Astronomy & Astrophysics

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$\begin{figure} \par\includegraphics[width=8cm,clip]{2615f1.eps} \end{figure}$ Figure 1: The four weighting schemes (A, B, C, and D) used in the Monte Carlo simulations in this paper. The figures illustrate a case where the sampling is improved in the centre of a spherically symmetric model cloud: background packages are sent preferentially towards the cloud centre (scheme A); more photon packages are created in the inner parts (scheme B); within the cloud photon packages are sent preferentially towards the cloud centre (scheme C); and scattered photon packages are directed preferentially towards the cloud centre (scheme D).
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{2615f2.eps} \end{figure}$ Figure 2: Convergence of calculated dust temperatures in model S₁ as the function of the number of photon packages per iteration and frequency. Results are shown for both normal Monte Carlo calculations (filled squares) and calculations where weighting was applied to both dust emission and background emission (open circles; see text for details). The upper curves show the maximum relative error in any of the shells, and the lower curves show the rms-value of the relative errors summed over all shells. The dotted line shows the expected N^-0.5 convergence rate.
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{2615f3.eps} \end{figure}$ Figure 3: Convergence of calculated dust temperatures in model S₂ vs. number of photon packages per iteration and frequency. The symbols are as in Fig. 2.
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{2615f4.eps} \end{figure}$ Figure 4: Rms-value of relative dust temperature errors as function of the number of simulated photon packages per iteration and frequency. Results are shown for model N₁ for the normal Monte Carlo simulations (filled squares) and for weighting scheme B where relatively more photon packages are created close to the centre of the model (open circles; see text for details).
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{2615f5.eps} \end{figure}$ Figure 5: Convergence of dust temperature calculations in model N₂ as the function of the number of photon packages per iteration and frequency (symbols as in Fig. 4).
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In the text

$\begin{figure} \par\includegraphics[width=7.5cm,clip]{2615f6.eps} \end{figure}$ Figure 6: Convergence of dust temperatures in model N₂ in the case of normal iterations and with the use of the extrapolation method. The plotted values are rms-values of the relative temperature errors.
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In the text

$\begin{figure} \par\includegraphics[width=7.5cm,clip]{2615f7.eps} \end{figure}$ Figure 7: Convergence of dust temperatures in model N₂ for diagonal AMC-method (solid line), diagonal AMC with extrapolation (dotted line) and AMC with tridiagonal operator (dash-dotted line).
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In the text

$\begin{figure} \par\includegraphics[width=7.5cm,clip]{2615f8.eps} \end{figure}$ Figure 8: Convergence of dust temperatures in model N₁ with (filled symbols) and without (open symbols) the use of a reference field when the number of photon packages per iteration is equal. The rms-values of relative temperature errors are plotted as function of the number of iterations.
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In the text

$\begin{figure} \par\includegraphics[width=7.5cm,clip]{2615f9.eps} \end{figure}$ Figure 9: Convergence as function of the number of photon packages for model S₁ with normal (squares) and weighted Monte Carlo (circles). Results obtained with pseudo random numbers (open symbols) and quasi random numbers (filled symbols) are shown.
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In the text

$\begin{figure} \par\includegraphics[width=7.5cm,clip]{2615f10.eps} \end{figure}$ Figure 10: Comparison of run times between our program (solid squares) and the Bjorkman & Wood (2001) method (triangles), as implemented in the MC3D program of Wolf (2003). The figures show rms-errors of the computed dust temperatures as functions of CPU-time. The two frames correspond to models with central optical depth $\tau _{\rm V}=100$ and $\tau _{\rm V}=1000$ . The dotted lines indicate run times of MC3D program when the longer initialisation time (pre-calculation of Planck mean opacities etc.) has been subtracted.
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In the text

$\begin{figure} \par\includegraphics[width=7.5cm,clip]{2615f11.eps} \end{figure}$ Figure 11: Dust temperatures in a model with a central black body source and a surrounding dust cloud with an optical depth $\tau=10~000$ at all frequencies. The solid line shows the results when emission was simulated only from those regions where distance to the closest cell boundary was below $\tau _{\rm abs}=15$ (see text). The solid squares correspond to a run with normal weighting (scheme B) and a factor of 4 larger number of photon packages. A direct numerical solution for a case where energy transfer across a surface is $\propto$ $A \times T_{{\rm dust}}^4$ (infinite optical depth) is drawn with a dashed line.
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{2615f3.eps} \end{figure}$	Figure 3: Convergence of calculated dust temperatures in model S₂ vs. number of photon packages per iteration and frequency. The symbols are as in Fig. 2.
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$\begin{figure} \par\includegraphics[width=8.8cm,clip]{2615f4.eps} \end{figure}$	Figure 4: Rms-value of relative dust temperature errors as function of the number of simulated photon packages per iteration and frequency. Results are shown for model N₁ for the normal Monte Carlo simulations (filled squares) and for weighting scheme B where relatively more photon packages are created close to the centre of the model (open circles; see text for details).
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$\begin{figure} \par\includegraphics[width=8.8cm,clip]{2615f5.eps} \end{figure}$	Figure 5: Convergence of dust temperature calculations in model N₂ as the function of the number of photon packages per iteration and frequency (symbols as in Fig. 4).
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$\begin{figure} \par\includegraphics[width=7.5cm,clip]{2615f6.eps} \end{figure}$	Figure 6: Convergence of dust temperatures in model N₂ in the case of normal iterations and with the use of the extrapolation method. The plotted values are rms-values of the relative temperature errors.
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$\begin{figure} \par\includegraphics[width=7.5cm,clip]{2615f7.eps} \end{figure}$	Figure 7: Convergence of dust temperatures in model N₂ for diagonal AMC-method (solid line), diagonal AMC with extrapolation (dotted line) and AMC with tridiagonal operator (dash-dotted line).
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$\begin{figure} \par\includegraphics[width=7.5cm,clip]{2615f8.eps} \end{figure}$	Figure 8: Convergence of dust temperatures in model N₁ with (filled symbols) and without (open symbols) the use of a reference field when the number of photon packages per iteration is equal. The rms-values of relative temperature errors are plotted as function of the number of iterations.
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$\begin{figure} \par\includegraphics[width=7.5cm,clip]{2615f9.eps} \end{figure}$	Figure 9: Convergence as function of the number of photon packages for model S₁ with normal (squares) and weighted Monte Carlo (circles). Results obtained with pseudo random numbers (open symbols) and quasi random numbers (filled symbols) are shown.
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