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Mass modelling of spherical systems through internal motions is hampered bythe mass/velocity anisotropy (VA) degeneracy inherent in the Jeans equation, aswell as the lack of techniques that are both fast and adaptable to realisticsystems. A new fast method, called MAMPOSSt, which performs a maximumlikelihood fit of the distribution of observed tracers in projected phasespace, is developed and thoroughly tested. MAMPOSSt assumes a shape for thegravitational potential, but instead of postulating a shape for thedistribution function in terms of energy and angular momentum, or supposingGaussian line-of-sight velocity distributions, MAMPOSSt assumes a VA profileand a shape for the 3D velocity distribution, here Gaussian. MAMPOSSt requiresno binning, differentiation, nor extrapolation of the observables. Tests oncluster-mass haloes from LambdaCDM cosmological simulations show that, with 500tracers, MAMPOSSt is able to jointly recover the virial radius, tracer scaleradius, dark matter scale radius and outer or constant VA 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 someparameters are frozen, and even worse when the virial radius is set to its truevalue, which appears to be the consequence of halo triaxiality. The accuracy ofMAMPOSSt depends weakly on the adopted interloper removal scheme, including anefficient iterative Bayesian scheme that we introduce here, which can directlyobtain the virial radius with as good precision as MAMPOSSt. Our tests showthat MAMPOSSt with Gaussian 3D velocities is very competitive with, and up to1000x faster than other methods. Hence, MAMPOSSt is a very powerful and rapidtool for the mass and anisotropy modeling of systems such as clusters andgroups of galaxies, elliptical and dwarf spheroidal galaxies.

MAMPOSSt: Modelling Anisotropy and Mass Profiles of Observed Spherical Systems. I. Gaussian 3D velocities