Table 1: Two mass density/shape models considered in this study. For model 9 1, an exact expression for both $\breve{\kappa}_Z$ and g_Z can be derived analytically from Eqs. (8) and (9) provided $H=\rm const.$ (see the Appendix B, Eq. (B.2); the kernel and field are finite for $k^\pm \ne 1$ ). Model 2 is a vertically parabolic profile. In this case, there is no solution in a closed form, whatever the shape of the system. Note that $\rho _1 \propto 1/a^2$ for $R \ll a$ whereas $\rho _1 \sim \rm const.$ for $R \gg a$ .
Model Properties

1 mass density 9 $\rho_1=\rho_0 \sqrt{\frac{R}{a}}\frac{k}{E(k)}$

(virtual) shape $H=\rm const.\;$ (flat system)

secondary kernels $\breve{\kappa}^{\rm ref.}_R$ and $\breve{\kappa}^{\rm ref.}_\Psi$ unknown

$\breve{\kappa}^{\rm ref.}_Z = G\rho_0\ln \left( \frac{1-{k^+}^2}{1-{k^-}^2} \right)$

radial field unknown

vertical field g_Z known; see Eq. (B.2)

potential unknown

2 mass density $\rho_2=\rho_0(a) \left[ 1-\left(\frac{z}{H} \right)^2 \right]$

(realistic) shape any

secondary kernels unknown

field unknown

potential unknown

**Table 1:** Two mass density/shape models considered in this study. For model 9 1, an exact expression for both $\breve{\kappa}_Z$ and g_Z can be derived analytically from Eqs. (8) and (9) provided $H=\rm const.$ (see the Appendix B, Eq. (B.2); the kernel and field are finite for $k^\pm \ne 1$ ). Model 2 is a vertically parabolic profile. In this case, there is no solution in a closed form, whatever the shape of the system. Note that $\rho _1 \propto 1/a^2$ for $R \ll a$ whereas $\rho _1 \sim \rm const.$ for $R \gg a$ .
Model	Properties
1	mass density 9	$\rho_1=\rho_0 \sqrt{\frac{R}{a}}\frac{k}{E(k)}$
(virtual)	shape	$H=\rm const.\;$ (flat system)
	secondary kernels	$\breve{\kappa}^{\rm ref.}_R$ and $\breve{\kappa}^{\rm ref.}_\Psi$ unknown
		$\breve{\kappa}^{\rm ref.}_Z = G\rho_0\ln \left( \frac{1-{k^+}^2}{1-{k^-}^2} \right)$
	radial field	unknown
	vertical field	g_Z known; see Eq. (B.2)
	potential	unknown
2	mass density	$\rho_2=\rho_0(a) \left[ 1-\left(\frac{z}{H} \right)^2 \right]$
(realistic)	shape	any
	secondary kernels	unknown
	field	unknown
	potential	unknown

Source LaTeX | All tables | In the text

	1 Introduction
	2 The field: Theoretical grounds
	3 Vertical integration of the kernel associated with the field: "Density splitting method''
	4 Field values
	5 Numerical treatment of the self-gravitating potential: The "double splitting'' method
	6 Resolutions and efficiency
	7 Concluding remarks
	Appendix A: Values of $\vec{(\delta \rho)} \vec{\kappa}$ at the coincident point
	Appendix B: Density/secondary kernels test-pairs
	Appendix C: Input parameters for the radial sampling
	Appendix D: Analytical formula
	References