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 Burkert (1995):

\(\rho\left(r\,|\,r_{0},\rho_{0}\right)= \frac{\rho_0}{\left(1+\frac{r}{r_0}\right) \times \left[1+\left(\frac{r}{r_0}\right)^2\right]}\),

with \(r_{-2} \approx 1.5213797068 \times r_0\) and \(\rho_0 = \rho(r=0)\).
kEINASTO Navarro et al. (2004), Springel et al. (2008):

\(\rho\left(r\,|\,r_{-2},\rho_{-2};\alpha\right)=\rho_{-2}\, \exp\left\{-\frac{2}{\alpha}\left[\left(\frac{r}{r_{-2}}\right)^\alpha -1\right]\right\}\)
kEINASTO_N Merritt et al. (2006), Graham et al. (2006):

\(\rho\left(r\,|\,r_{\rm e},\rho_{\rm e};n\right) = \rho_{\rm e} \, \exp\left\{-d_n \times \left[ \left(\frac{r}{r_{\rm e}}\right)^{1/n} - 1 \right]\right\}\),

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 Hernquist (1990) and Zhao (1996):

\(\rho\left(r\,|\,r_{\rm s},\rho_{\rm s};\alpha,\beta,\gamma\right)=\frac{2^{\frac{\beta-\gamma}{\alpha}}\,\times\,\rho_{\rm s}}{\left(\frac{r}{r_{\rm s}}\right)^\gamma \times \left[1+\left(\frac{r}{r_{\rm s}}\right)^\alpha\right]^{\frac{\beta-\gamma}{\alpha}}}\) , 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 Gao et al. (2004), Madau et al. (2008):

\(\rho\left(r\,|\,r_{200};ac,\alpha,\beta\right) \propto \frac{ (1+ac) \left(\frac{r}{r_{200}}\right)^{\beta -3} \left[\beta + ac (\beta - \alpha) \left(\frac{r}{r_{200}}\right)^\alpha\right]}{\left[1+ ac \left(\frac{r}{r_{200}}\right)^\alpha\right]^2}\)

To use only to describe \({\rm d}{\cal P}_V/{\rm d}V\).
kDPDV_SPRINGEL08_ANTIBIASED Springel et al. (2008):

\(\rho\left(r\,|\,r_{\rm e},\rho_{\rm e};\alpha\right) = \left(\frac{r}{r_{\rm e}}\right) \times \rho^{\rm EINASTO}(r,r_{\rm e},\rho_{\rm e},\alpha)\),

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 Springel et al. (2008):

\(\frac{\overline\rho_{\rm sub}}{\overline\rho_{\rm tot}}\left(r\,|\,r_{50};\alpha,\beta,\gamma\right) = \exp \left[\gamma + \beta \ln\left(\frac{r}{r_{50}} \right) + 0.5 \alpha \ln^2\left(\frac{r}{r_{50}} \right)\right]\),

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 Ishiyama (2014): 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 \(\int_{0}^{R_{vir}} \frac{{\rm d}{\cal P}_V(r)}{{\rm d}V}\, {\rm d}^{3}\vec{r} = 1\).