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• W69464779 abstract "We apply the theory of algebraic polynomials to analytically investigate the transonic properties of hydrodynamic accretion onto non-rotating astrophysical black holes. We first construct the equation describing the space gradient of the dynamical flow velocity of accreting matter. Such equation is isomorphic to a first order autonomous dynam ical system. Application of the fixed point condition enables us to construct an nth degree algebraic equation for the space variable along which the flow streamlines are defined to posse ss certain first integrals of motion. The constant coefficients for each term in that equatio n are functions of certain specified initial boundary conditions. Such initial boundary condit ions span over a certain domain on the real line R ‐ effectively, as individual sub-domain of R×R (spherical flow) and R×R×R (accretion disc) for the polytropic accretion, and of R (spherical flow) and R×R (accretion disc) for isothermal accretion. The solution of aforesaid e quation would then provide the critical (and consequently, the sonic) point rc. The critical points itself are permissible only within a certain open interval ]rg, L!1[, where rg is the radius of the event horizon and L!1 is the physically acceptable maximally allowed limit on the value of a critical point. For polynomials of degree n > 4, analytical solutions are not available. We use the Sturm’s theorem (a corollary of Sylvester’s theorem), to co nstruct the Sturm’s chain algorithm, which can be used to calculate the number of real roots (lying within a certain subdomain of R) for a polynomial of any countably finite arbitrarily large i ntegral n, subjected to certain sub-domains of constant co-efficients. The problem now reduces to identify the polynomials in rc with the Sturm’s sequence, and to find out the maximum number o f physically acceptable solution an accretion flow with certain geometri c configuration, space-time metric, and equation of state can have, and thus to investigate its multi-critical properties completely analytically, where the polynomials in rc are of n > 4 (for complete general relativistic axisymmetric flow, for example, where n = 14), and thus, for which the critical points can not be computed analytically. Our work, as we believe, has significant importance, because for the first time in the literature, we provide a purely analytical method, by applying certain powerful theorem of algebraic polynomials in pure mathematics, to check whether certain astrophysical hydrodynamic accretion may undergo more than one sonic transitions. Our work can be generalized to analytically calculate the maximal number of equilibrium points certain autonomous dynamical system can have in general." @default.
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• W69464779 date "2009-07-27" @default.
• W69464779 modified "2023-09-27" @default.
• W69464779 title "Transonicity in black hole accretion -- A mathematical study using the generalized Sturm chains" @default.
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