The Main Sequence Mass Fraction

The main sequence mass fraction, as the name suggests, is the fraction of a single stellar population’s initial mass that is still in the form of main sequence stars. Because this calculation does not concern evolved stars, neither a model for the post main sequence lifetime nor an initial-final remnant mass relation is needed; it is thus considerably simpler than the cumulative return fraction. This quantity is instead specified entirely by the IMF and the mass-lifetime relation.

It’s analytic form is given by:

\[h(t) = \int_l^{m_\text{to}(t)} m\frac{dN}{dm} dm \left[ \int_l^u m\frac{dN}{dm} dm \right]^{-1}\]

which for a power-law IMF \(dN/dm \sim m^{-\alpha}\) becomes

\[h(t) = \left[\frac{1}{2 - \alpha}m^{2 - \alpha}\Bigg|_l^{m_\text{to}(t)}\right] \left[\frac{1}{2 - \alpha}m^{2 - \alpha}\Bigg|_l^u\right]^{-1}\]

It may be tempting to cancel the factor of \(1/(2 - \alpha)\), but more careful consideration must be taken for piece-wise IMFs like Kroupa [26]:

\[h(t) = \left[ \sum_i \frac{1}{2 - \alpha_i} m^{2 - \alpha_i} \right]_l^{m_\text{to}(t)} \left(\left[ \sum_i \frac{1}{2 - \alpha_i} m^{2 - \alpha_i} \right]_l^u\right)^{-1}\]

where the summation is over the relevant mass ranges with different power-law indeces \(\alpha_i\). In the case of kroupa \(\alpha\) = 2.3, 1.3, and 0.3 for \(m\) > 0.5, 0.08 \(\leq m \leq\) 0.5, and \(m\) < 0.08, respectively.

Here we plot \(h\) as a function of the stellar population’s age assuming the mass-lifetime relation of Hurley, Pols & Tout (2000) [27] (see discussion here). By 10 Gyr, \(h(t)\) is as low as \(\sim0.45\) for the Kroupa IMF and \(\sim0.65\) for the Salpeter [28] IMF. In comparison, the cumulative return fraction \(r(t) \approx 0.45\) for the Kroupa IMF and \(\sim0.28\) for the Salpeter IMF. This suggests that the approximation \(h(t) \approx 1 - r(t)\) fails at the \(\sim5-10\%\) level, depending on the choice of IMF. This suggests that for old stellar populations, a non-negligible portion of the mass is contained in evolved stars and stellar remnants. VICE therefore differentiates between these two quantities in its implementation.

In reality, the rate of the stellar mass evolving off of the main sequence is given by \(\dot{h}M_*\) where \(M_*\) is the initial mass of the stellar population. However, the quantization of timesteps in VICE allows each timestep to represent a single stellar population which will eject mass \(M_*dh\) in a time interval \(dt\). For that reason, VICE is implemented with a calculation of \(h(t)\) rather than \(\dot{h}\).

In calculations of \(h(t)\) with the built-in Kroupa and Salpeter IMFs, the analytic solution is calculated. In the case of a user-customized IMF, VICE solves the equation numerically using quadrature with the methods described in chapter 4 of Press, Teukolsky, Vetterling & Flannery (2007) [29].

Relevant source code:

  • vice/src/ssp/msmf.c

  • vice/src/yields/integral.c