parameters were adjusted to yield a similar total circular speed curve. Model 2 only forms flocculent spiral structure due to the dominant bulge and halo relative to the disc. \[fig:circspeed\]](fig1.pdf){width="\columnwidth"}

Two Milky Way-like Models
-------------------------

We generate the initial conditions for our $N$-body simulations using <span style="font-variant:small-caps;">galactics</span> [@kuijken1995; @widrow2008], which allows one to construct axisymmetric disc-bulge-halo systems that are in approximate dynamical equilibrium. The phase space distribution function (DF) for the disc in these models, $f_d$, is a function of two exact integrals of motion, the energy and the component of the angular momentum about the symmetry axis, and an approximate third integral that corresponds to the vertical energy. By construction, the disc density obtained by integrating $f_d$ over all velocities is well-approximated by the function

$$\label{eq:discdensity}
\rho_d\left (R,\,z\right ) = \frac{M_d}{4\pi R_d^2 h} e^{-R/R_d} {\rm
  sech}^2(z/h)\,C\left (\left (R-R_t\right )/\delta R_t\right ),$$

where $R$ and $z$ are cylindrical coordinates and $C$ is a truncation function that goes smoothly from unity to zero at $R\simeq R_t$ over a width of order $\delta R_t$. The radial velocity dispersion is an exponentially decreasing function of $R$:

$$\sigma_R(R) = \sigma_{R0} \exp{\left (-R/2R_d\right )},$$

where we assume that the radial scale length for the square 