4. Results and discussion

The Planck mean cross-sections for various prolate and oblate spheroids and spheres of different sizes and chemical compositions have been calculated. Then the cross-sections were averaged over the grain orientations and both the radiation pressure force and the dust drift velocities were estimated.

The radial component of the radiation pressure force determines the dust drift velocity and as a result the radiation momentum transfer to the gas and the radial gas outflow velocity. To evaluate the grain shape effects we calculated the ratios of the forces and of the velocities for spheroidal and spherical particles of the same volume. The Planck mean factors for spheres presented in tables can be used for estimations of the absolute values of drift velocities. To illustrate its relative strength the transversal component of the radiation pressure force is compared with the radial one .

4.1. Radial radiation pressure

The values of and for subsonic motion (n=1 in Eq. (21 (click here))) are given in Table 1 (click here) for the particles consisting of amorphous carbon and astronomical silicate. The data are presented for two cases of dynamically aligned grains ( and ), the ratios for other angles lie between these values. For 3D-orientation, the approximation

works sufficiently well and the results obtained are not shown in Table 1 (click here).

  
Figure 2: The shape dependence of the ratio of radiation pressure forces for prolate (solid curves; a)) and oblate (dashed curves; b)) spheroids with 3D-orientation and spheres. The composition of absorbing materials is indicated

 

Prolate spheroid Oblate spheroid Sphere a/b

0.05 1.30 1.04 1.03 1.07 1.40 0.99 0.88 1.25  0.071

0.10 1.29 1.01 1.02 1.04 1.39 0.97 0.87 1.22 0.23

0.20 1.13 0.97 0.90 1.00 1.20 0.96 0.76 1.21 0.89

0.30 1.06 0.98 0.84 1.01 1.10 0.98 0.69 1.23 1.48

0.50 1.06 0.98 0.84 1.00 1.09 0.95 0.69 1.20 1.93

0.05 1.78 1.21 1.12 1.12 1.99 1.19 0.79 1.89

0.10 1.70 1.09 1.07 1.01 1.91 1.09 0.76 1.72

0.20 1.31 0.95 0.82 0.88 1.48 0.95 0.59 1.51

0.30 1.16 0.96 0.73 0.89 1.25 0.95 0.49 1.51

0.50 1.16 1.00 0.73 0.93 1.17 0.97 0.46 1.53

0.05 1.17 0.99 0.93 1.02 1.27 0.96 0.80 1.21  0.016

0.10 1.23 0.95 0.97 0.98 1.37 0.90 0.86 1.14  0.064

0.20 1.15 0.93 0.91 0.95 1.30 0.88 0.82 1.11 0.31

0.30 1.03 0.94 0.82 0.96 1.10 0.92 0.69 1.16 0.63

0.50 0.99 0.95 0.78 0.98 0.96 0.95 0.61 1.20 1.06

0.05 1.38 1.04 0.87 0.96 1.60 1.02 0.64 1.62

0.10 1.42 0.91 0.89 0.84 1.70 0.87 0.68 1.38

0.20 1.26 0.82 0.80 0.76 1.56 0.76 0.62 1.21

0.30 1.07 0.82 0.67 0.76 1.22 0.78 0.48 1.24

0.50 0.98 0.86 0.62 0.79 0.90 0.84 0.36 1.33

Table 1: The ratios of the radiation pressure force and the drift velocity for spheroids and spheres having an equal volume

 

As follows from Table 1 (click here) and Fig. 2 (click here), the ratio usually grows with the increase of the aspect ratio a/b and the absorptive capacity of the grain material. A very significant effect is found for small metallic spheroids (Fig. 2 (click here)). It was firstly mentioned by Il'in (1994) and can be explained by extremely large absorption of radiation with the electric vector oscillating along the major axis of a particle (TM mode for a prolate spheroid and TE mode for a oblate one). This fact is well known in the theory of electromagnetic radiation scattering (van de Hulst 1957) and is related to different behaviour of surface waves for conductors. It is obvious that the maximum effect occurs when the major semiaxis of a particle is perpendicular to the wave-vector, i.e. in the case of 2D-orientation and .

If we consider the silicate grains, the ratio reaches a maximum with the growth of the particle size ( if  K). After the maximum, the ratio becomes below 1, if the major axis of a rotating grain is not perpendicular to the wave-vector (i.e. ). This is connected with the growing role of the scattered radiation that leads to an increase of the angle between and (see Fig. 1 (click here)). Therefore, the projection of on the radial direction decreases.

The common feature of the grain motion is that spheroids with size are blown slightly faster than equivolume spheres. For strongly absorbing grains the effect can be extremely large. For instance, the drift velocity of even nonaligned iron spheroids and spheres may differ in or more times.

For large aligned particles, the difference in the final velocities of spheres and spheroids mainly depends on the ratio of the average geometrical cross-sections (see Appendix A). However, a difference in radiation pressure is present as well. For example, for oblate particles, the ratio of their velocities (see two last columns of Table 1 (click here)) is not equal to as it should be without this influence.

It should be also noted that in the case of 2D-orientation the grain shape effects are more pronounced for oblate particles in comparison with prolate ones (if we compare the particles of the same volume with ). They enlarge with the growth of the aspect ratio a/b and become maximal for needle-like particles.

4.2. Transversal radiation pressure

The appearance of the transversal component of the radiation pressure force is explained by the azimuthal asymmetry of the radiation recoil for a non-spherical particle. The strength of this component is tightly connected with the particle orientation relative to the wave-vector of incident radiation and also strongly depends on the aspect ratio, size and composition of grains. It should be noted that we study the scattered radiation in a Cartesian coordinate system in which the z-axis coincides with the rotation axis of a spheroid. For oblate particles, this leads to the change of the direction of in Fig. 1 (click here). Formally, it means that the second term in Eq. (10 (click here)) becomes larger than the first one, i.e. .

The transversal component can be very large and reaches up to 30% or more of the radial one in the cases considered (Table 2 (click here), Fig. 3 (click here)).

  
Figure 3: The angular dependence of the transversal component of the radiation pressure force for dynamically aligned prolate (solid curves) and oblate (dashed curves) spheroids consisting of amorphous carbon a) and astronomical silicate b). The values of the aspect ratio a/b are indicated

 

Prolate spheroid Oblate spheroid

a/b=1.1 1.5 2.0 4.0 1.1 1.5 2.0 4.0

0.10 0.15 0.51 0.83 1.61 -0.19 -1.05 -1.85    -3.29

0.20 0.54 1.97 3.31 6.35 -0.62 -3.48 -6.31 -12.1

0.30 0.64 2.45 4.22 8.51 -0.78 -3.90 -6.82 -13.9

0.50 0.37 1.84 3.52 8.13 -0.58 -2.23 -3.67    -9.20

0.10 0.33 1.16    1.90    3.75 -0.39    -2.04    -3.44    -6.12

0.20 1.37 5.01    8.07 13.3 -1.73    -8.59 -14.3 -23.7

0.30 1.84 7.33 12.1 19.2 -2.74 -12.0 -19.4 -32.3

0.50 1.31 7.38 13.0 20.8 -2.76 -10.9 -18.2 -28.6

Table 2: The ratio of the transversal and radial components of the radiation pressure force for dynamically aligned spheroidal grains,

 

The growth of the ratio with the particle size reflects the increasing azimuthal asymmetry of phase function and particle's albedo. After a peak value (near ), the ratio slightly decreases because of more symmetric azimuthal scattering of radiation by larger particles.

As follows from Fig. 3 (click here), the values of at which the ratio reaches a maximum depend on the particle size and are in the interval . Note that if the particle shape is not axially symmetric particles, the angular dependence of the transversal component and its strength may differ significantly from that presented in Fig. 3 (click here).

  
Figure 4: The shape dependence of the transversal component of the radiation pressure force for dynamically aligned () prolate (solid curves) and oblate (dashed curves) spheroids. The composition of silicate material is indicated

The higher absorptive capacity (or dirtiness) of silicates leads to the smaller ratios of . This fact is illustrated by Fig. 4 (click here), where the distinction between three silicate species is mainly related to the difference in particle's albedo.

The role of the transversal component of the radiation pressure force may be as follows. First, as the force is not parallel to the incident radiation wave-vector (and hence the radius-vector), the more is , the less is the radial drift velocity of grains. Second, the path length of a non-spherical dust grain through the shell (and therefore the number of collisions with gas particles) should be larger in comparison with the spherical particle of the same volume. Third, provided there is a preferable stable orientation of grains, a non-radial outflow of gas may be initiated.

4.3. Porous grains

With the growth of the fraction of vacuum f the grain porosity increases and the effective refraction index of a particle reduces. This leads to a decrease of the ratios and for metallic grains as the radiation pressure is mainly determined by absorption.

For silicate particles, the situation is more complicated: both the radial and transversal components of the radiation pressure force can decrease with increasing porosity of circumstellar grains as it is seen from Table 3 (click here). But the effect appears only for aggregates containing more than % of vacuum.

 

1.0 1.07 0.82   19.2% 0.63

0.9 1.07 0.82 19.3 0.52

0.7 1.07 0.83 19.2 0.33

0.5 1.07 0.84 18.1 0.18

0.3 1.07 0.86 15.7 0.07

0.1 1.04 0.91    8.63  0.012

Table 3: The behaviour of radiation pressure force for porous prolate spheroidal grains consisting of astronomical silicate, 2D-orientation, , a/b=4

 

It should be also mentioned that the approach used (Sect. 2.2.4) implies the uniform distribution of vacuum in a particle. But if a grain has one or several bubbles, the optical properties of such hollow particle may strongly differ from those of both the porous and compact particles of the same volume. The radiation pressure on a hollow dielectric particle may exceed that on a compact one because of the total internal reflection. Such an effect for infinite cylinders has been demonstrated by Kokody (1989).

4.4. Effects of stellar temperature variations and
polarization

A decrease of the stellar temperature is usually accompanied with an increase of the radial component of the radiation pressure force acting on a non-spherical particle. This fact is illustrated by Fig. 5 (click here), where the ratio of the drift velocities is plotted. Note that the values given for the effective temperatures as low as 500 K are related to the possible conditions in the outer layers of the optically thick envelopes. It is seen that the distinction is not very large and is more emphasized for carbon grains and larger values of the aspect ratio. The ratio is independent of when the grain size .

  
Figure 5: The temperature dependence of the drift velocity for prolate (solid curves) and oblate (dashed curves) spheroids from amorphous carbon a) and astronomical silicate b) in the case of 3D-orientation. The values of the aspect ratio a/b are indicated

Any increase of the stellar temperature reflects in a rise of the transversal component (Fig. 6 (click here)). This is due to the growth of the effective size parameter of the particle for hotter stars and hence the increase of the role of scattering.

  
Figure 6: The temperature dependence of the transversal component of the radiation pressure force for dynamically aligned prolate (solid curves) and oblate (dashed curves) spheroids. The values of the aspect ratio a/b are indicated

Since many late-type giants possess brightness variations, the common feature should be an increase of the radial component and a decrease of the transversal one during brightness minima in comparison with maxima.

Aligned non-spherical grains in the inner parts of the envelopes can polarize stellar radiation. In principle, the local polarization might be rather large. Then the polarized radiation incident on grains in the outer parts can to strengthen or weaken the shape effects discussed above.

4.5. Optically thick shells

Equation (4) used above to evaluate the radiation pressure force is strictly applicable only for optically thin shells. In the case of large optical thickness, one should insert into it the integration over all solid angles and take into account the angular dependence of the flux that might be obtained from radiative transfer modelling.

When the anisotropy of the flux is not pronounced both components of the radiation pressure force should be limited. On the other hand, even if has a sharp maximum but mainly includes thermal radiation of dust, the transversal component should be small because of low temperature of the radiation source (Fig. 6 (click here)).

For small particles, the shape effectthe difference of the radiation pressure forces acting on a spheroid and a equivolume sphere should be essential also in the case of optically thick shells as the effect is weakly affected by the source temperature (see Fig. 5 (click here)). Thus, if has an anisotropy, radiation pressure will occur and the shape effect will depend on the orientation of non-spherical particles relative to the direction of maximum flux.