Redshift of electromagnetic spectrum of radiation of the source that moves in the field of rotating black hole

In this paper, we obtained the equations of motion of a binary star in external gravitational field of a supermassive black hole. Using these equations, we developed a method for calculating the redshift of the spectrum of electromagnetic radiation, emitted by the source in a binary star that moves in external gravitational field of black hole, as a function of proper time of an earth observer. The case of a single point-like source is investigated firstly. Unlike widespread works, the proposed in this paper method uses solution of boundary value problem for isotropic geodesics. This problem is solved using the system of non-linear ordinary equations that are based on the analytical expressions of isotropic geodesics in Kerr metric. This gives possibility to find the impact parameters of isotropic geodesic more accurately. The validity of approximations that are used in the method is discussed in general and on the basis of numerical examples.Also the case of the source in a binary star that moves in external Kerr field is discussed. From the mathematical point of view the last problem can be considered as a certain generalization of the first problem.The proposed method is illustrated on the mathematical model of binary star with realistic parameters of the objects that are observed in the vicinity of the Center of our Galaxy. The obtained results are discussed.