Linking planetesimal and dust content in protoplanetary disks via a local toy model

Context. If planetesimal formation is an efficient process, as suggested by several models involving gravitational collapse of pebble clouds, then, not before long, a significant part of the primordial dust mass should be absorbed in many km-sized objects. A good understanding of the total amount of solids in the disk around a young star is crucial for planet formation theory. However, as the mass of particles above the mm size cannot be assessed observationally, one must ask how much mass is hidden in bigger objects. Aims: We performed 0-d local simulations to study how the planetesimal to dust and pebble ratio evolves in time and to develop an understanding of the potentially existing mass in planetesimals for a certain amount of dust and pebbles at a given disk age. Methods: We performed a parameter study based on a model considering dust growth, planetesimal formation, and collisional fragmentation of planetesimals, while neglecting radial transport processes. Results: While at early times, dust is the dominant solid particle species, there is a phase during which planetesimals make up a significant portion of the total mass starting at approximately 104-106 yr. The time of this phase and the maximal total planetesimal mass strongly depend on the distance to the star R, the initial disk mass, and the efficiency of planetesimal formation ‚àä. Planetesimal collisions are more significant in more massive disks, leading to lower relative planetesimal fractions compared to less massive disks. After approximately 106 yr, our model predicts planetesimal collisions to dominate, which resupplies small particles. Conclusions: In our model, planetesimals form fast and everywhere in the disk. For a given ‚àä, we are able to relate the dust content and mass of a given disk to its planetesimal content, providing us with some helpful basic intuition about mass distribution of solids and its dependence on underlying physical processes.