Interestingly, the presence of such an underdense librating region at the leading side of the planet has also been observed in simulations of giant planets which undergo inward runaway migration (Artymowicz 2004; D'angelo & Lubow 2008). Here, it is worth noting that the low-density region does not result from the gap opening process but is rather due to radiative effects. Never the less, using the terminology employed in the framework of Type III migration, Fig. 3 shows that radiative effects can give rise to a coorbital mass deficit given by (Masset & Papaloizou 2003)

(8)

\begin{equation}
\delta \mathrm{m}=4\pi a x_\mathrm{s}\left(\Sigma _\mathrm{s}-\Sigma _h\right),
\end{equation}

where Σs is the surface density at the upstream separatrix and Σh the mean surface density in the librating region. In order to provide an estimation for the coorbital mass deficit in the librating region, we make in the following the assumption that the only contribution to the density perturbation in the librating region results from entropy advection. In reality, however, the density perturbation features an additional contribution linked to the production of vortensity at the outgoing separatrix (Paardekooper et al. 2011). Ignoring this contribution gives the following expression for the surface density Σ− in the librating region (Baruteau & Masset 2008; Paardekooper et al. 2011):

(9)

\begin{equation}
\Sigma _-=\Sigma _\mathrm{s}\left(1-2\frac{\mid \xi \mid x}{\gamma a}\right) \quad \mathrm{for} \quad 0<x<x_\mathrm{s}.
\end{equation}

Here, we have also ignored the radial density gradient and assumed that the corotation torque was initially unsaturated. This gives a mean surface density in the librating region:

(10)

\begin{equation}
\Sigma _h=\Sigma _\mathrm{s}(1-\frac{\mid \xi \mid x_\mathrm{s}}{\gamma a}).
\end{equation}

The coorbital mass deficit then becomes:

(11)

\begin{equation}
\delta \mathrm{m} = 4\pi \mid \xi \mid x_\mathrm{s}^2 \Sigma _\mathrm{s}/\gamma ,
\end{equation}

which shows that the coorbital mass deficit scales with the entropy gradient and the width of the horseshoe region. We note in passing that at the boundary corresponding to $\dot{a} \,{\sim }\, \dot{a_f}$, we expect the corotation torque exerted on the planet to be given by (e.g. Papaloizou et al. 2007):

(12)

\begin{equation}
\Gamma _{\mathrm{CR}}=\frac{1}{2}\dot{a_f}\Omega _\mathrm{p}a \delta \mathrm{m}.
\end{equation}

Using the expression for the coorbital mass deficit given by equation (11), the previous equation then becomes:

(13)

\begin{equation}
\Gamma _{\mathrm{CR}}=\frac{3}{4}\frac{\xi }{\gamma }\Sigma _\mathrm{s} x_\mathrm{s}^4\Omega _\mathrm{p}^2,
\end{equation}

so that we recover the standard expression for the entropy-related horseshoe drag (Baruteau & Masset 2008). For $\dot{a} > \dot{a_f}$, however, the expression for the unsaturated corotation torque given by equation (12) is no longer valid and should be modified (Papaloizou et al. 2007). In that case, we expect the corotation torque to be prevented from saturation since the horseshoe region does not longer extend to the full 2π in azimuth, so that phase-mixing cannot occur. In isothermal discs with a large-scale vortensity gradient, Ogilvie & Lubow (2006) found that drift rates with $\dot{a} > \dot{a_f}$ can even lead to torques that are much larger than the unsaturated value in absence of migration. They showed that this arises due to corotational torques caused by a vortensity asymmetry in the coorbital region between gas on the leading and trailing sides of the planet. In the limit of low viscosity, the trapped material tends to conserve its initial vortensity as the planet migrates, leading to a possibly significant vortensity contrast between the librating region and the local disc. In that case, the impact on migration depends in fact on the sign of the vortensity deficit (Paardekooper 2014) which is defined as

(14)

\begin{equation}
\delta (\omega /\Sigma )=1-\frac{\omega _a/\Sigma _a}{\omega _{0}/\Sigma _{0}},
\end{equation}

where ωa/Σa is the local disc vortensity at the position of the planet and ω0/Σ0 the vortensity at the initial location of the planet. In the case where the planet migrates in the direction set by the vortensity-related horseshoe drag, the vortensity deficit is positive and this yields a positive feedback on migration, whereas if the planet migrates in the opposite direction to that set by the corotation torque, the vortensity deficit is negative and this gives rise to a negative feedback on migration (Paardekooper 2014).
Citation: Baruteau & Masset 2008
Citation function: Uses; Uses
Citation: Baruteau & Masset 2008
Citation function: Uses