# 10.2. Profiles $$\rho_{\rm DM}$$ and $${\rm d}{\cal P}_V/{\rm d}V$$¶

Possible keyword choices for gXXX_FLAG_PROFILE variables:

 Keyword Description kHOST Enforce the DM subhalo profile (or spatial distribution of haloes in the host) matching the host DM density distribution. kBURKERT $$\rho\left(r\,|\,r_{0},\rho_{0}\right)= \frac{\rho_0}{\left(1+\frac{r}{r_0}\right) \times \left[1+\left(\frac{r}{r_0}\right)^2\right]}$$, with $$r_{-2} \approx 1.5213797068 \times r_0$$ and $$\rho_0 = \rho(r=0)$$. kEINASTO $$\rho\left(r\,|\,r_{-2},\rho_{-2};\alpha\right)=\rho_{-2}\, \exp\left\{-\frac{2}{\alpha}\left[\left(\frac{r}{r_{-2}}\right)^\alpha -1\right]\right\}$$ kEINASTO_N $$\rho\left(r\,|\,r_{\rm e},\rho_{\rm e};n\right) = \rho_{\rm e} \, \exp\left\{-d_n \times \left[ \left(\frac{r}{r_{\rm e}}\right)^{1/n} - 1 \right]\right\}$$, with $$d_n\approx 3n-\frac{1}{3}+\frac{0.0079}{n}$$ (see Merritt et al. (2006)) and $$r_{-2} = r_{\rm e} \times \left(\frac{2n}{d_n}\right)^n$$ kZHAO $$\rho\left(r\,|\,r_{\rm s},\rho_{\rm s};\alpha,\beta,\gamma\right)=\frac{2^{\frac{\beta-\gamma}{\alpha}}\,\times\,\rho_{\rm s}}{\left(\frac{r}{r_{\rm s}}\right)^\gamma \times \left[1+\left(\frac{r}{r_{\rm s}}\right)^\alpha\right]^{\frac{\beta-\gamma}{\alpha}}}$$ , with $$r_{-2} = r_{\rm s} \times \left( \frac{\beta-2}{2-\gamma} \right)^{-{1}/{\alpha}}$$. Note that we use the description where $$\rho_{\rm s} = \rho(r_{\rm s})$$. kDPDV_GAO04 $$\rho\left(r\,|\,r_{200};ac,\alpha,\beta\right) \propto \frac{ (1+ac) \left(\frac{r}{r_{200}}\right)^{\beta -3} \left[\beta + ac (\beta - \alpha) \left(\frac{r}{r_{200}}\right)^\alpha\right]}{\left[1+ ac \left(\frac{r}{r_{200}}\right)^\alpha\right]^2}$$ To use only to describe $${\rm d}{\cal P}_V/{\rm d}V$$. kDPDV_SPRINGEL08_ANTIBIASED $$\rho\left(r\,|\,r_{\rm e},\rho_{\rm e};\alpha\right) = \left(\frac{r}{r_{\rm e}}\right) \times \rho^{\rm EINASTO}(r,r_{\rm e},\rho_{\rm e},\alpha)$$, with $$r_{-2} = \left(\frac{3}{2}\right)^{1/\alpha} r_{\rm e}$$. To use only to describe $${\rm d}{\cal P}_V/{\rm d}V$$. kDPDV_SPRINGEL08_FIT $$\frac{\overline\rho_{\rm sub}}{\overline\rho_{\rm tot}}\left(r\,|\,r_{50};\alpha,\beta,\gamma\right) = \exp \left[\gamma + \beta \ln\left(\frac{r}{r_{50}} \right) + 0.5 \alpha \ln^2\left(\frac{r}{r_{50}} \right)\right]$$, Springel et al. (2008) use $$\alpha = 0.36$$, $$\beta = 0.87$$, $$\gamma = 1.31$$. To use only to describe $${\rm d}{\cal P}_V/{\rm d}V$$. kISHIYAMA14 NFW profile with a mass dependent inner slope $$\gamma=-0.123\log(M_{\rm vir}/10^{-6}M_\odot)+1.461$$ if $$\alpha>1$$, or set to 1 otherwise. To use only to describe (subhalo) density profiles.

Note

All above profiles are spherically symmetric as a function of the radial coordinate $$r$$.
Additionally, they depend on a characteristic scale length, a density normalisation [1], and up to three dimensionless shape parameters. In the above notation,

$\rho = \rho\left(\,r\,|\,\rm scale\, length,\,density\, normalisation;\,shape\, parameters\right).$
• A triaxial distortion can be applied to all profiles as explained in Section 6.5.4.

• For the Milky Way halo, the scale length and density normalisation are set via the Sun’s distance from the Galactic centre and the local DM density,

$\tt{gMW\_RSOL} \quad \sf{and} \quad \tt{gMW\_RHOSOL}.$
• User-defined extragalactic or dSph haloes are defined in a definition file (as, e.g., the template file \$CLUMPY/data/list_generic.txt), where the scale length and density normalisation are specified as $$\tt rs$$ and $$\tt rhos$$, respectively.

• The shape parameters are controlled via the

$\tt gXXX\_SHAPE\_PARAMS\_0,\, gXXX\_SHAPE\_PARAMS\_1,\, gXXX\_SHAPE\_PARAMS\_2$

input parameters (when parsed via the command line or the global parameter file) or as #1, #2, #3 (in a halo definition file) according to the order as above.

• The density normalisation and scale radius can also be obtained by providing the total halo mass $$M_{\Delta}$$ of a halo with the corresponding overdensity definition $${\Delta}$$ and a concentration-mass relation $$c_\Delta(M_\Delta)$$. In CLUMPY, the density profiles of all subhaloes are defined this way.

• Using a profile as $${\rm d}{\cal P}_V/{\rm d}V$$ function (see Section 6.5.5.4) does not need a density normalisation to be provided, as the probability distribution is internally normalised to one. The scale radii of $${\rm d}{\cal P}_V/{\rm d}V$$ profiles are defined by the user relative to the host halo scale radius, via the

$\tt gXXX\_SUBS\_DPDV\_RSCALE\_TO\_RS\_HOST$

input variables.

 [1] Despite the profiles only used to describe $${\rm d}{\cal P}_V/{\rm d}V$$, where the normalisation is automatically chosen such that $$\int_{0}^{R_{vir}} \frac{{\rm d}{\cal P}_V(r)}{{\rm d}V}\, {\rm d}^{3}\vec{r} = 1$$.