Fig. 6.11 The solution of the Lane-Emden equation for the polytropic index n = 3. The first zero is at z0 6.8

6.5 Conclusive Remarks on Stellar Equilibrium

Differently from Newtonian theory, Einstein theory foresees a critical lower limit for the ratio between the radius and the mass of a star encoded in (6.3.67). Below such a limit no pressure can sustain the star into equilibrium and gravitational collapse continues up to the formation of a black-hole. Just above that limit there are two equilibrium states for exhausted stars, where gravitational attraction is balanced by a quantum phenomenon, namely the degeneracy pressure of a Fermi gas: the state of White Dwarf and that of Neutron Star. We can therefore conclude that, depending on the initial size of a normal star, there are three possible destinies at the end of its life-cycle. Supermassive stars end their life as black-holes, massive stars as Neutron Stars and medium size ones as White Dwarfs.

The existence of an upper mass-limit for Neutron-Stars and White-Dwarfs which is determined purely in terms of fundamental constants of Nature is of the utmost theoretical relevance. For instance, as we shall discuss in later chapters on Cosmology, supernovae of type Ia, which explode when a white-dwarf member of a binary system reaches the Chandrasekhar mass-limit by sucking material from the companion star, constitute a unique instance of very luminous and precise standard candles that played an essential role in measuring cosmological parameters opening up, at the beginning of the XXI century, entirely new perspectives on our understanding of the physical Universe.

Altogether Man’s understanding of stellar equilibrium, which is so essential to frame our present picture of the world and of its evolution, was drastically changed and not only marginally corrected in the first decades of the XX century by the inputs of General Relativity and of Quantum Mechanics. This showed that the typical stellar mass M, is not a randomly chosen number but it is explained by the fundamental Laws of Nature encoded in the values of fundamental constants.

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