disk-satellite interaction

basic conception

1. epicyclic frequency:

   a 'Rayleigh discriminant' to examine the 'boundaries' of an accretion disc with differential rotation $\mathrm{\Omega }$ is given by

   $${\kappa }^2=\frac{2\mathrm{\Omega }}{R}\frac{\mathrm{d}}{\mathrm{d}R}(R^2\mathrm{\Omega })$$

   where $R$ is the radial coordinate.

   derivation (Lagrange's equation & perturbation method)

   An astrophysical disk can be modeled as a fluid with negligible mass compared to the central object (e.g. a star) and with negligible pressure. We can suppose an axial symmetry such that $\mathrm{\Phi }(r,z)=\mathrm{\Phi }(r,-z)$.

   The Lagrangian in cylindrical coordiantes is

   $$L=\frac{m}{2}({\dot{r}}^2+r^2{\dot{\theta }}^2+{\dot{z}}^2)-m\mathrm{\Phi }$$

   Then we obtain the Lagrange's equation:

   $$\begin{array}{rl}\ddot{r}-r{\dot{\theta }}^2& =-{\partial }_r\mathrm{\Phi }\\ r\ddot{\theta }+2\dot{r}\dot{\theta }& =0\\ \ddot{z}& =-{\partial }_z\mathrm{\Phi }\end{array}$$

   The second line implies that the specific angular momentum is conserved. We can then define an effective potential ${\displaystyle {\mathrm{\Phi }}_{eff}=\mathrm{\Phi }+\frac{1}{2}{r}^2{\dot{\theta }}^2=\mathrm{\Phi }+\frac{h^2}{2r^2}}$ and so:

   $$\begin{array}{rl}\ddot{r}& =-{\partial }_r{\mathrm{\Phi }}_{eff}\\ \ddot{z}& =-{\partial }_z{\mathrm{\Phi }}_{eff}\end{array}$$

   We can apply a small perturbation $\delta \mathit{r}=\delta r{\mathit{e}}_r+\delta z{\mathit{e}}_z$ to the circular orbit:

   $$\mathit{r}=r_0{\mathit{e}}_r+\delta \mathit{r}$$

   So,

   $$\ddot{\mathit{r}}+\delta \ddot{\mathit{r}}=-\mathrm{\nabla }{\mathrm{\Phi }}_{eff}(\mathit{r}+\delta \mathit{r})\approx -\mathrm{\nabla }{\mathrm{\Phi }}_{eff}(\mathit{r})-{\partial }_r^2{\mathrm{\Phi }}_{eff}(\mathit{r})\delta r-{\partial }_z^2{\mathrm{\Phi }}_{eff}(\mathit{r})\delta z$$

   And thus:

   $$\begin{array}{r}\delta \ddot{r}=-{\partial }_r^2{\mathrm{\Phi }}_{eff}{\delta }_r=-{\mathrm{\Omega }}_r^2\delta r\\ \delta \ddot{z}=-{\partial }_z^2{\mathrm{\Phi }}_{eff}{\delta }_z=-{\mathrm{\Omega }}_z^2\delta z\end{array}$$

   We then note

   $${\kappa }^2={\mathrm{\Omega }}_r^2={\partial }_r^2{\mathrm{\Phi }}_{eff}={\partial }_r^2\mathrm{\Phi }+\frac{3h^2}{r^4}$$

   In a circular orbit $h_c^2=r^3{\partial }_r\mathrm{\Phi }$, ${\displaystyle {\mathrm{\Omega }}_c^2=\frac{1}{r}{\partial }_r\mathrm{\Phi }}$, finally yields:

   $${\kappa }^2={\partial }_r^2\mathrm{\Phi }+\frac{3}{r}{\partial }_r\mathrm{\Phi }=4{\mathrm{\Omega }}_c^2+2r{\mathrm{\Omega }}_c\frac{\mathrm{d}{\mathrm{\Omega }}_c}{\mathrm{d}r}=\frac{2{\mathrm{\Omega }}_c}{r}\frac{\mathrm{d}}{\mathrm{d}r}(r^2{\mathrm{\Omega }}_c)$$

2. mean-motion resonance (MMR):

   A mean-motion resonance occurs when multiple bodies have orbital periods or mean motions (orbital frequencies) that are simple integer ratios of each other.

GT79

some key points from GT79 the excitation of density waves at the Lindblad and corotaion resonances by an external potential

• main content:

  the linear response of a differentially rotating two-dimensional gas disk to a rigidly rotating external potential.

II. Basic Equations

• equation (3): the unperturbed angular velocity $\mathrm{\Omega }(r)$

  $$\begin{array}{}\text{(3)}& {\mathrm{\Omega }}^2(r)=\frac{1}{r}\frac{\mathrm{d}}{\mathrm{d}r}({\phi }_0+{\eta }_0)\end{array}$$

  derivation (enthalpy represents pressure)

  the Eular equation in radial direction of cylindrical coordinates for stable axial symmetry dics ( $\partial /\partial t=0,\partial /\partial \theta =0$ ):

  $$-\frac{v_{\theta }^2}{r}=-\frac{\partial \mathrm{\Phi }}{\partial r}-\frac{1}{\rho }\frac{\partial p}{\partial r}$$

  define the two-dimensional enthalpy $\eta$

  $$\mathrm{d}\eta =\frac{\mathrm{d}p}{\sigma }\Rightarrow \frac{\mathrm{d}\eta }{\mathrm{d}r}=\frac{1}{\sigma }\frac{\mathrm{d}p}{\mathrm{d}r}$$

  where $\sigma$ is the surface mass density.

  so we have

  $$-r{\mathrm{\Omega }}^2=-\frac{\mathrm{d}\phi }{\mathrm{d}r}-\frac{\mathrm{d}\eta }{\mathrm{d}r}$$$$\Rightarrow {\mathrm{\Omega }}^2(r)=\frac{1}{r}\frac{\mathrm{d}}{\mathrm{d}r}({\phi }_0+{\eta }_0)$$

• the linear perturbation equations (5,6,7,8)

  $$\begin{array}{}\text{(5)}& \frac{\partial {\mathit{v}}_1}{\partial t}+({\mathit{v}}_0\cdot \mathrm{\nabla }){\mathit{v}}_1+({\mathit{v}}_1\cdot \mathrm{\nabla }){\mathit{v}}_0& =-\mathrm{\nabla }({\phi }_1+{\phi }_1^D+{\eta }_1),\text{(6)}& \frac{\partial {\sigma }_1}{\partial t}+\mathrm{\nabla }\cdot ({\sigma }_0{\mathit{v}}_1)+\mathrm{\nabla }\cdot ({\sigma }_1{\mathit{v}}_0)& =0,\text{(7)}& {\eta }_1& =c_0^2({\sigma }_1/{\sigma }_0),\text{(8)}& {\mathrm{\nabla }}^2{\phi }_1^D& =4\pi G{\sigma }_1\delta (z),\end{array}$$

  where ${\mathit{v}}_0=r\mathrm{\Omega }(r){\hat{e}}_{\theta }$ and $\delta (z)$ is the Dirac delta function.

  derivation

  in brief,

  Eq.	derivation
  (5,6)	the linear perturbation method
  (7)	the definition of enthalpy
  (8)	the Poisson's equation of gravitational potential

  1. denoting the unpertubated gravitational potential ${\varphi }_0$, surface density ${\sigma }_0$, the Euler equation and continuity equation reads,

  $$\frac{\partial {\mathit{v}}_0}{\partial t}=({\mathit{v}}_0\cdot \mathrm{\nabla }){\mathit{v}}_0=\mathrm{\nabla }({\phi }_0+{\eta }_0)$$$$\frac{\partial {\sigma }_0}{\partial t}+\mathrm{\nabla }({\sigma }_0{\mathit{v}}_0)=0$$

  then we introduce the external perturbation potential ${\phi }_1$ and surface density perturbation ${\sigma }_1$ which generate additional potential ${\phi }_1^D$, hence we obtain the pertubation equations (5,6) by substituing ${\mathit{v}}_0+{\mathit{v}}_1$,${\phi }_0+{\phi }_1+{\phi }_1^D$,${\eta }_0+{\eta }_1$ to ${\mathit{v}}_0$,${\phi }_0$,${\eta }_0$

  $$\begin{array}{}\text{(5)}& \frac{\partial {\mathit{v}}_1}{\partial t}+({\mathit{v}}_0\cdot \mathrm{\nabla }){\mathit{v}}_1+({\mathit{v}}_1\cdot \mathrm{\nabla }){\mathit{v}}_0=-\mathrm{\nabla }({\phi }_1+{\phi }_1^D+{\eta }_1)\end{array}$$$$\begin{array}{}\text{(6)}& \frac{\partial {\sigma }_1}{\partial t}+\mathrm{\nabla }\cdot ({\sigma }_0{\mathit{v}}_1)+\mathrm{\nabla }\cdot ({\sigma }_1{\mathit{v}}_0)=0\end{array}$$

  2. from the paper, we know the sound speed $c_0$ satisfies (Eq.1 & 2)

  $$c_0^2=\frac{\mathrm{d}{p}_0}{\mathrm{d}{\sigma }_0}=\frac{\mathrm{d}}{\mathrm{d}{\sigma }_0}K{\sigma }_0^{\gamma }=K\gamma {\sigma }_0^{\gamma -1}$$

  with the definition of enthalpy $\eta$

  $$\eta =\int \frac{\mathrm{d}p}{\sigma }=\int \frac{1}{\sigma }\frac{\mathrm{d}p}{\mathrm{d}\sigma }\mathrm{d}\sigma =\int \frac{c^2}{\sigma }\mathrm{d}\sigma =\int \frac{K\gamma {\sigma }^{\gamma -1}}{\sigma }\mathrm{d}\sigma =\frac{K\gamma }{\gamma -1}{\sigma }^{\gamma -1}$$$$\Rightarrow ({\eta }_0+{\eta }_1)=\frac{K\gamma }{\gamma -1}({\sigma }_0+{\sigma }_1{)}^{\gamma -1}=\frac{K\gamma }{\gamma -1}[{\sigma }_0^{\gamma -1}+(\gamma -1){\sigma }_0^{\gamma -2}{\sigma }_1+\mathcal{O}({\sigma }_1^2)]$$$$\Rightarrow {\eta }_1=K\gamma {\sigma }_0^{\gamma -2}{\sigma }_1$$$$\begin{array}{}\text{(7)}& \Rightarrow {\eta }_1=c_0^2({\sigma }_1/{\sigma }_0)\end{array}$$

  3. as we mention at 1., the additional potential ${\phi }_1^D$ is arised by the surface density pertubation ${\sigma }_1$, so

  $$\begin{array}{}\text{(8)}& {\mathrm{\nabla }}^2{\phi }_1^D=4\pi G{\sigma }_1\delta (z)\end{array}$$

  this is the Poisson's equation of the gravitational potential.

• the response of the disk to ${\phi }_1$ is completely described by equation (7,8,13)

  $$\begin{array}{c}\{\frac{{\mathrm{d}}^2}{\mathrm{d}{r}^2}+[\frac{\mathrm{d}}{\mathrm{d}r}\ln \left(\frac{\sigma r}{D}\right)]\frac{\mathrm{d}}{\mathrm{d}r}+\frac{2m\mathrm{\Omega }}{r(m\mathrm{\Omega }-\omega )}[\frac{\mathrm{d}}{\mathrm{d}r}\ln \left(\frac{\sigma \mathrm{\Omega }}{D}\right)]-\frac{m^2}{r^2}\}\hfill \\ \text{(13)}& \hfill ({\phi }_1+{\phi }_1^D+{\eta }_1)=\frac{D{\eta }_1}{c^2}\end{array}$$

  where the subscripts on ${\sigma }_0$ and $c_0$ have been dropped.

  where $D={\kappa }^2-(m\mathrm{\Omega }-\omega {)}^2$

  derivation

  in brief, substitute the component form of equation (5) and equation (7) into the continuity equation (6).

  1. the component form of momentum equation (5):

     write each perturbation variable $X$ in the form $X=X(r)\exp i(m\theta -\omega t)$, and denote ${\mathit{v}}_1=u_1{\hat{e}}_r+v_1{\hat{e}}_{\theta }$

     • ${\displaystyle \frac{\partial {\mathit{v}}_1}{\partial t}}$
     $$\frac{\partial {\mathit{v}}_1}{\partial t}=-i\omega {\mathit{v}}_1=-i\omega u_1{\hat{e}}_r-i\omega v_1{\hat{e}}_{\theta }$$
     • $({\mathit{v}}_0\cdot \mathrm{\nabla }){\mathit{v}}_1$
     $$\{\begin{array}{l}{\mathit{v}}_0\cdot \mathrm{\nabla }=r\mathrm{\Omega }{\hat{e}}_{\theta }\cdot ({\displaystyle \frac{\partial }{\partial r}}{\hat{e}}_r+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial }{\partial \theta }}{\hat{e}}_{\theta })=\mathrm{\Omega }{\displaystyle \frac{\partial }{\partial \theta }}\\ {\displaystyle \frac{\partial {\mathit{v}}_1}{\partial \theta }}={\displaystyle \frac{\partial u_1}{\partial \theta }}{\hat{e}}_r+u_1{\displaystyle \frac{\partial {\hat{e}}_r}{\partial \theta }}+{\displaystyle \frac{\partial v_1}{\partial \theta }}+v_1{\displaystyle \frac{\partial {\hat{e}}_{\theta }}{\partial \theta }}=(im{u}_1-v_1){\hat{e}}_r+(u_1+im{v}_1){\hat{e}}_{\theta }\end{array}$$$$\Rightarrow ({\mathit{v}}_0\cdot \mathrm{\nabla }){\mathit{v}}_1=\mathrm{\Omega }{\displaystyle \frac{\partial {\mathit{v}}_1}{\partial \theta }}=\mathrm{\Omega }(im{u}_1-v_1){\hat{e}}_r+\mathrm{\Omega }(u_1+im{v}_1){\hat{e}}_{\theta }$$
     • $({\mathit{v}}_1\cdot \mathrm{\nabla }){\mathit{v}}_0$
     $${\mathit{v}}_0=r\mathrm{\Omega }{\hat{e}}_{\theta }\Rightarrow \{\begin{array}{l}{\displaystyle \frac{\partial {\mathit{v}}_0}{\partial r}}=(\mathrm{\Omega }+r{\displaystyle \frac{\mathrm{d}\mathrm{\Omega }}{\mathrm{d}r}}){\hat{e}}_{\theta },\\ {\displaystyle \frac{\partial {\mathit{v}}_0}{\partial \theta }}=-r\mathrm{\Omega }{\hat{e}}_r\end{array}$$$$\begin{array}{rl}({\mathit{v}}_1\cdot \mathrm{\nabla }){\mathit{v}}_0=u_1{\displaystyle \frac{\partial {\mathit{v}}_0}{\partial r}}+{\displaystyle \frac{v_1}{r}}{\displaystyle \frac{\partial {\mathit{v}}_0}{\partial \theta }}& =-\mathrm{\Omega }{v}_1{\hat{e}}_r+(\mathrm{\Omega }+r{\displaystyle \frac{\mathrm{d}\mathrm{\Omega }}{\mathrm{d}r}})u_1{\hat{e}}_{\theta }\\ & =-\mathrm{\Omega }{v}_1{\hat{e}}_r+(2B-\mathrm{\Omega })u_1{\hat{e}}_{\theta }\end{array}$$
     • $-\mathrm{\nabla }({\phi }_1+{\phi }_1^D+{\eta }_1)$
     $$\begin{array}{rl}-\mathrm{\nabla }({\phi }_1+{\phi }_1^D+{\eta }_1)& =-({\displaystyle \frac{\partial }{\partial r}}{\hat{e}}_r+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial }{\partial \theta }}{\hat{e}}_{\theta })({\phi }_1+{\phi }_1^D+{\eta }_1)\\ & =(-{\displaystyle \frac{\mathrm{d}}{\mathrm{d}r}}{\hat{e}}_r-{\displaystyle \frac{im}{r}}{\hat{e}}_{\theta })({\phi }_1+{\phi }_1^D+{\eta }_1)\end{array}$$

     above all, in radial component ${\hat{e}}_r$ and azimuthal component ${\hat{e}}_{\theta }$, we have

     $$\begin{array}{}\text{(10)}& \begin{array}{l}i(m\mathrm{\Omega }-\omega )u_1-2\mathrm{\Omega }{v}_1=-{\displaystyle \frac{\mathrm{d}}{\mathrm{d}r}}({\phi }_1+{\phi }_1^D+{\eta }_1)\\ 2B{u}_1+i(m\mathrm{\Omega }-\omega )v_1=-{\displaystyle \frac{im}{r}}({\phi }_1+{\phi }_1^D+{\eta }_1)\end{array}\end{array}$$

  2. the solution of systems of linear equations are easy to obtain by Linear Algebra

     $$\begin{array}{}\text{(11)}& \begin{array}{l}{u}_1=-{\displaystyle \frac{i}{D}}[(m\mathrm{\Omega }-\omega ){\displaystyle \frac{\mathrm{d}}{\mathrm{d}r}}+{\displaystyle \frac{2m\mathrm{\Omega }}{r}}]({\phi }_1+{\phi }_1^D+{\eta }_1),\\ v_1={\displaystyle \frac{1}{D}}[2B{\displaystyle \frac{\mathrm{d}}{\mathrm{d}r}}+{\displaystyle \frac{m}{r}}(m\mathrm{\Omega }-\omega )]({\phi }_1+{\phi }_1^D+{\eta }_1),\end{array}\end{array}$$

     where $D={\kappa }^2-(m\mathrm{\Omega }-\omega {)}^2$

  3. substituting above into the continuity equation (6) I won't try to calculate it again, you can try that as you wish. (╥﹏╥)

  resonance

  there are singularities in the coefficients of equation (13):

  • $D=0$: Lindblad resonance, $r=r_L$
  • $m\mathrm{\Omega }-\omega =0$: corotaion resonance, $r=r_c$

• the general solution of the perturbation equations away from the resonances:

  contain a homogeneous free-wave solutions and inhomogeneous non-wave solutions.

  1. the inhomogeneous non-wave solution ( ${\phi }_1^D\approx 0$ )
  $$\begin{array}{c}\frac{{\sigma }_1}{\sigma }=\frac{{\eta }_1}{c^2}=\hfill \\ \text{(15)}& \hfill \frac{1}{D}\{\frac{{\mathrm{d}}^2}{\mathrm{d}{r}^2}+[\frac{\mathrm{d}}{\mathrm{d}r}\ln \left(\frac{\sigma r}{D}\right)]\frac{\mathrm{d}}{\mathrm{d}r}+\frac{2m\mathrm{\Omega }}{r(m\mathrm{\Omega }-\omega )}[\frac{\mathrm{d}}{\mathrm{d}r}\ln \left(\frac{\sigma \mathrm{\Omega }}{D}\right)]-\frac{m^2}{r^2}\}{\phi }_1\end{array}$$
  2. the homogeneous free-wave solution ( ${\phi }_1=0$ )
  $$\begin{array}{}\text{(16)}& {\phi }_1^D(r)=\mathrm{\Phi }(r)\exp [i{\int }^{r}k(s)\mathrm{d}s]\end{array}$$$$\begin{array}{}\text{(17)}& {\sigma }_1=\frac{i\text{sgn}(k)}{2\pi G{r}^{1/2}}\frac{\mathrm{d}}{\mathrm{d}r}(r^{1/2}{\phi }_1^D)\end{array}$$

  here equation (17) is the WKB solution of Poisson's equation (8)

• the total angular momentum flux:

  $$\begin{array}{}\text{(30)}& F=F_G+F_A=-\text{sgn}(k)\frac{mr{\mathrm{\Phi }}^2}{4G}(1-\frac{c^2|k|}{\pi G\sigma })\end{array}$$

  derivation

  spiral density waves transport angular momentum by gravitational torques and by advective transport.

  1. the angular momentum flux across a cylinder of radius $r$ due to gravitational torques is

  $$\begin{array}{}\text{(23)}& F_G& =\frac{1}{4\pi G}{\int }_0^{2\pi }\mathrm{d}\theta {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\mathrm{d}zr\text{Re}\left[\frac{\partial {\phi }_1^D}{\partial \theta }\right]\text{Re}\left[\frac{\partial {\phi }_1^D}{\partial r}\right]\text{(25)}& & =\text{sgn}(k)\frac{mr{\mathrm{\Phi }}^2(r)}{4G}\end{array}$$

  2. the flux due to advective transport

  $$\begin{array}{}\text{(26)}& F_A& =r^2\sigma {\int }_0^{2\pi }\mathrm{d}\theta \text{Re}(u)\text{Re}(v)\text{(29)}& & =-\frac{\pi mr\sigma k}{D}{(1-\frac{c^2|k|}{2\pi G\sigma })}^2{\mathrm{\Phi }}^2\end{array}$$

  combining equation (25),(29), finally yields

  $$\begin{array}{}\text{(30)}& F=F_G+F_A=-\text{sgn}(k)\frac{mr{\mathrm{\Phi }}^2}{4G}(1-\frac{c^2|k|}{\pi G\sigma })\end{array}$$

III. Lindblad Resonances

• the long trailing wave excited at a Lindblad resonance has an angular momentum flux,

  $$\begin{array}{}\text{(46)}& F=-m{\pi }^2{\left\{\left|\frac{\sigma }{r\mathrm{d}D/\mathrm{d}r}\right|{[r\frac{\mathrm{d}{\phi }_1}{\mathrm{d}r}+\frac{2m\mathrm{\Omega }}{(m\mathrm{\Omega }-\omega )}{\phi }_1]}^2\right\}}_{r_L}\end{array}$$

  derivation

  computed from equations (15),(19),(30),(32), and (34)

  • the inhomogeneous non-wave solution (15)

  $$\begin{array}{c}\frac{{\sigma }_1}{\sigma }=\frac{{\eta }_1}{c^2}=\hfill \\ \text{(15)}& \hfill \frac{1}{D}\{\frac{{\mathrm{d}}^2}{\mathrm{d}{r}^2}+[\frac{\mathrm{d}}{\mathrm{d}r}\ln \left(\frac{\sigma r}{D}\right)]\frac{\mathrm{d}}{\mathrm{d}r}+\frac{2m\mathrm{\Omega }}{r(m\mathrm{\Omega }-\omega )}[\frac{\mathrm{d}}{\mathrm{d}r}\ln \left(\frac{\sigma \mathrm{\Omega }}{D}\right)]-\frac{m^2}{r^2}\}{\phi }_1\end{array}$$

  • the solutions of the dispersion relation (19)

  $$\begin{array}{}\text{(19)}& |k|=\frac{\pi G\sigma }{c^2}+ϵ{[{\left(\frac{\pi G\sigma }{c^2}\right)}^2-\frac{D}{c^2}]}^{1/2}\end{array}$$

  • the total angular momentum flux (30)

  $$\begin{array}{}\text{(30)}& F=F_G+F_A=-\text{sgn}(k)\frac{mr{\mathrm{\Phi }}^2}{4G}(1-\frac{c^2|k|}{\pi G\sigma })\end{array}$$

  • replace $D$ with $\mathcal{D}x$ (32)

  $$\begin{array}{}\text{(32)}& \mathcal{D}=[r(\mathrm{d}D/\mathrm{d}r){]}_{r_L}\end{array}$$$$\begin{array}{}\text{(31)}& x=(r-r_L)/r_L\end{array}$$

  • the potential (34)

  $$\begin{array}{}\text{(34)}& \mathrm{\Psi }={(r\frac{\mathrm{d}{\phi }_1}{\mathrm{d}r}+\frac{2m\mathrm{\Omega }}{(m\mathrm{\Omega }-\omega )}{\phi }_1)}_{r_L}\end{array}$$

IV. Corotaion Resonances

a) Disks without Self-Gravity

• the entire flux$$\begin{array}{}\text{(55)}& F_A=-\frac{{\pi }^2}{4}m\text{sgn}(x){\left[\frac{{\phi }_1^2}{\mathrm{d}\mathrm{\Omega }/\mathrm{d}r}\frac{\mathrm{d}}{\mathrm{d}r}\left(\frac{\sigma }{B}\right)\right]}_{r_c}\exp (-|qx|)\end{array}$$

b) Disks with Self-Gravity

• the torque exerted by the external potential$$\begin{array}{}\text{(93)}& T_c=\frac{m{\pi }^2}{2}{\left[\frac{{\phi }_1^2}{\mathrm{d}\mathrm{\Omega }/\mathrm{d}r}\frac{\mathrm{d}}{\mathrm{d}r}\left(\frac{\sigma }{B}\right)\right]}_{r_c}\end{array}$$

Appendix

a) Lindblad Resonances

while solving the Lindblad resonance, we define

$$\begin{array}{}\text{(34)}& \mathrm{\Psi }={(r\frac{\mathrm{d}{\phi }_1}{\mathrm{d}r}+\frac{2m\mathrm{\Omega }}{(m\mathrm{\Omega }-\omega )}{\phi }_1)}_{r_L}\end{array}$$$$\begin{array}{}\text{(36)}& \alpha =(2\pi G\sigma r/c^2{)}_{r_L}\text{sgn}(k)\end{array}$$$$\begin{array}{}\text{(38)}& \beta =(r/c{)}_{r_L}^2\mathcal{D}\end{array}$$

• the perturbed solution of velocity reads

  $$\begin{array}{}\text{(A6)}& u_1=-\frac{\kappa }{r_L|\mathcal{D}|}\mathrm{\Psi }{\int }_0^{\mathrm{\infty }}\mathrm{d}t\exp \left[i(xt-\frac{\alpha t^2}{2\beta }+\frac{t^3}{3\beta })\right]\end{array}$$$$\begin{array}{}\text{(A5)}& v_1=i\text{sgn}(\mathcal{D})\frac{\kappa }{2\mathrm{\Omega }}{u}_1\end{array}$$

• the torque exerted on the disk by the external potential in the neighborhood of a Lindblad resonance

  $$\begin{array}{}\text{(A7)}& T=-\pi m{\int }_0^{\mathrm{\infty }}\mathrm{d}rr{\phi }_1(r)\text{Im}{\sigma }_1(r)\end{array}$$$$\begin{array}{}\text{(A8)}& T=\pi m\text{Im}\left\{{\int }_0^{\mathrm{\infty }}\mathrm{d}r[\frac{m{\phi }_1\sigma v_1}{(m\mathrm{\Omega }-\omega )}+ir\sigma u_1\frac{\mathrm{d}}{\mathrm{d}r}\left(\frac{{\phi }_1}{m\mathrm{\Omega }-\omega }\right)]\right\}\end{array}$$$$\begin{array}{}\text{(A9)}& T_L=\frac{\pi m\sigma r_L\mathrm{\Psi }}{\kappa }\text{sgn}(\mathcal{D}){\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\mathrm{d}x\text{Re}(u_1)f(x)\end{array}$$

GT80

some key points from GT80 disk-satellite interactions

• main content:

  the interactions between a disk and a satellite which orbit the same central mass.

II. Steady-State Interactions at Resonances

a) The Disk

• Oort's parameters $A(r),B(r)$ and the epicyclic frequency $\kappa (r)$$$\begin{array}{}\text{(1)}& A(r)=\frac{r}{2}\frac{\mathrm{d}\mathrm{\Omega }}{\mathrm{d}r},B(r)=\mathrm{\Omega }(r)+A(r),{\kappa }^2(r)=4B(r)\mathrm{\Omega }(r)\end{array}$$

b) The Satellite

• the position of satellite when $e\ll 1$ in the first order in $e$ is

  $$\begin{array}{}\text{(2)}& r_s=a(1-e\cos {\kappa }_{s}t),{\theta }_s={\mathrm{\Omega }}_{s}t+\frac{2{\mathrm{\Omega }}_{s}e}{{\kappa }_s}\sin {\kappa }_{s}t\end{array}$$

  derivation

  see dynamics of planetary systems: the epicycle approximation

  considering the relation ${\kappa }_{\varphi }={\mathrm{\Omega }}_s$, ${\kappa }_R={\kappa }_s$, and the integration constants $\eta =\pi$, ${\varphi }_0=0$, the equation (a) there is the same as equation (2) here.

• the perturbation potential due to a satellite of mass $M_s$ reads

  $$\begin{array}{}\text{(4)}& {\varphi }^s(r,\theta ,t)=-\frac{G{M}_s}{|\mathit{r}-{\mathit{r}}_s|}+\frac{M_s}{M_p}{\mathrm{\Omega }}^2(r){\mathit{r}}_s\cdot \mathit{r}\end{array}$$

  derivation (the inertial potential)

  the acceleration of the central mass arised by the gravitation of the satellite reads

  $${\mathit{a}}_p=\frac{G{M}_s}{|{\mathit{r}}_s{|}^3}{\mathit{r}}_s$$

  for a unit mass point in this coordinate system, the inertial force satisfies

  $${\mathit{F}}_{iner}=-\mathrm{\nabla }{\varphi }_{iner}=-{\mathit{a}}_p$$

  so the inertial potential satisfies

  $$\mathrm{\nabla }{\varphi }_{iner}=\frac{G{M}_s}{|{\mathit{r}}_s{|}^3}{\mathit{r}}_s$$$$\Rightarrow {\varphi }_{iner}=\frac{G{M}_s}{|{\mathit{r}}_s{|}^3}{\mathit{r}}_s\cdot \mathit{r}+C$$

  in the binary system, we have

  $$G\frac{M_p{M}_s}{|{\mathit{r}}_s{|}^2}=M_s{\mathrm{\Omega }}^2|{\mathit{r}}_s|\Rightarrow \frac{G{M}_s}{|{\mathit{r}}_s{|}^3}=\frac{M_s}{M_p}{\mathrm{\Omega }}^2$$

  applying the relation above, we finally obtain

  $${\varphi }_{iner}=\frac{M_s}{M_p}{\mathrm{\Omega }}^2{\mathit{r}}_s\cdot \mathit{r}$$

• expand ${\varphi }^s$ in a Fourier series:

  $$\begin{array}{}\text{(5)}& {\varphi }^s(r,\theta ,t)=\sum _{l=-\mathrm{\infty }}^{\mathrm{\infty }}\sum _{m=0}^{\mathrm{\infty }}{\varphi }_{l,m}^s(r)\cos \{m\theta -[m{\mathrm{\Omega }}_s+(l-m){\kappa }_s]t\}\end{array}$$

  to first order in $e$, the only nonvanishing components are

  $$\begin{array}{}\text{(7)}& {\varphi }_{m,m}^s=-\frac{G{M}_s}{2a}(2-{\delta }_{m,0})(b_{1/2}^m-f\beta {\delta }_{m,1})\end{array}$$$$\begin{array}{}\text{(8)}& {\varphi }_{m+1,m}^s=-\frac{G{M}_s}{2a}e(2-{\delta }_{m,0})[(\frac{1}{2}+\frac{m{\mathrm{\Omega }}_s}{{\kappa }_s}+\frac{\beta }{2}\frac{\mathrm{d}}{\mathrm{d}\beta })b_{1/2}^m-f\beta (\frac{3}{2}-\frac{2B_s}{{\mathrm{\Omega }}_s}+\frac{{\mathrm{\Omega }}_s}{{\kappa }_s}){\delta }_{m,1}]\end{array}$$$$\begin{array}{}\text{(9)}& {\varphi }_{m+1,m}^s=-\frac{G{M}_s}{2a}e(2-{\delta }_{m,0})[(\frac{1}{2}-\frac{m{\mathrm{\Omega }}_s}{{\kappa }_s}+\frac{\beta }{2}\frac{\mathrm{d}}{\mathrm{d}\beta })b_{1/2}^m-f\beta (\frac{3}{2}-\frac{2B_s}{{\mathrm{\Omega }}_s}-\frac{{\mathrm{\Omega }}_s}{{\kappa }_s}){\delta }_{m,1}]\end{array}$$

  here $f={\mathrm{\Omega }}_s^2{a}^3/G{M}_p$, $\beta =r/a$, ${\delta }_{m,n}$ is the Kronecker delta function, and $b_{1/2}^m(\beta )$ is the Laplace coefficient,

  $$\begin{array}{}\text{(10)}& b_{1/2}^m(\beta )=\frac{2}{\pi }{\int }_0^{\pi }\frac{\cos m\varphi \mathrm{d}\varphi }{(1-2\beta \cos \varphi +{\beta }^2{)}^{1/2}}\end{array}$$

• the pattern speed of the $l,m$ potential component (in a Fourier series) is

  $${\mathrm{\Omega }}_{l,m}={\mathrm{\Omega }}_s+\frac{(l-m)}{m}{\kappa }_s$$

  the difference of mark

  the pattern speed is denoted by $\omega$ in GT79, both of them are the coefficients of time $t$.

c) Torques at Resonances

• the condition of Lindblad and corotation resonances:

  1. the Lindblad resonance:
  $$\begin{array}{}\text{(11)}& \mathrm{\Omega }(r)+ϵ\frac{\kappa (r)}{m}={\mathrm{\Omega }}_{l,m},ϵ=\pm 1,m>0\end{array}$$
  2. the corotation resonance:
  $$\begin{array}{}\text{(12)}& \mathrm{\Omega }(r)={\mathrm{\Omega }}_{l,m},m>0\end{array}$$

  derivation

  see the equation (13) in GT79

• the torque

  1. at a Lindblad resonance $r=r_L$,
  $$\begin{array}{}\text{(13)}& T_{l,m}^L=-m{\pi }^2{\left[\mathrm{\Sigma }{\left(\frac{r\mathrm{d}D}{\mathrm{d}r}\right)}^{-1}{(\frac{r\mathrm{d}{\varphi }_{l,m}^s}{\mathrm{d}r}+\frac{2\mathrm{\Omega }}{\mathrm{\Omega }-{\mathrm{\Omega }}_{l,m}}{\varphi }_{l,m}^s)}^2\right]}_{r_L}\end{array}$$

  where $D={\kappa }^2-m^2(\mathrm{\Omega }-{\mathrm{\Omega }}_{l,m}{)}^2$. Note that the sign of $T_{l,m}^L$ is opposite to that of $\mathrm{d}D/\mathrm{d}r$. Thus, angular momentum is removed from the disk at an inner Lindblad resonance and added to it at an outer Lindblad resonance.

  2. at a corotation resonance $r=r_c$,
  $$\begin{array}{}\text{(14)}& T_{l,m}^C=\frac{m{\pi }^2}{2}{[{\left(\frac{\mathrm{d}\mathrm{\Omega }}{\mathrm{d}r}\right)}^{-1}\frac{\mathrm{d}}{\mathrm{d}r}\left(\frac{\mathrm{\Sigma }}{B}\right)({\varphi }_{l,m}^s{)}^2]}_{r_c}\end{array}$$

• the modified Bessel function of order $\nu$ is denoted $K_{\nu }$

e) Orbital Variations

• the change rate of $a$ and $e$ are

  $$\begin{array}{}\text{(26)}& \frac{\mathrm{d}a}{\mathrm{d}t}=-\frac{2\mathrm{\Omega }{T}_r}{a{\kappa }^2{M}_s}\end{array}$$$$\begin{array}{}\text{(27)}& \frac{\mathrm{d}e}{\mathrm{d}t}=-[({\mathrm{\Omega }}_p-\mathrm{\Omega })-2e^2\mathrm{\Omega }(1+\frac{\mathrm{d}\ln \kappa }{\mathrm{d}\ln r})]\frac{T_r}{M_{s}e(a\kappa {)}^2}\end{array}$$

  derivation

  The derivation of equations (26) and (27) from equations (22)-(25) in Goldreich & Tremaine (1980) is as follows. The equations are:

  • Angular momentum:

    $$\begin{array}{}\text{(22)}& H=M_s{a}^2\mathrm{\Omega }\end{array}$$

  • Energy:

    $$\begin{array}{}\text{(23)}& E=\frac{1}{2}{M}_s[(a\mathrm{\Omega }{)}^2+(ea\kappa {)}^2]+M_s\mathrm{\Phi }(a)\end{array}$$

  • Torque relations:

    $$\begin{array}{}\text{(24)}& \frac{\mathrm{d}H}{\mathrm{d}t}& =-T_r,\text{(25)}& \frac{\mathrm{d}E}{\mathrm{d}t}& =-{\mathrm{\Omega }}_p{T}_r\end{array}$$

  • Identities:

    $$a{\mathrm{\Omega }}^2=\frac{\mathrm{d}\mathrm{\Phi }}{\mathrm{d}r},{\kappa }^2-3{\mathrm{\Omega }}^2=\frac{{\mathrm{d}}^2\mathrm{\Phi }}{\mathrm{d}{r}^2}\text{(evaluated at }r=a\text{)}$$

  • (a) Derivation of Equation (26) for $\mathrm{d}a/\mathrm{d}t$

  1. Differentiate equation (22):

     $$\frac{\mathrm{d}H}{\mathrm{d}t}=M_s(2a\frac{\mathrm{d}a}{\mathrm{d}t}\mathrm{\Omega }+a^2\frac{\mathrm{d}\mathrm{\Omega }}{\mathrm{d}t})$$

     Since $\mathrm{\Omega }=\mathrm{\Omega }(a)$, apply the chain rule:

     $$\frac{\mathrm{d}\mathrm{\Omega }}{\mathrm{d}t}=\frac{\mathrm{d}\mathrm{\Omega }}{\mathrm{d}a}\frac{\mathrm{d}a}{\mathrm{d}t}$$

     Substitute:

     $$\frac{\mathrm{d}H}{\mathrm{d}t}=M_s(2a\mathrm{\Omega }\frac{\mathrm{d}a}{\mathrm{d}t}+a^2\frac{\mathrm{d}\mathrm{\Omega }}{\mathrm{d}a}\frac{\mathrm{d}a}{\mathrm{d}t})=M_{s}a\frac{\mathrm{d}a}{\mathrm{d}t}(2\mathrm{\Omega }+a\frac{\mathrm{d}\mathrm{\Omega }}{\mathrm{d}a})$$

  2. Use Oort's parameters (equation 1):

     $$A=\frac{r}{2}\frac{\mathrm{d}\mathrm{\Omega }}{\mathrm{d}r},B=\mathrm{\Omega }+A$$

     At $r=a$:

     $$a\frac{\mathrm{d}\mathrm{\Omega }}{\mathrm{d}a}=2A$$

     Substitute:

     $$\frac{\mathrm{d}H}{\mathrm{d}t}=M_{s}a\frac{\mathrm{d}a}{\mathrm{d}t}(2\mathrm{\Omega }+2A)=2M_{s}aB\frac{\mathrm{d}a}{\mathrm{d}t}$$

  3. Apply torque equation (24):

     $$\frac{\mathrm{d}H}{\mathrm{d}t}=-T_r$$

     So:

     $$-T_r=2M_{s}aB\frac{\mathrm{d}a}{\mathrm{d}t}⟹\frac{\mathrm{d}a}{\mathrm{d}t}=-\frac{T_r}{2M_{s}aB}$$

  4. Use ${\kappa }^2=4B\mathrm{\Omega }$ (from equation 1):

     $$B=\frac{{\kappa }^2}{4\mathrm{\Omega }}$$

     Substitute:

     $$\frac{\mathrm{d}a}{\mathrm{d}t}=-\frac{T_r}{2M_{s}a\cdot ({\kappa }^2/(4\mathrm{\Omega }))}=-\frac{2\mathrm{\Omega }{T}_r}{a{\kappa }^2{M}_s}$$

     This matches equation (26).

  • (b) Derivation of Equation (27) for $\mathrm{d}e/\mathrm{d}t$

  1. Differentiate equation (23):

     $$\frac{\mathrm{d}E}{\mathrm{d}t}=M_s[\frac{1}{2}\cdot 2a\frac{\mathrm{d}a}{\mathrm{d}t}{\mathrm{\Omega }}^2+\frac{1}{2}{a}^2\cdot 2\mathrm{\Omega }\frac{\mathrm{d}\mathrm{\Omega }}{\mathrm{d}t}+\frac{1}{2}\cdot 2e{a}^2{\kappa }^2\frac{\mathrm{d}e}{\mathrm{d}t}+\frac{1}{2}{e}^2\cdot 2a{\kappa }^2\frac{\mathrm{d}a}{\mathrm{d}t}+\frac{1}{2}{e}^2{a}^2\cdot 2\kappa \frac{\mathrm{d}\kappa }{\mathrm{d}t}+\frac{\mathrm{d}\mathrm{\Phi }}{\mathrm{d}t}]$$

     Simplify:

     $$\frac{\mathrm{d}E}{\mathrm{d}t}=M_s[a{\mathrm{\Omega }}^2\frac{\mathrm{d}a}{\mathrm{d}t}+a^2\mathrm{\Omega }\frac{\mathrm{d}\mathrm{\Omega }}{\mathrm{d}t}+e{a}^2{\kappa }^2\frac{\mathrm{d}e}{\mathrm{d}t}+e^{2}a{\kappa }^2\frac{\mathrm{d}a}{\mathrm{d}t}+e^2{a}^2\kappa \frac{\mathrm{d}\kappa }{\mathrm{d}t}+\frac{\mathrm{d}\mathrm{\Phi }}{\mathrm{d}a}\frac{\mathrm{d}a}{\mathrm{d}t}]$$

     From $a{\mathrm{\Omega }}^2=\mathrm{d}\mathrm{\Phi }/\mathrm{d}r$, so $\mathrm{d}\mathrm{\Phi }/\mathrm{d}a=a{\mathrm{\Omega }}^2$:

     $$\frac{\mathrm{d}E}{\mathrm{d}t}=M_s[a{\mathrm{\Omega }}^2\frac{\mathrm{d}a}{\mathrm{d}t}+a^2\mathrm{\Omega }\frac{\mathrm{d}\mathrm{\Omega }}{\mathrm{d}t}+e{a}^2{\kappa }^2\frac{\mathrm{d}e}{\mathrm{d}t}+e^{2}a{\kappa }^2\frac{\mathrm{d}a}{\mathrm{d}t}+e^2{a}^2\kappa \frac{\mathrm{d}\kappa }{\mathrm{d}t}+a{\mathrm{\Omega }}^2\frac{\mathrm{d}a}{\mathrm{d}t}]$$

     Combine terms:

     $$\frac{\mathrm{d}E}{\mathrm{d}t}=M_s[\underset{\text{terms with }\mathrm{d}a/\mathrm{d}t}{\underbrace{2a{\mathrm{\Omega }}^2\frac{\mathrm{d}a}{\mathrm{d}t}+a^2\mathrm{\Omega }\frac{\mathrm{d}\mathrm{\Omega }}{\mathrm{d}t}+e^{2}a{\kappa }^2\frac{\mathrm{d}a}{\mathrm{d}t}+e^2{a}^2\kappa \frac{\mathrm{d}\kappa }{\mathrm{d}t}}}+e{a}^2{\kappa }^2\frac{\mathrm{d}e}{\mathrm{d}t}]$$

  2. Apply chain rule to $\mathrm{d}\mathrm{\Omega }/\mathrm{d}t$ and $\mathrm{d}\kappa /\mathrm{d}t$:

     $$\frac{\mathrm{d}\mathrm{\Omega }}{\mathrm{d}t}=\frac{\mathrm{d}\mathrm{\Omega }}{\mathrm{d}a}\frac{\mathrm{d}a}{\mathrm{d}t},\frac{\mathrm{d}\kappa }{\mathrm{d}t}=\frac{\mathrm{d}\kappa }{\mathrm{d}a}\frac{\mathrm{d}a}{\mathrm{d}t}$$

     Substitute:

     $$\frac{\mathrm{d}E}{\mathrm{d}t}=M_s[(2a{\mathrm{\Omega }}^2+a^2\mathrm{\Omega }\frac{\mathrm{d}\mathrm{\Omega }}{\mathrm{d}a}+e^{2}a{\kappa }^2+e^2{a}^2\kappa \frac{\mathrm{d}\kappa }{\mathrm{d}a})\frac{\mathrm{d}a}{\mathrm{d}t}+e{a}^2{\kappa }^2\frac{\mathrm{d}e}{\mathrm{d}t}]$$

  3. Apply torque equation (25):

     $$\frac{\mathrm{d}E}{\mathrm{d}t}=-{\mathrm{\Omega }}_p{T}_r$$

     So:

     $$-{\mathrm{\Omega }}_p{T}_r=M_s[(2a{\mathrm{\Omega }}^2+a^2\mathrm{\Omega }\frac{\mathrm{d}\mathrm{\Omega }}{\mathrm{d}a}+e^{2}a{\kappa }^2+e^2{a}^2\kappa \frac{\mathrm{d}\kappa }{\mathrm{d}a})\frac{\mathrm{d}a}{\mathrm{d}t}+e{a}^2{\kappa }^2\frac{\mathrm{d}e}{\mathrm{d}t}]$$

  4. Substitute $\mathrm{d}a/\mathrm{d}t$ from equation (26):

     $$\frac{\mathrm{d}a}{\mathrm{d}t}=-\frac{T_r}{2M_{s}aB}$$

     Insert:

     $$-{\mathrm{\Omega }}_p{T}_r=M_s[(2a{\mathrm{\Omega }}^2+a^2\mathrm{\Omega }\frac{\mathrm{d}\mathrm{\Omega }}{\mathrm{d}a}+e^{2}a{\kappa }^2+e^2{a}^2\kappa \frac{\mathrm{d}\kappa }{\mathrm{d}a})(-\frac{T_r}{2M_{s}aB})+e{a}^2{\kappa }^2\frac{\mathrm{d}e}{\mathrm{d}t}]$$

  5. Simplify and solve for $\mathrm{d}e/\mathrm{d}t$:

     $$-{\mathrm{\Omega }}_p{T}_r=-\frac{T_r}{2B}(2{\mathrm{\Omega }}^2+a\mathrm{\Omega }\frac{\mathrm{d}\mathrm{\Omega }}{\mathrm{d}a}+e^2{\kappa }^2+e^{2}a\kappa \frac{\mathrm{d}\kappa }{\mathrm{d}a})+M_{s}e{a}^2{\kappa }^2\frac{\mathrm{d}e}{\mathrm{d}t}$$

     Divide by $T_r$:

     $$-{\mathrm{\Omega }}_p=-\frac{1}{2B}(2{\mathrm{\Omega }}^2+a\mathrm{\Omega }\frac{\mathrm{d}\mathrm{\Omega }}{\mathrm{d}a}+e^2{\kappa }^2+e^{2}a\kappa \frac{\mathrm{d}\kappa }{\mathrm{d}a})+\frac{M_{s}e{a}^2{\kappa }^2}{T_r}\frac{\mathrm{d}e}{\mathrm{d}t}$$

     Rearrange:

     $$\frac{M_{s}e{a}^2{\kappa }^2}{T_r}\frac{\mathrm{d}e}{\mathrm{d}t}=-{\mathrm{\Omega }}_p+\frac{1}{2B}(2{\mathrm{\Omega }}^2+a\mathrm{\Omega }\frac{\mathrm{d}\mathrm{\Omega }}{\mathrm{d}a}+e^2{\kappa }^2+e^{2}a\kappa \frac{\mathrm{d}\kappa }{\mathrm{d}a})$$$$\frac{\mathrm{d}e}{\mathrm{d}t}=\frac{T_r}{M_{s}e{a}^2{\kappa }^2}[-{\mathrm{\Omega }}_p+\frac{1}{2B}(2{\mathrm{\Omega }}^2+a\mathrm{\Omega }\frac{\mathrm{d}\mathrm{\Omega }}{\mathrm{d}a}+e^2{\kappa }^2+e^{2}a\kappa \frac{\mathrm{d}\kappa }{\mathrm{d}a})]$$

  6. Simplify using identities:

     • From ${\kappa }^2=4B\mathrm{\Omega }$, so $2{\mathrm{\Omega }}^2+a\mathrm{\Omega }\frac{\mathrm{d}\mathrm{\Omega }}{\mathrm{d}a}=\frac{{\kappa }^2}{2}$ (derived from ${\kappa }^2=4{\mathrm{\Omega }}^2+2a\mathrm{\Omega }\frac{\mathrm{d}\mathrm{\Omega }}{\mathrm{d}a}$).
     • With $\frac{\mathrm{d}\ln \kappa }{\mathrm{d}\ln r}=\frac{\mathrm{d}\ln \kappa }{\mathrm{d}r}\frac{1}{\frac{\mathrm{d}\ln r}{\mathrm{d}r}}=\frac{\mathrm{d}\ln \kappa }{\mathrm{d}r}r=\frac{\mathrm{d}\ln \kappa }{\mathrm{d}\kappa }\frac{\mathrm{d}\kappa }{\mathrm{d}r}r=\frac{r}{\kappa }\frac{\mathrm{d}\kappa }{\mathrm{d}r}{|}_{r=a}=\frac{a}{\kappa }\frac{\mathrm{d}\kappa }{\mathrm{d}a}$.
       Substitute:
     $$\frac{\mathrm{d}e}{\mathrm{d}t}=\frac{T_r}{M_{s}e{a}^2{\kappa }^2}[-{\mathrm{\Omega }}_p+\frac{1}{2B}(\frac{{\kappa }^2}{2}+e^2{\kappa }^2(1+\frac{\mathrm{d}\ln \kappa }{\mathrm{d}\ln r}))]$$$$=\frac{T_r}{M_{s}e{a}^2{\kappa }^2}[-{\mathrm{\Omega }}_p+\underset{\mathrm{\Omega }}{\underbrace{\frac{{\kappa }^2}{4B}}}+\frac{e^2{\kappa }^2}{2B}(1+\frac{\mathrm{d}\ln \kappa }{\mathrm{d}\ln r})]$$$$=\frac{T_r}{M_{s}e{a}^2{\kappa }^2}[-{\mathrm{\Omega }}_p+\mathrm{\Omega }+\frac{e^2{\kappa }^2}{2B}(1+\frac{\mathrm{d}\ln \kappa }{\mathrm{d}\ln r})]$$

     Since ${\kappa }^2=4B\mathrm{\Omega }$, then ${\kappa }^2/(2B)=2\mathrm{\Omega }$:

     $$\frac{\mathrm{d}e}{\mathrm{d}t}=\frac{T_r}{M_{s}e{a}^2{\kappa }^2}[-({\mathrm{\Omega }}_p-\mathrm{\Omega })+2e^2\mathrm{\Omega }(1+\frac{\mathrm{d}\ln \kappa }{\mathrm{d}\ln r})]$$

     Rearrange to match equation (27):

     $$\frac{\mathrm{d}e}{\mathrm{d}t}=-[({\mathrm{\Omega }}_p-\mathrm{\Omega })-2e^2\mathrm{\Omega }(1+\frac{\mathrm{d}\ln \kappa }{\mathrm{d}\ln r})]\frac{T_r}{M_{s}e(a\kappa {)}^2}$$

     This is equation (27), where $(a\kappa {)}^2=a^2{\kappa }^2$.

  • Key Steps Recap:
    • For $\mathrm{d}a/\mathrm{d}t$: Differentiated angular momentum $H$, used torque $\mathrm{d}H/\mathrm{d}t=-T_r$, and substituted $B={\kappa }^2/(4\mathrm{\Omega })$.
    • For $\mathrm{d}e/\mathrm{d}t$: Differentiated energy $E$, used torque $dE/dt=-{\mathrm{\Omega }}_p{T}_r$, substituted $\mathrm{d}a/\mathrm{d}t$, and applied identities for ${\kappa }^2$ and logarithmic derivatives.
    • Note: All derivatives ($\mathrm{\Omega },\kappa ,\mathrm{\Phi }$) are evaluated at $r=a$, and $\mathrm{d}\ln \kappa /\mathrm{d}\ln r$ is the logarithmic derivative of $\kappa$ with respect to $r$.

W88

some key points from W88 On disk-planet interactions and orbital eccentricities

• main content:

  consider the co-orbital material, using vertically averaging method to obtain the potential

I. Introduction

The result in GT80 of corotation and Lindblad resonances was derived for a perturber orbiting outside a disk or within a gap, now consider an object embedded in a continuous disk without a gap.

II. Ring and Disk Torques

This section reviews the established theory for calculating torque ( $T$ ) on a perturber from disk resonances and its effect on eccentricity $e$.

• Lindblad Resonances (Eq.4, 8): They always excite eccentricity. Both inner and outer Lindblad resonances contribute to this excitation.
• Corotation Resonances (Eq.9, 12): For a perturber in a gap or outside a disk, they leading to net damping. However, for an object embedded in a smooth, continuous disk, the interior and exterior corotation torques largely cancel each other out. Their net contribution to eccentricity change becomes very small and of second order, depending on the second derivative of the vortensity (Eq. 18).
• Implications: Consider the role of co-orbital resonance (resonances at the same orbital distance as the perturber), which were previously sidestepped.

III. Co-orbital Torques

This is the paper's central section, where Ward develops a model to handle the mathematically tricky co-orbital resonances.

• The Problem: Standard potential expansions diverge at $r=a$ (the planet's orbit). Ward addresses this by vertically averaging the gravitational potential over the disk's scale height ($h$), which removes the singularity and acknowledges the 3D nature of the disk.

  $$\begin{array}{}\text{(20a)}& b_{1/2}^m(\gamma ,z)=\frac{2}{\pi }{\int }_0^{\pi }\frac{\cos m\lambda \mathrm{d}\lambda }{\sqrt{1-2\gamma \cos \lambda +{\gamma }^2+(z/r_p{)}^2}},\end{array}$$$$\begin{array}{}\text{(20b)}& b_{1/2}^m(\gamma ,z)\approx \frac{2}{\pi }{K}_0(\sqrt{m^2(1-\gamma {)}^2+(mz/r_p{)}^2}).\end{array}$$

• Key Findings:

  1. Distant Lindblad Resonances: ( $\alpha =4/3$ )

     $$\begin{array}{}\text{(27)}& ⟨{\mathrm{\Psi }}_{m-\epsilon ,m}⟩=-e\frac{G{M}_p}{a}\frac{m}{\sqrt{\pi }}\left(\frac{a\mathrm{\Omega }}{c}\right)\{{\mathcal{F}}_2(4/3,\xi )+4{\mathcal{F}}_1(4/3,\xi )+4{\mathcal{F}}_0(4/3,\xi )\},\end{array}$$$$\begin{array}{}\text{(29)}& {\left(\frac{1}{e}\frac{\mathrm{d}e}{\mathrm{d}t}\right)}^L=\frac{\pi }{3}m{\left(\frac{a\mathrm{\Omega }}{c}\right)}^2\mu \mathrm{\Omega }{\left(\frac{\sigma a^2}{M_{\odot }}\right)}^2\{{\mathcal{F}}_2+4{\mathcal{F}}_1+4{\mathcal{F}}_0{\}}_{\alpha =4/3}^2\end{array}$$

  2. Distant Corotation Resonances: ( $\alpha =2/3$ )

     $$\begin{array}{}\text{(28)}& ⟨{\varphi }_{m-\epsilon ,m}⟩=\epsilon e\frac{G{M}_p}{a}\frac{1}{\sqrt{\pi }}\left(\frac{a\mathrm{\Omega }}{c}\right)\{\mathcal{F}(2/3,0)+2\mathcal{F}(2/3,\xi )\}.\end{array}$$$$\begin{array}{}\text{(30)}& {\left(\frac{1}{e}\frac{\mathrm{d}e}{\mathrm{d}t}\right)}^C=-\epsilon \frac{4\pi }{3}{\left(\frac{a\mathrm{\Omega }}{c}\right)}^2\mu \left(\frac{a^3{\mathrm{\Omega }}^2}{M_{\odot }}\right)\left(\frac{\mathrm{d}}{\mathrm{d}r}\frac{\sigma }{\mathrm{\Omega }}\right)\{{\mathcal{F}}_1+2{\mathcal{F}}_0{\}}_{\alpha =2/3}^2\end{array}$$

  3. Co-orbital Lindblad Resonances: These are a new class of resonances that occur for material sharing the planet's orbit. Their forcing function is derived (Eq. 52):

     $$\begin{array}{rl}⟨{\mathrm{\Psi }}_{m+\epsilon ,m}⟩& ={\mathrm{\Psi }}_d+{\mathrm{\Psi }}_a\\ \text{(52)}& & =-e\frac{G{M}_{p}m}{a\sqrt{\pi }}\left(\frac{a\mathrm{\Omega }}{c}\right)[{\mathcal{F}}_2(0,\xi )-4{\mathcal{F}}_0(0,\xi )]+\frac{4}{\pi }\frac{G{M}_p}{h\sqrt{\pi }}\sin f_c\end{array}$$

     and the resulting torque is calculated (Eq. 60).

     • Effect: Crucially, all co-orbital Lindblad resonances damp eccentricity ($(1/e)(\mathrm{d}e/\mathrm{d}t{)}_{\text{co-orbital}}^L<0$). This is the opposite of distant Lindblad resonances.
     • Formula (Eq. 65)

  4. Co-orbital Corotation Resonances: These are also present and are strong (zeroth order in $e$). However, because their pattern speed equals the planet's mean motion (${\mathrm{\Omega }}_{ps}={\mathrm{\Omega }}_p$), the term $({\mathrm{\Omega }}_{ps}-{\mathrm{\Omega }}_p)$ in the fundamental eccentricity equation (Eq. 1) is zero. This severely reduces their ability to change eccentricity (Eq. 69). Their effect is much weaker ($\propto 1/m$) than that of co-orbital Lindblad torques ($\propto m$).

Questions about distant & co-orbital

I:

I'm still confused that the Eq(43)(44) obtained from the vertically average the disturbing function Eq.(21), and the Eq(27)(28) also come from Eq.(21), but why the Eq.(43)(44) represent the co-orbital material while the Eq(27)(28) represent the perturber(planet)?

Deepseek:

In Ward's paper, both Equation (27)/(28) and Equation (43)/(44) are derived from the vertically averaged disturbing function (Eq. (21)), but they represent different aspects of the disk-planet interaction due to distinct evaluation points and mathematical treatments. Here's a detailed explanation of the difference:

1. Equation (27) and (28): Representing the Perturber's Potential at Distant Resonances

• Origin: These equations are derived from Eq. (21) by evaluating the vertically averaged potential at specific resonance locations away from the planet's orbit. For Lindblad resonances (Eq. (27)), the evaluation uses $\alpha =m|1-\gamma |=4/3$, which corresponds to the approximate location of Lindblad resonances in a Keplerian disk (where $\gamma =r/a$). Similarly, for corotation resonances (Eq. (28)), $\alpha =2/3$ is used.
• Physical Meaning: Eqs. (27) and (28) quantify the strength of the planet's gravitational potential as experienced by disk material at these distant resonances. They are used to calculate the torque exerted by the planet on the disk (or vice versa) at these locations. Specifically:
• Eq. (27) gives the forcing function $⟨{\mathrm{\Psi }}_{m-ϵ,m}⟩$ for Lindblad resonances, which is used in the torque formula (Eq. (4)).
• Eq. (28) gives the potential amplitude $⟨{\varphi }_{m-ϵ,m}⟩$ for corotation resonances, used in the torque formula (Eq. (9)).
• Key Point: These equations focus on resonances that are not co-orbital; they are interior or exterior to the planet's orbit. Thus, they represent the planet's influence on distant disk material.

2. Equation (43) and (44): Representing the Potential for Co-orbital Material

• Origin: These equations are also derived from Eq. (21), but through a different process:
• Expansion around $r=a$: The vertically averaged potential is expanded in a Taylor series around the co-orbital point $r=a$ (see Eq. (38)), which includes terms like ${\mathcal{F}}_0(0,\xi )$, ${\mathcal{F}}_2(0,\xi )$, and the sign function $\text{sgn}(1-\gamma )$.
• Fourier Decomposition in Time: Due to the planet's eccentric motion, the potential varies periodically with the true anomaly $f$. This expansion is Fourier decomposed to isolate components with pattern speeds that resonate with co-orbital material (e.g., ${\mathrm{\Omega }}_p$ for corotation and ${\mathrm{\Omega }}_p\pm {\kappa }_p/m$ for Lindblad resonances). This decomposition results in the coefficients $A_k$ (given in Appendix A).
• Physical Meaning: Eqs. (43) and (44) represent the amplitudes of the gravitational potential components that drive resonances in the co-orbital region. Specifically:
• Eq. (43) gives $⟨{\varphi }_{m,m}⟩$ for the co-orbital corotation resonance (pattern speed ${\mathrm{\Omega }}_p$).
• Eq. (44) gives $⟨{\varphi }_{m\pm 1,m}⟩$ for the co-orbital Lindblad resonances (pattern speeds ${\mathrm{\Omega }}_p\pm {\kappa }_p/m$).
• Key Point: These potentials are not directly from the planet; instead, they arise from the disk's response to the planet's gravity. However, in this context, they are derived from the planet's potential but are used to describe how the co-orbital material is forced. The process captures the time-dependent effects due to eccentricity, which are crucial for co-orbital dynamics.

Why the Difference?

• Evaluation Point: Eqs. (27)/(28) are evaluated at resonance locations away from $r=a$, while Eqs. (43)/(44) are derived from an expansion at $r=a$.
• Mathematical Treatment: Eqs. (27)/(28) use a fixed $\alpha$ value based on resonance conditions, whereas Eqs. (43)/(44) involve a Fourier series in time to account for the planet's orbital motion, which is essential for co-orbital material that experiences periodic forcing.
• Physical Context: The co-orbital region involves material that shares the planet's orbit, so the potential must be handled with care to avoid singularities and include horseshoe dynamics. The Fourier decomposition achieves this by identifying the relevant components for resonance.

In summary, while both sets of equations originate from the same disturbing function (Eq. (21)), Eqs. (27)/(28) represent the planet's potential at distant resonances, and Eqs. (43)/(44) represent the potential components that drive resonances in the co-orbital region, derived through a specialized expansion and decomposition. This distinction is crucial for understanding how co-orbital torques differ from distant torques in eccentricity evolution.

• the formula of $⟨{\varphi }_{m,m}⟩$:$$\begin{array}{}\text{(43)}& ⟨{\varphi }_{m,m}⟩=\frac{1}{2}{A}_0\end{array}$$$$\begin{array}{}\text{(A3)}& A_0=-\frac{4G{M}_p}{\pi r_p\sqrt{\pi }}{\xi }^{-1}\{\pi {\mathcal{F}}_0-me{\gamma }_0[(\pi -2f_c)\cos f_c+2\sin f_c]+\frac{\pi }{2}(me{\gamma }_0{)}^2{\mathcal{F}}_2(\frac{1}{2}+{\cos }^2{f}_c)\}\end{array}$$

• Conclusion of Section III: For an embedded perturber, the primary source of eccentricity damping comes from co-orbital Lindblad resonances, not from the distant corotation resonances emphasized in previous work.

yang-li-2024

the theory part of yang-li-2024 mean-motion resonances with interfering density waves

• main content:

  an object 'A' moving on a fixed outer circular orbit and an object 'B' moving along an inner eccentric orbit, the system is assumed to be locked in a $m:m-1$ resonance such that

  $$\begin{array}{}\text{(1)}& m{\mathrm{\Omega }}_A=m{\mathrm{\Omega }}_B-{\kappa }_B\end{array}$$

summary

the main analyze (in interfering density waves):

• the normal torque due to gravitational torque and advective transport (migration torque)
• the inner and outer Lindblad resonance
• the corotation resonance

2 Interfering density waves and their backreaction

• the Fourier series of the individual gravitational field for object A or B ( $s=A,B$ )

  $$\begin{array}{}\text{(2)}& {\phi }_s(r,\varphi ,t)=\sum _{l_s=-\mathrm{\infty }}^{\mathrm{\infty }}\sum _{m_s=0}^{\mathrm{\infty }}{\phi }_{s,l_s,m_s}\cos \{m_s\varphi -[m_s{\mathrm{\Omega }}_s+(l_s-m_s){\kappa }_s]t\}\end{array}$$

  where the pattern frequency of the $l_s,m_s$ component is

  $$\begin{array}{}\text{(3)}& {\mathrm{\Omega }}_{l_s,m_s}={\mathrm{\Omega }}_s+\frac{l_s-m_s}{m_s}{\kappa }_s\end{array}$$

• the location of the inner and outer Lindblad resonances and corotation resonance is

  $$\begin{array}{}\text{(4)}& m(\mathrm{\Omega }-{\mathrm{\Omega }}_{l_s,m_s})=\pm \kappa ,m>0\end{array}$$$$\begin{array}{}\text{(6)}& \mathrm{\Omega }={\mathrm{\Omega }}_{l,m}\end{array}$$

• the location of the inner $m:m-1$ Lindblad resonance of object A should be close to the radius of object B.

  derivation

  1. the object A and B are in $m:m-1$ mean-motion resonance, the orbital velocities satisfy,

  $$m{\mathrm{\Omega }}_A=(m-1){\mathrm{\Omega }}_B$$$$\Rightarrow r_B=r_A{\left(\frac{{\mathrm{\Omega }}_A}{{\mathrm{\Omega }}_B}\right)}^{2/3}=r_A{\left(\frac{m-1}{m}\right)}^{2/3}$$

  2. the location of the inner Lindblad resonance of object A (for Keplerian disc, $\kappa \approx \mathrm{\Omega }$ )

  $$m({\mathrm{\Omega }}_{in}-{\mathrm{\Omega }}_A)=\kappa \approx {\mathrm{\Omega }}_{in}$$$$\Rightarrow m{\mathrm{\Omega }}_A\approx (m-1){\mathrm{\Omega }}_{in}$$$$\Rightarrow r_{in}=r_A{\left(\frac{{\mathrm{\Omega }}_A}{{\mathrm{\Omega }}_{in}}\right)}^{2/3}\approx r_A{\left(\frac{m-1}{m}\right)}^{2/3}$$

• the total angular flux contains the cross-term is (the migration torque)

  $$\begin{array}{}\text{(9)}& {\mathcal{F}}_{J\times }=-\text{sgn}(k)\frac{mr}{2G}(1-\frac{c^2|k|}{\pi G\sigma })\text{Re}({\mathrm{\Phi }}_A{\mathrm{\Phi }}_B{e}^{i(Q_0+C_{AB})})\end{array}$$

  derivation

  begin from the gravitational potential perturbation from object A and B

  $$\begin{array}{}\text{(7)}& \begin{array}{c}{\phi }_A^D(r)e^{im\varphi -i\omega t}={\mathrm{\Phi }}_A{e}^{i{\int }_A^{r}k\mathrm{d}r}{e}^{im\varphi -i\omega t}\\ {\phi }_B^D(r)e^{im\varphi -i\omega t}={\mathrm{\Phi }}_B{e}^{i{\int }_B^{r}k\mathrm{d}r}{e}^{im\varphi -i\omega t}{e}^{i{Q}_0}\end{array}\end{array}$$

  the total potential perturbation

  $${\phi }_D={\phi }_A^D+{\phi }_B^D$$$$\Rightarrow |{\phi }_D{|}^2=|{\phi }_A^D{|}^2+|{\phi }_B^D{|}^2+2\text{Re}({\phi }_A^D{\phi }_B^D)$$

  the cross-term $2\text{Re}({\phi }_A^D{\phi }_B^D)$ represents interference

  $$\begin{array}{rl}{\phi }_A^D{\phi }_B^{D\ast }& ={\mathrm{\Phi }}_A{\mathrm{\Phi }}_B\exp \left[i({\int }_A^{r}k\mathrm{d}r-{\int }_B^{r}k\mathrm{d}r)\right]e^{i{Q}_0}\\ & ={\mathrm{\Phi }}_A{\mathrm{\Phi }}_B\exp \left[i({\int }_A^{B}k\mathrm{d}r+Q_0)\right]\end{array}$$$$\text{Re}({\phi }_A^D{\phi }_B^{D\ast })=\text{Re}\left({\mathrm{\Phi }}_A{\mathrm{\Phi }}_B{e}^{i(Q_0+C_{AB})}\right)$$

  where ${\int }_A^{B}k\mathrm{d}r=C_{AB}$ is the phase shift due to wave propagation between resonances.

  substituting into equation (8), which is equation (30) in GT79

  $$\begin{array}{}\text{(8)}& {\mathcal{F}}_J=-\text{sgn}(k)\frac{mr|{\phi }_D{|}^2}{4G}(1-\frac{c^2|k|}{\pi G\sigma })\end{array}$$

  so the cross-term in the total angular momentum flux is

  $$\begin{array}{}\text{(9)}& {\mathcal{F}}_{J\times }=-\text{sgn}(k)\frac{mr}{2G}(1-\frac{c^2|k|}{\pi G\sigma })\text{Re}({\mathrm{\Phi }}_A{\mathrm{\Phi }}_B{e}^{i(Q_0+C_{AB})})\end{array}$$

• the resonant angle $Q$ is

  $$Q=m{\lambda }_A-(m-1){\lambda }_B-{\varpi }_B$$

  where ${\dot{\lambda }}_s={\mathrm{\Omega }}_s$ and ${\dot{\varpi }}_s={\mathrm{\Omega }}_s-{\kappa }_s$

2.1 Computing the torque

• the torque of the external potentials ${\varphi }_A={\phi }_{A,m,m},{\varphi }_B={\phi }_{B,m-1,m}$ acting on the disc is

  $$\begin{array}{}\text{(10)}& T=-m\pi {\int }_0^{\mathrm{\infty }}\mathrm{d}rr[{\varphi }_A(r)+{\varphi }_B(r)]\text{Im}[{\sigma }_A(r)+{\sigma }_B(r)]\end{array}$$

• the backreaction of the density wave produced by object A on object B should be

  $$\begin{array}{rl}{T}_B& =m\pi {\int }_0^{\mathrm{\infty }}\mathrm{d}rr{\varphi }_B(r)\text{Im}{\sigma }_A(r)\\ \text{(11)}& & =-\pi m\text{Im}\left\{{\int }_0^{\mathrm{\infty }}\mathrm{d}r[\frac{m{\varphi }_B\sigma v_A}{m\mathrm{\Omega }-\omega }+ir\sigma u_A\frac{\mathrm{d}}{\mathrm{d}r}\left(\frac{{\varphi }_B}{m\mathrm{\Omega }-\omega }\right)]\right\}\end{array}$$

2.1.1 Lindblad resonances

• the torque near the Lindblad resonance

  $$\begin{array}{}\text{(16)}& T_B=-\frac{\pi m\sigma r_L}{\kappa }\text{sgn}(\mathcal{D}){\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\mathrm{d}x{\mathrm{\Psi }}_{B,l,m}\text{Re}(u_A)f(x)\end{array}$$

  where the window function $f(x)$ is chosen such that $f=1$ near the resonance location and $f\to 0$ for $|x|\to \mathrm{\infty }$ to eliminate oscillatory contributions far away.

• the inner Lindblad resonance of object A operate the torque on object B

  $$\begin{array}{}\text{(21)}& T_{B,in}& =\frac{m\pi \sigma }{\mathcal{D}}{\mathrm{\Psi }}_{A,m,m}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\mathrm{d}xf(x)({\mathrm{\Psi }}_d+{\mathrm{\Psi }}_c\sin f_c)\times \text{Re}\{{\int }_0^{\mathrm{\infty }}\mathrm{d}t\exp \left[i(tx-\frac{\alpha t^2}{2\beta }+\frac{t^3}{3\beta }-iQ)\right]\}\text{(25)}& & =\frac{{\pi }^{2}m\sigma {\mathrm{\Psi }}_{A,m,m}}{\mathcal{D}}\{{\mathrm{\Psi }}_d\cos Q+\frac{e_B}{2}{\mathrm{\Psi }}_c\text{Re}[W(\alpha ,\beta )e^{-iQ}]\}\end{array}$$

• the outer one

  $$\begin{array}{}\text{(29)}& T_{B,out}=\frac{{\pi }^{2}m\sigma {\mathrm{\Psi }}_{A,m,m}{\mathrm{\Psi }}_{B,m-1,m}}{\mathcal{D}}\cos Q\end{array}$$

2.1.2 Corotation resonance

• the corotation torque:$$\begin{array}{}\text{(33)}& T_{B,co}=-\frac{m{\pi }^2}{2}{[{\left(\frac{\mathrm{d}\mathrm{\Omega }}{\mathrm{d}r}\right)}^{-1}\frac{\mathrm{d}}{\mathrm{d}r}\left(\frac{\sigma }{B_0}\right){\phi }_{A,m,m}{\phi }_{B,m-1,m}\cos Q]}_{r_C}\end{array}$$

2.1.3 Alternative smoothing scheme

• an alternative smoothing scheme in numerical simulation to modify the gravitational potential produced by planets$$\begin{array}{}\text{(36)}& {\mathrm{\Phi }}_s=-\frac{G{\mu }_{s}M}{(|{\mathit{r}}_{p,s}-\mathit{r}{|}^2+{ϵ}^2{)}^{1/2}}+{\mu }_s{\mathrm{\Omega }}_{p,s}^2{\mathit{r}}_{p,s}\cdot \mathit{r}\end{array}$$

2.2 Change rate of orbital quantities

$$\begin{array}{}\text{(42)}& \frac{\mathrm{d}{e}_B}{\mathrm{d}t}=-\frac{1}{m}\frac{T_{B,in}}{M_B{e}_B{a}_B^2{\mathrm{\Omega }}_B}:=-\frac{\cos (Q-Q_0)}{{\tau }_0}\end{array}$$$$\begin{array}{}\text{(44)}& \frac{{\dot{n}}_B}{n_B}=-3(GM{)}^{-2/3}{n}^{1/3}{M}_B^{-1}{T}_{B,in}:=-\frac{3m{e}_B}{\cos (Q-Q_0)}{\tau }_0\end{array}$$

some errors?

1. equation (14)

   $$\begin{array}{}\text{(14)}& v_A=i\text{sgn}(\mathcal{D})\frac{2\mathrm{\Omega }}{\kappa }{u}_A\end{array}$$

   is different from that in GT79

   $$\begin{array}{}\text{(A5)}& v_1=i\text{sgn}(\mathcal{D})\frac{\kappa }{2\mathrm{\Omega }}{u}_1\end{array}$$

2. equation (44)

   $$-\frac{3m{e}_B}{\cos }(Q-Q_0){\tau }_0$$

todo:

• the choice of $m_A=l_A=m,m_B=m,l_B=m-1$:

  keep the fixed phase difference between the Fourier series of A and B (the phase is: $m_s\varphi -[m_s{\mathrm{\Omega }}_s+(l_s-m_s){\kappa }_s]t$ )

  here for A:

  $$m_A\varphi -[m_A{\mathrm{\Omega }}_A+(l_A-m_A){\kappa }_A]t=m\varphi -m{\mathrm{\Omega }}_{A}t$$

  for B:

  $$m_B\varphi -[m_B{\mathrm{\Omega }}_B+(l_B-m_B){\kappa }_B]t=m\varphi -[m{\mathrm{\Omega }}_B-{\kappa }_B]t$$

  and the phase difference is time-independent while $m{\mathrm{\Omega }}_A=m{\mathrm{\Omega }}_B-{\kappa }_B$

  BUT it also work for parameters $m_A=l_A=km,m_B=km,l_B=km-k(k\in \mathbb{Z})$

• the relation between section 2 and 4,5? what does GS14 mainly talk?

3 Hydrodynamical simulations

3.1 Numerical set-ups

FARGO3D code

3.2 Simulations results

3.2.1 Non-resonant pairs

3.2.2 Planet pairs with 2:1 MMR

3.2.3 Other parameter dependence

In summary, the comparison between the numerically extracted torque and analytically predicted torque is:

• the location of the torque jump which locates at the inner Lindblad resonance
• the torque amplitudes: the linear dependence on $e_B$, planet mass, and the sinusoidal-like pattern of $Q$

4 Modified resonant dynamics

the equations of motion including the effect of interfering density waves for $n={\dot{\lambda }}_B,e$ are

$$\begin{array}{}\text{(45)}& \begin{array}{l}\dot{n}=3(m-1){\beta }_0{\mu }^{\prime }e{n}^2\sin Q-{\displaystyle \frac{n}{{\tau }_n}}+p{\displaystyle \frac{e^{2}n}{{\tau }_e}}-{\displaystyle \frac{6en\cos Q}{{\tau }_0}}\\ \dot{e}={\beta }_0{\mu }^{\prime }n\sin Q-{\displaystyle \frac{e}{{\tau }_e}}-{\displaystyle \frac{\cos Q}{{\tau }_0}}\end{array}\end{array}$$

• the normal migration torques: ${\tau }_e,{\tau }_n$-related terms
• the torques of interfering density waves: ${\tau }_0$-related terms

4.1 Without constant migration torques

4.2 With migration torques

• without the interfering density waves terms

• considering the interfering density waves terms