> 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/week6.md).

# Week6: Orbits

## Non-axisymmetric potentials

### Logarithmic potential

$$
\Phi\_L(x,y)=\frac12v\_0^2\ln\left(R\_c^2+x^2+\frac{y^2}{q^2}\right),\quad 0\<q<1
$$

Here, the angular momentum no longer conserves, though we still have a conserved Hamiltonian

* $$
  x^2+\frac{y^2}{q^2}\ll R\_C^2\Rightarrow \Phi\_L(x,y)=v\_0^2\ln R\_c+\frac{v\_0^2}{2R\_c^2}\left(x^2+\frac{y^2}{q^2}\right)
  $$

  This is just the potential of an harmonic oscillator, and when $q$ is irrational, the orbit does not close
* $$
  x^2+\frac{y^2}{q^2}\gg R\_C^2\Rightarrow \Phi\_L(x,y)=\frac12v\_0^2\ln\left(x^2+\frac{y^2}{q^2}\right)
  $$
  * if $q=1$, $\Phi\_L=v\_0^2\ln R$, and the circular speed is a constant - consistent with the flat circular-speed curves of many disk galaxies

In general, there are two kinds of closed orbits, namely **box orbits** (above) and **loop orbits** (below)

![](https://1509032923-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-MPMxe8Bu9WDT3p-DA_8%2Fsync%2Fc8e2508b48c77eee106d4d345e01b0ab727a1e05.png?generation=1608873188797729\&alt=media)

* **Box orbits**
  * $R\ll R\_c$ (oscillator) + could be distorted when $R\gtrsim R\_c$,&#x20;
  * Zero time-averaged angular momentum
  * Two integrals (two independent oscillations parallel to the coordinate axes)
* **Loop orbits**
  * $R\gg R\_c$
  * **Closed loop orbit**
    * closes on itself after one revolution
    * zero annular width

### Rotating logarithmic potential

Let the frame of reference in which the potential $\Phi$ is static rotate steadily at angular velocity $\vec\Omega\_b$ - **pattern speed**

* Lagragian

  $$
  \mathcal{L}=\frac12\left|\dot{\vec x}+\vec\Omega\_b\times\vec x\right|^2-\Phi(\vec x)
  $$
* Momentum

  $$
  \vec p=\frac{\partial \mathcal L}{\partial \dot{\vec x}}=\dot{\vec x}+\vec\Omega\_b\times\vec x
  $$
* Hamiltonian

  $$
  \begin{align\*}
  H\_J&=\vec p\cdot\dot{\vec x}-\mathcal{L}\\
  &=\vec p\cdot\left(\vec p-\vec\Omega\_b\times \vec x\right)-\frac12p^2+\Phi(\vec x)\\
  &=\frac12p^2+\Phi(\vec x)-\vec\Omega\_b\cdot(\vec x\times\vec p)\\
  &=H-\vec\Omega\_b\times\vec L
  \end{align\*}
  $$

  $H\_J$ has no explicit time dependence and is thus an integral, called the **Jacobi integral**

  $$
  E\_J=\frac12\left|\dot{\vec x}\right|^2+\Phi\_{eff}(\vec x)
  $$

  where

  $$
  \Phi\_{eff}(\vec x)\equiv \Phi(\vec x)-\frac12\left|\vec\Omega\_b\times\vec x\right|^2=\Phi(\vec x)-\frac12\left\[\Omega\_b^2x^2-\left(\vec\Omega\_b\cdot \vec x\right)^2\right]
  $$

  is the sum of gravitational potential and a **centrifugal potential**.
* Hamilton's equations

  $$
  \left{\begin{array}{l}
  \dot{\vec p}=-\nabla\Phi(\vec x)-\vec\Omega\_b\times\vec p\\
  \dot{\vec x}=\vec p-\vec\Omega\_b\times \vec x
  \end{array}\right.
  \Rightarrow \ddot{\vec x}=-\nabla\Phi(\vec x)-2\vec\Omega\_b\times\dot{\vec x}-\vec\Omega\_b\times\left(\vec\Omega\_b\times\vec x\right)
  $$

  thus

  $$
  \ddot{\vec x}=-\nabla\Phi(\vec x)-2\vec\Omega\_b\times\dot{\vec x}+\Omega\_b^2\vec x-\left(\vec\Omega\_b\cdot\vec x\right)\vec\Omega\_b
  $$

  * **Coriolis force**

    $$
    -2\vec\Omega\_b\times\dot{\vec x}
    $$
  * **Centrifugal force**

    $$
    -\vec\Omega\_b\times\left(\vec\Omega\_b\times\vec x\right)
    $$

  In the meantime

  $$
  \ddot{\vec x}=-\nabla\Phi\_{eff}(\vec x)-2\vec\Omega\_b\times\dot{\vec x}
  $$
* Lagrange points

  $$
  \nabla\Phi\_{eff}=0
  $$

  When we expand $\nabla \Phi\_{eff}$ around one of these points $\vec x\_L=(x\_L,y\_L)$ in powers of $(x-x\_L)$ and $(y-y\_L)$, we have

  $$
  \Phi\_{eff}(x\_L,y\_L)=\Phi\_{eff}(x\_L,y\_L)+\frac12\frac{\partial^2\Phi\_{eff}}{\partial x^2}\Bigg|*{\vec x\_L}(x-x\_L)^2\\
  +\frac{\partial^2\Phi*{eff}}{\partial x\partial y}\Bigg|*{\vec x\_L}(x-x\_L)(y-y\_L)+\frac12\frac{\partial^2\Phi*{eff}}{\partial y^2}\Bigg|\_{\vec x\_L}(y-y\_L)^2+\cdots
  $$

  For any bar-like potential whose principal axes lie along the coordinate axes, by symmetry, $\partial^2\Phi\_{eff}/\partial x\partial y$ at $x\_L$. Hence, if we retain only quadratic terms and define

  $$
  \xi\equiv x-x\_L,\quad \eta\equiv y-y\_L
  $$

  and

  $$
  \Phi\_{xx}=\frac{\partial^2\Phi\_{eff}}{\partial x^2}\Bigg|*{\vec x\_L},\quad \Phi*{yy}=\frac{\partial^2\Phi\_{eff}}{\partial y^2}\Bigg|\_{\vec x\_L}
  $$

  the equations of motion become

  $$
  \ddot \xi=2\Omega\_b\dot\eta-\Phi\_{xx}\xi,\quad \ddot \eta=-2\Omega\_b\dot\xi-\Phi\_{yy}\eta
  $$

  This is a pair of linear differential equations with constant coefficients!

  Let

  $$
  \xi=Xe^{\lambda t},\quad \eta=Ye^{\lambda t}
  $$

  we have

  $$
  \left{\begin{array}{l}
  \left(\lambda^2+\Phi\_{xx}\right)X-2\lambda\Omega\_b Y=0\\
  2\lambda\Omega\_b X+\left(\lambda^2+\Phi\_{yy}\right)Y=0
  \end{array}
  \right.
  $$

  The simultaneous equations have a non-trivial solution only if the determinant

  $$
  \left|\begin{array}{cc}
  \lambda^2+\Phi\_{xx}&-2\lambda\Omega\_b\\
  2\lambda\Omega\_b&\lambda^2+\Phi\_{yy}
  \end{array}\right|=0
  $$

  $$
  \Leftrightarrow \left(\lambda^2+\Phi\_{xx}\right)\left(\lambda^2+\Phi\_{yy}\right)+4\lambda^2\Omega\_b^2=0
  $$

  This is the **characteristic equation** for $\lambda$. If any of the four roots has non-zero real part, $\xi$ and $\eta$ will grow exponetially in time, and the Lagrangian point is said to be **unstable**. When all roots are pure imaginary, say $\lambda=\pm i\alpha$ or $\pm i\beta$, with $0\le \alpha\le \beta$ real, the general solution is

  $$
  \xi=X\_1\cos\left(\alpha t+\phi\_1\right)+X\_2\cos\left(\beta t+\phi\_2\right)\\
  \eta=Y\_1\sin\left(\alpha t+\phi\_1\right)+Y\_2\sin\left(\beta t+\phi\_2\right)
  $$

  and the Lagrangian point is **stable** since $\xi$ and $\eta$ oscillate. Each orbit is a superposition of motion at frequencies $\alpha$ and $\beta$ around two ellipses.

  Since

  $$
  \left(\alpha^2-\Phi\_{xx}\right)\left(\alpha^2-\Phi\_{yy}\right)-4\alpha^2\Omega\_b^2=0\\
  \left(\beta^2-\Phi\_{xx}\right)\left(\beta^2-\Phi\_{yy}\right)-4\beta^2\Omega\_b^2=0
  $$

  $X\_1$ & $Y\_1$, $X\_2$ & $Y\_2$ are related by

  $$
  Y\_1=\frac{\Phi\_{xx}-\alpha^2}{2\Omega\_b\alpha}X\_1=\frac{2\Omega\_b\alpha}{\Phi\_{yy}-\alpha^2}X\_1\\
  Y\_2=\frac{\Phi\_{xx}-\beta^2}{2\Omega\_b\beta}X\_2=\frac{2\Omega\_b\beta}{\Phi\_{yy}-\beta^2}X\_2
  $$

  To ensure $\lambda^2$ to be real and negative, there are three conditions

  * $$
    \lambda\_1^2\lambda\_2^2=\Phi\_{xx}\Phi\_{yy}>0
    $$
  * $$
    \lambda\_1^2+\lambda\_2^2=-\left(\Phi\_{xx}+\Phi\_{yy}+4\Omega\_b^2\right)<0
    $$
  * $$
    \Delta=\left(\Phi\_{xx}+\Phi\_{yy}+4\Omega\_b^2\right)^2-4\Phi\_{xx}\Phi\_{yy}>0
    $$

  Now we can analyse the **stability** of Lagrange points.

  ![](https://1509032923-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-MPMxe8Bu9WDT3p-DA_8%2Fsync%2F128b41fd8219dd3df32488fe03350faf2d9b75af.png?generation=1608873188582559\&alt=media)

  * $L\_1$ and $L\_2$ - saddle points - unstable

    $\Ph&#x69;*{xx}\Phi*{yy}<0$
* $L*3$ - minimum of $\Phi*{eff}$ - stable

  $\Ph&#x69;*{xx}>0$ and $\Phi*{yy}>0$, so the first two conditions are naturally satisfied. We can rewrite the third condition

  $$
  \left(\Phi\_{xx}-\Phi\_{yy}\right)^2+8\left(\Phi\_{xx}+\Phi\_{yy}\right)\Omega\_b^2+16\Omega\_b^4>0
  $$

  which is also satisfied.

  Without loss of generality, we let

  $$
  \Phi\_{xx}<\Phi\_{yy}
  $$

  since $x$-axis is the major axis of the potential.

  Now consider the motion about $L\_3$.

  * Since $\alpha^2<\Phi\_{xx}$ and $\alpha\ge 0$, we have $Y\_1/X\_1>0$, thus the star's motion around the $\alpha-$ellipse has the same sense as the rotation of the potential. Such an orbit is said to be **prograde** or **direct**.

    When $\Omega*b^2\ll |\Phi*{xx}|$, $\alpha^2\sim\Phi\_{xx}$, so $X\_1\gg Y\_1$ and this prograde motion runs almost parallel to the long axis of the potential.
* While $\beta^2>\Ph&#x69;*{yy}$ and $\beta>0$, we have $Y\_2/X\_2<0$, and the orbital motion is known as **retrograde**. When $\Omega\_b^2\ll |\Phi*{yy}|$, similarly $|X\_2|\ll|Y\_2|$, and the $\beta-$ellipse orbit goes over into a short-axis orbit.
  * A general **prograde** orbit around $L\_3$ is made up of motion on the $\beta-$ellipse (**retrograde**) around a guiding center moving around the $\alpha-$ellipse (**prograde**), and conversely for **retrograde** orbits.
* $L*4,L\_5$ - maximum of $\Phi*{eff}$ - depends on the details of the potential

  For the Logarithmic potential

  $$
  \Phi\_{eff}(x,y)=\frac12v\_0^2\ln\left(R\_c^2+x^2+\frac{y^2}{q^2}\right)-\frac12\Omega\_b^2(x^2+y^2),\quad 0\<q<1
  $$

  $L\_4,L\_5$ Occur at $(0,\pm y\_L)$, where

  $$
  0=\frac{\partial}{\partial y}\Phi\_{eff}(0,\pm y\_L)=\pm\left(\frac{v\_0^2}{q^2R\_c^2+y\_L^2}-\Omega^2\_b\right)y\_L
  $$

  $$
  \Rightarrow y\_L=\sqrt{\frac{v\_0^2}{\Omega^2\_b}-q^2R\_c^2}
  $$

  Thus $L\_4,L\_5$ are present only if $\Omega\_b\<v\_0/(qR\_c)$. Differentiating the effective potential again we find

  $$
  \Phi\_{xx}(0,y\_L)=-\Omega\_b^2(1-q^2)\\
  \Phi\_{yy}(0,y\_L)=-2\Omega\_b^2\left\[1-q^2\left(\frac{\Omega\_bR\_c}{v\_0}\right)^2\right]
  $$

  Hence $\Ph&#x69;*{xx}\Phi*{yy}>0$. Deciding whether the other stability conditions hold is tedious in the general case, but straightforward in the limit of negligible core radius,

  $$
  \frac{\Omega\_bR\_c}{v\_0}\ll 1
  $$

  $$
  \Rightarrow \Phi\_{xx}+\Phi\_{yy}+4\Omega\_b^2=\Omega\_b^2(1+q^2),\quad \Phi\_{xx}\Phi\_{yy}=2\Omega\_b^4(1-q^2)
  $$

  Hence

  $$
  \Phi\_{xx}+\Phi\_{yy}+4\Omega\_b^2>0\\
  \Delta=\Omega\_b^4\left(q^4+10q^2-7\right)>0\iff q^2>4\sqrt{2}-5\approx0.810^2
  $$

  For future use we note that for small $R\_c$, and to leading order in the ellipticity $\epsilon=1-q$, we have

  $$
  \Phi\_{xx}+\Phi\_{yy}+4\Omega\_b^2=\Omega\_b^2(1+q^2)\approx\Omega\_b^2(2-2\epsilon)\\
  \quad \Phi\_{xx}\Phi\_{yy}=2\Omega\_b^4(1-q^2)\approx 4\Omega\_b^4\epsilon\\
  $$

  $$
  \Rightarrow \Delta\approx4\Omega\_b^4\left(1-6\epsilon\right),\quad \sqrt\Delta\approx2\Omega\_b^2\left(1-3\epsilon\right)
  $$

  $$
  \alpha^2=\frac{\Phi\_{xx}+\Phi\_{yy}+4\Omega\_b^2-\sqrt{\Delta}}{2}\approx2\epsilon\Omega\_b^2\approx-\Phi\_{xx}\\
  \beta^2=\frac{\Phi\_{xx}+\Phi\_{yy}+4\Omega\_b^2+\sqrt{\Delta}}{2}\approx2(1-2\epsilon)\Omega\_b^2=2\Omega\_b^2+\mathcal{O}(\epsilon)
  $$

  In this way, when $\epsilon$ and $R\_c$ are both small,

  $$
  Y\_1\approx\frac{-4\epsilon\Omega\_b^2}{2\sqrt{2\epsilon}\Omega\_b^2}X\_1\approx-\sqrt{2\epsilon}X\_1\Rightarrow Y\_1\gg X\_1
  $$

  so the $\alpha-$ellipse is highly elongated in the $x-$ direction, while

  $$
  Y\_2\approx\frac{-\left\[2\epsilon+2(1-2\epsilon)\right]\Omega\_b^2}{2\sqrt{2(1-2\epsilon)}\Omega\_b^2}X\_2\approx-X\_2/\sqrt2
  $$