MAMPOSSt (Mamon, Biviano & Boué 2013) performs mass / orbit modeling by fitting the distribution of tracers in projected phase space (PPS: projected radius, line-of-sight velocity), assuming a mass profile, a velocity anisotropy profile and a 3D velocity distribution (here Gaussian). The algorithm performs very well on mock clusters (Mamon, Biviano & Boué 2013; Old et al. 2014), and in particular finished 2nd among 25 algorithms on determining M₂₀₀ of groups and clusters from a mock built by a semi-analytical model of galaxy formation (Old et al. 2015).

MAMPOSSt was run on the 8 non-tangential models of the default mock data suite. For each (6D) mock, the (2+1D) PPS was extracted for a distant observer placed along the z axis, using 3 random subsets of 1000 stars (among 10,000) and 10 random subsets of 100 stars. The outermost star considered was 5 times the radius of slope –2 of the true number density profile. MAMPOSSt was run in 3 increasing levels of difficulty:

Easy: The tracer number density profile is known. The shape of the velocity anisotropy profile is known, but for OM anisotropy, the radius of transition is a free parameter. The mass density profile is assumed to be a generalized NFW (where the inner slope, radius of slope –2 and normalization are free parameters). Hence, there are 3 to 4 free parameters depending on whether the system has isotropic or OM velocities, respectively.

Hard: Like Easy, but the velocity anisotropy is assumed to be of the Tiret+07 (with β₀=0, i.e. generalized Mamon-Lokas 2005b) form: β(r) = β_∞ r / (r+r_–2), where the transition radius is set to the radius of slope –2 of the number density profile. This anisotropy model increases more gradually from inner isotropy to more radial outer anisotropy than does the OM model, and is more consistent with the anisotropy profiles in ΛCDM halos. The number of free parameters is 4.

Very Hard: Like Hard, but the tracer number density profile is unknown, assumed to be a generalized Plummer profile where the scale radius and inner slope are free paramaters. The number of free parameters is thus 6.

Altgether, the total number of MAMPOSST runs was 8 x (3+10) x 3 = 312. The units used are kpc for scales, km/s for velocities, and M_Sun for masses. In all cases, the inner slope of the dark matter is allowed to vary from –2 to 0.

Presentation (30 Oct 2014): in http://www.iap.fr/users/gam/GAIACHALLENGE/Mamon.pdf

Diagnostic plots of results (for each of the 312 MAMPOSSt runs) are here.

These models are a development of http://arxiv.org/abs/1303.6099. The marginal likelihood $p(D|\Phi)$ is calculated by representing the DF by an arbitrary number of Gaussians of arbitrary weight/location/covariance in action space and marginalising the parameters describing the Gaussians. So, no assumption is made about the light profile (except that it is axisymmetric) or the orbit anisotropy. The implementation is for now still limited to error-free data though.

The results below assume the correct double power-law form for potential with scale radius held at correct value. Unknown parameters are then the inner slope $\gamma$ and scale densty $\rho_0$.

These use the standard “extended Schwarzschild” approach, but using big blocks of orbits in $(E,L^2)$ (or equiv actions) instead of single orbits. The models are fed $(X,Y,V_X,V_Y)$ from PlumCuspIso with 2% errors in $(V_X,V_Y)$ – and infinite uncertainty in $(Z,V_Z)$

Noise in result is probably due to finite resolution of block DF, plus the deficiency of the maximum-likelihood approach used by such “extended-Schwarzschild” methods. And there might be some implementation errors too.

No error:

With error:

Best fit mass and concetration for the 3D triaxial cusp model:

For this case, mass seems to be over-estimated compared with the true value

<latex>

  $1.171\times 10^9 M_\odot$.

</latex>

Discrete framework from Watkins et al. (2013). Models are spherical JAM Cappellari 2008 using LOS and PMs.

Runs with 10000/1000/100 stars.

Three levels:

• “easy_free” : Easy: 3 free parameters: rho0, r_dark, gamma_dark
• “aniso_free” : Medium: 4 free parameters: rho0, r_dark, gamma_dark, r_aniso
• “hard” : Hard: 6 free parameters: rho0, r_dark, gamma_dark, r_aniso, r_star, gamma_star

Some bonus runs/tests… “easy_core” (fit for rho0 and r_dark, assuming gamma_dark=0); “easy_cusp” (fit for rho0 and r_dark, assuming gamma_dark=1); “aniso” (fir for r_aniso only).

all results on dropbox