Context. Around 30 per cent of the observed exoplanets that orbit M dwarf stars are gas giants that are more massive than Jupiter. These planets are prime candidates for formation by disc instability. Aims. We want to determine the conditions for disc fragmentation around M dwarfs and the properties of the planets that are formed by disc instability. Methods. We performed hydrodynamic simulations of M dwarf protostellar discs in order to determine the minimum disc mass required for gravitational fragmentation to occur. Different stellar masses, disc radii, and metallicities were considered. The mass of each protostellar disc was steadily increased until the disc fragmented and a protoplanet was formed. Results. We find that a disc-to-star mass ratio between ~0.3 and ~0.6 is required for fragmentation to happen. The minimum mass at which a disc fragment increases with the stellar mass and the disc size. Metallicity does not significantly affect the minimum disc fragmentation mass but high metallicity may suppress fragmentation. Protoplanets form quickly (within a few thousand years) at distances around ~50 AU from the host star, and they are initially very hot; their centres have temperatures similar to the ones expected at the accretion shocks around planets formed by core accretion (up to 12 000 K). The final properties of these planets (e.g. mass and orbital radius) are determined through long-term disc-planet or planet–planet interactions. Conclusions. Disc instability is a plausible way to form gas giant planets around M dwarfs provided that discs have at least 30% the mass of their host stars during the initial stages of their formation. Future observations of massive M dwarf discs or planets around very young M dwarfs are required to establish the importance of disc instability for planet formation around low-mass stars.

Protostellar discs may become massive enough to fragment producing secondary lowmass objects: planets, brown dwarfs and low-mass stars. We study the effect of radiative feedback from such newly-formed secondary objects using radiative hydrodynamic simulations. We compare the results of simulations without any radiative feedback from secondary objects with those where two types of radiative feedback are considered: (i) continuous, and (ii) episodic. We find that: (i) continuous radiative feedback stabilizes the disc and suppresses further fragmentation, reducing the number secondary objects formed; (ii) episodic feedback from secondary objects heats and stabilises the disc when the outburst occurs, but shortly after the outburst stops, the disc becomes unstable and fragments again. However, fewer secondary objects are formed compared to the the case without radiative feedback. We also find that the mass growth of secondary objects is mildly suppressed due to the effect of their radiative feedback. However, their mass growth also depends on where they form in the disc and on their subsequent interactions, such that their final masses are not drastically different from the case without radiative feedback. We find that the masses of secondary objects formed by disc fragmentation are from a few M J to a few 0.1 M . Planets formed by fragmentation tend to be ejected from the disc. We conclude that planetary-mass objects on wide orbits (wide-orbit planets) are unlikely to form by disc fragmentation. Nevertheless, disc fragmentation may be a significant source of free-floating planets and brown dwarfs.

Radiative transfer is an important component of hydrodynamic simulations as it determines the thermal properties of a physical system. It is especially important in cases where heating and cooling regulate significant processes, such as in the collapse of molecular clouds, the development of gravitational instabilities in protostellar discs, disc-planet interactions, and planet migration. We compare two approximate radiative transfer methods which indirectly estimate optical depths within hydrodynamic simulations using two different metrics: (i) the gravitational potential and density of the gas (Stamatellos et al.), and (ii) the pressure scaleheight (Lombardi et al.). We find that both methods are accurate for spherical configurations e.g. in collapsing molecular clouds and within clumps that form in protostellar discs. However, the pressure scale-height approach is more accurate in protostellar discs (low and high-mass discs, discs with spiral features, discs with embedded planets). We also investigate the βcooling approximation which is commonly used when simulating protostellar discs, and in which the cooling time is proportional to the orbital period of the gas. We demonstrate that the use of a constant β cannot capture the wide range of spatial and temporal variations of cooling in protostellar discs, which may affect the development of gravitational instabilities, planet migration, planet mass growth, and the orbital properties of planets.