Hydrocode Comparison for a Disc-Planet System

Torques

The formula used to obtain the torques in this calculation is:

do i=1,nr
   do j=1,nphi
      torq = torq + mplanet*rho(i,j)*y*r(i)*dr*dphi/(dist*dist+eps*eps)**1.5
   enddo
enddo

in units where rho₀=1 and the coordinates of the planet are (x,y)=(1,0). The sign convention gives the torque acting on the planet.

Torques time series

The plots show the moving average of the torques over 10 periods in units where a=1, Period=2*pi and M_star=1-mu.

The different contributions from inner disk, outer disk and total torques are defined excluding the Roche lobe.

The torque evolution is plotted over 500 orbital periods.

Inviscid, Jupiter:
Viscous, Jupiter (viscosity = 10e-5):
Inviscid, Neptune:
Viscous, Neptune (viscosity = 10e-5):

The same quantities are plotted for the first 200 orbits.

Inviscid, Jupiter:
Viscous, Jupiter (viscosity = 10e-5):
Inviscid, Neptune:
Viscous, Neptune (viscosity = 10e-5):

Inviscid Jupiter

Code	Total torque

Inviscid Neptune

Code	Total torque

Angular momentum time evolution

The total angular momentum calculated frome the snapshots is plotted as a funtion of time.

Plots of dT/dr profiles

The figures show the specific torque calculated from the snapshots excluding the contribution from the Roche lobe.

The thick solid line is the torque from the Lindblad resonances. The torque density is zero in the region |r-a|<2H_p/3 where the flow is subsonic in the frame rotating with the planet (|omega-omega_p| < c/r).

Low resolution (n_rxn_phi=128x384)
Mid resolution (n_rxn_phi=256x768)
Higher resolution (n_rxn_phi=512x1536)

Comparison with Lindblad torques

The figures show the torque from inner and outer disk excluding the Roche lobe.

In the second figure the solid, dotted and dashed horizontal lines are the integrated torques from the Lindblad resonances.

Low resolution (n_rxn_phi=128x384)
Mid resolution (n_rxn_phi=256x768)
Higher resolution (n_rxn_phi=512x1536)

T_m profiles

The figures show the torque as a function of order m calculated as
T_m = (dT/dr)/|dm/dr|

where m is defined as a real number (e.g. Ward 1997).
m = sqrt(kappa^2/(omega-omega_p)^2-c^2/r^2)

The contribution from the Roche lobe is excluded and we normalize by the factor
$T_o = mu^2 pi Sigma_p r_p^4 h^{-3}$

The solid (dotted) line shows the torque from the Lindblad resonances in the outer (inner) disk using the same normalization.

Low resolution (n_rxn_phi=128x384)
Mid resolution (n_rxn_phi=256x768)
Higher resolution (n_rxn_phi=512x1536)