Harvard Astronomy 201b

Does the IMF come from the CMF?

In Uncategorized on March 31, 2011 at 2:55 am

Does the IMF come from the CMF?

The IMF (Initial Mass Function) is the distribution of masses at which stars were formed. It has been measured empirically for stars in a variety of environments. This excerpt from McKee and Ostriker 2007 describes our current knowledge of the IMF (bolding is mine):

How is the distribution of stellar masses, or IMF, established? This is one of most basic questions a complete theory of star formation must answer, but also one of the most difficult. Current evidence suggests that the IMF is quite similar in many different locations throughout the Milky Way, with the possible exception of star clusters formed very near the Galactic Center (Scalo 1998b presents evidence for significant variations in the IMF, but Elmegreen 1999 argues that much, if not all, of this is consistent with the expected statistical variations). The standard IMF of Kroupa (2001) is a three-part power-law with breaks at 0.08 M and 0.5 M; i.e., with α = 1.3 for 0.5 < m*/M < 50, α = 0.3 for 0.08 < m*/M < 0.5, and α = −0.7 for 0.01 < m*/M < 0.08. The slope of the IMF at m* M was originally identified by Salpeter (1955), who found α = 1.35. Up to 1 M, a log-normal functional form provides a smooth fit for the observed mass distribution (Miller & Scalo 1979), with Chabrier (2005) finding that mc ≈ 0.2 M and σ ≈ 0.55 apply both for individual stars in the disk and in young clusters; the system IMF (i.e., counting binaries as single systems) has mc = 0.25 M. Thus, the main properties of the IMF that any theory must explain are (a) the Salpeter power-law slope at high mass, (b) the break and turnover slightly below 1 M, (c) the upper limit on stellar masses 150 M (Elmegreen 2000, Figer 2005, Oey & Clarke 2005), and (d) the universality of these features over a wide range of star-forming environments, apparently independent of the mean density, turbulence level, magnetic field strength, and to large extent also metallicity. Theory predicts that there should be a lower limit on (sub)stellar masses (Low & Lynden-Bell 1976), but this has not been confirmed observationally.

Alves et al. 2007 emphasize:

The origin of the stellar IMF remains one of the major unsolved problems of modern astrophysics.

The IMF may itself originate from the CMF (Core Mass Function), the distribution of masses of dense molecular cores in a cloud. Alves et al. 2007 measure a CMF in various regions of the Pipe dark cloud.

Left: Pipe Nebula (APOD 2009 May 22), Right: Fig. 1 of Alves et al. 2007. Caption (taken from paper): Dust extinction map of the Pipe nebula molecular complex from Lombardi et al. (2006). This map was constructed from near-infrared observations of about 4 million stars in the background of the complex. Approximately 160 individual cores are identified within the cloud and are marked by an open circle proportional to the core radius. Most of these cores appear as distinct, well separated entities.

What sets the CMF?

The CMF is set by the physics of fragmentation in a molecular cloud.

Image Credit: Jill Bechtold, scanned from The Cosmic Perspective, 4th Edition (2006), by Bennett, Donahue, Schneider & Voit
If the CMF follows a log-normal distribution, that might indicate that turbulence plays an important role.

Relation between the CMF and the IMF

The dense core from which the proto-star forms is the “reservoir” of material. Therefore we might expect that mass of the proto-star is the mass of the core, with some efficiency factor. However, one could imagine that the mass of the proto-star depends only weakly on its reservoir mass, or that the reservoir mass only matters below a certain threshold, with some other process limiting the mass above that threshold (such as feedback from outflows, Shu et al. 1987). Competitive accretion might also weaken the mapping between the reservoir mass and the mass of the star that forms.

Does the form of the IMF match the CMF?

One piece of evidence that could support or repute the mapping of the CMF to IMF is if the distributions had the same shape. Early CO observations found that, unlike the IMF, the CMF was a scale-free single power law dN(m)/d, as opposed to the IMF, which has a peak from 0.08 M and 0.5 M, which means there is a special characteristic mass of about ~0.2. After its second break, the IMF power law is steeper than the CMF power law (exponent of -0.7 for CMF vs. -1.3 IMF). Thus there is a greater fraction of large mass cores than large mass stars. Although processes limiting the masses of stars could cause this (see previous section), it would not explain the peak. Moreover, the IMF is consistent with log-normal up to 1 solar mass, but the CMF is not.

This plot from Heithausen et al. 1998 shows the power law scaling of the CMF:

However, recall that we should trust the dust and sass the gas.

Observations using dense gas tracers instead of CO found that the CMF and IMF are similar (

Observations using dust emission instead of CO found that the slope after 1 stellar mass is similarly steep to the IMF and to flatten at 0.5 stellar masses (see references within Alves et al. 2007 ).

Using a dust extinction map of the Pipe nebula, Alves et al. 2007 derived a dense core mass function. They find that the shape is consistent with the IMF, scaled by a factor of 4:

Image caption from paper: Figure 2: Mass function of dense molecular cores plotted as filled circles with error bars. The grey line is the stellar IMF for the Trapezium cluster (Muench et al. 2002). The dashed grey line represents the stellar IMF in binned form matching the resolution of the data and shifted to higher masses by about a factor of 4. The dense core mass function is similar in shape to the stellar IMF function, apart from a uniform star formation efficiency factor.

Is the CMF the “provenance” of the IMF?

Does the consistency of the form of the CMF with the IMF necessarily imply that the IMF comes from the CMF? A recent paper Anathpindika 2011 proposes that the shape of the IMF is set by the accretion rate of protostars and that IMF is insensitive to the shape of the CMF; the similarity in shapes is a coincidence. The following figure shows the resulting IMF starting from a uniform CMF:

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