The mass limit for Brown dwarfs is 0.08 solar masses. Initial mass fraction was described my Chabrier in 2003 as a combination of an exponential cut-off for lower masses and a power-law for higher masses:
\frac{dn_{star}}{d\ln m} = \begin{cases} \exp\left[ -\frac{(\log_{10} m - \log_{10} 0.22)^2}{2 \times 0.57^2} \right], m < 1 M_\odot\\ \exp\left[ -\frac{(-\log_{10} 0.22)^2}{2 \times 0.57^2} \right] m^{-1.35}, m \ge 1 M_\odot \end{cases}
Where n_star is the number of stars.
Calculate the mass fraction, a_BD, for brown dwarfs as well as the number fraction, n_BD, of brown dwarfs compared to the number of stars.
Hint: the mass of stars in an interval [a:b] is given as the integral from a to b of m with respect to n_star.
