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This object implements the NcHaloDensityProfile class for a Hernquist density profile.
As described NcHaloDensityProfile, we just need to implement the dimensionless 3D density $\hat{\rho}(x)$
[which refers to the virtual function nc_halo_density_profile_eval_dl_density()].
In particular, the Hernquist profile is given by
\begin{equation}
\hat{\rho}(x) = \frac{1}{x(1 + x)^3},
\end{equation}
where $x = r/r_s$ and $r_s$ is the scale radius.
Both the mass $M_\Delta$ and the scale profile $\rho_s$ are written in terms of the integral
$I_{x^2\hat\rho}(c_\Delta)$ [virtual function nc_halo_density_profile_eval_dl_spher_mass()].
The respective Hernquist implementation provides
\begin{equation}
I_{x^2\hat\rho}(x) = \frac{x^2}{2(1 + x)^2}.
\end{equation}
The Hernquist dimensionless surface mass density [virtual function nc_halo_density_profile_eval_dl_2d_density()] is
\begin{equation}
\hat{\Sigma}(X) = \frac{1}{\left( X^2 - 1 \right)^2} \left[\frac{\left(2 + X^2 \right) \arctan \left(\sqrt{X^2 - 1} \right)}{\sqrt{\left( X^2 - 1 \right)}} - 3\right].
\end{equation}
For $X^2 - 1 < 0$ the equation above can be written in terms of $\mathrm{arctanh}(\sqrt{\vert X^2 - 1 \vert})$.
If $\vert X - 1 \vert < 10^{-6}$ or $X < 10^{-6}$, $\hat{\Sigma} (X)$ is computed using
the Taylor series expansion at $1$ or $0$ respectively (with sufficient terms in order to obtain double precision).
The Hernquist enclosed mass is [virtual function nc_halo_density_profile_eval_dl_cyl_mass()]
\begin{equation}
\hat{\overline{\Sigma}} (< X) = X^2 / (X^2 - 1) * \eft[1 - 2 * \arctan \left(\sqrt{\vert X - 1 \vert / (X + 1)\right)\right];,
\end{equation}
Similar expressions in terms of $\mathrm{arctanh}$ and approximations, as described above, are used here.
References: , arxiv:1712.04512.
NcHaloDensityProfileHernquist * nc_halo_density_profile_hernquist_new (const NcHaloDensityProfileMassDef mdef,const gdouble Delta);
This function returns the NcHaloDensityProfileHernquist implementation of
NcHaloDensityProfile setting “mass-def” to mdef
and “Delta” to Delta
.