<h2>What is it?</h2>
<p>An IMF, <span class="math-container">$\Phi(m)$</span>, is defined such as <span class="math-container">$\Phi(m){\rm d}m$</span> gives the fractions of stars with a mass between <span class="math-container">$m - {\rm d}m/2$</span> and <span class="math-container">$m + {\rm d}m/2$</span>, and with a normalized distribution</p>
<p><span class="math-container">$$\int_{m_{\rm min}}^{m_{\rm max}}m\Phi(m){\rm d}m = 1\ M_{\odot}.$$</span></p>
<p>Note that these boundaries (<span class="math-container">$m_{\rm min}$</span> and <span class="math-container">$m_{\rm max}$</span>) are ill-defined, but typically of the order of 0.1 <span class="math-container">$M_{\odot}$</span> and 100 <span class="math-container">$M_{\odot}$</span>, respectively.</p>
<h2>IMFs</h2>
<p>The various IMF used are the following, with their main characteristics:</p>
<ul>
<li>the <a href="http://adsabs.harvard.edu/abs/1955ApJ...121..161S" rel="noreferrer">Salpeter's IMF</a>, that is a parametrization of the IMF by a simple power-law, of the form
<span class="math-container">$$\Phi(m){\rm d}m \propto m^{-\alpha}{\rm d}m;$$</span></li>
<li>the <a href="http://adsabs.harvard.edu/abs/1979ApJS...41..513M" rel="noreferrer">Miller &amp; Scalo's IMF</a>, that is a parametrization of the IMF by a log-normal distribution of the form
<span class="math-container">$$\xi\left(\log(m)\right) = A_0 + A_1 \log(m) + A_2\left(\log(m)\right)^2;$$</span></li>
<li>the <a href="http://adsabs.harvard.edu/abs/2001MNRAS.322..231K" rel="noreferrer">Kroupa's IMF</a>, that is a parametrization of the IMF by a broken power-law;</li>
<li>the <a href="http://adsabs.harvard.edu/abs/2003PASP..115..763C" rel="noreferrer">Chabrier's IMF</a> and <a href="http://adsabs.harvard.edu/abs/2005ASSL..327...41C" rel="noreferrer">Chabrier's system IMF</a>, that is a combination of log-normal distribution (for low mass stars with masses less than 1 <span class="math-container">$M_{\odot}$</span>) and and a power-law distribution (for larger masses). The difference between the IMF and the system IMF is to merge resolved objects into multiple systems to compute the magnitude of systems instead of individual stars.</li>
</ul>
<h2>Determination</h2>
<p>As you see, all these IMF are parametrization, deduced from observations. In general, the observations used to infer these mass functions come from star clusters in our galaxy. All you need to do it is to find a mass-magnitude relationship to deduce, from an observed luminosity, a mass function. In general, the number density distribution per mess interval, <span class="math-container">${\rm d}n/{\rm d}m$</span>, takes the following form
<span class="math-container">$$\frac{{\rm d}n}{{\rm d}m}\left(m\right)_\tau = \left(\frac{{\rm d}n}{{\rm d}M_\lambda(m)}\right) \times \left(\frac{{\rm d}m}{{\rm d}M_\lambda(m)}\right)^{-1}_\tau,$$</span>
for a given age <span class="math-container">$\tau$</span> and an observed magnitude <span class="math-container">$M_\lambda$</span>. Then, it is just a matter of parametrization, but also of how well it can arise from a proper theory.</p>
<p>For this matter, the Chabrier's IMF is probably the one that is best back up by theoretical arguments. It relies on a gravo-turbulent theory, taking into account all the possible supports (thermal support, turbulent support and magnetic support) plus the dual nature of turbulence, that both favors star formation by compressing the gas, and impedes star formation, by dispersing the fluid. All the dirty details are given in <a href="http://adsabs.harvard.edu/abs/2008ApJ...684..395H" rel="noreferrer">Hennebelle &amp; Chabrier (2008)</a> and <a href="http://adsabs.harvard.edu/abs/2009ApJ...702.1428H" rel="noreferrer">Hennebelle &amp; Chabrier (2009)</a>, showing how you can analytically deduce an IMF from these theoretical considerations.</p>
<h2>Applications</h2>
<p>As far as I know, these IMF are more or less used for every type of population. However, you won't favor Salpeter's IMF if you have enough resolution to resolve low-mass objects, that are not at all well-taken into account with this IMF. You should also favor the Chabrier's <em>system</em> IMF in case of unresolved objects.</p>
<p>To know if all these IMFs are really well-suited to any kind of population is an open and difficult question (the so-called question of the universality of the IMF), in particular because you need to resolve individual stars in clearly identified clusters to deduce an IMF. There are some papers investigating the question (for example, you could have a look at <a href="http://adsabs.harvard.edu/abs/2012Natur.484..485C" rel="noreferrer">Cappellari et al. (2012)</a> for a recent discussion of the problem).</p>

CC BY-SA 3.0
2013-12-11T15:43:45.263Z
2013-12-11T15:43:45.263Z
2020-06-17T09:47:40.653Z
MBR
<p>The <a href="http://en.wikipedia.org/wiki/Initial_mass_function">Initial Mass Function (IMF)</a> is the empirical function which describes the initial masses of a population of stars. My questions are,</p>

<p>1) What are the various IMF's which are used?</p>

<p>2) For each, what type of population do they describe? (e.g. - galaxy, dwarf galaxy, globular cluster, etc..)</p>

<p>3) And how are they actually calculated? (meaning, do they come from simulations/observations, and what assumptions are made about each?) </p>

<p>Entire answers and pieces of answers are all welcome. Formulae (in latex, please) are encouraged.</p>

<stellar-astrophysics><initial-mass-function>
How exactly is the Initial Mass Function (IMF) calculated?
astromax
# How exactly is the Initial Mass Function (IMF) calculated?

(Asked by: username_0 on 2013-11-08)

The [Initial Mass Function (IMF)](http://en.wikipedia.org/wiki/Initial_mass_function) is the empirical function which describes the initial masses of a population of stars. My questions are,

1) What are the various IMF's which are used?

2) For each, what type of population do they describe? (e.g. - galaxy, dwarf galaxy, globular cluster, etc..)

3) And how are they actually calculated? (meaning, do they come from simulations/observations, and what assumptions are made about each?)

Entire answers and pieces of answers are all welcome. Formulae (in latex, please) are encouraged.

**Tags:** stellar-astrophysics, initial-mass-function


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## 1 answers

### Answer by username_1 on 2013-12-11. Score: 8

## What is it?

An IMF, $\Phi(m)$, is defined such as $\Phi(m){\rm d}m$ gives the fractions of stars with a mass between $m - {\rm d}m/2$ and $m + {\rm d}m/2$, and with a normalized distribution

$$\int\_{m\_{\rm min}}^{m\_{\rm max}}m\Phi(m){\rm d}m = 1\ M\_{\odot}.$$

Note that these boundaries ($m\_{\rm min}$ and $m\_{\rm max}$) are ill-defined, but typically of the order of 0.1 $M\_{\odot}$ and 100 $M\_{\odot}$, respectively.

## IMFs

The various IMF used are the following, with their main characteristics:

* the [Salpeter's IMF](http://adsabs.harvard.edu/abs/1955ApJ...121..161S), that is a parametrization of the IMF by a simple power-law, of the form
  $$\Phi(m){\rm d}m \propto m^{-\alpha}{\rm d}m;$$
* the [Miller & Scalo's IMF](http://adsabs.harvard.edu/abs/1979ApJS...41..513M), that is a parametrization of the IMF by a log-normal distribution of the form
  $$\xi\left(\log(m)\right) = A\_0 + A\_1 \log(m) + A\_2\left(\log(m)\right)^2;$$
* the [Kroupa's IMF](http://adsabs.harvard.edu/abs/2001MNRAS.322..231K), that is a parametrization of the IMF by a broken power-law;
* the [Chabrier's IMF](http://adsabs.harvard.edu/abs/2003PASP..115..763C) and [Chabrier's system IMF](http://adsabs.harvard.edu/abs/2005ASSL..327...41C), that is a combination of log-normal distribution (for low mass stars with masses less than 1 $M\_{\odot}$) and and a power-law distribution (for larger masses). The difference between the IMF and the system IMF is to merge resolved objects into multiple systems to compute the magnitude of systems instead of individual stars.

## Determination

As you see, all these IMF are parametrization, deduced from observations. In general, the observations used to infer these mass functions come from star clusters in our galaxy. All you need to do it is to find a mass-magnitude relationship to deduce, from an observed luminosity, a mass function. In general, the number density distribution per mess interval, ${\rm d}n/{\rm d}m$, takes the following form
$$\frac{{\rm d}n}{{\rm d}m}\left(m\right)\_\tau = \left(\frac{{\rm d}n}{{\rm d}M\_\lambda(m)}\right) \times \left(\frac{{\rm d}m}{{\rm d}M\_\lambda(m)}\right)^{-1}\_\tau,$$
for a given age $\tau$ and an observed magnitude $M\_\lambda$. Then, it is just a matter of parametrization, but also of how well it can arise from a proper theory.

For this matter, the Chabrier's IMF is probably the one that is best back up by theoretical arguments. It relies on a gravo-turbulent theory, taking into account all the possible supports (thermal support, turbulent support and magnetic support) plus the dual nature of turbulence, that both favors star formation by compressing the gas, and impedes star formation, by dispersing the fluid. All the dirty details are given in [Hennebelle & Chabrier (2008)](http://adsabs.harvard.edu/abs/2008ApJ...684..395H) and [Hennebelle & Chabrier (2009)](http://adsabs.harvard.edu/abs/2009ApJ...702.1428H), showing how you can analytically deduce an IMF from these theoretical considerations.

## Applications

As far as I know, these IMF are more or less used for every type of population. However, you won't favor Salpeter's IMF if you have enough resolution to resolve low-mass objects, that are not at all well-taken into account with this IMF. You should also favor the Chabrier's *system* IMF in case of unresolved objects.

To know if all these IMFs are really well-suited to any kind of population is an open and difficult question (the so-called question of the universality of the IMF), in particular because you need to resolve individual stars in clearly identified clusters to deduce an IMF. There are some papers investigating the question (for example, you could have a look at [Cappellari et al. (2012)](http://adsabs.harvard.edu/abs/2012Natur.484..485C) for a recent discussion of the problem).


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