4:15 PM - 4:30 PM
[PPS06-24] Linear Calculations of Density Waves Excited by a Giant Planet in a Disk Gap
Keywords:Planet formation, Protoplanetary disks, Disk-planet interactions
Density waves excited by planets excite in protoplanetary disks, have a significant impact on planet formation and gas disk evolution, resulting in planetary migration, gap formation and disk heating. Many linear calculations have been done to investigate density wave excitation (e.g. Goldreich & Tremaine 1979). However, all linear calculations assumed a disk with a flat surface density distribution. When there is a steep density gap created by a giant planet, only hydrodynamical simulations have investigated density wave excitation, and it is uncertain how the excitation differs from the case of a flat surface density distribution. The conventional view is that the torque due to the density waves simply decreases in proportion to the surface density in the gap. To understand the density wave excitation by a giant planet with a gap, we performed linear calculations of the density wave excitation in an unperturbed disk with a steep surface density gap.
To do this, we derived the perturbation equations describing density waves in the presence of a gap. The surface density gradient of the disk gap changes the rotation law of the unperturbed disk. This significantly changes the epicycle frequency, and also the position of the density wave excitation by the Lindblad resonance. We used the perturbation equations that include the effect of this change in the rotation law.
In the first gap model, we adopted a simple density distribution combining tanh(x) for a gap structure. We also investigated the case of a hypothetical steep surface density gap with a Rayleigh unstable rotation law. We obtain density waves with a shape similar to that of a flat density distribution as long as the gap is Rayleigh stable. On the other hand, a Rayleigh unstable gap can excite strong density waves with a completely different shape (figure 1).
Next, we used a realistic gap model developed by Kanagawa et al. (2017). We quantitatively compared the angular momentum flux of the excited density wave with the results of 2D hydrodynamical simulations by Kanagawa et al. (2016). We considered two cases: a planet with a Jupiter mass or half Jupiter mass. Both cases are in agreement with the results of the hydrodynamical simulations (figure 2). On the other hand, conventional estimates of the angular momentum fluxes of the density waves, which simply decrease with the gap surface densities, deviate from the results of the hydrodynamical simulations by the factor of 2 for a Jupiter-mass planet. This suggests that the effect of varying the rotation law in the gap, which is included in our linear calculations, allows us to reproduce the angular momentum fluxes of the density waves with high accuracy.
To do this, we derived the perturbation equations describing density waves in the presence of a gap. The surface density gradient of the disk gap changes the rotation law of the unperturbed disk. This significantly changes the epicycle frequency, and also the position of the density wave excitation by the Lindblad resonance. We used the perturbation equations that include the effect of this change in the rotation law.
In the first gap model, we adopted a simple density distribution combining tanh(x) for a gap structure. We also investigated the case of a hypothetical steep surface density gap with a Rayleigh unstable rotation law. We obtain density waves with a shape similar to that of a flat density distribution as long as the gap is Rayleigh stable. On the other hand, a Rayleigh unstable gap can excite strong density waves with a completely different shape (figure 1).
Next, we used a realistic gap model developed by Kanagawa et al. (2017). We quantitatively compared the angular momentum flux of the excited density wave with the results of 2D hydrodynamical simulations by Kanagawa et al. (2016). We considered two cases: a planet with a Jupiter mass or half Jupiter mass. Both cases are in agreement with the results of the hydrodynamical simulations (figure 2). On the other hand, conventional estimates of the angular momentum fluxes of the density waves, which simply decrease with the gap surface densities, deviate from the results of the hydrodynamical simulations by the factor of 2 for a Jupiter-mass planet. This suggests that the effect of varying the rotation law in the gap, which is included in our linear calculations, allows us to reproduce the angular momentum fluxes of the density waves with high accuracy.