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)

• Box orbits

  • $R\ll R_c$ (oscillator) + could be distorted when $R\gtrsim R_c$,

  • 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.

  

  • $L_1$ and $L_2$ - saddle points - unstable

    $\Phi{xx}\Phi{yy}<0$

• $L3$ - minimum of $\Phi{eff}$ - stable

  $\Phi{xx}>0$ 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 $\Omegab^2\ll |\Phi{xx}|$, $\alpha^2\sim\Phi_{xx}$, so $X_1\gg Y_1$ and this prograde motion runs almost parallel to the long axis of the potential.

• While $\beta^2>\Phi{yy}$ 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.

• $L4,L_5$ - 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 $\Phi{xx}\Phi{yy}>0$. 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

  $$Y_2\approx\frac{-\left[2\epsilon+2(1-2\epsilon)\right]\Omega_b^2}{2\sqrt{2(1-2\epsilon)}\Omega_b^2}X_2\approx-X_2/\sqrt2$$