  Physical conditions deep within planets and exoplanets have yet to be
measured directly, but indirect methods can calculate them. The polytropic
models are one possible solution to this problem. In the present paper, we
assume that the interiors of planets follow a polytropic equation of state.
Hydrostatic equilibrium conditions are used to determine the overall structural
properties of the constituent matter. In the frame of the conformable
fractional derivatives, we use polytropic gas spheres to model the density
profiles, pressure profiles, temperature distributions, and the mass-radius
relations for the interiors of the initial stage of exoplanets. Planets of
single chemical composition were used to study the behavior of the mass-radius
relation, pressure distributions, and temperature distribution variation with
the fractional parameter. We calculated 72 fractional models for the mass of
protoplanets of 1MJ, 3MJ, and 10MJ (MJ is the mass of Jupiter), and the values
of the polytropic index are n=0, 0.5, 1, 1.5, and the fractional parameter rang
0.75-1.
