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An analytical study on the multi-critical behaviour and related bifurcation phenomena for relativistic black hole accretion

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Abstract

We apply the theory of algebraic polynomials to analytically study the transonic properties of general relativistic hydrodynamic axisymmetric accretion onto non-rotating astrophysical black holes. For such accretion phenomena, the conserved specific energy of the flow, which turns out to be one of the two first integrals of motion in the system studied, can be expressed as a 8th degree polynomial of the critical point of the flow configuration. We then construct the corresponding Sturm’s chain algorithm to calculate the number of real roots lying within the astrophysically relevant domain of \({\mathbb{R}}\). This allows, for the first time in literature, to analytically find out the maximum number of physically acceptable solution an accretion flow with certain geometric configuration, space-time metric, and equation of state can have, and thus to investigate its multi-critical properties completely analytically, for accretion flow in which the location of the critical points can not be computed without taking recourse to the numerical scheme. This work can further be generalized to analytically calculate the maximal number of equilibrium points certain autonomous dynamical system can have in general. We also demonstrate how the transition from a mono-critical to multi-critical (or vice versa) flow configuration can be realized through the saddle-centre bifurcation phenomena using certain techniques of the catastrophe theory.

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Authors and Affiliations

  1. Faculty of Science, Banaras Hindu University, Varanasi, 221005, India

    Shilpi Agarwal

  2. Harish Chandra Research Institute, Allahabad, 211 019, India

    Tapas K. Das & Rukmini Dey

  3. Sarojini Naidu College for Women, Kolkata, 700028, India

    Sankhasubhra Nag

Authors
  1. Shilpi Agarwal
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  2. Tapas K. Das
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  3. Rukmini Dey
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  4. Sankhasubhra Nag
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Correspondence to Sankhasubhra Nag.

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Agarwal, S., Das, T.K., Dey, R. et al. An analytical study on the multi-critical behaviour and related bifurcation phenomena for relativistic black hole accretion. Gen Relativ Gravit 44, 1637–1655 (2012). https://doi.org/10.1007/s10714-012-1358-z

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  • DOI: https://doi.org/10.1007/s10714-012-1358-z

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