The scattering of Dirac spinors in rotating spheroids
Gao Zhi Fu, Chen Ci Xing, Wang Na
Abstract
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, $$\sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>σ</mml:mi></mml:math> , and stellar matter density, $$\mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>μ</mml:mi></mml:math> . It is found that the sensitivity of $$\sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>σ</mml:mi></mml:math> to the change in $$\mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>μ</mml:mi></mml:math> is proportional to the density $$\mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>μ</mml:mi></mml:math> . 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 $$10^{5}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mn>10</mml:mn><mml:mn>5</mml:mn></mml:msup></mml:math> – $$10^{6}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mn>10</mml:mn><mml:mn>6</mml:mn></mml:msup></mml:math> g cm $$^{-3}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow/><mml:mrow><mml:mo>-</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:math> , corresponding to a range of mass of about $$0.21{-}0.61 M_{\bigodot }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>0.21</mml:mn><mml:mo>-</mml:mo><mml:mn>0.61</mml:mn><mml:msub><mml:mi>M</mml:mi><mml:mo>⨀</mml:mo></mml:msub></mml:mrow></mml:math> and a range of radius of about $$6000{-}10,000$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>6000</mml:mn><mml:mo>-</mml:mo><mml:mn>10</mml:mn><mml:mo>,</mml:mo><mml:mn>000</mml:mn></mml:mrow></mml:math> km.