Example of apsidal precession, γ, in a self-gravitating disk. Here the disk is assumed to contain 50   M _⊕  between 16 and 32 AU, characteristic of a typical post-formation debris disk in the Nice model of solar system formation (Tsiganis et al. 2005). The solid curve shows the precession rate predicted by Eqs. (1)–(3), as a function of semi major axis. The dots and error bars show the results of a numerical calculation, integrating 3000 equal-mass particles with a softening parameter of ϵ ≈ 0.005   AU to smooth the effects of their mutual close encounters. The disk was binned into 100 annuli in a and the mean frequency of the longitude of pericenter was measured from the time-series of ϖ of the particles in each bin (dots) as well as its variance (error bars). Note that the precession frequency of a self-gravitating disk is negative.