Convolution background
In the above sections we have focussed mostly on the aspects of evolving stars (synthetically), and sampling distributions properly to reflect realistic populations of stars. However, in order to actually create realistic synthetic populations, with hard predictions of the numbers of (particular types of) e.g. stars, rates, or integrated nucleosynthetic yields, we still need to perform two important steps.
The first step is to assign a ``normalized yield’’, \(Y_{i}\), to each of the systems, which encodes the number of actual star-systems represented by each system \textit{i} that are formed for every solar mass formed in stars. This normalized yield is calculated as,
where \(\left<M\right>\) represents the \textit{average} mass of \textit{any} stellar system (regardless of its multiplicity).
In traditional MC sampling \(Y_{i}\) is the same for each star as \(p_{i}\) is just 1 over the number of stars in the synthetic population, but for grid-based sampling or adaptive-importance sampling it can be different for different systems, as \(p_{i}\) is assigned to the systems beforehand based on the initial distributions, and possibly the adapted distribution. % evolution splitting Another possible cause of unequal probabilities is the use of ``evolution splitting’’, which is the process of splitting a stellar evolution process into a number of child processes that each sample a stochastic process, like supernova natal-kicks, differently.
The second step is to combine the normalized yields of the systems with a star formation history, which is often expressed as a star-formation rate (SFR) in \(M_{\odot}\ yr^{-1}\) at a given lookback-time \(t_{\mathrm{lookback}}\). We determine the total mass formed between \(t_{\mathrm{lookback}}-\Delta t/2\) and \(t_{\mathrm{lookback}}+\Delta/2\) then as \(SFR(t_{\mathrm{lookback}})*\Delta t\). To determine the total number of stars like system \(i\), with a certain age \(t_{\mathrm{age}}\), in a current-day population given a star formation history we combine the total mass formed at that the lookback time corresponding to the age of the stars with the normalized yield as,
- :nbsphinx-math:`begin{equation}
label{eq:number_of_certain_age} N_{i,,t_{mathrm{age}}} (t=0) = Y_{i} mathrm{H}(t_{mathrm{lifetime},,i} - t_{mathrm{age}}) times mathrm{SFR}(t_{mathrm{age}})Delta t,
end{equation}`
where \(t_{\mathrm{lifetime},\,i}\) is the maximum lifetime of stellar system \(i\), and H(..) is a heavyside stepfunction that takes care to remove the system if it would have died by that age. We can calculate the total number of any star system with age \(t_{\mathrm{age}}\) expanding the above equation and summing over all star systems as,
- :nbsphinx-math:`begin{equation}
label{eq:all_stars_of_certain_age} N_{t_{mathrm{age}}} (t=0) = sum_{i}^{n} Y_{i} mathrm{H}(t_{mathrm{lifetime},,i} - t_{mathrm{age}}) times mathrm{SFR}(t_{mathrm{age}})Delta t,
end{equation}`
where \(n\) is the total number of star systems in the original simulation.
Finally, if we are interested in all the stars of any age that exist at a given time \(t\), we perform a summation over all ages as,
- :nbsphinx-math:`begin{equation}
label{eq:all_stars} N(t) = sum_{m} left[sum_{i}^{n} Y_{i} mathrm{H}(t_{mathrm{lifetime}, ,i} - t_{m}) times mathrm{SFR}(t_{m})Delta t right],
end{equation}`
where \(t_{m} = t+\Delta t \times m\) and \(m = t_{\mathrm{Universe}}-t/\Delta t\).
These calculations can be modified for transient events instead of total numbers (useful for e.g. binary black hole merger rates) , or for integrated nucleosynthetic yields (useful for e.g. chemodynamical galaxy evolution codes).
Alternatively, we can view part of the above two steps as follows. With just the probability \(p_{i}\), we can calculate, given a total number of stars in a population, how many stars like system \(i\) are around as,
- :nbsphinx-math:`begin{equation}
label{eq:alternative} N_{i} = p_{i} times N_{mathrm{tot}}
end{equation}`
To determine \(N_{\mathrm{tot}}\) we take the star formation rate at some time \(t\) to calculate the total mass formed into stars at that time, and use the average mass of a star system to convert that into a total number of stars formed,
- :nbsphinx-math:`begin{equation}
label{eq:2} N_{i} = p_{i} times frac{SFR(t) * Delta t}{left<Mright>}
end{equation}`