4 Distribution of $\gamma$ and its simulation

The distribution $f(\gamma )$ is shown in the Fig. 2 as a thick line. The negative and positive values of $\gamma$ are folded into a $0\hbox{$^\circ$ }$- $180\hbox{$^\circ$ }$ interval. The bin size is $30\hbox{$^\circ$ }$ given the $\epsilon_\gamma$ range 0- $30\hbox{$^\circ$ }$. In contrast to the uniform distribution obtained by BDS92, our $f(\gamma )$ is bell-shaped. The difference from the uniform distribution of $\gamma$ is significant at the level of 99.95% so the corresponding $f(e)\!=\!2e$ does not fit our data.

Although our data are limited (only 6 bins in the histogram), we performed some Monte-Carlo simulations to restore the possible shape of f(e). For this, an artificial set of $N\!=\!10\,000$ binaries was generated using the software provided by A. Tokovinin. The following types of the distribution of e were tested: fixed $e\!=\!\rm const.$, linear $f(e)\!=\!2e$ (with eccentricity cut-off: $e\!<\!e^*$) and bell-like $f(e)\!=\!(\pi\!/2)\sin (\pi e)$. The orbits had random orientation and phase. Then the resulting angles $\gamma$ and their histograms $f(\gamma )$ were computed. The criterion $\chi^2$ was used to reject the distributions of $\gamma$ which are not compatible with our data.

As a result (see plotted curves in Fig. 2), the good fit was obtained for any artificial sample with single eccentricity e<0.5. At the 99%-level of significance all the single e>0.6 are rejected. The linear relation $f(e)\!=\!2e$ does not fit our data for the cutoff values e^*>0.85 at the same level of significance. Meanwhile, the uniform and bell-shaped distributions of eccentricities give an $f(\gamma )$ very similar to the observed one. In general, all the distributions populated mostly by low-to-moderate values of eccentricity cannot be rejected.

Figure 2: Distribution $f(\gamma )$ for 174 MSC wide subsystems whose movement is still significant after a correction for internal movement was applied, i.e. $\epsilon _{\gamma }<30\hbox {$^\circ $ }$. Results of Monte-Carlo simulations are also shown. Designations: Stair-case: observed $f(\gamma )$; short- and long-dashed lines are for $e\!=\!0.4$  ( $\chi^2\!=\!4$) and bell-shaped distribution $f(e)\!\propto\!\sin (\pi e)$  ( $\chi^2\!=\!8$). Incompatible distributions: dotted line for high-e orbits ($e\!=\!0.9$ is shown; $\chi^2\!=\!177$); dot-dashed and grey thick lines - for linear $f(e)\!=\!2e$ with cutoff $e^*\!=\!0.99$  ( $\chi^2\!=\!38$) and with stability-constrained rejection ($\chi ^2=23$, see Discussion).