a stellar disk in a near-equilibrium state based on a Maxwellian approximation [@Hernquist1993], and we subsequently allowed the disk to evolve for 6 Gyr under the constraint of axisymmetry [@McMillanDehnen2007; @Fujii+2011].

We modeled a fixed potential of the dark matter (DM) halo whose density profile follows the NFW profile [@Navarro+1997]: $$\rho_{\rm h}(r)=\frac{\rho_0}{r/r_{\rm s}(1+r/r_{\rm s})^2},$$ where $$\rho_0 = \frac{M_{\rm h}}{4\pi {R_{\rm h}}^3}
\frac{{C_{\rm NFW}}^3}{\ln(1+C_{\rm NFW})+C_{\rm NFW}/(1+C_{\rm NFW})},$$ $$C_{\rm NFW} = R_{\rm h}/r_{\rm s},$$ an the terms $C_{\rm NFW}$, $M_{\rm h}$, and $R_{\rm h}$ denote the halo concentration parameter of the halo profile, the DM halo mass, and the virial radius, respectively. The model parameters adopted here are summarized in Table \[tbl:models\]. The circular velocity, velocity dispersions, $Q$-values and $\Gamma$-profile of the stellar disk are shown in Figure \[fig:rotcurve\]. Here, $\Gamma$ denotes a dimensionless shear rate of the disk as defined by the expression $-d\ln\Omega/d\ln R$, where $\Omega$ denotes the angular frequency.

We used the simulation code [ASURA]{} [@Saitoh+2008; @Saitoh+2009], where the self-gravity of stellar particles is calculated by the Tree with the GRAPE method [@Makino1991]. We used a software emulator of GRAPE, the Phantom-GRAPE [@Tanikawa+2012]. The opening angle was set to $0.5$ with the center-of-mass approximation. To perform a time integration, 