RECOGNITION AND CHARACTERIZATION OF HIERARCHICAL INTERSTELLAR STRUCTURE .1. CORRELATION-FUNCTION

We consider the problem of the quantitative description of multiscale structure in interstellar cloud complexes and gravitational collapse calculations, with an emphasis on the recognition and characterization of hierarchical fragmentation structure. The present paper discusses the response of the two-point correlation function to a variety of analytical models for density structure, for simple clustering of pointlike clouds to more complex models involving clouds with a distribution of sizes and densities and hierarchical substructure. By expressing the density distribution as the superposition of individual clouds, it is shown that the correlation function generates two types of terms: those involving each cloud's density convolved with itself ("self terms") and those involving pairs of different clouds ("cross terms"). The cross terms contain the relational information concerning spatial structure, but only as the frequency distribution of cloud pair separations. A self-similar hierarchical distribution of clouds will result in cross-term contributions to the correlation function which are periodic in the logarithm of the spatial lag. However, it is shown that these features are smeared and obscured by the finite sizes of the clouds involved in the cross terms, by the self terms, by edge effects, and by stochastic variations in cloud positions relative to a perfect geometrical hierarchy. It is emphasized that major distortions of the correlation function are introduced by the presence of any image features with size scales a significant fraction of the image size, such as gradients or a small number of large clouds. Such features, which are necessarily present for hierarchical systems, will control the shape of the correlation function at small separations and produce anticorrelations at intermediate lags and recorrelations at large lags. This behavior only reflects the gradients and separations of large clouds, which are statistically irrelevant and can be much more easily studied by visual inspection of the original density map. Most published attempts to interpret the correlation function for interstellar structures, except those that are restricted to the detection of clustering of pointlike sources, appear to be dominated by these effects.