  We experimentally determine the equation of state of a laser cooled gas. By
employing the Lane-Emden formalism, widely used in astrophysics, we derive the
equilibrium atomic profiles in large magneto optical traps where the
thermodynamic effects are cast in a polytropic equation of state. The effects
of multiple scattering of light are included, which results in a generalized
Lane-Emden equation for the atomic profiles. A detailed experimental
investigation reveals an excellent agreement with the model, with a two-fold
significance. In one hand, we can infer on the details of the equation of state
of the system, from an ideal gas to a correlated phase due to an effective
electrical charge for the atoms, which is accurately described by a
microscopical description of the effective electrostatic interaction. On the
other hand, we are able map the effects of multiple scattering onto directly
controllable experimental variables, which paves the way to subsequent
experimental investigations of this collective interaction.
