How Large Can a Grain Grow Before Collisions Break It? Fragmentation and Drift Size Limits Across the Disc

Turbulent collisions set a ceiling on grain size: grow too large, and relative velocities exceed the fragmentation threshold and collisions shatter instead of stick. Using the standard fragmentation-limit formula with turbulence alpha = 1e-3, this paper computes the ceiling across the disc for bare silicate grains (fragmentation velocity 1 m per second) and icy grains (10 m per second), and compares it with the size at which radial drift removes grains as fast as they grow. At 1 AU the silicate ceiling is 0.2 cm and the icy ceiling 24.4 cm; at 10 AU they fall to 0.02 cm and 2.4 cm. The drift limit is far above both through most of the disc — 20 m at 1 AU and 20.0 cm at 10 AU — and falls below the icy fragmentation ceiling only beyond about 82 AU, where both limits have shrunk to millimetres (2.0 mm against 2.4 mm at 100 AU). Fragmentation, not drift, is therefore the binding size limit in the planet-forming region. The Stokes number at the fragmentation ceiling is only 0.0003 for silicates at 1 AU but reaches 0.107 for ice at 10 AU, close to the range where streaming concentration becomes possible. The composition of a grain decides how large it can grow and which dynamical regime it can enter. Computed from stated assumptions, not measured; disc densities sit at 2.5e-67 of the gravitational frame's ceiling, so the limits are collisional, not gravitational.