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We study the stability of three analytical solutions of the Einstein's fieldequations for spheres of fluid. These solutions are suitable to describecompact objects including white dwarfs, neutron stars and supermassive starsand they have been extensively employed in the literature. We re-examine therange of stability of the Tolman VII solution, we focus on the stability of theBuchdahl solution which is under contradiction in the literature and we examinethe stability of the Nariai IV solution. We found that all the mentionedsolutions are stable in an extensive range of the compactness parameter. Wealso concentrate on the effect of the adiabatic index on the instabilitycondition. We found that the critical adiabatic index, depends linearly on theratio of central pressure over central energy density $P_c/{\cal E}_c$, up tohigh values of the compactness. Finally, we examine the possibility to imposeconstraints, via the adiabatic index, on realistic equations of state in orderto ensure stable configurations of compact objects.

The stability of relativistic stars and the role of the adiabatic index