Correlation among extinction efficiency and other parameters in an aggregate dust model
Tanuj Kumar Dhar Himadri Sekhar Das


   


  2017 Vol. X No. XX, 000–000
   


   




   Department of Physics, Assam University, Silchar 788011, India; hsdas13@gmail.com



Received 2017 Feb XX; accepted 2017 XXXX


We study the extinction properties of highly porous BCCA dust aggregates in a wide range of complex refractive indices (1.4 ≤ n ≤ 2.0, 0.001 ≤ k ≤ 1.0) and wavelength (0.11μ m ≤λ≤ 3.4μ m). An attempt has been made for the first time to investigate the correlation among extinction efficiency (Q_ext), the composition of dust aggregates (n, k), the wavelength of radiation (λ) and size parameter of the monomers (x). If k is fixed at any value between 0.001 and 1.0, Q_ext increases with increase of n from 1.4 to 2.0. Q_ext and n are correlated  via linear regression when the cluster size is small whereas the correlation is quadratic at moderate and higher sizes of the cluster. This feature is observed at all wavelengths (UV to optical to infrared). We also find that the variation of Q_ext with n is very small when λ is high. When n is fixed at any value between 1.4 and 2.0, it is observed that  Q_ext and k are correlated  via polynomial regression equation (of degree 1, 2, 3 or 4), where the degree of the equation depends on the cluster size, n and λ. The correlation  is linear for small size and quadratic/cubic/quartic for moderate and higher sizes. We have also found that Q_ext and x are correlated via a polynomial regression (of degree 3,4 or 5) for all values of n. The degree of regression is found to be n and k-dependent. The set of relations obtained from our work can be used to model interstellar extinction for dust aggregates in a wide range of wavelengths and complex refractive indices.



   Dhar & Das             
   Correlation among extinction efficiency and other parameters  
   Correlation among extinction efficiency and other parameters in an aggregate dust model
    Tanuj Kumar Dhar and Himadri Sekhar Das
   
    December 30, 2023
=======================================================================================








§ INTRODUCTION

The studies of cometary and interplanetary dust indicate that cosmic dust grains are likely to be fluffy, porous and
composites of many small grains fused together, due to dust-gas interactions, grain-grain collisions,
and various other processes <cit.>. Porous, composite aggregates are often modelled as cluster of small spheres (known as “monomers"), agglomerated under various aggregation rules. Here grain aggregates are assumed to be fluffy sub structured collections of very small particles loosely attached to one another. Each particle is assumed to consist of a single material, such as silicates or carbon, as formed in the  various separate sources of cosmic dust. Extinction generally takes place whenever electromagnetic radiation propagates through a medium containing small particles.  The spectral dependence of extinction, or extinction curve, is a function of the structure, composition, and size distribution of the particles. The study of interstellar extinction provides us useful information for understanding the properties of the dust.

It is now well accepted from observation and laboratory analysis of interplanetary dust particles that cosmic dust grains are fluffy aggregates or porous with irregular shapes <cit.>. Using Discrete Dipole Approximation (DDA) technique, several investigators studied the extinction properties of the composite grains <cit.>. <cit.> studied optical properties of composite grains as grain aggregates of amorphous carbon and astronomical silicates, using the Superposition transition matrix approach. Recently, <cit.> studied the light scattering properties of aggregate particles in a wide range of complex refractive indices and wavelengths to investigate the correlation among different parameters e.g., the positive polarization maximum, the amplitude of the negative polarization, geometric albedo, refractive indices and wavelength. The simulations were performed using the Superposition T-matrix code with Ballistic ClusterCluster Aggregate (BCCA) particles of 128 monomers and Ballistic Aggre-
gates (BA) particles of 512 monomers.

The extinction efficiency of dust aggregates depends on aggregate size, composition and wavelength of incident radiation. The dependence of complex refractive index (n, k) on Q_ext was studied by many groups in past for spherical and irregular particles using different scattering theories (Mie theory, DDA approach, T-matrix theory etc.). But no correlation equations were reported earlier by any group. In this paper, we study the extinction properties of randomly oriented porous dust aggregates with a wide range of complex refractive indices and wavelength of incident radiation. An attempt has been made for the first time to investigate the correlation among extinction efficiency (Q_ext), complex refractive indices (m = n + ik), wavelength (λ) and the size parameter of monomer (x).



§ NUMERICAL COMPUTATIONS

We have constructed the aggregates using ballistic
aggregation procedure <cit.> using two different
models of cluster growth. First via single-particle aggregation
and then through cluster-cluster aggregation. These aggregates are
built by random hitting and sticking particles together. The first
one is called Ballistic Particle-Cluster Aggregate (BPCA) when the
method allows only single particles to join the cluster of
particles. If the method allows clusters of particles to stick
together, the aggregate is called Ballistic Cluster-Cluster
Aggregate (BCCA). Actually, the BPCA clusters are more compact
than BCCA clusters <cit.>. The porosity of BPCA and
BCCA particles of 128 monomers has the values 0.90 and 0.94,
respectively. The fractal dimensions of BPCA and BCCA particles are given by D  3 and 2, respectively (Meakin 1984). A systematic explanation on dust aggregate model is
already discussed in our previous work <cit.>. It is to be noted that the
structure of these aggregates are similar to those of Interplanetary Dust Particles (IDP) collected in the stratosphere of Earth <cit.>.
It is also well understood from the laboratory diagnosis that the particle coagulation in the solar nebula grows under BCCA process <cit.>.

The general extinction A_λ is given by <cit.>:

    A_λ = -2.5 log[F(λ)/F_0(λ)] = 1.086   N_d  Q_ext σ _d,

where F(λ) and F_0(λ) are the observed and expected fluxes, N_d is the dust column density, Q_ext is the extinction efficiency factor determined from Superposition T-matrix code, and σ _d is the geometrical cross-section of a single particle.

The interstellar extinction curve (i.e. the variation of extinction with wavelength) is usually expressed by the ratio
A_λ/E(B-V) versus 1/λ. The extinction curve covers the wavelength range 0.11 to 3.4 μ m. The entire range consists of UV (ultra violet), visible and IR (infrared) regions. The IR range  corresponds to near infrared i.e. 0.750 to 2.5 μ m, the Visible range (0.38 to 0.76 μ m) and the UV range (the last part of violet in visible spectrum to 0.11 μ m).

The radius of an aggregate particle can be described by the radius of a sphere of equal volume given by a_v = a_mN^1/3, where N is the
number of monomers in the aggregate and a_m is the monomer's radius of aggregates . We have found from literature survey that most of the work related to interstellar extinction considered a normal size range of 0.001 to 0.250 micron, with a size distribution (mainly MRN distribution) (Jones 1988, Whittet 2003, Vaidya et al. 2007, Das et al. 2010). They found an `optimum' for the range of the cluster size generally used. The above size range of the monomer is more or less capable of evaluating average observed interstellar extinction curve. If we consider N = 64 and a_m in the range 0.001 to 0.065 micron (with a step size of 0.004 micron), then a_v will be 0.004 to 0.26 micron. This size range is almost comparable to the size range used by other investigators.

We use JaSTA-2 (Second version of the Java Superposition T-matrix Application) <cit.>,  which is an upgraded version of JaSTA <cit.>, to execute our computations which is based on <cit.>'s Superposition T-matrix code. All versions of JaSTA are freely available to download from . The computations with the T-matrix code is fast and this technique gives rigorous solutions for randomly oriented ensembles of  spheres. It is to be noted that the results obtained from the Discrete Dipole Approximation (DDA) approach and the T-matrix approach are almost same. <cit.> showed the results with aggregates using the DDA and the T-matrix code, and found almost same results with both the code. We perform the computations  with a wide range of complex refractive indices (n = 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0 and k = 0.001, 0.01, 0.05, 0.1, 0.3, 0.5, 0.7, 1.0) and wavelengths (0.11, 0.12, 0.13, 0.16, 0.175, 0.185, 0.20, 0.207, 0.22, 0.23, 0.26, 0.30, 0.365, 0.40, 0.55, 0.6, 0.7, 0.8, 0.90 and 3.4 μ m). In general, the range of n and k which is considered in our work almost covers the range of the complex refractive indices of silicate and carbon at different wavelengths. The numerical computation in the present work has been executed with BCCA cluster of 64 monomers.

We present the results for a_m = 0.001μ m, 0.017μ m, 0.041μ m and 0.065μ m where a_v is given by 0.004μ m, 0.068μ m, 0.16μ m and 0.26μ m. The monomer size parameter (x = 2 π a_m/λ) is taken in a range from 0.01 to 1.6. This study is mainly concentrated on investigation of correlation among Q_ext, (n, k), λ, and x.



§ RESULTS



 §.§ Dependence on monomer size (a_m)




  §.§.§ Correlation between Q_ext and n

We perform the computations with a wide range of complex refractive indices (1.4 ≤ n ≤   2.0 and 0.001 ≤ k ≤ 1.0)  and wavelength of incident radiation (0.11 ≤λ≤ 3.4μ m). To study the dependence of n and k on the extinction efficiency (Q_ext), we can either plot Q_ext versus k by keeping n fixed or plot Q_ext versus n by keeping k fixed. We first show the results for moderate size of the cluster ,i.e., at a_v = 0.16μ m. The results at other three sizes are also presented thereafter.

We plot Q_ext versus n at a_v = 0.16μ m, for k = 0.001, 0.01, 0.1, 0.5 and 1.0 respectively (although we have executed the code with all eight values of k mentioned above), in a single frame, for λ = 0.11, 0.20, 0.30 and 0.90μ m, which is shown in Fig.<ref>. It is to be noted that the plots are shown for four wavelengths although the computations have been performed for all the wavelengths.

It is observed from Fig.<ref> that if k is fixed at any value between 0.001 and 1.0, Q_ext increases with increase of n from 1.4 to 2.0 at all wavelengths. But a small decrease in Q_ext is also noticed at λ = 0.11μ m when n > 1.7 and k is low. The variation of Q_ext with n is small when k ≥ 0.5. The value of Q_ext is small when λ is large, i.e. Q_ext decreases when size parameter of the monomer (x = 2 π a_m/λ) decreases. We have also investigated  that the variation of Q_ext with n is very small when λ≥ 0.70μ m.

We have found that Q_ext and n can be fitted by quadratic regression where coefficient of determination[The coefficient of determination is a key output of regression analysis which is interpreted as the proportion of the variance in the dependent variable which ranges from 0 to 1. A higher coefficient is an indicator of a better goodness of fit for the observations.] (R^2) for each equation is ≈ 0.99. The best fit equation is given by


    2
    Q_ext = A_k n^2 + B_k n + C_k   ,     1.4 ≤ n ≤ 2.0,        0.001 ≤ k ≤ 1,

where, A_k, B_k and C_k are k-dependent coefficients of equation (2).

The coefficients obtained for different values of k (only five values of k are shown) are depicted in Table-<ref>. If we plot coefficients A_k, B_k and C_k  versus k (where k = 0.001, 0.01, 0.05, 0.1, 0.3, 0.5, 0.7, 1.0), we find that the best fit curves correspond to cubic regression, which have R^2 ≈ 0.99. We do  not show any figures in this case.

The coefficients are correlated with k by the relations:

    2a        A_k =    D_1 k^3 + D_2 k^2 + D_3 k + D_4  ,
           2b        B_k =    E_1 k^3 + E_2 k^2 + E_3 k + E_4  ,
           2c        C_k =    F_1 k^3 + F_2 k^2 + F_3 k + F_4  ,


All coefficients of equation 2(a-c)  are shown in Table-<ref>. Thus knowing the coefficients, the extinction efficiency (Q_ext) can be calculated for any value of n and k from the equation (2).















In Fig.<ref>, we report the results for a_v = 0.004μ m at λ = 0.11, 0.20, 0.30 and 0.90 μ m. A strong linear correlation between Q_ext and n is seen at this size for all wavelengths from 0.11 to 3.4 μ m. In Fig.<ref>, we plot Q_ext versus n for a_v = 0.068μ m at λ = 0.11μ m and 0.60 μ m. We have found that the nature is quadratic at all wavelengths from 0.11 to 3.4 μ m. Finally, we show the results for a_v = 0.26μ m at λ = 0.11μ m and 0.60 μ m, shown in Fig.<ref>. We have noticed that the Q_ext and n is correlated via a cubic regression at 0.11 μ m whereas the dependence is quadratic at other higher wavelengths. We do not show any equation or table in the above three cases. In summary, we can conclude that the correlation between Q_ext and n is linear when the cluster size is small whereas the correlation is quadratic at moderate and higher sizes of cluster.














  §.§.§ Correlation between Q_ext and k


We now plot Q_ext versus k for n = 1.4, 1.5, 1.6, 1.7, 1.8, 1.9 and 2.0 respectively, at a_v = 0.16μ m where λ is taken in between 0.11 μ m and 3.4 μ m. When λ is between 0.11 μ m and 0.26 μ m, we have found that Q_ext and k can be fitted via a polynomial regression equation where the degree of equation depends on the value of n and λ. In Fig.<ref> we show the plots for λ = 0.11, 0.16, 0.20 and 0.26μ m. At λ = 0.11μ m, we find that Q_ext and k are correlated by (i) quartic regression when n = 1.4, (ii) cubic regression when n = 1.5 and (iii) quadratic regression when n = 1.6, 1.7, 1.8, 1.9 & 2.0. Further at λ = 0.16μ m, the correlation is cubic when n = 1.5, 1.6, 1.7 & 1.8 and quadratic when n = 1.4, 1.9 & 2.0. The correlation at λ = 0.20μ m is cubic when n = 1.6, 1.7 & 1.8 and quadratic when n = 1.4, 1.5, 1.9 & 2.0. We also note that the correlation at 0.26 μ m is quadratic when n = 1.4, 1.5, 1.6, 1.7 & 1.8 and cubic when n = 1.9 & 2.0. At low values of n, Q_ext increases with increase of k whereas the trend is exactly opposite when n is high ,e.g., at λ = 0.20μ m, Q_ext increases with k when n ≤ 1.6, but it decreases when n ≥ 1.7. The vertical range of Q_ext in the plot also decreases when k increases. This range is maximum at k = 0.001  and minimum at k = 1.0.

In Fig.<ref>, we show the plots for 0.30 ≤λ≤ 3.4μ m and we have found that the fit is quadratic for all values of n. An increase in Q_ext with k is noticed at almost all wavelengths. The vertical range of Q_ext also decreases when k increases. Further, the plot of Q_ext with k is same at all values of n when λ is high (please see Fig.<ref>(iv)).

The best fit equation in the wavelength range 0.30 μ m to 3.4 μ m is given by


    3
    Q_ext = A_n k^2 + B_n k + C_n  ,   1.4 ≤ n ≤ 2.0,    0.001 ≤ k ≤ 1

where, A_n, B_n and C_n are n-dependent coefficients of equation (3).

The coefficients obtained for different values of k are shown in Table-<ref>. If we plot coefficients A_n, B_n and C_n  versus n (figures are not shown), we note that the best fit curves correspond to quadratic regression, which have R^2 ≈ 0.99.

Thus coefficients are given by

    3a        A_n =    D'_1 k^2 + D'_2 k + D'_3  ,
           3b        B_n =    E'_1 k^2 + E'_2 k + E'_3  ,
           3c        C_n =    F'_1 k^2 + F'_2 k + F'_3  ,


All coefficients of equation 3(a-c)  are shown in Table-<ref>. Thus, knowing the coefficients of equation 3(a-c), the extinction efficiency (Q_ext) can be also estimated for any value of n and k from the equation (3).






























In Fig.<ref>, we plot Q_ext against k for a_v = 0.004μ m at λ = 0.11, 0.30, 0.60 and 0.90 μ m, although the computations have been performed for wide range of wavelengths from 0.11 to 3.4 μ m. We observe the linear dependence at this size for all wavelengths. This linear nature becomes quadratic when the size of cluster is a_v = 0.068μ m. Fig.<ref> shows the results for a_v = 0.068μ m at λ = 0.11 μ m and 0.60 μ m. In Fig.<ref>, the results obtained for a_v = 0.26μ m are plotted. In this case, the nature of dependence looks similar with a_v = 0.16μ m. Q_ext and k are correlated  via polynomial regression equation (of degree 2, 3 or 4) where the degree of equation depends on the real  part of the refractive index (n) at (λ = 0.11μ m) (please see caption of Fig.<ref>). The nature is quadratic when λ > 0.11μ m. In summary, we can conclude that the dependence of Q_ext on k depends on the cluster size. The correlation is linear for small size and quadratic/cubic for moderate and higher sizes.


It is important to mention that the real part of the complex index of refraction (n) controls the effective phase speed of electromagnetic waves propagating through the medium, while the imaginary part k describes the rate of absorption of the wave. In any material, n and k are not free to vary independently of one another but rather are tightly coupled to one another via the so-called Kramer-Kronig relations. Therefore, the results presented above are quite expectable.




 §.§ Dependence on wavelength of radiation (λ)

We now study the dependence of Q_ext on λ for a particular set of (n, k) in case of a_v = 0.16μ m only. We observe the following results:

  
  (i) For n = 1.4, 1.5 and 1.6, Q_ext versus λ can be fitted via a quartic regression for k = 0.001, 0.01, 0.05, 0.1, 0.3, 0.5, 0.7 and 1.0 (please see Fig.<ref>).

  
  (ii) For n = 1.7, 1.8, 1.9 and 2.0, Q_ext versus λ can be fitted via a quartic regression in the wavelength range 0.11 - 0.40μ m [Fig.<ref>(i,iv) and Fig.<ref>(i,iv)] and a quadratic regression in the wavelength range 0.55 - 0.90μ m [Fig.<ref>(ii,v) and Fig.<ref>(ii,v)] for k = 0.001, 0.01, 0.05, 0.1 & 0.3.  Further, Q_ext versus λ can be fitted via a quartic regression in the wavelength range 0.11 - 0.90μ m for higher values of k = 0.5, 0.7 and 1 [Fig.<ref>(iii,vi) and Fig.<ref>(iii,vi)]. We did not include the plots for λ = 3.4μ m.


It is noticed from Figs.<ref>, <ref> and <ref> that Q_ext decreases with increase of λ when n ≤ 1.6. When n ≥ 1.7, Q_ext initially increases with increase of λ and reaches a maximum value, then it starts decreasing if λ is increased further. We also observe that Q_ext is maximum at k = 0.001 and minimum at k = 1.0 when λ = 0.11μ m. But this trend changes at a critical value of wavelength (λ_c) where exactly opposite nature is noticed. We notice that λ_c is (i) 0.16 μ m at n = 1.4, (ii) 0.185 μ m at n = 1.5, (iii) 0.207 μ m at n = 1.6, (iv) 0.22 μ m at n = 1.7, (v) 0.26 μ m at n = 1.8, (vi) 0.26 μ m at n = 1.9 and (vii) 0.30 μ m at n = 1.4. The values of Q_ext is maximum at k = 1.0 when λ > λ_c. We do not show any equations and tables in this case.




















 §.§ Dependence on the size parameter of monomer (x)

We now study the dependence of Q_ext on the size parameter of monomer (x = 2π a_m/λ)  for n = 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0 and k = 0.001, 0.01, 0.05, 0.1, 0.3, 0.5, 0.7, 1.0.  A wide range of size parameter, 0.01 ≤ x ≤ 1.6 (where, N = 64), is considered to investigate the correlation between Q_ext and x. The results are plotted in Figs. <ref>, <ref>, <ref> and <ref>  for k = 0.001, 0.01, 0.1 and 1.0, respectively. It can be seen from figures that if x is fixed at any value between 0.01 and 1.6, Q_ext increases with increase of n. This increase is prominent when x > 0.2. The vertical range of Q_ext also increases with increase of n from 1.4 to 2.0. This range is maximum  (i) at x = 1.4 in case of k = 0.001 (where, Q_ext = [9.34, 3.45]), (ii) at x = 1.4 in case of k = 0.01 (where, Q_ext = [9.26, 3.50]), (iii) at x = 1.4 in case of k = 0.1 (where, Q_ext = [8.56, 3.90]), and (iv) at x = 1.12 in case of k = 1.0 (where, Q_ext(max) = [6.74, 5.88]). The slope of Q_ext versus x curve increases with the increase of n from 1.4 to 2.0 which is noted for all values of k. It is also interesting to notice that the vertical range of Q_ext at x = 1.6 decreases with the increase of k and is lowest at k = 1.0. Further, Q_ext value does not depend much on n for highly absorptive particles (k = 1.0) when x < 0.5. The variation is also small when x > 0.5.

We have found that Q_ext and x can be fitted by a cubic regression for all values of n except 2.0, in case of k = 0.001, 0.01, and 0.1 respectively. The coefficient of determination (R^2) for each equation is ≈ 0.99. The best fit equation is given by

    4
        Q_ext = α_1 x^3 + α_2 x^2 + α_3 x + α_4,


where, α_1, α_2, α_3 and α_4 are n-dependent coefficients of equation (4). The coefficients are presented in Table-<ref>.

However, the best fit equation in case of n = 2.0 corresponds to a quartic regression for k = 0.001, 0.01, and 0.1, which is given by

    5
        Q_ext = β_1 x^4 + β_2 x^3 + β_3 x^2 + β_4 x + β_5,


where, β_1, β_2, β_3, β_4 and β_5 are n-dependent coefficients, shown in Table-<ref>.


The correlation between Q_ext and x is found to be quintic regression for all values of n at k = 1.0. The best fit equation is given by

    6
        Q_ext = γ_1 x^5 + γ_2 x^4 + γ_3 x^3 + γ_4 x^2 + γ_5 x + γ_6,

where, γ_1, γ_2, γ_3, γ_4, γ_5, and γ_6 are n-dependent constants of equation (6). The constants are given in Table-<ref>.

Equations (4), (5) and (6) are very useful in estimating Q_ext, where one can generate a large data set for Q_ext for selected set of n, k, a_m, and λ.
























§ RESULTS FROM CORRELATION EQUATIONS

In the previous sections, we have obtained a set of correlation equations  which can be used to calculate the extinction efficiency of dust aggregates with a wide range of size of aggregates and wavelength of radiation. We first calculate Q_ext from relations (2) and (3) for BCCA particles with N = 64 and a_m = 0.041μ m, for selected values of n, k and λ. The calculated values are compared with the computed values obtained using the Superposition T-matrix code. The results are shown in Table-<ref> and <ref>. We also estimate Q_ext using relations (4), (5) and (6) for selected values of x, n and k, and is shown in Table-<ref>.  It can be seen that the values obtained from computed values match well with the results obtained from correlation equations.

    In general, to model the interstellar extinction, one need to execute the light scattering code with different values
	of a_m (or a_v) and wavelength (λ), which is very time consuming. Using a size distribution for aggregates, it is possible to obtain the average extinction curves for silicate and graphite (and/or amorphous carbon) particles. With a suitable mixing among them, extinction curve against
	different wavelengths can be generated which can be fitted well with observed extinction
	curve. Some good pieces of work on modeling were already done for aggregate particles.
	Some preliminary results on modeling of interstellar extinction using aggregate dust model were already reported by <cit.>.
The present study shows that it is possible to study the extinction properties of interstellar dust aggregates for a given size of the particles and wavelength using relations (4), (5) and (6). The set of  correlation equations can be used to estimate the general extinction A_λ using equation (1) for a given size distribution which will help to model the interstellar extinction curve. At this stage, we are not interested in modeling as this study is primarily projected to investigate the dependency of extinction efficiency on size, wavelength and composition of particles. We show how this dependency can be framed with some correlation equations to study the extinction properties of interstellar dust.
	
<cit.> experimentally investigated the
	morphological effects on the extinction band in the infrared region for
	amorphous silica (SiO_2) agglomerates. They also compared the measured band profiles
	with calculations for five cluster shapes applying Mie, T-matrix and DDA codes. Our	
	correlations will be also helpful to study the experimental data. But it is
	also important to check the input parameters (size parameter, composition etc.) of the experimental setup before using the correlation equations, because the 	 relations are based on some selected set of parameters.















§ SUMMARY


  
  * We have first studied the dependency of Q_ext on size of the aggregates (a_m) to investigate the correlations between Q_ext and complex refractive indices (n, k) at a particular size. Computations are performed at four different sizes (a_v = 0.004, 0.068, 0.16 and 0.26 μ m).

If k is fixed at any value between 0.001 and 1.0, Q_ext increases with increase of n from 1.4 to 2.0. Q_ext and n are correlated  via linear regression when the cluster size is small whereas the correlation is quadratic at moderate and higher sizes of cluster. This feature is observed at all wavelengths (UV to optical to infrared). We have also found that the variation of Q_ext with n is very small when λ is high.

We have observed that  Q_ext and k are correlated  via polynomial regression equation (of degree 2,3 or 4) where the degree of equation depends on the cluster size, real  part of the refractive index of the particles (n) and wavelength (λ) of incident radiation. At a_v = 0.16 μ m, Q_ext and k is found to be correlated with a polynomial regression equation (of degree 2 or 3) when λ is between 0.11 μ m and 0.26 μ m. However, when λ > 0.26μ m,  we have found that the correlation between them is quadratic for all values of n. The vertical range of Q_ext in the plot also decreases when k increases. This range is maximum at k = 0.001  and minimum at k = 1.0. If we include results for four different cluster sizes, we can summarize that the correlation of Q_ext and k depends mainly on cluster size. The correlation is linear for small size and quadratic/cubic/quartic for moderate and higher sizes.


  * We study the dependence of Q_ext on λ for a_v = 0.16μ m. Q_ext decreases with increase of λ when n ≤ 1.6. When n ≥ 1.7, Q_ext initially increases with increase of λ and reaches a maximum value, then it starts decreasing if λ is increased further. For n = 1.4, 1.5 and 1.6, Q_ext versus λ can be fitted via a quartic regression for all values of k. For other values of n, the correlation is polynomial regression where the degree of equation depends on the value of n, k and λ.


  * We have found that Q_ext and x are correlated via a polynomial regression (of degree 3,4 or 5) for all values of n. The degree of regression is found to be n and k-dependent.


   
  * The correlation equations can be used to model interstellar extinction for dust aggregates in a wide range of size of the aggregates, wavelengths and complex refractive indices.









§ ACKNOWLEDGEMENT

We acknowledge Daniel Mackowski and Michael Mishchenko, who made their Multi-sphere T-matrix (MSTM) code  publicly available. We also acknowledge Prithish Halder for help on the execution of JaSTA-2 software package. The reviewer of this paper is highly acknowledged for useful comments and suggestions.


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