Broken Power Law (Kroupa 2001)
Implemented in
This is implemented in the BrokenPowerLawIMF class, where the user is free to pick whatever parameters they like or use the default (same as above).
Integrals
As mentioned in power law integrals, there are special case values of \(\alpha\), that are not explicity mentioned here. I will also only consider the case \(M_\textrm{min} < M_\textrm{transition} < M_\textrm{max}\). If this is not the case, simply ignore the term from that branch.
Total number of stars
\[\begin{split}\begin{align*}
\int^{M_\textrm{max}}_{M_\textrm{min}} \xi(M)\mathrm{d}M
&= \int^{M_\textrm{transition}}_{M_\textrm{min}} \xi_0 M^{\alpha_1}\textrm{d}M + \int^{M_\textrm{max}}_{M_\textrm{transition}} \xi_0\xi_\textrm{continuity}M^{\alpha_2} \textrm{d}M \\
&= \left[\xi_0 \frac{M^{\alpha_1+1}}{\alpha_1+1}\right]^{M_\textrm{transition}}_{M_\textrm{min}} + \left[\xi_0\xi_\textrm{continuity}\frac{M^{\alpha_2+1}}{\alpha_2+1}\right]^{M_\textrm{max}}_{M_\textrm{transition}} \\
&= \xi_0 \frac{M_\textrm{transition}^{\alpha_1+1} - M_\textrm{min}^{\alpha_1+1}}{\alpha_1 + 1} +
\xi_0 \xi_\textrm{continuity} \frac{M_\textrm{max}^{\alpha_2+1} - M_\textrm{transition}^{\alpha_2+1}}{\alpha_2 + 1}.
\end{align*}\end{split}\]
Total mass of stars
\[\begin{split}\begin{align*}
\int^{M_\textrm{max}}_{M_\textrm{min}} M\xi(M)\mathrm{d}M
&= \int^{M_\textrm{transition}}_{M_\textrm{min}} \xi_0 M^{\alpha_1+1}\textrm{d}M + \int^{M_\textrm{max}}_{M_\textrm{transition}} \xi_0\xi_\textrm{continuity}M^{\alpha_2+1} \textrm{d}M \\
&= \left[\xi_0 \frac{M^{\alpha_1+2}}{\alpha_1+2}\right]^{M_\textrm{transition}}_{M_\textrm{min}} + \left[\xi_0\xi_\textrm{continuity}\frac{M^{\alpha_2+2}}{\alpha_2+2}\right]^{M_\textrm{max}}_{M_\textrm{transition}} \\
&= \xi_0 \frac{M_\textrm{transition}^{\alpha_1+2} - M_\textrm{min}^{\alpha_1+2}}{\alpha_1 + 2} +
\xi_0 \xi_\textrm{continuity} \frac{M_\textrm{max}^{\alpha_2+2} - M_\textrm{transition}^{\alpha_2+2}}{\alpha_2 + 2}.
\end{align*}\end{split}\]