Σ is the density inside the gap, B is the second Oort constant. The function $$ f(P) = \left\{ \beginarrayl@\quadl (P-0.541)/4 & \mboxif $P<2.4646$\\ \\ 1-\exp(-P^0.75/3) & \mbox if $ P \geq 2.4646$ \\ \endarray \right . $$describes the gap depth expressed as the ratio between the gap surface density

and the unperturbed density at r  + . The variable P is defined by $$ P=\frac3H4R_H+\frac50(m_J/M) R \lesssim 1 $$where R is the Reynolds number and m J is the gas giant mass. In this way we are able to take into account the torque exerted on the outer disc by the gas in the gap and the corotation torque. The migration time can be estimated by $$ \tau_II = \frac(GM)^1/2m_Jr_J^1/22\Gamma. $$ (9) PD98059 Both types of migration (Types I and II) has been verified by numerical hydrodynamical calculations and good agreement has been found in the respective mass regimes. Type III Migration For intermediate-mass planets which open the gap only partially, it has been proposed the type III migration (Masset and Papaloizou 2003). This type of migration occurs if the disc mass is much higher than the mass of the planet. The corotation torques are responsible for this type of migration. This

migration can be very fast (Artymowicz 2004) and this is why it is called also “the runaway migration”. Resonance Capture It has been recognized that resonant structures may form as a result of the large scale orbital migration in young planetary systems discussed in Section “Planetary Migration”.

So resonant structures might be the indicators of the particular migration scenario Y-27632 research buy which took place in the past. The massive objects that we expect to find in forming planetary systems will migrate with different rates depending on their masses. Combining the expected differential Ceramide glucosyltransferase migration speeds described in the previous subsection with the strength of the commensurabilities given by Quillen (2006) and Mustill and Wyatt (2011), one can predict if the capture will take place or not. The resonant capture for the first order resonances in the restricted three body problem occurs when $$ \frac1\frac1\tau_I-\frac1\tau_II \geq \frac3 \pi \dot\eta_\rm crit \Omega_J $$ (10)where \(\dot \eta _\rm crit\) is the critical mean motion drift rate and Ω J is the angular velocity of the Jupiter-like planet. In the case of an internal 2:1 resonance \(\dot\eta_\rm crit=22.7~(\mathrmm_J/M)^4/3\), while for a 3:2 commensurability \(\dot\eta_\rm crit=126.4~(\mathrmm_J/M)^4/3\) (Quillen 2006). From Mustill and Wyatt (2011) it can be easily determined whether capture occurs for planet migrating in Types I or II regimes. For planets migrating through a gaseous disc, a non-zero eccentricity before the capture can cause the large libration amplitudes as it is observed in the HD 128311 system. Thus, when the eccentricities of the Jupiter-like planets are larger than 0.