Calculating the mass of a galaxy bulge

[Jules Winnfield]

I'm looking for a usable method to calculate the bulge mass of, say, the Milky Way or Andromeda. Most of the literature I've read agree that a de Vaucouleurs profile is the way to go. However, I'm unable to get an exact solution using Mathematica. The general form of the integral is:$$\Sigma_b(r)=\Sigma_{be}exp\left(-\kappa(\frac{r}{a_b})^{\frac{1}{4}}-1 \right)$$$$M_b=2\pi\int_{0}^{\infty}r\Sigma_b(r) dr=22.665 a_b^2 \Sigma_{be}$$ $$\rho_b(r)=\int_r^\infty\frac{d\Sigma_b(x)}{dx}\frac{1}{\sqrt{x^2-r^2}}dx$$$$M_b(R)=4\pi\int_0^R\rho_b(r)r^2 dr$$Attempting these integrals in Mathematica leaves me with meaningless results. Has anyone got an 'industry standard' method for doing this calculation. I've been using Young's tables until now and I was hoping there was a better method than just a table lookup.

 

[Chronos]

This may provide a useful example; https://arxiv.org/abs/astro-ph/0301144, Disk-Bulge-Halo Models for the Andromeda Galaxy. Additional approaches are addressed here; https://arxiv.org/abs/1608.08350, Rotation and Mass in the Milky Way and Spiral Galaxies.