Week6: Orbits

Non-axisymmetric potentials

Logarithmic potential

ΦL(x,y)=12v02ln⁡(Rc2+x2+y2q2),0<q<1\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

  • x2+y2q2≪RC2⇒ΦL(x,y)=v02ln⁡Rc+v022Rc2(x2+y2q2)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

  • x2+y2q2≫RC2⇒ΦL(x,y)=12v02ln⁡(x2+y2q2)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

    L=12∣x⃗˙+Ω⃗b×x⃗∣2−Φ(x⃗)\mathcal{L}=\frac12\left|\dot{\vec x}+\vec\Omega_b\times\vec x\right|^2-\Phi(\vec x)

  • Momentum

    p⃗=∂L∂x⃗˙=x⃗˙+Ω⃗b×x⃗\vec p=\frac{\partial \mathcal L}{\partial \dot{\vec x}}=\dot{\vec x}+\vec\Omega_b\times\vec x

  • Hamiltonian

    HJ=p⃗⋅x⃗˙−L=p⃗⋅(p⃗−Ω⃗b×x⃗)−12p2+Φ(x⃗)=12p2+Φ(x⃗)−Ω⃗b⋅(x⃗×p⃗)=H−Ω⃗b×L⃗\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

    EJ=12∣x⃗˙∣2+Φeff(x⃗)E_J=\frac12\left|\dot{\vec x}\right|^2+\Phi_{eff}(\vec x)

    where

    Φeff(x⃗)≡Φ(x⃗)−12∣Ω⃗b×x⃗∣2=Φ(x⃗)−12[Ωb2x2−(Ω⃗b⋅x⃗)2]\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

    {p⃗˙=−∇Φ(x⃗)−Ω⃗b×p⃗x⃗˙=p⃗−Ω⃗b×x⃗⇒x⃗¨=−∇Φ(x⃗)−2Ω⃗b×x⃗˙−Ω⃗b×(Ω⃗b×x⃗)\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

    x⃗¨=−∇Φ(x⃗)−2Ω⃗b×x⃗˙+Ωb2x⃗−(Ω⃗b⋅x⃗)Ω⃗b\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

      −2Ω⃗b×x⃗˙-2\vec\Omega_b\times\dot{\vec x}

    • Centrifugal force

      −Ω⃗b×(Ω⃗b×x⃗)-\vec\Omega_b\times\left(\vec\Omega_b\times\vec x\right)

    In the meantime

    x⃗¨=−∇Φeff(x⃗)−2Ω⃗b×x⃗˙\ddot{\vec x}=-\nabla\Phi_{eff}(\vec x)-2\vec\Omega_b\times\dot{\vec x}

  • Lagrange points

    ∇Φeff=0\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

    Φeff(xL,yL)=Φeff(xL,yL)+12∂2Φeff∂x2∣x⃗L(x−xL)2+∂2Φeff∂x∂y∣x⃗L(x−xL)(y−yL)+12∂2Φeff∂y2∣x⃗L(y−yL)2+⋯\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

    ξ≡x−xL,η≡y−yL\xi\equiv x-x_L,\quad \eta\equiv y-y_L

    and

    Φxx=∂2Φeff∂x2∣x⃗L,Φyy=∂2Φeff∂y2∣x⃗L\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

    ξ¨=2Ωbη˙−Φxxξ,η¨=−2Ωbξ˙−Φyyη\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

    ξ=Xeλt,η=Yeλt\xi=Xe^{\lambda t},\quad \eta=Ye^{\lambda t}

    we have

    {(λ2+Φxx)X−2λΩbY=02λΩbX+(λ2+Φyy)Y=0\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

    ∣λ2+Φxx−2λΩb2λΩbλ2+Φyy∣=0\left|\begin{array}{cc} \lambda^2+\Phi_{xx}&-2\lambda\Omega_b\\ 2\lambda\Omega_b&\lambda^2+\Phi_{yy} \end{array}\right|=0
    ⇔(λ2+Φxx)(λ2+Φyy)+4λ2Ωb2=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

    ξ=X1cos⁡(αt+ϕ1)+X2cos⁡(βt+ϕ2)η=Y1sin⁡(αt+ϕ1)+Y2sin⁡(βt+ϕ2)\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

    (α2−Φxx)(α2−Φyy)−4α2Ωb2=0(β2−Φxx)(β2−Φyy)−4β2Ωb2=0\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

    Y1=Φxx−α22ΩbαX1=2ΩbαΦyy−α2X1Y2=Φxx−β22ΩbβX2=2ΩbβΦyy−β2X2Y_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

    • λ12λ22=ΦxxΦyy>0\lambda_1^2\lambda_2^2=\Phi_{xx}\Phi_{yy}>0
    • λ12+λ22=−(Φxx+Φyy+4Ωb2)<0\lambda_1^2+\lambda_2^2=-\left(\Phi_{xx}+\Phi_{yy}+4\Omega_b^2\right)<0
    • Δ=(Φxx+Φyy+4Ωb2)2−4ΦxxΦyy>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

    (Φxx−Φyy)2+8(Φxx+Φyy)Ωb2+16Ωb4>0\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

    Φxx<Φyy\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

    Φeff(x,y)=12v02ln⁡(Rc2+x2+y2q2)−12Ωb2(x2+y2),0<q<1\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=∂∂yΦeff(0,±yL)=±(v02q2Rc2+yL2−Ωb2)yL0=\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
    ⇒yL=v02Ωb2−q2Rc2\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

    Φxx(0,yL)=−Ωb2(1−q2)Φyy(0,yL)=−2Ωb2[1−q2(ΩbRcv0)2]\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,

    ΩbRcv0≪1\frac{\Omega_bR_c}{v_0}\ll 1
    ⇒Φxx+Φyy+4Ωb2=Ωb2(1+q2),ΦxxΦyy=2Ωb4(1−q2)\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

    Φxx+Φyy+4Ωb2>0Δ=Ωb4(q4+10q2−7)>0  ⟺  q2>42−5≈0.8102\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

    Φxx+Φyy+4Ωb2=Ωb2(1+q2)≈Ωb2(2−2ϵ)ΦxxΦyy=2Ωb4(1−q2)≈4Ωb4ϵ\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\\
    ⇒Δ≈4Ωb4(1−6ϵ),Δ≈2Ωb2(1−3ϵ)\Rightarrow \Delta\approx4\Omega_b^4\left(1-6\epsilon\right),\quad \sqrt\Delta\approx2\Omega_b^2\left(1-3\epsilon\right)
    α2=Φxx+Φyy+4Ωb2−Δ2≈2ϵΩb2≈−Φxxβ2=Φxx+Φyy+4Ωb2+Δ2≈2(1−2ϵ)Ωb2=2Ωb2+O(ϵ)\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,

    Y1≈−4ϵΩb222ϵΩb2X1≈−2ϵX1⇒Y1≫X1Y_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

    Y2≈−[2ϵ+2(1−2ϵ)]Ωb222(1−2ϵ)Ωb2X2≈−X2/2Y_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
PreviousWeek7: OrbitsNextWeek5: Orbits

Last updated