© The Authors 2024

1 Introduction

Dust is a common component in astronomic environments from galaxies to stellar disks. It therefore plays a crucial role in observations. Most astronomers come into contact with dust because of dust extinction, which affects almost all observations (Draine 2003). Calzetti et al. (2000) and Bernstein et al. (2002) estimated that 30% or more of the energy that is produced by stars in the Universe is absorbed by dust and is reemitted in the infrared. The dust helps to cool the gas, which facilitates gravitational collapse (Omukai et al. 2005) and also plays a major role in later stages of star formation. It also constitutes the first stage in the formation of planets (Dullemond et al. 2008; Liu & Ji 2020). Rapidly rotating dust grains are the common source of polarized light and carry information about the magnetic field orientation (see e.g. Lazarian 2007; Andersson et al. 2015, for review). The catalysis of molecular hydrogen (Hollenbach & Salpeter 1971; Cazaux & Tielens 2002) and other astrochemical reactions in the interstellar medium (ISM) (Dishoeck 2014) is also impacted by grain rotation (Hoang & Tram 2020). The first models trying to explain the observed dust alignment, that is, paramagnetic relaxation (Davis & Greenstein 1951) or the Gold mechanism (Gold 1952), both encountered problems and were ultimately unable to account for the observed alignment. A great leap in the study of alignment was made when Purcell (1979) realized that grains could experience systematic torques acting in a specific direction in the grain coordinates for long time spans compared to the timescale that it takes gas collisions to randomize the grain rotation. With these systematic torques, the grain can spin up to much faster angular velocities than the typical thermal rotation rate. With suprathermal rotation, a grain can remain in a given orientation for much longer than the gas-collision timescale. This allows an alignment of the angular momentum with the maximum moment of inertia via inelastic relaxation or the Barnett relaxation, which was discovered by Purcell (1979) and is based on the Barnett (1917) effect, which causes magnetization along the rotation axis in a paramagnetic body. When the grain rotation axis is not aligned with a principal axis of inertia, the rotation axis and therefore the magnetization axis will precess in body coordinates, dissipating energy and aligning the rotation axis with the axis of the maximum moment of inertia. In the presence of an external magnetic field, the Davis-Greenstein process can align the grain with the magnetic field (Purcell 1979).

A one-to-one mapping of dust alignment to the magnetic field lines would be of great use to the study of magnetic fields. Polarization observations of the object in question could then always be used to obtain information about the magnetic field morphology. Uncertainties about the completeness of the model remained, however, such as the lack of an increase in polarization with extinction in cold dark clouds (Goodman et al. 1995) and the alignment of grains in cometary dust (Harwit & Vanýsek 1971). The exploration of other alignment mechanisms therefore continued.

The idea of systematic radiative torques (RATs) as the cause for rapid grain rotation had already been advanced decades before (Dolginov 1972; Dolginov & Mytrophanov 1976), but it only became an integral part of dust alignment theory with the work of Draine & Weingartner (1996). Using the discrete dipole approximation (DDA) method, the authors showed that scattering and absorption of light from irregular grains results in torques that act on timescales that are much longer than random gas collisions. The calculations were made on a irregular shape of intersecting cuboids, and the authors found that radiative torques strongly depend on the grain size. Additional grain shapes were investigated in the following work (Draine & Weingartner 1997) in the context of anisotropic radiation, H₂-formation torques, and paramagnetic relaxation. In most cases, the grain reached long-term stable conditions by aligning the principal grain axis with the magnetic field. Lazarian & Hoang (2007a) (LH07 hereafter) later introduced an analytic model (AMO) for a spherical grain that is attached to a flat rectangular mirror via a weightless noninteracting rod. In this way, they captured the helical properties of irregular grains. The AMO was compared to DDA simulations of irregular grain models that corresponded to the grain shapes in Draine & Weingartner (1997), and a good agreement was found. The alignment of the principal grain axis with the direction of incoming radiation was confirmed when magnetic field effects were subdominant. The parameterization of the RAT spin-up efficiency was found to be a power law with an exponent of about −3. Physical implications of RATs were explored in following studies (Lazarian & Hoang 2008; Hoang & Lazarian 2008, 2009, 2016), and the model was expanded by the addition of superparamagnetic relaxation due to iron inclusions. This increased the range of parameters in which grains can efficiently align with the magnetic field, which is necessary to explain high degrees of polarization from observations (Planck Collaboration XII 2020). The AMO was used successfully to model synthetic dust polarization observations as long as free alignment parameters could be used to adjust the model (Bethell et al. 2007; Hoang & Lazarian 2014; Reissl et al. 2016). Herranen et al. (2019) then made a step toward testing more realistic grain shapes: They used Gaussian ellipsoids, oblate spheroids, prolate spheroids, and spheres with random deformations and found a shallower slope of −2.6 for the power law for the RAT parameterization. In this work, we take one more step toward more realistic grain shapes by simulating random typically fractal aggregate grains composed of differently sized monomers. We compare their RATs to previous results. The motivation is to model the grain growth ab initio using the current understanding that dust is originally formed in AGB stellar winds or supernovae (Todini & Ferrara 2001; Schneider et al. 2014; Nanni et al. 2013). As the dust-forming materials in the winds cool down, heavy elements condense into spherical particles (in the following called monomers), which are the building blocks for the larger dust grains that are formed by monomer accretion and coagulation of grains (Gobrecht et al. 2016; Norris et al. 2012). A more realistic grain modeling may result in differing interactions with radiation. One example was given by Arnold et al. (2019), who found significant differences in dust blowout sizes between compact and agglomerated grains.

The aim of this work is to improve the understanding of RATs by simulating individual grains composed of monomers for different sources of radiation. The properties of grains and the way they are modeled is presented in Sect. 2. The theoretical basis for our simulations is presented in Sect. 3, and models for the sources are described in Sect. 4. The simulation codes are explained in Sect. 5, the results are shown in Sect. 6 and discussed in Sect. 7, and a summary and outlook are given in Sect. 8.

2 Ballistic dust grain aggregation

We modeled dust grains as conglomerates consisting of spherical particles of a single material, called monomers. The monomers stay attached to each other due to the intermolecular forces at their contact area (see e.g. Johnson et al. 1971). The growth of larger dust grains follows an aggregation process by means of ballistic collision and sticking of individual monomers (Dominik & Tielens 1997). To mimic the grain-growth process of ballistic aggregation (BA), we followed the procedure as outlined in Shen et al. (2008), modified to work with monomers of variable sizes (Reissl et al. 2024). Starting with two or three connected monomers, new monomers with radii drawn from a log-normal distribution between 10 nm and 100 nm were injected into the simulation volume from random directions so that they stuck to the grain upon collision.

Grains with an effective radius a_eff¹ from 50 up to 250 nm in steps of 50 nm were created. To reach an exact effective radius, the sizes of the final monomers added to the grain were sampled in a biased way (we lowered the upper bound on the monomer size), until the desired a_eff was reached exactly. In addition to BA, where each monomer remains at its initial sticking position, we modeled different monomer densities or porosities of aggregates by means of monomer migration. After hitting the forming grain, the added monomer rolls along the surface of the monomer it impacted until it touches a second monomer of the grain (BAM1). Here, all monomers are connected to at least two other monomers. For the BAM2 case each monomer rolls twice until it touches a third monomer. For BAM2, all monomers are connected to at least three other monomers. This is the densest way in which a random grain can be packed with monomers using the migration process. It follows that BA grains are the least dense/most porous and BAM2 grains are the most dense/least porous, an exemplary grain of each monomer density is shown in Fig. 1.

To recapitulate, the three grain properties are the material from which the grain is made (astrosilicate or hydrogenated amorphous carbon), which differs in the optical properties; the monomer density of the grain (BA, BAM1, or BAM2), which determines how closely the monomers are packed; and finally, the grain size given by the effective radius (50–250 nm in steps of 50 nm). For every combination of these three properties, 30 different grains were created, leading to a total of 900 individual grains. This method of modeling ballistic grain aggregates was previously used in Reissl et al. (2023).

Fig. 1 Exemplary BA (left), BAM1 (middle), BAM2 (right) grains with an effective radius of aeff = 250 nm. The rotation axis ${\hat{a}}_1$ is depicted as the red arrow and is associated with the direction of the maximum moment of inertia Imax.

3 Maximum grain rotation

In the presence of a directed radiation field, irregularly shaped dust grains can experience momentum transfer from photon interactions. In general, the time-averaged forces acting on such a grain may result in a net torque, accelerating the rotation. This radiative torque (RAT) can be evaluated by (see e.g. Draine & Weingartner 1996, and LH07) $${\mathbf{\Gamma }}_{\mathrm{R}\mathrm{A}\mathrm{T}}=\frac{1}{2}\gamma a_{\mathrm{e}\mathrm{f}\mathrm{f}}^2\int \lambda u_{\lambda }{\mathit{Q}}_{\mathrm{\Gamma }}\mathrm{d}\lambda ,$$(1)

where γ describes the anisotropy of the radiation field, λ is the wavelength, and the quantity u_λ = F_λ/c is the spectral energy density of a radiation field with the spectral flux F_λ, whereas c is the speed of light. These factors determine the overall strength of the torque, independent of the grain shape and orientation.

The effect of the individual grain shape and material is contained within the dimensionless spin-up efficiency vector Q_Γ of the RAT. A perfectly spherical grain would experience no net torque at all because of the symmetry between collisions causing clockwise and anticlockwise torques. However, for grains with irregular shapes, the contributions of individual radiation-dust interactions lead to a nonzero Q_Γ vector, which modulates the orientation, that is, the sign as well as the magnitude of Γ_RAT. In previous works, a power-law parameterization for Q_Γ = |Q_Γ| was suggested (LH07) $$Q_{\mathrm{\Gamma }}=\{\begin{array}{ll}0.4& \text{ if }\lambda \le 1.8a_{\mathrm{e}\mathrm{f}\mathrm{f}}\\ 0.4{\left(\frac{\lambda }{1.8a_{\mathrm{e}\mathrm{f}\mathrm{f}}}\right)}^{\alpha }& \text{ otherwise }\end{array}.$$(2)

We refer to this relation as the canonical parameterization. Here, the constant part transitions into a power law at wavelengths equal to 1.8 times the effective radius of the grain. The power-law exponent typically lies between α = −2.6 (Herranen et al. 2019) and α = −4 (LH07).

For our grain dynamics modeling, we assumed a stable alignment of the grain rotation with the direction of incoming radiation. For a single grain that stably rotates about its principal axis, the maximum possible angular momentum ω_RAT caused by RATs can be calculated as $${\omega }_{\mathrm{R}\mathrm{A}\mathrm{T}}=\frac{{\tau }_{\mathrm{d}\mathrm{r}\mathrm{a}\mathrm{g}}{\mathrm{\Gamma }}_{\mathrm{R}\mathrm{A}\mathrm{T}}}{I_{max}},$$(3)

where τ_drag is the drag timescale, and I_max is the highest moment of inertia, defining the a₁-axis in the body-centric system of the grain (see e.g. Draine & Weingartner 1996, as well as Fig. 1). The total drag acting on the grain is composed of the collisions with the surrounding gas and the loss of angular momentum by emitting infrared photons. The total drag timescale can be evaluated from the gas-drag timescale τ_gas and photon-drag timescale τ_IR with $${\tau }_{\text{drag }}=\frac{1}{{\tau }_{\text{gas }}^{-1}+{\tau }_{\text{IR }}^{-1}},$$(4)

where a shorter timescale results in a stronger drag. Here, the gas-drag timescale $${\tau }_{\mathrm{g}\mathrm{a}\mathrm{s}}=\frac{I_{\mathrm{m}\mathrm{a}\mathrm{x}}}{n_{\mathrm{g}\mathrm{a}\mathrm{s}}{m}_{\mathrm{g}\mathrm{a}\mathrm{s}}{v}_{\mathrm{t}\mathrm{h}}{a}_{\mathrm{e}\mathrm{f}\mathrm{f}}^4{Q}_{\mathrm{g}\mathrm{a}\mathrm{s}}},$$(5)

depends on the number density of the gas n_gas as well as on the gas mass m_gas per particle and the mean thermal velocity v_th of the gas (see e.g. Draine & Weingartner 1996). The shape of the grain affects the gas drag, with some shapes increasing the ability of the gas to exert torque on the grain. This influence of the grain shape is quantified by the dimensionless gas-drag efficiency Q_gas. We explored two environments with the same gas number densities n_gas and different temperatures T_gas, as listed in Table 1. The photon-drag timescale $${\tau }_{\text{IR}}=\frac{c^2{I}_{\text{max}}}{a_{\text{eff}}^2{f}_{\text{IR}}\left(T_{\text{dust}}\right)}$$(6)

depends on the integral over the photon emission, $$f_{\mathrm{I}\mathrm{R}}\left(T_{\text{dust }}\right)=4\pi {{\displaystyle \int }}_0^{\mathrm{\infty }}{\lambda }^2{B}_{\lambda }\left(T_{\text{dust }}\right)Q_{\text{abs }}(\lambda )\mathrm{d}\lambda .$$(7)

Here, B_λ (T_dust) is the Planck function modulated by the dimensionless efficiency of the absorption Q_abs(λ), and T_dust is the dust temperature. For simplicity, the dust is illuminated by a single source without extinction. For the dust heating, we assumed thermal equilibrium. Therefore, the temperature of a single dust grain can be evaluated via $${{\displaystyle \int }}_0^{\mathrm{\infty }}{F}_{\lambda }{Q}_{\mathrm{a}\mathrm{b}\mathrm{s}}(\lambda )\mathrm{d}\lambda =4\pi {{\displaystyle \int }}_0^{\mathrm{\infty }}{B}_{\lambda }\left(T_{\mathrm{d}\mathrm{u}\mathrm{s}\mathrm{t}}\right)Q_{\mathrm{a}\mathrm{b}\mathrm{s}}(\lambda )\mathrm{d}\lambda$$(8)

by equating the source flux F_λ that is absorbed by the grain with the characteristic emitted flux B_λQ_abs(λ) of each grain.

4 The radiation field

Typical sources of radiation were chosen to evaluate the RATs they cause by irradiating the grains. The different sources included two O-type stars labeled S1 and S2, with a flux of $$F_{\lambda }=\pi \frac{R^2}{d^2}{B}_{\lambda },$$(9)

where R is the stellar radius, and d is the distance from the star (1 pc). The emission of an active galactic nucleus (AGN) was modeled as $$L_{\lambda }^{\mathrm{A}\mathrm{G}\mathrm{N}}=\frac{C}{\lambda }\cdot \{\begin{array}{ll}6.3\cdot {10}^{-4}& \text{ if }0.01\mathrm{\mu }\mathrm{m}<\lambda \le 0.1\mathrm{\mu }\mathrm{m},\\ 2\cdot {10}^{-4}{\left(\frac{\lambda }{1\mathrm{\mu }\mathrm{m}}\right)}^{-0.5}& \text{ if }0.1\mathrm{\mu }\mathrm{m}<\lambda \le 5\mathrm{\mu }\mathrm{m},\\ 0.011{\left(\frac{\lambda }{1\mathrm{\mu }\mathrm{m}}\right)}^{-3}& \text{ otherwise }\end{array}$$(10)

following Giang et al. (2022) and Nenkova et al. (2008), where C is a constant chosen such that the integration of $L_{\lambda }^{\mathrm{A}\mathrm{G}\mathrm{N}}$ equals the typical AGN bolometric luminosity of L_bol = 10¹⁰L_⊙.

Finally, the spectral flux of the interstellar radiation field (ISRF) was parameterized as $$F_{\lambda }^{\mathrm{I}\mathrm{S}\mathrm{R}\mathrm{F}}=4\pi \{\begin{array}{ll}0& \text{ if }\lambda \le 0.0912\mathrm{\mu }\mathrm{m},\\ 3069{\left(\frac{\lambda }{1\mathrm{\mu }\mathrm{m}}\right)}^{3.4172}& \text{ if }0.0912\mathrm{\mu }\mathrm{m}<\lambda \le 0.11\mathrm{\mu }\mathrm{m},\\ 1.627& \text{ if }0.11\mathrm{\mu }\mathrm{m}<\lambda \le 0.134\mathrm{\mu }\mathrm{m},\\ 0.0566{\left(\frac{\lambda }{1\mathrm{\mu }\mathrm{m}}\right)}^{-1.6678}& \text{ if }0.134\mathrm{\mu }\mathrm{m}<\lambda \le 0.25\mathrm{\mu }\mathrm{m},\\ {10}^{-14}{B}_{\lambda }(7500\mathrm{K})+& \\ {10}^{-13}{B}_{\lambda }(4000\mathrm{K})+& \\ 4\cdot {10}^{-13}{B}_{\lambda }(3000\mathrm{K})& \text{ otherwise},\end{array}$$(11)

following Camps et al. (2015), which is based on Mathis et al. (1983) and Weingartner & Draine (2001). The applied parameters of the radiation sources are listed in Table 2, and the resulting spectral flux of the sources is shown in Fig. 2.

5 Optical and mechanical dust properties

To test the predictions of the RAT theory (LH07; Hoang et al. 2014), we considered silicates (SC) and carbonaceous (CA) grains. These materials are typically present in the ISM (see e.g. Draine 2003, for review). The optical properties of the materials are represented by the wavelength-dependent complex refractive index $\overline{n}=n+ik$, with n being the real part describing the refraction of light that interacts with the dust grain, and the imaginary part k governs the light attenuation. For the SC, we took the (Astro)silicate model data as presented in Draine & Lee (1984); Laor & Draine (1993), and the optical properties of hydrogenated amorphous carbon were outlined in Jones (2012)². Ice mantles were not considered here because high-intensity radiation environments would not allow them. Assuming condensation of all ice species at a temperature above 150 K (Zhang et al. 2015) and comparing with dust temperatures in Table 3, we would expect CA grains to be ice-free for the stellar sources as well as both CA and SC grains for the AGN source. The ice-free assumption may therefore generally not be valid for the ISRF and SC grains around the stellar sources. Additionally, a comparison with these dust temperatures assumes that there is no significant difference in the dust temperature when an icy surface is considered instead of the silicate or carbonaceous surface.

The radiative torque efficiency Q_Γ was calculated using the DDA code DDSCAT (version 7.3.3; see Draine & Flatau 1994, 2008; Flatau & Draine 2012). The grain aggregates were initialized and tabulated as outlined in Sect. 2. To create input shape files for the DDSCAT simulations, we filled the volume enclosed by the monomers with a cubic 3D grid, each grid point representing a distinct dipole, with properties determined by the complex refractive index. DDSCAT calculations of Q_Γ were performed in the external coordinate system $\{{\hat{e}}_1,{\hat{e}}_2,{\hat{e}}_3\}$ of incoming radiation, $${\mathit{Q}}_{\mathrm{\Gamma }}(\mathrm{\Theta })=Q_{\mathrm{e}1}(\mathrm{\Theta }){\hat{e}}_1+Q_{\mathrm{e}2}(\mathrm{\Theta }){\hat{e}}_2+Q_{\mathrm{e}3}(\mathrm{\Theta }){\hat{e}}_3,$$(12)

where ${\hat{e}}_1$ is antiparallel to the wave vector of incoming light. For simplicity, we assumed that the grain was aligned with the direction of radiation, that is, Θ = 0°. Subsequently, the direction of radiation and rotation coincide (${\hat{e}}_1\parallel {\hat{a}}_1$), and the RAT efficiency factor becomes ${\mathit{Q}}_{\mathrm{\Gamma }}\equiv Q_{\mathrm{e}1}(\mathrm{\Theta }){\hat{e}}_1$. Since RATs are most efficient for Θ = 0°, the data presented in this paper merely represent an upper limit for the modeling of the spin-up process of ballistic aggregates. However, the focus on the special case of Θ = 0° does not impact our comparison with previous attempts of modeling the efficiency of RATs. The simulation results were averaged over 21 equidistant values of the rotation angle of the grain around ${\hat{a}}_1$. The results were computed for 50 wavelengths between 0.05 μm and 150 μm within equidistant bins in log-space. Each combination of material, monomer density, size, and shape (see Sect. 2) was simulated with five different resolutions, that is, the separation of two nearest dipoles. They were combined using an average weighted by the dipole resolution. In addition to the RAT efficiency factor Q_Γ, DDSCAT also provides the efficiency of the absorption Q_abs that is required to evaluate the photon-drag timescale τ_IR and the dust temperature T_dust. The gas-torque efficiency Q_gas needed to evaluate the gas-drag timescale τ_gas was calculated with a Monte Carlo-based (MC) gas-dust collision approach as outlined in Reissl et al. (2023). The corresponding gas parameters³ are listed in Table 1.

Fig. 2 Spectral flux of sources ISRF, S1, S2, and AGN that were used to irradiate the grain aggregates.

Fig. 3 Drag timescales over the effective radius aeff dependent on the grain model (BA, BAM1, or BAM2). Left panel: gas drag timescale for different gas densities ngas for all grain materials. The gas temperature is at a constant value of Tgas = 100 K. Center panel: IR drag timescale for the CA grain material for different dust temperatures Tdust. Right panel: same as the center panel, but for the SC grain material.

6 Results

6.1 Drag timescales

We show the resulting timescales for the gas and photon drag for different gas densities and dust temperatures in Fig. 3. The gas-drag timescale τ_gas is presented in the left panel as an average of SC and CA materials. We emphasize that the gas-drag efficiency Q_gas is only dependent on the grain shape in our modeling and is independent of the material. Denser gas equals a lower τ_gas, in agreement with the literature (see, e.g., Draine & Weingartner 1997 and Reissl et al. 2024). The dependence on the monomer density is marginal, with the BA grains having a stronger gas drag than the BAM1 and BAM2 grains. This is due to the dependence of Q_gas on the surface area.

In detail, the BA grains present a larger surface area for interaction with gas particles. Following Eq. (5), the expected size dependence of τ_gas would be linear $(I_{\mathrm{m}\mathrm{a}\mathrm{x}}\propto a_{\text{eff }}^5)$. It is almost constant above 100 nm, however, which is due to the linear increase in Q_gas with size. In principle, the gas drag efficiency Q_gas should only depend on the shape of the grain and not on the size. Q_gas nonetheless increases with size, and the reason for this may be that a_eff is poorly suited to describe the grain size in this situation. The calculation of Q_gas used the simulated gas-drag-torque, which depends on the angular momentum lost by the grain when a gas particle leaves the surface. This in turn scales linearly with the distance from the axis of rotation of the grain from which the gas particle is ejected. For porous grains, this distance is often larger than a_eff. The torque simulated in this way contains information about the true shape of the grain, but in the formula for Q_gas, the size a_eff is used, which creates the size dependence of Q_gas.

The photon-drag timescale is shown in the middle for CA grains and on the right for SC grains in Fig. 3. As the efficiency of absorption Q_abs is higher overall for SC, more photons are absorbed and emitted, which causes a stronger photon drag. The number of emitted photons also depends on the dust temperature, with hotter dust experiencing stronger drag (see also Table 3). The dependence on the monomer density is marginal, and this is related to the fact that the moment of inertia is higher in the case of BA grains than for the more compact BAM2 grains for the same effective radius (see Appendix A in Reissl et al. 2024).

Fig. 4 Torque efficiency QΓ as a function of wavelength λ for an exemplary CA BA grain of size aeff = 200 nm. The spectra of the two stellar sources S1 and S2 are shown in blue, normalized by their maximum value. We emphasize that this particular grain has two changes in sign, resulting in a reduced net torque over wavelengths.

6.2 Torque efficiency

As we assumed that the principal grain axis ${\hat{a}}_1$ is parallel to ${\hat{e}}_1$, the direction of the incoming radiation, only the Q₁ component needs to be evaluated to obtain the total torque efficiency Q_Γ.

The results of the efficiency Q_Γ calculated with DDSCAT may possess positive (negative) values depending on the wavelength (see Fig. 4), corresponding to a direction of the angular velocity ω_RAT that is parallel (antiparallel) to ${\hat{e}}_1$. Hence, different wavelength regimes create opposing torques on the grain aggregates. When the sign change is subsequently taken into account, the net RAT for individual aggregates may considerably lower. In order to evaluate the average efficiency for different grain aggregates, we evaluated Γ_RAT over λ (see Eq. (1)) for each grain individually, including canceling effects of positive and negative spectral contribution of Q_Γ(λ), and we also calculated Γ_RAT using the absolute value |Q_Γ(λ)|, which ignores the opposing spectral contributions to the net RAT. We find that averaging over the DDSCAT data with the absolute value of |Q_Γ| resulted in only marginally higher RATs compared to when the sign of Q_Γ was taken into account. We emphasize that the difference is insignificant compared to the variation between individual grain shapes. Hence, we only used the absolute value |Q_Γ| in the following analysis so that data of different grain shapes can be combined.

The resulting dependence of the magnitude of Q_Γ on the wavelength for CA BA and SC BA grains is presented in Fig. 5. The values of Q_Γ are compared with the refractive index of the corresponding grain material. We report a feature for SC at λ > 8 μm that follows the form of the imaginary part k of the refractive index. This feature of SC was also noted in previous studies (see, e.g., LH07).

6.3 Parameterization for the RAT efficiency

Since RATs depend on the wavelength λ relative to the grain radius a_eff, it is convenient to evaluate |Q_Γ| as a function of λ/a_eff (see Fig. 6) instead of λ alone. Here, the slopes of the different grain sizes overlap tightly, showing the self-similar nature that allowed the canonical parameterization (Eq. (2)) for solid grains in the first place. Our results using grain aggregates show a comparable constant part for λ/a_eff < 1.8, but do not follow the overall power-law parameterization for λ/a_eff > 1.8. Instead, both materials SC and CA reveal a steeper power law that overlaps for all sizes from λ/a_eff ≈ 3 and then evolves into a shallower power law for λ/a_eff ≳ 10. A similar behavior of a broken power law was reported in previous studies, for instance, in LH07, or for Gaussian dust grains in Herranen et al. (2019), but is much more apparent here. Consequently, we suggest a modified parameterization for ballistic grain aggregates with two connected consecutive power laws. We performed least-square fits for the constant part (called c in this study) and the two power laws for all data points of the grains that shared the same material and monomer density. The fits of c were performed for individual grain sizes in the interval λ/a_eff < 3. All sizes were fit jointly for the first and second power law with exponents q₁ and q₂ in the intervals 3 < (λ/a_eff) < 12 and (λ/a_eff) > 20. In practice, the power laws were fit as linear functions to the logarithm of the data.

In order to conveniently combine the three regions into a continuous curve, we chose the following definition: $$\left|Q_{\mathrm{\Gamma }}\right|=\{\begin{array}{ll}c\left(a_{\mathrm{e}\mathrm{f}\mathrm{f}}\right)& \text{ if }\left(\frac{\lambda }{a_{\mathrm{e}\mathrm{f}\mathrm{f}}}\right)\le a,\\ c{\left(\frac{\lambda }{a_{\mathrm{e}\mathrm{f}\mathrm{f}}}/a\right)}^{q_1}& \text{ if }a\le \left(\frac{\lambda }{a_{\mathrm{e}\mathrm{f}\mathrm{f}}}\right)\le b,\\ c{\left(\frac{b}{a}\right)}^{q_1}{\left(\frac{\lambda }{a_{\mathrm{e}\mathrm{f}\mathrm{f}}}/b\right)}^{q_2}& \text{ otherwise. }\end{array}$$(13)

We joined the individual fits by calculating the intersection points a and b, where a is the intersection of c with the steep power law, and b is the intersection of both power laws. Therefore, like c, there will be a different value of a for each a_eff. The resulting parameterization is shown in Fig. 7 for the exemplary data set of carbon BA grains of size 200 nm. All resulting fit parameters of Eq. (13) are given in Appendix B.

Larger grains interact more efficiently with the radiation field, which results in a higher value c, as shown in Fig. 6. The BAM1 and BAM2 grain models have comparable curves for |Q_Γ|, where the values differ most in the constant part on a scale that is similar to the difference between two grain sizes. The more porous the grain, the larger c. The two materials differ in their values of c, which are higher for SC. Both power laws are steeper for CA, most likely due to interference by the feature in the SC material.

6.4 Net torques and angular velocities

The exact difference in RAT using the canonical or the modified parameterization, and subsequently, the magnitude of the angular velocity of the grains, cannot a priori be quantified because the RATs depend on the spectral characteristics of the radiation field. Therefore, we calculated the maximum angular velocity ω_RAT induced by RATs using both parameterizations and compared the results to those we obtained using the DDSCAT data of individual grains.

If the torque efficiency Q_Γ(λ) for a single grain exhibits some sign changes over wavelength (see Fig. 4), this leads to a lower torque overall as different wavelengths exert RAT components in opposite directions. As mentioned above, the effect of positive and negative components of Q_Γ canceling out appears to be small compared to the overall variation in the strength of Q_Γ between different grains. The torque along ${\hat{a}}_1$ was calculated for each grain following Eq. (1), where Q_Γ was used without taking the absolute value. Then, the maximum angular velocity was calculated with Eq. (3), where τ_drag and I_max were also evaluated individually for each grain.

Figure 8 shows the distribution of angular velocities and torques for individual SC BA grains and the most luminous source, that is, the AGN. The distributions are broken down by grain size. We report that the resulting values of the RATs may span a range of up to two orders of magnitude, with a larger spread below the mean. The mean of the distribution increases with grain size because larger grains have higher Q_Γ values in the constant part. The distributions of the different grain sizes are more distinct in Γ_RAT and overlap significantly for ω_RAT because of the drag timescale (see Eq. (3)).

A parameterization that describes the average value for an ensemble of grains means that we do not need to calculate ω_RAT for every single grain. We therefore compared the average values obtained from individual grain calculations (shown in Fig. 8 as solid lines) with the values obtained for the canonical parameterization for Q_Γ (dotted lines). This parameterization does not model the means of the distributions well. It overestimate the values of Γ_RAT and ω_RAT, especially for the smaller grains.

To quantify how the canonical parameterization of LH07 and the modified parameterization of our study compare with the distribution means for all combinations of radiation sources, materials, and monomer densities, we show the distribution means of the RATs in Fig. A.1 as a function the grain size a_eff. The CA material in the left column, the SC material in the right column, the monomer densities in the rows, and the four radiation sources are shown with different colors in each panel. The difference between the sources is due to the linear dependence of Γ_RAT on the energy density u and anisotropy factor γ, where larger RATs are reached for stronger radiation sources (see Eq. (1) and Table 2). The RAT of the S1 source should be larger by a factor of 7 due to γ and by a factor of 5.6 due to u. The difference seen in Fig. A.1 is larger than that with approximately two orders of magnitude. This is a result of the spectral energy distribution of this source. The integral in Eq. (1) places a much larger weight on u at wavelengths shorter than ≈1 μm, as Q_λ drops by about four orders of magnitude for longer wavelengths. In this range, the stellar sources outshine the ISRF.

In Fig. A.2 we present the mean angular velocities ω_RAT corresponding to the RATs shown in Fig. A.1. Here, the variations in ω_RAT between individual sources are different from the variation in Γ_RAT because ω_RAT also depends on τ_drag, which itself depends indirectly on u via the temperature dependence of τ_IR. Using the dust temperatures in Table 3, we can estimate τ_IR for the given temperature from Fig. 3. The gas-drag timescale is independent of u and about τ_gas ≳ 10¹² s for the chosen gas density of n_gas = 5 · 10⁷ m⁻³. The drag timescale is dominated by τ_gas for the lower dust temperatures of the ISRF and S1 sources, both timescales are about equal for S2, and τ_IR dominates for the AGN, where the dust is hottest.

Therefore, the difference in ω_RAT only depends on Γ_RAT for the ISRF and S1, but it includes τ_IR for S2 and AGN. The AGN therefore is much closer to S2 in ω_RAT (1 dex) than in Γ_RAT (3 dex). Even though the torque is much stronger for the AGN source, the grains do not rotate that much faster because they also become much hotter, which increases the IR drag. The difference in τ_IR between the two materials also leads to a lower angular velocity for S2 and AGN for SC compared to CA, but there is no difference between the two materials for S1 and ISRF because τ_gas dominates there. The difference between monomer densities is a decrease in Γ_RAT from BA to BAM2 that is smaller than one order of magnitude because Q_Γ decreases with monomer density. The resulting decrease with monomer density in ω_RAT is smaller because I_max also decreases with monomer density.

As expected, RATs strongly scale with a_eff, and this does not change much between sources, materials, and monomer densities. However, for the angular velocity ω_RAT, the results are less uniform, as shown in Fig. A.2. Here, there is no scaling with grain size for the ISRF source and a weak scaling for the AGN, which flattens toward the larger grains. For the stellar sources, ω_RAT first increases and then flattens for SC, and it even decreases toward larger grains for S1. In the case of CA, ω_RAT even scales inversely with grain size for the stellar sources. We remark that the scaling of ω_RAT with grain size depends on the interplay between multiple factors: The spectral shape of the source and the scaling of the drag timescale, which depends on the material and structure of the grain, and whether τ_IR or τ_gas dominates for the given parameters. As outlined in the pioneering work of Hoang et al. (2019) and Hoang (2019), RATs may spin up dust grains to a characteristic angular velocity ω_disr, where the internal grain structure can no longer resist the disruption because of centrifugal forces (RATD). The predicted order of magnitude for RATD to occur is ω_disr ≈ 10⁹ s⁻¹. Later, ab initio numerical N-body simulations of the disruption of rotating BA aggregates revealed that aggregates larger than a_eff = 300 nm are more resistant against RATD because they are deformed. We emphasize that the most luminous sources, AGN and S2 may spin up the dust aggregates beyond the limit of ω_RAT > ω_disr (see Fig. A.2), even though RATs are less efficient for ballistic aggregates. Hence, most of the grains may become disrupted by RATD in the proximity of an AGN or S2-like star. This means that ballistic aggregates are more likely to grow deep inside of dense molecular clouds, where they are shielded from a strong radiation field (Bethell et al. 2007).

As presented in Fig. A.2, the canonical parameterization gives more accurate results for the larger grain aggregates because they experience the strongest torques. We point out that the canonical parameterization overestimates the mean angular velocity ω_RAT overall, with differences of up to one order of magnitude. In comparison, the two-power-law parameterization resembles the mean of the DDSCAT data more precisely and performs equally well for all sources, materials, monomer densities, and sizes.

A way to quantify the efficiency of RATs of porous aggregates was proposed by Tatsuuma & Kataoka (2021) (TK21 here-after). The TK21 parameterization takes the porosity (monomer density) of the grain aggregates into account via the volume filling factor ϕ, where ϕ = 1 represents a solid grain, and ϕ = 0 represents completely empty space. Here, the volume-filling factor ϕ was added to the canonical parameterization as follows: $$\left|{\mathit{Q}}_{\mathrm{\Gamma }}\right|=\{\begin{array}{ll}0.4& \text{ if }\lambda \le 1.8a_{\mathrm{e}\mathrm{f}\mathrm{f}}\cdot \varphi ,\\ 0.4{\left(\frac{\lambda }{1.8a_{\mathrm{e}\mathrm{f}\mathrm{f}}\cdot \varphi }\right)}^{\alpha }& \text{ otherwise. }\end{array}$$(14)

Fig. A.3 shows the average angular velocity ω_RAT in comparison with the TK21 parameterization and canonical parameterization of LH07. The model proposed by TK21 clearly improves upon the canonical parameterization of LH07 for the less porous BAM1 and BAM2 grains, but now slightly underpredicts ω_RAT for the weaker sources in BAM1. For the more porous BA grains, however, TK21 clearly underpredicts ω_RAT because in this regime, the canonical parameterization performed best. As shown in Fig. A.3, merely introducing ϕ to take the grain porosity into account does not bring the canonical parameterization into agreement with our results. This indicates that complex grain shapes differ in their interaction with radiation compared to compact grains in more fundamental ways than just by making the grains porous.

Consequently, we may summarize that even though there is a large spread in ω_RAT between individual grains (see Fig. 8) due to random variation in the grain shape, it is still valid to use a parameterization of Q_Γ to model the average ω_RAT for an ensemble of ballistic grain aggregates.

For the three parameterizations shown in this work, the two-power-law parameterization is the most accurate choice for this task.

Fig. 5 Average torque efficiency |QΓ| over wavelengths for CA (left) and SC (right) BA aggregates of different grain sizes aeff (color-coded) in comparison with the real part n (dotted line) and imaginary part k (dashed line) of the corresponding refractive indices. We note that the characteristic features of QΓ follow the shape of k.

Fig. 6 Same as Fig. 5, but for the average torque efficiency QΓ over λ/aeff in comparison with the conical power-law parameterization (dotted red line), as suggested in LH07.

Fig. 7 Two-power-law parameterization shown as a red line for CA (left) and SC (right) BA grains in comparison with the full set of the numerical DDSCAT simulations (black dots) for grains of size aeff = 200 nm, where the absolute value of the torque efficiency |QΓ| is plotted as a function of λ/aeff. The constant part c is fitted for each grain size up to a value of λ/aeff = 3, the steeper power-law is fit as a linear function to the logarithm of the data for all sizes combined in the interval 3 < λ/aeff < 10, and the same for the shallow power law in the interval λ/aeff > 20. Finally, the two intersection points a and b were calculated and used to create a continuous parameterization for |QΓ| (see Eq. (13)).

Fig. 8 Histograms representing the distribution of the RAT ΓRAT (left) and angular velocity ωRAT (right) for SC BA grains of different sizes and the AGN as the radiation source. The continuous lines show the mean for each distribution, and the dotted lines show the value calculated using the canonical parameterization (LH07). The mean values for ωRAT are calculated from the mean values of ΓRAT shown in the lower panel as well as averages for τdrag and Imax. We note that the RATs by the canonical parameterization are consistently higher than the mean values of the dust aggregates.

7 Discussion and caveats

An underlying assumption in our RAT calculations using grain aggregates is the anisotropy of incoming radiation. As DDSCAT only simulates anisotropic radiation, we used the anisotropy factor γ to treat a certain percentage of the source radiation as anisotropic. For instance, a grain in a molecular cloud around a forming star receives a portion γ of the stellar radiation directly (anisotropically) and the rest isotropically from scattering on surrounding particles. These processes are not fully accounted for in this work, but we do not expect torques from isotropic radiation to dominate torques from anisotropic radiation, especially for our grain model without variations in the optical properties across the grain’s surface.

We also assumed perfect alignment of the grain rotation axis with the direction of the radiation, that is, Θ = 0. This is based on the fact that grains preferentially align with the direction of incoming radiation in the absence of magnetic fields (see LH07), even though magnetic fields are present in the typical environments probed here and the grains may not reach suprathermal angular velocities. For a more detailed picture, DDSCAT simulations that sample the whole range of Θ angles are required that are subsequently used to calculate phase-space trajectories of grains in Θ and ω, and they would also need to include typical magnetic fields. This approach may allow us to verify whether the assumption of a fixed orientation was justified, and if not, to gauge how large the deviation is. However, this is beyond the scope of our current study.

It is still possible to use a parameterization to model the average angular velocity with the more detailed grain model, even including positive and negative values of Q_Γ(λ) for individual grains. If needed, it might also be possible to parameterize the distribution of ω_RAT values (see Fig. 8), or other quantities using larger grain ensembles.

As we were limited to a small range of grain sizes due to the computational restrictions of DDSCAT, future simulation setups should be designed to test whether the two-power-law parameterization for |Q_Γ| also holds for larger grains. If this is the case and the two-power-law shape is also seen for general orientations of the grain, it would stand to reason that the form of |Q_Γ| following two power laws is intrinsic to the nature of grains composed of monomers, in the same way as the single power law of the canonical parameterization is for compact grains. It might be convenient to generalize the parameters of the parameterization and give them a functional form dependent on monomer density and grain size. Fig. 9 shows the constant part c of the parameterization as a function of grain size for different monomer densities. The linear increase in c with grain size has a different slope for each monomer density, and the slope roughly doubles from BAM2 to BAM1 and triples from BAM2 to BA. It is therefore possible that for each material there is a functional form of c as a linear function, with the slope depending on the monomer density. To determine whether and how the exponents q₁ and q₂ depend on the material or monomer density requires a larger sample of monomer densities and refractive indices (materials).

We have shown that aggregated grains have a different functional form of Q_Γ(λ) than compact grains, and they therefore do not reach angular velocities as high as compact grains would in the same circumstances, with differences of up to an order of magnitude for the smallest grains. However, our results and the canonical parameterization agree for the least dense and largest grains (BA, 250 nm). If the trend of increasing c for a larger grain size and lower monomer density (see Fig. 9) holds true, we would expect Γ_RAT to further increase for less dense and larger grains. The relation of a lower monomer density that leads to higher torques makes intuitive sense considering that it increases the total surface area of the grain and therefore the cross section for an interaction with radiation, just as increasing the grain size does. While the torque increases, ω_RAT does not increase further for larger grains, however, and in some cases, it even decreases again for the largest grains. Aggregate grains larger than those simulated here may potentially not reach much higher angular velocities, as the drag also increases with size.

How much ω_RAT increases for higher radiation energy densities strongly depends on the optical properties and therefore on the composition of the grains because different drag effects dominate in different environments. The significantly increased surface area and more complex geometry of aggregated grains has to be taken into account.

In environments with highly ionized gas and charged grains, such as an AGN, plasma drag in addition to gas drag and photon drag might additionally play a role for grain dynamics (Draine & Lazarian 1998). However, the charging of BA dust adds an additional level of complexity, where individual monomers may carry different charges and even undergo some stochastic fluctuation over time (see e.g. Matthews et al. 2013; Ma et al. 2013; Matthews et al. 2018). It is beyond the scope of this paper to include the plasma drag in the context of BA. However, the difference of the total grain charge of a BA compared to an equivalent sphere should be smaller than one order of magnitude, as shown in Ma et al. (2013) and Matthews et al. (2018). Hence, we estimate that the canonical parameterization of RAT would still overestimate the resulting angular velocity ω_RAT for ballistic aggregates in the proximity of an AGN. The dust grains were assumed to have no relative velocity to the gas, but grains close to strong radiation fields might experience linear acceleration by the radiation pressure and reach high drift velocities relative to the surrounding gas. We emphasize that the resulting rotation of the BA grain would then also be influenced by mechanical torques (METs, see e.g. Lazarian & Hoang 2007b; Das & Weingartner 2016; Hoang et al. 2018; Reissl et al. 2023), which work in a way similar to RATs, with gas collisions instead of photon collisions. However, it is still subject of debate how METs enhance the grain rotation in the presence of RATs (see also Lazarian & Hoang 2021).

We emphasize once again that our modeling of RATs did not include the dynamical processes associated with an external magnetic field. As noted in LH07 for an analytical RAT model, the appearance of attractors in the Θ-ω phase space is tightly connected to the ratio q = max(Q_e1)/max(Q_e2) when an external field is present. This finding was later confirmed by Herranen et al. (2019) and Herranen (2020) based on grain shapes that were modeled as Gaussian ellipsoids. We note, however, that a similar parameter q for a pure mechanical alignment was not found (Das & Weingartner 2016; Reissl et al. 2023). Hence, it remains to be seen whether the ratio q is a reliable proxy to predict the long-term stable alignment with the direction of the magnetic field, considering RATs as a spin-up mechanism of ballistic aggregates. Moreover, in the presence of a magnetic field, this is not only a matter of the efficiency of the RATs. Due to internal dissipation processes in paramagnetic and ferromagnetic materials, a fraction of the angular velocity ω is transferred into internal heat (see e.g. Davis & Greenstein 1951; Dolginov & Mytrophanov 1976; Purcell 1979). Consequently, the ω reported in our study may be lower, depending on the magnetic field strength and grain orientation. A similar effect of a reduction in ω for METs was shown by Reissl et al. (2023). However, to which degree a magnetic field would reduce the maximum possible angular velocity ω_RAT of BAM aggregates remains to be addressed in forthcoming publications.

Fig. 9 Constant part of the two-power-law parameterization c vs. grain size for CA (left) and SC (right). The dotted lines show the linear regression for each grain model.

8 Summary and outlook

We presented a numerical study of the impact of the grain shape on the radiative torque (RAT) efficiency and its dependence on the magnitude of the radiation field as well as the material and porosity of the dust grains. As a source of radiation, we considered O-type stars, an AGN, and the ISRF typical for the Milky Way. The dust grains were modeled by means of ballistic aggregation and migration (BAM) of monomers of various sizes. The aggregation process was biased to control the size and number of connections of the grain aggregates. We calculated the optical dust grain properties in the discrete dipole approximation using the DDSCAT code, considering the refractive indices of pure carbonaceous and silicate grains. Based on the optical properties of the dust aggregates and the radiation energy density, we calculated the maximum angular velocity ω_RAT according to the RAT alignment theory. We note again that the conclusions presented here are still preliminary in nature, as a stable alignment of the grain rotation axis and the direction of incoming radiation was assumed and no magnetic fields were included. Our major findings are listed below.

• We report that the ISRF produced average angular velocities ω_RAT = 10⁶ s⁻¹ for both carbon and silicate grains, whereas the O-type stars and AGN sources produced much larger ω_RAT in a range of 10⁸ s⁻¹ to 10¹⁰ s⁻¹ for carbon grains and 10⁷ s⁻¹ to 10⁹ s⁻¹ for silicate grains.

• Compared to the O-type stars, the angular velocity ω_RAT of grains for the AGN did not increase in a similar manner because the photon emission drag forces on the grains increase. The reason is that the dust temperatures in the AGN environment are higher.

• Considering an entire ensemble of BAM grain aggregates, we proposed a new parameterization for the RAT efficiency that follows a broken power law with a constant part for shorter wavelengths followed by two distinct power laws to cover longer wavelengths. The proposed parameterization describes the average angular velocities ω_RAT reached by the ensembles of grains more precisely than the canonical parameterization derived for compact grains.

• According to our DDSCAT calculations, individual grain aggregates may have sign changes in the RAT efficiency over the entire wavelength regime. This reduces the net torque integrated over wavelength. However, the appearance of this sign changes affects the average grain spin-up behavior of the entire ensemble only marginally.

• The stronger stellar source and the AGN produce angular velocities that are potentially high enough for a rotational disruption of the grains. The results for the angular velocities obtained here could be used to study the rotational disruption of ballistic aggregate grains on the level of individual grains in the future.

Our results for a parameterization with two power laws for grains composed of monomers, potentially improved by expansion to a parameterization for the distribution of ω_RAT values, can facilitate detailed studies of grain rotation including grain size, porosity, and material by replacing computationally expensive DDSCAT simulations of individual grains. Further improvements could be made by dropping the assumption of unobscured source radiation, including isotropic radiation, surface effects, H₂ formation, and other additional drag effects. Especially in order to apply the results to grain alignment, a general orientation of the grains as well as magnetic fields should be added to complement the analysis.

Acknowledgements

We thank the anonymous referee for a constructive report. Special thanks go to Thiem Hoang for numerous enlightening discussions. R.S.K. also thanks the Harvard-Smithsonian Center for Astrophysics and the Radcliffe Institute for Advanced Studies for their hospitality during his sabbatical, and the 2024/25 Class of Radcliffe Fellows for highly interesting and stimulating discussions. S.R., J.A.J., and R.S.K. acknowledge financial support from the Heidelberg cluster of excellence (EXC 2181 - 390900948) “STRUCTURES: A unifying approach to emergent phenomena in the physical world, mathematics, and complex data”, specifically via the exploratory project EP 4.4. S.R. and R.S.K. also thank for support from Deutsche Forschungsgemeinschaft (DFG) via the Collaborative Research Center (SFB 881, Project-ID 138713538) ‘The Milky Way System’ (subprojects A01, A06, B01, B02, and B08). And we thank for funding form the European Research Council in the ERC synergy grant “ECOGAL – Understanding our Galactic ecosystem: From the disk of the Milky Way to the formation sites of stars and planets” (project ID 855130). The project made use of computing resources provided by The Länd through bwHPC and by DFG through grant INST 35/1134-1 FUGG. Data are in part stored at SDS@hd supported by the Ministry of Science, Research and the Arts and by DFG through grant INST 35/1314-1 FUGG.

Appendix A Figures

In this section we present the full scope of our simulation results in Figs.A.1–A.3.

Fig. A.1 RATs ΓRAT (solid line) per grain size aeff calculated with the two-power-law parametrization is shown in comparison to the RATs predicted by the canonical parametrization suggested in LH07 (dotted line) as well as the average of the DDSCAT simulation data (diamonds). Each color represents one of the different sources i.e. ISRF (gray), S1 (dark blue), S2 (light blue), or AGN (red), respectively, utilizing the CA grain material (left column) and the SC material (right column) with monomer-densities BA (top row), BAM (middle row), or BAM2 (bottom row). This figure is to be compared with Fig. 8.

Fig. A.2 Same as Fig. A.1 but for angular velocity ωRAT. We emphasize that the canonical parametrization may over-predict the magnitude of ωRAT up to one order of magnitude higher for BAM grain aggregates as the two-power-law parametrization.

Fig. A.3 Same as Fig. A.2 but showing the canonical parametrization of LH07 (dotted lines) in comparison with the parametrization for porous grains suggested in TK21 (dashed lines).

Appendix B Two power-law parametrization fit parameters

For completeness we provide the full set of fitted parameters of the two-power-law RAT parametrization in table B.1 for CA grains and in table B.2 for SC grains.

References

All Tables

All Figures

	Fig. 1 Exemplary BA (left), BAM1 (middle), BAM2 (right) grains with an effective radius of aeff = 250 nm. The rotation axis ${\hat{a}}_1$ is depicted as the red arrow and is associated with the direction of the maximum moment of inertia Imax.
In the text

	Fig. 2 Spectral flux of sources ISRF, S1, S2, and AGN that were used to irradiate the grain aggregates.
In the text

Fig. 3 Drag timescales over the effective radius aeff dependent on the grain model (BA, BAM1, or BAM2). Left panel: gas drag timescale for different gas densities ngas for all grain materials. The gas temperature is at a constant value of Tgas = 100 K. Center panel: IR drag timescale for the CA grain material for different dust temperatures Tdust. Right panel: same as the center panel, but for the SC grain material.

In the text

	Fig. 4 Torque efficiency QΓ as a function of wavelength λ for an exemplary CA BA grain of size aeff = 200 nm. The spectra of the two stellar sources S1 and S2 are shown in blue, normalized by their maximum value. We emphasize that this particular grain has two changes in sign, resulting in a reduced net torque over wavelengths.
In the text

	Fig. 5 Average torque efficiency |QΓ| over wavelengths for CA (left) and SC (right) BA aggregates of different grain sizes aeff (color-coded) in comparison with the real part n (dotted line) and imaginary part k (dashed line) of the corresponding refractive indices. We note that the characteristic features of QΓ follow the shape of k.
In the text

	Fig. 6 Same as Fig. 5, but for the average torque efficiency QΓ over λ/aeff in comparison with the conical power-law parameterization (dotted red line), as suggested in LH07.
In the text

Fig. 7 Two-power-law parameterization shown as a red line for CA (left) and SC (right) BA grains in comparison with the full set of the numerical DDSCAT simulations (black dots) for grains of size aeff = 200 nm, where the absolute value of the torque efficiency |QΓ| is plotted as a function of λ/aeff. The constant part c is fitted for each grain size up to a value of λ/aeff = 3, the steeper power-law is fit as a linear function to the logarithm of the data for all sizes combined in the interval 3 < λ/aeff < 10, and the same for the shallow power law in the interval λ/aeff > 20. Finally, the two intersection points a and b were calculated and used to create a continuous parameterization for |QΓ| (see Eq. (13)).

In the text

Fig. 8 Histograms representing the distribution of the RAT ΓRAT (left) and angular velocity ωRAT (right) for SC BA grains of different sizes and the AGN as the radiation source. The continuous lines show the mean for each distribution, and the dotted lines show the value calculated using the canonical parameterization (LH07). The mean values for ωRAT are calculated from the mean values of ΓRAT shown in the lower panel as well as averages for τdrag and Imax. We note that the RATs by the canonical parameterization are consistently higher than the mean values of the dust aggregates.

In the text

	Fig. 9 Constant part of the two-power-law parameterization c vs. grain size for CA (left) and SC (right). The dotted lines show the linear regression for each grain model.
In the text

Fig. A.1 RATs ΓRAT (solid line) per grain size aeff calculated with the two-power-law parametrization is shown in comparison to the RATs predicted by the canonical parametrization suggested in LH07 (dotted line) as well as the average of the DDSCAT simulation data (diamonds). Each color represents one of the different sources i.e. ISRF (gray), S1 (dark blue), S2 (light blue), or AGN (red), respectively, utilizing the CA grain material (left column) and the SC material (right column) with monomer-densities BA (top row), BAM (middle row), or BAM2 (bottom row). This figure is to be compared with Fig. 8.

In the text

	Fig. A.2 Same as Fig. A.1 but for angular velocity ωRAT. We emphasize that the canonical parametrization may over-predict the magnitude of ωRAT up to one order of magnitude higher for BAM grain aggregates as the two-power-law parametrization.
In the text

	Fig. A.3 Same as Fig. A.2 but showing the canonical parametrization of LH07 (dotted lines) in comparison with the parametrization for porous grains suggested in TK21 (dashed lines).
In the text