Discourse on product life responsibility

Consider M_d = 0.025M_∗. In the self-gravitating disc, it takes longer for the inner vortex to build up (e.g. comparing t = 88.4P₀ in Fig. 3.25), disrupt the coorbital region and flow across the gap. This delays vortex-induced rapid migration, by about 50P0 compared to the non-self-gravitating case.

The extent of rapid migration is unaffected by self-gravity. Notice also the increased oscillations inrp(t) when self-gravity is included. This is because of the sustained multi-vortex configuration at the outer gap edge causing large oscillations in disc-planet torques, whereas without self-gravity these vortices would have merged.

After the first scattering event, migration stalls while the planet opens a gap at its new orbital radius and vortex formation recurs. With self-gravity included, the re-formation of a single vortex takes longer and it is narrower in radial extent than the non self-gravitating case. When the thinner self-gravitating vortex passes by the planet, little vortex material splits off from the main vortex and flows across the gap, unlike the non-self-gravitating vortex where material breaks off more easily (Chapter 2). This is probably due to the self-gravity of the vortex. Hence, there is a longer stalling period when self-gravity is included. Thus the net effect of self-gravity is to slow the migration in this example.

For the setup used in Chapter 2, a second fast migration episode was not seen within the simulated time when self-gravity was included, but it may eventually occur.

The total practical simulation time of a few hundred orbits is still very short compared to disc lifetimes. However, for the Qo = 4 disc model used in previous parts of this Chapter, two episodes of rapid migration occurs (Fig. 3.26). Self-gravity does not change the physical nature of vortex-induced, non-smooth migration.

3.7 Summary and discussion

The effect of disc self-gravity on vortex-forming instabilities associated with planetary gaps were explored. Analysis shows vortex modes are stabilised by self-gravity through its effect on the linear mode when the background remains fixed. This aspect has been confirmed by linear calculations. Linear calculations showed that the vortex forming modes with the highest growth rate shift to higher m with increasing disc mass. This is due to the combined effect of self-gravity through the response and through the background state.

Hydrodynamic simulations with and without self-gravity for a range of disc masses were performed. More vortices form as the disc mass increased in accordance with linear calculations. However, for sufficiently strong self-gravity, the vortex modes are

Figure 3.26: Repeated episodes of vortex-induced migration in a self-gravitating disc, for the Qo= 4 disc model used in previous sections.

suppressed and global spiral modes develop instead. Self-gravity delays vortex merging and multi-vortex configurations can be sustained for longer with increasing disc mass, allowing vortices to evolve individually.

The nature of post-merging vortices is also affected by gravity. With weak self-gravity ( i.e. M_d≤0.024M_∗), a single vortex, extended in azimuth forms and circulates at the outer gap edge. For M_d = 0.031M_∗ the final configuration is a vortex pair at the outer gap edge, each localised in azimuth. In this case a vortex, containing on the order of 20M_⊕, is gravitationally bound. The effect of their gravitational influence on the rest of the disc is to redistribute mass radially whereas for lower disc masses, redistribution is restricted to being azimuthal.

The internal flow in these self-gravitating, localised vortices adjusts so that they are not destroyed by the background shear, taking on a structure similar to that of a vortex resulting from perturbing the disc with the Kida solution (Kida, 1981). Such vortices form in discs much less massive than that required for direct disc fragmentation.

Vortices can trap dust particles due to their association with pressure maxima (Bracco et al., 1999; Chavanis, 2000). Increasing self-gravity leads to vortices of stronger density contrast (Fig. 3.16), so self-gravitating vortices may be more effective at collecting solid particles, and assist planetesimal formation.

3.7 Summary and discussion

Supplementary simulations of Kida-like vortices were performed to understand the effect of self-gravity on inter-vortex interactions. In a self-gravitating disc, coorbital vortex pairs behave like coorbital planets and can execute mutual horseshoe turns. As a consequence there exists a minimum inter-vortex distance. Merging is then avoided if this minimal separation is still larger than a critical separation below which vortex merging occurs.

The effect of self-gravity found here, should also applicable to other types of struc-tured features in a protoplanetary disc that could support vortex forming instabilities.

Such a possibility is the boundary region between a dead zone and active region of the disc (Lyra et al., 2009).

The basics of vortex-induced migration remains unchanged by self-gravity. With self-gravity, the resistance to forming a single large vortex results in such migration being delayed. Self-gravitating vortices are are less effective in scattering the planet because they do not disrupt the co-orbital region as significantly as their non-self-gravitating counterparts. It can be said that in the regime of disc masses where vortices form and are significantly affected by self-gravity, vortex-induced migration is slowed down.

Chapter 4

Edge modes in self-gravitating discs

Numerical experiments Chapter 3 showed that global spiral disturbances associated with gap edges can develop in massive discs. They arise from a linear instability associated with both the self-gravity of the disc and local vortensity maxima which coincide with gap edges. They can be physically interpreted as disturbances localised around a vortensity maximum that further perturb gravitationally the smooth parts of the disc by exciting waves at Lindblad resonances. The angular momentum carried away reacts back on the edge disturbance so as to destabilise it.

These edge modes are studied in this Chapter. The disc-planet models are identical to that used in Chapter 3, but briefly reproduced in §4.1 for ease of reference. In

§4.2, the existence of edge modes and its basic features are illustrated with a numerical simulation. An analytic discussion and interpretation of edge modes is then presented in§4.3—§4.4. In §4.5, linear calculations for various disc models confirm the existence of edge dominated modes for low values of the azimuthal mode number. The physical picture of edge modes presented in the analytical discussion is consistent with these numerical solutions.

In §4.6 results from hydrodynamic simulations for a range of disc masses are dis-cussed. These are all stable in the absence of the planet. However, they exhibit lowm edge modes once a planet-induced gap is present. The form and behaviour of these is found to be in accord with linear theory. The spiral arms associated with the edge in-stabilities are shown to produce fluctuating torques acting on the planet. Fast inwards type III migration may occur, but outwards migration due to interaction with spiral arms is also observed. §4.8 concludes this Chapter.

4.1 Disc model

To apply angular momentum conservation in the analytical description of edge modes, three-dimensional disc models are required. This is because although the disc material may be considered to be confined to a thin sheet, the gravitational potential it generates is still three-dimensional. Thus, it is important to consider the self-gravity part as fully three-dimensional. This is governed by the Poisson equation

∇²Φ = 4πGρ,

where ρ is the three dimensional mass density and Φ is the associated gravitational potential which can be found via

Φ(r) =−G Z

D

ρ(r^′)d³r^′

|r−r^′| , (4.1)

whereDis the domain whereρ is non zero. This coincides with the disc domain when this is not separated from external material. The full 3D hydrodynamic equations are listed in §1.4 and is the basis for analytical work. In analytic discussions, fluids with a general barotropic equation of state are considered, for whichp=p(ρ) anddp/dρ=c²_s.

4.1.1 Razor thin discs

For numerical work, the disc is considered two-dimensional and only its gravitational potential in the plane is used. The governing equations are therefore those listed in

§1.4.6, which were used throughout Chapter 3. In numerical experiments, the locally isothermal equation of state is used with fixed aspect-ratio h = 0.05. The physical setup and notation is the same as Chapter 3, but simulations will be presented for a range of viscosity and softening parameters.

The discs occupy r = [r_i, r_o] = [1,10] and are parametrised by Q_o, the Keplerian ToomreQvalue at the outer boundary. The initial surface density profile is given by Eq.

3.1 (Chapter 3). Recall that 2D self-gravity requires the use of a gravitational softening length (see §1.4.6, Eq. 1.21) to prevent a singularity and approximately account for the vertical dimension. The softening length prescription ǫ_g(r) = ǫ_g0hr is used, and ǫg0 = 0.3 set as a fiducial value, but values up to 1.0 will be considered.

The initial azimuthal velocity is found by assuming the centrifugal force balances forces due to stellar gravity, the disc’s self-gravity and pressure gradient. Thus the