The focus of (Sikivie et al. 1997) was on indentifying velocity streams from non spherically-symmetric angular momentum distributions. Although the same geometry as used here was employed, a (FG84, Bertschinger 1985)-type one particle integration was used which assumes strict self-similar phase mixing and so is insensitive to phase space instability.

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The approach from the force equation in (Sikivie et al. 1997) yields a slightly different self-similar constraint for the angular momentum:

$\begin{displaymath}j^{2}_{\rm Sikivie}=\frac{J^{2}_{\rm Sikivie}S^{4}_{\rm T}a^{... ...}_{\rm T}GMa}{\xi _{\rm a}^{2}}=\frac{j^{2}}{\xi _{\rm a}^{2}},\end{displaymath}$

where $\xi _{\rm a}(x)=\left. \xi \right\vert _{t=0}$ is obtained with the combination of Eqs. (10), (13) and S=x/a and we identify our conventions with $J^{2}_{\rm Sikivie}S_{\rm T}^{4}\equiv J^{2}.$

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...

The NFW profile in terms of critical density, density and radial scales reads

$\begin{displaymath}\frac{\rho }{\rho_{\rm c}}=\frac{\delta _{\rm c}}{\frac{r}{r_{\rm s}}\left( 1+\frac{r}{r_{\rm s}}\right) ^{2}},\end{displaymath}$

with $r_{\rm s}=r_{200}/c$ defining the concentration factor and the two scales correlated through (see (NFW))

$\begin{displaymath}\delta _{\rm c}=\frac{c^{3}}{\left[ \ln (1+c)-\frac{c}{1+c}\right] }\cdot\end{displaymath}$

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... off

An example calculation: the constant mass of one shell is given in the simulation's unit as $m_{1\rm shell}=2\times10^{-2}.$ The maximum density contrast in Fig. 2 can be taken as $\frac{\delta \rho }{\rho _{0}}=4\times10^{13}.$ Because of its high value, the density contrast, denoted $\delta$ in this note, can be identified with the density itself

$\begin{displaymath}\frac{\delta \rho }{\rho _{0}}=4\times10^{13}\gg 1\Leftrightarrow \frac{\delta \rho }{\rho _{0}}\equiv \delta \simeq \rho .\end{displaymath}$

Thus, the volume of innermost shells can be evaluated as

$\begin{displaymath}\delta \simeq \rho =\frac{m_{1\rm shell}}{V_{1\rm shell}}\Rightarrow V_{1\rm shell}\simeq \frac{m_{1\rm shell}}{\delta },\end{displaymath}$

so the characteristic length scale of a shell in the centre is given by

$\begin{eqnarray*}L_{1\rm shell}&\simeq& ^{3}\sqrt{\frac{m_{1\rm shell}}{\delta }... ...mes10^{-16}}=^{3}\sqrt{0.5}\times10^{-5}\simeq 7.9\times 10^{-6}.\end{eqnarray*}$

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...Chen & Jing 2002)

After we completed this work, the results from Hiotelis (2002) have come to our attention, that are similar to ours in the case of a same value for $\lambda$ .

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