A&A 466, 413420 (2007)
DOI: 10.1051/00046361:20066899
Closedform expressions for particle relative velocities induced by turbulence
(Research Note)
C. W. Ormel^{1}

J. N. Cuzzi^{2}
1  Kapteyn Astronomical Institute, University of Groningen, PO box 800, 9700 AV Groningen, The Netherlands
2 
Ames Research Center, NASA, Mail Stop 2453, Moffett Field, CA 94035, USA
Received 8 December 2006 / Accepted 1 February 2007
Abstract
In this note we present complete, closedform expressions for random relative velocities between colliding particles of arbitrary size in nebula turbulence. These results are exact for very small particles (those with stopping times much shorter than the large eddy overturn time) and are also surprisingly accurate in complete generality (that is, also apply for particles with stopping times comparable to, or much longer than, the large eddy overturn time). We note that some previous studies may have adopted previous simple expressions, which we find to be in error regarding the size dependence in the large particle regime.
Key words: turbulence  dust, extinction  planetary systems: protoplanetary disks
Gas in astrophysical environments is often in a turbulent state of motion,
constantly affected by temporally and spatially varying accelerations from
eddies having a variety of scales. A particle, due to its inertia, does not
instantaneously follow the gas motions but requires a certain time in order to
align with the gas motion. The particle's interaction with the gas is captured in the definition of the stopping time of the particle (sometimes also referred to as friction time),

(1) 
where
and
are, respectively, the sound speed and the volume mass density of the gas, and m and
the mass and projected surface area of the particle. Due to this inertial lag, a particle develops a relative
velocity with respect to the gas. In addition, these lags also cause particles to acquire relative velocities among themselves.
While the general problem of calculating these relative velocities has received
considerable attention in the basic fluid dynamics community (see Cuzzi & Hogan 2003, for references; henceforth CH03), the formalism most frequently used in the astrophysics community
was developed by Völk et al. (1980) and Markiewicz et al. (1991, henceforth MMV). In these works the final results are given in terms of integrals that were not solved analytically. Some workers have used simple fits to these numerical results in their models of dust coagulation; however, simple closedform expressions for
particleparticle relative velocities would help streamline these models (e.g. Ormel et al. 2007; Nomura & Nakagawa 2006; Dullemond & Dominik 2005; Suttner & Yorke 2001).
Recently, CH03 obtained closedform expressions from
the MMV model for particle velocities in inertial space ,
for particlegas relative velocities
,
and for relative velocities between two identical particles
,
but did not extend their results to the general case of two particles of different stopping times. Moreover,
CH03 stressed the validity of their analytical results for particles with stopping times much shorter than the large eddy turnover time. In this note we generalize the
approach and results of CH03 to obtain closedform
expressions for relative velocities between particles of arbitrary, and unequal, size.
In Sect. 2 we define important quantities and review previous work. In
Sect. 3 we
present two independent approaches for obtaining the desired closedform solutions.
In Sect. 4 we give our conclusions and a summary.
2 Definitions and previous work
Nebula gas turbulence is generally described as being composed of eddies having a
range of spatial scales
and spatial frequencies
,
with an energy
spectrum
and total energy
per unit mass providing
the normalization condition

(2) 
from which
.
The largest, or integral scale, eddies have spatial scale
,
and the smallest, or
Kolmogorov scale, eddies have spatial scale
.
The form of E(k)
given above is the inertial range expression most often assumed, with E(k)=0 for
or
.
Völk et al. (1980) used a spectrum
P(k)=2 E(k)
and stipulated no smallest scale
for the turbulence, but Weidenschilling (1984)
and MMV noted that a finite value for
had profound effects on the
particle velocities, especially the relative velocities
for small particles. Each eddy
wavenumber k has a characteristic velocity
and overturn time
.
Our standard definition of the particle Stokes number is
,
where
is the overturn time of the largest eddy, generally
taken to be the local orbit period. The local turbulent intensity is described by its Reynolds number, Re, defined as the ratio between the turbulent and the molecular kinematic viscosities,
.
The values for
and t at the integral scale then follow from Re, e.g.,
and
.
These expressions bring Re into the final expressions for particle velocities as a limit on certain integrals (cf.
CH03 for more detail). In the notation of astrophysical "models'',
where ,
,
and
are the sound speed, vertical scale height, and kinematic viscosity of the nebula gas and
is the orbital frequency.
Völk et al. (1980) introduced the concept of "eddy
classes''. Class I eddies vary slowly enough that a particle, upon entering a
class I eddy, will forget its initial motion and align itself to the gas
motions of the eddy before the eddy decays or the particle leaves the eddy.
Class II eddies, on the other hand, have fluctuation times shorter than the
particle's stopping time ,
and fluctuate too rapidly to provide more than
a small perturbation on the
particle. The timescale on which an eddy decays is
given by t_{k}, while the
eddycrossing timescale is
,
with
the relative velocity between
a grain and an eddy. For an eddy
to be of class I both t_{k} and
must be larger than the
particle's stopping time. The boundary between these classes occurs at k=k^{*} (or at
t_{k} = t^{*}) which can be defined as
(Völk et al. 1980, MMV):

(3) 
It is important to realize that k^{*} (or t^{*}) is a function of stopping time ,
that is, the boundary separating the two classes is different for each particle.
The different treatment for the two eddy classes k<k^{*} and k>k^{*} forms the
core of the derivation of the turbulenceinduced particle velocities.
All turbulent velocities in this note are statistical, rootmeansquare, averaged quantities. The average inertial space particle velocity
is given by
Eq. (6) of MMV.



(4) 
in which
.
The K^{2} term in the first integral results from the more recently preferred "n=1'' gas velocity autocorrelation function (MMV and CH03). The functions
and
with
were first obtained by Völk et al. (1980).
CH03 noted that, for very small particles with
or
,
the second integral becomes negligible, leaving only the first integral which is analytically solvable and for which the upper limit can be extended to
with negligible error.
Here, to generalize the approach of CH03 to particles of arbitrary size, we approximate
for all particle sizes (see CH03 Sect. 2.2.3 for supporting logic). Numerical calculations of
and
validate this approximation to order unity (see Appendix A), and we gain further confidence in it from a posteriori comparison with exact numerical model results. The general expression for
is then the same as in the
regime, and the same analytical result is obtained,
i.e. CH03,
CH03 did not give this explicit result for ,
but merely noted that it was
straightforward to derive it from their Eq. (19) for
and the general
relationship
;
however we will use it explicitly here.
Comparison of the predictions of this simple expression with detailed numerical
results (MMV, CH03) show that it is indeed a good approximation for arbitrary .
A more accurate approximation to Eq. (4), in which the g and h functions are approximated as powerlaws in k^{*}/k, is outlined in Appendix A. Unless
,
we can neglect
the Reynolds number term in Eq. (6) and obtain
,
a well known result (Völk et al. 1980; Cuzzi et al. 1993; Schräpler & Henning 2004) which describes the diffusivity of large particles in turbulence.
3 Results
3.1 kspace approach
MMV (their Eq. (7)) expressed the relative velocities
between particles of different stopping times t_{1} and t_{2} as

(7) 
Having already derived
(i=1,2) above, we can determine
by evaluating the cross term
;
this paper presents analytical solutions of this problem obtained in two separate ways. In this subsection we retain the wavenumber dependence; in the next subsection we transform to time variables. In Eq. (8) of MMV the cross term is given as a sum over the two particle sizes involved, which we separate here, writing
,
where



(8) 
Changing variable to
,
substituting for E(k), and converting stopping time t_{i} to Stokes number
:

(9) 
where we have taken, without loss of generality,
.
The first integral is trivial and the second integral can be solved exactly as in Eqs. (17)(19) of CH03. In evaluating the specific value of the integrals above, we need a closed form for the upper limit
.
A simple prescription is readily found by inspection of Fig. 3 of CH03:
.
That is, the boundary eddy for particles with stopping time t_{1} is that for which
until
,
beyond which it remains constant. This is merely a convenient mathematical shorthand to keep everything in closed form. Then, repeating the analytical solution of CH03 (Eqs. (17)(19)) we obtain

(10) 
This solution for the cross term is easily combined with Eq. (6) to obtain expressions for particleparticle relative velocities
.
Further manipulation of these expressions may be possible, but the important point here is that
can be expressed in closed form as function of
,
,
,
and Re. With a few minutes of algebra, simpler expressions can be found in the limiting regimes of interest (
etc.) which agree well with those which we present in the next section, for analytical solutions obtained in the time domain instead of the wavenumber domain, and where an analytical solution for the boundary k^{*}(t^{*}) is used rather than the form for x_{1}^{*} adopted above. It should be recalled that, for very small particles
,
x_{1}^{*} has an upper limit of
(see, e.g. CH03 Fig. 3).
The integrals expressing
and
are transformed into a simpler
form by changing variables from k to t_{k}. Since
and
E(k) = A k^{5/3} for a Kolmogorov
power spectrum (where A is the normalization factor), we obtain that
.
Now,
from
the normalization of the turbulent spectrum (Eq. (2)),
with
the velocity of the largest eddy, and
also by normalizing the power spectrum (see CH03).
We then end up with

(11) 
which can be substituted into all the integrals, putting them into a simpler form.
For instance, Eq. (5) becomes for particle i

(12) 
Similarly, the cross term becomes
With
)
and
since t^{*} refers to an eddy's turnover time. We now solve for
by splitting
the integral in Eq. (12) at t^{*}_{12} and subtract the corresponding
terms from Eq. (14) to get



(15) 
where the
symbol indicates interchange between particles 1 and 2. With further manipulation and cancellation
of terms, the previous expression simplifies slightly to
This is perhaps the most concise way to write the expressions for
.
The first term we call
since this term involves class
II (fast) eddies. If
(heavy particles) all eddies are fast and only this term
remains. Conversely, if
(small particles) the contribution from
vanishes and the second term,
,
determines relative velocities. In
the intermediate regime,
,
both terms contribute. Written
in terms of the Stokes numbers these terms becomes
Note again that since
we also have that
.
Below, we will first solve
for
,
and then consider solutions for
in various
limiting cases of the particle stopping times.
3.2.1 Solving for t^{*}
The relative velocity between a particle with stopping time
and an eddy k, is given by Völk et al. (1980), Eq. (15):

(19) 
is any systematic velocity component not driven by turbulence  such as due to pressuregradient driven azimuthal headwind, the ensuing radial drift, or vertical settling under solar gravity. We can integrate this equation in the same fashion as Eq. (14) and arrive at
in which
.
Also, using the definition for t_{k} (see text above Eq. (11)), k^{*} can be expressed as
.
Inserting the expressions for k^{*} and
into Eq. (3), assuming that
for simplicity (see however Sect. 3.3), we obtain:

Figure 1:
The function
.
If
and
(no systematic velocity drifts; see Sect. 3.3) this equation is equal to zero
and we find a solution
.
On the other hand, for
,
the RHS of Eq. (24) is 0.5 and
. 
Open with DEXTER 
where we have defined
and
.
The LHS of Eq. (24) is plotted in Fig. 1. If
for small particles, the RHS of Eq. (24) is negligible and the numerical solution
for y^{*} becomes
,
or
.
On the other hand, when
nears ,
the
term causes the RHS of Eq. (24) to drop to 0.5,
and
.
For
we always have that
;
i.e., for such a particle all eddies
are of class 2. In Fig. 2 we compare the exact solution (dashed line)
for t^{*} with the
approximation (solid line; in both cases
is simply enforced), and the empirical function
(dotted line; see Sect. 3.1).
The exact solution for
(Eq. (16)) is given in Fig. 3 both for
(solid curve) and for particles of equal
stopping times (dashed curve). A Reynolds number of
has been adopted.
3.3 The role of V:
eddycrossing effects
Systematic velocities
due to vertical settling, and pressuregradient headwinds and drifts, will occur (e.g. Nakagawa et al. (1986)). Because particles drift through eddies, their transit time is affected (because
is larger) and the boundary between class I and II eddies shifts. Cuzzi et al. (1993) include this effect, due to vertical settling, in their model of particle diffusion (their Eq. (43)). The model presented here offers a generalized way of treating this effect, which we will only sketch here.

Figure 2:
Three different assumptions for t^{*} (or the related k^{*}) are shown here. 
Open with DEXTER 

Figure 3:
Exact solution to Eq. (16) for
in
the case of identical particles (dashed line) and
(solid line) for a Reynold number
of 10^{8}.
The dotted curves are approximations to Eq. (16) given by Eqs. (27)(29). 
Open with DEXTER 
Repeating the procedure outlined in Sect. 3.2.1 but retaining the V_{0} term in
(Eq. (20)), we end up with Eq. (24) including a correction term

(22) 
where we have substituted
.
The correction term can be roughly constrained using an estimate of the systematic drift velocity
,
where
is the Keplerian velocity
at distance a from the Sun,
is the orbit frequency, and
is a radial pressure gradient parameter; also we take
(see,
e.g., Nakagawa et al. 1986 or Cuzzi & Weidenschilling 2006). Then
and Eq. (22) becomes,

(24) 
Normally
is assumed (Cuzzi et al. 1993; Nakagawa et al. 1986), but its real value, and that of ,
are not well known. Equation (24) shows that for a given value of ,
F(y^{*}) increases with increasing
.
Consequently,
is also higher (see Fig. 1). The boundary between the class I and II eddies therefore shifts to higher values of t^{*}, that is, there are less class I eddies for high
and the
upper limit (when
is reached at lower Stokes numbers. Inserting the definition of F(y^{*}) (LHS of Eq. (22)) into Eq. (24) with
and solving for ,
we find that the Stokes number at which
occurs at

(25) 
For example, for
,
reaches its upper limit at
.
In the small particle regime (
), however, the exact value of
is unimportant since F(y^{*}) is always close to zero, and the
approximation is justified. It is only for
that the RHS of
Eq. (24) starts to becomes significant and
.
This is the weaklyturbulent or nonturbulent regime where class II eddies dominate even for small particles. In practise, however, it means that eddy crossing effects are important only if turbulence is very weak and we will not treat them
further in this paper.
As intuitionbuilding examples we obtain simple, closedform expressions for
in various limiting
regimes from the tspace solutions; similar results are easily obtained from the kspace
solutions (Sect. 3.1). Without loss of generality we take particle 1 to have the largest
stopping time, i.e.,
and
t_{12}^{*}=t_{1}^{*}. Moreover, we assume that
;
i.e., that
and there is an extended inertial
range of eddies. Recall again that
for
,
and that
will not exceed 1.
In this limit all eddies are of class I and
.
For each particle, the second term on the RHS of Eq. (17) is negligible; thus

(26) 
In the very small particle regime (
),
and

(27) 
Since
,
this
expression transforms directly to
,
in
good agreement with the heuristic, although physically motivated, expression
of Weidenschilling (1984).
If t_{1} (the stopping time of the larger particle) approaches the Kolmogorov
scale, two changes
occur. First, the
term
in Eq. (17) now becomes linear with
,
since
grows proportional to
(the second term is
still negligible throughout most of this regime). Relative velocities therefore
increase as the squareroot of stopping time. Second, class II eddies also
contribute to
(Eq. (18)). This contribution scales
also with
,
but is significantly larger and does not
disappear when t_{1} = t_{2}. From a physical point of view, class II eddies act
as small, random kicks to the particle trajectory, while two
particles captured by a class I eddy are subject to the same,
systematic, change in motion. Class II eddies are therefore much more effective
in generating velocity differences for similarsized particles.
In the "fully intermediate regime'', i.e.,
,
we can also
ignore the
terms in Eq. (18). In addition, the
approximation holds. Upon writing
,
Eqs. (17), (18) become linear with
and we can write
as (see Appendix B)

(28) 
where
is the ratio between the stopping times and
.
For
we then find that
,
while for equal
particles the numerical factor goes down to 2.0. Written in terms of stopping
times the relative velocities become,
.
This also compares well with Weidenschilling (1984) fits
for this regime (who gives prefactors of 2.1 and 3.0, respectively). Note,
however, that our full expressions for
(Eqs. (16)(18)) also capture the behavior near the
and
"turning points'' (see Fig. 3).

Figure 4:
Contour plots of particleparticle, turbulence induced, relative velocities
normalized to .
A) Numerical results of Markiewicz et al. (1991), without inner scale (
). B) Analogous result from our closedform expressions with the fixed
approximation
(Sect. 3.2.1). C) Like B, but with an exact solution for y^{*} and with
.
D) Using the CH03 formula for k^{*},
,
and also with
.
Contours are drawn twice per logarithmic decade (at
and at 10^{i}) with an additional contour at 0.8 and 1.15. 
Open with DEXTER 
If
,
and there is no contribution from class
I eddies (Eq. (17)). Also, we can neglect the
terms in
Eq. (18) and the relative velocity simply becomes

(29) 
This result can, of course, directly be obtained from the
terms
(Eq. (12)) since the crossterm vanishes in this regime. For small
relative velocities are still ;
however, if both
Stokes numbers are large, the relative velocity decreases roughly with
the square root of the smallest particle stopping time. Note that the linear fit of
Weidenschilling (1984) in this regime (his Eq. (15)) is inappropriate
(see, however, Völk et al. 1980, Weidenschilling 1988, Weidenschilling & Cuzzi 1993, Cuzzi et al. 1993, in which a squareroot fall off is advocated).
Since an explicit, closedform solution to the Völk et al. (1980) and MMV expressions for
has not previously been
available, many dust coagulation models (e.g., Ormel et al. 2007; Dullemond & Dominik 2005; Suttner & Yorke 2001) have relied on the
Weidenschilling (1984) fits to calculate relative velocities.
Turbulent motions and relative velocities for particles in the
regime have therefore been underestimated in these calculations.
However, concerning these works, we also think no major conclusions have been affected, since the error is introduced only for large dust particles, that is, if the system is already well evolved.
3.5 Contour plots
Following Völk et al. (1980) and MMV we also
present our results as contour plots. Figure 4A shows, for comparison, the results of MMV, obtained by numerical evaluation of the integrals involved without an inner turbulent scale (
). The next three panels of Fig. 4 show the result using our closedform expressions derived from Eq. (16). In panel B, the
approximation has been used and, like Fig. 2 of MMV (panel A), the inner scale of the turbulence is extended to infinity so that Eqs. (28), (29) apply.
Somewhat systematically higher values for
when compared to MMV can be explained by the CH03 approximation for
(see Eq. (6)) but these discrepancies are less than 10%.
In panels C and D we show the contour plots corresponding to the other formulations for k^{*} (see Fig. 2), i.e., the exact solution for y^{*} (panel C) and the CH03 empirical approximation (panel D).
The differences between these three methods for determining k^{*} differ around the
point (see Fig. 2) and are reflected in the contour plots. For
,
panel C compares best to the numerical result of MMV, but no significant errors are made when using the
approximation or the CH03 formula for k^{*}.
In panels C and D of Fig. 4, a Reynolds number of
has been adopted. For
,
therefore, velocities are greatly suppressed since only class I eddies remain to generate relative velocities and relative velocities disappear completely for equal friction times. Also, the contours are much closer spaced since in this limit the velocity
is proportional to
(see Eq. (26)).
4 Conclusions
We have extended and, essentially, completed the work of Cuzzi & Hogan (2003),
who derived explicit, closedform expressions for particle velocities in
turbulence based on the physics originally developed by
Völk et al. (1980) and Markiewicz et al. (1991). Within the
framework of this physics, the only approximations used here are in Eq. (6)
for the particle velocities (where a posteriori comparisons with exact numerical solutions indicate the approximation is well justified) and in Eqs. (20) et seq where the
systematic velocity
is neglected to simplify calculating the boundary between eddy
classes (generalizing this step should be straightforward, however). The full analytic
expression for
is given by Eq. (16) (or by the sum of
Eqs. (17), (18)), but more simple, explicit expressions apply in restricted
regimes (provided
):
 Equation (27), in the very small particle limit
(
);
 Equation (28), in the "fully intermediate''
regime, i.e., for
;
 Equation (29), for
.
Near the
and
turning points the behavior is more complex (see Fig. 3) and for accurate analytical approximations
one has to revert to the full expressions for
given by
Eqs. (16)(18).
Acknowledgements
The authors thank Robert Hogan for computational assistance in creating Fig. A.1 and the referee, H. Völk, for positive feedback. C.W.O. acknowledges support from the Netherlands Organisation for Scientific Research (NWO) that made this work possible. J.N.C.'s contributions were
supported by a grant from NASA's Planetary Geology and Geophysics Program.
Appendix A: more accurate closedform solution for
and all related velocities, using powerlaw approximations to the functions g and h
In Sect. 2 the very simple approximation
was introduced for all stokes numbers
and eddy scales k. It proves to be quite adequate for most purposes; however, as noted in Sect. 3.5, small inaccuracies remain at the 10% level because the approximation overestimates the contributions of fast eddies to
and other velocity components. Figure A.1 shows the detailed behavior of the functions g
and h for
for Stokes numbers of
,
0.1, and 1.0. The inflection point for all three values of
is at k=k^{*} (recall that
). For k > k^{*}, the functions are well approximated by powerlaws of 1/3 and 3/4, respectively, i.e.,
g(k)=(k/k^{*})^{1/3} and
h(k)=(k/k^{*})^{3/4}. The success of the approximation of Sect. 2 is due to the fact that the power in the weighting function k E(k) (dotted line; we multiply with k since we compare logarithmically) decreases rapidly with increasing k; thus by the time the assumption
g(k)=h(k)=1 becomes really bad, the relative contribution of successive terms has become small. For small ,
the weighted contribution of eddy power has already become very small even before
(the logic of CH03). For
or larger, the weighting function has dropped by nearly an order of magnitude by the time h(k) (the fasterdecreasing function) has dropped to 0.3 (dasheddotted line), and this seems to account for the success of our simple assumption.

Figure A.1:
The functions g (solid lines) and h (dashed lines) plotted for three different Stokes numbers at a Reynolds number of 10^{8}. After k>k^{*} the functions show powerlaw behavior. The power spectrum (weighted by k) is plotted by the dotted line. 
This behavior can be understood from the definition of
.
For ,
and
are both constant. Then, because
,
scales as
and becomes large at large k. Since
for large ,
we get that
.
Similarly,
,
which is a bit shallower than the 3/4 exponent observed over most regions of interest (Fig. A.1). While the 2/3 exponent is reached at large k, the 3/4 exponent seems more appropriate at intermediate k. Yet, in our subsequent analysis, we will use the largek limit for this exponent (2/3) because it simplifies the math. Thus, we approximate the g and h behavior as follows: unity for k<k^{*}, and power laws in k/k^{*} with exponents of 1/3 and 2/3 for k>k^{*}. Then Eq. (4) becomes



(A.1) 
Where we have that
,
such that in the case of very small or very large particles one of the integrals vanishes (Sect. 3.2.1). Since the approximation g=h=1 still holds for k<k^{*} (or for t>t^{*}) the velocities resulting due to class 1 eddies (Eq. (17)) are not affected; the new approximation only affects Eq. (18). By writing
,
with
the solution to Eq. (A.1) involves integrals of the form

(A.2) 
with n=2 for the K^{2} term and p=1/3 or 2/3. These integrals can be solved analytically. Going to "tspace'', however, gives somewhat cleaner solutions and we will from here on follow that approach and show how it affects relative velocities, i.e.,
.
After the change of variables (
t_{k}/t^{*} = (k/k^{*})^{2/3}) the second term of Eq. (A.1) becomes

(A.3) 
We now introduce the dimensionless variable
(cf. Eq. (20)). Then
t_{k}/t^{*} = y/y^{*} with
.
Also K=1/(1+y) and
1K=y/(1+y) and
Eq. (A.3) becomes



(A.4) 



(A.5) 
in which the functions
and
are defined as
The expressions for
now consist of several contributions. First,
and
are evaluated at both the upper (y^{*}) and lower ()
limits. This must be done for both particles 1 and 2,
because the
term (Eq. (18)) has separate contributions from each
particle. For the particle of highest friction time (say this is t_{1}) the powerlaw approximation for g and h holds over the range
is calculated, i.e.,
.
However, for the second particle the powerlaw approximation only holds for
,
while for the remaining range over which the integral in
is evaluated, i.e.,
,
the g=h=1 approximation applies.
This gives us several terms that contribute to
.
Collecting these terms, the new expression for
becomes
Although still fully analytical, this more accurate expression for
is also more complicated and we did not present it in the main body of the paper.
Equation (A.8) is useful, however, for readers whose applications demand this higher level of accuracy.
Appendix B: Derivation of Eq. (28)
We consider the limiting case of
.
The
approximation for
then holds, i.e.,
with
.
We will now argue that we can neglect the
terms in Eq. (18). For particle 1 this is obvious since
.
The last term (where
is in the denominator) then becomes simply
.
However, for the interchange term a similar approximation

(B.1) 
is not that obvious since we have not put a constraint on
.
For example, if
the
term dominates the denominator. However, in that case this term and its approximation are small anyway compared to
,
such that by making the approximation in Eq. (B.1) our final result is notaffected. Similarly, if
,
Eq. (B.1) (which goes to
)
or its approximation (
)
are insignificant since
.
Only if
,
i.e.,
,
does the
term
matter, but then the approximation in
Eq. (B.1) is well justified.
All terms in Eq. (18) are then linear in Stokes and we can reduce it to

(B.2) 
with
.
Similarly, Eq. (17) becomes

(B.3) 
Combining these expressions and collecting the
and
terms then gives Eq. (28).
 Cuzzi, J. N.,
& Hogan, R. C. 2003, Icarus, 164, 127 [NASA ADS] [CrossRef] (In the text)
 Cuzzi, J. N.,
Dobrovolskis, A. R., & Champney, J. M. 1993, Icarus,
106, 102 [NASA ADS] [CrossRef]
 Cuzzi, J. N.,
& Weidenschilling, S. J. 2006, ParticleGas Dynamics and
Primary Accretion, Meteorites and the Early Solar System II,
353
(In the text)
 Dullemond,
C. P., & Dominik, C. 2005, A&A, 434, 971 [NASA ADS] [CrossRef] [EDP Sciences]
 Markiewicz,
W. J., Mizuno, H., & Völk, H. J. 1991, A&A,
242, 286 [NASA ADS] (In the text)
 Nakagawa, Y., Sekiya,
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