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Keywords:

  • accretion, accretion discs;
  • black hole physics;
  • gravitation;
  • hydrodynamics;
  • MHD

ABSTRACT

We present an analytical solution for thin-disc accretion on to a Kerr black hole that extends the standard Novikov–Thorne α-disc in three ways: (i) it incorporates non-zero stresses at the inner edge of the disc; (ii) it extends into the plunging region; and (iii) it uses a corrected vertical gravity formula. The free parameters of the model are unchanged. Non-zero boundary stresses are included by replacing the Novikov–Thorne no-torque boundary condition with the less strict requirement that the fluid velocity at the innermost stable circular orbit is the sound speed, which numerical models show to be the correct behaviour for luminosities below ∼30 per cent Eddington. We assume the disc is thin so we can ignore advection. Boundary stresses scale as αh and advection terms scale as h2 [where h is the disc opening angle (h=H/r)], so the model is self-consistent when h < α. We compare our solution with slim-disc models and general relativistic magnetohydrodynamic disc simulations. The model may improve the accuracy of black hole spin measurements.