4 Correlation analysis

In order to check if the perturbations of the gravitational system induce characteristic correlations in phase-space, the indices D and $\delta$ of the mass-size relation and the velocity-dispersion-size relation are determined.

The index of the mass-size relation can be found via,

$$D(r)=\left(\frac{{\rm d}\log M}{{\rm d}\log r}\right)(r),$$ (12)

where M is the sum of the masses of all particles having a relative distance r. The mass size relation is then

$$M(r)\propto r^{D(r)},$$ (13)

where r can be considered as the scale. For a homogeneous mass distribution and for a hierarchical fragmented, fractal structure the index is a constant. For the latter case the index is not necessarily an integer and lies in the interval, $D_{\rm T}<D<D_{\rm S}$, where $D_{\rm T}$ and $D_{\rm S}$ are the topological dimension and the dimension of the embedding space, respectively (Mandelbrot 1982).

The index of the velocity-dispersion-size relation is determined by,

$$\delta(r)=\left(\frac{{\rm d}\log\sigma}{{\rm d}\log r}\right)(r),$$ (14)

where $\sigma$ is the velocity-dispersion of all particles having a relative distance r. Observations of the interstellar medium suggest a constant index $\delta=\delta_{\rm L}$ on scales ${\cal O}(0.1)-{\cal O}(100)$ pc with . This is expressed by Larson's law (e.g. Larson 1981; Scalo 1985; Falgarone & Perault 1987; Myers & Goodman 1988)

$$\sigma\propto r^{\delta_L}.$$ (15)

In order to preclude the effect of boundary conditions on the scaling relations, upper and lower cutoffs have to be taken into account. The lower cutoff is given by the softening length. An upper cutoff arises from the final system size. Thus, the scope of application is for the velocity-dispersion-size and the mass-size relation, $\epsilon <r<2R_{90}$, and $\epsilon <r<R_{90}/2$, respectively, where R₉₀ is the radius of the sphere centered at the origin which contains $90\%$ of the mass (see Huber & Pfenniger 2001a).