Cylindrically symmetric spiralling and radial inviscid accretion in power-law and logarithmic potentials

We study cylindrically symmetric steady state accretion of polytropic matter spiralling onto the symmetry axis in power-law and logarithmic potentials. The model allows to qualitatively understand the accretion process in a symmetry different from that of the classical Bondi accretion. We study the integral curves as level lines of some Hamiltonian and apply this method also to Bondi accretion. The isothermal solutions in power-law potentials (as well as in any radius-dependent potential) can be expressed in exact form in terms of the Lambert W function, while in the case of logarithmic potential exact solutions can be found for any polytropic exponent.