idPIE profile description
dPIE summary
A summary on the dual Pseudo-Isothermal Elliptical matter distribution (dPIE) may be found here, and this type of gravitational potential is described at length in Elìasdòttir et al. (2007, Appendix A)
. It is identified in lenstool by id: 81.
Assuming we neglect ellipticity in this documentation, dPIE profiles write:
\rho_{\mathrm{dPIE}}(r) = \frac{\rho_0}{\left[ 1 + \left( \frac{r}{s} \right)^2 \right] \left[ 1 + \left( \frac{r}{a} \right)^2 \right]}
where \rho_0 is the density normalisation, a the core radius, and s the cut radius.
A sum of dPIE profiles may be assumed to represent the total matter density
\rho_m
(baryons + dark matter) in the lens:
\rho_m = \sum_i \rho_{\mathrm{dPIE}, i}.
Thus the profile of the gravitational potential
\Phi may be deduced from the dPIE sum:
\Phi(r) = - 4 \pi G \sum_i \int_0^r \mathrm{d}s s^{-2} \int_0^s \mathrm{d}t t^2 \rho_{\mathrm{dPIE}, i}(t).
For one dPIE profile
\rho_{\mathrm{dPIE}}(r), the potential writes:
\Phi_{\mathrm{dPIE}}(r) = \frac{a^2 s^2}{a^2 - s^2} \left[ \frac{s}{r} \arctan \frac{r}{s} - \frac{a}{r} \arctan \frac{r}{a} + \frac{1}{2} \ln \left( \frac{r^2 + s^2}{r^2 + a^2} \right) \right].
Hydrostatic idPIE n_e ICM density profile
If we assume the intra-cluster medium (ICM) to be in hydrostatic equilibrium, we may simplify the Navier-Stokes equation to:
\frac{\partial_r \left( n_e T_e \right)}{n_e} = \frac{\mu_g m_a}{k_B} \partial_r \Phi,
where n_e is the ICM electron number volume density, T_e the ICM electron temperature, \mu_g \approx 0.60 the mean molecular weight of the ICM gas, m_a \approx 1.66 \times 10^{-27} kg the unified atomic mass, and k_B the Boltzmann constant.
Assuming the temperature T_e to be a function of the electronic density, we can integrate this expression to:
\mathcal{J}_z (n_e) = \int_0^{n_e} \mathrm{d} n \frac{n T_e (n)}{n} = \frac{\mu_g m_a}{k_B} \Phi (r),
where
\mathcal{J}_z is a bijection, as long as the radial density profile
\rho_m is a sum of dPIE potentials.
Using a self-similar polytropic temperature profile, the
\mathcal{J}_z integral only depends on redshift
z.
Bijections being invertible functions, we may revert the previous equation, thus yielding the idPIE density profile:
n_e = \mathcal{J}^{-1}_z \left( \frac{\mu_g m_a}{k_B} \Phi (r) \right).
ICM profile optimisation with idPIE profile
Given the n_e ICM electron density, we can compute S_X, the X-ray surface brightness:
S_X (x, y, \Delta E) = \frac{1}{4 \pi (1 + z)^4} \frac{\mu_e}{\mu_H} \int_{\mathrm{l.o.s.}} n_e^2 (x, y, l) \Lambda (\Delta E (1 + z), T_e, Z) \mathrm{d}l,
where \Delta E is the observed energy band, z is the cosmological redshift of the lens, \mu_e and \mu_H are respectively the mean molecular weight of electron and hydrogen, and \Lambda is the normalised cooling function (in \mathrm{J.m}^3.\mathrm{s}^{-1}) for an ICM electron temperature T_e and metallicity Z. Here, we assume the metallicity to be constant throughout the cluster Z = 0.3 Z_{\odot}.
Once the model surface brightness map computed, it is compared to observations of Chandra or XMM-Newton X-ray satellites.
Note
TODO: See section on statistics for more details.