All Figures

$\begin{figure} \par\includegraphics[width=16.9cm,clip]{8702fg01.eps}\end{figure}$ Figure 1: The functions U and Q divided by the protoplanet to central star mass ratio are plotted for $\rho _0 \propto r^{-1/2},$ b=0.03 and m=2 in dimensionless units. Real parts are plotted with dashed lines, imaginary parts with full lines. Curves for the Landau parameters $\epsilon = 10^{-5}$ $\epsilon = 3.\times 10^{-2}$ are shown. The response is very similar for these cases but the latter case is slightly more damped which is manifest by somewhat smaller amplitude swings.
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In the text

$\begin{figure} \par\includegraphics[width=17cm,clip]{8702fg02.eps}\end{figure}$ Figure 2: The density response divided by the protoplanet to central star mass ratio, for $\rho _0 \propto r^{1/2},$ b=0.03 and m=2 in dimensionless units. Real part dashed line, imaginary part full line. For the left panel the Landau parameter $\epsilon = 10^{-5}$ and for the right panel $\epsilon = 6$ $\times$ 10^-2. In the former case the amplitudes of the real and imaginary parts reach extreme values of order 10⁵.
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In the text

$\begin{figure} \par\includegraphics[width=16.8cm,clip]{8702fg03.eps}\end{figure}$ Figure 3: The cumulative torque in units of $q^2 \Sigma (r_{\rm p})\Omega _{\rm K}^2r_{\rm p}^4$ acting on the protoplanet plotted as a function of dimensionless radius for $\rho _0 \propto r^{-1/2}$ , b=0.03 and m=2. For the left panel plots are given for the Landau parameters $\epsilon = 10^{-5}$ (dashed line) $\epsilon = 10^{-4}$ (dotted line) $\epsilon = 3$ $\times$ 10^-2 (full line). For the right panel plots are given for the dimensionless thermal diffusivities $D_{\rm e}= 10^{-6}$ (dotted line) and $D_{\rm e}= 10^{-5}$ (full line). The cumulative torque is defined to be zero at the outer boundary and equal the total torque at the inner boundary. In all of these cases the total torque is positive.
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In the text

$\begin{figure} \par\includegraphics[width=17cm,clip]{8702fg04.eps}\end{figure}$ Figure 4: The cumulative torque in units of $q^2 \Sigma (r_{\rm p})\Omega _{\rm K}^2r_{\rm p}^4$ acting on the protoplanet as a function of m for $\rho _0 \propto r^{-1/2},$ b=0.03 ( left panel) and b=0.01 ( right panel). For both panels plots are given for the Landau parameters $\epsilon = 10^{-5}$ (full line), $\epsilon = 10^{-4}$ (dotted line), $\epsilon = 10^{-3}$ (dashed line), $\epsilon = 3$ $\times$ 10^-2 (dot dashed line).
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In the text

$\begin{figure} \par\includegraphics[width=16.9cm,clip]{8702fg05.eps}\end{figure}$ Figure 5: The cumulative torque in units of $q^2 \Sigma (r_{\rm p})\Omega _{\rm K}^2r_{\rm p}^4$ acting on the protoplanet as a function of m for $\rho _0 \propto r^{-3/2}$ and b=0.03 ( left panel) and for $\rho _0 \propto r^{-1/2}$ , b=0.03 but with thermal diffusion ( right panel). For the left panel plots are given for the Landau parameters $\epsilon = 10^{-5}$ (full line), $\epsilon = 10^{-4}$ (dotted line), $\epsilon = 10^{-3}$ (dashed line), $\epsilon = 3$ $\times$ 10^-2 (dot dashed line). In the right panel plots are given for the dimensionless thermal diffusivities $D_{\rm e}= 10^{-6}$ (dotted line) and $D_{\rm e}= 10^{-5}$ (full line). The curves almost coincide for these cases.
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In the text

$\begin{figure} \par\includegraphics[width=8cm,clip]{8702fg06.eps}\end{figure}$ Figure 6: Contour plot of the full linear density response for $\rho _0 \propto r^{-1/2}$ and thermal diffusivity $D_{\rm e}= 10^{-6}$ in dimensionless units. The linear theory is only likely to be valid for planet masses significantly less than an earth mass in this case (see Eq. (28)). Note the narrow high and low density regions on the corotation circle near the protoplanet. These, also seen in the non-linear simulations, respectively lead and trail the protolanet giving rise to a positive corotation torque. However, the effects of this torque are not strong enough to counteract the negative Lindblad torques contributed by the high density wakes.
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In the text

$\begin{figure} \par\includegraphics[width=8.4cm,clip]{8702fg07.eps}\end{figure}$ Figure 7: Total torque in units of $q^2 \Sigma (r_{\rm p})\Omega _{\rm K}^2r_{\rm p}^4$ as a function of time for two different planet masses and two density profiles for the disc. The disc was kept locally isothermal, and a viscosity $\nu =10^{-5}$ $r_{\rm p}^2 \Omega _{\rm p}$ . The dotted lines indicate the results of our linear calculations.
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In the text

$\begin{figure} \par\includegraphics[width=17.2cm,clip]{8702fg08.eps}\end{figure}$ Figure 8: Overview of $\rho r^{1/2}$ for a q=1.26 $\times$ 10^-5 planet (4 ${{M}_\oplus }$ around a Solar mass star) after 20 orbits. Left panel: locally isothermal equation of state. Right panel: adiabatic equation of state.
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In the text

$\begin{figure} \par\includegraphics[width=8.4cm,clip]{8702fg09.eps}\end{figure}$ Figure 9: Total torque in units of $q^2 \Sigma (r_{\rm p})\Omega _{\rm K}^2r_{\rm p}^4$ as a function of time for a q=1.26 $\times$ 10^-5 planet (4 ${{M}_\oplus }$ around a Solar mass star), using a locally isothermal equation of state (solid line) and solving the full energy equation, for two different radial entropy gradients $\xi={\rm d}\log(P/\rho^{\gamma})/{\rm d}\log r$ (dotted line: $\xi =0$ , dashed line: $\xi =-1.2$ ). All models have $\alpha =-1/2$ .
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In the text

$\begin{figure} \par\includegraphics[width=8.3cm,clip]{8702fg10.eps}\end{figure}$ Figure 10: The cumulative torque in units of $q^2 \Sigma (r_{\rm p})\Omega _{\rm K}^2r_{\rm p}^4$ acting on the protoplanet plotted as a function of dimensionless radius for $\rho _0 \propto r^{1/2},$ $\xi =-1.2$ and m=2 at different times. The cumulative torque is defined to be zero at the outer boundary and equal the total torque at the inner boundary.
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In the text

$\begin{figure} \par\includegraphics[width=8.3cm,clip]{8702fg11.eps}\end{figure}$ Figure 11: Total torque in units of $q^2 \Sigma (r_{\rm p})\Omega _{\rm K}^2r_{\rm p}^4$ as a function of time for 3 different planet masses and an adiabatic equation of state with $\xi =-1.2$ and $\alpha =-1/2$ . The torque is divided by q² to filter out purely linear effects.
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In the text

$\begin{figure} \par\includegraphics[width=7.4cm,clip]{8702fg12.eps}\end{figure}$ Figure 12: Real (dashed lines) and imaginary (solid lines) parts of the m=2Fourier component of the density perturbation due to a q=3.17 $\times$ 10^-6 planet after 20 (black) and 10 (red) orbits in an adiabatic disc with $\xi =-1.2$ and $\alpha =-1/2$ . The inset shows a close-up on the corotation region. A comparison with results obtained from linear theory indicates the form of a linear response apart from in the corotation region.
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In the text

$\begin{figure} \par\includegraphics[width=8.3cm,clip]{8702fg13.eps} \end{figure}$ Figure 13: Imaginary parts of the m=2 Fourier component of the density perturbation due to a q=1.25 $\times$ 10^-5 planet for different radial entropy profiles with $\alpha =-1/2$ . For the negative entropy slope, we show three different planet masses.
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In the text

$\begin{figure} \par\includegraphics[width=8.1cm,clip]{8702fg14.eps}\end{figure}$ Figure 14: Imaginary parts of the m=2 Fourier component of the density perturbation due to a q=1.25 $\times$ 10^-5 planet for different resolutions with $\xi =-1.2$ and $\alpha =-1/2$ , together with a run with a steeper radial density profile ( $\alpha =1/2$ , with the same value for $\gamma$ ).
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In the text

$\begin{figure} \par\includegraphics[width=8.3cm,clip]{8702fg15.eps}\end{figure}$ Figure 15: Total torque in units of $q^2 \Sigma (r_{\rm p})\Omega _{\rm K}^2r_{\rm p}^4$ as a function of time for $\rho _0 \propto r^{-1/2}$ and q=1.26 $\times$ 10^-5. The solid line shows an isothermal run, the dotted line an adiabatic run.
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In the text

$\begin{figure} \par\includegraphics[width=8.5cm,clip]{8702fg16.eps}\end{figure}$ Figure 16: Total torque in units of $q^2 \Sigma (r_{\rm p})\Omega _{\rm K}^2r_{\rm p}^4$ as a function of time on a q=3.17 $\times$ 10^-6 planet for different values of the heat diffusion coefficient $D_{\rm e}$ , in a disc with $\alpha =-1/2$ and $\xi =-1.2$ .
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In the text

$\begin{figure} \par\includegraphics[width=8.3cm,clip]{8702fg17.eps}\end{figure}$ Figure 17: Real (dashed lines) and imaginary (solid lines) parts of the m=2 Fourier component of the density perturbation due to a q=3.17 $\times$ 10^-6 planet after 40 orbits, for different values of the heat diffusion coefficient $D_{\rm e}$ , in a disc with $\alpha =-1/2$ and $\xi =-1.2$ . The inset shows a close-up on the corotation region.
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In the text

$\begin{figure} \par\includegraphics[width=8.3cm,clip]{8702fg18.eps}\end{figure}$ Figure 18: Total torque in units of $q^2 \Sigma (r_{\rm p})\Omega _{\rm K}^2r_{\rm p}^4$ as a function of time for $\rho _0 \propto r^{-1/2}$ and two different planet masses. For the q=3.17 $\times$ 10^-6 planet, we have used $D_{\rm e}=7$ $\times$ 10^-7, for the q=1.26 $\times$ 10^-5 planet we have used $D_{\rm e}=5.6$ $\times$ 10^-6.
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In the text

$\begin{figure} \par\includegraphics[width=8.3cm,clip]{8702fg19.eps}\end{figure}$ Figure 19: Density, multiplied by r^3/2, after 10 orbits of a q=1.26 $\times$ 10^-5 planet embedded in a disc with $\rho _0 \propto r^{-3/2}$ , $T \propto r^{-1}$ and $\xi =-0.85$ . Approximate locations of the streamlines are indicated by the black lines.
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In the text

$\begin{figure} \par\includegraphics[width=17.2cm,clip]{8702fg20.eps}\end{figure}$ Figure 20: Left panel: total torque, in units of $q^2 \Sigma (r_{\rm p})\Omega _{\rm K}^2r_{\rm p}^4$ , on a q=1.26 $\times$ 10^-5 planet embedded in a disc with $\rho _0 \propto r^{-3/2}$ , $T \propto r^{-1}$ and $\xi =-0.85$ . A small thermal diffusion coefficient $D_{\rm e}=10^{-7}$ was used to regularise the solution near corotation. Results for two values of the softening parameter b are shown in black and red, together with a simulation with b=0.03 and $D_{\rm e}=0$ (green line). Right panel: azimuthal variation of the density at r=0.99 at three different times for b=0.03 and $D_{\rm e}=10^{-7}$ .
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In the text

$\begin{figure} \par\includegraphics[width=8.3cm,clip]{8702fg21.eps}\end{figure}$ Figure 21: Long-term evolution of the total torque, in units of $q^2 \Sigma (r_{\rm p})\Omega _{\rm K}^2r_{\rm p}^4$ , on a q=1.26 $\times$ 10^-5 planet in a disc with $\rho _0 \propto r^{-3/2}$ , $T \propto r^{-1}$ and $\xi =-0.85$ , for three different thermal diffusivities.
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In the text

$\begin{figure} \par\includegraphics[width=17.2cm,clip]{8702fg22.eps}\end{figure}$ Figure 22: Density, multiplied by r^3/2, for three models for q=1.26 $\times$ 10^-5, $\rho _0 \propto r^{-3/2}$ , $T \propto r^{-1}$ and $\xi =-0.85$ after 160 orbits, focusing on the horseshoe turn leading the planet. Left panel: $D_{\rm e}= 10^{-6}$ , $\nu =0$ . Middle panel: $D_{\rm e}= 10^{-5}$ , $\nu =0$ . Right panel: $D_{\rm e}=\nu =10^{-6}$ .
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In the text

$\begin{figure} \par\includegraphics[width=8.5cm,clip]{8702fg23.eps}\end{figure}$ Figure 23: Long-term evolution of the total torque, in units of $q^2 \Sigma (r_{\rm p})\Omega _{\rm K}^2r_{\rm p}^4$ , on a q=1.26 $\times$ 10^-5 planet in a disc with $\rho _0 \propto r^{-3/2}$ , $T \propto r^{-1}$ and $\gamma =1.1$ , for three different thermal diffusivities. All models have a small kinematic viscosity of $\nu =10^{-6}$ $r_{\rm p}^2 \Omega _{\rm p}$ .
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In the text

$\begin{figure} \par\includegraphics[width=8.4cm,clip]{8702fg07.eps}\end{figure}$	Figure 7: Total torque in units of $q^2 \Sigma (r_{\rm p})\Omega _{\rm K}^2r_{\rm p}^4$ as a function of time for two different planet masses and two density profiles for the disc. The disc was kept locally isothermal, and a viscosity $\nu =10^{-5}$ $r_{\rm p}^2 \Omega _{\rm p}$ . The dotted lines indicate the results of our linear calculations.
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$\begin{figure} \par\includegraphics[width=17.2cm,clip]{8702fg08.eps}\end{figure}$	Figure 8: Overview of $\rho r^{1/2}$ for a q=1.26 $\times$ 10^-5 planet (4 ${{M}_\oplus }$ around a Solar mass star) after 20 orbits. Left panel: locally isothermal equation of state. Right panel: adiabatic equation of state.
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$\begin{figure} \par\includegraphics[width=8.3cm,clip]{8702fg10.eps}\end{figure}$	Figure 10: The cumulative torque in units of $q^2 \Sigma (r_{\rm p})\Omega _{\rm K}^2r_{\rm p}^4$ acting on the protoplanet plotted as a function of dimensionless radius for $\rho _0 \propto r^{1/2},$ $\xi =-1.2$ and m=2 at different times. The cumulative torque is defined to be zero at the outer boundary and equal the total torque at the inner boundary.
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$\begin{figure} \par\includegraphics[width=8.3cm,clip]{8702fg11.eps}\end{figure}$	Figure 11: Total torque in units of $q^2 \Sigma (r_{\rm p})\Omega _{\rm K}^2r_{\rm p}^4$ as a function of time for 3 different planet masses and an adiabatic equation of state with $\xi =-1.2$ and $\alpha =-1/2$ . The torque is divided by q² to filter out purely linear effects.
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$\begin{figure} \par\includegraphics[width=8.3cm,clip]{8702fg13.eps} \end{figure}$	Figure 13: Imaginary parts of the m=2 Fourier component of the density perturbation due to a q=1.25 $\times$ 10^-5 planet for different radial entropy profiles with $\alpha =-1/2$ . For the negative entropy slope, we show three different planet masses.
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$\begin{figure} \par\includegraphics[width=8.1cm,clip]{8702fg14.eps}\end{figure}$	Figure 14: Imaginary parts of the m=2 Fourier component of the density perturbation due to a q=1.25 $\times$ 10^-5 planet for different resolutions with $\xi =-1.2$ and $\alpha =-1/2$ , together with a run with a steeper radial density profile ( $\alpha =1/2$ , with the same value for $\gamma$ ).
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$\begin{figure} \par\includegraphics[width=8.3cm,clip]{8702fg15.eps}\end{figure}$	Figure 15: Total torque in units of $q^2 \Sigma (r_{\rm p})\Omega _{\rm K}^2r_{\rm p}^4$ as a function of time for $\rho _0 \propto r^{-1/2}$ and q=1.26 $\times$ 10^-5. The solid line shows an isothermal run, the dotted line an adiabatic run.
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$\begin{figure} \par\includegraphics[width=8.5cm,clip]{8702fg16.eps}\end{figure}$	Figure 16: Total torque in units of $q^2 \Sigma (r_{\rm p})\Omega _{\rm K}^2r_{\rm p}^4$ as a function of time on a q=3.17 $\times$ 10^-6 planet for different values of the heat diffusion coefficient $D_{\rm e}$ , in a disc with $\alpha =-1/2$ and $\xi =-1.2$ .
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$\begin{figure} \par\includegraphics[width=8.3cm,clip]{8702fg17.eps}\end{figure}$	Figure 17: Real (dashed lines) and imaginary (solid lines) parts of the m=2 Fourier component of the density perturbation due to a q=3.17 $\times$ 10^-6 planet after 40 orbits, for different values of the heat diffusion coefficient $D_{\rm e}$ , in a disc with $\alpha =-1/2$ and $\xi =-1.2$ . The inset shows a close-up on the corotation region.
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$\begin{figure} \par\includegraphics[width=8.3cm,clip]{8702fg18.eps}\end{figure}$	Figure 18: Total torque in units of $q^2 \Sigma (r_{\rm p})\Omega _{\rm K}^2r_{\rm p}^4$ as a function of time for $\rho _0 \propto r^{-1/2}$ and two different planet masses. For the q=3.17 $\times$ 10^-6 planet, we have used $D_{\rm e}=7$ $\times$ 10^-7, for the q=1.26 $\times$ 10^-5 planet we have used $D_{\rm e}=5.6$ $\times$ 10^-6.
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$\begin{figure} \par\includegraphics[width=8.3cm,clip]{8702fg19.eps}\end{figure}$	Figure 19: Density, multiplied by r^3/2, after 10 orbits of a q=1.26 $\times$ 10^-5 planet embedded in a disc with $\rho _0 \propto r^{-3/2}$ , $T \propto r^{-1}$ and $\xi =-0.85$ . Approximate locations of the streamlines are indicated by the black lines.
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$\begin{figure} \par\includegraphics[width=8.3cm,clip]{8702fg21.eps}\end{figure}$	Figure 21: Long-term evolution of the total torque, in units of $q^2 \Sigma (r_{\rm p})\Omega _{\rm K}^2r_{\rm p}^4$ , on a q=1.26 $\times$ 10^-5 planet in a disc with $\rho _0 \propto r^{-3/2}$ , $T \propto r^{-1}$ and $\xi =-0.85$ , for three different thermal diffusivities.
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$\begin{figure} \par\includegraphics[width=17.2cm,clip]{8702fg22.eps}\end{figure}$	Figure 22: Density, multiplied by r^3/2, for three models for q=1.26 $\times$ 10^-5, $\rho _0 \propto r^{-3/2}$ , $T \propto r^{-1}$ and $\xi =-0.85$ after 160 orbits, focusing on the horseshoe turn leading the planet. Left panel: $D_{\rm e}= 10^{-6}$ , $\nu =0$ . Middle panel: $D_{\rm e}= 10^{-5}$ , $\nu =0$ . Right panel: $D_{\rm e}=\nu =10^{-6}$ .
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$\begin{figure} \par\includegraphics[width=8.5cm,clip]{8702fg23.eps}\end{figure}$	Figure 23: Long-term evolution of the total torque, in units of $q^2 \Sigma (r_{\rm p})\Omega _{\rm K}^2r_{\rm p}^4$ , on a q=1.26 $\times$ 10^-5 planet in a disc with $\rho _0 \propto r^{-3/2}$ , $T \propto r^{-1}$ and $\gamma =1.1$ , for three different thermal diffusivities. All models have a small kinematic viscosity of $\nu =10^{-6}$ $r_{\rm p}^2 \Omega _{\rm p}$ .
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