Numerical computation of gravitational field of infinitely-thin axisymmetric disc with arbitrary surface mass density profile and its application to preliminary study of rotation curve of M33

We developed a numerical method to compute the gravitational field of an infinitely-thin axisymmetric disc with an arbitrary surface mass density profile. We evaluate the gravitational potential by a split quadrature using the double exponential rule and obtain the acceleration vector by numerically differentiating the potential by Ridders' algorithm. By using the new method, we show the rotation curves of some non-trivial discs: (i) truncated power-law discs, (ii) discs with a non-negligible center hole, (iii) truncated Mestel discs with edge-softening, (iv) double power-law discs, (v) exponentially-damped power-law discs, and (vi) an exponential disc with a sinusoidal modulation of the density profile. Also, we present a couple of model fittings to the observed rotation curve of M33: (i) the standard deconvolution by assuming a spherical distributin of the dark matter and (ii) a direct fit of infinitely-thin disc mass with a double power-law distribution of the surface mass density.