Harvard Astronomy 201b

Nearby Galaxies & the Kennicutt-Schmidt Relation

In Uncategorized on April 19, 2011 at 1:59 pm

(AG’s handwritten notes will be merged into this post after the class meeting on 4/19/11.)

Galaxy/ISM “Evolution”

Figure 1 of Galametz et al. 2011

Kennicutt-Schmidt Relations

Good intro: from TAUVEX web site.

Schmidt 1959 Paper on “The Rate of Star Formation”

Kennicutt 1998 Review Paper on (see Figure 9, shown below)

from Kennicutt 1998

Current Understanding of Kennicutt-Schmidt Relations

(reproduced from Goodman & Rosolowsky NSF Proposal, 2008)

The very last line of Marten Schmidt’s 1959 paper reads: “the mean density of a galaxy may determine, as a first approximation, its present gas content and thus its evolution stage.”  And, as a “first approximation,” the relationship between the star formation rate and the gas density,

that Schmidt put forward has held up remarkably well  (Kennicutt 2007).  We seek here to see how far beyond “first approximation” studies of nearby star forming regions can presently help us go in the extragalactic context, and how much a more refined view could help in future studies of galaxy evolution.

The modern version of the Schmidt Law is largely due to the work of Kennicutt and collaborators, who study the relation between “star formation rate” and “surface density” in nearby galaxies.  Many different indicators are used by Kennicutt et al., and by others, to measure each of these quantities, and we focus on the vagaries of their interpretation below.  Suffice it to say here that the “Kennicutt Law” holds over more than 4 orders of magnitude in surface density, and has a scatter about the relationship of roughly 1-2 orders of magnitude (Figure 6).  The Kennicutt Law is:

where a is a proportionality constant and the exponent q is typically of order 1.4 (Kennicutt 2008). If gas scale heights are assumed to not vary much from galaxy to galaxy, then re-writing the Schmidt law (eq. , using volume densities) in terms of surface densities, gives exponent q=1.5, making it the same as the Kennicutt Law to within uncertainties (Krumholz & Thompson 2007).

The K-S relation effectively implies that the efficiency function with which stars form from gas in galaxies is unchanging to within two orders of magnitude, after the many billions of years the galaxies in the sample have had to evolve.   That “efficiency function” however, is not linear, in that q ≠ 1.

This non-linearity has led others to propose two kinds of revisions to K-S ideas, both of which rely upon the important fact (Lada 1992) that studies of local (Milky Way) star forming regions clearly show that stars only form in the densest regions of molecular clouds.

The first kind of “revised” K-S relation investigates how the SFR depends on the surface density of gas above a higher density threshold than just the ~100 cm-3 needed to produce 12CO emission[1]. Gao & Solomon (2004) observed HCN, which is excited at densities above a few x 104 cm-3, in a large sample of galaxies, and they derived the relation:

This empirical relation has (slightly) less scatter than one using CO only, and, perhaps more importantly, has a linear power-law slope, suggesting that the surface density of HCN may be a linear determinant of the star formation rate.  The Gao & Solomon work inspired Wu et al. (2005) to test the relationship between LIR and LHCN in Milky Way molecular clouds, and the linear relationship was shown to continue right down to the scale of local GMCs.  Many in the extragalactic community rejoiced at these results, which could have meant that the quest for the “perfect” tracer of star-forming gas in a galaxy ended at “whatever it is that emits in HCN.”  But, as we explain at the close of this section, the story is, alas, not that simple.

The second kind of revised K-S relation acknowledges that density may not be the only determinant of fecundity in molecular gas. Blitz, Rosolowsky, Wong and collaborators have put forward the idea that pressure, rather than density, is likely to be more fundamental (Blitz & Rosolowsky 2006 and references therein).  The idea that pressure is critical (cf. Bertoldi & McKee 1992) is supported by analysis of nearby star-forming regions where it is clear that dense star-forming cores are often pressure-bound by the weight of the cloud around them (Lada et al. 2008) rather than only being confined only by their own self-gravity.[2]

In §2.4 below, we lay out a plan for measuring properties of Milky Way clouds that should be able to test both of these physically-motivated “revisions” to empirical K-S laws, as well as other ideas.  First, though, let us consider what modern theory predicts.  Krumholz and McKee (2005) can “predict” (explain) the observed K-S relations with three premises they state as:

  1. star formation occurs in virialized molecular clouds that are supersonically turbulent;
  2. the density distribution within these clouds is log-normal, as expected for supersonic isothermal turbulence; and
  3. stars form in any subregion of a cloud that is so overdense that its gravitational potential energy exceeds the energy in turbulent motions.

Our own work long ago as well as many others’ (cf. Larson 1981) has shown that #1 is clearly true.  Our recent work has shown that #2 is true for at least one well-studied local star-forming region (see §1.2, and Figure 1).  Our work on dendrograms allows us to find the “subregions” #3 is talking about, and to quantify the ratio of turbulent to gravitational energy with a virial parameter (see §1.3.2, and Figures 4 and 5).

Additional recent theoretical work, motivated by Gao & Solomon’s HCN results, predicts not only the origin of the K-S relations seen in a 12CO, but also in a host of other spectral lines.  Krumholz and Thompson (2007) and Narayanan et al. (2008) have investigated how the shapes of K-S relations change based on the molecular line tracer used to probe gas surface density.  Both groups’ work points out a very key, but somewhat subtle, feature of molecular line observations that is often ignored in the “K-S” community (but not in the Milky Way star-formation community!).  The relationship between observed emission in a spectral line and the density of the emitting region depends on how far above or below the “critical density” required to excite the transition the emitting region is.

Narayanan et al. (2008) clearly show that emission from a region which is nearly all above the critical density (e.g. CO under nearly any conditions) will give K-S slopes q>1, and emission from regions where much material is below the critical density of the tracer used will give K-S relations with slopes q<1, due to the inclusion of significant amounts of sub-thermally excited matter.

Krumholz & Thompson give an intuitive explanation of how HCN gives a slope of unity.  If a K-S relation has a slope 1.5, then a factor of (Sgas)1 comes from the amount of gas available for star formation, and a factor of (Sgas)0.5 comes from the dependence of free fall time on density.  Krumholz & McKee’s (2005) and several others’, models of the K-S relationship assumes that all “bound” gas (#3 above) collapses on a free-fall time, so that over time, that process gives an exponent of q=1.5 in equation 1, for a galaxy with a finite reservoir of gas and a constant efficiency of turning gas into stars.  Krumholz & Thompson argue that if a tracer (like HCN) has its critical density near or above the average density of star-forming gas, then the “free-fall” factor goes away, leaving Gao & Solomon’s linear relation, because the emission is coming from regions that all have the same free-fall time.

Very recently, Bussmann et al. (2008) found q=0.79±0.09 for a sample of more than 30 nearby galaxies observed in the (high critical density) HCN (3-2) line: that sub-linear slope was predicted in advance by the models of Narayanan et al. 2008.  However, a soon-to-be-published extensive observational study of massive-star-forming clumps within the Milky Way by Wu et al. (2008) finds a more linear (q≈1) relation for high-density tracers, including HCN (3-2).  Wu et al. also find that the five different dense gas tracers for which they construct K-S relations within the Milky Way rise steeply (and not really as  power-law) below an infrared luminosity threshold of ~10^4.5 L, and that above this threshold each gives slightly different (near-unity) slope and offset.


[1] Usually, CO is taken to be an indicator of “gas” in K-S relations.  Kennicutt (Kennicutt 1998) and others have experimented with HI+CO and find slightly tighter correlations,  but others (Blitz & Rosolosky 2006) find systematic effects to be at the origin of the tightened correlations.

[2] We note that this question, about how exactly cores “connect” to their environment is so interesting on its own that an entirely separate NSF proposal from this one has been submitted to address it.

References  for Above, beyond Schmidt & Kennicutt

  • Krumholz, M. R. & McKee, C. F. 2005, A General Theory of Turbulence-regulated Star Formation, from Spirals to Ultraluminous Infrared Galaxies, ApJ, 630, 250-268
  • Krumholz, M. R. & Thompson, T. A. 2007, The Relationship between Molecular Gas Tracers and Kennicutt-Schmidt Laws, ApJ, 669, 289-298
  • Narayanan, D., Cox, T. J., Shirley, Y., Dave, R., Hernquist, L. & Walker, C. K. 2008, Molecular Star Formation Rate Indicators in Galaxies, ApJ, 684, 996-1008
  • Wu, J., Evans, N. J., II, Gao, Y., Solomon, P. M., Shirley, Y. L. & Vanden Bout, P. A. 2005,Connecting Dense Gas Tracers of Star Formation in our Galaxy to High-z Star Formation, ApJ, 635, L173-L176
  • Wu, J., Evans, N. J., Shirley, Y. L. & Knez, C. 2010, The Properties of Massive, Dense Clumps: Mapping Surveys of HCN and CS, ApJS, 188, 313.

Additional Sample Relevant Recent K-S work:

  • “CARMA Survey Toward Infrare-Bright Nearby Galaxies (STING): Molecular Gas Star Formation Law in NGC 4254,” Rahman et al. 2011 (SeeFigure 7 for example of inter-comparision of star formation tracers)
  • “On the relation between the Schmidt and Kennicutt–Schmidt star formation laws and its implications for numerical simulations”, Schaye & Dalla Vecchia 2007. (Different K-S laws can be derived based on assuming different effective equations of state, but authors conclude that this does not give deep physical insight.)
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