G. A. Mamon, A. Biviano, __G. Boué__

**Abstract**

Mass modelling of spherical systems through internal kinematics is hampered by the mass–velocity anisotropy degeneracy inherent in the Jeans equation, as well as the lack of techniques that are both fast and adaptable to realistic systems. A new fast method, called Modelling Anisotropy and Mass Profiles of Observed Spherical Systems (MAMPOSSt), is developed and thoroughly tested. MAMPOSSt performs a maximum-likelihood fit of the distribution of observed tracers in projected phase space (projected radius and line-of-sight velocity). As in other methods, MAMPOSSt assumes a shape for the gravitational potential (or equivalently the total mass profile). However, instead of postulating a shape for the distribution function in terms of energy and angular momentum, or supposing Gaussian line-of-sight velocity distributions, MAMPOSSt assumes a velocity anisotropy profile and a shape for the 3D velocity distribution. The formalism is presented for the case of a Gaussian 3D velocity distribution. In contrast to most methods based on moments, MAMPOSSt requires no binning, differentiation, nor extrapolation of the observables. Tests on cluster-mass haloes from ?CDM dissipationless cosmological simulations indicate that, with 500 tracers, MAMPOSSt is able to jointly recover the virial radius, tracer scale radius, dark matter scale radius and outer or constant velocity anisotropy with small bias (<10% on scale radii and <2% on the two other quantities) and inefficiencies of 10%, 27%, 48% and 20%, respectively. MAMPOSSt does not perform better when some parameters are frozen, and even particularly worse when the virial radius is set to its true value, which appears to be the consequence of halo triaxiality. The accuracy of MAMPOSSt depends weakly on the adopted interloper removal scheme, including an efficient iterative Bayesian scheme that we introduce here, which can directly obtain the virial radius with as good precision as MAMPOSSt. Additional tests are made on the number of tracers, the stacking of haloes, the chosen aperture, and the density and velocity anisotropy models. Our tests show that MAMPOSSt with Gaussian 3D velocities is very competitive with other methods that are either currently restricted to constant velocity anisotropy or 3 orders of magnitude slower. These tests suggest that MAMPOSSt can be a very powerful and rapid method for the mass and anisotropy modelling of systems such as clusters and groups of galaxies, elliptical and dwarf spheroidal galaxies.

**Keywords**

methods: analytical - galaxies: clusters: general - galaxies: haloes - galaxies: kinematics and dynamics - dark matter

**Monthly Notices of the Royal Astronomical Society**

Volume 429, Issue 4, Page 3079

2013 March