L3 (Maschberger 2013)

Form

Maschberger (2013) suggest an IMF that can capture the high and low mass behaviour we observe, while being continuous, and suggest the following form

\[\xi(M) = \xi_0 \left(\frac{M}{\mu}\right)^{-\alpha}\left(1 + \left(\frac{M}{\mu}\right)^{1-\alpha}\right)^{-\beta}\]

With the values

\[\begin{split}\begin{gather*}\alpha=2.3\\ \beta=1.4\\ \mu=0.2\end{gather*}\end{split}\]

to mimic a Kroupa (2001) or Chabrier (2003) IMF.

Note

We typically define power laws like \(M^\alpha\), with \(\alpha < 0\), as opposed to having \(\alpha > 0\) and \(M^{-\alpha}\). This is an exception, as it would potentially be more confusing to switch all the signs compared to the original paper.

Implemented in

This is implemented in the L3IMF class, where the user is free to pick whatever parameters they like or use the default (same as above).

Integrals

The integrals of interest are provided in the original paper, Maschberger (2013), modulo some manipulation. As such we do not derive the integrals from first principles, like other IMFs, but rather describe how to convert them into a form we are interested in.

For this we will use the auxillary function:

\[G(M) = \left( 1 + \left( \frac{M}{\mu} \right)^{1-\alpha} \right)^{1-\beta}.\]

It is informative to first take the derivative

\[\frac{\mathrm{d}G}{\mathrm{d}M} = \frac{(1-\alpha)(1-\beta)}{\mu}\left(\frac{M}{\mu}\right)^{-\alpha}\left(1 + \left(\frac{M}{\mu}\right)^{1-\alpha}\right)^{-\beta}\]

and note the relation to the IMF

\[\xi(M) = \xi_0\frac{\mu}{(1-\alpha)(1-\beta)}\frac{\mathrm{d}G}{\mathrm{d}M}.\]

Total number of stars

For this integral we use the form of the IMF directly above. See also eqn 21 of Maschberger (2013).

\[\begin{split}\begin{align} \int^{M_\textrm{max}}_{M_\textrm{min}}\xi(M)\mathrm{d}M &= \xi_0\frac{\mu}{(1-\alpha)(1-\beta)}\int^{M_\textrm{max}}_{M_\textrm{min}}\frac{\mathrm{d}G}{\mathrm{d}M}\mathrm{d}M \\ &= \xi_0\frac{\mu}{(1-\alpha)(1-\beta)}\left[G(M)\right]^{M_\textrm{max}}_{M_\textrm{min}} \\ &= \xi_0\frac{\mu}{(1-\alpha)(1-\beta)}\left\{G(M_\textrm{max}) - G(M_\textrm{min})\right\}. \end{align}\end{split}\]

Total mass of stars

For this integral we use the parameterisation in eqns 24–25 of Maschberger (2013). This requires the incomplete beta function. Under the hood we will use scipy.special.beta and scipy.special.betainc. Define the incomplete beta function \(B(a, b, t)\) as it is by scipy:

\[ B(a, b, t) = \frac{1}{\mathtt{beta}(a, b)}\int_0^t x^{a-1}(1-x)^{b-1}\mathrm{d}x, \]

with the fact \(\int^{b}_{a}f(x)\mathrm{d}x = \int^{b}_{0}f(x)\mathrm{d}x - \int^{a}_{0}f(x)\mathrm{d}x\) we can write the integral with arbitrary limits

\[\int_{t_1}^{t_2} x^{a-1}(1-x)^{b-1}\mathrm{d}x = \mathtt{beta}(a, b)\{B(a, b, t_2) - B(a, b, t_1)\}\]

We will also use the substitution and it’s derivative

\[\begin{split} t(M) = \left(\frac{M}{\mu}\right)^{1-\alpha}\left(1 + \left(\frac{M}{\mu}\right)^{1-\alpha}\right)^{-1} \\ \frac{\mathrm{d}t}{\mathrm{d}M} = \frac{1-\alpha}{\mu}\left(\frac{t}{1-t}\right)^{\frac{-\alpha}{1-\alpha}}(1-t)^2 .\end{split}\]

\[\begin{split}\begin{align} \int^{M_\textrm{max}}_{M_\textrm{min}}M\xi(M)\mathrm{d}M &= \xi_0\int^{M_\textrm{max}}_{M_\textrm{min}}\left(\frac{\mu}{\mu}\right)M\left(\frac{M}{\mu}\right)^{-\alpha}\left(1 + \left(\frac{M}{\mu}\right)^{1-\alpha}\right)^{-\beta}\mathrm{d}M \\ &= \mu\xi_0\int^{M_\textrm{max}}_{M_\textrm{min}}\left(\frac{M}{\mu}\right)^{1-\alpha}\left(1 + \left(\frac{M}{\mu}\right)^{1-\alpha}\right)^{-\beta}\mathrm{d}M \\ &= \xi_0\frac{\mu^2}{1-\alpha}\int^{t(M_\textrm{max})}_{t(M_\textrm{min})}t^{a-1}(1-t)^{b-1}\mathrm{d}t \\ &= \xi_0\frac{\mu^2}{1-\alpha}\mathtt{beta}(a, b)\{B(a, b, t(M_\textrm{max})) - B(a, b, t(M_\textrm{min}))\} \end{align}\end{split}\]

with \(a = \frac{2-\alpha}{1-\alpha}, b = \beta - a\).