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Spiral Galaxies - classical description of spiral arms and rotational velocity pattern - toy model

We propose an explanation of features of spiral galaxies: spiral arms and observed flat rotation curves, without the presence of an exotic form of matter. The formalism is based on Boltzmanns transport equation for the collisional matter and the very-low-velocity post-Newtonian approximation of the general relativity equations expressed in the Maxwell-like form. The Maxwell-like formulation provides the base for the explanation of the above phenomena in the language of dynamically created gravitoelectromagnetic fields by the movement of mass streams in the plane of the galaxy disc. In the model we use radical simplifications expressed as neglect of the gravitational interaction between neighbors and approximation of the incompressible mass flow. In this frame we show that if the galaxy disc is fuelled constantly by collisional mass carriers, then the amplification of the gravitomagnetic field can be large enough to create the rotational velocity pattern and spiral arms. In this framework the collisional part of the mass gas in the galaxy disc plane, i.e. molecules and atoms, is responsible for the creation of the gravitomagnetic field. According to the model the spiral pattern of arms is static and determined by a direction of the mass flow. The model reproduces qualitatively the observed spiral arms and reproduces well the shape of the rotational velocity pattern. As an example of the usability of the proposed mechanism, we reproduce qualitatively the above features for the IC 342 and NGC 4321 galaxies.