> For the complete documentation index, see [llms.txt](https://slowdiveptg.gitbook.io/notes/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://slowdiveptg.gitbook.io/notes/galactic-dynamics/week7.md).

# Week7: Orbits

## Non-axisymmetric potentials

### Weak Bars

We choose a polar coordinate $(R,\phi)$ in which the line $\phi=0$ coincides with the long axis of the potential. Since we assume the bar is weak, we may write

$$
\Phi(R,\phi)=\Phi\_0(R)+\Phi\_1(R,\phi)
$$

where $\Phi\_1/\Phi\_0\ll1$ (Pertubation).

* Lindblad resonances (with a steady pattern speed $\Omega\_b$)

  We seek to represent a general loop orbit as a superposition of the circular motion of a guiding center and small oscillations around this guiding center.

  Then the Lagrangian is

  $$
  \mathcal{L}=\frac12\dot R^2+\frac12\left\[R(\dot\phi+\Omega\_b)\right]^2-\Phi(R,\phi)
  $$

  So the equations of motion are

  $$
  \ddot R=R(\dot\phi+\Omega\_b)^2-\frac{\partial\Phi(R,\phi)}{\partial R}\\
  \frac{\text d}{\text dt}\left\[R^2(\dot\phi+\Omega\_b)\right]=-\frac{\partial\Phi(R,\phi)}{\partial \phi}
  $$

  since we regard the weak bar as perturbation, we also divide $R$ and $\phi$ into zeroth- and first-order parts

  $$
  R(t)=R\_0+R\_1(t),\quad \phi(t)=\phi\_0(t)+\phi\_1(t)
  $$

  For the zeroth-order

  $$
  R\_0(\dot\phi\_0+\Omega\_b)^2=\left(\frac{\text d\Phi\_0}{\text dR}\right)\_{R\_0},\quad \ddot\phi\_0=0
  $$

  which is the usual equation for centrifugal equilibrium at $R\_0$. We further define $\Omega\_0\equiv\Omega(R\_0)$, where

  $$
  \Omega(R)=\pm\sqrt{\frac1R\frac{\text d\Phi\_0}{\text dR}}
  $$

  is the circular frequency at $R$ in the potential $\Phi\_0$. The angular speed for the circular motion then becomes

  $$
  \dot\phi\_0=\Omega\_0-\Omega\_b
  $$

  we choose the origin of time so that

  $$
  \phi\_0(t)=\left(\Omega\_0-\Omega\_b\right)t
  $$

The equations of motion yield

$$
\ddot R\_1=\left(R\_0+R\_1\right)\left(\dot \phi\_0+\dot \phi\_1+\Omega\_b\right)^2-\left\[\frac{\partial}{\partial R}\left(\Phi\_0+\Phi\_1\right)\right]\_{R\_0+R\_1}
$$

$$
\frac{\text d}{\text dt}\left\[(R\_0+R\_1)(\dot\phi\_0+\dot\phi\_1+\Omega\_b)\right]=-\left\[\frac{\partial}{\partial \phi}\left(\Phi\_0+\Phi\_1\right)\right]\_{\phi\_0+\phi\_1}
$$

Since

$$
\begin{align\*}
\left\[\frac{\partial}{\partial R}\left(\Phi\_0+\Phi\_1\right)\right]*{R\_0+R\_1}
&=\left\[\frac{\partial}{\partial R}\left(\Phi\_0+\Phi\_1\right)\right]*{R\_0}+\left\[\frac{\partial^2}{\partial R^2}\left(\Phi\_0+\Phi\_1\right)\right]*{R\_0}R\_1\\
&=\left(\frac{\partial\Phi\_0}{\partial R}\right)*{R\_0}+\left\[\frac{\partial\Phi\_1}{\partial R}+\frac{\text d^2\Phi\_0}{\text d R^2}R\_1\right]\_{R\_0}+\mathcal{O(R\_1^2)}\\
\end{align\*}
$$

$$
\begin{align\*}
\left\[\frac{\partial}{\partial \phi}\left(\Phi\_0+\Phi\_1\right)\right]*{\phi\_0+\phi\_1}
&=\left(\frac{\partial\Phi\_0}{\partial \phi}\right)*{\phi\_0}+\left\[\frac{\partial\Phi\_1}{\partial \phi}+\frac{\partial^2\Phi\_0}{\partial \phi^2}\phi\_1\right]*{R\_0}+\mathcal{O(\phi\_1^2)}\\
&=\left(\frac{\partial\Phi\_1}{\partial \phi}\right)*{R\_0}+\mathcal{O(\phi\_1^2)}
\end{align\*}
$$

for the first-order, we have

$$
\begin{align\*}
\ddot R\_1
&=\left(R\_0+R\_1\right)\left(\dot \phi\_0+\dot \phi\_1+\Omega\_b\right)^2-\left(\frac{\partial\Phi\_0}{\partial R}\right)*{R\_0}-\left\[\frac{\partial\Phi\_1}{\partial R}+\frac{\text d^2\Phi\_0}{\text d R^2}R\_1\right]*{R\_0}\\
&=2R\_0\dot\phi\_1\left(\dot\phi\_0+\Omega\_b\right)+R\_1\left(\dot \phi\_0+\dot \phi\_1+\Omega\_b\right)^2-\left\[\frac{\partial\Phi\_1}{\partial R}+\frac{\text d^2\Phi\_0}{\text d R^2}R\_1\right]*{R\_0}\\
&\equiv2R\_0\Omega\_0\dot\phi\_1+R\_1\left(\Omega^2-\frac{\text d^2\Phi\_0}{\text d R^2}\right)*{R\_0}-\left(\frac{\partial\Phi\_1}{\partial R}\right)\_{R\_0}
\end{align\*}
$$

$$
\iff \ddot R\_1-2R\_0\Omega\_0\dot\phi\_1+R\_1\left(\frac{\text d^2\Phi\_0}{\text d R^2}-\Omega^2\right)*{R\_0}=-\left(\frac{\partial\Phi\_1}{\partial R}\right)*{R\_0}
$$

and

$$
\begin{align\*}
0
&=\frac{\text d}{\text dt}\left\[(R\_0+R\_1)^2(\dot\phi\_0+\dot\phi\_1+\Omega\_b)\right]+\left\[\frac{\partial}{\partial \phi}\left(\Phi\_0+\Phi\_1\right)\right]*{\phi\_0+\phi\_1}\\
&=2(R\_0+R\_1)\dot R\_1\Omega\_0+(R\_0+R\_1)^2\ddot\phi\_1 +\left(\frac{\partial\Phi\_1}{\partial \phi}\right)*{R\_0}\\
&=2R\_0\dot R\_1\Omega\_0+R\_0^2\ddot\phi\_1 +\left(\frac{\partial\Phi\_1}{\partial \phi}\right)\_{R\_0}
\end{align\*}
$$

$$
\iff \ddot\phi\_1+2\Omega\_0\frac{\dot R\_1}{R\_0}=-\frac1{R\_0^2}\left(\frac{\partial\Phi\_1}{\partial \phi}\right)\_{R\_0}
$$

To proceed further we must select a specific form of $\Phi\_1$

$$
\Phi\_1(R,\phi)=\Phi\_b(R)\cos(m\phi),\quad \Phi\_b(R)<0
$$

where $m\in N^*$, since any potential that is an **even function** of $\phi$ can be expanded as a Fourier series. If $m=2$ the potential is \*barred*.

At first we assume that $\phi\_1\ll1$ and hence $\phi(t)$ always remains close to $(\Omega\_0-\Omega\_b)t$. With this assumption we have

$$
\ddot R\_1-2R\_0\Omega\_0\dot\phi\_1+R\_1\left(\frac{\text d^2\Phi\_0}{\text d R^2}-\Omega^2\right)*{R\_0}\equiv-\left(\frac{\text d\Phi\_b}{\text d R}\right)*{R\_0}\cos\left\[m\left(\Omega\_0-\Omega\_b\right)t\right]
$$

$$
\ddot\phi\_1+2\Omega\_0\frac{\dot R\_1}{R\_0}=\frac{m\Phi\_b(R\_0)}{R\_0^2}\sin\left\[m\left(\Omega\_0-\Omega\_b\right)t\right]
$$

$$
\Rightarrow \dot \phi\_1=-2\Omega\_0\frac{R\_1}{R\_0}-\frac{\Phi\_b(R\_0)}{R\_0^2\left(\Omega\_0-\Omega\_b\right)}\cos\left\[m\left(\Omega\_0-\Omega\_b\right)t\right]+Const
$$

Then we can eliminate $\dot\phi\_1$ from the equations

$$
\ddot R\_1+4R\_1\Omega\_0^2+\frac{2\Omega\_0\Phi\_b(R\_0)}{R\_0\left(\Omega\_0-\Omega\_b\right)}\cos\left\[m\left(\Omega\_0-\Omega\_b\right)t\right]+R\_1\left(\frac{\text d^2\Phi\_0}{\text d R^2}-\Omega^2\right)*{R\_0}\equiv-\left(\frac{\text d\Phi\_b}{\text d R}\right)*{R\_0}\cos\left\[m\left(\Omega\_0-\Omega\_b\right)t\right]+Const
$$

$$
\iff \ddot R\_1+\left(\frac{\text d^2\Phi\_0}{\text d R^2}+3\Omega^2\right)*{R\_0}R\_1=-\left\[\frac{\text d\Phi\_b}{\text d R}+\frac{2\Omega\Phi\_b}{R\_0\left(\Omega-\Omega\_b\right)}\right]*{R\_0}+Const
$$

We define

$$
\kappa^2=\frac{\text d^2\Phi\_0}{\text d R^2}+3\Omega^2
$$

and

$$
\kappa\_0^2=\left(\frac{\text d^2\Phi\_0}{\text d R^2}+3\Omega^2\right)*{R\_0}=\left(R\frac{\text d\Omega^2}{\text d R}+4\Omega^2\right)*{R\_0}\quad\left(\kappa\_0^2<4\Omega\_0^2\right)
$$

since

$$
\frac{\text d^2\Phi\_0}{\text d R^2}
\=\frac{\text d(R\Omega^2)}{\text d R}=R\frac{\text d\Omega^2}{\text d R}+\Omega^2
$$

$\kappa\_0$ is just the usual **epicycle frequency**. The constant in the equation is not important as it can be absorbed into $R\_1$. Then we have got an equation of motion of a harmonic oscillator of natural frequency of $\kappa\_0$ that is driven at a frequency of $m\left(\Omega\_0-\Omega\_b\right)$. The general solution is

$$
R\_1(t)=C\_1\cos(\kappa\_0t+\alpha)-\left\[\frac{\text d\Phi\_b}{\text d R}+\frac{2\Omega\Phi\_b}{R\_0\left(\Omega-\Omega\_b\right)}\right]\_{R\_0}\frac{\cos\left\[m\left(\Omega\_0-\Omega\_b\right)t\right]}{\Delta}
$$

where $C\_1$ and $\alpha$ are arbitrary constants, and

$$
\Delta\equiv \kappa\_0^2-m^2\left(\Omega\_0-\Omega\_b\right)^2
$$

We may also eliminate $t$ from this solution with the equation $\phi\_0=\left(\Omega\_0-\Omega\_b\right)t$

$$
R\_1(\phi\_0)=C\_1\cos\left(\frac{\kappa\_0t}{\Omega\_0-\Omega\_b}+\alpha\right)+C\_2\cos(m\phi\_0)
$$

where

$$
C\_2=-\frac{1}{\Delta}\left\[\frac{\text d\Phi\_b}{\text d R}+\frac{2\Omega\Phi\_b}{R\_0\left(\Omega-\Omega\_b\right)}\right]\_{R\_0}
$$

If $C\_1=0$, $R\_1(\phi\_0)$ becomes periodic in $\phi\_0$ with period $2\pi/m$, so the orbit is a closed loop one. Orbits with $C\_1\neq0$ are non-closed loop ones parented by this closed loop.

For the $C\_1=0$ case, there are two sorts of singular points

* $\Omega\_0-\Omega\_b=0$
* $\Delta=0$

  The former, of which $\Omega\_0=\Omega\_b$, corresponds to the case of **corotation resonance**. The guiding center simply corotates with the potential.

  The latter, of which $m\left(\Omega\_0-\Omega\_b\right)=\pm\kappa\_0$, is known as **Lindblad resonances**. Radii at which such resonances occur are called **Lindblad radii**.
* Inner Lindblad resonance - plus sign - larger $\Omega\_0$
* Outer Lindblad resonance - minus sign - smaller $\Omega\_0$
* Orbits trapped at resonance

  When $R\_0$ approches the radius of either a Linblad resonance or the corotation resonance, the value of $R\_1$ predicted with linear approximation diverges, thus we should modify the analysis.

  In the previous analysis we have assumed $\phi\_1\ll1$, which is not neccessarily satisfied. If the bar strength $\Phi\_1$ is proportional to some small parameter $\epsilon$, we assume $\phi\_1$ is of order unity, $R\_1$ is of order $\epsilon^{1/2}$, and the time derivative of any quantity is smaller than that quantity by of order $\epsilon^{1/2}$. When we further put the guiding center at $L\_5$, the equations of motion go like

  $$
  \ddot R\_1+\left(\kappa\_0^2-4\Omega\_0^2\right)R\_1-2R\_0\Omega\_0\dot\phi\_1=-\left(\frac{\partial\Phi\_1}{\partial R}\right)\_{R\_0}
  $$

  $$
  \ddot\phi\_1+2\Omega\_0\frac{\dot R\_1}{R\_0}=-\frac1{R\_0^2}\left(\frac{\partial\Phi\_1}{\partial \phi}\right)\_{R\_0}
  $$

  According to our ordering, in the first equation, $\ddot R\_1$ is of order $\epsilon^{3/2}$, $\Phi\_1$ is of order unity, while other terms are of order $\epsilon^{1/2}$. In the second equation, each term is of order $\epsilon^{1/2}$. Thus

  $$
  \left(\kappa\_0^2-4\Omega\_0^2\right)R\_1-2R\_0\Omega\_0\dot\phi\_1=0,\quad \ddot\phi\_1+2\Omega\_0\frac{\dot R\_1}{R\_0}=-\frac1{R\_0^2}\left(\frac{\partial\Phi\_1}{\partial \phi}\right)\_{R\_0}
  $$

  $$
  \Rightarrow\ddot\phi\_1\left(\frac{\kappa\_0^2}{\kappa\_0^2-4\Omega\_0^2}\right)=-\frac1{R\_0^2}\left(\frac{\partial\Phi\_1}{\partial \phi}\right)\_{\left(R\_0,\phi\_0+\phi\_1\right)}
  $$

  Recall $\Phi\_1=\Phi\_1(R,\phi)=\Phi\_b(R)\cos(m\phi)$ and let $m=2$

  $$
  \ddot\phi\_1=-\frac{2\Phi\_b}{R\_0^2}\left(\frac{4\Omega\_0^2-\kappa\_0^2}{\kappa\_0^2}\right)\sin\left\[2\left(\phi\_0+\phi\_1\right)\right]
  $$

  where $\Phi\_b<0$. At $L\_5$ we have $\phi\_0=\pi/2$, so

  $$
  \ddot\phi\_1=\frac{2\Phi\_b}{R\_0^2}\left(\frac{4\Omega\_0^2-\kappa\_0^2}{\kappa\_0^2}\right)\sin\left(2\phi\_1\right)
  $$

  $$
  \iff \ddot\psi=-p^2\sin\psi
  $$

  where $\psi=2\phi\_1$, and

  $$
  p^2=-\frac{4\left|\Phi\_b\right|}{R\_0^2}\left(\frac{4\Omega\_0^2-\kappa\_0^2}{\kappa\_0^2}\right)
  $$

  This is simply the equation of a pendulum. Here the singularity appeared at corotation ($\Omega\_0=\Omega\_b$) no longer exists in this more careful analysis. Also, the stable equilibrium point of the pendulum, $\phi\_1 = 0$, is at the maximum, not the minimum, of the potential $\Phi\_1$ (the donkey effect).

  * The donkey effect - *in azimuth stars behave like donkeys, slowing down when pulled forwards and speeding up when held back* - Lynden–Bell & Kalnajs (1972)

  If the integral of motion

  $$
  E\_p=\frac12\dot\psi^2-p^2\cos\psi
  $$

  is less than $p^2$, the star oscillates slowly or **librates** about the Lagragian point, whereas if $E\_p>p^2$, the star is not trapped by the bar but **circulates** about the center of the galaxy.

  For small oscillations, the libration frequency is $p$.

  We may also obtain the shape of the orbit

  $$
  R\_1=\frac{R\_0\Omega\_0}{\kappa\_0^2-4\Omega\_0^2}\dot\psi=\pm \frac{R\_0\Omega\_0}{\kappa\_0^2-4\Omega\_0^2}\sqrt{2\left\[E\_p+p^2\cos\left(2\phi\_1\right)\right]}
  $$