Planet Formation

7 Early Evolution of Planetary Systems

7.1 Migration in Gaseous Disks

7.1.1 Planet-Disk Torque in the Impulse Approximation

• our model:

  a planet and a gaseous disk orbiting a central object

• the key derivation:

  1. the change to the perpendicular velocity of the free particle in the gaseous disk that occurs during the encounter
  $$\begin{array}{}\text{(7.1)}& |\delta v_{\mathrm{\perp }}|=\frac{2G{M}_p}{b\mathrm{\Delta }v}\end{array}$$
  derivation (Eq. 5.58)

  

  $$\begin{array}{}\text{(5.56)}& F_{\mathrm{\perp }}=\frac{G{m}^2}{d^2}\cos \theta \end{array}$$$$\begin{array}{}\text{(5.57)}& F_{\mathrm{\perp }}=\frac{G{m}^2}{b^2}{[1+{\left(\frac{\sigma t}{b}\right)}^2]}^{-3/2}\end{array}$$$$\begin{array}{}\text{(5.58)}& |\delta v_{\mathrm{\perp }}|={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{F_{\mathrm{\perp }}}{m}\mathrm{d}t=\frac{2Gm}{b\sigma }\end{array}$$
  2. considering the conservative kinetic energy of the gas particle, we can obtain the change of the parallel velocity
  $$\begin{array}{}\text{(7.2)}& \mathrm{\Delta }{v}^2=|\delta v_{\mathrm{\perp }}{|}^2+(\mathrm{\Delta }v-\delta v_{\parallel }{)}^2\end{array}$$$$\begin{array}{}\text{(7.3)}& \Rightarrow \delta v_{\parallel }\simeq \frac{1}{2\mathrm{\Delta }v}{\left(\frac{2G{M}_p}{b\mathrm{\Delta }v}\right)}^2\end{array}$$
  3. denoting the semi-major axis $a$, the implied angular momentum change per unit mass of the gas is
  $$\begin{array}{}\text{(7.4)}& \mathrm{\Delta }j=\frac{2G^2{M}_p^{2}a}{b^2\mathrm{\Delta }{v}^3}\end{array}$$
  4. calculate for the total torque due to its interactions with the gas outside the orbit:

  denoting the surface density of the gaseous disk $\mathrm{\Sigma }$, the mass in the disk between $b$ and $b+\mathrm{d}b$ is

  $$\begin{array}{}\text{(7.5)}& \mathrm{d}m\approx 2\pi a\mathrm{\Sigma }\mathrm{d}b\end{array}$$

  all of the gas within the annulus will encounter the planet in a time interval

  $$\begin{array}{}\text{(7.6)}& \mathrm{\Delta }t=\frac{2\pi }{|\mathrm{\Omega }-{\mathrm{\Omega }}_p|}\end{array}$$

  approximating $|\mathrm{\Omega }-{\mathrm{\Omega }}_p|$ as

  $$\begin{array}{}\text{(7.7)}& |\mathrm{\Omega }-{\mathrm{\Omega }}_p|\simeq \frac{3{\mathrm{\Omega }}_p}{2a}b\end{array}$$
  approximation

  in the Keplerian disk, the orbital angular velocity is

  $$\frac{GMm}{r^2}=m{\omega }^{2}r\Rightarrow \omega =\sqrt{GM}{r}^{-3/2}$$

  letting $r=a$, $\mathrm{d}r=b$, ${\mathrm{\Omega }}_p=\sqrt{GM}{a}^{-3/2}$, yields

  $$\begin{array}{rl}|\mathrm{\Omega }-{\mathrm{\Omega }}_p|& =|\mathrm{d}\omega |\\ & =\left|\sqrt{GM}\frac{-3}{2}{r}^{-5/2}\mathrm{d}r\right|\\ & =\frac{3\sqrt{GM}}{2a^{5/2}}b\\ & =\frac{3{\mathrm{\Omega }}_p}{2a}b\end{array}$$

  the total torque is

  $$\begin{array}{}\text{(7.8)}& \frac{\mathrm{d}J}{\mathrm{d}t}=-{\int }_0^{\mathrm{\infty }}\frac{\mathrm{\Delta }j\mathrm{d}m}{\mathrm{\Delta }t}=-{\int }_0^{\mathrm{\infty }}\frac{8G^2{M}_p^{2}a\mathrm{\Sigma }}{9{\mathrm{\Omega }}_p^2}\frac{\mathrm{d}b}{b^4}\end{array}$$$$\begin{array}{}\text{(7.9)}& \frac{\mathrm{d}J}{\mathrm{d}t}=-\frac{8}{27}\frac{G^2{M}_p^{2}a\mathrm{\Sigma }}{{\mathrm{\Omega }}_p^2{b}_{\text{min}}^3}\end{array}$$

7.1.2 Physics of Gas Disk Torques

the physical effects that can lead to migration torques:

• the impulse approximation
• at Lindblad resonances
• at co-orbital (corotation) resonances
• the thermal effects from the vicinity of the planet

7.1.3 Torque Formulae

for a disk with a power-law scaling of surface density with radius,

$$\begin{array}{}\text{(7.14)}& \mathrm{\Sigma }(r)\propto r^{-\alpha }\end{array}$$

• the net Lindblad torque on a planet of mass $M_p$ in a circular orbit at distance $a$ from the star is given in linear theory as

$$\begin{array}{}\text{(7.15)}& {\mathrm{\Gamma }}_{\text{LR}}=-(2.34-0.1\alpha ){\mathrm{\Gamma }}_0,\end{array}$$

where ${\mathrm{\Gamma }}_0$, the reference torque, is,

$$\begin{array}{}\text{(7.16)}& {\mathrm{\Gamma }}_0={\left(\frac{M_p}{M_{\ast }}\right)}^2{\left(\frac{h}{r}\right)}^{-2}\mathrm{\Sigma }{a}^4{\mathrm{\Omega }}_K^2\end{array}$$

• reference migration time scale for a planet with orbital angular momentum $L$ as

$$\begin{array}{}\text{(7.17)}& {\tau }_0=\frac{L}{{\mathrm{\Gamma }}_0}\propto M_{\ast }^{-3/2}{M}_p^{-1}{\left(\frac{h}{r}\right)}^2{\mathrm{\Sigma }}^{-1}{a}^{-1/2}\end{array}$$

e.g.

for a one sun-mass star system ( $M_{\ast }=M_{\odot }$ )

$$\begin{array}{rl}{\tau }_0& \approx 2.0\times {10}^5{\left(\frac{M_p}{3M_{\oplus }}\right)}^{-1}{\left(\frac{h/r}{0.02}\right)}^2{\left(\frac{\mathrm{\Sigma }}{{10}^3{\text{g cm}}^{-2}}\right)}^{-1}{\left(\frac{a}{0.1\text{AU}}\right)}^{-1/2}\text{yr},\\ & \approx 4.2\times {10}^6{\left(\frac{M_p}{0.1M_{\oplus }}\right)}^{-1}{\left(\frac{h/r}{0.03}\right)}^2{\left(\frac{\mathrm{\Sigma }}{{10}^3{\text{g cm}}^{-2}}\right)}^{-1}{\left(\frac{a}{\text{AU}}\right)}^{-1/2}\text{yr},\\ \text{(7.18)}& & \approx 7.4\times {10}^5{\left(\frac{M_p}{5M_{\oplus }}\right)}^{-1}{\left(\frac{h/r}{0.05}\right)}^2{\left(\frac{\mathrm{\Sigma }}{{10}^2{\text{g cm}}^{-2}}\right)}^{-1}{\left(\frac{a}{10\text{AU}}\right)}^{-1/2}\text{yr}\end{array}$$

7.1.4 Gas Disk Migration Regimes