The role of the drag force in the gravitational stability of dusty planet forming disc -- I. Analytical theory

Recent observations show that planet formation is already underway in young systems, when the protostar is still embedded into the molecular cloud and the accretion disc is massive. In such environments, the role of self gravity (SG) and gravitational instability (GI) is crucial in determining the dynamical evolution of the disc. In this work, we study the dynamical role of drag force in self-gravitating discs as a way to form planetesimals in early protoplanetary stages. We obtain the dispersion relation for density-wave perturbations on a fluid composed of two phases (gas and dust) interacting through the common gravitation field and the mutual drag force, and we find that the stability threshold is determined by three parameters: the local dust-to-gas density ratio, the dust relative temperature and the relevant Stokes number. In a region of parameters space, where young protoplanetary discs are likely to be found, the instability can be dust driven, occurring at small wavelengths. In this regime, the Jeans mass is much smaller than the one predicted by the standard gravitational instability model. This mechanism can be a viable way to form planetary cores in protostellar discs, since their predicted mass is about 10 Earth masses.