This paper explores the effect of the LMC on the mass estimates obtained from the timing argument. We show that accounting for the presence of the LMC systematically lowers the Local Group mass ($M_{rm LG}$) derived from the relative motion of the Mi
lky Way--Andromeda pair. Motivated by this result we apply a Bayesian technique devised by Pe~narrubia et al. (2014) to simultaneously fit (i) distances and velocities of galaxies within 3~Mpc and (ii) the relative motion between the Milky Way and Andromeda derived from HST observations, with the LMC mass ($M_{rm LMC}$) as a free parameter. Our analysis returns a Local Group mass $M_{rm LG}=2.64^{+0.42}_{-0.38}times 10^{12}M_odot$ at a 68% confidence level. The masses of the Milky Way, $M_{rm MW}=1.04_{-0.23}^{+0.26}times 10^{12}M_odot$, and Andromeda, $M_{rm M31}=1.33_{-0.33}^{+0.39}times 10^{12}M_odot$, are consistent with previous estimates that neglect the impact of the LMC on the observed Hubble flow. We find a (total) LMC mass $M_{rm LMC}=0.25_{-0.08}^{+0.09}times 10^{12}M_odot$, which is indicative of an extended dark matter halo and supports the scenario where this galaxy is just past its first pericentric approach. Consequently, these results suggest that the LMC may induce significant perturbations on the Galactic potential.
This paper uses dynamical invariants to describe the evolution of collisionless systems subject to time-dependent gravitational forces without resorting to maximum-entropy probabilities. We show that collisionless relaxation can be viewed as a specia
l type of diffusion process in the integral-of-motion space. In time-varying potentials with a fixed spatial symmetry the diffusion coefficients are closely related to virial quantities, such as the specific moment of inertia, the virial factor and the mean kinetic and potential energy of microcanonical particle ensembles. The non-equilibrium distribution function (DF) is found by convolving the initial DF with the Green function that solves Einsteins equation for freely diffusing particles. Such a convolution also yields a natural solution to the Fokker-Planck equations in the energy space. Our mathematical formalism can be generalized to potentials with a time-varying symmetry, where diffusion extends over multiple dimensions of the integral-of-motion space. The new probability theory is in many ways analogous to stochastic calculus, with two significant differences: (i) the equations of motion that govern the trajectories of particles are fully deterministic, and (ii) the diffusion coefficients can be derived self-consistently from microcanonical phase-space averages without relying on ergodicity assumptions. For illustration we follow the cold collapse of $N$-body models in a time-dependent logarithmic potential. Comparison between the analytical and numerical results shows excellent agreement in regions where the potential evolution does not depart too strongly from the adiabatic regime.
