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Abstract There are many stars that are rotating spheroids in the Universe, and studying them is of very important significance. Since the times of Newton, many astronomers and physicists have researched gravitational properties of stars by considering the moment equations derived from Eulerian hydrodynamic equations. In this paper we study the scattering of spinors of the Dirac equation, and in particular investigate the scattering issue in the limit case of rotating Maclaurin spheroids. Firstly we give the metric of a rotating ellipsoid star, then write the Dirac equation under this metric, and finally derive the scattering solution to the Dirac equation and establish a relation between differential scattering cross-section, $$\sigma $$ σ , and stellar matter density, $$\mu $$ μ . It is found that the sensitivity of $$\sigma $$ σ to the change in $$\mu $$ μ is proportional to the density $$\mu $$ μ . Because of the weak gravitational field and the constant mass density, our results are reasonable. The results can be applied to white dwarfs, main sequence stars, red giants, supergiant stars and so on, as long as their gravitational fields are so weak that they can be treated in the Newtonan approximations, and the fluid is assumed to be incompressible. Notice that we take the star’s matter density to be its average density and the star is not taken to be compact. Obviously our results cannot be used to study neutron stars and black holes. In particular, our results are suitable for white dwarfs, which have average densities of about $$10^{5}$$ 105 –$$10^{6}$$ 106 g cm$$^{-3}$$ -3 , corresponding to a range of mass of about $$0.21{-}0.61 M_{\bigodot }$$ 0.21-0.61M⨀ and a range of radius of about $$6000{-}10,000$$ 6000-10,000 km.

There are many rotating spheroids in the universe, and many astronomers and physicists have used theoretical methods to study the characteristics of stellar gravity since Newton's time. This paper derives the solutions of eight scattering states (ϕ(0),χ(0),ϕ(1),χ(1),ϕ(2)$$ \Big({\phi}^{(0)},{\chi}^{(0)},{\phi}^{(1)},{\chi}^{(1)},{\phi}^{(2)} $$,χ(2),ϕ(3)$$ {\chi}^{(2)},{\phi}^{(3)} $$, andχ(3))$$ {\chi}^{(3)}\Big) $$ for the Dirac equation with positive‐energy E=im$$ E= im $$, and establishes the relationship between the differential scattering cross section σi(p,θ,φ)$$ {\sigma}_i\left(p,\theta, \varphi \right) $$ and the stellar density μ$$ \mu $$. It is found that: (1) For the eight scattering states, their average scattering cross‐sections σi‾$$ \overline{\sigma_i} $$ are proportional to μ2$$ {\mu}^2 $$, and depend on the star's radius, and the higher the stellar density μ$$ \mu $$, the greater the sensitivity of σi‾$$ \overline{\sigma i} $$ to the change of μ$$ \mu $$; (2) For the four scattering states χ(i),i=0,1,2,3$$ {\chi}^{(i)},i=0,1,2,3 $$, their average scattering amplitudes f‾(p,θ)$$ \overline{f}\left(p,\theta \right) $$ and σ‾(p,θ)$$ \overline{\sigma}\left(p,\theta \right) $$ depend on the mass m$$ m $$ of the particles; while for the other four scattering states ϕ(i)$$ {\phi}^{(i)} $$, i=0,1,2,3$$ i=0,1,2,3 $$, then f‾$$ \overline{f} $$ and σ‾$$ \overline{\sigma} $$ are independent of m$$ m $$. This study links the gravitational characteristics of stars with the scattering cross section, creating a new method for studying the gravitational characteristics, which helps to reveal the mystery of the gravity of rotating ellipsoidal stars. [ABSTRACT FROM AUTHOR]