9.3. Close-to-circular orbits: the epicycle approximation¶
9.3.1. General considerations¶
For orbits that are close to circular, we can explicitly derive the separation of the potential in Equation (9.8) by expanding the effective potential into a Taylor series around its minimum at the guiding center radius \(R_g\equiv R_g(L_z)\) and \(z=0\): \begin{equation}\label{eq-diskorbits-epicycleapprox} \Phi_\mathrm{eff}(R,z;L_z) \approx \Phi_\mathrm{eff}(R_g,0;L_z) + \frac{1}{2}\left(\frac{\partial^2 \Phi_\mathrm{eff}}{\partial R^2}\right)\Bigg|_{(R_g,0)}\,(R-R_g)^2 +\frac{1}{2}\left(\frac{\partial^2 \Phi_\mathrm{eff}}{\partial z^2}\right)\Bigg|_{(R_g,0)}\,z^2\,, \end{equation} up to terms that are third order in \((R-R_g)\) and \(z\) (the first order terms are zero, because we are at the minimum of the effective potential). The equations of motion (9.4) and (9.5) in this approximation then become \begin{align} \ddot{R} = \ddot{R}-\ddot{R_g} &= -\left(\frac{\partial^2 \Phi_\mathrm{eff}}{\partial R^2}\right)\Bigg|_{(R_g,0)}\,(R-R_g)\,;\quad \ddot{z} = -\left(\frac{\partial^2 \Phi_\mathrm{eff}}{\partial z^2}\right)\Bigg|_{(R_g,0)}\,z\,, \end{align} (because \(\ddot{R_g} = 0\)). These are the equations of a decoupled harmonic oscillator in \((R-R_g,z)\), with frequencies \begin{align}\label{eq-eom-epicycle-1} \kappa^2(R_g) & = \left(\frac{\partial^2 \Phi_\mathrm{eff}}{\partial R^2}\right)\Bigg|_{(R_g,0)}= \left(\frac{\partial^2 \Phi}{\partial R^2}\right)\Bigg|_{(R_g,0)}+3\,\frac{L_z^2}{R_g^4}\,,\\ \nu^2(R_g) & = \left(\frac{\partial^2 \Phi_\mathrm{eff}}{\partial z^2}\right)\Bigg|_{(R_g,0)}= \left(\frac{\partial^2 \Phi}{\partial z^2}\right)\Bigg|_{(R_g,0)}\,.\label{eq-eom-epicycle-2} \end{align} The first of these, \(\kappa\), is known as the epicycle frequency or as the radial frequency, while the second is known as the vertical frequency. In the second equality in each line above, we have substituted in the equation for the effective potential (Equation 9.2). A third important frequency is the circular frequency \(\Omega(R_g)\), which is the azimuthal frequency of the circular orbit at \(R_g\) and is therefore \(\Omega(R_g) = v_c(R_g)/R_g = L_z/R_g^2\). Using that \(\Omega^2(R_g) = (\partial \Phi/\partial R)|_{R_g}/R_g\), we can also derive the following relation between \(\kappa\) and \(\Omega\) \begin{equation}\label{eq-epifreqkapp-omega} \kappa^2(R_g) = \left(R\,\frac{\mathrm{d}\Omega^2}{\mathrm{d}R}+4\,\Omega^2\right)\Bigg|_{R_g}\,, \end{equation} which we can also write as \begin{equation} \frac{\kappa^2}{\Omega^2}(R_g) = 2\left(\frac{\mathrm{d}\ln [\Omega\,R^2]}{\mathrm{d}\ln R}\right)\Bigg|_{R_g}\,. \end{equation}
As we discussed in Chapter 2.4.4, rotation curves of galaxies go between the extremes of a Keplerian rotation curve \(v_c(R) \propto R^{-1/2}\) and that corresponding to a constant density sphere, \(v_c(R) \propto R\). For these extremes we have that \(\Omega\,R^2 \propto R^{1/2}\) and \(\Omega\,R^2 \propto R^2\), respectively. These correspond to \(\kappa/\Omega = 1\) and \(\kappa/\Omega = 2\) and thus \begin{equation}\label{eq-epicycle-kappaomegarange} \Omega \lesssim \kappa \lesssim 2\Omega\,, \end{equation} in galaxies. This is the analogous relation to that for the range in \(T_r/T_\phi\) that we derived for spherical potentials in Equation (4.43). Therefore, we see that the ratio \(\kappa/\Omega\) is informative about the mass distribution.
Because they are the equations of a decoupled harmonic oscillator in \((R-R_g,z)\), it is straightforward to solve the equations of motion (9.12) and (9.13). The vertical oscillation is \begin{equation}\label{eq-epicycle-vertical-sol} z = z_\mathrm{max}\,\cos(\nu\,t+\zeta)\,. \end{equation} where \(z_\mathrm{max}\) is the maximum height above the mid-plane that the orbit reaches and \(\zeta\) is a phase parameter (the phase along the orbit at \(t=0\)). Because disk galaxies have rotation curves that are close to flat, they have the interesting property that the vertical frequency is essentially determined by the local density of matter. To obtain this result, we use the Poisson equation in cylindrical coordinates (Equation 7.21) and substitute \(v_c^2 = R\,\partial \Phi / \partial R\): \begin{align} 4\pi\,G\,\rho & = \frac{1}{R}\,\frac{\partial}{\partial R}\left(R\,\frac{\partial \Phi}{\partial R}\right) + \frac{\partial^2 \Phi}{\partial z^2} \approx \frac{1}{R}\,\frac{\mathrm{d} v_c^2}{\mathrm{d} R} + \frac{\partial^2 \Phi}{\partial z^2} \approx \nu^2\,, \end{align} where we use that \(\mathrm{d}v_c / \mathrm{d}R \approx 0\) in galaxies with a flat rotation curve. For the second equality to hold at \(z\neq 0\), we need that \(v_c^2 \approx R\,\partial \Phi(R,z) / \partial R\), but this is the case for typical disk mass distributions up to multiple kpc from the mid-plane (shown under very general conditions by Bovy & Tremaine 2012).
Because \(\nu \propto \sqrt{\rho}\), the vertical frequency can only be considered to be a constant when the density is well approximated as being constant. Because galactic disks are thin with a steep vertical density gradient, this clearly cannot be the case for most of the disk. The solution in Equation (9.17) is therefore only applicable to the small number of objects (e.g., young stars, open clusters) whose orbits do not stray far from the mid-plane (\(z_\mathrm{max} \lesssim 100\,\mathrm{pc}\)).
The motion in \((R-R_g)\) is given by \begin{equation}\label{eq-epicycle-radial-sol} R(t)-R_g = X\,\cos\left(\kappa\, t + \alpha\right)\,, \end{equation} with \(X\) and \(\alpha\) constants to be determined from the initial conditions. To obtain the full orbit, we also need to find \(\phi(t)\) from the conservation of angular momentum: \(R^2\,\dot{\phi} = L_z\). Therefore we have that \begin{equation} \dot{\phi} = \frac{L_z}{R^2(t)} = \frac{L_z}{(R_g+[R-R_g])^2}\,. \end{equation} Because \(R-R_g \ll R_g\) for a close-to-circular orbit, we can Taylor expand the numerator and get \begin{equation} \dot{\phi} = \frac{L_z}{R_g^2}\,\left(1-2\,\frac{R(t)-R_g}{R_g}\right)\,. \end{equation} Substituting in the radial motion from Equation (9.19), using that \(\Omega = L_z/R_g^2\), and finally integrating this equation, we find the \(\phi(t)\) solution \begin{equation}\label{eq-epicycle-angle-sol} \phi(t) = \phi_0+\Omega\,t-2\,\frac{\Omega}{\kappa}\,\frac{X}{R_g}\,\sin\left(\kappa\,t+\alpha\right)\,. \end{equation}
What is the motion that corresponds to the solution in Equations (9.19) and (9.22)? When we subtract out the motion of the guiding center itself, which is \((R,\phi)[t] = (R_g,\Omega\,t+\phi_0)\), we are left with the motion \begin{align}\label{eq-epicycle-orbit} R(t)-R_g & = X\,\cos\left(\kappa\,t+\alpha\right)\,;\quad R_g\,\left(\phi(t) - \Omega\,t-\phi_0\right) = -2\,\frac{\Omega}{\kappa}\,X\,\sin\left(\kappa\,t+\alpha\right)\,. \end{align} This solution corresponds to an ellipse centered on the guiding center with axis ratio given by \(\gamma = 2\Omega/\kappa\). From the range in \(\kappa/\Omega\) in Equation (9.16), this axis ratio is \(1/2 \lesssim \gamma \lesssim 1\). This ellipse is known as the epicycle and this explains why we call for \(\kappa\) the epicycle frequency: \(\kappa\) is the frequency at which a star travels around its epicycle ellipse.
9.3.2. The epicycle approximation and the Oort constants¶
We can relate the epicycle approximation to the Oort constants that describe the local velocity field. This allows us to determine how the Oort constants affect the local stellar kinematics beyond the mean field that we discussed in Chapter 8.4.
For an axisymmetric disk, we can relate the frequencies of close-to-circular orbits to the Oort constants. In Equation (8.34), we expressed \(A\) in terms of \(\Omega(R)\). Similarly, from Equation (8.33) for \(B\) and using that \(v_c(R) = \Omega(R)\,R\), we find \begin{equation}\label{eq-diskorbits-B-asOmega} 2B = -2\Omega(R_0) -R_0\,\frac{\mathrm{d}\Omega}{\mathrm{d}R}\Bigg|_{R_0}\,. \end{equation} Using Equation (9.14), we can then express \(\kappa(R_0)\) in terms of the Oort constants \begin{equation}\label{eq-diskorbits-kappa-asOort} \kappa^2(R_0) = -4B\,(A-B)\,. \end{equation} Using the measured values for \(A\) and \(B\) from Equation (8.36), we therefore find that in the solar neighborhood \begin{equation} \kappa(R_0) = 33.7\,\,\mathrm{km\,s}^{-1}\,\mathrm{kpc}^{-1}\,, \end{equation} and \(\kappa(R_0)/\Omega(R_0) \approx 1.25\). Stars in the solar neighborhood therefore execute about 5 radial oscillations for every 4 orbits around the Galactic center.
In Chapter 8.4, we discussed how the mean velocities, as line-of-sight velocities or proper motions, of stars in the solar neighborhood are given in terms of the Oort constants. We can go beyond the mean velocity and determine how the dispersion in velocities is related to the Oort constants when we assume axisymmetry. For that we start from Equation (9.23) and take the derivative to get expressions for the velocity of a star in the epicycle approximation that is currently located at \(R \approx R_0\) \begin{align}\label{eq-epicycle-v} v_R(t) & = -X\,\kappa\sin\left(\kappa\,t+\alpha\right)\,;\quad v_\phi(t) = R_0\,\Omega(R_g) -2\,\Omega(R_g)\,X\,\cos\left(\kappa\,t+\alpha\right)\,, \end{align} where \(X\) is again the amplitude of the star’s epicycle motion in the radial direction, \(R_g\) is its guiding-center radius, and \(R_0\,\Omega(R_g)\) is the contribution to the velocity from the guiding-center’s angular motion. Note that the guiding-center radius \(R_g\) for stars near the Sun is not \(R_0\); in general, each star has its own guiding-center radius in the epicycle approximation.
Assuming that the Sun is on a circular orbit at \(R_0\) (that is, assuming that we have corrected the Sun’s motion to that of the LSR; see Chapter 10.3.1), we can subtract the Sun’s motion from Equations (9.27) to get the relative velocity \begin{align} \Delta v_R(t) = v_R(t) & = -X\,\kappa\sin\left(\kappa\,t+\alpha\right)\,,\\ \Delta v_\phi(t) = v_\phi(t) - v_c(R_0) & = R_0\,\left[\Omega(R_g) - \Omega(R_0)\right]-2\,\Omega(R_g)\,X\,\cos\left(\kappa\,t+\alpha\right)\,, \end{align} For stars near the Sun, we can Taylor expand \(\Omega(R_g) - \Omega(R_0)\) in terms of \(R_g-R_0\) in the second equation and we only keep terms up to first order in \(X\), which gives \begin{align} \Delta v_\phi(t) & = R_0\,\frac{\mathrm{d} \Omega}{\mathrm{d} R}\Bigg|_{R_0}\,(R_g-R_0) -2\,\Omega(R_g)\,X\,\cos\left(\kappa\,t+\alpha\right)\nonumber\\ & = -\left(R_0\,\frac{\mathrm{d} \Omega}{\mathrm{d} R}\Bigg|_{R_0}+2\Omega(R_0)\right)\,X\,\cos\left(\kappa\,t+\alpha\right) \,, \end{align} where we have used the epicycle approximation for \(R_g-R(t)\) applied to the current location \(R(t) = R_0\). We can then express both components of the relative velocity of local stars measured with respect to the LSR in terms of the Oort constants using Equations (9.24) and (9.25) \begin{align} \Delta v_R(t) & = -\sqrt{-4B\,(A-B)}\,X\,\sin\left(\kappa\,t+\alpha\right)\,;\quad \Delta v_\phi(t) = 2\,B\,X\,\cos\left(\kappa\,t+\alpha\right)\,.\label{eq-diskorbits-dvT-epicycle} \end{align} For a steady-state distribution, we may assume that we are seeing a random mix of phases in a local sample and that the phase and epicycle amplitude are uncorrelated. If the phases are randomly distributed then we have that \(\langle \sin^2\left(\kappa\,t+\alpha\right)\rangle = \langle\cos^2\left(\kappa\,t+\alpha\right)\rangle = 1/2\). Thus, if we average the squared velocities of a sample of local stars, we average over all phases and find the velocity dispersions \begin{align} \sigma^2_R(t) & = -2\,B\,(A-B)\,\langle X^2\rangle\,;\quad \sigma^2_\phi(t) = 2\,B^2\,\langle X^2 \rangle\,. \end{align}
If we know the Oort constants, then the radial and tangential velocity dispersion tell us about the typical epicycle amplitude of a sample of stars. For example, intermediate-age stars near the Sun have \(\sigma_R \approx 30\,\mathrm{km\,s}^{-1}\) and therefore have epicycle amplitudes of \(\sqrt{\langle X^2\rangle} \approx \sqrt{\sigma_R^2 / [-2\,B\,(A-B)]}\approx 1.4\,\mathrm{kpc}\).
The ratio of the radial and tangential velocity dispersion of local stars only depends on the Oort constants \begin{equation}\label{eq-sr-st-epicycle} \frac{\sigma^2_\phi}{\sigma^2_R} = \frac{B}{B-A}\,. \end{equation} In the solar neighborhood therefore, we expect \(\sigma_\phi/\sigma_R \approx 2/3\) from the measured values of the Oort constants. Because \(B = -A\) for a flat rotation curve, in the case of a flat rotation curve we would find \(\sigma_\phi/\sigma_R = 1/\sqrt{2}\).
Equation (9.33) is a useful relation because it shows how the dispersions in the velocities of local stars are directly related to the Galactic potential. However, because for intermediate-age stars the typical epicycle amplitude \(\sqrt{\langle X^2\rangle} \approx 1.4\,\mathrm{kpc}\) is quite large, corrections to Equation (9.33) due to non-circular orbits are in fact quite large (Kuijken & Tremaine 1991). Equation (9.33) therefore really applies only to the youngest populations of stars near the Sun, but for those stars the assumption of dynamical equilibrium is suspect, because they may not have had enough time to phase mix. To go beyond the description of the velocity dispersion using the epicycle approximation, we need to build equilibrium models for disk populations, which is the subject of the next chapter.
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