Gravitational Instability in Irradiated Discs

Short-Lived Gravitational Instability in Isolated Irradiated Discs

Sahl Rowther, Daniel J. Price, Christophe Pinte, Rebecca Nealon, Farzana Meru, Richard Alexander

Abstract

Irradiation from the central star controls the temperature structure in protoplanetary discs. Yet simulations of gravitational instability typically use models of stellar irradiation with varying complexity, or ignore it altogether, assuming heat generated by spiral shocks is balanced by cooling, leading to a self-regulated state. In this paper, we perform simulations of irradiated, gravitationally unstable protoplanetary discs using 3D hydrodynamics coupled with live Monte-Carlo radiative transfer.

We find that the resulting temperature profile is approximately constant in time, since the thermal effects of the star dominate. Hence, the disc cannot regulate gravitational instabilities by adjusting the temperatures in the disc. In a 0.1 Solar mass disc, the disc instead adjusts by angular momentum transport induced by the spiral arms, leading to steadily decreasing surface density, and hence quenching of the instability. Thus, strong spiral arms caused by self-gravity would not persist for longer than ten thousand years in the absence of fresh infall, although weak spiral structures remain present over longer timescales. Using synthetic images at 1.3mm, we find that spirals formed in irradiated discs are challenging to detect.

In higher mass discs, we find that fragmentation is likely because the dominant stellar irradiation overwhelms the stabilising influence of PdV work and shock heating in the spiral arms.

Disc Evolution

Figure 1 compares the evolution of gravitational instabilities in a 0.1 Solar mass disc modelled with live radiative transfer (left panel) and \(\beta \)-cooling (right panel).

\(\beta \)-cooling -- In the absence of irradiation, the disc continuously cools until strong spiral structures develop. Eventually the disc reaches a steady state (in terms of spiral structures) as PdV work and shock heating balance the cooling.

Radiative Transfer -- The inclusion of irradiation essentially fixes the temperatures in the disc since the thermal effects of the star dominates. Hence in an irradiated disc, changes in the surface density instead drive the evolution of the disc. Angular momentum transport by the spiral arms reduces the surface density in the outer gravitationally unstable regions of the disc. Thus, unlike with \(\beta \)-cooling the disc becomes more stable over time.

Figure 1: A video showing the evolution of gravitational instabilities in a 0.1\(M_\odot\) disc when modelled with live radiative transfer (left panel) and with \(\beta\)-cooling (right panel). Although both discs become gravitationally unstable, the inclusion of stellar irradiation results in weaker spiral structures.

The Importance of the Star

Figure 2 compares the temperature structure for a 0.1 Solar mass disc modelled with live radiative transfer (left panels) and \(\beta \)-cooling (right panels).

\(\beta \)-cooling -- In the absence of irradiation, the spiral arms are the only source of heat through PdV work and shock heating. Hence, the temperature structure is identical to the density structure.

Radiative Transfer -- The temperature is now controlled by the irradiation of the star as evident by the smooth temperature profile. In contrast to the density structure, the spiral arms are barely visible in the temperature structure when stellar irradiation is included. PdV work and shock heating is negligible compared to the irradiation of the star.

⭐ Hence, an irradiated gravitationally unstable disc cannot regulate gravitational instabilities by adjusting the temperatures in the disc due to the radiation of the star being the dominant (and constant) source of energy.

**Figure 2:** Cross-section slices of the face-on and edge-on views of the temperature structure of a 0.1\(M_\odot\) disc at 9 outer orbits. In contrast to the density structure, spirals are barely visible in temperature with irradiation (top left panel). PdV work and shock heating is negligible compared to the radiation of the star. Whereas with \(\beta \)-cooling, the temperature structure is strongly correlated to the density structure (top right panel) since the spirals are the only source of heating. The irradiated disc is also thermally stratified and varies with radius, consistent with observations (bottom left panel). The vertical temperature structure with \(\beta\)-cooling (bottom right panel) is unusual due to the disc also spreading vertically. Thus, heating due to artificial viscosity is also relevant for the upper-most regions which is more poorly resolved. In the midplane, the disc temperature is uniform with radius unlike with radiative transfer. The bright 'x' shape in the inner-most regions is also due to a large inner cavity with very few particles.

Fragmentation in Higher Mass Discs

Figure 3 shows the final state of a 0.25 Solar mass disc modelled with live radiative transfer. Despite gravitational instabilities being weaker, the disc has fragmented, forming clumps.

In a \(\beta\)-cooled disc, fragmentation could be prevented as the disc could regulate its temperatures through PdV work and shock heating in the spiral arms.

However as shown above, this is negligible in an irradiated disc where the temperatures are set by the star since the thermal effects of stellar irradiation dominates. The dominance of stellar irradiation prevents the disc from stabilising due to PdV work and shock heating in the spiral arms.

⭐ Hence, fragmentation remains a possible fate for irradiated gravitationally unstable discs, confirming the predictions in Kratter & Murray-Clay (2011).