  We open the paper with introductory considerations describing the motivations
of our long-term research plan targeting gravitomagnetism, illustrating the
fluid-dynamics numerical test case selected for that purpose, that is, a
perfect-gas sphere contained in a solid shell located in empty space
sufficiently away from other masses, and defining the main objective of this
study: the determination of the gravitofluid-static field required as initial
field ($t=0$) in forthcoming fluid-dynamics calculations. The determination of
the gravitofluid-static field requires the solution of the isothermal-sphere
Lane-Emden equation. We do not follow the habitual approach of the literature
based on the prescription of the central density as boundary condition; we
impose the gravitational field at the solid-shell internal wall. As the
discourse develops, we point out differences and similarities between the
literature's and our approach. We show that the nondimensional formulation of
the problem hinges on a unique physical characteristic number that we call
gravitational number because it gauges the self-gravity effects on the gas'
fluid statics. We illustrate and discuss numerical results; some peculiarities,
such as gravitational-number upper bound and multiple solutions, lead us to
investigate the thermodynamics of the physical system, particularly entropy and
energy, and preliminarily explore whether or not thermodynamic-stability
reasons could provide justification for either selection or exclusion of
multiple solutions. We close the paper with a summary of the present study in
which we draw conclusions and describe future work.
