2 Notations

The planet orbital radius is $r_{\rm p}$, and its orbital frequency  $\Omega_{\rm p}$. The position of a fluid element in the disk is represented by its polar coordinates radius r and azimuth $\theta$, counted rotation-wise, with its origin in the planet direction. The distance of a fluid element to the planet orbit is $x=r-r_{\rm p}$, and its dimensionless counterpart is $\hat x= x/r_{\rm p}$. The Keplerian frequency is $\Omega_{\rm K}(r)$, and the disk material orbital frequency is $\Omega(r)$. The disk kinematic viscosity is $\nu$ and its dimensionless counterpart is $\hat\nu=\nu/(r_{\rm p}^2\Omega_{\rm p})$. The $\alpha$ parameterization (Shakura & Syunyaev 1973) will also be used, with $\nu=\alpha Hc_{\rm s}$, H being the disk vertical scale length and $c_{\rm s}$ being the sound speed. The disk surface density is denoted with $\Sigma$, while its unperturbed uniform value is with $\Sigma_0$. The planet mass is $m_{\rm p}$, the central object mass is M_*, and the ratio of both is $q=m_{\rm p}/M_*$. The torque exerted on the planet is denoted $\Gamma$, and its non-dimensional counterpart is $\hat\Gamma=\Gamma/\Gamma_0$where $\Gamma_0=\pi r_{\rm p}^4\Omega_{\rm p}^2q^2\Sigma_0/h^3$, h being the aspect ratio. The perturbed velocity has radial and azimuthal components v_r and $v_\theta$.