Galactokinetics

Abstract

Galactic disks lie at the heart of many of the most pressing astrophysical puzzles. There are sophisticated kinetic theories that describe some aspects of galaxy disk dynamics, but extracting quantitative predictions from those theories has proven very difficult, meaning they have shed little light on observations/simulations of galaxies. Here, we begin to address this issue by developing a tractable theory describing fluctuations and transport in thin galactic disks. Our main conceptual advance is to split potential fluctuations into asymptotic wavelength regimes relative to orbital guiding radius and epicyclic amplitude (similar to plasma gyrokinetics), and then to treat separately the dynamics in each regime. As an illustration, we apply our results to quasilinear theory, calculating the angular-momentum transport due to a transient spiral. At each stage we verify our formulae with numerical examples. Our approach should simplify many important calculations in galactic disk dynamics. In a follow-up paper, we apply these ideas to the theory of linear spiral structure in stellar disks.

Figures (8)

• Figure 1: Two orbits in the logarithmic potential. The peri/apocenter distances $(R_{\mathrm{p}}, R_{\mathrm{a}})$ for each orbit are shown with red dashed circles, and the initial location is shown with a small blue star. Full details of the orbital parameters are given in Table \ref{['table:three_orbits']}. The top row shows the orbits as viewed in an inertial $(X,Y)$ frame. The bottom row shows the same orbits in the frame rotating azimuthally with frequency $\Omega_\varphi$ (equation \ref{['eqn:azi_freq']}). In green we show what the orbits would look like if we had ignored the Dehnen drift correction in \ref{['eqn:azi_freq']} and just rotated at $\Omega$.
• Figure 2: Diagram illustrating the short, intermediate and long wavelength regimes for various epicyclic amplitudes $a_R$, or equivalently velocity dispersion of the stellar population $\sigma$ (assuming $a_R=a$, see equations \ref{['eqn:rms_epicyclic_amplitude']}-\ref{['eqn:rms_radial_velocity']}). This plot assumes $R_\mathrm{g}=8\,$kpc, and a flat rotation curve ($\gamma=\sqrt{2}$) with circular velocity $V_0=220$ km s$^{-1}$. We also indicate the typical wavelengths of potential fluctuations due to various astrophysical perturbations.
• Figure 3: Orbit (b) from Figure \ref{['fig:three_orbits_rotating']} (which has $\epsilon_R=0.11$), superimposed on top of the contours of a logarithmic spiral potential \ref{['eqn:Phi_Spiral']} with $m$ arms and pitch angle $\alpha$. The values of $m$ and $\alpha$ are chosen such that panels (a), (b) and (c) exhibit potential fluctuations characteristic of the long ($ka_R\sim\epsilon_R$), intermediate ($ka_R\sim 1$) and short ($ka_R\sim\epsilon_R^{-1}$) wavelength regimes (equations \ref{['eqn:def_long_wavelength_regime']}-\ref{['eqn:def_short_wavelength_regime']}) respectively. We caution that the spirals are painted on this figure for illustrative purposes only; we have not chosen any pattern speed for them, and we do not mean to imply that the orbits shown are corotating with the spirals.
• Figure 4: (Real part of the) Fourier components of the potential fluctuations $\delta \phi_{\bm{n}}$ calculated for an $m=4$ logarithmic spiral potential perturbation \ref{['eqn:Phi_Spiral']}, embedded in a logarithmic background potential \ref{['eqn:log_halo']}, at the action-space location of the orbit shown in Figure \ref{['fig:three_orbits_rotating']}a (for which $\epsilon_R = 0.053$). From left to right we decrease the spiral's pitch angle and hence increase its wavenumber $k$ (see equation \ref{['eqn:logspiral_wavenumber_modulus']}). Panels (a)-(c) show the result for $n_\varphi = 4$ and for $n_R = 0$, $1$ and $2$ respectively. The black line shows the result of evaluating the right-hand side of the expression \ref{['eq:potential_expansion_epicyclic']} numerically, while the red and blue dotted lines show the asymptotic results \ref{['eq:deltaphi_cold_expansion']} and \ref{['eqn:very_short_perturbations']} respectively. See text for more details.
• Figure 5: As in Figure \ref{['fig:deltaphi_dependence_k']} except we plot the absolute difference between the numerical solution (the black line in Figure \ref{['fig:deltaphi_dependence_k']}) and the result of the long wavelength approximation (equation \ref{['eq:deltaphi_cold_expansion']}, red), the short wavelength approximation (equation \ref{['eqn:deltaPhi_WKB_general']}, blue), the same equation with the Dehnen terms dropped (equation \ref{['eqn:deltaPhi_WKB']}, magenta), and the tight-winding approximation (equation \ref{['eqn:very_short_perturbations']}, gold). We also indicate the magnitudes $\epsilon_R$ and $\epsilon_R^2$ with dotted and dashed black horizontal lines respectively.