Accepted for publication in the Astrophysical Journal, July 16, 2007
Preprint typeset using LATEX style emulateapj v. 08/22/09
THE RELATIONSHIP BETWEEN MOLECULAR GAS TRACERS AND KENNICUTT-SCHMIDT LAWS
Mark R. Krumholz
and Todd A. Thompson
Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544
Accepted for publication in the Astrophysical Journal, July 16, 2007
ABSTRACT
We provide a model for how Kennicutt-Schmidt (KS) laws, which describe the correlation between
star formation rate and gas surface or volume density, depend on the molecular line chosen to trace
the gas. We show that, for lines that can be excited at low temperatures, the KS law depends on how
the line critical density compares to the median density in a galaxy’s star-forming molecular clouds.
High critical density lines trace regions with similar physical properties across galaxy types, and this
produces a linear correlation between line luminosity and star formation rate. Low critical density
lines probe regions whose properties vary across galaxies, leading to a star formation rate that varies
superlinearly with line luminosity. We show that a simple model in which molecular clouds are treated
as isothermal and homogenous can quantitatively reproduce the observed correlations between galactic
luminosities in far infrared and in the CO(1 → 0) and HCN(1 → 0) lines, and naturally explains why
these correlations have different slopes. We predict that IR-line luminosity correlations should change
slope for galaxies in which the median density is close to the line critical density. This prediction
may be tested by observations of lines such as HCO+(1 → 0) with intermediate critical densities, or
by HCN(1 → 0) observations of intensely star-forming high redshift galaxies with very high densities.
Recent observations by Gao et al. hint at just such a change in slope. We argue that deviations from
linearity in the HCN(1 → 0)−IR correlation at high luminosity are consistent with the assumption of
a constant star formation efficiency.
Subject headings: ISM: clouds — ISM: molecules — stars: formation — galaxies: ISM — radio lines:
1. INTRODUCTION
Schmidt (1959, 1963) first proposed that the rate at
which a gas forms stars might follow a simple power
law correlation of the form ρ̇∗ ∝ ρNg , where ρ̇∗ is the
star formation rate per unit volume, ρg is the gas den-
sity, and N is generally taken to be in the range 1 − 2.
In the decades since, observations have revealed two
strong correlations that appear to be evidence for this
hypothesis. First, galaxy surveys reveal that the in-
frared luminosity of a galaxy, which traces the star for-
mation rate, varies with its luminosity in the CO(1 → 0)
line, which traces the total mass of molecular gas, as
LFIR ∝ L1.4−1.6CO (Gao & Solomon 2004a,b; Greve et al.
2005; Riechers et al. 2006a). Kennicutt (1998a,b) iden-
tified the closely-related correlation between gas surface
density Σg and star formation rate surface density Σ̇∗,
Σ̇∗ ∝ Σ1.4±0.15g , a relation that has come to be known
as the Kennicutt Law. Since over the bulk of the dy-
namic range of Kennicutt’s data galaxies are predomi-
nantly molecular, this is effectively a correlation between
molecular gas, as traced by CO(1 → 0) line emission,
and star formation. Spatially resolved observations of
galaxies confirm that, at least for molecule-rich galax-
ies where resolved CO(1 → 0) observations are possible,
star formation is more closely coupled with gas traced by
CO(1 → 0) than with atomic gas (Wong & Blitz 2002;
Heyer et al. 2004; Komugi et al. 2005; Kennicutt et al.
2007)
Electronic address: krumholz@astro.princeton.edu, thomp@astro.princeton.edu
1 Hubble Fellow
2 Lyman Spitzer Jr. Fellow
Second, Gao & Solomon (2004a,b) find that there is a
strong correlation between the IR luminosity of galax-
ies and emission in the HCN(1 → 0) line, which mea-
sures the mass at densities significantly greater than
that probed by CO(1 → 0). However, they find that
their correlation, which covers nearly three decades in
total galactic star formation rate, is linear: LFIR ∝
LHCN. Wu et al. (2005) show that this correlation ex-
tends down to individual star-forming clumps of gas in
the Milky Way, provided that their infrared luminosities
are >∼ 104.5 L⊙. Interestingly, however, Gao et al. (2007)
find a deviation from linearity in the IR-HCN correlation
for a sample of intensely star-forming high redshift galax-
ies. These sources show small but significant excesses of
infrared emission for their observed HCN emission.
The difference in power law indices between the LFIR−
LCO and LFIR −LHCN correlations is statistically signif-
icant, and, on its face, puzzling. An index near N = 1.5
seems natural if one supposes that a roughly constant
fraction of the gas present in molecular clouds will be
converted into stars each free-fall time. In this case one
expects ρ̇∗ ∝ ρ1.5g (Madore 1977; Elmegreen 1994). If
gas scale heights do not vary strongly from galaxy to
galaxy, this implies Σ̇∗ ∝ Σ1.5g as well, which is consis-
tent with the observed Kennicutt law. More generally,
since the dynamical timescale in a marginally Toomre-
stable (Q ≈ 1; see Martin & Kennicutt 2001) galactic
disk is of order Ω−1 ∝ (Gρg)−1/2, where Ω is the angular
frequency of the disk, an index close to N = 1.5 is ex-
pected if star formation is regulated by any phenomenon
that converts a fixed fraction of the gas into stars on this
time scale (Elmegreen 2002).
http://arxiv.org/abs/0704.0792v2
mailto:krumholz@astro.princeton.edu, thomp@astro.princeton.edu
On the other hand, Wu et al. (2005) suggest a simple
interpretation of the linear IR-HCN correlation. They ar-
gue that the individual HCN-emitting molecular clumps
that they identify in the Milky Way represent a funda-
mental unit of star formation. The linear correlation
between star formation rate and HCN luminosity across
galaxies arises because a measurement of the HCN lu-
minosity for a galaxy simply counts the number of such
structures present within it, each of which forms stars at
some roughly fixed rate regardless of its galactic environ-
ment. However, in this interpretation it is unclear why
the structures traced by HCN(1 → 0) emission should
form stars at the same rate in any galaxy. After all, one
could equally well argue that molecular clouds traced by
CO(1 → 0) are fundamental units of star formation, but
the non-linear IR-CO correlation clearly shows that these
objects do not form stars at a fixed rate per unit mass.
Moreover, the evidence presented by Gao et al. (2007)
that the linear IR-HCN correlation varies in extremely
luminous high redshift galaxies suggests that the rela-
tionship between HCN emission and star-formation may
be somewhat more complex.
In this paper we attempt to explain the origin of the
difference in slope between the CO and HCN correla-
tions with star formation rate, and more generally to
give a theoretical framework for understanding how cor-
relations between star formation rate and line luminosity,
which we generically refer to as Kennicutt-Schmidt (KS)
laws, depend on the tracer used to define them. Our cen-
tral argument is conceptually quite simple, and in some
sense represents a combination of the intuitive arguments
for CO and HCN given above.
Consider an observation of a galaxy in a molecular
tracer with critical density ncrit, which essentially mea-
sures the mass of gas at densities of ncrit or more, i.e. the
gas that is dense enough for that particular transition to
be excited. In galaxies where the median density of the
molecular gas is significantly larger than ncrit, this means
that the observation will detect the majority of the gas,
and the bulk of the emission will come from gas whose
density is near the median density. Since the gas den-
sity will vary from galaxy to galaxy, the star formation
rate per unit gas mass will vary as roughly ρ1.5g , with one
factor of ρg coming from the amount of gas available for
star formation, and an additional factor of ρ0.5g coming
from the dependence of the free-fall or dynamical time
on the density.
On the other hand, in galaxies where the median gas
density is small compared to the critical density for the
chosen transition, observations will pick out only high
density peaks. Since the density in these peaks is set by
ncrit, and not by the conditions in the galaxy, these peaks
are at essentially the same density in any galaxy where
they are observed, and the corresponding free-fall times
in these regions are constant as well. As a result, the star
formation rate per unit mass of gas traced by that line
is approximately the same in every galaxy, because the
corresponding free-fall time is the same in every galaxy.
In the rest of this paper, we give a quantitative version
of this intuitive argument, and then discuss its conse-
quences. In § 2 we develop a simple formalism to com-
pute the star formation rate and the molecular line lu-
minosity of galaxies, and in § 3 we use this formalism to
predict the correlation between star formation rate and
luminosity. We show that our predictions provide a very
good fit for a variety of observations, and make predic-
tions for future observations. We discuss the implications
of our work and its limitations in § 4, and summarize our
conclusions is § 5.
2. STAR FORMATION RATES AND LINE LUMINOSITIES
2.1. Cloud Properties
Consider a galaxy in which the star-forming molecu-
lar clouds have a volume-averaged mean molecular hy-
drogen number density n = ρg/µH2 , where ρg is the
volume-averaged mass density of the molecular clouds in
the galaxy and µH2 = 3.9×10−24 g is the mean mass per
hydrogen molecule for a gas of standard cosmic composi-
tion. Observations indicate that n varies by two to three
decades over the galaxies for which the Kennicutt and
Gao & Solomon correlations are measured, from n ≈ 50
cm−3 in normal spirals like the Milky Way (McKee
1999) up to n ≈ 104 cm−3 in the strongest starburst
systems in the local universe (e.g. Downes & Solomon
1998). There is strong evidence that densities in molec-
ular clouds follow a lognormal probability distribution
function (PDF; see reviews by Mac Low & Klessen 2004
and Elmegreen & Scalo 2004)
d lnx
(lnx− lnx)2
, (1)
where x = n/n is the molecular hydrogen number den-
sity n relative to the average density, σ is the width of
the lognormal, and lnx = −σ2/2. For this distribution
the median density is nmed = n exp(σ
2/2). Numerical
experiments show that for supersonic isothermal turbu-
lence σ2 ≈ ln
1 + 3M2/4
, where M is the 1D Mach
number of the turbulence (Nordlund & Padoan 1999;
Ostriker et al. 1999; Padoan & Nordlund 2002). Mach
numbers in star-forming molecular clouds range from
M ∼ 30 (McKee 1999) in normal spirals to M ∼ 100
in strong starbursts (Downes & Solomon 1998), imply-
ing that median densities in molecular clouds range from
∼ 103 cm−3 in normal spirals to ∼ 106 cm−3 in star-
bursts. Star forming clouds within a galaxy are approx-
imately isothermal, except very near strong sources of
stellar radiation, so we assume a fixed temperature T
for the clouds. Observationally, T ranges from roughly
10 K in normal spirals (McKee 1999) up to as much
about 50 K in strong starbursts (Downes & Solomon
1998; Gao & Solomon 2004b).
2.2. Star Formation Rates
First let us ask how quickly stars form in such a
medium. Krumholz & McKee (2005) give a model for
star formation regulated by supersonic turbulence in
which a population of molecular clouds of total mass Mcl
form stars at a rate Ṁ∗ = SFRffMcl/tff(n), where tff(n)
is the free-fall time evaluated at the mean density and
SFRff is a number of order 10
−2 that depends weakly on
M. We therefore estimate the star formation rate per
unit volume as a function of the mean density given by
ρ̇∗ ≈ SFRff
32Gµ3H2n
. (2)
Molecular Gas and Kennicutt-Schmidt Laws 3
We adopt the Krumholz & McKee result SFRff ≈
0.014(M/100)−0.32 for clouds with a fiducial virial ra-
tio of αvir = 1.3.
Alternately, Krumholz & Tan (2007) point out that
observed correlations between the star formation rate
and the luminosity in different density tracers imply that
over a 3− 4 decade range in density n,
Ṁ∗ ≈ 10−2
Mcl(> n)
tff(n)
, (3)
where Mcl(> n) is the mass of gas with a density of n
or higher, and Mcl = Mcl(> 0). For a given choice of
n this provides an alternative estimate of the star for-
mation rate which is purely empirical, and independent
of any particular theoretical model. However, the differ-
ence between the star formation rates predicted by (2)
and (3) is small. For gas with a lognormal PDF,
Mcl(> n) =
1 + erf
−2 lnx+ σ2
23/2σ
, (4)
and using this to evaluate equation (3) indicates that, for
Mach numbers in the observed range, the two prescrip-
tions (2) and (3) give about the same star formation rate
over a very broad range in x. For example, at M = 30
the two estimates agree to within a factor of 3 for den-
sities in the range 0.2 < x < 4 × 104. Given the scatter
inherent in observational estimates of the star formation
rate, a factor of 3 difference is not particularly signifi-
cant, so it matters little which prescription we adopt. In
practice, we will use equation (2).
2.3. Line Luminosities
Now we must compute the luminosity of molecular line
emission from the galaxy. Even for a cloud that is not
in local thermodynamic equilibrium (LTE), for optically
thin emission this calculation is straightforward. How-
ever, the molecular lines used most often in galaxy sur-
veys are generally optically thick. To handle the effect of
finite optical depth on molecule level populations and line
luminosities, we adopt an escape probability approxima-
tion and treat clouds as homogeneous spheres. This is
not fully consistent with our assumption that clouds have
lognormal density PDFs, since the escape probability for-
malism assumes a uniform level population throughout
the cloud, and the essence of our argument in this paper
turns on how the level population varies with density.
However, this approach gives us an approximate way of
incorporating the optical thickness of star-forming clouds
into our model, the only alternative to which for turbu-
lent media is full numerical simulation (e.g. Juvela et al.
2001). We therefore proceed by treating clouds as ho-
mogeneous in order to determine their escape probabili-
ties, and we then relax the assumption of homogeneity,
while keeping the escape probabilities fixed, in order to
determine level populations and cloud luminosities as a
function of density.
Consider a cloud of radius R in statistical equilibrium
but not necessarily in LTE. In the escape probability
approximation, the fraction fi of molecules of species S
in state i is given implicitly by the linear system
(nqji + βjiAji) fj =
(nqij + βijAij)
 fi (5)
fi=1, (6)
where qij is the collision rate for transitions from state
i to state j, Aij is the Einstein spontaneous emission
coefficient for this transition, βij is the cloud-averaged
escape probability for photons emitted in this transition,
the sums are over all quantum states, and we understand
that Aij = 0 for i ≤ j and qij = 0 for i = j.
Equations (5) and (6) allow us to compute the level
populations fi for given values of βij . To completely
specify the system, we must add an additional consis-
tency condition relating the values of βij to the level
populations. For a homogeneous spherical cloud, the es-
cape probability for a given line is related to the optical
depth from the center to the edge of the cloud τij by
(B. Draine, 2007, private communication)
βij ≈
1 + 0.5τij
, (7)
where τij is computed at the central frequency of the
line. In turn, the optical depth is related to the level
populations by
τij =
4(2π)3/2Mcs
nX(S)fjR
, (8)
where λij is the wavelength of transition i → j, gi and
gj are the statistical weights of states i and j, cs is
the isothermal sound speed of the gas, and X(S) is the
abundance of molecules of species S. Note that this
equation implicitly assumes that the cloud has a uni-
form Maxwellian velocity distribution with 1D disper-
sion Mcs, consistent with our treatment of the clouds
as homogeneous spheres. One additional complication is
that we do not directly know cloud radii for most exter-
nal galaxies, where observations cannot resolve individ-
ual molecular clouds. However, we often can diagnose
the optical depths of transitions by comparing line ratios
of molecular isotopomers of different abundances. We
therefore take τ10, the optical depth of the transition be-
tween the first excited state and the ground state, as
known. For a given level population this fixes the value
of R.
We solve equations (5)–(8) using Newton-Raphson it-
eration. In this procedure, we guess an initial set of es-
cape probabilities βij , and solve the linear system (5) and
(6) to find the corresponding initial level populations fi.
We then compute the optical depths τij from equation
(8). The guessed escape probabilities βij and the corre-
sponding optical depths τij generally will not satisfy the
consistency condition (7), so we then iterate over βij val-
ues using a Newton-Raphson approach, seeking βij for
which the level populations give optical depths τij such
that all elements of the matrix βij − 1/(1 + 0.5τij) are
equal to zero within some specified tolerance. We use
the LTE level populations and escape probabilities for
our initial guess, so that the iteration converges rapidly
when the system is close to LTE.
Once we have determined the escape probabilities βij ,
we compute the luminosity by holding the βij values fixed
but allowing the level populations to vary with density,
then integrating over the PDF. Thus, the total luminos-
TABLE 1
Model Parameters
Parameter Normal galaxy Intermediate Starburst Reference
T 10 20 50 1–4
M 30 50 80 1–4
X(CO) 2× 10−4 4× 10−4 8× 10−4 5
X(HCO+) 2× 10−9 4× 10−9 8× 10−9 6, 7
X(HCN) 1× 10−8 2× 10−8 4× 10−8 6–8
τCO(1→0) 10 20 40 9
τHCO+(1→0) 0.5 1.0 2.0 6, 7
τHCN(1→0) 0.5 1.0 2.0 6, 7
OPR 0.25 0.25 0.25 10
Note. — OPR = H2 ortho- to para-ratio. References:
1 – Solomon et al. (1987), 2 – Gao & Solomon (2004b), 3 –
Downes & Solomon (1998), 4 – Wu et al. (2005), 5 – Black (2000), 6 –
Nguyen et al. (1992), 7 – Wild et al. (1992), 8 – Lahuis & van Dishoeck
(2000), 9 – Combes (1991), 10 – Neufeld et al. (2006)
ity per unit volume in a particular line is
Lij = X(S)βijAijhνij
d lnx
d lnx, (9)
where νij is the line frequency, fi is an implicit function
of n given by the solution to equations (5) and (6), and
we assume that the abundance X(S) is independent of
n. The line luminosity per unit mass is Lij/(µH2n).
An IDL code that implements this calcu-
lation is available for public download from
http://www.astro.princeton.edu/∼krumholz/ astron-
omy.html.
3. CORRELATIONS AND KENNICUTT-SCHMIDT LAWS
3.1. Lines and Parameters
Using the formalism of § 2, we can now predict the
correlation between the star formation rate and the
luminosity of a galaxy in molecular lines. We make
these predictions for three representative molecular lines:
CO(1 → 0), HCO+(1 → 0), and HCN(1 → 0). For
the first and last of these transitions, there are exten-
sive observational surveys. We select HCO+(1 → 0)
in addition to these two because there is some obser-
vational data for it, and because its critical density of
ncrit = βHCO+4.6 × 104 cm−3 makes it intermediate
between CO(1 → 0), with ncrit = βCO560 cm−3, and
HCN(1 → 0), with ncrit = βHCN2.8 × 105 cm−3.3 Here
βS is the escape probability for the 1 → 0 transition of
species S. These critical densities are for T = 20 K. All
molecular data are taken from the Leiden Atomic and
Molecular Database4 (Schöier et al. 2005).
We make our calculations for three sets of fiducial pa-
rameters which we summarize in Table 1. The three sets
correspond roughly to typical conditions in normal disk
galaxies like the Milky Way, to starburst galaxies like
Arp 220, and to a case intermediate between the two. We
have selected parameters for each case to roughly model
the systematic variation of ISM parameters as one moves
3 Note that our critical density for HCN(1 → 0) is somewhat
larger than the value quoted by Gao & Solomon (2004a,b), proba-
bly because their calculation is based on somewhat different as-
sumptions about how to extrapolate from calculated rate coef-
ficients for HCN collisions with He to collisions with H2. See
Schöier et al. (2005) for details.
4 http://www.strw.leidenuniv.nl/∼moldata/
from normal disk galaxies to starbursts. Thus, we vary
the ISM temperature from 10− 50 K and the molecular
cloud Mach number from 30−80 as we move from Milky
Way-like molecular clouds to temperatures and Mach
numbers typical of starbursts (e.g. Downes & Solomon
1998). Similarly, starbursts, which preferentially occur
at galactic centers, have systematically larger metallici-
ties than galaxies like the Milky Way (e.g. Zaritsky et al.
1994; Yao et al. 2003; Netzer et al. 2005). To explore this
effect, we use abundances and 1 → 0 optical depths are
twice and four times as large for our intermediate and
starburst models, respectively, as for our normal galaxy
model.
3.2. Kennicutt-Schmidt Laws
We first plot, in Figure 1, the quantities L−1[dL(<
n)/d lnn] (solid lines) and M−1[dM(< n)/d lnn] (dot-
ted lines) as a function of density n for galaxies with
mean densities n = 102, 103, and 104 cm−3, for the
tracers CO(1 → 0), HCO+(1 → 0), and HCN(1 →
0), and for the Mach number and temperature corre-
sponding to our intermediate case in Table 1. Here
L(< n) and M(< n) are the luminosity and mass per
unit volume contributed by gas of density n or less,
i.e. L(< n) = X(S)βijAijhνij
∫ lnn
fin(dp/d lnn)d lnn,
M(< n) =
∫ lnn
µH2n(dp/d lnn)d lnn, L = L(< ∞), and
M = M(< ∞). Physically, L−1[dL(< n)/d lnn] and
M−1[dM(< n)/d lnn] represent the fractional contribu-
tion to the total line luminosity and the total mass that
comes from each unit interval in the logarithm of den-
sity. The plot shows what density range provides the
dominant contribution to the line luminosity in differ-
ent lines and for galaxies of differing mean densities, and
how the gas contributing light compares to the gas con-
tributing mass. Because the mass distribution is entirely
specified by n and M, the dotted lines are the same in
each of the three panels. Additionally, because of our
choice M = 50 (Table 1), the median density (the den-
sity corresponding to the peak in M−1[dM(< n)/d lnn])
is nmed ≈ 43n. In each panel, the critical density for
each molecule is identified by a vertical dashed line.
The top panel clearly shows that for the CO line, the
light and the mass track one another very closely, even
at the lowest densities. Thus, because nmed > ncrit, the
http://www.astro.princeton.edu/~krumholz/
Molecular Gas and Kennicutt-Schmidt Laws 5
Fig. 1.— Fractional contribution to the total luminosity
L−1[dL(< n)/d lnn] (solid lines) and mass M−1[dM(< n)/d lnn]
(dotted lines) versus density n for the lines CO(1 → 0) (top
panel), HCO+(1 → 0) (middle panel), and HCN(1 → 0)
(bottom panel). The three curves show the cases n = 102
cm−3, 103 cm−3, and 104 cm−3, from leftmost to rightmost.
We also show the critical density of each molecule, corrected
for radiative trapping (dashed vertical lines). These calcula-
tions use the parameters for the intermediate case listed in Table 1.
solid lines move in lock-step with the dashed lines as n
increases. In contrast, for HCN most of the luminosity
comes from densities near the critical density regardless
of the mass distribution. For the lowest n this means that
the line luminosity is entirely dominated by the high den-
sity tail of the mass distribution. As the median density
nmed varies by a factor of 100 (from 4.3× 103− 4.3× 105
cm−3), the peak of L−1[dL(< n)/d lnn] moves by just
a factor of a few in n. The HCO+ line is intermedi-
ate between CO and HCN. For n = 102 cm−3 and 103
cm−3, nmed
<∼ ncrit, and as with HCN most of the emis-
sion comes from near the critical density. For n = 104
cm−3, nmed > ncrit, and the light starts to follow the
mass, in a pattern similar to that for CO. Although Fig-
ure 1 shows only the intermediate case, the normal galaxy
and starburst cases give qualitatively identical results.
This confirms the intuitive argument given in § 1: high
critical density transitions trace regions of similar den-
sity in every galaxy, while low critical density transitions
trace regions whose density is close to the median density.
Now consider how the luminosity in a given line corre-
lates with the star formation rate in galaxies of varying
mean densities. For a given n, we can compute the vol-
ume density of star formation from equation (2) and the
line luminosity density from (9). To facilitate compari-
son with observations, rather than considering the total
line luminosity, we use the quantity L′ (Solomon et al.
1997), which is related to the luminosity L by
8πkBν2
L, (10)
converted to the units K km s−1 pc2.
Similarly, we can estimate the far infrared luminosity
from the star formation rate. There is a tight correla-
tion between far-IR emission and star formation, par-
ticularly for dense, dusty galaxies like those that make
up most of the dynamic range of the Kennicutt (1998a)
sample. To the extent that most or all of the light from
young stars is re-processed by dust before escaping the
galaxy, the bolometric luminosity integrated over the
wavelength range 8− 1000 µm, which we define as LFIR,
simply provides a calorimetric measurement of the total
energy output by young stars, and is therefore an excel-
lent tracer of recent star formation (Sanders & Mirabel
1996; Rowan-Robinson et al. 1997; Kennicutt 1998a,b;
Hirashita et al. 2003; Bell 2003; Iglesias-Páramo et al.
2006). We therefore estimate the FIR luminosity from
the star formation rate via
LFIR = ǫṀ∗c
2, (11)
where ǫ is an IMF-dependent constant. For consistency
with Kennicutt (1998a,b), we take ǫ = 3.8× 10−4. To be
precise and to facilitate comparison with observations, we
adopt the Sanders & Mirabel (1996) definition of LFIR
as a weighted sum of the luminosity in the 60 and 100
µm IRAS bands. This definition of the infrared luminos-
ity generically underestimates the total infrared luminos-
ity [8 − 1000]µm by a factor of 1.5 − 2 (Calzetti et al.
2000; Dale et al. 2001; Bell 2003). However, we use the
ǫ value appropriate for LFIR rather than for the total
IR luminosity because some of the observations to which
we wish to compare our model (see § 3.3) provide only
LFIR. Note that this choice for the connection between
the star formation rate and the infrared luminosity is not
fully consistent with our choice of the gas temperature
for the three sets of parameters — normal, intermediate,
and starburst — listed in Table 1, an issue we discuss in
more detail in § 4.3.
We plot the ratio of star formation rate to line lumi-
nosity, and infrared luminosity to line luminosity, as a
function of n in Figure 2. First consider the top panel,
which shows all three lines computed for the interme-
diate case. This again confirms our intuitive argument.
Since the luminosity per unit volume in the CO line is
roughly proportional to the mass density, and the star
formation rate / IR luminosity is proportional to mass
density to the 1.5 power, the ratios Ṁ∗/L
′ and LFIR/L
vary roughly as n0.5. A powerlaw fit to the data over the
range shown in Figure 2 gives an index of 0.57. In con-
trast, the ratio of star formation density to HCN luminos-
ity density is nearly constant for galaxies with n < 103
cm−3, and varies quite weakly with n up to densities
of 104 cm−3, values found in the densest starbursts. A
powerlaw fit from 10 cm−3 to 104 cm−3 gives an index of
0.17; from 10 cm−3 and 103 cm−3, the best fit powerlaw
index is 0.08. As in Figure 1, the slope of the Ṁ∗/L
curve for HCO+ represents an intermediate case, with
a roughly constant ratio of Ṁ∗/L
′ and LFIR/L
′ at low
n, rising to a slope comparable to that for CO at high
values of n.
Now consider the bottom three panels in Figure 2.
Each panel shows the ratio of star formation rate and
infrared luminosity to line luminosity for a single line,
computed for each of the three galaxy models. The most
Fig. 2.— Ratio of star formation rate or infrared luminosity to
line luminosity, as a function of mean density n. In the top panel
we show the lines CO(1 → 0) (solid line), HCO+(1 → 0) (dot-
dashed line), and HCN(1 → 0) (dashed line) for the intermediate
case in Table 1. In the next three panels we show the CO(1 → 0),
HCO+(1 → 0), and HCN(1 → 0) lines for the normal galaxy case
(dot-dashed line), intermediate case (solid line), and starburst case
(dashed line).
important point to take from these plots is that the choice
of galaxy model has little effect in most cases. The largest
differences are for HCN, where at n = 10 cm−3 the IR to
line ratio predicted for the intermediate case differs from
the normal galaxy case by a factor of 6.1, and from the
starburst case by a factor of 4.1. This variation comes
primarily from changes in the Mach number and the op-
tical depth between models. The higher Mach number of
the starburst model significantly increases the amount of
mass in the high overdensity tail of the probability dis-
tribution, while the higher optical depth lowers the ef-
fective critical density. Both of these effect increase the
amount of mass dense enough to emit in HCN(1 → 0)
and reduce Ṁ∗/L
′. At higher mean densities these effects
become less important and the models converge, so that
by n = 104 cm−3 the range in Ṁ∗/L
′ from the normal
to the starburst case is only a factor of 3.5.
Most importantly, our central conclusion that
Ṁ∗/L
HCN is roughly constant across galaxies, while
Ṁ∗/L
CO rises as roughly [L
0.5, still holds when we
consider how conditions vary across galaxies. Galaxies
with low mean densities n are generally closest to the
normal galaxy case, while those with high mean densi-
ties should be closest to the starburst case, and this sys-
tematic variation in galaxy properties with n still leaves
Ṁ∗/L
′ relatively flat for HCN, and varying with a slope
close to 0.5 for CO. From the normal galaxy case at
n = 10 cm−3 to the starburst case at n = 104 cm−3,
the value of Ṁ∗/L
′ varies by more than a factor of 50
for the CO(1 → 0), but by less than a factor of 3 for the
HCN(1 → 0).
3.3. Comparison with Observations
The calculations illustrated in Figure 2 demonstrate
the basic argument that one expects a roughly constant
star formation rate per unit line luminosity for high den-
sity tracers (e.g., HCN), and a star formation rate per
unit luminosity that rises like luminosity to the ∼ 0.5
for low density tracers (e.g., CO). However, in large sur-
veys one cannot always determine the mean density in
a galaxy, which would be required to construct an ob-
servational analog to Figure 2. Instead, we can use our
calculated dependence of star formation rate and line
luminosity on density to compare to observations as fol-
lows. Equation (9) gives the total molecular line lumi-
nosity per unit volume and equation (2) gives the star
formation rate, which we convert to an IR luminosity
via equation (11). For fixed assumed volume of molecu-
lar star-forming gas (Vmol) we can then predict the ex-
pected correlations between L′ in a given molecular line
and LFIR. The three panels of Figure 3 show our results
for LFIR as a function of L
CO, L
HCN, and L
for the
intermediate model (see Table 1) and for several values
of Vmol. Figure 4 shows how are results vary as a func-
tion of the assumed T and M. There, for fixed Vmol,
we compare our predictions for the intermediate model
with the normal and starburst models. In both figures we
compare our models to data culled from the literature.
From the work of Gao & Solomon (2004a,b),
Greve et al. (2005, their Fig. 7), Riechers et al. (2006b,
their Fig. 5), and Gao et al. (2007), as well as the
theoretical arguments in the preceding sections, we
expect a strong, but not linear, correlation between
the CO luminosity and the star formation rate —
as measured by LFIR — with the approximate form
LFIR ∝ L
CO . The left panel of Figure 3 shows the
CO data, the approximate correlation expected (solid
line segment; offset from the data for clarity) and the
theoretical prediction (solid lines) for a total volume of
molecular gas of Vmol = 10
7, 108, and 109 pc3. Because
at fixed LFIR, galaxies exhibit a dispersion in Vmol we
expect there to be intrinsic scatter in this correlation,
roughly bracketed by the range of Vmol plotted.
The middle and right panels of Figure 3 show the same
prediction for L′
and L′HCN. In these cases, because
the molecular line luminosity is nearly linearly propor-
tional to LFIR, the dependence on Vmol is much weaker
than for L′CO. However, systematic changes or differences
in the fiducial parameters for the calculation (see Table
1) introduce uncertainty and scatter into the correlation.
Figure 4 assesses this dependence. It compares the pre-
dictions of our model for normal (dot-dashed lines), inter-
mediate (solid lines), and starburst (dashed lines) galax-
ies, as defined in Table 1, for fixed Vmol = 10
8 pc3. Our
simple model reproduces the data rather well, and it pre-
dicts that generically there may be more intrinsic scatter
in the L′CO−LFIR correlation than in either L
HCN−LFIR
or L′
− LFIR.
Molecular Gas and Kennicutt-Schmidt Laws 7
Fig. 3.— LFIR (L⊙) versus L
(1 → 0) (left panel), L′
(1 → 0) (middle panel), and L′
(1 → 0) (K km s−1 pc2; right panel).
The lines in each panel derive from the model presented in this paper with a constant total volume of molecular material of 107, 108, and
109 pc3 (lowest to highest). The thick solid line segment shows power-law slopes to guide the eye. Data in the left and right panels are
from Gao & Solomon (2004a,b) (circles) and Gao et al. (2007) (open squares for detections, arrows for upper limits). The middle panel
combines data from Nguyen et al. (1992) (small circles with lines), Graciá-Carpio et al. (2006) (big circles), and Riechers et al. (2006b)
(open square; using the Gao et al. (2007) FIR luminosity and magnification factor). For all data, LFIR is defined based on a weighted sum
of the galaxy luminosity in the 60 and 100 µm IRAS bands, as described by Sanders & Mirabel (1996). For the Nguyen et al. (1992) data,
the uncertainties in LHCO+ indicated by the lines arise because Nguyen et al. provide both HCN(1 → 0) and HCO
+(1 → 0) intensities,
but the values for L′
derived from their work generally fall a factor of 2 − 3 below the L′
from Gao & Solomon (2004a,b) for the
same systems. This is probably because Nguyen et al. use a single beam pointing rather than integrating fully over extended sources, and
therefore miss some of the flux. We therefore show two values of L′
, connected by a line, for each Nguyen et al. data point: a smaller
value calculated directly from the data listed in their Table 2, and a larger value obtained by multiplying the L′HCN value of Gao & Solomon
for that galaxy by the ratio IHCO+/IHCN measured by Nguyen et al. If this ratio is constant over the source, this estimate should correctly
account for the flux outside the beam in the Nguyen et al. HCO+ observation.
Note that in both the middle and right panels of Fig-
ures 3 and 4, one expects a turn upward in the corre-
lation at high LFIR, a deviation from linearity. This
follows from the fact that in our model, at fixed Vmol,
systems with higher LFIR have higher average densities.
At sufficiently high LFIR we thus expect L
HCN−LFIR and
−LFIR to steepen, in analogy with the L′CO−LFIR
correlation. The data points with very high LFIR in Fig-
ures 3 and 4, which might be used to test this prediction
of our model, are gravitationally lensed, at high redshift,
and contaminated by bright AGN. It is therefore un-
clear if the deviation from linearity implied particularly
by the upper limits in L′HCN in the right panels of Fig-
ures 3 and 4 is a result of enhanced LFIR, caused by the
AGN emission (Carilli et al. 2005), or is instead a result
of less molecular line emission per unit star formation,
as our model implies (Fig. 2). Gao et al. (2007) note,
however, that in the three systems for which the contri-
bution from the AGN has been estimated (F10214+4724,
D. Downes & P. Solomon 2007, in preparation; Clover-
leaf, Weiß et al. 2003; APM 08279+5255, Weiß et al.
2005, 2007) the corrections are only significant for APM
08279+5255. This suggests that the data are so far con-
sistent with our interpretation, but clearly much more
data at high LFIR — or, more precisely for our purposes,
at high density — is required to test our predictions. We
discuss the issue of AGN contamination further in § 4.3.
As a final note, the data so far do support the utility
of HCO+ as a useful tracer of dense gas. Papadopoulos
(2007) has argued against the utility of HCO+ as a faith-
ful tracer of mass in starbursts on the basis that, since it
is an ion, its abundance is strongly dependent on the free-
electron abundance and might therefore vary strongly
between galaxies with different ionizing radiation back-
grounds. We cannot rule out this possibility given the
limited data set available, but we see no strong evi-
dence in favor of it from the data shown in Figures 3
and 4. As we have argued, HCO+(1 → 0) is particu-
larly useful because its critical density is between that
of CO(1 → 0) and HCN(1 → 0) and, thus, as Fig-
ure 3 and 4 show, the correlation between LFIR and
L′HCO+ should steepen from linear to super-linear over
the range of galaxies presented in the CO panels. A care-
ful, large-scale HCO+(1 → 0) survey similar to the work
of Gao & Solomon (2004a) on HCN(1 → 0) should reveal
these trends. Lines with similarly low excitation temper-
atures and intermediate critical densities like CS(1 → 0)
should behave analogously.
4. DISCUSSION
4.1. Implications for Kennicutt-Schmidt Laws and Star
Formation Efficiencies
Our results suggest that KS laws in different tracers
naturally fall into two regimes, although there is a broad
range of molecular tracers that are intermediate between
the two extremes. Tracers for which the critical density is
small compared to the median density in a galaxy repre-
sent one limit. In these tracers, the light faithfully follows
the mass, so the KS law measures a relationship between
total mass and star formation. In any model in which
star formation occurs at a roughly constant rate per dy-
namical time, this must produce a KS law in which the
star formation rate rises with density to a power of near
1.5, and the ratio of star formation to luminosity rises
as density to the 0.5 power. In terms of surface rather
than volume densities, this implies Σ̇∗ ∝ Σ
−1/2. If
we further add the observation that the scale heights h of
the star-forming molecular layers of galaxies are roughly
constant across galaxy types, one form of the observed
Kennicutt (1998a,b) star formation law follows immedi-
ately (Elmegreen 2002). Moreover, in a galactic disk,
Fig. 4.— The same as Figure 3, but with constant Vmol = 10
8 pc3, and for the model parameters corresponding to “starburst” (dashed),
“intermediate” (solid), and “normal” (dot-dashed) (see Table 1). Therefore, the middle solid line in each panel of Figure 3 is the same as
the solid line in each panel in this Figure.
h ∝ Σg/n and n ∝ Ω2/Q (e.g. Thompson et al. 2005);
since in star-forming disks the Toomre-Q is about unity
(Martin & Kennicutt 2001), substituting for h immedi-
ately gives Σ̇∗ ∝ ΣΩ, the alternate form of the Kennicutt
(1998a,b) law.
The other limit is tracers for which the critical density
is large compared to the median galactic density. These
tracers pick out a particular density independent of the
mean or median density in the galaxy, and thus all the
regions they identify have the same dynamical time re-
gardless of galactic environment. In this case the star
formation rate will simply be proportional to the total
mass of the observed regions, yielding a constant ratio of
star formation rate to mass, as is observed for HCN in
the local universe (Fig. 3, right panel; Gao & Solomon
2004a,b; Wu et al. 2005).
We predict that there should be a transition between
linear and super-linear scaling of LFIR with line luminos-
ity at the point where galaxies transition from median
densities that are smaller than the line critical density
to median densities larger than the critical density. The
HCO+(1 → 0) line, and other lines with similar critical
densities, e.g. CS(1 → 0) and SO(1 → 0), should show
this behavior for galaxies in the local universe. The ob-
served correlation between LHCO+ and LFIR appears to
be consistent with our prediction, although at present the
data are not of sufficient quality to distinguish between
a break and a single powerlaw relation. There are hints
that the very highest luminosity star-forming galaxies,
which all reside at high redshift and may well reach ISM
densities not found in any local systems, show such a
break in the IR-HCN correlation.
One important point to emphasize in this analysis is
that we have been able to explain the observed correla-
tions between line and infrared luminosities, and hence
between gas masses at various densities and star for-
mation rates, without resorting to the hypothesis that
the star formation process is fundamentally different in
galaxies of different properties. Although uncertainties
in both our model and the observations do not preclude
an order-unity change in the star formation efficiency or
SFRff as a function of LFIR, there is currently no evi-
dence for such a change in the data, contrary to claims
made by, e.g. Graciá-Carpio et al. (2006). In fact, all
of the observational trends are predicted by our sim-
ple model with constant star formation efficiency. This
is consistent with other lines of evidence that the frac-
tion of mass at a given density that turns into stars
is roughly 1% per free-fall time independent of density
(Krumholz & Tan 2007).
4.2. Does Star Formation Have a Fundamental Size or
Density Scale?
Based on the linear correlation between HCN(1 → 0)
luminosity and star formation rate, seen both in ex-
ternal galaxies and in individual molecular clumps in
the Milky Way, Gao & Solomon (2004a,b) and Wu et al.
(2005) propose that HCN(1 → 0) emission traces a fun-
damental unit of star formation. They explain the linear
IR-HCN correlation as a product of this; in their model,
HCN luminosity correlates linearly with star formation
rate because HCN luminosity simply counts the number
of such units.
Based on our analysis, we argue that this hypothesis is
only partially correct. We concur with Gao & Solomon
and Wu et al. that the HCN(1 → 0) luminosity of a
galaxy does simply reflect the mass of gas that is dense
enough to excite the HCN(1 → 0) line. However, our
analysis shows that this does not necessarily imply that
this density represents a special density in the star for-
mation process, or that objects traced by HCN(1 → 0)
represent a physically distinct class. We show that a
linear correlation between star formation rate and line
luminosity is expected for any line with a critical density
comparable to or larger than the median molecular cloud
density in the galaxies used to define the correlation. It
is possible that HCN(1 → 0)-emitting regions represent
a physically distinct scale of star formation as Wu et al.
propose, but one can explain the linear IR-HCN correla-
tion equally well if they are just part of the same contin-
uous medium as the regions traced by CO(1 → 0) and
by other transitions. Even the star-forming clouds them-
selves may simply be parts of a continuous distribution
of ISM structures occupying the entire galaxy, as argued
by Wada & Norman (2007). In this case there need be
no special density scales other than the mean and me-
dian densities for the star-forming clouds on their largest
scales, and the density at which star formation becomes
rapid, converting the mass into stars in of order a free-
Molecular Gas and Kennicutt-Schmidt Laws 9
fall time. This transition scale is unknown, but must
be considerably larger than the density traced by HCN
(Krumholz & Tan 2007).
4.3. Limitations and Cautions
4.3.1. Self-Consistency
As mentioned in § 3.2, our approach of leaving the gas
temperature T and Mach number M as free parame-
ters is not entirely consistent with our calculation of the
IR luminosity, since the IR luminosity and temperature
are of course related. In principle, with a model of how
the energy output from stars heats the dust and gas, to-
gether with a structural model connecting the energy and
momentum output from stars to the generation of turbu-
lence, it should be possible to self-consistently compute
both the gas temperature and the Mach number from the
volumetric star formation rate (see, e.g., Thompson et al.
2005). Such a model would return T andM as a function
of n and possibly other galaxy properties, while simulta-
neously predicting a set of Kennicutt-Schmidt laws.
If the line luminosity depended strongly on T or M,
or if one required knowledge of the temperature to com-
pute the infrared luminosity of a galaxy, we would would
have no alternative to constructing such a model if we
wished to explain the observed IR-line luminosity cor-
relation. However, we can avoid this by relying on the
observationally-calibrated star formation-IR correlation,
and because, as we show in Figures 2, 3, and 4, the line
luminosity varies quite weakly over a reasonable range of
T and M for our chosen lines. For this reason, any model
for computing T and M as a function of galaxy proper-
ties, if it were consistent with observations, would not
significantly alter the IR-line luminosity correlation we
derive. This is true, however, only for lines that require
low temperatures to excite. As we discuss in § 4.3.2,
lines that require higher temperatures to excite do de-
pend sensitively on the temparature in the galaxy, and a
model capable of predicting the IR-line luminosity cor-
relation for these lines must also include a calculation of
the temperature structure of the galaxy.
4.3.2. Isothermality
Our assumption of isothermality means that our anal-
ysis will only apply to molecular lines for which the tem-
perature Tup corresponding to the upper state energy is
< 10 K, low enough to be excited even in the coolest
molecular clouds in normal spiral galaxies. The reason
for this is that at temperatures larger than Tup, the lu-
minosity in a line generally varies at most linearly with
the temperature. As the similarity between the results
with our different galaxy models illustrates, changing the
temperature within the range of ∼ 10 − 50 K produces
only a factor of a few change in the luminosity of the
lines we have studied. In contrast, line luminosity re-
sponds exponentially to temperature changes when the
temperature is below the value corresponding to the up-
per state’s energy. This means that lines sensitive to
high temperatures pick out primarily the regions that
are warm enough for the line to be excited. Density has
only a secondary effect. The emission will therefore re-
flect the temperature distribution in star-forming clouds
more than the density distribution, an effect that our
isothermal assumption precludes us from treating. KS
laws in high temperature tracers are likely to find lin-
ear relationships between star formation rate and mass
regardless of the critical density of the molecule in ques-
tion because they will simply be correlating the mass
of dust warmed to >∼ 100 Kelvin, which is essentially
what is measured by LFIR, with the mass of gas warmed
to temperatures above Tup. However, our model will
not apply to these lines, and for this reason we do not
attempt to compare to observations using higher tran-
sitions of CO (3 → 2, 4 → 3, 5 → 4, 6 → 5, and
7 → 6, which have Tup = 33, 55, 83, 116, and 154 K,
respectively; Greve et al. 2005, Solomon & Vanden Bout
2005), CS(5 → 4) (Tup = 35 K; Plume et al. 1997), or
other high temperature tracers.
4.3.3. Molecular Abundances
We have not considered density-dependent variations
in molecule abundances. One potential source of vari-
ation in molecular abundance is freeze-out onto grain
surfaces at high densities and low temperatures (e.g.
Tafalla et al. 2004a,b). Chemodynamical models suggest
that freeze-out is not likely to become significant for ei-
ther carbonaceous or nitrogenous species until densities
n >∼ 106 cm−3 (Flower et al. 2006), but may become se-
vere at higher densities, so whether depletion is signif-
icant depends on what fraction of the total luminosity
would be contributed by gas of this density or higher
were there no freeze-out. Figure 1 suggests that freeze-
out is likely to modify the total galactic luminosity of
CO, HCO+, and HCN fairly little even at a mean ISM
density of n = 104 cm−3, but may have significant effects
for galaxies of larger mean densities or for lines for which
the critical densities is comparable to the freeze-out den-
sity. If freeze-out is significant, our conclusions will be
modified.
4.3.4. Atomic Gas
In the simple model developed here, we have neglected
the role of atomic gas entirely. Whether the density or
surface density of atomic gas plays a role in controlling
the star formation rate is subject to debate on both ob-
servational and theoretical grounds (Kennicutt 1998a,b;
Wong & Blitz 2002; Heyer et al. 2004; Komugi et al.
2005; Krumholz & McKee 2005; Kennicutt et al. 2007),
so it is unclear how much a limitation this omission re-
ally is. We can say with confidence that in molecule-rich
galaxies, which provide almost all the dynamic range of
both the Kennicutt (1998a,b) correlation and the cor-
relations illustrated in Figures 3 and 4, the atomic gas
plays almost no role simply because there is so little of
it. Thus, our predictions should be quite robust, except
perhaps at the very low luminosity ends of Figures 3 and
4.3.5. AGN Contributions
A final point is not so much a limitation of our work as
a cautionary note about comparing our model with ob-
servations. We have included in our model IR luminosity
only from star formation, and molecular line luminosity
only from molecules in cold star-forming clouds. How-
ever, an AGN may make a significant contribution to a
galaxy’s luminosity in the far infrared by direct heating of
dust grains, and in molecular lines via an X-ray dissocia-
tion region. Indeed, several of the systems with the high-
est IR luminosities in Figures 3 and 4 are contaminated
by AGN. As noted in §3.3, this complicates an assessment
of our prediction of an up-turn in the L′HCN − LFIR and
L′HCO+ − LFIR correlations at high luminosity. This de-
viation from linearity at high gas density (at fixed Vmol,
high LFIR) is an essential prediction of our model, but
testing it relies on a careful separation of the contribu-
tion of the AGN to both the IR and line luminosities (e.g.
Maloney et al. 1996). In fact, Carilli et al. (2005) discuss
the possibility that the AGN’s contribution to the IR lu-
minosity in these systems causes them to be above the
local linear L′HCN−LFIR correlation. Such a contamina-
tion would mimic the prediction of our model. However,
Gao et al. (2007) argue that the sub-millimeter galaxies
in their sample are not AGN dominated and that just
one of three quasars in their sample (APM 08279+5255)
has a large AGN IR component. See Gao et al. (2007)
for more discussion. For these reasons we contend that
although our model is consistent with the existing data,
the current evidence for a break in the L′HCN−LFIR cor-
relation should be viewed with caution and more data
in high density/luminosity systems is clearly required to
understand the role of AGN contamination in shaping
the correlation.
5. CONCLUSIONS
We provide a simple model for understanding how
Kennicutt-Schmidt laws, which relate the star formation
rate to the mass or surface density of gas as inferred from
some particular line, depend on the line chosen to define
the correlation. We show that for a turbulent medium
the luminosity per unit volume in a given line, provided
that line can be excited at temperatures lower than the
mean temperature in a galaxy’s molecular clouds, in-
creases faster than linearly with the density for molecules
with critical densities larger than the median gas density.
The star formation rate also rises super-linearly with the
gas density, and the combination of these two effects pro-
duces a close to linear correlation between star formation
rate and line luminosity. In contrast, the line luminosity
rises only linearly with density for lines with low critical
densities, producing a correlation between star formation
rate and line luminosity that is super-linear.
Based on this analysis, we construct a model for the
correlation between a galaxy’s infrared luminosity and its
luminosity in a particular molecular line. Our model is
extremely simple, in that it relies on an observationally-
calibrated IR-star formation rate correlation, it treats
molecular clouds as having homogenous density and ve-
locity distributions, temperatures, and chemical compo-
sitions, and it only very crudely accounts for variations
in molecular cloud properties across galaxies. Despite
these approximations, the model naturally explains why
some observed correlations between infrared luminosity
and line luminosity in galaxies are linear, and some are
super-linear. Using it, we are able to compute quantita-
tively the correlation between infrared and HCN(1 → 0)
line luminosity, and between IR and CO(1 → 0) line lu-
minosity. We show that our model provides a very good
fit to observations in these lines, and we are able to make
similar predictions for any molecular line that can be ex-
cited at low temperatures, as we demonstrate for the
example of HCO+(1 → 0). Moreover, we are able to ex-
plain the observed data without recourse to the hypoth-
esis that the star formation process is somehow different,
either more or less efficient, in different types of galaxies
or for media of different densities. Instead, our model is
able to explain the observed correlations using a simple,
universal star formation law.
One strong prediction of our model is that there should
be a break from linear to non-linear scaling in the HCN-
IR correlation at very high IR luminosity, and a similar
break in the HCO+-IR correlation at somewhat lower lu-
minosity. The data for HCO+ are consistent with this
prediction but do not yet strongly favor a break over pure
powerlaw behavior. However, there is some preliminary
evidence for a break in the IR-HCN correlation in high
redshift galaxies more luminous than any found in the
local universe, although with these high redshift obser-
vations it is difficult to rule out the alternative explana-
tion of the break as arising due to a progressively rising
AGN contribution to the IR luminosity (see §3.3 and
§4.3.5). Future galaxy surveys both in the local universe
and at high redshift may be used to test our predictions
for HCO+(1 → 0), HCN(1 → 0), and other molecular
lines.
We thank L. Blitz, B. Draine, A. Leroy, E. Rosolowsky,
and A. Socrates for helpful discussions, N. Evans and
the anonymous referee for useful comments on the
manuscript, and R. Kennicutt for kindly providing a
preprint of his submitted paper. We thank Y. Gao for
providing LFIR for the systems used in Figures 3 and 4.
MRK acknowledges support from NASA through Hubble
Fellowship grant #HSF-HF-01186 awarded by the Space
Telescope Science Institute, which is operated by the As-
sociation of Universities for Research in Astronomy, Inc.,
for NASA, under contract NAS 5-26555. TAT acknowl-
edges support from a Lyman Spitzer, Jr. Fellowship.
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420, 87

  We provide a model for how Kennicutt-Schmidt (KS) laws, which describe the
correlation between star formation rate and gas surface or volume density,
depend on the molecular line chosen to trace the gas. We show that, for lines
that can be excited at low temperatures, the KS law depends on how the line
critical density compares to the median density in a galaxy's star-forming
molecular clouds. High critical density lines trace regions with similar
physical properties across galaxy types, and this produces a linear correlation
between line luminosity and star formation rate. Low critical density lines
probe regions whose properties vary across galaxies, leading to a star
formation rate that varies superlinearly with line luminosity. We show that a
simple model in which molecular clouds are treated as isothermal and homogenous
can quantitatively reproduce the observed correlations between galactic
luminosities in far infrared and in the CO(1->0) and HCN(1->0) lines, and
naturally explains why these correlations have different slopes. We predict
that IR-line luminosity correlations should change slope for galaxies in which
the median density is close to the line critical density. This prediction may
be tested by observations of lines such as HCO^+(1->0) with intermediate
critical densities, or by HCN(1->0) observations of intensely star-forming high
redshift galaxies with very high densities. Recent observations by Gao et al.
hint at just such a change in slope. We argue that deviations from linearity in
the HCN(1->0)-IR correlation at high luminosity are consistent with the
assumption of a constant star formation efficiency.

Accepted for publication in the Astrophysical Journal, July 16, 2007
Preprint typeset using LATEX style emulateapj v. 08/22/09
THE RELATIONSHIP BETWEEN MOLECULAR GAS TRACERS AND KENNICUTT-SCHMIDT LAWS
Mark R. Krumholz
and Todd A. Thompson
Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544
Accepted for publication in the Astrophysical Journal, July 16, 2007
ABSTRACT
We provide a model for how Kennicutt-Schmidt (KS) laws, which describe the correlation between
star formation rate and gas surface or volume density, depend on the molecular line chosen to trace
the gas. We show that, for lines that can be excited at low temperatures, the KS law depends on how
the line critical density compares to the median density in a galaxy’s star-forming molecular clouds.
High critical density lines trace regions with similar physical properties across galaxy types, and this
produces a linear correlation between line luminosity and star formation rate. Low critical density
lines probe regions whose properties vary across galaxies, leading to a star formation rate that varies
superlinearly with line luminosity. We show that a simple model in which molecular clouds are treated
as isothermal and homogenous can quantitatively reproduce the observed correlations between galactic
luminosities in far infrared and in the CO(1 → 0) and HCN(1 → 0) lines, and naturally explains why
these correlations have different slopes. We predict that IR-line luminosity correlations should change
slope for galaxies in which the median density is close to the line critical density. This prediction
may be tested by observations of lines such as HCO+(1 → 0) with intermediate critical densities, or
by HCN(1 → 0) observations of intensely star-forming high redshift galaxies with very high densities.
Recent observations by Gao et al. hint at just such a change in slope. We argue that deviations from
linearity in the HCN(1 → 0)−IR correlation at high luminosity are consistent with the assumption of
a constant star formation efficiency.
Subject headings: ISM: clouds — ISM: molecules — stars: formation — galaxies: ISM — radio lines:
1. INTRODUCTION
Schmidt (1959, 1963) first proposed that the rate at
which a gas forms stars might follow a simple power
law correlation of the form ρ̇∗ ∝ ρNg , where ρ̇∗ is the
star formation rate per unit volume, ρg is the gas den-
sity, and N is generally taken to be in the range 1 − 2.
In the decades since, observations have revealed two
strong correlations that appear to be evidence for this
hypothesis. First, galaxy surveys reveal that the in-
frared luminosity of a galaxy, which traces the star for-
mation rate, varies with its luminosity in the CO(1 → 0)
line, which traces the total mass of molecular gas, as
LFIR ∝ L1.4−1.6CO (Gao & Solomon 2004a,b; Greve et al.
2005; Riechers et al. 2006a). Kennicutt (1998a,b) iden-
tified the closely-related correlation between gas surface
density Σg and star formation rate surface density Σ̇∗,
Σ̇∗ ∝ Σ1.4±0.15g , a relation that has come to be known
as the Kennicutt Law. Since over the bulk of the dy-
namic range of Kennicutt’s data galaxies are predomi-
nantly molecular, this is effectively a correlation between
molecular gas, as traced by CO(1 → 0) line emission,
and star formation. Spatially resolved observations of
galaxies confirm that, at least for molecule-rich galax-
ies where resolved CO(1 → 0) observations are possible,
star formation is more closely coupled with gas traced by
CO(1 → 0) than with atomic gas (Wong & Blitz 2002;
Heyer et al. 2004; Komugi et al. 2005; Kennicutt et al.
2007)
Electronic address: krumholz@astro.princeton.edu, thomp@astro.princeton.edu
1 Hubble Fellow
2 Lyman Spitzer Jr. Fellow
Second, Gao & Solomon (2004a,b) find that there is a
strong correlation between the IR luminosity of galax-
ies and emission in the HCN(1 → 0) line, which mea-
sures the mass at densities significantly greater than
that probed by CO(1 → 0). However, they find that
their correlation, which covers nearly three decades in
total galactic star formation rate, is linear: LFIR ∝
LHCN. Wu et al. (2005) show that this correlation ex-
tends down to individual star-forming clumps of gas in
the Milky Way, provided that their infrared luminosities
are >∼ 104.5 L⊙. Interestingly, however, Gao et al. (2007)
find a deviation from linearity in the IR-HCN correlation
for a sample of intensely star-forming high redshift galax-
ies. These sources show small but significant excesses of
infrared emission for their observed HCN emission.
The difference in power law indices between the LFIR−
LCO and LFIR −LHCN correlations is statistically signif-
icant, and, on its face, puzzling. An index near N = 1.5
seems natural if one supposes that a roughly constant
fraction of the gas present in molecular clouds will be
converted into stars each free-fall time. In this case one
expects ρ̇∗ ∝ ρ1.5g (Madore 1977; Elmegreen 1994). If
gas scale heights do not vary strongly from galaxy to
galaxy, this implies Σ̇∗ ∝ Σ1.5g as well, which is consis-
tent with the observed Kennicutt law. More generally,
since the dynamical timescale in a marginally Toomre-
stable (Q ≈ 1; see Martin & Kennicutt 2001) galactic
disk is of order Ω−1 ∝ (Gρg)−1/2, where Ω is the angular
frequency of the disk, an index close to N = 1.5 is ex-
pected if star formation is regulated by any phenomenon
that converts a fixed fraction of the gas into stars on this
time scale (Elmegreen 2002).
http://arxiv.org/abs/0704.0792v2
mailto:krumholz@astro.princeton.edu, thomp@astro.princeton.edu
On the other hand, Wu et al. (2005) suggest a simple
interpretation of the linear IR-HCN correlation. They ar-
gue that the individual HCN-emitting molecular clumps
that they identify in the Milky Way represent a funda-
mental unit of star formation. The linear correlation
between star formation rate and HCN luminosity across
galaxies arises because a measurement of the HCN lu-
minosity for a galaxy simply counts the number of such
structures present within it, each of which forms stars at
some roughly fixed rate regardless of its galactic environ-
ment. However, in this interpretation it is unclear why
the structures traced by HCN(1 → 0) emission should
form stars at the same rate in any galaxy. After all, one
could equally well argue that molecular clouds traced by
CO(1 → 0) are fundamental units of star formation, but
the non-linear IR-CO correlation clearly shows that these
objects do not form stars at a fixed rate per unit mass.
Moreover, the evidence presented by Gao et al. (2007)
that the linear IR-HCN correlation varies in extremely
luminous high redshift galaxies suggests that the rela-
tionship between HCN emission and star-formation may
be somewhat more complex.
In this paper we attempt to explain the origin of the
difference in slope between the CO and HCN correla-
tions with star formation rate, and more generally to
give a theoretical framework for understanding how cor-
relations between star formation rate and line luminosity,
which we generically refer to as Kennicutt-Schmidt (KS)
laws, depend on the tracer used to define them. Our cen-
tral argument is conceptually quite simple, and in some
sense represents a combination of the intuitive arguments
for CO and HCN given above.
Consider an observation of a galaxy in a molecular
tracer with critical density ncrit, which essentially mea-
sures the mass of gas at densities of ncrit or more, i.e. the
gas that is dense enough for that particular transition to
be excited. In galaxies where the median density of the
molecular gas is significantly larger than ncrit, this means
that the observation will detect the majority of the gas,
and the bulk of the emission will come from gas whose
density is near the median density. Since the gas den-
sity will vary from galaxy to galaxy, the star formation
rate per unit gas mass will vary as roughly ρ1.5g , with one
factor of ρg coming from the amount of gas available for
star formation, and an additional factor of ρ0.5g coming
from the dependence of the free-fall or dynamical time
on the density.
On the other hand, in galaxies where the median gas
density is small compared to the critical density for the
chosen transition, observations will pick out only high
density peaks. Since the density in these peaks is set by
ncrit, and not by the conditions in the galaxy, these peaks
are at essentially the same density in any galaxy where
they are observed, and the corresponding free-fall times
in these regions are constant as well. As a result, the star
formation rate per unit mass of gas traced by that line
is approximately the same in every galaxy, because the
corresponding free-fall time is the same in every galaxy.
In the rest of this paper, we give a quantitative version
of this intuitive argument, and then discuss its conse-
quences. In § 2 we develop a simple formalism to com-
pute the star formation rate and the molecular line lu-
minosity of galaxies, and in § 3 we use this formalism to
predict the correlation between star formation rate and
luminosity. We show that our predictions provide a very
good fit for a variety of observations, and make predic-
tions for future observations. We discuss the implications
of our work and its limitations in § 4, and summarize our
conclusions is § 5.
2. STAR FORMATION RATES AND LINE LUMINOSITIES
2.1. Cloud Properties
Consider a galaxy in which the star-forming molecu-
lar clouds have a volume-averaged mean molecular hy-
drogen number density n = ρg/µH2 , where ρg is the
volume-averaged mass density of the molecular clouds in
the galaxy and µH2 = 3.9×10−24 g is the mean mass per
hydrogen molecule for a gas of standard cosmic composi-
tion. Observations indicate that n varies by two to three
decades over the galaxies for which the Kennicutt and
Gao & Solomon correlations are measured, from n ≈ 50
cm−3 in normal spirals like the Milky Way (McKee
1999) up to n ≈ 104 cm−3 in the strongest starburst
systems in the local universe (e.g. Downes & Solomon
1998). There is strong evidence that densities in molec-
ular clouds follow a lognormal probability distribution
function (PDF; see reviews by Mac Low & Klessen 2004
and Elmegreen & Scalo 2004)
d lnx
(lnx− lnx)2
, (1)
where x = n/n is the molecular hydrogen number den-
sity n relative to the average density, σ is the width of
the lognormal, and lnx = −σ2/2. For this distribution
the median density is nmed = n exp(σ
2/2). Numerical
experiments show that for supersonic isothermal turbu-
lence σ2 ≈ ln
1 + 3M2/4
, where M is the 1D Mach
number of the turbulence (Nordlund & Padoan 1999;
Ostriker et al. 1999; Padoan & Nordlund 2002). Mach
numbers in star-forming molecular clouds range from
M ∼ 30 (McKee 1999) in normal spirals to M ∼ 100
in strong starbursts (Downes & Solomon 1998), imply-
ing that median densities in molecular clouds range from
∼ 103 cm−3 in normal spirals to ∼ 106 cm−3 in star-
bursts. Star forming clouds within a galaxy are approx-
imately isothermal, except very near strong sources of
stellar radiation, so we assume a fixed temperature T
for the clouds. Observationally, T ranges from roughly
10 K in normal spirals (McKee 1999) up to as much
about 50 K in strong starbursts (Downes & Solomon
1998; Gao & Solomon 2004b).
2.2. Star Formation Rates
First let us ask how quickly stars form in such a
medium. Krumholz & McKee (2005) give a model for
star formation regulated by supersonic turbulence in
which a population of molecular clouds of total mass Mcl
form stars at a rate Ṁ∗ = SFRffMcl/tff(n), where tff(n)
is the free-fall time evaluated at the mean density and
SFRff is a number of order 10
−2 that depends weakly on
M. We therefore estimate the star formation rate per
unit volume as a function of the mean density given by
ρ̇∗ ≈ SFRff
32Gµ3H2n
. (2)
Molecular Gas and Kennicutt-Schmidt Laws 3
We adopt the Krumholz & McKee result SFRff ≈
0.014(M/100)−0.32 for clouds with a fiducial virial ra-
tio of αvir = 1.3.
Alternately, Krumholz & Tan (2007) point out that
observed correlations between the star formation rate
and the luminosity in different density tracers imply that
over a 3− 4 decade range in density n,
Ṁ∗ ≈ 10−2
Mcl(> n)
tff(n)
, (3)
where Mcl(> n) is the mass of gas with a density of n
or higher, and Mcl = Mcl(> 0). For a given choice of
n this provides an alternative estimate of the star for-
mation rate which is purely empirical, and independent
of any particular theoretical model. However, the differ-
ence between the star formation rates predicted by (2)
and (3) is small. For gas with a lognormal PDF,
Mcl(> n) =
1 + erf
−2 lnx+ σ2
23/2σ
, (4)
and using this to evaluate equation (3) indicates that, for
Mach numbers in the observed range, the two prescrip-
tions (2) and (3) give about the same star formation rate
over a very broad range in x. For example, at M = 30
the two estimates agree to within a factor of 3 for den-
sities in the range 0.2 < x < 4 × 104. Given the scatter
inherent in observational estimates of the star formation
rate, a factor of 3 difference is not particularly signifi-
cant, so it matters little which prescription we adopt. In
practice, we will use equation (2).
2.3. Line Luminosities
Now we must compute the luminosity of molecular line
emission from the galaxy. Even for a cloud that is not
in local thermodynamic equilibrium (LTE), for optically
thin emission this calculation is straightforward. How-
ever, the molecular lines used most often in galaxy sur-
veys are generally optically thick. To handle the effect of
finite optical depth on molecule level populations and line
luminosities, we adopt an escape probability approxima-
tion and treat clouds as homogeneous spheres. This is
not fully consistent with our assumption that clouds have
lognormal density PDFs, since the escape probability for-
malism assumes a uniform level population throughout
the cloud, and the essence of our argument in this paper
turns on how the level population varies with density.
However, this approach gives us an approximate way of
incorporating the optical thickness of star-forming clouds
into our model, the only alternative to which for turbu-
lent media is full numerical simulation (e.g. Juvela et al.
2001). We therefore proceed by treating clouds as ho-
mogeneous in order to determine their escape probabili-
ties, and we then relax the assumption of homogeneity,
while keeping the escape probabilities fixed, in order to
determine level populations and cloud luminosities as a
function of density.
Consider a cloud of radius R in statistical equilibrium
but not necessarily in LTE. In the escape probability
approximation, the fraction fi of molecules of species S
in state i is given implicitly by the linear system
(nqji + βjiAji) fj =
(nqij + βijAij)
 fi (5)
fi=1, (6)
where qij is the collision rate for transitions from state
i to state j, Aij is the Einstein spontaneous emission
coefficient for this transition, βij is the cloud-averaged
escape probability for photons emitted in this transition,
the sums are over all quantum states, and we understand
that Aij = 0 for i ≤ j and qij = 0 for i = j.
Equations (5) and (6) allow us to compute the level
populations fi for given values of βij . To completely
specify the system, we must add an additional consis-
tency condition relating the values of βij to the level
populations. For a homogeneous spherical cloud, the es-
cape probability for a given line is related to the optical
depth from the center to the edge of the cloud τij by
(B. Draine, 2007, private communication)
βij ≈
1 + 0.5τij
, (7)
where τij is computed at the central frequency of the
line. In turn, the optical depth is related to the level
populations by
τij =
4(2π)3/2Mcs
nX(S)fjR
, (8)
where λij is the wavelength of transition i → j, gi and
gj are the statistical weights of states i and j, cs is
the isothermal sound speed of the gas, and X(S) is the
abundance of molecules of species S. Note that this
equation implicitly assumes that the cloud has a uni-
form Maxwellian velocity distribution with 1D disper-
sion Mcs, consistent with our treatment of the clouds
as homogeneous spheres. One additional complication is
that we do not directly know cloud radii for most exter-
nal galaxies, where observations cannot resolve individ-
ual molecular clouds. However, we often can diagnose
the optical depths of transitions by comparing line ratios
of molecular isotopomers of different abundances. We
therefore take τ10, the optical depth of the transition be-
tween the first excited state and the ground state, as
known. For a given level population this fixes the value
of R.
We solve equations (5)–(8) using Newton-Raphson it-
eration. In this procedure, we guess an initial set of es-
cape probabilities βij , and solve the linear system (5) and
(6) to find the corresponding initial level populations fi.
We then compute the optical depths τij from equation
(8). The guessed escape probabilities βij and the corre-
sponding optical depths τij generally will not satisfy the
consistency condition (7), so we then iterate over βij val-
ues using a Newton-Raphson approach, seeking βij for
which the level populations give optical depths τij such
that all elements of the matrix βij − 1/(1 + 0.5τij) are
equal to zero within some specified tolerance. We use
the LTE level populations and escape probabilities for
our initial guess, so that the iteration converges rapidly
when the system is close to LTE.
Once we have determined the escape probabilities βij ,
we compute the luminosity by holding the βij values fixed
but allowing the level populations to vary with density,
then integrating over the PDF. Thus, the total luminos-
TABLE 1
Model Parameters
Parameter Normal galaxy Intermediate Starburst Reference
T 10 20 50 1–4
M 30 50 80 1–4
X(CO) 2× 10−4 4× 10−4 8× 10−4 5
X(HCO+) 2× 10−9 4× 10−9 8× 10−9 6, 7
X(HCN) 1× 10−8 2× 10−8 4× 10−8 6–8
τCO(1→0) 10 20 40 9
τHCO+(1→0) 0.5 1.0 2.0 6, 7
τHCN(1→0) 0.5 1.0 2.0 6, 7
OPR 0.25 0.25 0.25 10
Note. — OPR = H2 ortho- to para-ratio. References:
1 – Solomon et al. (1987), 2 – Gao & Solomon (2004b), 3 –
Downes & Solomon (1998), 4 – Wu et al. (2005), 5 – Black (2000), 6 –
Nguyen et al. (1992), 7 – Wild et al. (1992), 8 – Lahuis & van Dishoeck
(2000), 9 – Combes (1991), 10 – Neufeld et al. (2006)
ity per unit volume in a particular line is
Lij = X(S)βijAijhνij
d lnx
d lnx, (9)
where νij is the line frequency, fi is an implicit function
of n given by the solution to equations (5) and (6), and
we assume that the abundance X(S) is independent of
n. The line luminosity per unit mass is Lij/(µH2n).
An IDL code that implements this calcu-
lation is available for public download from
http://www.astro.princeton.edu/∼krumholz/ astron-
omy.html.
3. CORRELATIONS AND KENNICUTT-SCHMIDT LAWS
3.1. Lines and Parameters
Using the formalism of § 2, we can now predict the
correlation between the star formation rate and the
luminosity of a galaxy in molecular lines. We make
these predictions for three representative molecular lines:
CO(1 → 0), HCO+(1 → 0), and HCN(1 → 0). For
the first and last of these transitions, there are exten-
sive observational surveys. We select HCO+(1 → 0)
in addition to these two because there is some obser-
vational data for it, and because its critical density of
ncrit = βHCO+4.6 × 104 cm−3 makes it intermediate
between CO(1 → 0), with ncrit = βCO560 cm−3, and
HCN(1 → 0), with ncrit = βHCN2.8 × 105 cm−3.3 Here
βS is the escape probability for the 1 → 0 transition of
species S. These critical densities are for T = 20 K. All
molecular data are taken from the Leiden Atomic and
Molecular Database4 (Schöier et al. 2005).
We make our calculations for three sets of fiducial pa-
rameters which we summarize in Table 1. The three sets
correspond roughly to typical conditions in normal disk
galaxies like the Milky Way, to starburst galaxies like
Arp 220, and to a case intermediate between the two. We
have selected parameters for each case to roughly model
the systematic variation of ISM parameters as one moves
3 Note that our critical density for HCN(1 → 0) is somewhat
larger than the value quoted by Gao & Solomon (2004a,b), proba-
bly because their calculation is based on somewhat different as-
sumptions about how to extrapolate from calculated rate coef-
ficients for HCN collisions with He to collisions with H2. See
Schöier et al. (2005) for details.
4 http://www.strw.leidenuniv.nl/∼moldata/
from normal disk galaxies to starbursts. Thus, we vary
the ISM temperature from 10− 50 K and the molecular
cloud Mach number from 30−80 as we move from Milky
Way-like molecular clouds to temperatures and Mach
numbers typical of starbursts (e.g. Downes & Solomon
1998). Similarly, starbursts, which preferentially occur
at galactic centers, have systematically larger metallici-
ties than galaxies like the Milky Way (e.g. Zaritsky et al.
1994; Yao et al. 2003; Netzer et al. 2005). To explore this
effect, we use abundances and 1 → 0 optical depths are
twice and four times as large for our intermediate and
starburst models, respectively, as for our normal galaxy
model.
3.2. Kennicutt-Schmidt Laws
We first plot, in Figure 1, the quantities L−1[dL(<
n)/d lnn] (solid lines) and M−1[dM(< n)/d lnn] (dot-
ted lines) as a function of density n for galaxies with
mean densities n = 102, 103, and 104 cm−3, for the
tracers CO(1 → 0), HCO+(1 → 0), and HCN(1 →
0), and for the Mach number and temperature corre-
sponding to our intermediate case in Table 1. Here
L(< n) and M(< n) are the luminosity and mass per
unit volume contributed by gas of density n or less,
i.e. L(< n) = X(S)βijAijhνij
∫ lnn
fin(dp/d lnn)d lnn,
M(< n) =
∫ lnn
µH2n(dp/d lnn)d lnn, L = L(< ∞), and
M = M(< ∞). Physically, L−1[dL(< n)/d lnn] and
M−1[dM(< n)/d lnn] represent the fractional contribu-
tion to the total line luminosity and the total mass that
comes from each unit interval in the logarithm of den-
sity. The plot shows what density range provides the
dominant contribution to the line luminosity in differ-
ent lines and for galaxies of differing mean densities, and
how the gas contributing light compares to the gas con-
tributing mass. Because the mass distribution is entirely
specified by n and M, the dotted lines are the same in
each of the three panels. Additionally, because of our
choice M = 50 (Table 1), the median density (the den-
sity corresponding to the peak in M−1[dM(< n)/d lnn])
is nmed ≈ 43n. In each panel, the critical density for
each molecule is identified by a vertical dashed line.
The top panel clearly shows that for the CO line, the
light and the mass track one another very closely, even
at the lowest densities. Thus, because nmed > ncrit, the
http://www.astro.princeton.edu/~krumholz/
Molecular Gas and Kennicutt-Schmidt Laws 5
Fig. 1.— Fractional contribution to the total luminosity
L−1[dL(< n)/d lnn] (solid lines) and mass M−1[dM(< n)/d lnn]
(dotted lines) versus density n for the lines CO(1 → 0) (top
panel), HCO+(1 → 0) (middle panel), and HCN(1 → 0)
(bottom panel). The three curves show the cases n = 102
cm−3, 103 cm−3, and 104 cm−3, from leftmost to rightmost.
We also show the critical density of each molecule, corrected
for radiative trapping (dashed vertical lines). These calcula-
tions use the parameters for the intermediate case listed in Table 1.
solid lines move in lock-step with the dashed lines as n
increases. In contrast, for HCN most of the luminosity
comes from densities near the critical density regardless
of the mass distribution. For the lowest n this means that
the line luminosity is entirely dominated by the high den-
sity tail of the mass distribution. As the median density
nmed varies by a factor of 100 (from 4.3× 103− 4.3× 105
cm−3), the peak of L−1[dL(< n)/d lnn] moves by just
a factor of a few in n. The HCO+ line is intermedi-
ate between CO and HCN. For n = 102 cm−3 and 103
cm−3, nmed
<∼ ncrit, and as with HCN most of the emis-
sion comes from near the critical density. For n = 104
cm−3, nmed > ncrit, and the light starts to follow the
mass, in a pattern similar to that for CO. Although Fig-
ure 1 shows only the intermediate case, the normal galaxy
and starburst cases give qualitatively identical results.
This confirms the intuitive argument given in § 1: high
critical density transitions trace regions of similar den-
sity in every galaxy, while low critical density transitions
trace regions whose density is close to the median density.
Now consider how the luminosity in a given line corre-
lates with the star formation rate in galaxies of varying
mean densities. For a given n, we can compute the vol-
ume density of star formation from equation (2) and the
line luminosity density from (9). To facilitate compari-
son with observations, rather than considering the total
line luminosity, we use the quantity L′ (Solomon et al.
1997), which is related to the luminosity L by
8πkBν2
L, (10)
converted to the units K km s−1 pc2.
Similarly, we can estimate the far infrared luminosity
from the star formation rate. There is a tight correla-
tion between far-IR emission and star formation, par-
ticularly for dense, dusty galaxies like those that make
up most of the dynamic range of the Kennicutt (1998a)
sample. To the extent that most or all of the light from
young stars is re-processed by dust before escaping the
galaxy, the bolometric luminosity integrated over the
wavelength range 8− 1000 µm, which we define as LFIR,
simply provides a calorimetric measurement of the total
energy output by young stars, and is therefore an excel-
lent tracer of recent star formation (Sanders & Mirabel
1996; Rowan-Robinson et al. 1997; Kennicutt 1998a,b;
Hirashita et al. 2003; Bell 2003; Iglesias-Páramo et al.
2006). We therefore estimate the FIR luminosity from
the star formation rate via
LFIR = ǫṀ∗c
2, (11)
where ǫ is an IMF-dependent constant. For consistency
with Kennicutt (1998a,b), we take ǫ = 3.8× 10−4. To be
precise and to facilitate comparison with observations, we
adopt the Sanders & Mirabel (1996) definition of LFIR
as a weighted sum of the luminosity in the 60 and 100
µm IRAS bands. This definition of the infrared luminos-
ity generically underestimates the total infrared luminos-
ity [8 − 1000]µm by a factor of 1.5 − 2 (Calzetti et al.
2000; Dale et al. 2001; Bell 2003). However, we use the
ǫ value appropriate for LFIR rather than for the total
IR luminosity because some of the observations to which
we wish to compare our model (see § 3.3) provide only
LFIR. Note that this choice for the connection between
the star formation rate and the infrared luminosity is not
fully consistent with our choice of the gas temperature
for the three sets of parameters — normal, intermediate,
and starburst — listed in Table 1, an issue we discuss in
more detail in § 4.3.
We plot the ratio of star formation rate to line lumi-
nosity, and infrared luminosity to line luminosity, as a
function of n in Figure 2. First consider the top panel,
which shows all three lines computed for the interme-
diate case. This again confirms our intuitive argument.
Since the luminosity per unit volume in the CO line is
roughly proportional to the mass density, and the star
formation rate / IR luminosity is proportional to mass
density to the 1.5 power, the ratios Ṁ∗/L
′ and LFIR/L
vary roughly as n0.5. A powerlaw fit to the data over the
range shown in Figure 2 gives an index of 0.57. In con-
trast, the ratio of star formation density to HCN luminos-
ity density is nearly constant for galaxies with n < 103
cm−3, and varies quite weakly with n up to densities
of 104 cm−3, values found in the densest starbursts. A
powerlaw fit from 10 cm−3 to 104 cm−3 gives an index of
0.17; from 10 cm−3 and 103 cm−3, the best fit powerlaw
index is 0.08. As in Figure 1, the slope of the Ṁ∗/L
curve for HCO+ represents an intermediate case, with
a roughly constant ratio of Ṁ∗/L
′ and LFIR/L
′ at low
n, rising to a slope comparable to that for CO at high
values of n.
Now consider the bottom three panels in Figure 2.
Each panel shows the ratio of star formation rate and
infrared luminosity to line luminosity for a single line,
computed for each of the three galaxy models. The most
Fig. 2.— Ratio of star formation rate or infrared luminosity to
line luminosity, as a function of mean density n. In the top panel
we show the lines CO(1 → 0) (solid line), HCO+(1 → 0) (dot-
dashed line), and HCN(1 → 0) (dashed line) for the intermediate
case in Table 1. In the next three panels we show the CO(1 → 0),
HCO+(1 → 0), and HCN(1 → 0) lines for the normal galaxy case
(dot-dashed line), intermediate case (solid line), and starburst case
(dashed line).
important point to take from these plots is that the choice
of galaxy model has little effect in most cases. The largest
differences are for HCN, where at n = 10 cm−3 the IR to
line ratio predicted for the intermediate case differs from
the normal galaxy case by a factor of 6.1, and from the
starburst case by a factor of 4.1. This variation comes
primarily from changes in the Mach number and the op-
tical depth between models. The higher Mach number of
the starburst model significantly increases the amount of
mass in the high overdensity tail of the probability dis-
tribution, while the higher optical depth lowers the ef-
fective critical density. Both of these effect increase the
amount of mass dense enough to emit in HCN(1 → 0)
and reduce Ṁ∗/L
′. At higher mean densities these effects
become less important and the models converge, so that
by n = 104 cm−3 the range in Ṁ∗/L
′ from the normal
to the starburst case is only a factor of 3.5.
Most importantly, our central conclusion that
Ṁ∗/L
HCN is roughly constant across galaxies, while
Ṁ∗/L
CO rises as roughly [L
0.5, still holds when we
consider how conditions vary across galaxies. Galaxies
with low mean densities n are generally closest to the
normal galaxy case, while those with high mean densi-
ties should be closest to the starburst case, and this sys-
tematic variation in galaxy properties with n still leaves
Ṁ∗/L
′ relatively flat for HCN, and varying with a slope
close to 0.5 for CO. From the normal galaxy case at
n = 10 cm−3 to the starburst case at n = 104 cm−3,
the value of Ṁ∗/L
′ varies by more than a factor of 50
for the CO(1 → 0), but by less than a factor of 3 for the
HCN(1 → 0).
3.3. Comparison with Observations
The calculations illustrated in Figure 2 demonstrate
the basic argument that one expects a roughly constant
star formation rate per unit line luminosity for high den-
sity tracers (e.g., HCN), and a star formation rate per
unit luminosity that rises like luminosity to the ∼ 0.5
for low density tracers (e.g., CO). However, in large sur-
veys one cannot always determine the mean density in
a galaxy, which would be required to construct an ob-
servational analog to Figure 2. Instead, we can use our
calculated dependence of star formation rate and line
luminosity on density to compare to observations as fol-
lows. Equation (9) gives the total molecular line lumi-
nosity per unit volume and equation (2) gives the star
formation rate, which we convert to an IR luminosity
via equation (11). For fixed assumed volume of molecu-
lar star-forming gas (Vmol) we can then predict the ex-
pected correlations between L′ in a given molecular line
and LFIR. The three panels of Figure 3 show our results
for LFIR as a function of L
CO, L
HCN, and L
for the
intermediate model (see Table 1) and for several values
of Vmol. Figure 4 shows how are results vary as a func-
tion of the assumed T and M. There, for fixed Vmol,
we compare our predictions for the intermediate model
with the normal and starburst models. In both figures we
compare our models to data culled from the literature.
From the work of Gao & Solomon (2004a,b),
Greve et al. (2005, their Fig. 7), Riechers et al. (2006b,
their Fig. 5), and Gao et al. (2007), as well as the
theoretical arguments in the preceding sections, we
expect a strong, but not linear, correlation between
the CO luminosity and the star formation rate —
as measured by LFIR — with the approximate form
LFIR ∝ L
CO . The left panel of Figure 3 shows the
CO data, the approximate correlation expected (solid
line segment; offset from the data for clarity) and the
theoretical prediction (solid lines) for a total volume of
molecular gas of Vmol = 10
7, 108, and 109 pc3. Because
at fixed LFIR, galaxies exhibit a dispersion in Vmol we
expect there to be intrinsic scatter in this correlation,
roughly bracketed by the range of Vmol plotted.
The middle and right panels of Figure 3 show the same
prediction for L′
and L′HCN. In these cases, because
the molecular line luminosity is nearly linearly propor-
tional to LFIR, the dependence on Vmol is much weaker
than for L′CO. However, systematic changes or differences
in the fiducial parameters for the calculation (see Table
1) introduce uncertainty and scatter into the correlation.
Figure 4 assesses this dependence. It compares the pre-
dictions of our model for normal (dot-dashed lines), inter-
mediate (solid lines), and starburst (dashed lines) galax-
ies, as defined in Table 1, for fixed Vmol = 10
8 pc3. Our
simple model reproduces the data rather well, and it pre-
dicts that generically there may be more intrinsic scatter
in the L′CO−LFIR correlation than in either L
HCN−LFIR
or L′
− LFIR.
Molecular Gas and Kennicutt-Schmidt Laws 7
Fig. 3.— LFIR (L⊙) versus L
(1 → 0) (left panel), L′
(1 → 0) (middle panel), and L′
(1 → 0) (K km s−1 pc2; right panel).
The lines in each panel derive from the model presented in this paper with a constant total volume of molecular material of 107, 108, and
109 pc3 (lowest to highest). The thick solid line segment shows power-law slopes to guide the eye. Data in the left and right panels are
from Gao & Solomon (2004a,b) (circles) and Gao et al. (2007) (open squares for detections, arrows for upper limits). The middle panel
combines data from Nguyen et al. (1992) (small circles with lines), Graciá-Carpio et al. (2006) (big circles), and Riechers et al. (2006b)
(open square; using the Gao et al. (2007) FIR luminosity and magnification factor). For all data, LFIR is defined based on a weighted sum
of the galaxy luminosity in the 60 and 100 µm IRAS bands, as described by Sanders & Mirabel (1996). For the Nguyen et al. (1992) data,
the uncertainties in LHCO+ indicated by the lines arise because Nguyen et al. provide both HCN(1 → 0) and HCO
+(1 → 0) intensities,
but the values for L′
derived from their work generally fall a factor of 2 − 3 below the L′
from Gao & Solomon (2004a,b) for the
same systems. This is probably because Nguyen et al. use a single beam pointing rather than integrating fully over extended sources, and
therefore miss some of the flux. We therefore show two values of L′
, connected by a line, for each Nguyen et al. data point: a smaller
value calculated directly from the data listed in their Table 2, and a larger value obtained by multiplying the L′HCN value of Gao & Solomon
for that galaxy by the ratio IHCO+/IHCN measured by Nguyen et al. If this ratio is constant over the source, this estimate should correctly
account for the flux outside the beam in the Nguyen et al. HCO+ observation.
Note that in both the middle and right panels of Fig-
ures 3 and 4, one expects a turn upward in the corre-
lation at high LFIR, a deviation from linearity. This
follows from the fact that in our model, at fixed Vmol,
systems with higher LFIR have higher average densities.
At sufficiently high LFIR we thus expect L
HCN−LFIR and
−LFIR to steepen, in analogy with the L′CO−LFIR
correlation. The data points with very high LFIR in Fig-
ures 3 and 4, which might be used to test this prediction
of our model, are gravitationally lensed, at high redshift,
and contaminated by bright AGN. It is therefore un-
clear if the deviation from linearity implied particularly
by the upper limits in L′HCN in the right panels of Fig-
ures 3 and 4 is a result of enhanced LFIR, caused by the
AGN emission (Carilli et al. 2005), or is instead a result
of less molecular line emission per unit star formation,
as our model implies (Fig. 2). Gao et al. (2007) note,
however, that in the three systems for which the contri-
bution from the AGN has been estimated (F10214+4724,
D. Downes & P. Solomon 2007, in preparation; Clover-
leaf, Weiß et al. 2003; APM 08279+5255, Weiß et al.
2005, 2007) the corrections are only significant for APM
08279+5255. This suggests that the data are so far con-
sistent with our interpretation, but clearly much more
data at high LFIR — or, more precisely for our purposes,
at high density — is required to test our predictions. We
discuss the issue of AGN contamination further in § 4.3.
As a final note, the data so far do support the utility
of HCO+ as a useful tracer of dense gas. Papadopoulos
(2007) has argued against the utility of HCO+ as a faith-
ful tracer of mass in starbursts on the basis that, since it
is an ion, its abundance is strongly dependent on the free-
electron abundance and might therefore vary strongly
between galaxies with different ionizing radiation back-
grounds. We cannot rule out this possibility given the
limited data set available, but we see no strong evi-
dence in favor of it from the data shown in Figures 3
and 4. As we have argued, HCO+(1 → 0) is particu-
larly useful because its critical density is between that
of CO(1 → 0) and HCN(1 → 0) and, thus, as Fig-
ure 3 and 4 show, the correlation between LFIR and
L′HCO+ should steepen from linear to super-linear over
the range of galaxies presented in the CO panels. A care-
ful, large-scale HCO+(1 → 0) survey similar to the work
of Gao & Solomon (2004a) on HCN(1 → 0) should reveal
these trends. Lines with similarly low excitation temper-
atures and intermediate critical densities like CS(1 → 0)
should behave analogously.
4. DISCUSSION
4.1. Implications for Kennicutt-Schmidt Laws and Star
Formation Efficiencies
Our results suggest that KS laws in different tracers
naturally fall into two regimes, although there is a broad
range of molecular tracers that are intermediate between
the two extremes. Tracers for which the critical density is
small compared to the median density in a galaxy repre-
sent one limit. In these tracers, the light faithfully follows
the mass, so the KS law measures a relationship between
total mass and star formation. In any model in which
star formation occurs at a roughly constant rate per dy-
namical time, this must produce a KS law in which the
star formation rate rises with density to a power of near
1.5, and the ratio of star formation to luminosity rises
as density to the 0.5 power. In terms of surface rather
than volume densities, this implies Σ̇∗ ∝ Σ
−1/2. If
we further add the observation that the scale heights h of
the star-forming molecular layers of galaxies are roughly
constant across galaxy types, one form of the observed
Kennicutt (1998a,b) star formation law follows immedi-
ately (Elmegreen 2002). Moreover, in a galactic disk,
Fig. 4.— The same as Figure 3, but with constant Vmol = 10
8 pc3, and for the model parameters corresponding to “starburst” (dashed),
“intermediate” (solid), and “normal” (dot-dashed) (see Table 1). Therefore, the middle solid line in each panel of Figure 3 is the same as
the solid line in each panel in this Figure.
h ∝ Σg/n and n ∝ Ω2/Q (e.g. Thompson et al. 2005);
since in star-forming disks the Toomre-Q is about unity
(Martin & Kennicutt 2001), substituting for h immedi-
ately gives Σ̇∗ ∝ ΣΩ, the alternate form of the Kennicutt
(1998a,b) law.
The other limit is tracers for which the critical density
is large compared to the median galactic density. These
tracers pick out a particular density independent of the
mean or median density in the galaxy, and thus all the
regions they identify have the same dynamical time re-
gardless of galactic environment. In this case the star
formation rate will simply be proportional to the total
mass of the observed regions, yielding a constant ratio of
star formation rate to mass, as is observed for HCN in
the local universe (Fig. 3, right panel; Gao & Solomon
2004a,b; Wu et al. 2005).
We predict that there should be a transition between
linear and super-linear scaling of LFIR with line luminos-
ity at the point where galaxies transition from median
densities that are smaller than the line critical density
to median densities larger than the critical density. The
HCO+(1 → 0) line, and other lines with similar critical
densities, e.g. CS(1 → 0) and SO(1 → 0), should show
this behavior for galaxies in the local universe. The ob-
served correlation between LHCO+ and LFIR appears to
be consistent with our prediction, although at present the
data are not of sufficient quality to distinguish between
a break and a single powerlaw relation. There are hints
that the very highest luminosity star-forming galaxies,
which all reside at high redshift and may well reach ISM
densities not found in any local systems, show such a
break in the IR-HCN correlation.
One important point to emphasize in this analysis is
that we have been able to explain the observed correla-
tions between line and infrared luminosities, and hence
between gas masses at various densities and star for-
mation rates, without resorting to the hypothesis that
the star formation process is fundamentally different in
galaxies of different properties. Although uncertainties
in both our model and the observations do not preclude
an order-unity change in the star formation efficiency or
SFRff as a function of LFIR, there is currently no evi-
dence for such a change in the data, contrary to claims
made by, e.g. Graciá-Carpio et al. (2006). In fact, all
of the observational trends are predicted by our sim-
ple model with constant star formation efficiency. This
is consistent with other lines of evidence that the frac-
tion of mass at a given density that turns into stars
is roughly 1% per free-fall time independent of density
(Krumholz & Tan 2007).
4.2. Does Star Formation Have a Fundamental Size or
Density Scale?
Based on the linear correlation between HCN(1 → 0)
luminosity and star formation rate, seen both in ex-
ternal galaxies and in individual molecular clumps in
the Milky Way, Gao & Solomon (2004a,b) and Wu et al.
(2005) propose that HCN(1 → 0) emission traces a fun-
damental unit of star formation. They explain the linear
IR-HCN correlation as a product of this; in their model,
HCN luminosity correlates linearly with star formation
rate because HCN luminosity simply counts the number
of such units.
Based on our analysis, we argue that this hypothesis is
only partially correct. We concur with Gao & Solomon
and Wu et al. that the HCN(1 → 0) luminosity of a
galaxy does simply reflect the mass of gas that is dense
enough to excite the HCN(1 → 0) line. However, our
analysis shows that this does not necessarily imply that
this density represents a special density in the star for-
mation process, or that objects traced by HCN(1 → 0)
represent a physically distinct class. We show that a
linear correlation between star formation rate and line
luminosity is expected for any line with a critical density
comparable to or larger than the median molecular cloud
density in the galaxies used to define the correlation. It
is possible that HCN(1 → 0)-emitting regions represent
a physically distinct scale of star formation as Wu et al.
propose, but one can explain the linear IR-HCN correla-
tion equally well if they are just part of the same contin-
uous medium as the regions traced by CO(1 → 0) and
by other transitions. Even the star-forming clouds them-
selves may simply be parts of a continuous distribution
of ISM structures occupying the entire galaxy, as argued
by Wada & Norman (2007). In this case there need be
no special density scales other than the mean and me-
dian densities for the star-forming clouds on their largest
scales, and the density at which star formation becomes
rapid, converting the mass into stars in of order a free-
Molecular Gas and Kennicutt-Schmidt Laws 9
fall time. This transition scale is unknown, but must
be considerably larger than the density traced by HCN
(Krumholz & Tan 2007).
4.3. Limitations and Cautions
4.3.1. Self-Consistency
As mentioned in § 3.2, our approach of leaving the gas
temperature T and Mach number M as free parame-
ters is not entirely consistent with our calculation of the
IR luminosity, since the IR luminosity and temperature
are of course related. In principle, with a model of how
the energy output from stars heats the dust and gas, to-
gether with a structural model connecting the energy and
momentum output from stars to the generation of turbu-
lence, it should be possible to self-consistently compute
both the gas temperature and the Mach number from the
volumetric star formation rate (see, e.g., Thompson et al.
2005). Such a model would return T andM as a function
of n and possibly other galaxy properties, while simulta-
neously predicting a set of Kennicutt-Schmidt laws.
If the line luminosity depended strongly on T or M,
or if one required knowledge of the temperature to com-
pute the infrared luminosity of a galaxy, we would would
have no alternative to constructing such a model if we
wished to explain the observed IR-line luminosity cor-
relation. However, we can avoid this by relying on the
observationally-calibrated star formation-IR correlation,
and because, as we show in Figures 2, 3, and 4, the line
luminosity varies quite weakly over a reasonable range of
T and M for our chosen lines. For this reason, any model
for computing T and M as a function of galaxy proper-
ties, if it were consistent with observations, would not
significantly alter the IR-line luminosity correlation we
derive. This is true, however, only for lines that require
low temperatures to excite. As we discuss in § 4.3.2,
lines that require higher temperatures to excite do de-
pend sensitively on the temparature in the galaxy, and a
model capable of predicting the IR-line luminosity cor-
relation for these lines must also include a calculation of
the temperature structure of the galaxy.
4.3.2. Isothermality
Our assumption of isothermality means that our anal-
ysis will only apply to molecular lines for which the tem-
perature Tup corresponding to the upper state energy is
< 10 K, low enough to be excited even in the coolest
molecular clouds in normal spiral galaxies. The reason
for this is that at temperatures larger than Tup, the lu-
minosity in a line generally varies at most linearly with
the temperature. As the similarity between the results
with our different galaxy models illustrates, changing the
temperature within the range of ∼ 10 − 50 K produces
only a factor of a few change in the luminosity of the
lines we have studied. In contrast, line luminosity re-
sponds exponentially to temperature changes when the
temperature is below the value corresponding to the up-
per state’s energy. This means that lines sensitive to
high temperatures pick out primarily the regions that
are warm enough for the line to be excited. Density has
only a secondary effect. The emission will therefore re-
flect the temperature distribution in star-forming clouds
more than the density distribution, an effect that our
isothermal assumption precludes us from treating. KS
laws in high temperature tracers are likely to find lin-
ear relationships between star formation rate and mass
regardless of the critical density of the molecule in ques-
tion because they will simply be correlating the mass
of dust warmed to >∼ 100 Kelvin, which is essentially
what is measured by LFIR, with the mass of gas warmed
to temperatures above Tup. However, our model will
not apply to these lines, and for this reason we do not
attempt to compare to observations using higher tran-
sitions of CO (3 → 2, 4 → 3, 5 → 4, 6 → 5, and
7 → 6, which have Tup = 33, 55, 83, 116, and 154 K,
respectively; Greve et al. 2005, Solomon & Vanden Bout
2005), CS(5 → 4) (Tup = 35 K; Plume et al. 1997), or
other high temperature tracers.
4.3.3. Molecular Abundances
We have not considered density-dependent variations
in molecule abundances. One potential source of vari-
ation in molecular abundance is freeze-out onto grain
surfaces at high densities and low temperatures (e.g.
Tafalla et al. 2004a,b). Chemodynamical models suggest
that freeze-out is not likely to become significant for ei-
ther carbonaceous or nitrogenous species until densities
n >∼ 106 cm−3 (Flower et al. 2006), but may become se-
vere at higher densities, so whether depletion is signif-
icant depends on what fraction of the total luminosity
would be contributed by gas of this density or higher
were there no freeze-out. Figure 1 suggests that freeze-
out is likely to modify the total galactic luminosity of
CO, HCO+, and HCN fairly little even at a mean ISM
density of n = 104 cm−3, but may have significant effects
for galaxies of larger mean densities or for lines for which
the critical densities is comparable to the freeze-out den-
sity. If freeze-out is significant, our conclusions will be
modified.
4.3.4. Atomic Gas
In the simple model developed here, we have neglected
the role of atomic gas entirely. Whether the density or
surface density of atomic gas plays a role in controlling
the star formation rate is subject to debate on both ob-
servational and theoretical grounds (Kennicutt 1998a,b;
Wong & Blitz 2002; Heyer et al. 2004; Komugi et al.
2005; Krumholz & McKee 2005; Kennicutt et al. 2007),
so it is unclear how much a limitation this omission re-
ally is. We can say with confidence that in molecule-rich
galaxies, which provide almost all the dynamic range of
both the Kennicutt (1998a,b) correlation and the cor-
relations illustrated in Figures 3 and 4, the atomic gas
plays almost no role simply because there is so little of
it. Thus, our predictions should be quite robust, except
perhaps at the very low luminosity ends of Figures 3 and
4.3.5. AGN Contributions
A final point is not so much a limitation of our work as
a cautionary note about comparing our model with ob-
servations. We have included in our model IR luminosity
only from star formation, and molecular line luminosity
only from molecules in cold star-forming clouds. How-
ever, an AGN may make a significant contribution to a
galaxy’s luminosity in the far infrared by direct heating of
dust grains, and in molecular lines via an X-ray dissocia-
tion region. Indeed, several of the systems with the high-
est IR luminosities in Figures 3 and 4 are contaminated
by AGN. As noted in §3.3, this complicates an assessment
of our prediction of an up-turn in the L′HCN − LFIR and
L′HCO+ − LFIR correlations at high luminosity. This de-
viation from linearity at high gas density (at fixed Vmol,
high LFIR) is an essential prediction of our model, but
testing it relies on a careful separation of the contribu-
tion of the AGN to both the IR and line luminosities (e.g.
Maloney et al. 1996). In fact, Carilli et al. (2005) discuss
the possibility that the AGN’s contribution to the IR lu-
minosity in these systems causes them to be above the
local linear L′HCN−LFIR correlation. Such a contamina-
tion would mimic the prediction of our model. However,
Gao et al. (2007) argue that the sub-millimeter galaxies
in their sample are not AGN dominated and that just
one of three quasars in their sample (APM 08279+5255)
has a large AGN IR component. See Gao et al. (2007)
for more discussion. For these reasons we contend that
although our model is consistent with the existing data,
the current evidence for a break in the L′HCN−LFIR cor-
relation should be viewed with caution and more data
in high density/luminosity systems is clearly required to
understand the role of AGN contamination in shaping
the correlation.
5. CONCLUSIONS
We provide a simple model for understanding how
Kennicutt-Schmidt laws, which relate the star formation
rate to the mass or surface density of gas as inferred from
some particular line, depend on the line chosen to define
the correlation. We show that for a turbulent medium
the luminosity per unit volume in a given line, provided
that line can be excited at temperatures lower than the
mean temperature in a galaxy’s molecular clouds, in-
creases faster than linearly with the density for molecules
with critical densities larger than the median gas density.
The star formation rate also rises super-linearly with the
gas density, and the combination of these two effects pro-
duces a close to linear correlation between star formation
rate and line luminosity. In contrast, the line luminosity
rises only linearly with density for lines with low critical
densities, producing a correlation between star formation
rate and line luminosity that is super-linear.
Based on this analysis, we construct a model for the
correlation between a galaxy’s infrared luminosity and its
luminosity in a particular molecular line. Our model is
extremely simple, in that it relies on an observationally-
calibrated IR-star formation rate correlation, it treats
molecular clouds as having homogenous density and ve-
locity distributions, temperatures, and chemical compo-
sitions, and it only very crudely accounts for variations
in molecular cloud properties across galaxies. Despite
these approximations, the model naturally explains why
some observed correlations between infrared luminosity
and line luminosity in galaxies are linear, and some are
super-linear. Using it, we are able to compute quantita-
tively the correlation between infrared and HCN(1 → 0)
line luminosity, and between IR and CO(1 → 0) line lu-
minosity. We show that our model provides a very good
fit to observations in these lines, and we are able to make
similar predictions for any molecular line that can be ex-
cited at low temperatures, as we demonstrate for the
example of HCO+(1 → 0). Moreover, we are able to ex-
plain the observed data without recourse to the hypoth-
esis that the star formation process is somehow different,
either more or less efficient, in different types of galaxies
or for media of different densities. Instead, our model is
able to explain the observed correlations using a simple,
universal star formation law.
One strong prediction of our model is that there should
be a break from linear to non-linear scaling in the HCN-
IR correlation at very high IR luminosity, and a similar
break in the HCO+-IR correlation at somewhat lower lu-
minosity. The data for HCO+ are consistent with this
prediction but do not yet strongly favor a break over pure
powerlaw behavior. However, there is some preliminary
evidence for a break in the IR-HCN correlation in high
redshift galaxies more luminous than any found in the
local universe, although with these high redshift obser-
vations it is difficult to rule out the alternative explana-
tion of the break as arising due to a progressively rising
AGN contribution to the IR luminosity (see §3.3 and
§4.3.5). Future galaxy surveys both in the local universe
and at high redshift may be used to test our predictions
for HCO+(1 → 0), HCN(1 → 0), and other molecular
lines.
We thank L. Blitz, B. Draine, A. Leroy, E. Rosolowsky,
and A. Socrates for helpful discussions, N. Evans and
the anonymous referee for useful comments on the
manuscript, and R. Kennicutt for kindly providing a
preprint of his submitted paper. We thank Y. Gao for
providing LFIR for the systems used in Figures 3 and 4.
MRK acknowledges support from NASA through Hubble
Fellowship grant #HSF-HF-01186 awarded by the Space
Telescope Science Institute, which is operated by the As-
sociation of Universities for Research in Astronomy, Inc.,
for NASA, under contract NAS 5-26555. TAT acknowl-
edges support from a Lyman Spitzer, Jr. Fellowship.
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