scheme. We compute the center of mass of disc particles within a cylinder of radius 20 kpc and height 2 kpc centered in the $x$-$y$ plane and translate all disc particles to that frame. This procedure is repeated until a translation of less than 1 pc is achieved in all coordinate directions. Next, we account for any tilt of the disc relative to the $x$-$y$ plane by using a two-dimensional Newton-Raphson scheme to find the Euler angles about the $x$- and $y$- axes that minimize the root mean square vertical displacement of disc particles within a cylinder of radius 20 kpc and height 2 kpc centered in the $x$-$y$ plane. This method does a good job of diagonalizing the moment of inertia tensor of the disc.

In Fig. \[fig:surfacedensity\], we show the disc surface density relative to the equilibrium surface density for the two models at various times. The maps extend to $R=25 \, {\rm kpc}$, or about $9 R_d$, which is the radius at which we begin to truncate the disc. This radius also corresponds to the outermost radius at which the warp in the Milky Way’s stellar disc has been observed [@momany2006]. The decrease of $\sim 20 - 