Code paper

This is disc-bulge-halo model build with Eugene Vasiliev's particle generator.

Exponential disc

\[ f(J) = \frac{1}{(2\pi)^3} f_\varphi (J_\varphi) f_R (J_r, J_\varphi) f_z (J_z, J_\varphi)\ , \]

where

\[ f_\varphi = 4 \pi M_\textrm{disc} \frac{\Omega}{\kappa^2} \exp\left( -R/R_\textrm{disc} \right)\ , \\ f_r = \frac{\sigma_r^2} \kappa \exp\left( - \kappa J_r / \sigma_r^2 \right)\ , \\ f_z = \frac{\sigma_z^2} \nu \exp\left( - \nu J_z / \sigma_z^2 \right)\ . \]

Radial velocity dispersion:

\[ \sigma_r = \sigma_{r,0} \exp\left[ -R / ( R_\sigma ) \right]\ . \]

Vertical velocity dispersion:

\[ \sigma_z = \sqrt{2} h \nu\ , \]

where \(h\) is the disc thickness.

Bulge and halo

For bulge and halo we adopt SpheroidDensity model, which is an extension of the two-power profile (BT, p.70):

\[ \rho(r) = \rho_0 \left(\frac\mu a \right)^\gamma \left( 1+ \left(\frac\mu a\right)^\alpha \right)^{\displaystyle \frac{\gamma-\beta}\alpha } \exp\left[-\left( \frac{\mu}{r_\textrm{cut}}\right)^\xi \right] \]

where

\[ \mu^2 \equiv x^2 + (y/p)^2 + (z/q)^2\ . \]

If not stated, default values apply (usually 1).

Hernquist bulge

NFW halo

Potential multipole expansion

\[ \Phi(r, \theta) = \sum\limits_{l=0}^{l_\textrm{max}} \sqrt{2l+1} \Phi_{l,0} (r) P_l (\cos\theta)\ , \]

where \(\Phi_{l,0} (r)\) are tabulated in rotcurve_iter5_0, \(P_l(x)\) are Legendre polynomials:

\[ P_0(x) = 1 \ ,\\ P_1(x) = x \ ,\\ P_2(x) = (3x^2-1)/2 \ , \\ P_3(x) = (5x^2-3)x/2 \ , \\ P_4(x) = (35x^4-30x^2+3)/8 \ , \]

\(0\leq \theta \leq \pi\).

Particle distribution

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