On the Polish doughnut accretion disk via the effective potential approach

We revisit the Polish doughnut model of accretion disks providing a comprehensive analytical description of the Polish doughnut structure. We describe a perfect fluid circularly orbiting around a Schwarzschild black hole, source of the gravitational field, by the effective potential approach for the exact gravitational and centrifugal effects. This analysis leads to a detailed, analytical description of the accretion disk, its toroidal surface, the thickness, the distance from the source. We determine the variation of these features with the effective potential and the fluid angular momentum. Many analytical formulas are given. In particular it turns out that the distance from the source of the inner surface of the torus increases with increasing fluid angular momentum but decreases with increasing energy function defined as the value of the effective potential for that momentum. The location of torus maximum thickness moves towards the external regions of the surface with increasing angular momentum, until it reaches a maximum an then decreases. Assuming a polytropic equation of state we investigate some specific cases.