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6 Discussion

6.1 Dimensionless analysis

For the sake of analysis and discussion, we transform the dust moment equations derived in Sects. 5.1 and 5.2 into their dimensionless form by introducing reference values as $t\!\to\!t/t_{\rm ref}$, $\nabla\!\to\!l_{\rm ref}\nabla$, $\rho\!\to\!\rho/\rho_{\rm ref}$, ${\vec v}\!\to\!{\vec v}/v_{\rm ref}$, ${\vec g}\!\to\!{\vec g}/g_{\rm ref}$, $J(V_l)\!\to\!J(V_l)/J_{\rm ref}$, $L_j\!\to\!L_j/L_{j,\rm ref}$, $\chi^{\rm net}\!\to\!\chi^{\rm net}/\chi_{\rm ref}$, $\xi\!\to\!\xi/\xi_{\rm ref}$, $\mu_{\rm kin}\!\to\!\mu_{\rm kin}/\mu_{\rm kin}^{\rm ref}$following the procedure described in Paper I. The reference values are to be chosen according to the expected order of magnitude of the respective quantities and the length and time-scales under investigation. After this substitution, all quantities are dimensionless and can be compared by number.

This allows us to identify the leading terms in the equations, the inner and outer regions of a brown dwarf atmosphere. The remaining constants (products of the reference values) can be summarised into characteristic numbers which provide an efficient way to describe the qualitative behaviour of the dust component.

The dimensionless dust moment equations for nucleation, growth, evaporation, and equilibrium drift write for a subsonic free molecular flow ( ${Kn}\gg 1 \wedge \vec{v}_{\rm dr}^{\hspace{-0.6ex}^{\circ}}\!\ll\!c_{\rm T}$)

                  $\displaystyle \big[{S\!r}\big]\frac{{\rm\partial} }{{\rm\partial} t}\big(\rho L_j\big)$ + $\displaystyle \nabla ({\vec v}_{\rm gas}~\rho L_j)\;=\;
\big[{S\!r}\cdot{Da^{\rm nuc}_{\rm d}\!\cdot S\!e_{\rm j}}\big]~J({V_{\ell}})$  
  + $\displaystyle \left[{S\!r}\cdot{Da^{\rm gr}_{{\rm d}, lKn}}\right]\frac{j}{3}~
\chi^{\rm net}_{\rm lKn}~\rho L_{j-1}$  
  + $\displaystyle \left[\left(\frac{\pi\gamma}{32}\right)^{\!1/2}
\frac{S\!r\cdot M...
\xi_{\rm lKn} \nabla\!\left(\frac{L_{j+1}}{c_{\rm T}}~\vec{e}_{\rm r}\!\right)$ (77)

and for a laminar viscous flow ( ${Kn}\ll 1 \wedge Re_{\rm d}\!<\!1000$)
                    $\displaystyle \big[{S\!r}\big]\frac{{\rm\partial} }{{\rm\partial} t}\big(\rho L_j\big)$ + $\displaystyle \nabla ({\vec v}_{\rm gas}~\rho L_j)\;=\;
\big[{S\!r}\cdot{Da^{\rm nuc}_{\rm d}\!\cdot S\!e_{\rm j}}\big]~J({V_{\ell}})$  
  + $\displaystyle \left[{S\!r}\cdot{Da^{\rm gr}_{\rm d, sKn}}\right]\frac{j}{3}~
\chi^{\rm net}_{\rm sKn}~\rho L_{j-2}
  + $\displaystyle \left[\left(\frac{\pi\gamma}{288}\right)^{\!1/2}\!\!
...} \nabla\!\left(\frac{\rho L_{\rm j+2}}
{\mu_{\rm kin}}~\vec{e}_{\rm r}\right).$ (78)

The equations are valid for $j\!=\!0,1,2,...$ . The constants in squared brackets are written in terms of characteristic numbers, which are further explained in Table 2. All other quantities and terms in Eqs. (77) and (78) are of the order of unity for an appropriate choice of the reference values.

The following discussion is based on a typical structure of a brown dwarf atmosphere with solar abundances in the gas phase, i.e. neglecting the possible depletion due to dust formation (see Table 2). As underlying ($T,\rho$)-structure we refer to a brown dwarf model atmosphere with $T_{\rm eff}\!=\!1000~$K and $\log~g\!=\!5$ which has been kindly provided by T. Tsuji (2002)[*]. As exemplary dust species we consider solid SiO2(amorphous quartz), growing by the accretion of SiO and H2O (Eq. (38)). Since the nucleation of SiO2 seems dubious (the monomer is rather unstable as a free molecule and hence not very abundant in the gas phase) we consider nucleation of TiO2instead[*].


Table 2: Characteristic numbers and reference values used for the analysis of the dimensionless dust moment Eqs. (77) and (78). Hydro- and thermodynamic reference values are taken from a static brown dwarf model atmosphere (Tsuji 2002) with $T_{\rm eff}\!=\!1000$ K and $\log~g\!=\!5$, considering an inner ( $T_{\rm ref}\!=\!1700~$K) and an outer ( $T_{\rm ref}\!=\!1000$ K) layer, typical for the growth-dominated region and the region of effective nucleation, respectively (see text). We assume constant $\gamma\!=\!7/5$ and $\bar{\mu}\!=\!2.35~$amu ( $\rm 1~amu\!=\!1.6605\times10^{-24}~g$). In each layer, two reference length scales are considered: $l_{\rm
ref}\!=\!10^{~4}$cm (microscopic scale) and $l_{\rm
ref}\!=\!10^{~6}$cm ( $\vec{\approx H_p}$; macroscopic scale). The reference values for the dust complex have been adopted according to the experience of Paper I, considering an undepleted gas of solar abundances. The growth rates are calculated for SiO2 via addition of SiO and H2O (see Eq. (38)), assuming $S_{\rm r}\!\gg\!1$ and $\alpha_{\rm r}\!=\!1$. Further molecular and dust material quantities are listed in Table 3. Boldfaced numbers mark those values which are related to the proper Knudsen number case ( ${Kn}\!\ll\!1$ or ${Kn}\!\gg\!1$).
Name Characteristic   Value    
  Number inside   outside  
Mach number ${M }=\frac{v_{\rm ref}}{c_{\rm S}}$     $\approx 1/10$      
Froude number ${Fr}=\frac{v_{\rm ref}}{t_{\rm ref}}
\frac{1}{g_{\rm ref}}$ 0.842 $8.42\times10^{-3}$   0.495 $4.95\times10^{-3}$  
Strouhal number ${S\!r}= \frac{l_{\rm ref}}{t_{\rm ref}v_{\rm ref}}$     $\approx 1$      
hydrodyn. Knudsen number ${Kn}^{\rm ^{HD}}=
\frac{l_{\rm ref}}{2\langle a \rangle{\rm ref}}$ $5\times10^{+6}$ $5\times10^{+8}$   $5\times10^{+9}$ $5\times10^{+11}$  
Knudsen number (Eq. (6)) ${Kn} = \frac{\bar{\ell}_{\rm ref}}
{2\langle a \rangle{\rm ref}}$ $3.09\times10^{-3}$ ... $1.86\times10^{+1}$  
Drift number ${Dr}= \frac{\rho_{\rm d, ref}}{\rho_{\rm ref}}$ $8.83\times10^{+3}$ ... $5.30\times10^{+4}$  
combined drift number ( ${Kn}\!\gg\!1$) $\displaystyle\left(\frac{\pi\gamma}{32}\right)^{\!1/2}
\frac{S\!r\cdot M\cdot Dr}{Kn^{^{\rm HD}}\!\!\cdot Fr}$ $7.78\times10^{-5}$ ... $\bf 7.93\times10^{-7}$  
combined drift number ( ${Kn}\!\ll\!1$) $\displaystyle\left(\frac{\pi\gamma}{288}\right)^{\!1/2}
\frac{S\!r\cdot M\cdot Dr}{Kn\cdot Kn^{^{\rm HD}}\!\!\cdot Fr}$ $\bf 8.38\times10^{-3}$ ... $1.42\times10^{-8}$  
      $j\!=\!0$ : 1   $j\!=\!0$ : 1  
Sedlmaÿr number $(j \in \mathbb{N} _0)$ ${S\!e_{\rm j}} = \left( \frac{a_\ell}
{\langle a \rangle_{\rm ref}}\right)^{\rm j}$ $j\!=\!1$ :  $2.08\times10^{-4}$   $j\!=\!1$ :  $2.08\times10^{-1}$  
      $j\!=\!2$ :  $4.32\times10^{-8}$   $j\!=\!2$ :  $4.32\times10^{-2}$  
      $j\!=\!3$ :  $8.99\times10^{-12}$   $j\!=\!3$ :  $8.99\times10^{-3}$  
Damköhler no. of nucleation ${Da^{\rm nuc}_{\rm d}} =
\frac {t_{\rm ref} J_{\rm ref} }
{\rho_{\rm ref}L_{0, {\rm ref}}}$ 0 0   $3.24\times10^{+4}$ $3.24\times10^{+6}$  
Damköhler no. of growth ( ${Kn}\!\gg\!1$) ${Da^{\rm gr}_{\rm d, lKn}} =
\frac{t_{\rm ref}\chi_{\rm ref, lKn}}
{(\frac{4~\pi}{3}\langle a \rangle^3_{\rm ref})^{1/3}}$ 3.95 $3.95\times10^{+2}$   $\bf 6.58\times10^{+2}$ $\bf 6.58\times10^{+4}$  
Damköhler no. of growth ( ${Kn}\!\ll\!1$) ${Da^{\rm gr}_{\rm d, sKn}} =
\frac{t_{\rm ref}\chi_{\rm ref, sKn}}
{(\frac{4~\pi}{3}\langle a \rangle^3_{\rm ref})^{2/3}}$ $\bf 9.24\times10^{-2}$ $\bf 9.24$   $9.24\times10^{+4}$ $9.24\times10^{+6}$  
Name Physical Quantity   Reference Value    
      inside   outside  
temperature $T_{\rm ref}$ [K] 1700 ... 1000  
density $\rho_{\rm ref}$ [g/cm3] $3\times 10^{-4}$ ... $5\times 10^{-5}$  
thermal pressure $P_{\rm ref} = \frac{\rho_{\rm ref}kT_{\rm ref}}
{\bar{\mu}}$ [dyn/cm2] $1.80\times10^{+7}$ ... $1.77\times10^{+6}$  
velocity of sound $c_{\rm S}=\sqrt{\gamma\frac{P_{\rm ref}}
{\rho_{\rm ref}}}$ [cm/s] $2.90\times10^{+5}$ ... $2.23\times10^{+5}$  
velocity $v_{\rm ref}$ [cm/s]     $\approx c_{\rm S}/10$      
length $l_{\rm ref}$ [cm] 10+4 10+6   10+4 10+6  
hydrodyn. time $t_{\rm ref} = \frac{l_{\rm ref}}{v_{\rm ref}}$ [s] $\!3.45\times10^{-1}$ $3.45\times10^{+1}$   $4.49\times10^{-1}$ $4.49\times10^{+1}$  
gravitational acceleration $g_{\rm ref}$ [cm/s2]     10+5      
mean particle radius $\langle a \rangle_{\rm ref}$ [cm] 10-3 ... 10-6 $^{(\lozenge)}$
0th dust moment ( $=n_{\rm d}/\rho$) $L_{0, \rm ref}=\frac{\Delta V_{\rm SiO_2}
n_{\rm ref, SiO}}{\rho_{\rm ref}\frac{4\pi}{3}
\langle a \rangle_{\rm ref}^3}$ [1/g] $1.35\times10^{+5}$ ... $1.35\times10^{14}$ $^{(\Box)}$
nucleation rate $J_{\rm ref}/n_{\rm <H>, ref}$ [1/s] $0\;^{(\dagger)}$ ... $2.30\times10^{-5}\;^{(\ddagger)}$  
growth velocity ( ${Kn}\!\gg\!1$, Eq. (67)) $\chi^{\rm ref}_{\rm lKn}$ [cm/s] $1.85\times10^{-2}$ ... $\bf 2.36\times10^{-3}$ $^{(\vartriangle)}$
growth velocity ( ${Kn}\!\ll\!1$, Eq. (76)) $\chi^{\rm ref}_{\rm sKn}$ [cm2/s] $\bf 6.96\times10^{-7}$ ... $5.34\times10^{-7}$ $^{(\vartriangle)}$
diffusion constant (Eq. (26)) $D_{\rm ref}$ [cm2/s] $5.28\times10^{-1}$ ... 2.43 $^{(\vartriangle)}$
mean free path (Eq. (10)) $\bar{\ell}_{\rm ref}$ [cm] $6.19\times10^{-6}$ ... $3.71\times10^{-5}$  
total hydrogen number density $n_{\langle {\rm H} \rangle, {\rm ref}}
=\frac{\rho_{\rm ref}}{1.427~{\rm amu}}$ [1/cm3] $1.27\times10^{20}$ ... $2.11\times10^{19}$  
molecular number density $n_{\rm ref} = n_{\rm ref, SiO}$ [1/cm3] $4.49\times10^{15}$ ... $7.49\times10^{14}$ $^{(\vartriangle)}$

$^{(\lozenge)}$ : According to the experience of Paper I.     $^{(\Box)}$ : We choose the reference value of the dust-to-gas mass ratio $\rho_{\rm d}\!\cdot\!L_{3,\rm ref}$ by considering the case when all Si is bound in solid SiO2 and adapt the referece value for the total dust particle density $\rho_{\rm ref}L_{0,\rm ref}$ according to the assumed grain's reference size $\langle a_{\rm ref}\rangle$: $L_{j,\rm ref}\!=\!(\frac{4\pi}{3} \langle
a_{\rm ref}\rangle^{~3})^{j/3}L_{0,\rm ref}$ (Gail ${\rm\hspace*{0.7ex}\&\hspace*{0.7ex}}$Sedlmayr 1999).     $^{(\dagger)}$ : We assume vanishing nucleation rates in this region around 1700 K, because the supersaturation ratios $S\!\ga\!1$ are either too small for efficient nucleation or the nucleation species have already been consumed by growth.     $^{(\ddagger)}$ : The reference value for the nucleation rate $J_{\rm ref}$is chosen by considering homogeneous nucleation of TiO2 according to Paper I.     $^{(\vartriangle)}$ : SiO is considered as key growth species.

6.2 Hierarchy of nucleation, growth and drift

An analysis of the characteristic numbers in front of the source terms in Eqs. (77) and (78) (see Table 2) reveals a hierarchy of nucleation $\to$ growth $\to$ drift:

In the cool outer layers, the gas is strongly supersaturated ($S\!\gg\!1$) and nucleation is effective ( $J({V_{\ell}})\!=\!J_\star\!>\!0$). The products of the Damköhler number of nucleation $Da_{\rm d}^{\rm nuc}$ with the Strouhal number $S\!r$ and the Sedlmaÿr numbers $S\!e$ are as large as the products of the Damköhler numbers of growth $Da_{\rm d}^{\rm gr}$ with the Strouhal number $S\!r$ (even larger for small j) and much larger than the combined drift numbers, indicating that the nucleation provides an important source term in Eqs. (77) and (78). Consequently, the condensable elements will be quickly consumed by the process of nucleation before the particles can grow much further. Hence, the dust particles remain very small in this layer of effective nucleation.
In the warmer layers, the gas is almost saturated ($S\!\ga\!1$) and nucleation is not effective ( $J({V_{\ell}})\!\to\!0$). The products of the Damköhler numbers of growth $\rm Da_d^{gr}$ with the Strouhal number $S\!r$ are large and the growth term (the second term on the r.h.s.) is the leading source term in the dust moment Eqs. (77) and (78). In comparison, the influence of the drift term is small as quantified by the combined drift numbers in Table 2. Consequently, the dust growth process will substantially be completed before the dust grains start to settle gravitationally. In these growth-dominated layers, a few existing particles will quickly consume all condensable elements from the gas phase and, thus, will reach much larger particle sizes. Since these particles cannot be created via nucleation here, they must have been formed elsewhere and transported into these layers by winds or drift. The growth will either be terminated by element consumption or by the loss due to gravitational settling when the particles reach their maximum size as introduced in Sect. 3.1.
At the cloud base, the gas is hot and saturated ($S\!=\!1$). Consequently, nucleation and growth vanish and the only remaining source term on the r.h.s. of Eqs. (77) and (78) is the additional advection of the particles due to their drift motion (drift term). The combined drift numbers, however, are small indicating that the drift term has a smaller influence than the hydrodynamical advection term.

6.3 Scaling laws


Table 3: Additional material quantities used to calculate the reference values listed in Table 2.
Name     Value
dust material density $\rho_{\rm d, SiO_2}$ [g/cm3] 2.65
monomer volume $\Delta V_{\rm SiO_2}$ [cm3] $3.76\times10^{-23}$
lower dust grain radius $ a_{\ell, \rm SiO_2}$ [cm] $2.08\times10^{-7}$
molecular radius $r_{\rm SiO}$ [cm] $2.0\times10^{-8}$


Table 4: Scaling of the physical processes, ordered by their appearance from left to right in Eqs. (77) and (78).
physical process ${Kn}\!\gg\!1$ ${Kn}\!\ll\!1$
time-derivative $\propto\frac{l_{\rm ref}}{v_{\rm ref}~t_{\rm ref}}$
advective term 1
nucleation term $^{(\vartriangle)}\!\!\!$ $\propto\frac{l_{\rm ref}~J_{\rm ref}}{v_{\rm ref}~\rho_{\rm ref}~
L_{0,\rm ref}~\langle a_{\rm ref}\rangle^j }$
growth term $^{(\vartriangle)}$ $\propto j\cdot\frac{l_{\rm ref}~\rho_{\rm ref}~\sqrt{T_{\rm ref}}}
{v_{\rm ref}~\langle a_{\rm ref}\rangle}$ $\propto j\cdot\frac{l_{\rm ref}~\sqrt{T_{\rm ref}}}
{v_{\rm ref}~\langle a_{\rm ref}\rangle^2}$
drift term $\propto\frac{g_{\rm ref}~\rho_{\rm d,ref}~\langle a_{\rm ref}\rangle}
{v_{\rm ref}~\rho_{\rm ref}~\sqrt{T_{\rm ref}}}$ $\propto\frac{g_{\rm ref}~\rho_{\rm d,ref}~\langle a_{\rm ref}\rangle^2}
{v_{\rm ref}~\sqrt{T_{\rm ref}}}$
$^{(\vartriangle)}$ : the importance of this process depends on the considered dust moment j with the following
associated mean dust quantity: $j\!=\!0\to$ dust particle density, $j\!=\!1\to$ dust size,
$j\!=\!2\to$ dust surface area, $j\!=\!3\to$ dust mass density.

Table 4 shows some dependencies of the combined characteristic numbers (the squared brackets in Eqs. (77), (78)), which provides scaling laws for the importance of the different processes in the different regimes:

The drift term scales as $\propto\langle
a_{\rm ref}\rangle$ ( ${Kn}\!\gg\!1$) or $\propto\langle a_{\rm
ref}\rangle^2$ ( ${Kn}\!\ll\!1$) whereas the growth term scales as $\propto\!\langle a_{\rm ref}\rangle^{-p}$ with different $p\!\ge\!0$ for the different cases (Table 4). This means that at a certain large mean particle size, the drift term will start to dominate over growth, which in fact just occurs at the maximum particle size $a_{\rm max}$ introduced in Sect. 3.1.
Nucleation (always) and growth ( ${Kn}\!\gg\!1$) become increasingly important with increasing gas density $\rho_{\rm ref}$ (note that typically $J_{\rm ref}\!\propto\!\rho_{\rm
ref}^{2+p}$ with $p\!>\!0$), whereas the drift term ( ${Kn}\!\gg\!1$) diminishes for increasing $\rho_{\rm ref}$, i.e.the drift is generally more important in a thin gas. However, the particles are expected to remain smaller in such a thin gas situated in the upper layers, which has a stronger, opposite effect on the drift term.
The larger the considered length scale $l_{\rm ref}$, the more stiff the dust moment equations become because of the increasingly large nucleation and growth terms. This may actually cause some numerical difficulties for models on macroscopic scales. In comparison, the advective and the drift term are independent on $l_{\rm ref}$, which means that these terms gain importance on small scales in relation to the other terms.
The drift term scales as $\propto~g_{\rm ref}~\rho_{\rm d,ref}$, i.e.the drift is naturally more important for heavy grains in a strong gravitational field.

6.4 Control mechanisms

In Table 2, we have assumed $S\!r\!=\!1$, i.e.we have considered time-scales $t_{\rm ref}$ of the order of $l_{\rm
ref}/v_{\rm ref}$, appropriate for disordered (e.g.turbulent) velocity fields, where the l.h.s. terms of Eqs. (77) and (78) are of comparable importance. However, if large-scale systematic motions are stable for a long time (e.g.a circulating thunderstorm or a stable convection roll), the system may reach a quasi-stationary situation where $t_{\rm ref}$ becomes much larger and hence $S\!r\!\to\!0$. In this case, the first term on the l.h.s. of Eqs. (77) and (78) vanishes (see Table 4) whereas all other terms remain unaffected.

An interesting special case occurs if additionally $\vec{v}_{\rm
gas}\!\to0$, i.e.when the dust-forming system reaches the static case. In this case, also the advective terms in Eqs. (77) and (78) vanish and the source terms must balance each other. In the $S\!>\!1$ case, this means that the gain of dust by nucleation and growth must be balanced by the loss of dust due to rain-out, which means that the gas will be depleted. In the $S\!<\!1$ case, just the opposite is true, i.e.the loss by evaporation must be balanced by the gain of dust particles raining in from above. Consequently, the gas in such undersaturated layers will be enriched by the condensable elements liberated by the evaporating grains.

However, both control mechanisms (in the static limit) result in an efficient transport of condensable elements from the cool upper layers into the warm inner layers, which cannot last forever. We may conclude that if the brown dwarf's atmosphere is truly static for a long time, there is no other than the trivial solution for Eqs. (77) and (78) where the gas is saturated ( $S\!\equiv\!1$) and dust-free ( $L_j\!\equiv\!0$). This situation changes, however, if the brown dwarf's atmosphere is turbulent or, in particular, when it is convective. In that case, the replenishment of the atmosphere with fresh uncondensed gas from the deep interior will counteract the downward transport of condensable elements by the formation and gravitationally settling of dust grains. Simulations of this quasi-static balance will be the subject of the forthcoming paper in this series.

6.5 Outlook

A dynamical modelling of the dust component in brown dwarf atmospheres by means of a moment method as proposed in this paper - consistently coupled to hydrodynamics, radiative transfer and element depletion in the scope of hydrodynamical or classical stellar atmosphere calculations - seems straightforward as soon as two major problems can be solved:

Knudsen number fall differentiation:

The structure of the dust moment Eqs. (77) / (78) changes with changing Knudsen number. The size distribution function f(V), however, will generally include small and large grains simultaneously, which may possess large and small Knudsen numbers, respectively. A proper treatment of all dust grains by means of a moment method is hence possible only in one of the limiting cases ${Kn}\!\gg\!1$ or ${Kn}\!\ll\!1$, where all grains fall into one particular case. This situation may be relevant for the cool upper regions of a brown dwarf atmosphere, where the densities are small and the dust particles remain tiny. In general, however, a solution of this problem by means of a moment method can only be approximate in nature and will require additional numerical tricks, e.g.switching the case as soon as the Knudsen number according to the mean size of the particles reaches unity.

Open system of equations:

The number of unknowns in Eqs. (77) / (78) exceeds the number of equations, since the drift term on the r.h.s. involves a higher dust moment Lj+1 (for ${Kn}\!\gg\!1$) or even Lj+2 (for ${Kn}\!\ll\!1$). Therefore, in order to benefit from the moment Eqs. (77) / (78), we require a physically reliable closure condition, as e.g.

 \begin{displaymath}L_{j+1} ~=~ F~(L_0, L_1,~\ldots, L_j)\ .
\end{displaymath} (79)

Such closure conditions are fastidious problems. We are optimistic, nevertheless, to find such a closure term, because the size distribution f(V) is usually a very smooth function and the dust moments Lj often reveal a very simple functional dependency on the index j, indicating that the true number of the degrees of freedom for a parametric description of f(V) is actually quite small.

One idea to construct such a closure condition has been developed by Deufelhard ${\rm\hspace*{0.7ex}\&\hspace*{0.7ex}}$Wulkow (1989) and Wulkow (1992), studying the kinetics of polyreaction systems. The size distribution function f(V) is here approximated by a weight function $\Psi^\alpha(V)$, which describes the basic shape of f(V), and modified by a sum of orthogonal polynomials $\{p_k(V)\}$ (k=0,1,2, ... ,n) as

 \begin{displaymath}f(V,\vec{x},t) = \Psi^\alpha(V) \sum_{k=0}^{\rm n}
a_k(\vec{x},t)~p_k^\alpha(V) \ .
\end{displaymath} (80)

$\alpha$ is an additional parameter (dependent on $\vec{x},t$) of the weight function and ak are polynomial coefficients. The parameters $\alpha$ and ak $(k\!=\!0,1,2,~...~,n)$ are stepwise adjusted in order to fit the known moments Lj $(j\!=\!0,1,2,~...~,n)$exactly, using the orthogonality relation $\int
\Psi^\alpha(V)~p_i^\alpha(V)~p_k^\alpha(V)~dV=\delta_{ik}$. Once these coefficients are known, all missing moments of the size distribution function can be reconstructed from Eq. (80), again utilising the orthogonality relation. This procedure leads to reliable results, if the weight function $\Psi^\alpha(V)$ is already close to the actual size distribution function f(V), such that the polynomials only provide small corrections and the sum in Eq. (80) converges rapidly.

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