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Register a new userKinematic deprojection and mass inversion of spherical systems of known velocity anisotropy
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Traditionally, the mass / velocity anisotropy degeneracy (MAD) inherent in the spherical, stationary, non-streaming Jeans equation has been handled by assuming a mass profile and fitting models to the observed kinematical data. Here, the opposite approach is considered: the equation of anisotropic kinematic projection is inverted for known arbitrary anisotropy to yield the space radial velocity dispersion profile in terms of an integral involving the radial profiles of anisotropy and isotropic dynamical pressure. Then, through the Jeans equation, the mass profile is derived in terms of double integrals of observable quantities. Single integral formulas for both deprojection and mass inversion are provided for several simple anisotropy models (isotropic, radial, circular, general constant, Osipkov-Merritt, Mamon-Lokas and Diemand-Moore-Stadel). Tests of the mass inversion on NFW models with these anisotropy models yield accurate results in the case of perfect observational data, and typically better than 70% (in 4 cases out of 5) accurate mass profiles for the sampling errors expected from current observational data on clusters of galaxies. For the NFW model with mildly increasing radial anisotropy, the mass is found to be insensitive to the adopted anisotropy profile at 7 scale radii and to the adopted anisotropy radius at 3 scale radii. This anisotropic mass inversion method is a useful complementary tool to analyze the mass and anisotropy profiles of spherical systems. It provides the practical means to lift the MAD in quasi-spherical systems such as globular clusters, round dwarf spheroidal and elliptical galaxies, as well as groups and clusters of galaxies, when the anisotropy of the tracer is expected to be linearly related to the slope of its density.
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Mass modelling of spherical systems through internal motions is hampered by the mass/velocity anisotropy (VA) 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 MAMPOSSt, which performs a maximum likelihood fit of the distribution of observed tracers in projected phase space, is developed and thoroughly tested. MAMPOSSt assumes a shape for the gravitational potential, but 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 VA profile and a shape for the 3D velocity distribution, here Gaussian. MAMPOSSt requires no binning, differentiation, nor extrapolation of the observables. Tests on cluster-mass haloes from LambdaCDM cosmological simulations show that, with 500 tracers, MAMPOSSt is able to jointly recover the virial radius, tracer scale radius, 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 some parameters are frozen, and even 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. Our tests show that MAMPOSSt with Gaussian 3D velocities is very competitive with, and up to 1000x faster than other methods. Hence, MAMPOSSt is a very powerful and rapid tool for the mass and anisotropy modeling of systems such as clusters and groups of galaxies, elliptical and dwarf spheroidal galaxies.
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