Structure Formation in Astrophysics

We have heard a lot about probability functions p(M) at this meeting for mass M of planets or stars or clouds or clusters under various conditions. Since we have covered such an enormous range of masses, it is not surprising that power-law distributions close to the scale-invariant power have recurred so often. A power law differing from this distribution in the direction of favouring either low or high masses must of course have a turnover (or termination) towards this end to avoid a divergence. The physical reason for such a turnover is of interest, as is the question of continuity between the various types of objects. Bingelli and Hascher (PASP 119, 592, 2007) have followed this power-law continuity over 36 orders of magnitude in mass from asteroids to galaxy superclusters. It is instructive to look at similar probability distribution functions in quite different fields. I will give only the examples of two different kinds of human aggregates. One example, which has been discussed for more than a century or so, is the probability distribution for the size (i.e. the number of inhabitants) of a village, town or city. Near the end of the nineteenth century, the deviation from scale invariance was a slight increase towards the bottom end, i.e. overall slightly more people lived in a village of population 100-200 than in a city of 250 000 to 500 000.