dynamics of astrophysical discs

the lecture notes dynamics of astrophysical discs

results below are just reshowed from the lecture notes, the derivation could be find in the notes.

lecture 2: Orbital dynamics

2.1 Orbits in an axisymmetric potential

• the effective potential$${\mathrm{\Phi }}^{\text{eff}}=\frac{h^2}{2r^2}+\mathrm{\Phi }$$

for the family of circular orbits ( $r=$ constant ) in the midplane ( $z=0$ )

• the specific angular momentum

  $$h_c=\sqrt{r^3{\mathrm{\Phi }}_r(r,0)}$$

• the angular velocity

  $${\mathrm{\Omega }}_c=\frac{h_c}{r^2}$$

• the specific energy

  $${\epsilon }_c=\frac{h_c^2}{2r^2}+\mathrm{\Phi }(r,0)$$

• the orbital shear rate

  $$S=-r\frac{\mathrm{d}{\mathrm{\Omega }}_c}{\mathrm{d}r}$$

• the dimensionless orbital shear parameter

  $$q=-\frac{\mathrm{d}\ln {\mathrm{\Omega }}_c}{\mathrm{d}\ln r}=\frac{S}{{\mathrm{\Omega }}_c}$$

e.g.

In the case of a point-mass potential, the circular Keplerian orbits satisfy

$$\mathrm{\Phi }(r,0)=-\frac{GM}{r}$$$$h_c=\sqrt{GMr}$$$${\mathrm{\Omega }}_c=\sqrt{\frac{GM}{r^3}}$$$${\epsilon }_c=-\frac{GM}{2r}$$$$S=\frac{3}{2}{\mathrm{\Omega }}_c$$$$q=\frac{3}{2}$$

2.2 Oscillations and precession

• Oscillations:

  1. the radial frequency (often called the epicyclic frequency and denoted $\kappa$ )
  $${\mathrm{\Omega }}_r^2={\mathrm{\Phi }}_{rr}^{\text{eff}}(r,0)=4{\mathrm{\Omega }}_c^2+2r{\mathrm{\Omega }}_c\frac{\mathrm{d}{\mathrm{\Omega }}_c}{\mathrm{d}r}=2(2-q){\mathrm{\Omega }}_c^2$$$$\ddot{\delta r}=-{\mathrm{\Omega }}_r^2\delta r$$
  2. the vertical frequency (sometimes denoted $\nu$ )
  $${\mathrm{\Omega }}_z^2={\mathrm{\Phi }}_{zz}^{\text{eff}}(r,0)$$$$\ddot{\delta z}=-{\mathrm{\Omega }}_z^2\delta z$$

  Keplerian orbits satisfy

  $${\mathrm{\Omega }}_r={\mathrm{\Omega }}_z=\mathrm{\Omega }$$

  meaning that (slightly) eccentric or inclined orbits close after one turn.

• precession:

  1. the apsidal precession rate
  $$\mathrm{\Omega }-{\mathrm{\Omega }}_r$$
  2. the nodal precession rate
  $$\mathrm{\Omega }-{\mathrm{\Omega }}_z$$

2.3 Mechanics of accretion

• transfer of angular momentum:

  consider two particles in circular orbits in the midplane. the total angular momentum and energy are

  $$H=H_1+H_2=m_1{h}_1+m_2{h}_2,$$$$E=E_1+E_2=m_1{\epsilon }_1+m_2{\epsilon }_2$$

  in an infinitesimal exchange:

  $$\mathrm{d}H=\mathrm{d}{H}_1+\mathrm{d}{H}_2=m_1\mathrm{d}{h}_1+m_2\mathrm{d}{h}_2,$$$$\mathrm{d}E=\mathrm{d}{E}_1+\mathrm{d}{E}_2=m_1{\mathrm{\Omega }}_1\mathrm{d}{h}_1+m_2{\mathrm{\Omega }}_2\mathrm{d}{h}_2$$

  if $\mathrm{d}H=0$ then

  $$\mathrm{d}E=({\mathrm{\Omega }}_1-{\mathrm{\Omega }}_2)\mathrm{d}{H}_1$$

• transfer of angular momentum and mass:

  $$\begin{array}{rl}\mathrm{d}M& =\mathrm{d}{m}_1+\mathrm{d}{m}_2=0,\\ \mathrm{d}H& =\mathrm{d}{H}_1+\mathrm{d}{H}_2=0,\mathrm{d}{H}_i=m_i\mathrm{d}{h}_i+h_i\mathrm{d}{m}_i,\\ \mathrm{d}{E}_i& =m_i{\mathrm{\Omega }}_i\mathrm{d}{h}_i+{\epsilon }_i\mathrm{d}{m}_i\\ & ={\mathrm{\Omega }}_i\mathrm{d}{H}_i+({\epsilon }_i-h_i{\mathrm{\Omega }}_i)\mathrm{d}{m}_i,\\ \mathrm{d}E& =({\mathrm{\Omega }}_1-{\mathrm{\Omega }}_2)\mathrm{d}{H}_1+[({\epsilon }_1-h_1{\mathrm{\Omega }}_1)-({\epsilon }_2-h_2{\mathrm{\Omega }}_2)]\mathrm{d}{m}_1\end{array}$$

lecture 3: Global and local views

3.1 Local view of orbital dynamics

• local coordinates $(x,y,z)$

  $$r=r_0+x,\varphi ={\mathrm{\Omega }}_{0}t+\frac{y}{r_0},z=z.$$

• the tidal potential

  $$\begin{array}{rl}{\mathrm{\Phi }}_t& =\frac{1}{2}({\frac{{\partial }^2\mathrm{\Phi }}{\partial r^2}|}_{r_0}-{\mathrm{\Omega }}_0^2)x^2+\frac{1}{2}{\frac{{\partial }^2\mathrm{\Phi }}{\partial z^2}|}_{r_0}{z}^2\\ & =-{\mathrm{\Omega }}_0{S}_0{x}^2+\frac{1}{2}{\mathrm{\Omega }}_{z0}^2{z}^2\end{array}$$

• the local representation of the family of circular orbits in the midplane is

  $$x=\text{constant},\dot{y}=-S_{0}x,z=0$$
  derivation

  start from the definition of local coordinates $y$:

  $$\varphi ={\mathrm{\Omega }}_{0}t+\frac{y}{r_0}$$

  the first derivative of time above is:

  $${\mathrm{\Omega }}_c={\mathrm{\Omega }}_0+\frac{\dot{y}}{r_0}$$$$\begin{array}{rl}\Rightarrow \dot{y}& =r_0({\mathrm{\Omega }}_c-{\mathrm{\Omega }}_0)\\ & =r_{0}x\frac{{\mathrm{\Omega }}_c-{\mathrm{\Omega }}_c{|}_{r_0}}{r-r_0}\\ & =x{r}_0\frac{\mathrm{d}{\mathrm{\Omega }}_c}{\mathrm{d}r}\\ & =-x{S}_0\end{array}$$

  which can be interpreted as an orbital shear flow with shear rate $S_0=q_0{\mathrm{\Omega }}_0$

  

• the solutions of motion equations:

  $$\begin{array}{rl}x& =x_0+\text{Re}(A{e}^{-i{\mathrm{\Omega }}_{r0}t}),\\ y& =y_0-S_0{x}_{0}t+\text{Re}\left(\frac{2{\mathrm{\Omega }}_{0}A}{i{\mathrm{\Omega }}_{r0}}{e}^{-i{\mathrm{\Omega }}_{r0}t}\right),\\ z& =\text{Re}(B{e}^{-i{\mathrm{\Omega }}_{z0}t})\end{array}$$

• three conserved quantities

  $$\begin{array}{rl}{p}_y& =(2{\mathrm{\Omega }}_0-S_0)x_0,\\ {\epsilon }_{\text{h}}& =\frac{1}{2}{\mathrm{\Omega }}_{r0}^2(|A{|}^2-\frac{S_0}{2{\mathrm{\Omega }}_0}{x}_0^2),\\ {\epsilon }_{\text{v}}& =\frac{1}{2}{\mathrm{\Omega }}_{z0}^2|B{|}^2\end{array}$$

3.2 Symmetries of the local model