The Fractal Dimension of Projected Clouds

The interstellar medium seems to have an underlying fractal structure, which can be characterized through its fractal dimension. However, interstellar clouds are observed as projected two-dimensional images, and the projection of a three-dimensional fractal distorts its measured properties. Here we use simulated fractal clouds to study the relationship between the three-dimensional fractal dimension ( D f ) of modeled clouds and the dimension resulting from their projected images. We analyze different fractal dimension estimators: the correlation and mass dimensions of the clouds and the perimeter-based dimension of their boundaries ( D per ). We find the functional forms relating D f with the projected fractal dimensions, as well as their dependence on the image resolution, which allow us to estimate the "real" D f value of a cloud from its projection. The application of these results to Orion A indicates in a self-consistent way that 2.5 ≲ D f ≲ 2.7 for this molecular cloud, a value higher than the result D per + 1 ≃ 2.3 sometimes assumed in the literature for interstellar clouds.