ASTR 5610, Majewski [SPRING 2016]. Lecture Notes

## The Initial Mass Function (and Luminosity Function, briefly)

The Initial Mass Function

• A Mass Function is simply the distribution of stars by mass.

• The present Mass Function of a system reflects the Initial Mass Function (IMF) of formed stars plus the effects of stellar evolution.

Obviously, evolution acts to remove highest mass stars most quickly.

• Let's think about a single starburst first, and just after let the number of stars formed with masses in range (, + d) be:

Here, N0 is a normalization factor that depends on the burst strength, while ξ describes the shape of the IMF.
• (Note, in many treatments, the function ξ is characterized instead by the symbol ψ.)

• Regarding N0 -- it is typical to normalize ξ to the total mass, rather than the total number of new born stars because there is a large weighting in the total number of stars towards very low mass stars, which can be very difficult to enumerate.

Using the equation just above, this normalization is done so that:

Then

and N0 then becomes the number of solar masses in the SF burst.

• We have no a priori reason to think this, but it appears that ξ() is a remarkably universal function that applies to starbursts occurring in a variety of different circumstances, places, and times in the universe (at least for < ~ ).

• Typically assumed to be a simple power law (between limits!):

(Salpeter Mass Function)

x = 1.35 by Salpeter (1955)

##### Poster from a 2004 meeting showing the Salpeter mass function shape.
• Note, mass functions are not typically observable.

• Instead, we observe the luminosity function , Φ(M), which is the number of stars visible per absolute magnitude, M. (Note the different symbol M here for absolute magnitude.)

(Actually, we don't even observe that -- rather we observe apparent magnitudes and need to get distances too: a ball of wax I defer for now too!).
• We then must convert the LF to the MF with knowledge of the mass-luminosity relation appropriate to the age of the population (again, tricky).
• How are IMFs derived?

From observations of luminosity functions of course.

1. Easiest way is to look for very young open cluster and observe the Initial Luminosity Function, Φ0(M), directly.

Convert Φ0(M) to ξ() with ZAMS mass-luminosity relation.

But, if there has been any evolution, have to account for missing high mass (high luminosity) and missing low mass (pre-MS; see below) stars.
2. If we use field stars in the Milky Way, we can, for example, assume the disk had a constant SFR in the past (not strictly true!).

Measure present LF - Φ(M) of Main Sequence stars.

Then the Initial LF is:

where t = time since SF started.

Obviously, the proportion of stars whose MS lifetimes are longer than t is just proportional to what we observe - i.e., they are all still around.

But stars evolved away must be accounted for and the t / τMS corrects for the amount of time that could have contributed to present number of these stars still around.

(I.e., the only stars of magnitude M we see are those that were formed in the last (τMS / t ) fraction of the life of the population.)

We will return to the question of the local LF when we talk about starcounts.

• Once we have Φ0(M), then:

The relationship between mass and absolute magnitude can be gotten by:

1. Theoretical stellar atmospheres models coupled to models of main sequence structure at different masses -- works for most stellar masses but hard for <~ 0.5M because...

1. Low mass stars are slow to settle onto M.S. -- have to account for many low mass stars not being on true M.S. yet.

##### The CMD for a young open cluster, showing stars at the top of the M.S. evolving away while stars at the bottom of the M.S. are still getting to the M.S. From Carroll & Ostlie.
2. Late type stars have many molecular bands, stellar atmospheres are hard to calculate.

3. <~ 0.3 fully convective - don't have good models for convection.

2. Observe binary stars, from which masses can be derived using orbital dynamics (one of the few ways we can measure a stellar mass). But...

1. Requires a lot of work to get individual orbits monitored spectroscopically (get spectroscopic parallaxes and radial velocities) and astrometrically (get trig parallaxes -- if possible -- and the relative orbits of the two stars around each other, i.e., the relative masses of the stars to each other from 1a1= 2a2, and the orientation of the orbit on sky and the ellipticity).

With knowledge of the period, P, and the semimajor axis, a, get (1+ 2).

But unless you can determine the a1 and a2 separately, you cannot portion the relative mass in each component.

And then there is still the sin i (inclination of the orbit) ambiguity, which, unless we can resolve (i.e., know i), means we really only at best get 1sin3i and 2sin3i. This means we really only get a lower limit on the actual masses (because sin i < 1 ).

(Note that < sin3i > = 0.59 for randomly oriented orbital planes.)

2. Subject to significant observational uncertainty.

3. Required data for binary systems including stars of all masses hard to get - sparse statistics...

An example of a mass-absolute magnitude relationship from binary star data is given below:

3. Asteroseismology may be a promising avenue in the future, though currently limited to brighter stars.

• Armed with the mass-absolute magnitude relationship we can proceed to convert the ILF to the IMF.

But this process is hard because not only do we need M(), but its derivative, because

Data are just not good enough for this.

• The actual IMF is more complex than Salpeter with (Scalo 1986) flatter parts at ends and steeper parts in 1-10 range.

The peak at the low mass end may be illusory:

• A peak in the LF at low luminosities can occur due to a combination of incompleteness and the fact that near about the H-burning limit, 0.1, the luminosity of stars drops precipitously and the stars within a small range in mass are spread over wide range of magnitudes.

• Theory in this part of diagram of mass-magnitude relationship not good either.

• Just hard to work there!

##### Some various IMF shapes, including more recent papers. From Wikipedia.

• For most galaxies, it is safe to assume, to even high mass:

which is an average over the whole galaxy

Note: these high mass stars are somewhat missing from solar neighborhood, so this is different from what is shown in Fig. 5.12.

• However, for a number of reasons, people have debated the possibility that star formation may be bimodal, with a separate "high mass" mode that occurs in some extreme cases (place or time; Schwarzschild & Spitzer 1953).

• If formation of the very first stars favored very massive stars, then one could explain the "local dark matter problem" (i.e., that half of the mass in the solar neighborhood is not accounted for) as residing in the remnants (black holes and neutron stars) of an initial population of massive stars (Larson 1986, MNRAS, 218, 409).

Note that microlensing experiments, which would be sensitive to the existence of these "MACHOS" (Massive Compact Halo Objects), have pretty much ruled out that there are enough to account for the local dark matter.

• The observed paucity of low metallicity stars (the so-called "G-dwarf problem") might be explained by the fact that early star formation underproduced low mass to high mass stars compared to the present day IMF (Schmidt 1963).

##### Various versions of bimodal star formation models. (Left) The bimodal SF model from Larson (1986). (Right) Schematic of the bimodal IMF predicted by Nakamura & Umemura (2001). The shadings represent the different outcomes (remnants or enrichment through supernovae, etc.) for the various mass ranges. From http://www.maths.monash.edu.au/~scamp/Personal-Website/popIII_files/popIII_summ_formation.html .
• The colors of the bluest starburst galaxies might be more easily explained if they are dominated by high mass mode star formation (Charlot et al. 1993).

• Observations of metallicity in the intracluster media of some galaxy clusters indicates a large overabundance of [ α / Fe ], which might come about from a large number of Type II supernovae which blew their nucleosynthetic yield into the ICM.

• The differences in SF modes may relate to a strong dependence of the critical mass for cloud fragmentation on the gas temperature.

• Note: ~50% of "star systems" contain binary or multiple stars (so ~ 2/3 of all stars are in multiple/binary systems).

(Cecilia Payne-Gaposchkin: "Three out of every two stars is a binary"!)

From solar neighborhood (low mass stars mainly):

for stars

for systems

(equal mass binaries are preferred and binarity seems lower in M stars - ~35%).

• IMF for halo hard to establish:

• Globulars are bad for ξ () because of dynamical evolution -- low mass stars lost preferentially. (We will look at this when we discuss globular clusters.)

• Subdwarfs - proper motion selected (biases) ...

• dM / d complicated because of (non-solar) abundance variations.

• Overall, a somewhat steeper mass law is suggested:

All material copyright © 2003,2006,2008,2010,2012,2014,2061 Steven R. Majewski. All rights reserved. These notes are intended for the private, noncommercial use of students enrolled in Astronomy 551 and Astronomy 5610 at the University of Virginia.