All Figures

$\begin{figure} \scalebox{0.7} { \begin{picture} (0,0) \includegraphics{4717fig1... ...(50,32){\makebox(0,0){\strut{}\Large symmetry axis}} \end{picture}} \end{figure}$ Figure 1: Projection of an axisymmetric distribution of a physical quantity. The ellipse at the top marks the region where the kernel function corresponding to the projection along the line-of-sight is non-zero for fixed R and Z.
Open with DEXTER

In the text

$\begin{figure} \scalebox{0.45} { \begin{picture} (0,0) \includegraphics{4717fi... ... }}} \put(177,10){\makebox(0,0){\strut{}\Huge (b)}} \end{picture}} \end{figure}$ Figure 2: Ratio of the reconstructed to the original X-ray emissivity after n=30 iteration steps, $\phi _n/\phi$ . The halo centre is at the centre left of each plot. Panel a) shows the ratio obtained without regularisation. One can clearly see the spike-shaped artifacts of the reconstruction. Panel b) shows the ratio for the reconstruction including the regularisation. It is close to unity everywhere except very close to the centre, where the algorithm converges slowly and grid resolution becomes a problem.
Open with DEXTER

In the text

$\begin{figure} \scalebox{0.45} { \begin{picture} (0,0) \includegraphics{4717fi... ... }}} \put(177,10){\makebox(0,0){\strut{}\Huge (b)}} \end{picture}} \end{figure}$ Figure 3: Ratio of the reconstructed to the original density and temperature for the analytic halo after n=20 iterations. An inclination angle of $i=70^\circ$ was chosen and assumed to be known in performing the reconstruction. Panel a) shows the density ratio $\rho _n/\rho$ and panel b) the temperature ratio T_n/T. In the central region, the errors are of the order of $1\%$ . We plot the ratios for all R and Z values possible within the box used for the reconstruction (see Fig. 1). This causes the star-like shape of the perimeter of the plot in the R, Z plane. Close to that boundary, at large R or Z values, the ratios can significantly differ from unity. This is, however, expected because the ellipses used in the reconstruction of $\rho$ and T at those points lie mostly outside the maps of $\psi _{{\rm X-ray}}$ and $\psi _{{\rm SZ}}$ .
Open with DEXTER

In the text

$\begin{figure} \scalebox{0.7}{ \begin{picture} (0,0)% \includegraphics{4717fig4}... ...rature reconst., num. halo with noise}}% \end{picture}% \endgroup } \end{figure}$ Figure 4: Dependence of the quality of the deprojection on the inclination of the symmetry axis. For the analytic halo, which was deprojected as described above, we show the volume-weighted rms relative errors $(\rho _n-\rho )/\rho$ and (T_n-T)/T as functions of the inclination angle i after n=20 iterations. We also show the same errors after n=5 iterations for reconstructions of a numerical halo from maps with observational noise. A detailed description of these reconstructions is given in Sect. 4. The averages of the errors were computed within a sphere of radius $500~ h^{-1}~ {\rm kpc}$ around the halo centre. The quantity $1-\cos(i)$ shown on the abscissae is chosen such as to have a flat number-density distribution for randomly oriented halos. The inclination angle i was assumed to be known in performing the deprojection.
Open with DEXTER

In the text

$\begin{figure} \scalebox{0.7} { \begin{picture} (0,0)% \includegraphics{4717fi... ...\strut{} radius $[h^{-1} {\rm kpc}]$ }}% \end{picture}% \endgroup } \end{figure}$ Figure 6: Density and temperature profiles of the original and the reconstructed analytic halos. The upper panel shows the density (falling curves, left axis) and the temperature profiles (rising curves, right axis) of the original analytic halo, the halo reconstructed without observational noise (and without any smoothing), and the halo reconstructed from maps with observational noise to which the complete smoothing scheme was applied. The lower panel shows the profile of the ratio of the reconstructed density $\rho _n$ to the original density $\rho$ . The number of iterations used was n=20 in the case without noise and n=5 in the case with noise. An inclination of $i=70^\circ$ was chosen and assumed to be known in the reconstruction.
Open with DEXTER

In the text

$\begin{figure} \scalebox{0.45} { \begin{picture} (0,0) \includegraphics{4717fi... ... }}} \put(177,10){\makebox(0,0){\strut{}\Huge (b)}} \end{picture}} \end{figure}$ Figure 7: Reconstruction of the simulated cluster g51 without noise and smoothing. The ratios of the reconstructed to the original density and temperature are shown. The deprojection was done with n=5 iterations. An inclination angle of $i=68^\circ$ between the line-of-sight and the principal inertial axis of the cluster gas was chosen and assumed to be known in the reconstruction. Panel a) shows the density ratio $\rho _n/\rho$ and panel b) the temperature ratio T_n/T. In the central region, the errors are of the order of $10\%$ .
Open with DEXTER

In the text

$\begin{figure} \scalebox{0.45} { \begin{picture} (0,0) \includegraphics{4717fi... ... }}} \put(177,10){\makebox(0,0){\strut{}\Huge (b)}} \end{picture}} \end{figure}$ Figure 8: Reconstruction of the simulated cluster g51 without noise but with "radius-dependent smoothing''. The ratios of the reconstructed to the original density and temperature are shown. The deprojection was done with n=5 iterations. An inclination angle of $i=68^\circ$ between the line-of-sight and the principal inertial axis of the cluster gas was chosen and assumed to be known in the reconstruction. Panel a) shows the density ratio $\rho _n/\rho$ and panel b) the temperature ratio T_n/T. In the central region, the errors are of the order of $10\%$ .
Open with DEXTER

In the text

$\begin{figure} \scalebox{0.45} { \begin{picture} (0,0) \includegraphics{4717fi... ... }}} \put(177,10){\makebox(0,0){\strut{}\Huge (b)}} \end{picture}} \end{figure}$ Figure 9: Reconstruction of the simulated cluster g51 with noise and and the complete smoothing scheme applied. The ratios of the reconstructed to the original density and temperature are shown. The deprojection was done with n=5 iterations. An inclination angle of $i\approx 68^\circ$ between the line-of-sight and the principal inertial axis of the cluster gas was chosen and assumed to be known in the reconstruction. Panel a) shows the density ratio $\rho _n/\rho$ and panel b) the temperature ratio T_n/T. In the central region, the errors are of the order of $15\%$ .
Open with DEXTER

In the text

$\begin{figure} \scalebox{0.7} { \begin{picture} (0,0)% \includegraphics{4717fi... ...\strut{} radius $[h^{-1} {\rm kpc}]$ }}% \end{picture}% \endgroup } \end{figure}$ Figure 10: Gas density and temperature profiles of the original and the reconstructed cluster g51. The upper panel shows the density profiles (falling curves, left axis) and the temperature profiles (rising curves, right axis) of the original cluster, the cluster reconstructed without observational noise but with "radius-dependent smoothing'', reconstructed without observational noise and without any smoothing, reconstructed from maps with observational noise and the complete smoothing scheme applied, and reconstructed from maps with observational noise but without "radius-dependent smoothing''. The lower panel shows the profile of the ratio of the reconstructed density $\rho _n$ to the original density $\rho$ . The number of iterations used was n=5 in all cases. An inclination angle of $i\approx 68^\circ$ was chosen and assumed to be known in the reconstruction.
Open with DEXTER

In the text

$\begin{figure} \scalebox{0.7} { \begin{picture} (0,0)% \includegraphics{4717fi... ...alo with noise, but without r.-d. sm.}}% \end{picture}% \endgroup } \end{figure}$ Figure 11: Dependence of the quality of the density reconstruction on the number of iterations used. We show the volume-weighted relative rms error $\frac{\rho_n-\rho}{\rho}$ within spheres with r=500 h^-1 kpc radius and for different deprojections of our analytic halo model and of the numerically simulated cluster g51. The figure legend explains which halo, whether or not noise, and what kind of smoothing were used for the different reconstructions shown. Note that "photon-noise smoothing'' of the X-ray maps is always and only used for maps with noise.
Open with DEXTER

In the text

$\begin{figure} \hspace*{-2.5mm}% \par\scalebox{0.7} { \begin{picture} (0,0)% \... ...ox(0,0){\strut{} \Large $i'=40^\circ$ }} \end{picture}% \endgroup } \end{figure}$ Figure 12: Accuracy of emission-weighted temperature $T_{{\rm ew}}$ maps and densities of reconstructed analytic halos. The deprojections started from maps obtained by projecting the analytic halo along a line-of-sight with inclination angles of $i'=40^\circ$ and $i'=70^\circ$ . Inclination angles i between $0^\circ$ and $90^\circ$ were used for the reconstruction. As expected, the best reconstructions are obtained for $i\approx i'$ . The errors are shown for deprojections from maps without observational noise and from smoothed maps with observational noise, and were averaged within a region of radius 500 h^-1 kpc around the map or halo centre.
Open with DEXTER

In the text

$\begin{figure} \hspace*{-2.5mm}% \par\scalebox{0.7} { \begin{picture} (0,0)% \... ...ox(0,0){\strut{} \Large $i'=40^\circ$ }} \end{picture}% \endgroup } \end{figure}$ Figure 13: Accuracy of emission weighted temperature $T_{{\rm ew}}$ maps and densities for the reconstructed simulated halo g51. Maps obtained by projecting g51 along a line-of-sight with inclination angles of $i'=40^\circ$ and $i'=68^\circ$ were used. The reconstruction assumed inclination angles i between $0^\circ$ and $90^\circ$ . Best reconstructions are obtained for $i\approx i'$ . The errors are shown for deprojections from maps without observational noise but using "radius-dependent smoothing'', and from maps with observational noise on which the complete smoothing scheme was applied. The rms relative errors were obtained within a circle of radius 200 h^-1 kpc around the map centre for the T_ew maps and inside a sphere of radius 500 h^-1 kpc around the halo centre for the density reconstructions.
Open with DEXTER

In the text

$\begin{figure} \scalebox{0.7} { \begin{picture} (0,0) \includegraphics{4717fig1... ...(50,32){\makebox(0,0){\strut{}\Large symmetry axis}} \end{picture}} \end{figure}$	Figure 1: Projection of an axisymmetric distribution of a physical quantity. The ellipse at the top marks the region where the kernel function corresponding to the projection along the line-of-sight is non-zero for fixed R and Z.
Open with DEXTER