All Figures

$\begin{figure} \par\includegraphics[angle=270,width=8.8cm]{9291F01.ps}\end{figure}$ Figure 1: Gas density profiles after 400 orbits for a few planet eccentricities, and common initial profile (plain line).
Open with DEXTER

In the text

$\begin{figure} \par\includegraphics[angle=270,width=8.8cm]{9291F02.ps}\end{figure}$ Figure 2: Tapering function f( d) defined by Eq. (5) for four values of the parameter p.
Open with DEXTER

In the text

$\begin{figure} \par\includegraphics[angle=270,width=8.8cm]{9291F03.ps}\end{figure}$ Figure 3: Influence of the inner disc on a planet on a fixed orbit, as a function of the eccentricity. NR (+ symbols): in the non-rotating frame; CF ( $\times$ symbols): in the corotating frame; $\Omega$ F (open squares): in the frame rotating monotonically with the same period as the planet.
Open with DEXTER

In the text

$\begin{figure} \par\includegraphics[angle=270,width=8.8cm]{9291F04.ps}\end{figure}$ Figure 4: Specific torque ( left panel) and power ( right panel) acting on the planet during one orbit for a fixed semi-major axis and eccentricity, e=0.15. The different curves correspond to measures using different values of the p parameter in Eq. (5). The horizontal lines correspond to the average in each of the four cases $p\neq 0$ .
Open with DEXTER

In the text

$\begin{figure} \par\includegraphics[angle=270,width=8.8cm]{9291F05.ps}\end{figure}$ Figure 5: Evolution during one orbit of the vectors ${\vec r}_{\rm p}$ : position of the planet in the frame rotating at velocity $\Omega _0$ ( right); ${\vec v}_{\rm p}\vert _{\rm NR}$ : velocity of the planet in the non-rotating frame ( top); ${\vec v}_{\rm p}\vert _{\Omega\rm F}$ : velocity of the planet in the frame rotating at constant velocity $\Omega _0$ ( middle); $\vec{F}$ : force from the disc on the planet ( left, arbitrary unit).
Open with DEXTER

In the text

$\begin{figure} \par\includegraphics[width=8.8cm]{9291F06.ps}\end{figure}$ Figure 6: Density maps, in linear grey scale, in the frame rotating at constant velocity $\Omega _0$ at four different moments of the orbit corresponding to Fig. 4.
Open with DEXTER

In the text

$\begin{figure} \par\includegraphics[angle=270,width=8.8cm]{9291F07.ps}\end{figure}$ Figure 7: Density profiles at time 0 (dashed line), at the moment where the planets are released (solid line), and at the end of the simulation (after 250 years, dotted line).
Open with DEXTER

In the text

$\begin{figure} \par\includegraphics[angle=270,width=8.8cm]{9291F08.ps}\end{figure}$ Figure 8: Semi-major axis (blue, left y-axis) and eccentricity (red, right y-axis) evolution of GJ 876 b (light colour, a₁, e₁) and GJ 876 c (dark, a₂, e₂). The horizontal lines correspond to the observed values, as given by Table 1.
Open with DEXTER

In the text

$\begin{figure} \par\includegraphics[angle=270,width=8.8cm]{9291F09.ps}\end{figure}$ Figure 9: Resonant angles associated with the 2:1 resonance $\theta _1$ , $\theta _2$ , $\Delta \omega$ as functions of time.
Open with DEXTER

In the text

$\begin{figure} \par\includegraphics[angle=270,width=8.8cm]{9291F10.ps}\end{figure}$ Figure 10: Eccentricity evolution of GJ 876 b (light colour, e₂) and GJ 876 c (dark, e₁) in three cases: (i) standard simulation (same as Fig. 8, curves labelled ``ref''); (ii) same as (i) but the inner planet is not affected by the disc (curves labelled ``no inn.''); (iii) same as (i) but the p parameter of the tapering function f is 0.8 (curves labelled ``p = 0.8''). The horizontal lines correspond to the observed values, as given by Table 1.
Open with DEXTER

In the text

$\begin{figure} \par\includegraphics[angle=270,width=8.8cm]{9291F11.ps}\end{figure}$ Figure 11: Semi-major axis and eccentricity evolution of GJ 876 b and GJ 876 c, with disc clearing from time t=250 yr on. The horizontal lines correspond to the observed values, as given by Table 1.
Open with DEXTER

In the text

$\begin{figure} \par\includegraphics[width=8cm]{9291F12a.eps} \includegraphics[width=8cm]{9291F12b.eps} \end{figure}$ Figure 12: Behaviour of the semi-major axes and the eccentricities of the resonant giant planets in the system GJ 876. The horizontal lines correspond to the observed values of the eccentricities. Top: only the outer disc is taken into account, and therefore only the orbital evolution of the outer planet is affected, with the e-folding times $\tau _{a_2}=2\times 10^4$ years, $\tau _{e_2}=2\times 10^2$ years, so K₂=100. Bottom: same as above in the presence of outer and inner discs. In this case, $\tau _{a_1}=-2\times 10^4$ , $\tau _{e_1}=2.5\times 10^3$ , $\tau _{a_2}=2\times 10^4$ , and $\tau _{e_2}=2.5\times 10^3$ years (giving K₁=K₂=8).
Open with DEXTER

In the text

$\begin{figure} \par\includegraphics[width=8cm]{9291F13a.eps} \includegraphics[width=8cm]{9291F13b.eps} \end{figure}$ Figure 13: Behaviour of the semi-major axes and the eccentricities of the giant planets in the resonant system HD 82 943. The horizontal lines correspond to the observed values of the eccentricities. Top: only the motion of the outer planet is affected, with the e-folding times $\tau _{a_2}=2\times 10^4$ and $\tau _{e_2}=2.5\times 10^3$ years (K₂=8). Bottom: an inner disc is also assumed, with $\tau _{a_1}=-2\times 10^5$ , $\tau _{e_1}=5\times 10^4$ years e-folding times for the inner planet (K₁=4), which allows the use of larger $\tau _{e_2}=2.5\times 10^3$ years and thus lower ratio K₂ =4 for the outer planet.
Open with DEXTER

In the text

$\begin{figure} \par\includegraphics[width=8cm]{9291F14a.eps} \includegraphics[width=8cm]{9291F14b.eps} \end{figure}$ Figure 14: Behaviour of the semi-major axes and the eccentricities of the resonant giant planets in the resonant system HD 128 311. The horizontal lines correspond to the observed values of the eccentricities. Top: only the motion of the outer planet is damped by an outer disc, with $K_2=\tau _{a_2}/\tau _{e_2}=7$ . Bottom: an inner disc is also assumed. The e-folding times for the inner planet are $\tau _{a_1}=2\times 10^4$ , $\tau _{e_1}=10^4$ years (K₁=2), which allows to use much lower ratio K₂ =2 for the outer planet.
Open with DEXTER

In the text

$\begin{figure} \par\includegraphics[width=8cm,angle=270]{9291F15.ps}\end{figure}$ Figure 15: Pairs of migration parameters $\tau _{a_1}$ and $\tau _{e_1}$ which result in the observed state of the system GJ 876 for fixed $\tau _{a_2}=4\times 10^4$ years and $\tau _{e_2}=5\times 10^3$ years. Crosses: empirically found with N-body simulations. Dashed curve: analytical expression Eq. (16) (see Sect. 4.4). Solid line: constant ratio K₁=8, for comparison.
Open with DEXTER

In the text

$\begin{figure} \par\includegraphics[angle=270,width=8.8cm]{9291F01.ps}\end{figure}$	Figure 1: Gas density profiles after 400 orbits for a few planet eccentricities, and common initial profile (plain line).
Open with DEXTER

$\begin{figure} \par\includegraphics[angle=270,width=8.8cm]{9291F02.ps}\end{figure}$	Figure 2: Tapering function f( d) defined by Eq. (5) for four values of the parameter p.
Open with DEXTER

$\begin{figure} \par\includegraphics[angle=270,width=8.8cm]{9291F03.ps}\end{figure}$	Figure 3: Influence of the inner disc on a planet on a fixed orbit, as a function of the eccentricity. NR (+ symbols): in the non-rotating frame; CF ( $\times$ symbols): in the corotating frame; $\Omega$ F (open squares): in the frame rotating monotonically with the same period as the planet.
Open with DEXTER

$\begin{figure} \par\includegraphics[angle=270,width=8.8cm]{9291F04.ps}\end{figure}$	Figure 4: Specific torque ( left panel) and power ( right panel) acting on the planet during one orbit for a fixed semi-major axis and eccentricity, e=0.15. The different curves correspond to measures using different values of the p parameter in Eq. (5). The horizontal lines correspond to the average in each of the four cases $p\neq 0$ .
Open with DEXTER

$\begin{figure} \par\includegraphics[width=8.8cm]{9291F06.ps}\end{figure}$	Figure 6: Density maps, in linear grey scale, in the frame rotating at constant velocity $\Omega _0$ at four different moments of the orbit corresponding to Fig. 4.
Open with DEXTER

$\begin{figure} \par\includegraphics[angle=270,width=8.8cm]{9291F07.ps}\end{figure}$	Figure 7: Density profiles at time 0 (dashed line), at the moment where the planets are released (solid line), and at the end of the simulation (after 250 years, dotted line).
Open with DEXTER

$\begin{figure} \par\includegraphics[angle=270,width=8.8cm]{9291F08.ps}\end{figure}$	Figure 8: Semi-major axis (blue, left y-axis) and eccentricity (red, right y-axis) evolution of GJ 876 b (light colour, a₁, e₁) and GJ 876 c (dark, a₂, e₂). The horizontal lines correspond to the observed values, as given by Table 1.
Open with DEXTER

$\begin{figure} \par\includegraphics[angle=270,width=8.8cm]{9291F09.ps}\end{figure}$	Figure 9: Resonant angles associated with the 2:1 resonance $\theta _1$ , $\theta _2$ , $\Delta \omega$ as functions of time.
Open with DEXTER

$\begin{figure} \par\includegraphics[angle=270,width=8.8cm]{9291F11.ps}\end{figure}$	Figure 11: Semi-major axis and eccentricity evolution of GJ 876 b and GJ 876 c, with disc clearing from time t=250 yr on. The horizontal lines correspond to the observed values, as given by Table 1.
Open with DEXTER

$\begin{figure} \par\includegraphics[width=8cm,angle=270]{9291F15.ps}\end{figure}$	Figure 15: Pairs of migration parameters $\tau _{a_1}$ and $\tau _{e_1}$ which result in the observed state of the system GJ 876 for fixed $\tau _{a_2}=4\times 10^4$ years and $\tau _{e_2}=5\times 10^3$ years. Crosses: empirically found with N-body simulations. Dashed curve: analytical expression Eq. (16) (see Sect. 4.4). Solid line: constant ratio K₁=8, for comparison.
Open with DEXTER