Power Law (Salpeter 1955)
Form
Salpeter (1955) suggest a simple power law:
\[\xi(M) = \xi_0M^\alpha\]
with
\[\alpha = -2.35.\]
Implemented in
This is implemented in the PowerLawIMF class, where the user is free to pick whatever slope they like or use the default (same as above).
Integrals
Total number of stars
\[\begin{split}\begin{align}
\int^{M_\textrm{max}}_{M_\textrm{min}}\xi(M)\mathrm{d}M
&= \xi_0\int^{M_\textrm{max}}_{M_\textrm{min}}M^\alpha\mathrm{d}M \\
&= \xi_0\left[\frac{M^{\alpha+1}}{\alpha + 1}\right]^{M_\textrm{max}}_{M_\textrm{min}} \\
&= \xi_0\frac{M^{\alpha+1}_\textrm{max} - M^{\alpha+1}_\textrm{min}}{\alpha + 1}.
\end{align}\end{split}\]
Total mass of stars
\[\begin{split}\begin{align}
\int^{M_\textrm{max}}_{M_\textrm{min}}M\xi(M)\mathrm{d}M
&= \xi_0\int^{M_\textrm{max}}_{M_\textrm{min}}M^{\alpha+1}\mathrm{d}M \\
&= \xi_0\left[\frac{M^{\alpha+2}}{\alpha + 2}\right]^{M_\textrm{max}}_{M_\textrm{min}} \\
&= \xi_0\frac{M^{\alpha+2}_\textrm{max} - M^{\alpha+2}_\textrm{min}}{\alpha + 2}.
\end{align}\end{split}\]
These are only valid for \(\alpha \neq -1\) and \(\alpha \neq -2\), respectively. In either of these cases the integrals evaluate to
\[\xi_0\ln\left(\frac{M_\textrm{max}}{M_\textrm{min}}\right).\]