CHASSIS - Inverse Modelling of Relaxed Dynamical Systems

The state of a non-relativistic gravitational dynamical system is known at any time t if the dynamical rule, i.e. Newton’s equations of motion, can be solved; this requires specification of the gravitational potential. The evolution of a bunch of phase space coordinates w is deterministic, though generally non-linear. We discuss the novel Bayesian non-parametric algorithm CHASSIS that gives phase space pd f f (w) and potential Φ(x) of a relaxed gravitational system. CHASSIS is undemanding in terms of input requirements in that it is viable given incomplete, single-component velocity information of system members. Here x is the 3-D spatial coordinate and w = x + v where v is the 3-D velocity vector. CHASSIS works with a 2-integral f = f (E, L) where energy E = Φ+ v2/2, v2 = ∑3 i=1 v 2 i and the angular momentum is L = |r×v|, where r is the spherical spatial vector. Also, we assume spherical symmetry. CHASSIS obtains the f (·) from which the kinematic data is most likely to have been drawn, in the best choice forΦ(·), using an MCMC optimiser (Metropolis-Hastings). The likelihood function L is defined in terms of the projections of f (·) into the space of observables and the maximum in L is sought by the optimiser. The recovered solutions can be susceptible to large uncertainties given the dimensionality of the domain of the unconstrained f (·) and the typically small, observed velocity samples in distant astrophysical systems. This scenario is tackled by assuming f = f (E), i.e. we assume the phase space to be isotropic. However, this simplifying assumption of isotropy is addressed by undertaking a Bayesian test of significance that is developed to be used in the non-parametric context. A test based on the p-value estimates of the goodness of isotropy in the data was previously undertaken (Chakrabarty & Saha 2001). However, p-values are sensitive to sample sizes and obfuscate interpretation of analyses of differently sized kinematic samples. Thus, a Bayesian formalism is a better alternative, eg. Fully Bayesian Significance Test or FBST (Pereira & Stern 1999, Pereira, Stern & Wechsler 2008). The null hypothesis that we aim to test, is that the data are drawn from an isotropic f (·), i.e. H0 : f̂ = Ψ[E(v2/2+Φ(r))] where the data are drawn from f̂ and Ψ is some function: Ψ > 0 for E < 0 and Ψ = 0 otherwise. Within FBST, the evidence value (ev) in favour of H0 is obtained by first numerically spotting the most likely configuration (θ∗) that is compatible with H0 and then finding the volume of the tangential set T by numerical integration. Here T is the set of all configurations with posterior probability in excess of that of θ∗. We have developed the implementation of this scheme in the non-parametric context; in CHASSIS, the configurations are f (E) and Φ(r). To have configurations obeying H0, we perform sampling from the f (E) − Φ(r) pair identified upon convergence of a run of CHASSIS. From this sampling, the resulting f (E)−Φ(r) configuration corresponding to the highest L is compared to all the other f (E) −Φ(r) pairs, in order to obtain a measure of the volume of T . We discuss two distinct applications of the isotropic version of CHASSIS. In one, 2 distinct kinematic data sets of 2 distinct types of members of an example galaxy are analysed by CHASSIS under the assumption of isotropy. The f (·) and Φ(·) recovered from runs done with the two data sets are identified as distinct. Given that the same galaxy cannot be described by two different gravitational potentials, the risk involved in the very method of extracting the galactic potential from kinematic data of individual galactic members is demonstrated here for the first time. The goodness of the assumption of isotropy, given the 2 data sets, is quantified by our Bayesian test of hypothesis. In the second application, it is shown that once the amount of gravitational matter inside a fiduciary radius is pinned down from independent measurements, the recovered Φ(·) is unique, irrespective of the toy f (·), from which kinematic data samples are drawn as input for CHASSIS, as long as velocity dispersion values are measured at 1 or more different locations in the system. This consistent nature of Φ(·) is arrived at, notwithstanding varied forms for the assumed toy f (·), including isotropic as well as E & L dependent forms.