PIEMD¶

Set the potential profile to the type dPIE described in Eliasdottir et al. 2007 .

In set_lens.c:set_dynamics(), the impact parameter is computed as such:

$b_0 = \frac{6 \pi \sigma_0^2}{c^2}$

with $\frac{\pi}{c^2} = 7.2 10^{-6}$ arcsec / $(\rm km/\rm s)^2$. To obtain this value, $\pi$ is converted to 648,000 arcsec.

In file e_grad2.c, the 2nd derivatives of the gradient are computed as such with the core radius a and the cut radius s in arcsec:

$t05 = b_0 \frac{s}{s - a}$

$z = \sqrt{R^2 + a^2} - a - \sqrt{R^2 + s^2} +s$

$p = \left( 1 - \frac{1}{\sqrt{1 + \frac{R^2}{a^2}}} \right) \frac{a}{R^2} - \left( 1 - \frac{1}{\sqrt{1 + \frac{R^2}{s^2}}} \right) \frac{s}{R^2}$

$\partial^2_{xx} = b_0 \frac{s}{s - a} \left( p\frac{x^2}{R^2} + z \frac{y^2}{R^4} \right)$

$\partial^2_{yy} = b_0 \frac{s}{s - a} \left( p\frac{y^2}{R^2} + z \frac{x^2}{R^4} \right)$

$\partial^2_{xy} = b_0 \frac{s}{s - a} \left( p\frac{x\ y}{R^2} - z \frac{x\ y}{R^4} \right)$

From the 2nd derivatives, the convergence is computed in g_mass.c:computeKmass()

The theoretical expression taken from Eq. 7 in Limousin et al. 2005 is

$\Sigma(R) = \frac{\sigma_0^2}{2 G} \frac{s}{s -a} \left(\frac{1}{\sqrt{R^2 + a^2}} - \frac{1}{\sqrt{R^2 + s^2}} \right)$

and the critical mass is

$\Sigma_{crit} = \frac{c^2}{4 \pi G} \frac{D_{OS}}{D_{OL} D_{LS}}$

By taking the ratio with $\Sigma_0 = \frac{\sigma_0^2}{2G}$, we find

$\kappa_{th} = \frac{\Sigma_0}{\Sigma_{crit}} = \frac{2 \pi \sigma_0^2}{c^2} \frac{D_{OL} D_{LS}}{D_{OS}}$

Therefore the relation between Lenstool and the theory is

$\kappa_{th} = \frac{2}{3}\ \kappa_{lt}$

Which translates in velocity dispersion as

$\sigma_{0\ th} = \sqrt{\frac{3}{2}}\ \sigma_{0\ lt}$

Note that the mass keyword corrects for this factor internally, and returns the theoretical convergence map.