The fractional white dwarf hydrodynamical nonlinear differential equation and emergence of quark stars

In recent years, considerable interest has been stimulated by many applications of fractional calculus in astrophysics. Motivated by recent advances of the statistical mechanical description of degenerate matter gas and fractional statistical physics, we discussed the fractional formulation of the white dwarf stellar dynamical problem. Our approach is based on the familiar definition of the Riemann-Liouville fractional integral operator of order 0 < α < 1. After deriving the fractional equation of state in D-dimensions, we focused on the three-dimensional case and we derive the fractional Chandrasekhar or Lane-Emden non-linear differential equation (LENDE) by discussing the hydrostatic equilibrium. It was observed that the equation of states for both the non-relativistic and relativistic degenerate gas are strongly influenced by the fractional parameter α. Besides, for the ultra-relativistic case, it was observed the non-existence of a unique mass for relativistic white dwarfs and hence the Chandrasekhar mass law which states that “there exist a unique mass for relativistic white dwarfs, above which hydrostatic equilibrium cannot be maintained and the stars starts to collapse” is violated. This violation may be realized by hypothetical quark stars from non-perturbative QCD. Additional consequences are discussed in some details.