Effects of Aluminum/Carbon and Morphology on Optical Characteristics and Radiative Forcing of Alumina Clusters Emitted by Solid Rockets in the Stratosphere

Abstract

Alumina (Al₂O₃) particles, the primary combustion products of solid rockets, can accumulate in the stratosphere, changing the global radiative balance. These Al₂O₃ particles were usually treated as homogeneous spheres. However, they contain impurities and may form clusters during the combustion process. Models representing Al-containing and C-containing Al₂O₃ clusters were developed, denoted as Al₂O₃ shell model (ASM) and Al₂O₃ core model (ACM), respectively. The superposition T-matrix method (STMM) was applied to examine their optical characteristics. Subsequently, a method to obtain the top-of-atmosphere flux was proposed by integrating the models with the moderate resolution atmospheric transmission code (MODTRAN). With the addition of Al/C, the absorption cross-section enhances by several orders of magnitude at 0.55 μm and increases slightly at 10 μm. The equivalent sphere models will weaken their scattering ability. A 4Tg mass burden of Al₂O₃ produces radiative forcing of −0.439 Wm⁻². However, the addition of Al and C reduces the forcing by up to 15% and 12%, respectively. In summary, the optical characteristics and radiative forcing of Al₂O₃ clusters are sensitive to Al/C and morphology models. While our findings are impacted by various uncertainties, they contribute valuable insights into the radiative forcing of Al₂O₃ particles, potential climatic changes by space activities.

1. Introduction

The growth of human space activities and various projects driven by government agencies, such as the National Aeronautics and Space Administration (NASA) [1,2,3] and the European Space Agency (ESA) [4,5], as well as commercial organizations like SpaceX and Blue Origin [6], has attracted widespread concerns about the associated climate impacts [7,8]. Traditionally, stratospheric aerosols were believed to primarily originate from volcanic eruptions and biomass burning [9,10]. However, solid rockets, which provide powerful impulses and play an important role in the booster phase of space missions, emit their combustion products directly into the stratosphere, posing environmental challenges. Solid rockets usually use Al as an additive to improve impulse, leading to large amounts of Al₂O₃ particles in the exhaust plume [11]. These Al₂O₃ particles can accumulate in the stratosphere [12]. In the past, environmental assessments of Al₂O₃ particles mainly focused on ozone decomposition [13]. However, they also scatter and absorb solar radiation, as well as absorb thermal radiation from Earth, changing Earth’s radiative balance as shown in Figure 1. In other words, Al₂O₃ particles, as stratospheric aerosols, can offset the greenhouse effect, similar to sulfates and volcanic aerosols [14,15,16,17,18]. The corresponding impacts are poorly known, owing to the lack of related data [8,19,20]. With the increasing rate of launches and improved technology to obtain accurate information on Al₂O₃ from solid rockets, the climate effects of Al₂O₃ particles have become a topic of great interest [21,22].

The radiative forcing caused by aerosols is commonly employed to assess their climate impacts. Ross et al. [21] determined the radiative forcing by subtracting the integrated longwave absorption forcing from the integrate shortwave scattering forcing. They found that Al₂O₃ particles produce a positive radiative forcing of 4.8 mWm⁻², warming the stratosphere and cooling the Earth. Obviously, the radiative forcing of aerosols is closely linked to their scattering and absorption optical properties. Weisenstein et al. [22] explored the efficiency of Al₂O₃ particles as an alternative to sulfate aerosol for solar geoengineering. The results showed that Al₂O₃ particles were more efficient scatterers of sunlight than sulfate particles, and using Al₂O₃ particles for solar geoengineering could help overcome the problems associated with sulfate particles. According to these results, for 240 nm radius Al₂O₃ particles, an injection rate of 4 Tg yr⁻¹ produces a radiative forcing of −1.2 Wm⁻². The injection rate of Al₂O₃ particles exhibited a basically linear relationship with the corresponding radiative forcing of specific radius aerosols. However, it is crucial to note that both studies assumed that Al₂O₃ particles emitted by solid rockets were homogenous, uniform-distribution spheres. In reality, the chemico-physical properties of Al₂O₃ particles are complex, warranting accurate morphology models for further investigation.

Understanding the formation process of Al₂O₃ particles in the combustion chamber of solid rockets is crucial for establishing their morphology models. Al starts to burn at the surface, progressing inward, and tends to undergo incomplete oxidation due to a limited burning time [23]. That leads to Al₂O₃ shell outside and Al core inside particles as shown in the transmission electron micrograph, Figure 2a,b [24]. The observation of the green AlO flame bands and Al in the exhaust products supports this observation [25]. C, from the degradation of binder or ablation, may adhere to the surface of Al₂O₃, producing Al₂O₃ core and C shell particles, as shown in Figure 2c [26]. In Figure 2c, the yellow dots represent the specific points on the particle surface where measurements were taken. These measurements indicate that the combustion products are composed of aluminum, oxygen, carbon, and gold, thus proving the existence of carbon on the surface of Al₂O₃ particles [27]. These particles, with a non-uniform radius, may aggregate into clusters at high temperatures [23,28]. Apparently, Al-containing and C-containing Al₂O₃ clusters were directly emitted from solid rockets. Both Al and C are more efficient absorbers than Al₂O₃ [28,29]. Consequently, Al-containing and C-containing Al₂O₃ clusters are definitely chemico-physically distinct. Therefore, there is a need to develop Al/C-containing Al₂O₃ cluster models to accurately represent their unique characteristics.

To investigate the radiative forcing of Al₂O₃ clusters, their optical characteristics should be studied in advance [21,30]. Previous research has studied the effects of chemico-physical properties on the optical characteristics of Al₂O₃ particles. Pluchino et al. [26,31] calculated the infrared emissivity spectra for a single Al₂O₃ sphere coated with a thin layer of C. They found such a C shell could enhance the emissivity of Al₂O₃ particles. However, they treated the Al₂O₃ particle as a single sphere. Xu et al. [32] found the optical properties of Al₂O₃ particles varied during its transformation from the liquid to the alpha phase. Above all, the chemico-physical properties of Al₂O₃ particles are strongly related to the optical characteristics. It is reasonable to assume that Al/C impurities and the morphology of Al₂O₃ clusters may manifest distinct optical characteristics and radiative forcing. Nevertheless, to the best of our knowledge, no research has involved the investigation of optical characteristics and radiative forcing of Al-containing and C-containing Al₂O₃ clusters. Due to many uncertainties [8,21], our objective is mainly to assess the impacts of Al/C impurity and morphology on the optical characteristics and radiative forcing of Al₂O₃ clusters.

The structure of this paper is as follows. First, models of Al-containing and C-containing Al₂O₃ clusters were developed. Second, a method to calculate the radiative forcing by these clusters was proposed. Third, the optical characteristics and radiative forcing of Al-containing and C-containing Al₂O₃ clusters were investigated. The conclusions are summarized in the last section.

2. Materials and Methods

2.1. Models of Al-Containing and C-Containing Al₂O₃ Clusters

2.1.1. Monomers of Al-Containing and C-Containing Al₂O₃ Cluster

In this study, we focused on Al and C, two common impurities found in Al₂O₃ particles. We conducted a detailed morphology analysis of micrographs of Al₂O₃ particles from the combustion products of solid rockets, along with their formation mechanisms. This led to the development of a monomer model for Al-containing Al₂O₃ clusters, called the Al₂O₃ shell model (ASM). This monomer model is represented by two concentric spheres, consisting of an Al₂O₃ shell and an Al core, as shown in Figure 3a. Similarly, the monomer model for the C-containing Al₂O₃ clusters, called the Al₂O₃ core model (ACM), comprises two concentric spheres, with an Al₂O₃ core and a C shell, as depicted in Figure 3b. The volumetric fractions of Al and C within the particles were calculated as follows:

$$V_{\mathrm{C}}=\frac{r^3-r_{\mathrm{core}}^3}{r^3}$$

(1)

$$V_{\mathrm{Al}}=\frac{r_{\mathrm{core}}^3}{r^3}$$

(2)

where r and r_core are the radius of the monomer shell and the core, respectively. Given that the Al₂O₃ shell is prone to rupture with a high quantity of Al, V_Al was defined to range from 0 to 0.5. V_C is considered between 0~0.02, due to the small presence of C within the Al₂O₃ particles [33].

2.1.2. Al-Containing and C-Containing Al₂O₃ Cluster Models

Depending on the way a cluster forms, the diffusion-limited aggregation (DLA) algorithm includes particle–cluster aggregation and cluster–cluster aggregation [34]. We chose particle–cluster aggregation, due to its relative simplicity compared to cluster–cluster aggregation [35]. The construction and morphology of the fractal structure is described as:

$$N_s=k_{\mathrm{f}} {\left(\frac{R_g}{a}\right)}^{D_{\mathrm{f}}},$$

(3)

$$N_s=k_{\mathrm{f}} {\left(\frac{R_g}{a}\right)}^{D_{\mathrm{f}}},$$

(4)

$$a=\frac{1}{Ns}{\displaystyle \sum _{i=1}^{Ns}{r}_i},$$

(5)

where k_f is the fractal prefactor, N_s is the number of monomers in the cluster, R_g represents the radius of gyration, D_f is the fractal dimension, a is the mean radius of the monomer, l_j is the distance from monomer j to the center of the cluster, and r_i is the radius of the i_th monomer in the cluster. The DLA is used to simulate clusters with monomers of the same radius. However, the radius of Al₂O₃ follows the lognormal distribution [11]:

$$P_r(r)=\frac{1}{\sqrt{2\pi }r\ln ({\sigma }_{\mathrm{g}})}\exp \left[-{\left(\frac{\ln (r)-\ln (r_{\mathrm{g}})}{\sqrt{2}\ln ({\sigma }_{\mathrm{g}})}\right)}^2\right],$$

(6)

where σ_g = 1.5 × 10⁻³ μm is the geometrical standard deviation and r_g = 0.1 μm is the mean radius. An improved DLA, which creates clusters with monomers of non-uniform radius, was applied [36]. Finally, the Al-containing cluster model (ASM) and the C-containing Al₂O₃ cluster model (ACM) were proposed, as shown in Figure 3a,b.

2.1.3. Methods of Simplifying Cluster to Sphere

To reduce the amount of computation, Al₂O₃ clusters were usually simplified as homogeneous spheres [21,22]. The equivalent volume sphere model (EVSM) and equivalent surface sphere model (ESSM), as shown in Figure 3c,d, were used to represent these simplified sphere models. The effective radius of the equivalent sphere by EVSM and ESSM is described by:

$$r_{\mathrm{EVSM}}=\sqrt[3]{{\displaystyle \sum _{i=1}^{N\mathrm{s}}{r}_i^3}},$$

(7)

$$r_{\mathrm{ESSM}}=\sqrt[2]{{\displaystyle \sum _{i=1}^{N\mathrm{s}}{r}_i^2}},$$

(8)

where Ns is the monomer number and r_i is the radius of i_th monomer in the cluster. The effective refractive index of the equivalent sphere is calculated by effective-medium theory of Maxwell–Garnett [37] illustrated below:

$$\frac{\epsilon -{\epsilon }_{{\mathrm{Al}}_2{\mathrm{O}}_3}}{\epsilon +2{\epsilon }_{{\mathrm{Al}}_2{\mathrm{O}}_3}}=V_{\mathrm{Al}}\frac{{\epsilon }_{\mathrm{Al}}-{\epsilon }_{{\mathrm{Al}}_2{\mathrm{O}}_3}}{{\epsilon }_{\mathrm{Al}}+2{\epsilon }_{{\mathrm{Al}}_2{\mathrm{O}}_3}}$$

(9)

$$\frac{\epsilon -{\epsilon }_{\mathrm{C}}}{\epsilon +2{\epsilon }_{\mathrm{C}}}=V_{\mathrm{C}}\frac{{\epsilon }_{{\mathrm{Al}}_2{\mathrm{O}}_3}-{\epsilon }_{\mathrm{C}}}{{\epsilon }_{{\mathrm{Al}}_2{\mathrm{O}}_3}+2{\epsilon }_{\mathrm{C}}}$$

(10)

where ε is the effective permittivity, m is refractive index and $m=\sqrt{\epsilon }$, and ${\epsilon }_{{\mathrm{Al}}_2{\mathrm{O}}_3}$, ${\epsilon }_{\mathrm{Al}}$ and ${\epsilon }_{\mathrm{C}}$ are permittivity of Al₂O₃, Al, and C, respectively.

2.2. STMM

Traditional Mie theory is unsuitable for clusters. STMM was applied to obtain the optical characteristics of Al₂O₃ clusters. STMM is an improved T-matrix [38], and is a direct computer solver for the frequency-domain macroscopic Maxwell equations involving no approximations, and is extensively used to obtain the electromagnetic optical properties for composite particles [39]. Furthermore, STMM was extended to the arbitrary configuration of spheres located internally or externally to other spheres, which was suitable in this study. The incident and scattering field expansion are shown below [40]:

$${\mathrm{E}}_{\mathrm{inc}}(r)={\displaystyle \sum _{\mathrm{n}=1}^{\mathrm{L}}{\displaystyle \sum _{\mathrm{m}=-\mathrm{n}}^{\mathrm{n}}{\displaystyle \sum _{\mathrm{p}=1}^2{\mathrm{f}}_{\mathrm{mnp}}^0}}}{\mathrm{N}}_{\mathrm{nmp}}^{\left(1\right)}(kr)$$

(11)

$${\mathrm{E}}_{\mathrm{sca}}(r)={\displaystyle \sum _{\mathrm{n}=1}^{\mathrm{L}}{\displaystyle \sum _{\mathrm{m}=-\mathrm{n}}^{\mathrm{n}}{\displaystyle \sum _{\mathrm{p}=1}^2{\mathrm{a}}_{\mathrm{mnp}}^0}}}{\mathrm{N}}_{\mathrm{nmp}}^{\left(3\right)}(kr)$$

(12)

where ${\mathrm{N}}_{\mathrm{nmp}}^{\left(1\right)}(kr)$ and ${\mathrm{N}}_{\mathrm{nmp}}^{\left(3\right)}\left(kr\right)$ are vector spherical wave functions with degree m, order n, and mode p. $k$ is the wavenumber. $a_{mnp}^0$ is the scattered field coefficient and $f_{mnp}^0$ is the incident field coefficient. ${\mathrm{E}}_{\mathrm{inc}}(r)$ and ${\mathrm{E}}_{\mathrm{sca}}\left(r\right)$ are incident and scattered fields, respectively. Scattering and absorption cross-sections are described as [40]:

$$C_{\mathrm{ext}}^{{\mathrm{Al}}_2{\mathrm{O}}_3}=\frac{\pi }{k^2}{\displaystyle \sum _{n=1}^L{\displaystyle \sum _{m=-n}^n{\displaystyle \sum _{p=1}^2{a}_{\mathrm{mnp}}^0{f}_{\mathrm{mnp}}^{0_{*}}}}}$$

(13)

$$C_{\mathrm{sca}}^{{\mathrm{Al}}_2{\mathrm{O}}_3}=\frac{\pi }{k^2}{\displaystyle \sum _{n=1}^L{\displaystyle \sum _{m=-n}^n{\displaystyle \sum _{p=1}^2{\left|a_{\mathrm{mnp}}^0\right|}^2}}}$$

(14)

$$C_{\mathrm{ext}}^{{\mathrm{Al}}_2{\mathrm{O}}_3}=C_{\mathrm{abs}}^{{\mathrm{Al}}_2{\mathrm{O}}_3}+C_{\mathrm{sca}}^{{\mathrm{Al}}_2{\mathrm{O}}_3}$$

(15)

where $C_{\mathrm{ext}}^{{\mathrm{Al}}_2{\mathrm{O}}_3}$, $C_{\mathrm{sca}}^{{\mathrm{Al}}_2{\mathrm{O}}_3}$, and $C_{\mathrm{abs}}^{{\mathrm{Al}}_2{\mathrm{O}}_3}$ are extinction, scattering, and absorption cross-sections, respectively.

In the standard [I Q U V] representation of polarization state and angular distribution, the normalized Stokes scattering matrix has the block-diagonal structure [41]:

$$\tilde{P}(\theta )=\left[\begin{array}{cccc}{P}_{11}^{{\mathrm{Al}}_2{\mathrm{O}}_3}(\theta )& P_{12}(\theta )& 0& 0\\ P_{12}(\theta )& P_{22}(\theta )& 0& 0\\ 0& 0& P_{33}(\theta )& P_{34}(\theta )\\ 0& 0& -P_{34}(\theta )& P_{44}(\theta )\end{array}\right],$$

(16)

where $0\le \theta \le 180°$, $P_{11}^{{\mathrm{Al}}_2{\mathrm{O}}_3}(\theta )$ is called scattering phase function and describes the spatial distribution of the light scattering energy of the particle, and satisfies the normalization condition:

$$\frac{1}{2}{\displaystyle \underset{0}{\overset{\pi }{\int }}{P}_{11}^{{\mathrm{Al}}_2{\mathrm{O}}_3}(\theta )}\sin \theta d\theta =1.$$

(17)

A detailed description of STMM is provided by Mackowski [42]. Many studies averaged over five or ten realizations of cluster particles to reflect their general optical properties [43,44,45]. Therefore, by selecting ten clusters of Al₂O₃ clusters, we aimed to ensure a manageable computational load while capturing the fractal characteristics of the cluster particles.

2.3. Mass Burden and Accumulation Area of Al₂O₃ Particles in the Stratosphere

2.3.1. Mass Burden

Currently, the mass burden of Al₂O₃ particles in the stratosphere is smaller than that of sulfate particles. With the expected increase in space exploration and commercial launch activities, the quantity of Al₂O₃ particles will continue to grow. However, the detailed data on the mass burden of Al₂O₃ particles in the stratosphere are challenging to obtain, due to several reasons, such as the low frequency of launches, technical complexity, confidentiality, security, and challenges in data collection [19,20]. Therefore, the mass burden of Al₂O₃ particles has been estimated based on various assumptions in prior research. Danilin et al. [13] supposed the annual emission of Al₂O₃ particles includes nine space shuttles and four Titan IV launches, which was 1.3 times the actual emission of the past decade, releasing approximately 3.9 kt yr⁻¹ Al₂O₃ particles into the atmosphere, with 1.12 kt yr⁻¹ reaching the stratosphere directly. Ross et al. [21] formulated an equation to estimate the mass burden, which was related to the total mass of propellant used per launch, the annual launch frequency, the duration particles remain in the atmosphere, and the emission indices. According to their results, an accumulation of 1.2 kt of Al₂O₃ particles is in the stratosphere. Later, Weisenstein et al. [22] suggested an injection rate of 1~8 Tg yr⁻¹. Actually, Al₂O₃ is an important mineral, globally produced at a rate of approximately 300 Tg yr⁻¹ [46]. There is a large body of experience in making Al₂O₃ particles. For example, liquid-feed flame spray pyrolysis can be used to make Al₂O₃ particles greater than 1 kt yr⁻¹. This study focuses on how impurities and morphology affect the optical characteristics and radiative forcing of Al₂O₃, rather than on precise quantification. Therefore, this paper assumes an injection rate of 1 Tg yr⁻¹ of submicron Al₂O₃ clusters into the stratosphere, with an atmospheric lifetime of four years for these particles [12], implying a total stratospheric mass burden of 4 Tg. We ignored the Al₂O₃ clusters in the troposphere because particles in the troposphere are more quickly removed by gravitational settling and rainout/washout [13,21].

2.3.2. Accumulation Aera

Danilin et al. [13] utilized the Goddard Institute for Space Studies/University of California at Irvine three-dimensional chemistry-transport model to simulate the atmospheric accumulation of Al₂O₃ particles. Their simulation accounted for the mass density variations of Al₂O₃ particles by considering factors such as particle washout, gravitational sedimentation, and interaction with background sulfate aerosol. To narrow the scope of the assessment, the interaction of Al₂O₃ and sulfate is not considered here. Focusing on scenarios involving washout and gravitational sedimentation, labeled as case B in their research, they found that the space distribution of Al₂O₃ predominantly resides between 10 and 30 km altitude and within 10° N–90° N latitude. Additionally, they verified the plausibility of a uniform distribution of Al₂O₃ particles. They did this by comparing the outcomes of pulsed emissions, which were injected every 40 days, with continuous emissions throughout the year. Both methods maintained the same annual emission rate. Consequently, we assume that Al₂O₃ particles are uniformly distributed within the specified range of 10 to 30 km altitude and 10° N–90° N latitude.

2.4. Radiative Forcing

2.4.1. The Method for Radiative Forcing by Impurity-Containing Al₂O₃ Clusters

MODTRAN [47], provides simulations of spectral radiative flux through Earth’s atmosphere and takes into account the atmospheric composition, altitude, and the presence of clouds, was widely used in the field of remote sensing [48,49]. MODTRAN contains standard values of all input parameters, and it allows for the integration of experimental measurements to closely mirror actual atmospheric conditions. Consequently, we have proposed a methodology that integrates the Al/C-containing Al₂O₃ cluster models with the MODTRAN radiative transfer code. This integration aims to derive remote sensing data of flux at top-of-atmosphere, subsequently facilitating the calculation of the radiative forcing by Al₂O₃.

The algorithm for radiative forcing by Al₂O₃ clusters was described herein. The monochromatic radiant intensity $I\left(\mathsf{\Omega };\lambda \right)$ at spectral wavelength $\lambda$ and in the direction $\mathsf{\Omega }$ is the sum of two parts [50]:

$$I\left(\mathsf{\Omega };\lambda \right)=\exp \left[-\tau (\lambda )\right]I\left(\mathsf{\Omega };\lambda \right)+{\int }_0^{\tau (\lambda )}\exp \left[-\tau ({\lambda }^{\prime })\right]J\left(\mathsf{\Omega };{\lambda }^{\prime }\right)d\tau ({\lambda }^{\prime })\tau (\lambda )J\left(\mathsf{\Omega };\lambda \right)$$

(18)

where $\tau (\lambda )$ is optical depth and $J\left(\mathsf{\Omega };\lambda \right)$ is the atmospheric source term. These two terms are described in detail below.

The optical depth $\tau (\lambda )$ is determined by the path integrals over extinction coefficients, symbolized by ${\kappa }_{\mathrm{ext}}(\lambda )$. ${\kappa }_{\mathrm{ext}}(\lambda )$ is the product of an extinction cross-section $C_{\mathrm{ext}}(\lambda )$ and density $\rho$. Additionally, ${\kappa }_{\mathrm{ext}}(\lambda )$ can also be described as the sum of absorption coefficients ${\kappa }_{\mathrm{abs}}(\lambda )$ and scattering coefficients ${\kappa }_{\mathrm{sca}}(\lambda )$:

$$\tau (\lambda )={\int }_0^{\mathrm{top}\text{-}\mathrm{of}\text{-}\mathrm{atmosphere}}{\kappa }_{\mathrm{ext}}^{\mathcal{l}}(\lambda )d\mathcal{l}$$

(19)

$${\kappa }_{\mathrm{ext}}(\lambda )={\kappa }_{\mathrm{abs}}(\lambda )+{\kappa }_{\mathrm{sca}}(\lambda )$$

(20)

$${\kappa }_{\mathrm{ext}}(\lambda )=C_{\mathrm{ext}}(\lambda )\rho ,{\kappa }_{\mathrm{abs}}(\lambda )=C_{\mathrm{abs}}(\lambda )\rho ,{\kappa }_{\mathrm{sca}}(\lambda )=C_{\mathrm{sca}}(\lambda )\rho$$

(21)

Combined with Equation (15), the overall extinction cross-section $C_{\mathrm{ext}}(\lambda )$ can be expressed as follows:

$$\begin{array}{ll}{C}_{\mathrm{ext}}(\lambda )& =C_{\mathrm{ext}}^{{\mathrm{Al}}_2{\mathrm{O}}_3}(\lambda )+C_{\mathrm{ext}}^{\mathrm{gas}}(\lambda )\\ & =C_{\mathrm{abs}}^{{\mathrm{Al}}_2{\mathrm{O}}_3}(\lambda )+C_{\mathrm{sca}}^{{\mathrm{Al}}_2{\mathrm{O}}_3}(\lambda )+C_{\mathrm{ext}}^{\mathrm{gas}}(\lambda )\end{array}$$

(22)

The atmospheric source term $J\left(\mathsf{\Omega };\lambda \right)$ includes three distinct components:

$$J\left(\mathsf{\Omega };\lambda \right)=J_{em}\left(\mathsf{\Omega };\lambda \right)+J_{ss}\left(\mathsf{\Omega };\lambda \right)+J_{ms}\left(\mathsf{\Omega };\lambda \right)$$

(23)

the first term in Equation (23), the local thermal emission $J_{em}\left(\mathsf{\Omega };\lambda \right)$, is the product of the Planck blackbody function $B\left(T;\lambda \right)$ and the local emissivity, which is represented as a ratio of absorption to extinction coefficients. The second term, the directly transmitted single scattered solar irradiance $J_{ss}\left(\mathsf{\Omega };\lambda \right)$, is the product of the top-of-atmosphere solar irradiance $F^{sun}(\lambda )$, the scattering point to sun transmittance $\exp \left[-{\tau }_{sun}(\lambda )\right]$, the effective scattering phase function $P_{11}^{{\mathrm{Al}}_2{\mathrm{O}}_3}\left(\mathsf{\Omega },{\mathsf{\Omega }}^{sun};\lambda \right)$ as in Equation (16). The last term, the multiply scattered radiation $J_{ms}\left(\mathsf{\Omega };\lambda \right)$, is an integral over all directions (4π steradians) of the incoming diffuse radiation scattered into the path:

$$\begin{array}{l}{J}_{em}\left(\mathsf{\Omega };\lambda \right)=\frac{{\kappa }_{\mathrm{abs}}^{\mathrm{gas}}(\lambda )+{\kappa }_{\mathrm{abs}}^{{\mathrm{Al}}_2{\mathrm{O}}_3}(\lambda )}{{\kappa }_{\mathrm{ext}}(\lambda )}B\left(T;\lambda \right)\\ J_{ss}\left(\mathsf{\Omega };\lambda \right)=\frac{{\kappa }_{\mathrm{sca}}^{{\mathrm{Al}}_2{\mathrm{O}}_3}(\lambda )}{{\kappa }_{\mathrm{ext}}(\lambda )}{P}_{11}^{{\mathrm{Al}}_2{\mathrm{O}}_3}\left(\mathsf{\Omega },{\mathsf{\Omega }}^{sun};\lambda \right)\exp \left[-{\tau }_{sun}(\lambda )\right]F^{sun}(\lambda )\\ J_{ms}\left(\mathsf{\Omega };\lambda \right)=\frac{{\kappa }_{\mathrm{sca}}^{{\mathrm{Al}}_2{\mathrm{O}}_3}(\lambda )}{{\kappa }_{\mathrm{ext}}(\lambda )}\underset{4\pi }{{\displaystyle \int }}{P}_{11}^{{\mathrm{Al}}_2{\mathrm{O}}_3}\left(\mathsf{\Omega },{\mathsf{\Omega }}^{\prime };\lambda \right)I_{\lambda }\left({\mathsf{\Omega }}^{\prime };\lambda \right)d{\mathsf{\Omega }}^{\prime }\end{array}$$

(24)

Then, combining with Equation (18), the top-of-atmosphere flux $F$ can be obtained by:

$$F={\displaystyle {\int }_{\lambda }{\displaystyle {\int }_{\Omega }I\left({\Omega }^{\prime };{\lambda }^{\prime }\right)d{\Omega }^{\prime }d{\lambda }^{\prime }}}$$

(25)

Al₂O₃ clusters, as solid aerosols, have the capability to scatter and absorb incoming solar radiation. Al₂O₃ has a strong absorption band between 10 and 15 μm, enabling it to absorb upwelling infrared radiation from the Earth [51]. Thus, the instantaneous shortwave and longwave radiative forcing were both considered here, which is integrated over 0.34 μm to 30 μm. Rather than relaxed radiative forcing, which provides accurate climate forcing by considering the atmospheric response, instantaneous radiative forcing indicates the immediate perturbations by aerosols in the atmosphere, offering a straightforward approach to assessing their impact. The instantaneous radiative forcing (RF) at the top of the atmosphere was defined by [52]:

$$RF=(F_{{\mathrm{Al}}_2{\mathrm{O}}_3}^{\downarrow }-F_{{\mathrm{Al}}_2{\mathrm{O}}_3}^{\uparrow })-(F_0^{\downarrow }-F_0^{\uparrow })$$

(26)

where $F_{{\mathrm{Al}}_2{\mathrm{O}}_3}^{\downarrow }$ and $F_0^{\downarrow }$ are the downward flux with and without Al₂O₃ clusters at top of the atmosphere, respectively. $F_{{\mathrm{Al}}_2{\mathrm{O}}_3}^{\uparrow }$ and $F_0^{\uparrow }$ are the same quantities for the upward flux. It can be noticed that at the top of the atmosphere $F_{{\mathrm{Al}}_2{\mathrm{O}}_3}^{\downarrow }\equiv F_0^{\downarrow }$, combined with Equation (25), the RF is defined as:

$$\begin{array}{ll}RF& =F_0^{\uparrow }-F_{{\mathrm{Al}}_2{\mathrm{O}}_3}^{\uparrow }\\ & ={\displaystyle {\int }_{\lambda }{\displaystyle {\int }_{\Omega }{I}_0\left({\Omega }^{\prime };{\lambda }^{\prime }\right)d{\Omega }^{\prime }d{\lambda }^{\prime }}}-{\displaystyle {\int }_{\lambda }{\displaystyle {\int }_{\Omega }{I}_{{\mathrm{Al}}_2{\mathrm{O}}_3}\left({\Omega }^{\prime };{\lambda }^{\prime }\right)d{\Omega }^{\prime }d{\lambda }^{\prime }}}\end{array}$$

(27)

where $I_{{\mathrm{Al}}_2{\mathrm{O}}_3}\left(\Omega ;\lambda \right)$ is the radiant intensity with Al₂O₃ clusters in the stratosphere, and $I_0\left(\Omega ;\lambda \right)$ is the same quantity without Al₂O₃ clusters along the path, that means $C_{\mathrm{ext}}^{{\mathrm{Al}}_2{\mathrm{O}}_3}(\lambda )=0$ in Equations (18)–(24).

The radiative forcing is influenced by chemico-physical properties of Al₂O₃, like monomer number, phase, refractive index, impurity type, size distribution, morphology, and the mass burden [22]. Despite the complexity of these factors, our goal is to preliminarily estimate the effects of Al/C and morphology on their optical characteristics and radiative forcing. To focus on this objective and ensure that the equivalent radius of the Al₂O₃ cluster remains within the submicron range so that they can sustain in the stratosphere for long time, we assume the Al₂O₃ clusters are composed of five monomers. This assumption allows us to simplify the model while still capturing the essential features relevant to our study.

2.4.2. Validation of the Method

To illustrate the spectral behavior of the Al₂O₃, Ross presented the top-of-atmosphere flux comparisons, showing scenarios with and without Al₂O₃ particles, as depicted in Figure 4a. The refractive indexes of Al₂O₃ along 6.25 μm to 30 μm are in Begemann et al. [51], and the distribution properties of Al₂O₃ is in Danilin [53]. Utilizing the configuration provided by Ross, Figure 4b displays the simulation outcomes derived from the method we proposed for investigating the radiative forcing by impurity-containing Al₂O₃ clusters. The overall flux trend is consistent with that in Figure 4a. This consistency validates the approach we have proposed. The minor differences observed between the two figures in Figure 4 are attributed to certain unelaborated configurations in the study by Ross.

In our calculation, we assumed clear sky conditions. The 1976 US standard atmosphere model and the discrete ordinate algorithm with 16 streams were used to describe multiple scattering. The integration path for our calculation is set to extend vertically from the ground to the top of the atmosphere.

3. Results and Discussion

The optical characteristics and radiative forcing of Al-containing and C-containing Al₂O₃ clusters were investigated in this section. To understand the discrepancy produced by simplifying these clusters to homogenous sphere, EVSM and ESSM were calculated as well.

3.1. Optical Characteristics

Two classical wavelengths, 0.55 μm (shortwave) and 10 μm (longwave), were chosen to analyze the optical characteristics of impurity-containing Al₂O₃ clusters, the same as Frol’kis et al. [54]. This selection enables a preliminary evaluation of the general optical characteristics in shortwave and longwave regions. The refractive indices of Al₂O₃, Al and C at 0.55 μm and 10 μm are in

Table 1. The refractive index of Al₂O₃, Al and C at 0.55 and 10 μm.

Wavelength (μm)		0.55	10
Refractive	Al2O3	1.77 + 1.65 × 10−7i	0.994 + 0.28i
	Al	1.015 + 6.63i	25 + 85.96i
	C	1.917 + 0.41i	3.67 + 3.03i

[28,29,51]. These refractive indexes can vary in the literature depending on the measurement techniques and conditions. However, it is well-established that the imaginary part of the refractive index for Al₂O₃ is significantly lower than that of Al and C. This indicates that Al₂O₃ has much lower absorption characteristics compared to the other two materials.

3.1.1. Optical Characteristics of Al-containing Al₂O₃ Clusters

Figure 5a presents the optical properties of Al-containing Al₂O₃ clusters. At 0.55 μm, C_abs for ASM increases by approximately five orders of magnitude as V_Al rises from 0 to 0.1, then stabilizes as V_Al continues to increase. Conversely, at 10 μm, C_abs initially rises and then slightly decreases with increasing V_Al. For pure Al₂O₃ clusters (V_Al = 0), the C_abs at 10 μm is higher than at 0.55 μm, indicating predominant absorption in the longwave band. However, this reverses when V_Al exceeds 0.1, highlighting a significant boost in absorption at 0.55 μm due to Al impurities. The C_sca is shown in the second row of Figure 5. C_sca at 0.55 μm considerably exceeds that at 10 μm, indicating that Al₂O₃’s scattering predominantly occurs in the shortwave band. The scattering at 0.55 μm is significantly higher, while it is almost negligible at 10 μm. Additionally, the equivalent sphere models, EVSM and ESSM, represented as dashed lines in Figure 5, show notable deviations from the ASM, and these discrepancies vary with V_Al. EVSM/ESSM models tend to underestimate the scattering ability at 0.55 μm and the absorption ability at 10 μm compared to ASM, highlighting the impact of equivalent sphere models.

The phase function ($P_{11}^{{\mathrm{Al}}_2{\mathrm{O}}_3}$) of ASM as a function of scattering angle at 0.55 and 10 μm are shown in Figure 6, displaying distinct behaviors at two wavelengths. At 0.55 μm, with increasing V_Al, the trend of $P_{11}^{{\mathrm{Al}}_2{\mathrm{O}}_3}$ initially shows a decrease and then an increase in forward scattering, while backward scattering demonstrates an opposite trend—initially increasing and then decreasing. The magnitude of forward scattering predominates over backward scattering, and the curve varies smoothly with the scattering angle, showing only slight changes with different V_Al. Conversely, at 10 μm, $P_{11}^{{\mathrm{Al}}_2{\mathrm{O}}_3}$ changes dramatically and regularly with the increase in V_Al. There is a consistent decrease in forward scattering and an increase in backward scattering as V_Al increases, with backward scattering eventually surpassing forward scattering. The different behavior of $P_{11}^{{\mathrm{Al}}_2{\mathrm{O}}_3}$ at 0.55 and 10 μm is due to their different chemico-physical properties. Obviously, the effects of Al on $P_{11}^{{\mathrm{Al}}_2{\mathrm{O}}_3}$ is stronger at 10 μm than that at 0.55 μm.

3.1.2. Optical Characteristics of C-Containing Al₂O₃ Clusters

Figure 7 and Figure 8 show the optical properties of C-containing Al₂O₃ clusters at wavelengths of 0.55 and 10 μm, encompassing C_abs, C_sca, and $P_{11}^{{\mathrm{Al}}_2{\mathrm{O}}_3}$. In Figure 7, C_abs of ACM is directly proportional to V_C. The addition of C significantly enhances the C_abs of ACM about four orders of magnitude at 0.55 μm, whereas at 10 μm, the increase is less pronounced, growing less than twice the initial value. The C_sca of ACM inversely correlates with V_C at 0.55 μm, whereas it shows a direct proportionality at 10 μm. Quantitatively, the C_sca at 0.55 μm is substantially higher, making the scattering at 10 μm negligible in comparison. The calculations for C_abs and C_sca using EVSM and ESSM also display distinctions from the ACM, particularly at 0.55 μm, where the C_sca is significantly underestimated by the equivalent sphere models.

Figure 8 illustrates $P_{11}^{{\mathrm{Al}}_2{\mathrm{O}}_3}$ of ACM as a function of scattering angle at wavelengths of 0.55 and 10 μm. The small figures in Figure 8 are an enlarged view of the scattering phase function at scattering angles of 0 degree (forward scattering) and 180 degree (backward scattering). The impact of C on $P_{11}^{{\mathrm{Al}}_2{\mathrm{O}}_3}$ is relatively minor when compared to Al. The detailed analysis, particularly focusing on the enlarged views of forward and backward scattering, reveals that $P_{11}^{{\mathrm{Al}}_2{\mathrm{O}}_3}$ changes consistently with V_C. For both 0.55 and 10 μm wavelengths, there is an increase in forward scattering and a decrease in backward scattering as V_C increases. This pattern suggests that the optical properties of Al₂O₃ clusters exhibit a sensitivity to the presence of C impurity.

Above all, the optical properties of Al₂O₃ particles are significantly influenced by Al/C impurities and morphology models, showing distinct behaviors at wavelength of 0.55 and 10 μm. The addition of Al and C notably enhances the absorption of Al₂O₃ clusters at 0.55 μm. Equivalent sphere models, EVSM and ESSM, differ markedly from actual cluster models. That is because of the difference in morphology between cluster and sphere. It is clear to understand it when we treat the cluster as a collection of dipoles. The radius of the equivalent sphere is larger than that of monomer in the cluster. The dipoles in a sphere are closer to each other than those in the cluster, therefore the interaction is significantly different, which leads to a different scattering pattern in the far field. Accurate assessment of how impurity and morphology affect optical characteristics is crucial for comprehending their impacts on radiative forcing.

3.2. Radiative Forcing

Following the methodology outlined in Section 2.4.1, the radiative forcing of Al-containing and C-containing Al₂O₃ clusters, alongside their equivalent sphere model counterparts, are calculated in this section. To elucidate the influence of impurities on radiative forcing, the relative difference (RD) is introduced as a metric, defined by equation:

$$RD=\frac{R{F}_{\mathrm{pure}}-R{F}_{\mathrm{impurity}}}{R{F}_{\mathrm{pure}}}\times 100\%$$

(28)

$R{F}_{\mathrm{pure}}$ represents the radiative forcing of pure Al₂O₃, where V_Al/C = 0. Conversely, $R{F}_{\mathrm{impurity}}$ denotes the radiative forcing of Al₂O₃ containing Al/C impurities, characterized by V_Al/C > 0.

3.2.1. Radiative Forcing of Al-Containing Al₂O₃ Clusters

Figure 9a depicts the radiative forcing of Al-containing Al₂O₃ clusters (ASM) and the corresponding equivalent sphere models (EVSM and ESSM) as a function of V_Al. The radiative forcing of ASM exhibits a nuanced response to changes of V_Al, initially weakening and then strengthening, with values ranging from −0.439 to −0.375 W/m². This is supported by the observations in Figure 5. The C_sca is a major contributor, dictating the scattering-driven forcing (negative value). The C_abs at the shortwave intensifies by five orders of magnitude as V_Al increases, offsetting the scattering forcing. However, beyond V_Al of 0.1, the rise in the C_abs slows, and the C_sca increases at 0.55 μm, thereby enhancing the scattering forcing. The presence of Al is evidently important in the analysis of radiative forcing of Al₂O₃ clusters. Moreover, the radiative forcing of EVSM and ESSM are markedly diverge from those of ASM, due to the morphological differences between cluster and sphere. Figure 9b reveals that the maximum RD of ASM reaches 14.6%. The RD for EVSM and ESSM exceed those for ASM, suggesting that sphere models might introduce greater discrepancies. Clearly, the radiative forcing of Al₂O₃ clusters is highly sensitive to both Al and the chosen morphological model.

3.2.2. Radiative Forcing of C-Containing Al₂O₃ Clusters

Similar to Figure 9, Figure 10a displays the radiative forcing for C-containing Al₂O₃ clusters (ACM) and the corresponding equivalent sphere models as a function of V_C. With an increase in V_C, the radiative forcing of ACM tends to weaken, with values ranging from −0.439 to −0.375 W/m². Negative forcing means the scattering radiation plays key roles. Referencing Figure 7, at 0.55 μm, the addition of C results in a decrease in C_sca and a substantial increase in C_abs, potentially leading to diminished scattering forcing. EVSM and ESSM are observed to generate stronger scattering forcing compared to ACM. The RD OF ESSM is highest, followed by EVSM, with ACM being the smallest. Overall, the influence of C and the choice of morphology model are crucial factors that cannot be overlooked.

Overall, the radiative forcing of Al₂O₃ clusters is notably influenced by the presence of Al/C impurities and the morphological model. There is a strong correlation between the clusters’ radiative forcing and their optical characteristics. The findings indicate that adding Al/C impurities tends to reduce the scattering forcing of the Al₂O₃ clusters. Even though both Al and C are effective absorbers, their radiative forcing shows significant differences. Additionally, comparing the equivalent sphere models with the precise cluster models reveals considerable disparities, not only in the magnitude of radiative forcing but also in their sensitivity to impurities. However, if simplification is necessary to reduce computational demands, EVSM is recommended, as it introduces relatively smaller discrepancies compared to ESSM.

4. Conclusions

To accurately simulate the optical characteristics and radiative forcing of Al₂O₃ clusters emitted by solid rockets in the stratosphere, precise morphology models of Al/C-containing Al₂O₃ clusters were developed. Then, by combining these models with the MODTRAN radiative transfer code, a method was proposed to estimate the radiative forcing of these clusters. The STMM was employed to calculate their optical properties. Additionally, equivalent sphere models were evaluated to assess their deviations from the actual cluster models. The findings underscore that both the optical characteristics and radiative forcing of Al₂O₃ clusters are significantly influenced by Al/C impurities and the morphology models.

The sensitivity factors influencing the optical characteristics of Al₂O₃ clusters include wavelength, impurities, and morphology models. The introduction of Al/C impurities significantly increases the C_abs of Al₂O₃ clusters at 0.55 μm by several orders of magnitude, whereas the increase is less pronounced at 10 μm. Conversely, the C_sca decreases at 0.55 μm but increases at 10 μm. This indicates wavelength-dependent variations in optical behavior. Obviously, Al/C greatly enhances the absorption ability and weakens the scattering ability at 0.55 μm. The optical characteristics of ASM and ACM, corresponding to Al-containing and C-containing Al₂O₃ clusters, exhibit distinct behaviors. In addition, the impact of Al/C impurities and morphology on radiative forcing is pronounced. A 4 Tg mass burden of Al₂O₃ clusters could result in a radiative forcing of −0.439 W/m². However, adding Al and C impurities reduces this forcing. The minimum values with Al and C impurities are −0.375 W/m² and −0.387 W/m², respectively. Ignoring these impurities could lead to a maximum RD of 15% in ASM and 12% in ACM. Additionally, EVSM and ESSM introduce significant deviations when analyzing the optical characteristics and radiative forcing of Al₂O₃ clusters. Therefore, it is essential to consider the impurities and morphology of Al₂O₃ clusters in studies of their optical properties and radiative forcing.

While the impurity-containing Al₂O₃ cluster models used in this study provide a closer representation of real Al₂O₃ particles, they do not capture the full complexity. Factors include varying phase states, multiple impurities, intricate morphology, and interactions with other atmospheric substances and so on. Therefore, due to the many uncertainties involved, the main goal of this research was to offer a preliminary analysis of how Al/C impurities and morphology affect the optical characteristics and radiative forcing of Al₂O₃ particles in the stratosphere, rather than to provide definitive results. Further exploration in this area is essential. The insights gained from this study provide valuable knowledge on the optical properties and radiative impact of Al₂O₃ particles. This understanding is important for examining the effects of space activities on climate change and for developing strategies to reduce environmental risks associated with rocket emissions.