  Mass estimates of galaxy clusters from X-ray and Sunyeav-Zel'dovich
observations assume the intracluster gas is in hydrostatic equilibrium with
their gravitational potential. However, since galaxy clusters are dynamically
active objects whose dynamical states can deviate significantly from the
equilibrium configuration, the departure from the hydrostatic equilibrium
assumption is one of the largest sources of systematic uncertainties in cluster
cosmology. In the literature there has been two methods for computing the
hydrostatic mass bias based on the Euler and the modified Jeans equations,
respectively, and there has been some confusion about the validity of these two
methods. The word "Jeans" was a misnomer, which incorrectly implies that the
gas is collisionless. To avoid further confusion, we instead refer these
methods as "summation" and "averaging" methods respectively. In this work, we
show that these two methods for computing the hydrostatic mass bias are
equivalent by demonstrating that the equation used in the second method can be
derived from taking spatial averages of the Euler equation. Specifically, we
identify the correspondences of individual terms in these two methods
mathematically and show that these correspondences are valid to within a few
percent level using hydrodynamical simulations of galaxy cluster formation. In
addition, we compute the mass bias associated with the acceleration of gas and
show that its contribution is small in the virialized regions in the interior
of galaxy cluster, but becomes non-negligible in the cluster outskirts of
massive clusters where materials are being actively accreted. We discuss
implications for understanding the origin of hydrostatic mass biases and for
characterizing them using hydrodynamical simulations as well as observations.
