Formation of spirals in early stage protoplanetary discs

Abstract

Class II protoplanetary discs feature numerous non-axisymmetric substructures like spirals and the underlying mechanisms for their formation are still highly debated. Coincidentally, early stage, massive discs are subject to the gravitational instability that causes them to collapse into denser substructures. However, like for most instabilities, real systems usually remain marginally stable, here with Toomre parameter $Q \gtrsim 1$. We study how the self-gravity of the gas triggers the growth of spiral structures in the disc. We specifically focus on discs that are considered stable, that is, with respect to the gravitational instability (with $Q > 1$), as these discs remain unstable to non-axisymmetric perturbations like spirals. After a linear stability analysis, we produce high-resolution 2D shearing sheet simulations with the GPU-accelerated code \idefix of self-gravitating discs. We probe different initial densities and thermodynamical models of Toomre-stable discs. The initial transient growth of the spiral wave matches the linear theory provided we take into account the time dependency of the amplification. The spirals are then rapidly non-linearly amplified with growth rate $\approx 10$ orbital time scale. After this time spiral large scale mode are amplified up to 1000 times more than linear theory predicts. At later times, low density discs reach a weak gravito-turbulent state with $α\approx 10^{-3}$ and discs with higher density undergo runaway collapse of the spiral arms. All discs exhibit dominant large-scale spirals.

Figures (17)

• Figure 1: Time evolution of the growth rate of spiral density modes. Top: Maximum amplification root $\omega_\pm$ at different times. We note how the shear moves the most unstable region across the wave-vector plane. The times of these panels correspond $t=0$ and to the vertical grey lines of the bottom panel. Bottom: Example of the effective amplification factor with the $\gamma_{+}(t)$ factor for the mode $(k_{x,0}, k_y) = (-2,1)$, represented by a black "x" on the top panels. The light-blue curve is the semi-analytical solution for that mode.
• Figure 2: Contour of the imaginary part of the roots of the effective growth rate after 10 local orbital times for $Q = 3/2$. Positive values (red) correspond to amplified modes, negative values (blue) correspond to damped modes, the black dotted line corresponds to constant amplitude modes. We note that the amplification region is smeared out along the $k_x$ direction like shown on Fig. \ref{['fig:ModeEvolution']}. We note that if one were to evaluate $\gamma_{\max}$ at later time, only a region of decreasing size would be damped.
• Figure 3: Density map of snapshots of the G1.10_S0.80 runs at different times. Before $\Omega t<10$ spiral arms have too low contrast to be seen with this normalisation. After $\Omega t>45$, the contrast is also increased when the spiral arms collapse. On the first panel, the spirals are in the linear growth regime; for the last two panels, where the spiral arms overlap in some places, they are in the non-linear (interaction) regime.
• Figure 4: Time evolution of the shearing-wave modes $(\ell,m)=(2,1)$ of G1.10_S0.80. The blue dots measure the shearing-wave mode amplitude, and the orange solid line corresponds to the analytical solution of Eqn. (\ref{['eq:linSW']}) for the corresponding mode.
• Figure 5: Mode amplification for the G1.10_S0.80 simulation. The axis scales are given both in terms of wave vector $k_{x,0},k_y$ and corresponding wave number $\ell,m$ for the box size of the simulation. Left: Measured amplification factor from the simulations. Right: Analytic amplification factor $\sim \sqrt{t}$ at large times computed from Eqn. \ref{['eq:gamma']}. We show the numerical quantity for the intermediate $\Gamma=1.1$ simulation, but this analytic amplification factor is computed under the isothermal assumption, as $\Gamma\neq1$ can not be linearised. Top: At $\Omega t = 5$. Bottom: At $\Omega t =15$.