TITIUS-BODE LAWS IN THE SOLAR-SYSTEM .2. BUILD YOUR OWN LAW FROM DISKMODELS

Simply respecting both scale and rotational invariance, it is easy to construct an endless collection of theoretical models predicting a Tit ius-Bode law, irrespective to their physical content. Due to the numer ous ways to get the law and its intrinsic arbitrariness, it is not an useful constraint on theories of solar system formation. To illustrate the simple elegance of scale-invariant methods, we explicitly cook up one of the simplest examples, an infinitely thin cold gaseous disk ro tating around a central object. In that academic case, the Titius-Bode law holds during the linear stage of the gravitational instability. T he time scale of the instability is of the order of a self-gravitating time scale, (G rho(d))(-1/2), where rho(d) is the disk density. This model links the separation between different density maxima with the r atio M(D)/M(C) of the masses of the disk and the central object; for i nstance, M(D)/M(C) of the order of 0.18 roughly leads to the observed separation between the planets. We discuss the boundary conditions and the limit of the WKB approximation.