# Star formation inefficiency and Kennicutt-Schmidt laws in Early-Type
Galaxies

Brian Jiang  Department of Physics, Columbia University, 550 West 120th St,
New York, NY 10027, USA  Luca Ciotti  Department of Physics and Astronomy,
University of Bologna, Bologna, Italy  Zhaoming Gan  New Mexico Consortium,
Los Alamos, NM 87544, USA  Department of Astronomy, Columbia University, 550
West 120th St, New York, NY 10027, USA  Jeremiah P. Ostriker  Department of
Astronomy, Columbia University, 550 West 120th St, New York, NY 10027, USA
Department of Astrophysical Sciences, Princeton University, Princeton, NJ
08544, USA.

###### Abstract

Star formation in disk galaxies is observed to follow the empirical Kennicutt-
Schmidt law, a power-law relationship between the surface density of gas
$\Sigma_{\rm gas}$ and the star formation rate per unit surface
$\dot{\Sigma}_{*}$. In contrast to disk galaxies, early-type galaxies (ETGs)
are typically associated with little to no star formation, and thus are
usually termed “quiescent”. Recent observations, however, have noted the
presence of massive gaseous cold disks in ETGs, raising the question as to why
the conversion of gas into stars is truly inefficient. With the aid of the
latest simulations, performed with our high-resolution hydrodynamic numerical
code MACER, we reevaluate the traditional classification of ETGs as quiescent,
dead galaxies. In fact, in the presence of moderate galaxy rotational support,
circumnuclear gaseous disks of kpc size form following cooling episodes of the
ISM, and are not destroyed by AGN feedback. The issue of star formation in
such disks is therefore unavoidable. In MACER we follow star formation by
considering two channels, that of 1) Toomre instability and 2) gas
cooling/Jeans instability. We find that the resulting Kennicutt-Schmidt laws
for the simulated ETGs reproduce the observed slope in disk galaxies, though
with considerable scatter and lower normalization by a factor of $\approx 2$
or more for the highest mass galaxies. The Toomre instability is the main
driver of the slope, while cooling/Jeans induced star formation dominates the
central regions. Observational checks of our star formation predictions are
thus essential for confirming the form of local star formation laws and
reassessing star formation inefficiency in ETGs. The process we describe is
similar to that of the star forming disks - of much lower luminosity - found
in the central spheroidal regions of local spirals, such as the MW and M31.

###### Subject headings:

galaxies: elliptical and lenticular; galaxies: star formation

##  1\. Introduction

Understanding how the physical properties of interstellar gas affect star
formation is important for developing models of galactic evolution and
possibly explaining the differences in star formation rate (SFR) across
different galaxy types. A robust empirical correlation between the SFR and gas
density in disk galaxies was first reported by Schmidt (1959), suggesting a
(volumetric) star formation law for the Milky Way well described by a power-
law $\dot{\rho}_{*}\propto\rho_{\rm gas}^{n}$, with $2<n<3$. Starting with
Kennicutt’s compilation of H$\alpha$ measurements to trace star formation and
HI and CO data to trace atomic and molecular gas, it has been found that both
global (Kennicutt 1998) and resolved ( Kennicutt 1989, hereafter K89) laws
relating SFR and gas surface densities read
$\dot{\Sigma}_{*}\propto\Sigma_{\rm gas}^{n}$, with $1<n<3$, and $n\approx
1.4$ as the most accepted value. This empirical law, universally known as the
Kennicutt-Schmidt law (hereafter KS) forms the critical basis for current
theoretical and numerical work on common disk galaxies. Here and in the
following, the global observables entail averaging the gas surface density and
the star formation rate within some prescribed radius in the galactic disk,
while the resolved values of $\Sigma_{\rm gas}$ and $\dot{\Sigma}_{*}$ are
just angular averages over annuli of prescribed inner and outer radii $R$ and
$R+\Delta R$.

Due to the importance of the subject, it is not surprising that a huge amount
of observational, numerical, and theoretical work has been done to elucidate
the physics behind the observed star formation rates. As the real phenomenon
involves turbulent hydrodynamics, gravitational instabilites, radiative
transport, magnetic fields, etc., in general a phenomenological approach is
used (with considerable success) in the attempt to capture the basic physical
principles driving star formation. In this framework, two main channels of
star formation at work in relatively isolated galaxies (i.e., galaxies where
gas rich merging events and other environmental effects such as near
encounters, tidal stripping and harassment, can be neglected) can be easily
identified. The first is related to local/global instabilities in rotating
gaseous disks (hereafter the “Toomre instability”, see e.g. Binney & Tremaine
2008). The second is the also well known criterion based on the comparison
between the cooling and the Jeans instability times (hereafter the
“cooling/Jeans” channel). Both channels are almost certainly at work in star
formation episodes hosted by rotating, massive cold gaseous disks. Regarding
the Toomre instability channel, theory and simulations have suggested that in
disk galaxies, gravitational instability criterions (Boissier et al., 2003;
Kennicutt, 1998) accurately replicate both the KS power-law (see Leroy et al.
2008 for qualification) and the cutoff threshold, while different disk
thickness (Bacchini et al., 2019), turbulence (Shetty & Ostriker, 2008), and
shear (Davis et al., 2014) can contribute to the observed scatter in $n$.
Additionally, efforts have also gone into examining modified KS relations with
terms including orbital velocities and velocity dispersions of stars (Schaye,
2004), with the aim of finding a universal relationship between gas and SFR
densities in all types of star-forming galaxies (see Sun et al. 2023).
Regarding the cooling/Jeans channel, it is obvious that in case of dense and
cold gas, the consequences of the associated short time scales are
inescapable, resulting in fast cooling, fragmentation, collapse of the ISM,
and consequent star formation. For example, numerical simulations restricted
to the cooling/Jeans channel (with no AGN feedback, and in absence of Toomre
instability) have shown that the resulting star formation rates are in nice
qualitative agreement with the observed slope of the KS empirical law Negri et
al. (2015).

Therefore, a first natural question is if the Toomre and the cooling/Jeans
channels are cooperative, or complementary, in the process of star formation
in rotating gaseous disks. A second, more observationally motivated question,
concerns the systematic differences of star formation as a function of the
morphological type of the host galaxy. In fact, in contrast to disk galaxies,
ETGs are typically classified as “red and dead” stellar systems, with quenched
star formation as a result of a post star-burst depletion of the molecular gas
reservoir (Cappellari et al., 2011; Baron et al., 2022), and successive
maintenance of galactic winds due to the cooperative effects of SNIa, and
recurrent AGN feedback events fueled by periodic cooling of the ISM produced
by the galaxy aging stellar population (e.g., Ciotti et. al 1991 and Ciotti &
Ostriker 1997; see also Kim & Pellegrini 2012). However, in sufficiently
massive ETGs, the SNIa and AGN feedback effects are not strong enough to
maintain the galaxy devoid of the gas produced by the stars, and so in such
systems some central star formation will occur. Our work (Ciotti et al. 2022,
hereafter C22) and that of others leads to the conclusion that for massive
ETGs ($M_{*}\approx 10^{9}$), roughly $10^{11}M_{\odot}$ of gas is ejected by
feedback processes (SNIa and AGN feedback) and a remaining $10^{9}M_{\odot}$
falls to a central kpc disk in what was initially termed “cooling flow” (e.g.,
see Fabian 1994, Matthews & Brighenti 2003) and references therein. There it
can fall to the center to feed the AGN of be transformed locally into a (top
heavy) population of new stars, the most massive of which will eject, by
dynamical processes, some fraction of the gas from which they were born. In
fact observations show that ETGs possess some reservoir of cold gas, with
approximately 50 percent of massive ETGS (stellar mass $M_{*}\gtrsim
10^{10}\;M_{\odot}$) containing $10^{6}-10^{9}\;M_{\odot}$ of cold gas in the
form of atomic and molecular hydrogen (e.g., see Cappellari et al. 2011, Li et
al. 2020): while the gas reservoirs for massive disk galaxies can reach
$\approx 10^{10}-10^{11}M_{\odot}$ (Zhou et al. 2021, Saintonge & Catinella
2022), validating the picture of overall quiescent ETGs, non-negligible star
formation seems inevitable given the two channels above. Quite remarkably, it
has been noted that ETGs - for given amount of cold gas available - lie on a
significantly lower regime compared to starburst and disk galaxies on the
global KS relations (Davis et al., 2014), implying that conversion from gas to
stars in ETGs is more inefficient per unit mass. This inefficiency of star
formation, despite non-negligible reservoirs of cold gas, has been identified
as one of the most persistent problems in the field of star formation (Peng et
al., 2015).

As numerical simulations have demonstrated (Negri et al. 2015, C22), ordered
rotation enhances ISM instabilities and radiative cooling; because the cooling
rate grows quadratically with density, the increasing ISM density triggers (as
noted above) a cooling flow. As the catastrophically cooling ISM looses its
thermal pressure, it accumulates onto a circumnuclear disk due to the angular
momentum barrier. Though the gaseous disk masses of ETGs are typically one to
two orders of magnitude less than what has been observed for spiral galaxies,
disk instabilities and density-dependent cooling rates dictate that nonzero
star formation is inevitable. ETGs, however, are noted to be significantly
less efficient in turning this cold gas to stars in comparison to spiral
galaxies. The reason why star formation is so inefficient in ETGs, however, is
not well understood; possible explanations involve the stabilizing influence
of stellar population on gaseous disks (Kawata et al., 2007), low disk self-
gravity and increased shear (Martig et al., 2013), and counter-rotation of
stellar and gaseous populations (Osman & Bekki, 2017). Confirmations of
theoretical predictions via observations of star formation in ETGs are
hampered by the lack of resolution of inner star-forming disks; recent far-
infrared observations by Baron et al. (2022) have suggested that a potential
resolution of this problem can be that much of the star formation is obscured.

In this paper we address the two problems mentioned above, tacking advantage
of the latest suite of simulations produced with the current version of our
high-resolution hydrodynamical simulation code MACER, which includes numerical
algorithms for the radiative cooling of the ISM, the formation of cold gaseous
disks in the presence of ordered rotation, and star formation following simple
feedback of various forms and robust input physics. We notice that the
resolution of MACER (parsec scale in the inner regions) greatly exceeds that
of most observations and typical cosmological simulations, enabling the
simulation and analysis of star forming disks with half mass radius close to
the galactic nucleus.

We are therefore in a good position to study the star formation inefficiency
problem in ETGs and the form of local star formation laws. Applying MACER
towards analyzing the formation of dense star-forming disks close to the
galactic nucleus, we have concluded that the observed inefficiency in SFR can
be partially explained due to an underestimation of total star formation and
the rapid ejection of cold gas. While the degree of star formation is
certainly less than that of disk galaxies, we note that the formation of cold,
centrifugally supported equatorial disks due to the conservation of angular
momentum in the ISM of rotating ETGs creates an environment conductive for
star formation following cooling flows (see C22). Dense stellar disks embedded
in these cold gaseous disks, if they exist, would lie close to the galactic
center beyond the resolution of what most resolved observations have
attempted, and would be vulnerable to destruction via late-time (dry) merging.
Gas lost from evolving stars is considerable and can be estimated at $\approx
10\%$ of the mass of the remaining stars (Kim & Pellegrini 2012).

This paper is organized as follows: in Section 2, we provide a brief summary
of the relevant numerical physics of MACER; in Section 3.1, we discuss the
position of ETGs on the global KS law with respect to the cooling flow
problem; and in Section 3.2-3, we discuss resolved KS law to examine the
dynamics of local SF, followed by a summary of results.

##  2\. The numerical code and input physics

In this paper we address some aspect of star formation in ETGs by a detailed
analysis of the suite of numerical simulations presented in C22, obtained with
the latest 2D version of our high-resolution code MACER (Massive AGN
Controlled Ellipticals Resolved), built upon Athena++ (version 1.0.0, see
Stone et al. 2020). With this code we solve the Eulerian hydrodynamical
equations for the ISM of ETGs with a central supermassive black hole, while
including the effects of circumgalactic (CGM) gaseous infall, dust grain
evolution, stellar feedback, AGN activity, and the metallic evolution of the
ISM and also providing an accurate treatment of stellar dynamics. At this
stage, simulations are performed under the assumption of axisymmetry, and
phenomena such as merging, and tidal interactions with other galaxies are not
taken into account. For a detailed description of the input physics, and other
technical aspects, see C22 and references therein.

For the models discussed here, the hydrodynamical equations are solved in
spherical $(r,\vartheta)$ coordinates, while allowing for rotation in the
$\varphi$ direction. The outer boundary of the computational domain is fixed
to $r=250$ kpc from the galactic center on a logarithmic radial grid, while
the inner boundary is set to either 2.5 pc or 25 pc, the latter of which is
used for a (relatively) fast parameter-space exploration; we notice that the
obtained resolution, also in the less resolved case, is significantly finer
than the $\approx 150$ pc inner boundary of typical cosmological simulations
and resolves the fiducial Bondi radius, allowing infall to the central BH to
be appropriately computed.

The galaxy models used are the axisymmetric JJe dynamical models consisting of
a moderately flat Jaffe ellipsoidal stellar distribution (corresponding to an
E3 galaxy) embedded in a dark matter halo, resulting in a spherical Jaffe
density profile (Ciotti et al. 2021, C22). The additional gravitational field
produced by a quasi-isothermal DM halo is also considered, to model the
effects of a group/cluster halo.

We analyze the star formation properties in three families of models, defined
by their initial stellar mass, and fully described in C22 (see Table 1
therein). In the family of high mass galaxy models (HM) the initial stellar
mass is $M_{*}=7.8\times 10^{11}\;M_{\odot}$, the edge-on circularized
effective radius is $<R_{\rm e}>=11.8$ kpc, and the stellar central velocity
dispersion $\sigma_{*0}=312$ km s-1. In the second family of medium mass
galaxy models (MM) the initial stellar mass is $M_{*}=3.35\times
10^{11}\;M_{\odot}$, the edge-on circularized effective radius is $<R_{\rm
e}>=7.04$ kpc, and the stellar central velocity dispersion $\sigma_{*0}=265$
km s-1. In the third family of low mass galaxy models (LM) the initial stellar
mass is $M_{*}=1.54\times 10^{11}\;M_{\odot}$, the edge-on circularized
effective radius is $<R_{\rm e}>=4.57$ kpc, and the stellar central velocity
dispersion $\sigma_{*0}=223$ km s-1. All the models are simulated over a time
interval of $\Delta t=12$ Gyr from $t_{0}=2$ Gyr, i.e., we start our
simulations after the galaxy formation epoch.

In the presence of some galaxy rotation, the angular momentum associated with
the gas injected into the ISM by the evolving stars (if not ejected by the
galaxy as a galactic wind) leads to the unavoidable formation of a cold and
rotationally supported gaseous disk in the galaxy equatorial plane, where star
formation occurs (in the simulations, the gravity of the resulting stellar
disk is also considered): of course, higher the rotation, bigger the gaseous
disk produced by ISM cooling, with important consequences for star formation.
It is therefore natural to explore how star formation depends on the galaxy
rotational structure while maintaining all the other properties of the model
fixed. 111Notice that changing rotational support at fixed structure affects
some aspect of the ISM evolution, changing the amount of thermalization of the
stellar ejecta due to stellar velocity dispersion and to the relative motion
of stellar streaming rotational field and ISM velocity field. For an extensive
discussion see Negri et al. (2014a) and Negri et al. (2014b); see also Kim &
Pellegrini (2012). Accordingly, among the models presented in C22, we focus
our study on two different rotational realizations for each of the LM, MM, and
HM galaxies. For each mass the two families are named constant-k (and the
models are indicated as HMc, MMc, and LMc), and exponential-k (HMe, MMe, and
LMe) for the following reasons. As is well known, the azimuthal velocity field
of a rotating stellar system can be (and usually is) split in its ordered and
dispersion components by adopting a generalised Satoh (1980) $k$-decomposition
(see equation 11 in C22). In the constant-$k$ rotation case the Satoh
parameter is constant over the galaxy body, with $k=0$ corresponding to a non-
rotating galaxy, with the flattening totally supported by tangential velocity
dispersion, while $k=1$ corresponds to a “fast” rotating galaxy (the isotropic
rotator), with the flattening fully supported by ordered rotation. In our HMc,
MMc, and LMc models we consider this latter possibility, i.e., they are
relatively fast spinning ETGs. In the exponential-$k$ rotation case we
considered instead the spatially-dependent Satoh parameter

$k_{\rm e}(r)={\rm e}^{-r/<R_{\rm e}>},$ (1)

so that the galaxy rotational support (and angular momentum injection from the
rotating stellar population) decreases significantly at large radii, while in
the central regions stars rotate almost as fast as an isotropic rotator.
Summarizing, in C22 and in the present study, angular momentum injection in
the ISM is maximal in constant $k>1$ models.

###  2.1. Star formation

Star formation is implemented in MACER as follows. A first, necessary
condition to activate the star formation routines entails checking, grid by
grid and at each time step, that the gas temperature is lower than $4\times
10^{4}$ K. Under this condition, $\dot{\rho}_{*}$ is computed by considering
the two channels mentioned in the Introduction: the Toomre instability
channel, and the cooling/Jeans channel. As we will see, in some circumstances
the two channels are both active; in this case $\dot{\rho}_{*}$ is defined as
the sum of the two rates (provided the total mass consumption rate in the
considered time step does not exceed a prescribed fraction of the available
mass in the considered grid).

The first channel of star formation is related to the Toomre instability
(e.g., see Toomre 1964, Binney & Tremaine 2008). As is well known, the local
stability of a rotating, self-gravitating disk can be understood in terms of
the relative importance of temperature (or velocity dispersion) vs. disk
gravitational field (i.e. surface density). Specifically, a self-gravitating
gaseous disk is locally stable when

$Q(R)\equiv{c_{\rm s}(R)\kappa(R)\over\pi G\Sigma_{\rm gas}(R)}>1,$ (2)

where in our case $R$ is the distance from the galaxy center in the disk
plane, $\Sigma_{\rm gas}$ is the gas surface density of the disk, $c_{\rm s}$
is the sound velocity, and $\kappa$ is the radial epicyclic frequency. As an
effect of disk instability, angular momentum of the rotating gas is
transferred outward due to a non-axisymmetric gravitational torques;
correspondingly, mass is transferred inwards and is eventually accreted on the
central SMBH222A semi-analytical algorithm mimicking this process is also
included in MACER, and is essential for producing SMBH accretion in presence
of ordered rotation.. Unstable over-densities in the infalling material (in
the present simulations, due to the imposed geometry, specifically gaseous
rings) trigger bursty phases of star formation, which then decrease the
surface density, increase $Q(R)$ above unity, and re-stabilize the ISM, in a
sort of self-regulation (e.g., see Bertin & Lodato 1999). In Gan et al. (2019)
the star formation rate $\dot{\rho}_{\rm*,Q}(R)$ associated with Toomre
instability is captured in a phenomenological way by

$\dot{\rho}_{\rm*,Q}=\eta_{\rm SF,Q}\Delta Q\rho_{\rm
gas}\Omega,\quad\quad\Delta Q=\max(1-Q,0),\quad\quad\eta_{\rm SF,Q}=0.02,$ (3)

where $\Omega(R)$ is the disk angular velocity profile.

The second channel of star formation, activated only when the gas density is
higher than $10^{5}\textrm{atom}\;\textrm{cm}^{-3}$, compares the gas cooling
timescale $\tau_{\rm cool}$ and the dynamical timescale $\tau_{\rm dyn}$ of
the ISM, and evaluates the associated star formation rate
$\dot{\rho}_{\rm*,C}$ defined as follows.

$\dot{\rho}_{\rm*,C}=\eta_{\rm SF,C}{\rho_{\rm gas}\over\tau_{\rm
SF,C}},\quad\quad\tau_{\rm SF,C}=\max(\tau_{\rm cool},\tau_{\rm
dyn}),\quad\quad\eta_{\rm SF,C}=0.01.$ (4)

The efficiency parameters $\eta$ are consistent with results from previous
simulations (Kim et al., 2021) and observations (Sun et al., 2023). The
cooling timescale is computed as the ratio of total thermal energy to the rate
at which the gas cools, the latter of which is determined by bremsstrahlung
emission, Compton heating and cooling, and recombination (Gan et al. 2019),
with allowance for heating due to the central flaring AGN. The dynamical
timescale is the minimum between the Jeans collapse timescale $\tau_{\rm
Jeans}=\sqrt{{3\pi}/{32G\rho_{\rm gas}}}$ and the rotational timescale
$\tau_{\rm rot}=2\pi R/v_{rot}$. If the dynamical time dominates, $\tau_{\rm
SF,C}\propto\rho_{\rm gas}^{-1/2}$, whereas if the cooling timescale
dominates, $\tau_{\rm SF,C}\propto\rho_{\rm gas}^{-1}$, with the resulting
star formation rate proportional to either $\rho_{\rm gas}^{3/2}$ or
$\rho_{\rm gas}^{2}$, respectively.

##  3\. Results

###  3.1. Gaseous and stellar disks from MACER simulations, and the global KS
law

A preliminary discussion of the global properties of the gaseous rotating cold
disks in ETGs (and of the associated stellar disks) produced by MACER
simulations can be found in C22 (see Table 2 therein); here we consider
several additional properties of the circumnuclear disks, providing all the
information needed to address the focus of this paper. A complete summary of
the results is contained in Table 1 and Table 2. In the Tables, the data are
given for increasing galaxy mass, and for each mass in order of increasing
amount of ordered rotation. In particular, in Table 1 we give the stellar and
gaseous properties of the circumnuclear disks, both at the end of the
simulations (corresponding to a simulation time of $\Delta t=12$ Gyr, and to a
final age of the galaxy of $14$ Gyr), and time-averaged over $\Delta t$. In
particular, time-averaged quantities are indicated with a bar over the
corresponding symbol, so that the time-averaged values of the quantity $f(t)$
is given by

$\overline{f}\equiv{1\over\Delta t}\int_{t_{0}}^{t_{0}+\Delta
t}f(t^{\prime})\,dt^{\prime},$ (5)

where, as assumed in MACER, at the initial time of simulation the age of the
galaxy is $t_{0}=2$ Gyr. Therefore, for each of the six models, the columns in
Table 1 give the total mass $\Delta M_{*}$ of new stars formed in the
circumnuclear disk, the time-average $\overline{R}_{\dot{M}_{*}/2}$ of the
instantaneous half-mass radius of star formation $R_{\dot{M}_{*}/2}(t)$, the
total mass $M_{\rm d*}$ of the circumnuclear stellar disk at the end of the
simulation (where the difference $\Delta M_{*}-M_{\rm d*}$ is accounted for
stellar evolution), and the final half-mass radius $R_{\rm d*/2}$ of the
stellar disk. Then, for the properties of the circumnuclear gaseous cold
(defined as in C22 as gas with $T\leq 5\times 10^{5}$ K) disk, for each model
we report the time-average $\overline{M}_{\rm gas}$ of the gas mass in the
circumnuclear disk, together with the time-averages $\overline{R}_{\rm gas/2}$
and $\overline{R}_{\rm gas}$ of the gaseous disk half-mass and truncation
radii, respectively. The last two columns of Table 1 finally give the values
at the end of simulation of the disk total cold gas mass $M_{\rm gas}$, and
the associated truncation radius $R_{\rm gas}$.

In Table 2 additional information about gas content and star formation in the
circumnuclear disks are reported to study the global properties of the
simulated KS relations. Accordingly, for the six models we construct the
equivalent surface star formation rate, defined from the quantities in Table 1
as

$<\dot{\Sigma}_{*}>\equiv{\Delta M_{*}\over
2\pi\overline{R}_{\dot{M}_{*}/2}^{2}\Delta t},$ (6)

i.e., the time-independent and spatially uniform star formation rate of a
fictitious starforming disk of half-mass radius
$\overline{R}_{\dot{M}_{*}/2}$, forming a total stellar mass $\Delta M_{*}$
over the time $\Delta t$. We then compute the time-average of the (spatially
averaged) surface density of the disk within the instantaneous half-mass
radius, given by

$\overline{<\Sigma_{\rm gas}>}\equiv\overline{{M_{\rm gas}(t)\over 2\pi R_{\rm
gas/2}^{2}(t)}}.$ (7)

We finally evaluate the global, time-averaged star formation efficiency,
naturally defined as

$\epsilon_{*}\equiv\overline{{\dot{M}_{*}(t)\over M_{\rm
gas}(t)}}\simeq{\Delta M_{*}\over\overline{M}_{\rm gas}\Delta t},$ (8)

and, for comparison with the observed KS law, we also estimate the global,
time-averaged star formation amplitude as

$<{\cal
A}_{*}>\equiv\log_{10}{<\dot{\Sigma}_{*}>}-1.4\log_{10}\overline{<\Sigma_{\rm
gas}>},$ (9)

where, following the common approach, $<\dot{\Sigma}_{*}>$ is in units of
$M_{\odot}{\rm kpc}^{-2}{\rm yr}^{-1}$, and $\overline{<\Sigma_{\rm gas}>}$ in
$M_{\odot}{\rm pc}^{-2}$.

Before focusing on the properties of the resolved KS laws for the six models,
we list the main global results, beginning with Table 1. We immediately note a
few robust trends. 1) The total mass of newly formed stars in the
circumnuclear disk $\Delta M_{*}$ is in the range of $10^{8}-10^{9}M_{\odot}$
and increases systematically with galaxy mass; at fixed galaxy mass, star
formation is also larger for isotropic rotators (subscript “c” in Column 1).
Both these trends are quite natural; in particular, the enhanced star
formation in case of substantial galaxy rotation confirms previous studies
(Negri et al. 2015, Gan et al. 2019) that angular momentum tends to increase
ISM cooling and successive instabilities. 2) The present day mass $M_{\rm d*}$
of the resulting stellar disks is obviously less than $\Delta M_{*}$, a
consequence of stellar evolution: notice that the quite large difference
between the two quantities (with a reduction factor slightly larger than
$\simeq 50\%$) in all the models is due to the adopted top-heavy IMF in the
simulations (see C22). 3) The present day half-mass radius $R_{\rm d*/2}$ of
the stellar disk is found, for all models, in the range $\simeq 100-300$ pc,
being obviously larger for the three isotropic rotator models: notice how well
$R_{\rm d*/2}$ matches $\overline{R}_{\dot{M}_{*}/2}$ in Column 2, the time-
averaged half-mass radius of star formation in the disk.

The properties of the stellar disks reflect nicely the properties of their
gaseous component, the source of ISM material for star formation: some
systematic differences are however important and should be noticed. 4) The
present-day mass $M_{\rm gas}$ of cold gas in the disk, in each of the three
different familiesof models, is significantly larger (with values as high as
$6\,10^{10}M_{\odot}$) in the isotropic rotators than in the corresponding
low-rotation cases, where in these latter cases $M_{\rm gas}$ is only slightly
larger than $M_{\rm d*}$. 5) A similar comment applies to the size of the
gaseous component of the disks, as can be see by comparing Columns (4) and
(7): in all models, the stellar disks are obviously restricted to the central
regions of the parent gaseous disks, that can easily reach the kpc scale.

All the previous results reflect in the global quantities concerning star
formation in the models, as reported in Table 2: in each of the three families
of galaxy models, the averaged surface density star formation rate
$<\dot{\Sigma}_{*}>$ is higher in the low-rotation models, while the opposite
happens for averaged cold gas surface density $\overline{<\Sigma_{\rm gas}>}$.
The corresponding global, time-averaged star formation efficiencies are in
general of few percents: therefore, the efficiency with which cold gas is
converted to stars in the numerical simulations is within the lower range
predicted by Kennicutt (1998) for spiral galaxies. The last, but particularly
important quantity, one of the focus points of the present paper, is the
amplitude of the simulated KS laws. From the last column in Table 2, we found
that $<{\cal A}_{*}>$ is in the range $(-5.1,-3.4)$; generally, these
amplitudes lie lower than, but not extremely so, the Kroupa IMF-corrected KS
amplitude of measured for spirals/starbursts (Davis et al., 2014); lower
amplitudes systematically correspond to models with low rotation of the
stellar population. Therefore, a reduction of the rotational support in ETGs
at fixed galaxy mass, appears to decrease the amplitude of the resulting KS
law, and one is tempted to conclude that specific angular momentum of the
stellar population could be the leading parameter determining the
normalization coefficient of the KS law.

Table 1Stellar and gas properties of the circumnuclear disks

Model | $\Delta M_{*}$ | $\overline{R}_{\dot{M}_{*}/2}$ | $M_{\rm d*}$ | $R_{\rm d*/2}$ | $\overline{M}_{\rm gas}$ | $\overline{R}_{\rm gas/2}$ | $\overline{R}_{\rm gas}$ | $M_{\rm gas}$ | $R_{\rm gas}$
---|---|---|---|---|---|---|---|---|---
| $(10^{8}M_{\odot})$ | (kpc) | $(10^{8}M_{\odot})$ | (kpc) | $(10^{8}M_{\odot})$ | (kpc) | (kpc) | $(10^{8}M_{\odot})$ | (kpc)
| (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9)
LMe | 4.8 | 0.1 | 2.1 | 0.1 | 7.3 | 0.2 | 0.7 | 2.4 | 0.7
LMc | 6.8 | 0.2 | 3.0 | 0.3 | 72.4 | 2.8 | 7.7 | 60.3 | 4.4
MMe | 12.2 | 0.1 | 5.3 | 0.1 | 14.0 | 0.3 | 1.0 | 11.0 | 0.5
MMc | 28.0 | 0.3 | 12.2 | 0.3 | 111.5 | 2.7 | 9.9 | 46.6 | 3.6
HMe | 28.9 | 0.1 | 12.5 | 0.1 | 27.2 | 0.5 | 1.85 | 12.3 | 0.6
HMc | 68.7 | 0.3 | 29.8 | 0.3 | 149.5 | 2.9 | 12.4 | 57.8 | 3.0

  * Star formation results for the 3 families of models in order of increasing galaxy stellar mass, with radially dependent (subscript “e”), and isotropic (subscript “c”) Satoh decomposition for the streaming rotational velocity of the stellar population; a bar over a quantity means time average over the simulation time $\Delta t=12$ Gyr defined as in equation (5). The columns give: (1) total mass of new stars formed; (2) time-average of the instantaneous half-mass radius of star formation $R_{\dot{M}_{*}/2}(t)$; (3) total mass of the circumnuclear stellar disk at the end of the simulation: the difference $\Delta M_{*}-M_{\rm d*}$ is accounted for stellar evolution; (4) final half-mass radius of the stellar disk; (5) time-average of the cold ($T\leq 5\times 10^{5}$ K) gas mass in the circumnuclear disk; (6) time-average of the half-mass radius of the cold gaseous disk; (7) time-average of the truncation radius of the cold gaseous disk; (8) final value of the cold gas mass in the circumnuclear disk; (9) final value of the truncation radius of the cold gaseous disk.

Table 2Global parameters of the simulated KS in ETGs

Model | $<\dot{\Sigma}_{*}>$ | $\overline{<\Sigma_{\rm gas}>}$ | $\epsilon_{*}$ | $<{\cal A}_{*}>$
---|---|---|---|---
| $(M_{\odot}{\rm kpc}^{-2}{\rm yr}^{-1})$ | $(M_{\odot}{\rm pc}^{-2})$ | (Gyr-1) |
| (1) | (2) | (3) | (4)
LMe | 0.5 | 2456.5 | 0.05 | -5.1
LMc | 0.2 | 153.7 | 0.01 | -3.8
MMe | 1.2 | 2580.3 | 0.07 | -4.7
MMc | 0.5 | 248.7 | 0.02 | -3.7
HMe | 3.6 | 1977.1 | 0.09 | -4.1
HMc | 1.4 | 332.5 | 0.04 | -3.4

  * Relevant parameters of the simulated KS law in the simulated ETGs reported in Table 1. The columns give: (1) the equivalent (time-independent and spatially uniform) surface star formation rate $<\dot{\Sigma}_{*}>$ of a fictitious starforming disk of half-mass radius $\overline{R}_{\dot{M}_{*}/2}$, forming a total stellar mass $\Delta M_{*}$ over the time $\Delta t$, defined as in equation (6); (2) time-average of the average surface density of cold gas in the disk $\overline{<\Sigma_{\rm gas}>}$, defined as in equation (7); (3) time-average of the global star formation efficiency $\epsilon_{*}$, defined as in equation (8); (4) amplitude of the KS star formation law, defined as in equation (9).

###  3.2. Resolved KS Laws

Even though in principle star formation can happen everywhere over the galaxy
model, the simulations revealed that all star formation occurs in rotationally
supported, thin, cold disks in the galaxy equatorial plane. To investigate the
radial dependence of the star formation rate $\dot{\Sigma}_{*}$ as a function
of the surface gas density $\Sigma_{\rm gas}$, i.e., the resolved KS law as
produced by the MACER simulations, for each of the six models we compute the
mass of cold ($T\leq 5\times 10^{5}$ K) ISM, and the star formation rate, over
radial annuli $(R,R+\Delta R)$, and then we divide the results for $2\pi
R\Delta R$. This computation is repeated, for each annulus, with a sampling
time of 1 Gyr over the time $\Delta t=12$ Gyr spanned by the simulations, so
that for each of the analyzed models we construct a large set of points
$(\Sigma_{\rm gas},\dot{\Sigma}_{*})$, corresponding to different radii in the
disk, and to different evolutionary times.

Figure 1.— Resolved star formation rate of the high-mass (HM) models for the
low-rotation case with exponentially declining Satoh $k(r)$ parameter with
(HMe, left column), and for fast rotating $k=1$ isotropic model (HMc, right
column). In the top panels local star formation rates
$\dot{\Sigma}_{*}=\dot{\Sigma}_{\rm*,Q}+\dot{\Sigma}_{\rm*,C}$ (in units of
$M_{\odot}$ kpc-2 yr-1) are given as a function of the local disk gas surface
density $\Sigma_{\rm gas}$ (in units of $M_{\odot}$ pc-2), with a sampling
time of 1 Gyr over the time span $\Delta t=12$ Gyr, so the plots represent
also the history of star formation in the model. For reference, the solid line
represents the Davis et al. (2014) observed KS law in equation (10), of slope
$1.4$ and amplitude ${\cal A}_{*}=-3.76$. In the bottom panels red and blue
points separate respectively the contribution of Toomre instability
($\dot{\Sigma}_{\rm*,Q}$) and cooling/Jeans instabilities
($\dot{\Sigma}_{\rm*,C}$) to the total star formation rate of each point in
the top panels.

With the aid of Figure 1 we begin by illustrating the results for the two
high-mass models HM, in the low rotation case with the exponentially
decreasing Satoh parameter $k(r)$ (HMe, left panels), and for the fast
rotating, isotropic rotator (HMc, right panels) with constant $k=1$. In the
top panels, all the points $(\Sigma_{\rm gas},\dot{\Sigma}_{*})$ are plotted
with a color scheme mapping the distance from the galaxy center, with
decreasing distance from red to blue. A few trends are apparent. The first is
just a confirmation that the cold rotating disk extends more in the fast
rotating isotropic HMc model than in the less rotating HMe model, as is
obvious from the tail of red points in the top-right panel, absent in the top-
left panel. The second important result is that the bulk of the points are
nicely aligned parallel to the heavy solid line showing the observed KS law

$\dot{\Sigma}_{*}\simeq 10^{-3.76}\,\Sigma_{\rm gas}^{1.4},$ (10)

as given by Davis et al. (2014), where the units are the same as in Table 2,
and the amplitude333Notice that the amplitude in equation (10) is not the
estimated time and space averaged amplitude $<{\cal A}_{*}>$ in equation (9).
In particular, the relative sizes of amplitudes in equation (9) for the low
mass models are reversed from those in Figure 4, suggesting that cold gas is
over counted for the exponential case - a rapid decrease in rotation leaves
much of the gas unable to produce stars. ${\cal A}_{*}=-3.76$, similar to what
found by other authors. For example Boquien et al. (2011) report observed
amplitudes in the range $-3.83\lesssim{\cal A}_{*}\lesssim-3.02$ for spiral
galaxies when surface densities/rates are computed in the same units as above.
The third result is that our resolved KS laws, albeit parallel to the observed
law, are clearly below it, with lower star formation rate at given surface
density of the cold gas in the disk, i.e. in the simulations ETGs gaseous
disks are predicted to be inefficient with respect to observed KS law observed
in disk galaxies by almost an order of magnitude. Notice that from Table 2
this is reflected by the time-averaged, global value of $<{\cal
A}_{*}>\simeq-4.1$ for the low-rotating HMe model, but not for the isotropic
rotator HMc (where $<{\cal A}_{*}>\simeq-3.4$). The fourth feature of the
simulated points to be noticed is the large scatter (in both models) of
$\Sigma_{\rm gas}$ at approximately constant $\dot{\Sigma}_{*}$, both in the
very central regions, and at the edge of the gaseous rotating disk.
Physically, the scatter implies that at the very high (center) and very low
(disk outskirsts) star formation rates, the gas surface density is only weakly
correlated with star formation, with significant differences in $\Sigma_{\rm
gas}$ at fixed $\dot{\Sigma}_{*}$ at the center, and with almost uncorrelated
$\Sigma_{\rm gas}$ and $\dot{\Sigma}_{*}$ at large radii. More quantitatively,
star formation rate becomes abruptly horizontal near the center, with
$\dot{\Sigma}_{*}\gtrsim 10^{9.5}M_{\odot}\textrm{kpc}^{-2}\textrm{Gyr}^{-1}$.
At larger radii ($\approx$ 2.5 kpc) we also see an abrupt threshold cutoff at
gas densities less than $10^{-3}\;\textrm{g}\;\textrm{cm}^{-2}$. K89 remarks
that a similar behavior is a consistently observed phenomena for disk
galaxies, and that this cutoff threshold is typically found to occur within
the range of $10^{-3}-10^{-4}\;\textrm{g}\;\textrm{cm}^{-2}$, which is
incidentally very similar to what we observe for our ETG model in Figure 1.
That this feature is also present in our simulations is encouraging regarding
the veracity of our star formation recipe and similarity of star-formation
laws across different galaxy types. Following K89, we argue that the universal
cause for this phenomena arises from stability considerations captured by the
simple Toomre star formation channel: at lower densities, the gas is stable
against large-scale perturbations, suppressing star formation. High resolution
observations of ETGs are essential, therefore, for confirming or refusing the
stunning similarity of star formation dependence on gas density between disk
and ETGs found in our work.

All these trends are sufficiently interesting to merit a further scrutiny. In
fact, we recall that in the MACER treatment of star formation,
$\dot{\Sigma}_{*}=\dot{\Sigma}_{\rm*,Q}+\dot{\Sigma}_{\rm*,C}$, different
possibilities are considered, i.e., star formation can be produced by Toomre
instability, by cooling/Jeans instabilities, or by a combination of the two,
depending on the cold gas state (see Section 2). Therefore, a natural question
is which, if either, of the two channels is dominant in different regions of
the $(\Sigma_{\rm gas},\dot{\Sigma}_{*})$ planes. For this reason, in the
bottom panels of Figure 1 we plot star formation resulting from different
formation channels, indicating the star formation rate due to Toomre
instability in red and the star formation rate due to cooling/Jeans
instabilities in blue; all the points in the top panels corresponding to star
formation with both channels active now split in pairs of red and blue points.
From the bottom panels, we conclude that the bulk of the KS law obtained by
MACER is due to Toomre instabilities, which are responsible for the large
scatter of the points at the edge of the cold disk. Conversely, star formation
in the very central regions is dominated by cooling/Jeans instabilities.
Notice the almost vertical strip of blue points at high values of $\Sigma_{\rm
gas}$: these points are not missing in the top panels, they just represent a
marginal star formation well below the Toomre instability channel, and they
“disappear” when added to the red points relative to the same numerical cell
and same sampling time. A complementary view of the relative importance of
Toomre and cooling/Jeans instabilities, where the dependence of the main
channels of star formation as a function of the distance from the center can
be clearly seen, is given in Figure 2, nicely confirming the conclusions
above.

The dependence of the surface density profile on disk scale height, which
rapidly evolves over the disk, accounts for the reason for the different
relevance of the two channels in different locations of the gaseous disk. We
identify the regime between $2.5\times 10^{-1}$ and $2.5$ kpc as dominated by
the Toomre channel, which produces a SFR identical to the KS power law. K89,
and more recently Boissier et al. (2003), have both remarked that the Toomre
instability criterion remarkably reproduces SFR power laws in spiral galaxies;
despite the different gas properties and disk structure of ETGs, the form of
local star formation is the same in our computations.

Figure 2.— Star formation channels for models HMe (left) and HMc (right),
color coded to show the position inside the rotating cold gaseous disk in the
equatorial plane of the models. From bottom panels it is apparent how
cooling/Jeans instability star formation is mainly confined to the very
central regions, and producing the highest values of star formation. Toomre
instabilities instead occur over all the disk.

Figures 3 and 4 nicely confirm and extend the global picture illustrated above
also to different galaxy masses, with simulated (resolved) star formation KS
laws parallel to the observed one, characterized by lower amplitude ${\cal
A}_{*}$, and by quite significant scatter in the central regions and at the
edge of the circumnuclear gaseous disks.

Figure 3.— Resolved star formation rate of the medium-mass (MM) models, for
the low-rotation case with exponentially declining Satoh $k(r)$ parameter with
(MMe, left column), and for fast rotating $k=1$ isotropic model (MMc, right
column). In the top panels local star formation rates
$\dot{\Sigma}_{*}=\dot{\Sigma}_{\rm*,Q}+\dot{\Sigma}_{\rm*,C}$ (in units of
$M_{\odot}$ kpc-2 yr-1) are given as a function of the local disk gas surface
density $\Sigma_{\rm gas}$ (in units of $M_{\odot}$ pc-2), with a sampling
time of 1 Gyr over the time span $\Delta t=12$ Gyr, so the plots represent
also the history of star formation in the model. For reference, the solid line
represents the Davis et al. (2014) observed KS law in equation (10), of slope
$1.4$ and amplitude ${\cal A}_{*}=-3.76$. In the bottom panels red and blue
points separate respectively the contribution of Toomre instability
($\dot{\Sigma}_{\rm*,Q}$) and cooling/Jeans instabilities
($\dot{\Sigma}_{\rm*,C}$) to the total star formation rate of each point in
the top panels. Figure 4.— Resolved star formation rate of the low-mass (LM)
models, for the low-rotation case with exponentially declining Satoh $k(r)$
parameter with (LMe, left column), and for fast rotating $k=1$ isotropic model
(LMc, right column). In the top panels local star formation rates
$\dot{\Sigma}_{*}=\dot{\Sigma}_{\rm*,Q}+\dot{\Sigma}_{\rm*,C}$ (in units of
$M_{\odot}$ kpc-2 yr-1) are given as a function of the local disk gas surface
density $\Sigma_{\rm gas}$ (in units of $M_{\odot}$ pc-2), with a sampling
time of 1 Gyr over the time span $\Delta t=12$ Gyr, so the plots represent
also the history of star formation in the model. For reference, the solid line
represents the Davis et al. (2014) observed KS law in equation (10), of slope
$1.4$ and amplitude ${\cal A}_{*}=-3.76$. In the bottom panels red and blue
points separate respectively the contribution of Toomre instability
($\dot{\Sigma}_{\rm*,Q}$) and cooling/Jeans instabilities
($\dot{\Sigma}_{\rm*,C}$) to the total star formation rate of each point in
the top panels. Figure 5.— Volumetric star formation law for the low-rotation,
high-mass model HMe (left), and the high-mass, fast rotating (isotropic) model
HMc (right), a figure analogous to the top panels in Figure 1. The total
volumetric star formation rates $\dot{\rho}_{*}$ are given as functions of
local gas density $\rho_{\rm gas}$, with sampling time of 1 Gyr throughout the
$\Delta t=12$ Gyr time spanned by the simulation. The heavy solid line shows a
fit to the $\dot{\rho}_{*}\propto\rho_{\rm gas}^{1.5}$ volumetric KS law,
while the heavy dashed line is the observed relation derived for spirals and
dwarf galaxies (equation 4.1 in Bacchini et al. 2020).

###  3.3. The volumetric KS law

As mentioned in §1, volumetric KS laws have also been proposed and
investigated as alternatives to the standard KS law built from surface
densities. Owing to the thinness of the stellar disks and difficulties of
measuring volume densities in galaxies, surface density star formation laws
are more frequently used. However, surface densities are frequently affected
by projection effects and the flaring of the disk thickness; thus, volumetric
density relations may offer more accurate insight into star formation laws
while also being more applicable for simulations. Due to these observational
difficulties, however, it is thought that conversion between surface densities
and volumetric densities is non-trivial (Bacchini et al. 2019, 2020). Our
numerical simulations allow us to study volumetric power laws while avoiding
observational 2D projection effects.

We study the radial behavior of volumetric star formation laws by averaging
SFR and gas densities over radial annuli over 1 Gyr timesteps throughout the
$\Delta t=12$ Gyr of evolution. The two panels of Figure 5 contain plots of
the fit (heavy solid line) of the obtained resolved volumetric KS law for the
high mass models. We observe that the overall volumetric densities follow a
remarkable correspondence with the surface density relationships in Figure 1,
and that the fit is improved in the low radii cooling regime with less
horizontal spread. Using Equation 4, a dynamical timescale on the order of
$\rho_{\rm gas}^{-1/2}$ explains the good $n\approx 1.5$ power law fit,
consistent with some observational work (Sun et al., 2023): we conclude that
in MACER simulations it is the longer dynamical time rather than the cooling
time that dominates the star formation in this spatial region, favoring
$n\approx 1.5$ rather than the occasionally observed $n\approx 2$ (Bacchini et
al. 2020, heavy dashed lines). In any case, also the resolved volumetric KS
laws obtained from the simulations confirm a lower efficiency of star
formation at given gas density in the circumnuclear disks of the simulated
ETGs. While the volumetric KS relation corresponds with our physical
prescription at small radii, the surface density KS relation matches less
well, as noted in §3.2. This can be attributed to a rapidly flaring scale
height, as seen in the left hand panel of Figure 5, resulting in projection
effects; we note, however, that Toomre instability channel matches well in
both cases. The threshold cutoff observed at $\approx$ 2.5 kpc once again is
attributed to Toomre instability. Ultimately, we propose that surface-density
SF laws, which are easier to measure, can be translated into volumetric-
density SF laws, which describe the form of a local SF, given the close
correspondence between the two. More importantly, we once again assert that
the same empirical SF laws in disk galaxies can be applied towards ETGs. High-
resolution observations of ETGs to confirm these power law fits will thus be
beneficial for applications towards semi-analytical star formation recipes in
simulations and for understanding the local analytic form of star formation
laws.

##  4\. Summary and Conclusions

In this paper we present a detailed study of star formation laws in in ETGs by
analysing the results of high resolution numerical simulations reported in C22
and performed with the latest version of our MACER code. For any axisymmetric
ETG with some level of ordered rotation in their stellar populations, both
numerical simulations and simple physical arguments have clearly established
that the ISM cools and develops large-scale instabilities, inevitably leading
to the formation of cold and rotating gaseous disks in the galaxy equatorial
plane. This is confirmed by recent observations demonstrating the existence of
massive gaseous disks in 50% of ETGs (Young et al., 2011). Theoretically, such
disks should be prone to star formation, raising the question as to why their
observed star formation is so inefficient in comparison to disk galaxies
(Negri et al., 2015).

Similar to what happens in rotating disk galaxy models, our simulations lead
to the formation of rotating, cold, gaseous disks with kpc scale in the
presence of galaxy rotation, gas cooling, and angular momentum. We implement
star formation over this cold gas by considering two different channels: one
based on Toomre instability, and another determined by the longer of the local
cooling and dynamical (Jeans) times of the cold ISM. When the local values of
density and temperature of the ISM are respectively larger and smaller than
two prescribed threshold values, both channels are active, and the star
formation rate is their sum. Remarkably, our star formation recipes result in
a SFR-gas scaling law quite similar to the observed KS relation.

The Toomre instability star formation channel reproduces an $n\approx 1.4$
resolved KS star formation power law with a lower cutoff threshold, which is
analogous to what has been observed in disk galaxies and can be attributed to
gravitational stability considerations. The cooling channel also obeys a
power-law density relationship, but deviates from the KS star formation law at
higher densities closer to the center of the galaxy, which may be attributed
to a rapidly flaring scale height. In preliminary 3D simulations more recently
performed, smaller scale, nonaxisymmetric fluctuations tend to remove this
small discrepancy. Volumetric star formation power laws are also explored in
effort to remove artificial effects of disk flaring on surface density
measurements, and we find a close correspondence between the volumetric and
surface density forms of the KS power law. Globally, we find that our
simulated ETGs lie approximately parallel but with a lower normalization to
the same KS relation as disk galaxies

The similarity of the observed star formation power-laws in disk galaxies with
our simple numerical implementation in ETGs is remarkable, implying that these
KS laws provide simple recipes that can be used in semi-analytical models of
ETGs as well. The prediction of the existence of small star-forming disks in
ETGs encourages high-resolution observations of galactic centers as well as
further work in understanding their dynamical properties and exploring the
form of local - rather than empirical - star formation laws to see if our
predictions correspond with reality. Therefore, observational checks of our
predictions are imperative for evaluating these conclusions and for
determining the evolution of cool gas with respect to the star formation
inefficiency problem. ETGs are not quiescent in our view and observations are
need to check our SFR estimates: cooling flows events that occur in rotating
ETGs (from the non ejected gas) should be observed as miniature versions of
the standard observed disk star forming regions in normal spiral systems, and
the central regions of the MW and M31.

We thank Cecilia Bacchini, Eve Ostriker, and an anonymous referee for
important comments that improved the paper content and presentation. ZG is
grateful for the financial support from Princeton University via the
subcontract to New Mexico Consortium.

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