All Figures

$\begin{figure} \par\includegraphics[width=7.45cm,clip]{5210fig1.ps}\end{figure}$ Figure 1: Ratio of the radiation pressure force to the gravitational force vs. grain size for different grain materials and porosities, P, calculated for an A5V ( $\beta$ Pictoris-like) star. The thick lines represent silicate Mg $_{\rm0.95}$ Fe $_{\rm0.05}$ SiO $_{\rm 3}$ . The thin lines are used for water ice. The solid lines are for solid grains, the dashed lines are for 50% porous grains, and the dash-dotted lines are for 80% porous grains.
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In the text

$\begin{figure} \par\includegraphics[height=4.25cm,width=4.25cm,angle=-90,clip]{5... ...cludegraphics[height=3.95cm,width=5.25cm,angle=-90,clip]{5210f2d.ps}\end{figure}$ Figure 2: Nominal case. Color-coded maps (log-scale) of the vertical optical thickness of avalanche grains, $\tau _{\rm\perp ,{\rm av}}$ , at different stages of the avalanche evolution (t= 0.6, 5, 10, 40 orbital periods at 20 AU). The planetesimal debris are released at t=0 at 20 AU from the star. Field particles are not included in the plots. The position of the star is marked by the white cross.
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In the text

$\begin{figure} \par\includegraphics[width=7.5cm,clip]{5210fig3.ps}\end{figure}$ Figure 3: Time evolution of the cross-sectional area amplification factor (the ratio of the total cross-sectional area of the avalanche grains within 500 AU to its initial value at t=0). Initial increase is due to dust production by outflowing planetesimal debris colliding with the disc material. When the grain removal (due to star radiation pressure) rate exceeds the grain production, the value of F drops (see text for more details).
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In the text

$\begin{figure} \par\includegraphics[width=7.8cm,clip]{5210fig4.ps}\end{figure}$ Figure 4: Ratio of the total number of grains produced by the avalanche by the time t, $N_{\rm tot}$ , to the initial number of released planetesimal debris, N₀.
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In the text

$\begin{figure} \par\includegraphics[width=7.65cm,clip]{5210fig5.ps}\end{figure}$ Figure 5: Maximum amplification factor as a function of the minimum size assumed for the initial planetesimal debris. The power-law index for the size distribution is equal to its nominal value $p_{\rm0,pl}=3.5$ . For smaller $p_{\rm0,pl}$ the variation with $s_{\rm min,c}$ is less pronounced.
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In the text

$\begin{figure} \par\includegraphics[width=7.65cm,clip]{5210fig6.ps}\end{figure}$ Figure 6: Dependence of the maximum amplification factor on the power-law index of the initial size-frequency distribution of the planetesimal debris (Eq. (8)), $s_{\rm min,pl}=0.1~\mu$ m.
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In the text

$\begin{figure} \par\includegraphics[width=7.6cm,clip]{5210fig7.ps}\end{figure}$ Figure 7: Dependence of the maximum amplification factor on the location of the primary planetesimal breakup.
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In the text

$\begin{figure} \par\includegraphics[width=7.85cm,clip]{5210fig8.ps}\end{figure}$ Figure 8: Radial velocities vs. distance from the star for grains with different $\beta$ values released by parent bodies on circular orbits for a $\beta$ Pic-like star. The release distances are 20 AU and 70 AU.
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In the text

$\begin{figure} \par\includegraphics[width=7.6cm,clip]{5210fig9.ps}\end{figure}$ Figure 9: Dependence of the maximum amplification factor on the minimum size for debris produced in avalanche collisions (Eq. (12) with $q_1=1.5, q_2=1.83, m_s=M_{\rm lf}/3$ ).
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In the text

$\begin{figure} \par\includegraphics[width=7.6cm,clip]{5210f10.ps}\end{figure}$ Figure 10: Dependence of the maximum amplification factor on the value q₁ in Eq. (12) for collisionally produced grains (q₂=1.83).
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In the text

$\begin{figure} \par\includegraphics[width=7.4cm,clip]{5210f11.ps}\end{figure}$ Figure 11: Maximum amplification factor as a function of porosity for pure silicate grains. The value of the threshold energy, Q^*, is assumed to be the same as in the nominal case.
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In the text

$\begin{figure} \par\includegraphics[width=7.6cm,clip]{5210f12.ps}\end{figure}$ Figure 12: Maximum amplification factor vs. value of the threshold energy, Q^*. Values for s₀=1 cm grains are denoted on the axis. For the other sizes the threshold energy is given by Eq. (11).
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In the text

$\begin{figure} \par\includegraphics[width=7.8cm,clip]{5210f13.ps}\end{figure}$ Figure 13: Different test radial distributions for $\tau _{\rm\perp }$ ( $\tau_{\Vert}$ being constant). The thick solid line is the distribution for the nominal case, taken from Augereau et al. (2001).
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In the text

$\begin{figure} \par\includegraphics[width=7.65cm,clip]{5210f14.ps}\end{figure}$ Figure 14: Maximum amplification factor as a function of $\tau_{\Vert}$ . The open circles are the results of our simulations. The dashed line is the theoretical prediction (Eq. (16)).
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In the text

$\begin{figure} \par {\hspace*{6mm}\includegraphics[angle=-90,width=6.4cm,clip]{5... ...*{6mm}}\vspace*{3mm} \includegraphics[width=7.7cm,clip]{5210f15b.ps}\end{figure}$ Figure 15: Top panel: face-on case: color-coded map of the ratio between the geometric surface densities of the avalanche grains and that of the "field'' population for the nominal case. Bottom panels: edge-on case: midplane fluxes (arbitrary units) for the nominal case at avalanche maximum, edge-on orientation. The 2 solid lines indicates the total midplane fluxes ("field''+avalanche) for each side of the disc (differences between the 2 sides are so small here that the 2 lines are almost indistinguishable). The dashed lines show the midplane fluxes for just the avalanche particles. Plot a) corresponds to the forward scattering function and b) to the isotropic case.
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In the text

$\begin{figure} \par {\hspace*{6mm}\includegraphics[angle=-90,width=6.5cm,clip]{5... ...s}\hspace*{6mm}} \par\includegraphics[width=7.7cm,clip]{5210f16b.ps}\end{figure}$ Figure 16: Same as Fig. 15, but for collisionally weaker grains with $Q^*_0(s_0=1~{\rm cm})=10^6$ erg/g (see Sect. 4.4).
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In the text

$\begin{figure} \par {\hspace*{6mm}\includegraphics[angle=-90,width=6.5cm,clip]{5... ...{6mm}}\vspace*{3mm} \includegraphics[width=7.95cm,clip]{5210f17b.ps}\end{figure}$ Figure 17: Same as Fig. 15, but for the case of a disc 5 times more massive than in the nominal case.
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In the text

$\begin{figure} \par {\hspace*{6mm}\includegraphics[angle=-90,width=6.5cm,clip]{5... ...*{6mm}}\vspace*{3mm} \includegraphics[width=7.7cm,clip]{5210f18b.ps}\end{figure}$ Figure 18: Same as Fig. 15, but for a vertically thinner disc, described by Eq. (19). The total mass of the disc is the same as in the nominal case, but the asymmetries become prominent.
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In the text

$\begin{figure} \par\includegraphics[width=7.7cm,clip]{5210f19.ps}\end{figure}$ Figure B.1: The total number of SPs as a function of time for a typical nominal case run.

In the text

$\begin{figure} \par\includegraphics[width=7.8cm,clip]{5210fig4.ps}\end{figure}$	Figure 4: Ratio of the total number of grains produced by the avalanche by the time t, $N_{\rm tot}$ , to the initial number of released planetesimal debris, N₀.
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$\begin{figure} \par\includegraphics[width=7.65cm,clip]{5210fig5.ps}\end{figure}$	Figure 5: Maximum amplification factor as a function of the minimum size assumed for the initial planetesimal debris. The power-law index for the size distribution is equal to its nominal value $p_{\rm0,pl}=3.5$ . For smaller $p_{\rm0,pl}$ the variation with $s_{\rm min,c}$ is less pronounced.
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$\begin{figure} \par\includegraphics[width=7.65cm,clip]{5210fig6.ps}\end{figure}$	Figure 6: Dependence of the maximum amplification factor on the power-law index of the initial size-frequency distribution of the planetesimal debris (Eq. (8)), $s_{\rm min,pl}=0.1~\mu$ m.
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$\begin{figure} \par\includegraphics[width=7.6cm,clip]{5210fig7.ps}\end{figure}$	Figure 7: Dependence of the maximum amplification factor on the location of the primary planetesimal breakup.
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$\begin{figure} \par\includegraphics[width=7.85cm,clip]{5210fig8.ps}\end{figure}$	Figure 8: Radial velocities vs. distance from the star for grains with different $\beta$ values released by parent bodies on circular orbits for a $\beta$ Pic-like star. The release distances are 20 AU and 70 AU.
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$\begin{figure} \par\includegraphics[width=7.6cm,clip]{5210fig9.ps}\end{figure}$	Figure 9: Dependence of the maximum amplification factor on the minimum size for debris produced in avalanche collisions (Eq. (12) with $q_1=1.5, q_2=1.83, m_s=M_{\rm lf}/3$ ).
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$\begin{figure} \par\includegraphics[width=7.6cm,clip]{5210f10.ps}\end{figure}$	Figure 10: Dependence of the maximum amplification factor on the value q₁ in Eq. (12) for collisionally produced grains (q₂=1.83).
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$\begin{figure} \par\includegraphics[width=7.4cm,clip]{5210f11.ps}\end{figure}$	Figure 11: Maximum amplification factor as a function of porosity for pure silicate grains. The value of the threshold energy, Q^*, is assumed to be the same as in the nominal case.
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$\begin{figure} \par\includegraphics[width=7.6cm,clip]{5210f12.ps}\end{figure}$	Figure 12: Maximum amplification factor vs. value of the threshold energy, Q^*. Values for s₀=1 cm grains are denoted on the axis. For the other sizes the threshold energy is given by Eq. (11).
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$\begin{figure} \par\includegraphics[width=7.8cm,clip]{5210f13.ps}\end{figure}$	Figure 13: Different test radial distributions for $\tau _{\rm\perp }$ ( $\tau_{\Vert}$ being constant). The thick solid line is the distribution for the nominal case, taken from Augereau et al. (2001).
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$\begin{figure} \par\includegraphics[width=7.65cm,clip]{5210f14.ps}\end{figure}$	Figure 14: Maximum amplification factor as a function of $\tau_{\Vert}$ . The open circles are the results of our simulations. The dashed line is the theoretical prediction (Eq. (16)).
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$\begin{figure} \par {\hspace{6mm}\includegraphics[angle=-90,width=6.5cm,clip]{5... ...s}\hspace{6mm}} \par\includegraphics[width=7.7cm,clip]{5210f16b.ps}\end{figure}$	Figure 16: Same as Fig. 15, but for collisionally weaker grains with $Q^*_0(s_0=1~{\rm cm})=10^6$ erg/g (see Sect. 4.4).
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$\begin{figure} \par {\hspace{6mm}\includegraphics[angle=-90,width=6.5cm,clip]{5... ...{6mm}}\vspace{3mm} \includegraphics[width=7.95cm,clip]{5210f17b.ps}\end{figure}$	Figure 17: Same as Fig. 15, but for the case of a disc 5 times more massive than in the nominal case.
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$\begin{figure} \par {\hspace{6mm}\includegraphics[angle=-90,width=6.5cm,clip]{5... ...{6mm}}\vspace*{3mm} \includegraphics[width=7.7cm,clip]{5210f18b.ps}\end{figure}$	Figure 18: Same as Fig. 15, but for a vertically thinner disc, described by Eq. (19). The total mass of the disc is the same as in the nominal case, but the asymmetries become prominent.
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