the surface’ ([63]).

## 3.4 Comparison with alternative vesicular models

It should be noted that the current model is essentially different from the numerical model of vesicular particles presented in [56]. Indeed, in the latter model, all the spherical vesicles are, by construction, fully embedded inside the particle. This makes the particle porous with only closed porosity, and the surface is smoother (ripples but no ‘craters’) than the BPSD particles generated by the present algorithm (ripples and ‘craters’).

The actual model is also different from the numerical model of ‘roughened particles’ presented in [92]. Indeed, in the latter model, all roughened particles exhibit smooth open porosity (‘craters’), but no closed porosity.

In this way, the BPSD model is versatile, including both the model of [56] (representing either spherical grains embedded in a matrix or fully coated vesicular particles) and the model of [92] (compact particles with rough surfaces). Using the terminology in [101] for irregularly shaped particles, the BPSD numerical model can simulate ‘agglomerated debris’ (where polydisperse vesicles are the debris inside a spherical matrix), ‘rough-surfaced spheres’, ‘strongly-damaged spheres’ and ‘debris of spheres’. With only three parameters, the BPSD model should then be very flexible to address realistic situations.

## 3.5 Remark about the definition of particle porosity

Alternative definitions – related to the radius of gyration rather than the circumradius – have been given to characterize the porosity of irregular particles (see e.g. [69, 16, 51]). However, the relation (1) is more natural for spheroidal particles since it relates the porosity to the particle mass, *m*, and to the overall particle radius, $a_{cir}$.

A slightly different definition of porosity has been proposed in [86] based on the approximation that the total pore volume inside a particle is about $4\pi(a_{cir}^3 - a_{eff}^3)/3$. It leads to porosity values close to (1) when particles are almost round in shape.

Other characterizations of particle porosity have been presented in [43] such as the inflation, *I* (a measure of solid filling factor and size), and dimensionality, *D* (a measure of shape). In the case of BPSD particles built on a regular cubic lattice, *D* is a slightly decreasing function of porosity, and the porosity itself defined in (1) is $f \approx 1 - 1/I$ (the difference between the two quantities is due to discretization of the underlying lattice).
<figure><img src="image_5.png" /><figcaption>Figure 4: Top: Scanning-electron microscope image of a silicate glass vesicular volcanic ash particle collected from Tolbachik volcano ([99]) ; Bottom: Scanning-electron microscope image of the section of a Fe-rich Antarctic micrometeorite ([17]).</figcaption></figure>

In the example shown in Fig. 3, 1000 independent BPSD particles of porosity f = 0.5 leads to the average values: *I* = 1.96 and *D* = 2.94, close to the expected values 2 and 3 respectively.

# 4 Optics of BPSD particles

Cometary coma is a typical context in which the relevant set of dust particles is extremely difficult to be specified due to the variety of particle shapes, sizes and compositions. A number of various numerical models have already been proposed to try to explain the observed polarization from cometary dust, and our purpose here is not to fine-tune parameters to perfectly fit the real data. Our aim is to present evidence that BPSD particles could be yet another possible type of cometary dust particles among others.

## 4.1 Typical features of BPSD polarization profiles

In [52], the authors collected cometary observational data. In particular, they discuss how the degree of linear polarization changes with wavelength of light at specific phase angles (for a given set of comets, and specially the comet 1P/Halley). General trends for dust-rich comets are as follows: 1) the degree of polarization *P*(10°) at phase angle α = 10° (negative branch) goes through a shallow minimum, *P*(10°) ≈ -3%, around λ ≈ 1.4 µm ; 2) just after the inversion point (phase angle α ≈ 20°), the degree of polarization *P*(30°) at phase angle α = 30° exhibits smooth linear increase with wavelength ; 3) around the phase angle 65° (positive branch), polarization *P*(65°) passes through a maximum, *P*(65°) ≈ 20 – 26% (depending on the type of comet) at λ – 1.5 ~ 2.0 µm.
<figure><img src="image_6.png" /></figure>

Figure 5: Top. Linear polarization degree from an ensemble of polydisperse porous BPSD particles (porosity $f = 0.5$) as a function of wavelength. The size-distribution of the BPSD particles behaves as ~$a^{-4}$ in between $a_{min} = 0.01$ µm and $a_{max} = 2$ µm. The BPSD particles here consist of pyroxene (Mg$_{0.5}$Fe$_{0.5}$SiO$_{3}$, empty symbols), or astronomical silicate (red continuous curves). The numerical data are obtained using Maxwell-Garnett effective medium theory for phase angles $10°$ and $30°$. For comparison, the observational data from 1P/Halley comet (as collected in [52]) about the negative polarization branch at phase angle $α = 10°$ are shown by the thick dashed curve, and at phase angle $α = 30°$ by the thin dashed curve. Bottom. Phase angle dependence of the linear polarization of the BPSD particle population as above (circles are pyroxene particles; red continuous curve is astronomical silicate particles), at the wavelength $λ = 1.2$ µm. Without adjusting any parameters of such populations of porous particles, these standard numerical data exhibit polarization characteristics comparable to polarization profiles from real cometary dust.
To address the question of whether a polydisperse population of BPSD particles can reproduce these behaviours, we consider the power-law distribution ([78]), ~ $a_{cir}^{\alpha}$, of the porous particle circumradius, $a_{cir}$, over the range ($a_{min}$, $a_{max}$). Such a size distribution is close to that obtained for comet dust *in situ* ([41]), and is expected to be generally relevant for cometary particles ([1]). It is worth noting that a power-law radius distribution implies that the number of smaller grains significantly dominates the number of larger ones when the index $\alpha$ is negative and large enough ([37]). Therefore, the upper bound, $a_{max}$, has generally little significance for the optical properties of the dust particle cloud. In contrast, decreasing the lower bound, $a_{min}$, increases the degree of positive polarization (more particles with smaller size parameter).

Typical BPSD results are shown on Fig. 5a and Fig. 5b. They correspond to a population of polydisperse BPSD particle with differential size-distribution power-law index $\alpha = -4$ ([25]). The smallest particle radius is set to $a_{min} = 0.01$ µm, and the largest particle radius to: $a_{max} = 2$ µm. The particles consist either of pyroxene MgₓFe₁₋ₓSiO₃ with $x = 0.5$ ([20]) (empty symbols) or of astronomical silicate ([21]) (filled symbols). These materials are selected here because of their ubiquitous presence in cosmic dust (e.g. in interstellar medium ([21, 32]), in comets near perihelion ([53]), in planetary nebulae ([10]), in the solar nebula ([5]), in protoplanetary disks ([82]). All particles are porous with the same porosity $f = 0.5$. The polarization data were obtained using the standard Maxwell-Garnett Effective Medium Theory. This theory is expected to give reasonably correct results in this case since a porous BPSD particle is a homogeneous spherical particle with inserted round void volumes, so the material and the void do not play the same role (unlike Bruggeman's effective theory ([64]).

Except for the value of the power-law index $\alpha$ (chosen in agreement with [25]), the values of the other parameters of the BPSD model are here selected as 'typical', that is: $x = 0.5$ (since $0 \le x \le 1$) and $f = 0.5$ (here too: $0 \le f < 1$), while the smallest particle radius is set to 10 nm and the largest particle radius to 2 µm because we plan to compare these effective medium numerical data with the DDSCAT results using simulated BPSD particles (not presented here).

Fig. 5a shows that the negative polarization of magnitude -3% observed around $\alpha = 10°$ for Comet 1P/Halley (the dashed curve in the figure) is approximately realized both for BPSD pyroxene or BPSD silicate particles with standard parameter values (filled circles).

The BPSD polarization at phase angle 30° shows a flat maximum, in contrast to the slight linear increase in the degree of polarization observed for comet 1P/Halley (not shown, see [52]). The numerical values appear to be closer (in magnitude and shape) to the values observed at phase angle 40° (the dotted curve in the figure).

The phase-angle dependence of the degree of linear polarization for BPSD particles correctly shows the negative polarization branch (here $\alpha < 12°$ for pyroxene particles and $\alpha < 15°$ for astronomical silicate particles). The usual maximum around $\alpha = 90°$ is also present in both cases. The amplitude of the maximum is however too large compared to cometary observations, but, as noted above, this maximum can be lowered decreasing the relative amount of small particles, and/or increasing the magnitude of the relative effective refractive index $|m(f) - 1|$. Also in reality, no astronomical
observations are available at phase angles close to 90° where polarization values are expected to show their maximum. Then this point is unessential.

In conclusion, in order to more accurately match cometary data, the material (here: pyroxene with $x = 0.5$ or astronomical silicate) should be redefined to take into account astronomical constraints ([32]), and the porosity value (which could depend on particle size) and the definite radius distribution should be adjusted finely. The two examples in Fig. 5 show, however, that a population of porous BPSD particles made of pyroxene or astronomical silicate seems to readily give optical properties consistent with the main features of the light polarization of 1P/Halley cometary dust. We then claim that these vesicular particles (made of a material to be adjusted) may be part of some coma dust. Indeed, these BPSD particles are expected to be the most naturally present particles in this context as compared to the other kind of dust particles, as BPSD particles are the only particles which take into account several disruptive processes possible in cometary environment.

## 5 Radiation pressure on BPSD particles

A natural – though underrated – question is whether such particles might be present with other types of particles thought to be part of a comet’s dust tail. To clarify this point, we discuss below the radiation pressure on two types of porous particles of the same mass, porosity, and radius, in the same physical situations, but with different pore sizes.

We consider two models of porous dust particles, namely: BPSD particles as defined in Section 3, and TD particles as defined in [86]. Briefly, (full details in [86]), a Touching-Dipole (TD) particle is constructed by randomly removing dipoles one after the other from a homogeneous (dense) particle of radius $a_{cir}$. The removal step is realized if the resulting structure remains connected (no particle fragmentation). The process stops when the actual porosity of the particle reaches the targeted porosity $f$. This formation process gives rise to labyrinth-shaped pore particles. Pores are of small (dipole-sized) typical diameter for these porous particles. Fig. 6 illustrates a TD particle of porosity $f = 0.5$. These particles have been shown to be possible candidates for non-fractal cometary dust particles ([11, 12]).

The radial force $\mathbf{F}_{rp}$ due to radiation pressure exerted on a randomly oriented particle is ([68]):

$$ \mathbf{F}_{rp} = \frac{S C_{rp}}{c} \mathbf{n}, \qquad (2) $$

$$ C_{rp} \equiv C_{ext} - C_{sca} \langle \cos \theta \rangle, \qquad (3) $$

where $c$ is the light velocity, $S$ is the energy flux density of the incident electromagnetic wave at the particle position, and $\mathbf{n}$ is the unit vector in the direction of the incident
<figure><img src="image_7.png" /><figcaption>Figure 6: Random TD particle with porosity $f = 0.5$ and circumradius $0.77 \mu m$. Top: 3D representation of the particle as an aggregate of 119 520 dipoles (each small sphere represents a dipole with diameter $0.02 \mu m$). Unlike BPSD particles (see Fig. 3), the length-scale of the surface irregularity is small, of the order of one dipole diameter. Bottom: central slice of the particle (material is white, void is black). The pores are organized into long, tortuous 3D tubular pores.</figcaption></figure>

wave. In the definition (3) of the radiation pressure cross section, the particle characteristics $C_{ext}$ and $C_{sca}$ are the orientation-averaged extinction and scattering cross sections respectively, and $\langle \cos \theta \rangle$ is the asymmetry parameter of the scattering angle $\theta$ ([8]).

For a given dust particle, the dimensionless ratio $\beta$ between the radiation pressure force, $F_{rp}$ and the gravitational force $F_{gr}$ acting on it, is independent of the distance from the source of radiation:

$$ \beta = \mathcal{K} \frac{C_{rp}}{a_{\text{eff}}^3}, \qquad (4) $$

with $\mathcal{K} = 3L_{\odot}/(16\pi^2GM_{\odot}c\rho)$ a typical length depending on the solar luminosity, $L_{\odot}$, and mass, $M_{\odot}$. $G$ is the gravitational constant. For pyroxene particles with density $\rho = 3.310^3 \text{ kg.m}^{-3}$, we have: $\mathcal{K} = 0.0557 \mu m$ in (4), when the cross section $C_{rp}$ is expressed in $\mu m^2$ and the particle effective radius, $a_{\text{eff}}$, in $\mu m$. From the point of view of particle dynamics, $1 - \beta$ is the factor describing the reduced gravitation field ([96]).

In principle, $C_{rp}$ in (4) should be understood as the integration of the radiation pressure cross section over all wavelengths emitted by the Sun ([9]). To simplify the discussion, we consider below (as in [54]) that all solar luminosity is emitted at wavelength $\lambda^* = 0.502 \mu m$ which is the wavelength where the solar irradiance spectrum is maximal ([77]). This approximation allows to work with the unique refractive index $m = 1.7 + i0.05$ of pyroxene at the wavelength $\lambda^*$.

The relationship between $\beta$ and particle porosity is not well understood. It has been suggested ([54]) that low-density particles might have significantly higher $\beta$ values than
denser particles of the same size. However, the authors in [54] also wrote: "Voids and other inhomogeneities can lead to either an increase or a decrease in β". Subsequent studies ([80]) confirmed that porous particles can indeed have lower β values under similar conditions, based on extensive scattering calculations ([89, 50]) that showed moderate β values for porous particles made of astronomical silicate generated by Ballistic Particle-Aggregation models.

It is important to understand the role of β. If different types of particles coexist in a comet's nucleus, variations in their β values can lead to complex patterns in the comet's dust tail ([28]). Systematic comparison of β values can help identify the types of particles present in specific regions of a comet's type-II dust tail.

Using DDSCAT 7.3 code, we calculated the β values for BPSD particles and TD particles, both with porosity *f* = 0.5, and plotted them in Fig. 7. For comparison, the β values for Mie particles with the same circumradius and effective refractive index given by Maxwell-Garnett formula ([64]) (using *f* = 0.5) is plotted as the blue continuous curve (EMT) on the same figure. Discussion of the data is given in the following Sections.

## 5.1 β values of EMT porous particles

The Effective Medium Theory gives a fiducial curve of β for spherical porous particles of given porosity *f*. It is represented by the blue continuous curve in Fig. 7. The β value reaches a maximum, $β_{max} = 0.29$, at $a_{eff} ≈ 0.115$ µm, that is: $x ≈ 1.8$ where the size parameter is defined as: $x = 2πa_{cir}/λ$.

Note the complicated profile of the β curve near the maximum. It is reminiscent of Mie resonances at specific values of $a_{cir}$. In contrast, the limit cases $x \ll 1$ (the electric dipole limit) and $x \gg 1$ (the geometrical optics limit) are simple. In the first case (very small particles), the value of β tends to a constant because $C_{rp} ≈ C_{ext} ∝ a_{cir}^3$ for Rayleigh particles ([8]). In the latter case (very large particles), β follows the geometrical optics law: $β ∝ 1/a_{eff}$ (the thin solid black line in Fig. 7) because the radiation pressure efficiency $Q_{rp}$ tends to a constant in this limit ([73, 49]).

## 5.2 β values of TD porous particles

In [93] it was demonstrated that Mie particles with an effective refractive index could reproduce the optical properties of porous particles with small (Rayleigh) pores. However, it was also noted ([76]) that interactions between pores are not properly accounted for in the EMT, which then leads to significant discrepancies between the EMT and porous particles with spatially correlated pores (such as pores in TD particles ([11]).

These points are illustrated in Fig. 7 where the values of β for porous TD particles (red triangles) are different but do not deviate much from the corresponding values obtained by EMT theory. Indeed it was noticed at the beginning of Section 5 and in Fig. 6 that the pores of the presented TD particles are essentially small in diameter
<figure><img src="image_8.png" /></figure>

Figure 7: Ratio $\beta$ of solar radiation pressure to solar gravity acting on dust particles of porosity $f = 0.5$ made of pyroxene (refractive index $m = 1.7 + i0.05$ and density $3.3 \text{ g.cm}^{-3}$). The wavelength of the incident light is $0.502 \mu\text{m}$. Semi-logarithmic plot with the particle effective radius, $a_{\text{eff}}$, in $\mu\text{m}$. Black circles: BPSD particles; red triangles: TD particles; blue continuous curve: spherical particles of radius $a_{\text{cir}} = a_{\text{eff}}/(1-f)^{1/3}$ with Maxwell-Garnett EMT refractive index. Lines between the symbols are guides for the eye. The thin dashed curve is the geometrical optics (GO) approximation $\sim 1/a_{\text{eff}}$. The thin dashed horizontal line for $a_{\text{eff}} < 0.03 \mu\text{m}$ is the limit Rayleigh value (R). Each point for BPSD or TD particles results from DDSCAT 7.3 calculations on independent simulated particles of given effective radius and porosity, averaged over 45 random target orientations and two incident polarizations. For comparison, the $\beta$ values for a dense particle of radius $a_{\text{eff}}$ (not shown on the figure) peak at $\beta = 1.07$ for $a_{\text{eff}} = 0.114 \mu\text{m}$.
(~ 0.02 µm) compared to the wavelength (= 0.502 µm), but must be spatially correlated (labyrinthine structure) to maintain a coherent structure.

## 5.3 β values of BPSD porous particles

In contrast to TD particles, the construction process of BPSD particles generates intrinsically large pores (here, average pore radius = 0.2 µm). Although the porosity is here identical in all three cases (BPSD, TD and EMT particles), the β values of BPSD particles are much higher (by a factor ~ 2 around the maximum) than the corresponding values for EMT or TD particles.

The data presented in Fig. 7 illustrate a case where pore morphology has a significant effect on values of β, namely: large pores are correlated with higher β-values than the EMT at the same porosity. Extrapolating this observation to fractal aggregates of fractal dimension $d_f < 3$ (in which voids of any size are present), it can be stated that similar correlation must lead to significantly high β-values of fractal aggregates. This is in agreement with the results on fractal Ballistic Cluster-Cluster Aggregation particles ($d_f \approx 2.0$) presented in [90].

This result can also be related to the failure of EMT to explain light scattering by large-pore dust particles ([7]). This point has been known for a long time ([93]) and an extended version of the effective medium approximation ([13]) has been proposed to analyse the optical extinction from particles with voids larger than the wavelength. But this extended theory remains complicated to use and DDSCAT is preferred for its versatility and the almost exact values of the optical characteristics of the particle.

# 6 Concluding remarks

We studied light scattering by vesicular particles using a new statistical shape model. This model generates irregular solid particles with internal spherical pores and cratered surfaces. To assess the influence of these features on light scattering, we modelled particles with polydisperse vesicles and, for comparison, examined particles with only small pores. Light scattering calculations were performed using the discrete dipole approximation.

In situations where solar radiation pressure is the dominant non-gravitational force acting on the particles (see [1] for an overview and discussion of the various forces affecting cometary dust particles), the coexistence of two different types of particles of the same mass at a given location in a type-II dust tail can only occur if their respective values of the β ratio are nearly identical. In fact, the motion of particle is governed by the net force difference between gravitational and radiation pressure forces, that is ([9]):

$$ \frac{d^2\mathbf{r}}{dt^2} = -(1 - \beta) \frac{GM_\odot}{r^2} \mathbf{n}. \qquad (5) $$

As a result, dust particles with different β values follow distinct trajectories, leading to different dust particles populations depending on their location in the comet's tail ([81]).
As previously discussed, while particle mass, material composition, and porosity are commonly considered in determining the value of the *β* parameter, these properties are not always sufficient to characterize the dynamics of porous dust particles. In particular, for particles made of pyroxene with diameter between approximately 0.2 and 1 micrometer, our simulations show that the internal structure – specifically, the size distribution of pores – can significantly influence the *β* value. This implies that two particles with the same mass, porosity, and composition may still respond differently to solar radiation pressure depending on their pore morphology.

This finding calls for caution when interpreting *β* values in terms of porosity alone (e.g. [85]), especially in the context of cometary dust dynamics. It also challenges the often-assumed trend that increasing porosity always leads to a decrease in *β* at constant particle mass ([89]).

The *β* parameter, which governs the acceleration (5) of dust particles under solar radiation pressure, is observable through the morphology of cometary tails ([84]) and, combined with light-scattering measurements, provides key insights into the physical properties of the dust. Our results highlight the importance of considering internal structural details – beyond bulk porosity – in modelling the dynamics of porous particles.

While the present study focuses on sub-micron pyroxene particles, we expect the sensitivity of *β* to pore size to be a general feature of porous particles, possibly occurring for other materials such as carbonaceous compounds or ices. However, the specific size range over which this effect occurs likely depends on the material and remains to be determined. This opens a pathway for further investigation and provides an additional constraint in the modelling of cometary dust.

## Acknowledgements

We thank the anonymous referee for helpful suggestions that led to improvements in the paper. R.B. gratefully acknowledges the hospitality received during his stay at the Department of Physics, Assam University, Silchar (India), where this study was initiated.

## Data Availability

The simulation data used for this work will be provided upon reasonable request to the corresponding author.

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