Resonant heating rates

The linearized Boltzmann equation is a linear partial differential equation in seven variables. Using action-angle variables, we can separate the equation and employ standard distribution functions constructed according to Jeans' theorem (Binney & Tremaine 1987). The explicit form of the linearized Boltzmann equation is

$${\partial f_1 \over \partial t}+{\partial f_1 \over \partia... ...0 \over \partial \bf I}{\partial H_1 \over \partial \bf w}=0,$$ (7)

where $\bf w$ is the vector of angles, and $\bf I$ are the conjugate actions. The quantities f₀ and H₀ depend on the actions alone. Making the assumption that the tidal force from the Galaxy is small, the perturbation may be separated into phase-space and time components, $H_1=\eta({\bf r})g(t)$, expanded in a Fourier series in action-angle variables (e.g. Tremaine & Weinberg 1984). Each term $f_{1\bf l}$ in the Fourier series is the solution of the following differential equation:

$${\partial f_{1\bf l} \over \partial t}+(i\bf l \cdot \Omega... ...v i{\bf l} \cdot{\partial f_0 \over \partial \bf I}H_{1\bf l},$$ (8)

where $\bf\Omega =\partial H_0/\partial{\bf I}$ and

$$V_{\bf l}(\bf I)={1\over (2\pi)^3}\int_{-\pi}^{\pi}\eta({\bf r})e^{-i\bf l \cdot w}d^3\bf w.$$ (9)

The quantity ${\bf l}$ is a vector of integers whose rank is the number of degrees of freedom; e.g. for the three dimensional problems considered here, ${\bf l}=(l_1,l_2,l_3)$. In practice, we usually confine our perturbation to a particular set of spherical harmonics or cylindrical harmonics which restricts two out of three to a finite set (see Tremaine & Weinberg 1984 for details).

The rate of change in energy or action arising from the perturbation follows from Hamilton's equations and is

$${\dot E} \sum_{\bf l=-\infty}^{\infty}i\bf l \cdot \Omega H_{1-\bf l}f_{\bf l}$$ =

$${\dot I}_j \sum_{\bf l=-\infty}^{\infty}iI_j H_{1-\bf l}f_{\bf l}.$$ = (10)

For periodic perturbations, the time dependent amplitude may be represented by a Fourier series,

$$g(t)=\sum_{n=-\infty}^{\infty}a_n e^{i n\omega t}.$$ (11)

With this form for g(t), equation (8) may be solved for the perturbed distribution function by Laplace transform. Finally, phase averaging the quantities in equations (10 yields the following time-asymptotic rates:

$$\langle \dot E \rangle -8\pi^4\sum_{\bf l=-\infty}^{\infty}(\bf l \cdot \Omega )({\bf l}... ...2 \sum_{n=-\infty}^{\infty}\vert a_n\vert^2\delta(n\omega-\bf l \cdot \Omega ).$$ = (12)

$$\langle \dot I_j \rangle -8\pi^4\sum_{\bf l=-\infty}^{\infty}l_j({\bf l} \cdot{\partial f_... ...2 \sum_{n=-\infty}^{\infty}\vert a_n\vert^2\delta(n\omega-\bf l \cdot \Omega ).$$ = (13)

Murali & Weinberg (1997a) show that this expansion, continued to the next order, results in an equation for the change in distribution function in terms of these rates which takes the form:

$$\langle {\dot f}_2\rangle \propto {\partial\over\partial{\bf I}} \cdot \langle{\bf {\dot I}}\rangle$$ (14)

which may be solved by standard flux-conserving finite-difference methods.