Regulation of star formation rates in multiphase galactic disks: a thermal/dynamical equilibrium model

TL;DR: In this article, the authors developed a model for the regulation of galactic star formation rates in disk galaxies, in which interstellar medium (ISM) heating by stellar UV plays a key role, by requiring that thermal and dynamical equilibrium are simultaneously satisfied within the diffuse gas, and that stars form at a rate proportional to the mass of the self-gravitating component.

Abstract: We develop a model for the regulation of galactic star formation rates ΣSFR in disk galaxies, in which interstellar medium (ISM) heating by stellar UV plays a key role. By requiring that thermal and (vertical) dynamical equilibrium are simultaneously satisfied within the diffuse gas, and that stars form at a rate proportional to the mass of the self-gravitating component, we obtain a prediction for ΣSFR as a function of the total gaseous surface density Σ and the midplane density of stars+dark matter ρsd. The physical basis of this relationship is that the thermal pressure in the diffuse ISM, which is proportional to the UV heating rate and therefore to ΣSFR, must adjust until it matches the midplane pressure value set by the vertical gravitational field. Our model applies to regions where Σ 100 M ☉ pc–2. In low-ΣSFR (outer-galaxy) regions where diffuse gas dominates, the theory predicts that . The decrease of thermal equilibrium pressure when ΣSFR is low implies, consistent with observations, that star formation can extend (with declining efficiency) to large radii in galaxies, rather than having a sharp cutoff at a fixed value of Σ. The main parameters entering our model are the ratio of thermal pressure to total pressure in the diffuse ISM, the fraction of diffuse gas that is in the warm phase, and the star formation timescale in self-gravitating clouds; all of these are (at least in principle) direct observables. At low surface density, our model depends on the ratio of the mean midplane FUV intensity (or thermal pressure in the diffuse gas) to the star formation rate, which we set based on solar-neighborhood values. We compare our results to recent observations, showing good agreement overall for azimuthally averaged data in a set of spiral galaxies. For the large flocculent spiral galaxies NGC 7331 and NGC 5055, the correspondence between theory and observation is remarkably close.

Summary

1. INTRODUCTION

• Star formation is regulated by many physical factors, with processes from sub-parsec to super-kiloparsec scales contributing to setting the overall rate (see, e.g., McKee & Ostriker 2007).
• Section 3 then compares to the observed data set previously presented in Leroy et al. (2008).

2.1. Model Concepts and Construction

• The authors construct a local steady-state model for the star formation rate in the disk, with independent variables the total surface density of neutral gas (Σ), the midplane stellar density (ρs), and the dark matter density (ρdm).
• The abundance of gravitationally bound, starforming clouds is nevertheless important for establishing an equilibrium state in the diffuse gas, because the FUV that heats the diffuse ISM originates in young OB associations.
• Thus, an equilibrium state, in which cooling balances heating and pressure balances gravity, can be obtained by a suitable division of the gas mass into star-forming (gravitationally bound) and diffuse components such that their ratio is proportional to the vertical gravitational field.
• 6 Dopita (1985) previously showed that assuming the pressure to be proportional to the star formation rate yields scaling properties similar to observed relationships.
• The authors then discuss, from a physical point of view, how the various feedback processes might act to adjust the system over time, steering it toward the equilibrium they have identified (Section 2.6).

2.2. Gas Components

• The authors divide the neutral ISM into two components.
• One component consists of the gas that is collected into gravitationally bound clouds (GBCs) localized near the galactic midplane, with mean surface density (averaged over ∼ kpc scales) of ΣGBC.
• Here, the authors use the term “diffuse” in the sense of being widely dispersed or scattered throughout the volume; the diffuse component may include both tenuous, volume-filling gas and small, dense cloudlets (see below).
• The authors treat the diffuse gas as a two-phase cloud–intercloud medium in thermal pressure equilibrium, with turbulent vertical velocity dispersion v2z assumed to be the same for warm and cold phases.
• Very recent numerical studies provide support for the quasi-equilibrium assumption—see C.-G. Kim et al. (2010, in preparation).

2.3. Vertical Dynamical Equilibrium of Diffuse Gas

• In the circumstance that the diffuse-gas scale height is much larger than that of the stars, gz ≈ 2πGΣs would be substituted for the gravity of the stellar component, yielding a contribution analogous to that in Equation (5) with ΣGBC → Σs .
• Note that if the Toomre parameter for the stellar disk and the vertical-to-horizontal velocity dispersion ratio are both constant with radius, then ρs ∝ ρdm ∝ (Vc/R)2.
• Numerical simulations in multiphase gas have shown that the magnetic field is amplified by the magnetorotational instability to a level B2/(8π ) = (1 − 2)Pth, independent of the mass fractions of cold and warm gas and the vertical gravitational field strength (Piontek & Ostriker 2005, 2007), while |Bz/Bφ| 1.

2.4. Thermal Equilibrium of Diffuse Gas

• As expressed by Equation (11), the thermal pressure in the diffuse gas must respond to the dynamical constraint imposed by vertical momentum conservation in the disk.
• Thus, the authors expect the midplane thermal pressure in the diffuse gas to be comparable to the two-phase value defined by the thermal equilibrium curve, Pth ≈ Ptwo-phase.
• For simplicity, the authors neglect variations in JFUV associated with the radiative transfer here; they shall simply assume JFUV ∝ ΣFUV ∝ ΣSFR.
• Equivalently, since Wolfire et al. (2003) predict the value of Pth,0/JFUV,0 from theory, their model depends on the measured ratio of local FUV intensity to local star formation rate, JFUV,0/ΣSFR,0.

2.5. The Equilibrium Star Formation Rate

• In normal galaxies, GBCs are identified with GMCs (the outer layers of which are in fact atomic—see below).
• These are mild (no more than a factor of 2–3 over the range Σ = 10–100 M pc−2) and almost all find a consistent normalization, with 2 Gyr being a typical timescale.
• Note that because CO is optically thick in clouds with NH,cloud ∼ 1022 cm−2 and normal metallicity, observed CO emission in unresolved clouds may in fact trace the atomic and “dark gas” portions of GBCs as well as the regions where CO is present, because these contribute to the gravitational potential and therefore the total CO linewidth.
• GBC is calibrated from observations in which star-forming clouds are primarily molecular (and observable in CO lines), their basic approach would remain unchanged for GBCs in different parameter regimes, provided that a well-defined value of tSF,GBC is known (from either observations with appropriate corrections for atomic and dark gas, or from theory).the authors.

2.6. Approach to Equilibrium

• Figure 1(a) shows an extreme case of higher-than-equilibrium fdiff , in which the midplane pressure is higher than Pmax,warm.
• The arrows in Figure 1(a) indicate how the midplane pressure and thermal equilibrium curve would evolve.
• The authors imagine a region with a bound-cloud proportion 1 − fdiff and ΣSFR/Σ above the self-consistent overall equilibrium values, such that heating associated with the high star formation rate makes the thermal equilibrium curve sit at high pressure (i.e., Ptwo-phase is high).
• With self-consistent numerical simulations, it will be possible to assess whether f̃w and Pth/Ptwo-phase secularly depend on Σ and ρsd.

2.7. Sample Solution for an Idealized Galaxy

• If the dark matter density dominates the stellar density in the outer disk, then since ρdm ∝ (Vc/R)2, constant Vc would imply ΣSFR/Σ ∝ R−1 in the outer disk.
• Second, there are two regimes evident for ΣSFR and Σdiff versus Σ: a high-surface-density regime in which the gas is mostly in self-gravitating clouds and the limiting solution ΣSFR = Σ/tSF is approached, and a low-surface-density regime in which ΣSFR has a steeper dependence.
• In adopting G′0 = ΣSFR/ΣSFR,0 for Equation (15), the authors have neglected optical depth effects.
• Thus, the current simple theory overestimates Σdiff in the central parts of galaxies, where τ⊥ becomes large.

2.8. Additional Considerations

• Finally, the authors remark on a few additional points related to assumptions behind and application of the theory presented above.
• The weak dependence on both ρsd and Σ2, and the fact that only their ratio appears so that variations will be partially compensated, implies that f̃w would indeed be expected to vary only modestly, at least in outer disks.
• The star formation rate could therefore decline to ∼ 0.002 times the local value, or 6 × 10−6 M kpc−2 yr−1, before the metagalactic UV becomes important; this occurs only in the far-outer regions of disks.
• An individual GBC is composed of a mixture of molecular gas and cold atomic gas that depends on shielding, and could be primarily atomic at sufficiently low metallicity.
• Thus, the internal dynamics of primarily atomic GBCs—including the processes that determine the internal star formation efficiency—are expected to be similar to those in primarily molecular GBCs, provided that their gravitational potentials and internal velocity dispersions are similar so that vturb vth,cold.

3. COMPARISON TO OBSERVATIONS

• The formulae derived above yield predictions for ΣSFR as a function of galactic gas and stellar properties, and can be compared with observations.
• The authors note that with slight adjustments of f̃w/α, the flaring of the stellar disk, or the dark matter density compared to the standard parameters and prescriptions, even closer agreement between predicted and observed ΣSFR can be obtained.
• These galaxies also have an irregular— and sublinear on average—relationship between ΣSFR and Σmol as inferred from CO.
• Blitz & Rosolowsky (2006) obtained an empirical fit relating the molecular-to-atomic-gas mass fractions to a midplane pressure estimate, Σmol/Σatom = (Ph/Ph,0)γ , for Ph = PBR ≡ Σ(2Gρs)1/2vg. (26) The empirical formula of Blitz & Rosolowsky (2006) produces somewhat more rapid decline in ΣSFR at low Σ (for large radii) than the prediction of their model.

4. SUMMARY AND DISCUSSION

• The authors have developed a theory for self-regulated star formation in multiphase galactic ISM disks in which stellar heating mediates the feedback.
• Here, the authors have not attempted to address these issues, but instead they have simply adopted an empirical solar-neighborhood value for the ratio of JFUV to ΣSFR to calibrate their relationships.
• The authors also discuss how the galactic environment variables (Σ, ρsd, Z) and the ISM model parameters control the transition between the diffuse-dominated and GBC-dominated regimes, and how presence of both regimes in a given annulus (due to spiral structure) affects estimates of the star formation rate.
• The authors start with the diffuse-gas thermal equilibrium Equation (18) relating the thermal pressure to the star formation rate ΣSFR = ΣGBC/tSF, which may be expressed as Pth = 1 φd Pth,0 ΣSFR,0 ΣGBC tSF .

Figures

Figure 7. Same as in Figure 3, for the galaxy NGC 5055. Also included is the comparison with the empirical formulae of Blitz & Rosolowsky (2006; blue plusses) and Leroy et al. (2008; green boxes), as described in the text.

Figure 6. Same as in Figure 5, except Hs ∝ 1/Σs is adopted for the model. (A color version of this figure is available in the online journal.)

Figure 5. Comparison between observed annular averages of ΣSFR (black squares) and model prediction (red triangles), for a set of spiral galaxies. For the model, Hs = const is adopted.

Figure 1. Schematic of approach to overall equilibrium. In (a), a higher-thanequilibrium value of the diffuse fraction fdiff makes the midplane pressure in the diffuse gas (dashed line) too high, while weak heating when the bound-cloud abundance 1 − fdiff is lower-than-equilibrium makes the thermal equilibrium curve (solid line) lie at too-low pressure. In (c), fdiff is lower-than-equilibrium, making the midplane pressure in the diffuse gas (dashed line) too low; at the same time, excess heating due to a high abundance of star-forming bound clouds situates the thermal equilibrium curve (solid line) too high. In (b), fdiff is appropriate for simultaneous dynamical and thermal equilibrium with Pth at the midplane equal to Ptwo-phase. Arrows in (a) and (c) indicate the direction of evolution of the midplane pressure level and thermal equilibrium curve for fdiff to approach the equilibrium value.

Figure 4. Same as in Figure 3, except a flaring stellar disk with Hs ∝ 1/Σs is adopted. (A color version of this figure is available in the online journal.)

Figure 3. Comparison between observed annular averages and model prediction for the galaxy NGC 7331. In (a), (b), and (d), filled squares with error bars show the observed values of ΣSFR or ΣSFR/Σ, where Σ is the total gas surface density; open triangles (red) show the predictions from the theory of Section 2. In (c), stars show the observed stellar surface density Σs , squares show total observed Σ, filled triangles show the observed atomic gas surface density, and open triangles (red) show the model prediction for the diffuse-gas surface density Σdiff . For the observations, error bars indicate scatter in the values within each azimuthal ring (systematic errors are larger). For the model prediction, Hs = const is adopted for the stellar scale height.

Figure 2. Sample solution of the self-consistent star formation rate for an idealized galactic model: (a) ΣSFR as a function of Σ (solid), together with the maximum possible rate Σ/tSF (dotted), and the outer-disk limiting solution given in Equation (22) (dashed); (b) the ratio ΣSFR/Σ as a function of Σs , together with the maximum value 1/tSF (dotted); (c) the input stellar surface density Σs (stars) and input total gas surface density Σ (squares) as a function of radius, together with the solution for Σdiff (triangles), all as a function of galactic radius R; (d) the solution for ΣSFR as a function of R. In (b) and (d), in addition to the case of constant stellar disk thickness (solid), results for a flaring disk Hs ∝ 1/Σs are shown as dashed lines.

Full text

The Astrophysical Journal, 721:975–994, 2010 October 1 doi:10.1088/0004-637X/721/2/975
C

2010. The American Astronomical Society. All rights reserved. Printed in the U.S.A.
REGULATION OF STAR FORMATION RATES IN MULTIPHASE GALACTIC DISKS:
A THERMAL/DYNAMICAL EQUILIBRIUM MODEL
Eve C. Ostriker
1
, Christopher F. McKee
2,3
, and Adam K. Leroy
4,5
1
Department of Astronomy, University of Maryland, College Park, MD 20742, USA; ostriker@astro.umd.edu
2
Departments of Physics and Astronomy, University of California, Berkeley, CA 94720, USA; cmckee@astro.berkeley.edu
3
LERMA-LRA, Ecole Normale Superieure, 24 rue Lhomond, 75005 Paris, France
4
National Radio Astronomy Observatory, 520 Edgemont Road, Charlottesville, VA 22903, USA; aleroy@nrao.edu
Received 2010 March 11; accepted 2010 July 30; published 2010 September 3
ABSTRACT
We develop a model for the regulation of galactic star formation rates Σ
SFR
in disk galaxies, in which interstel-
lar medium (ISM) heating by stellar UV plays a key role. By requiring that thermal and (vertical) dynamical
equilibrium are simultaneously satisfied within the diffuse gas, and that stars form at a rate proportional to the
mass of the self-gravitating component, we obtain a prediction for Σ
SFR
as a function of the total gaseous surface
density Σ and the midplane density of stars+dark matter ρ
sd
. The physical basis of this relationship is that the
thermal pressure in the diffuse ISM, which is proportional to the UV heating rate and therefore to Σ
SFR
, must adjust
until it matches the midplane pressure value set by the vertical gravitational field. Our model applies to regions
where Σ  100 M

pc
−2
.Inlow-Σ
SFR
(outer-galaxy) regions where diffuse gas dominates, the theory predicts that
Σ
SFR
∝ Σ
√
ρ
sd
. The decrease of thermal equilibrium pressure when Σ
SFR
is low implies, consistent with obser-
vations, that star formation can extend (with declining efficiency) to large radii in galaxies, rather than having a
sharp cutoff at a fixed value of Σ. The main parameters entering our model are the ratio of thermal pressure to total
pressure in the diffuse ISM, the fraction of diffuse gas that is in the warm phase, and the star formation timescale
in self-gravitating clouds; all of these are (at least in principle) direct observables. At low surface density, our
model depends on the ratio of the mean midplane FUV intensity (or thermal pressure in the diffuse gas) to the star
formation rate, which we set based on solar-neighborhood values. We compare our results to recent observations,
showing good agreement overall for azimuthally averaged data in a set of spiral galaxies. For the large flocculent
spiral galaxies NGC 7331 and NGC 5055, the correspondence between theory and observation is remarkably close.
Key words: galaxies: ISM – galaxies: spiral – ISM: kinematics and dynamics – galaxies: star formation –
turbulence
Online-only material: color figures
1. INTRODUCTION
Star formation is regulated by many physical factors, with
processes from sub-parsec to super-kiloparsec scales contribut-
ing to setting the overall rate (see, e.g., McKee & Ostriker
2007). One of the key factors expected to control the star forma-
tion rate is the available supply of gas. Over the whole range of
star-forming systems, from entire spiral galaxies to circumnu-
clear starbursts, the global average of the surface density of star
formation, Σ
SFR
, is observed to be correlated with the global
average of the neutral gas surface density Σ as Σ
SFR
∝ Σ
1+p
with 1 + p ≈ 1.4 (Kennicutt 1998). Recent observations at
high spatial resolution have made it possible to investigate
local, rather than global, correlations of the star formation
rate with Σ, using either azimuthal averages over rings, or
mapping with apertures down to  kpc scales (e.g., Wong
& Blitz 2002; Boissier et al. 2003, 2007; Heyer et al. 2004;
Komugi et al. 2005; Schuster et al. 2007; Kennicutt et al. 2007;
Dong et al. 2008; Bigiel et al. 2008; Blanc et al. 2009;Verley
et al. 2010). While power-law (Schmidt 1959, 1963) relation-
ships are still evident in these local studies, steeper slopes are
found for the outer, atomic-dominated regions of spiral galaxies
(as well as dwarf galaxies) compared to the inner, molecular-
dominated regions of spirals. In addition, measured indices
in the low-Σ,low-Σ
SFR
regime vary considerably from one
galaxy to another. Thus, no single Schmidt law characterizes the
5
Hubble Fellow.
regulation of star formation on local scales in the outer parts of
galaxies.
The nonlinearity of observed Schmidt laws implies that not
just the quantity of gas, but also its physical state and the sur-
rounding galactic environment, affect the star formation rate.
Indices p>0 imply that the star formation efficiency is higher
in higher-density regions, which are generally nearer the centers
of galaxies and have shorter dynamical times. Indeed, the expec-
tation based on theory and numerical simulations (Goldreich &
Lynden-Bell 1965;Kim&Ostriker2001) in thin, single-phase
gaseous disks is that gravitational instabilities leading to star
formation would grow only if the Toomre parameter (Toomre
1964) Q ≡ v
th
κ/(πGΣ) is sufficiently small. These instabilities
would develop over a timescale comparable to the galactic or-
bital time t
orb
≡ 2π/Ω, which corresponds to about twice the
two-dimensional Jeans time t
J,2D
≡ v
th
/(GΣ) when Q is near
critical. Here, v
th
is the thermal speed (v
2
th
≡ P
th
/ρ = kT/μ for
P
th
, ρ, and T the gas thermal pressure, density, and temperature),
and κ is the epicyclic frequency (κ
2
≡ R
−3
dΩ
2
/dR). While
the implied scaling Σ
SFR
∝ ΣΩ is roughly satisfied globally
(Kennicutt 1998), supporting the notion that galaxies evolve to-
ward states with Q roughly near critical (e.g., Quirk 1972), for
more local observations this does not provide an accurate pre-
diction of star formation (e.g., Leroy et al. 2008; Wong 2009).
In addition to galactic rotation and shear rates, an important
aspect of local galactic environment is the gravity of the stellar
component. The background stellar gravity compresses the disk
vertically (affecting the three-dimensional Jeans time t
J
≡
975

976 OSTRIKER, MCKEE, & LEROY Vol. 721
√
π/(Gρ ) ), and perturbations in the stellar density can act in
concert with gaseous perturbations in gravitational instabilities
(altering the effective Q). Thus, one might expect the stellar
surface density Σ
s
and/or volume density ρ
s
to affect the star
formation rate (see, e.g., Kim & Ostriker 2007 and references
therein). For example, if the stellar vertical gravity dominates
that of the gas (see Section 2 for a detailed discussion of this),
a scaling Σ
SFR
∝ Σ/t
J
would imply Σ
SFR
∝ Σ
3/2
(G
3
ρ
s
)
1/4
/v
th
for a constant-temperature gas disk. Although star formation
does appear to be correlated with Σ
s
(e.g., Ryder & Dopita
1994; Hunter et al. 1998; see also below), the simple scaling
∝ Σ/t
J
(taking into account both gaseous and stellar gravity,
and assuming constant v
th
, in calculating t
J
) does not in fact
provide an accurate local prediction of star formation rates (e.g.,
Abramova & Zasov 2008; Leroy et al. 2008; Wong 2009).
A likely reason for the inaccuracy of the simple star formation
prescriptions described above is that they do not account for
the multiphase character of the interstellar medium (ISM), in
which most of the volume is filled with low-density warm
(or hot) gas but much (or even most) of the mass is found
in clouds at densities two or more orders of magnitude greater
than that of the intercloud medium. For the colder (atomic and
molecular) phases, the turbulent velocity dispersions are much
larger than v
th
, so that the mean gas density ¯ρ averaged over
the disk thickness depends on the turbulent vertical velocity
dispersion. Even when multiphase gas and turbulence (and
stellar and gas gravity) are taken into account in simulations,
the simple estimate Σ
SFR
∝ Σ/t
J
∝ Σ
√
G ¯ρ (using ¯ρ directly
measured from the simulations) yields Schmidt-law indices
steeper than the true values measured in both the simulations and
in real galaxies (Koyama & Ostriker 2009a). Perhaps this should
not be surprising, since one would expect the proportions of
gas among different phases, as well as the overall vertical
distribution, to affect the star formation rate. If, for example,
most of the ISM’s mass were in clouds of fixed internal density
that formed stars at a fixed rate, then increasing the vertical
velocity dispersion of this system of clouds would lower ¯ρ but
leave Σ
SFR
unchanged.
The relative proportions of gas among different phases seems
difficult to calculate from first principles, because it depends
on how self-gravitating molecular clouds form and how they
are destroyed, both of which are very complex processes.
Intriguingly, however, analysis of recent observations of spiral
galaxies has shown that the surface density of the molecular
component averaged over ∼ kiloparsec annuli or local patches
shows a relatively simple overall behavior, increasing roughly
linearly with the empirically estimated midplane gas pressure
(Wong & Blitz 2002; Blitz & Rosolowsky 2004, 2006;Leroy
et al. 2008). The physical reason behind this empirical relation
has not, however, yet been explained.
In this paper, we use a simple physical model to analyze how
the gas is partitioned into diffuse and self-gravitating compo-
nents, based on considerations of dynamic and thermodynamic
equilibrium. We develop the idea that the midplane pressure in
the diffuse component must simultaneously satisfy constraints
imposed by vertical force balance, and by balance between heat-
ing (primarily from UV) and cooling. In particular, we pro-
pose that the approximately linear empirical relation between
molecular content and midplane pressure identified by Blitz &
Rosolowsky (2004, 2006) arises because the equilibrium gas
pressure is approximately proportional to the UV heating rate;
since the mean UV intensity is proportional to the star for-
mation rate and the star formation rate is proportional to the
molecular mass in normal spirals, the observed relationship nat-
urally emerges.
6
We use our analysis to predict the dependence
of the star formation rate on the local gas, stellar, and dark
matter content of disks, and compare our predictions with ob-
servations. The analysis, including our basic assumptions and
observational motivation for parameters that enter the theory,
is set out in Section 2. Section 3 then compares to the ob-
served data set previously presented in Leroy et al. (2008). In
Section 4, we summarize and discuss our main results.
2. ANALYSIS
2.1. Model Concepts and Construction
In this section, we construct a local steady-state model for the
star formation rate in the disk, with independent variables the
total surface density of neutral gas (Σ), the midplane stellar
density (ρ
s
), and the dark matter density (ρ
dm
). The latter
two quantities enter only through their effect on the vertical
gravitational field. To develop this model, we suppose that
the diffuse gas filling most of the volume of the ISM is in
an equilibrium state. The equilibrium in the diffuse ISM has
two aspects: force balance in the vertical direction (with a sum
of pressure forces offsetting a sum of gravitational forces),
and balance between heating and cooling (where heating is
dominated by the FUV). Star-forming clouds, because they
are self-gravitating entities at much higher pressure than their
surroundings, are treated as separate from the space-filling
diffuse ISM. The abundance of gravitationally bound, star-
forming clouds is nevertheless important for establishing an
equilibrium state in the diffuse gas, because the FUV that heats
the diffuse ISM originates in young OB associations. We assume
(consistent with observations and numerical simulations) that
the equilibrium thermal state established for the diffuse medium
includes both warm and cold atomic gas. This hypothesis leads
to a connection between the dynamical equilibrium state and
the thermal equilibrium state: there are two separate constraints
on the pressure that must be simultaneously satisfied. These
conditions are met by an appropriate partition of the available
neutral gas into diffuse and self-gravitating components.
The reason for the partition between diffuse and self-
gravitating gas can be understood by considering the physical
requirements for equilibrium. The specific heating rate (Γ)in
the diffuse gas is proportional to the star formation rate, which is
proportional to the amount of gas that has settled out of the ver-
tically dispersed diffuse gas and collected into self-gravitating
clouds. The specific cooling rate (nΛ) in the diffuse gas is pro-
portional to the density and hence to the thermal pressure, which
(if force balance holds) is proportional to the vertical gravity
and to the total surface density of diffuse gas. Thus, an equi-
librium state, in which cooling balances heating and pressure
balances gravity, can be obtained by a suitable division of the
gas mass into star-forming (gravitationally bound) and diffuse
components such that their ratio is proportional to the vertical
gravitational field. If too large a fraction of the total surface
density is in diffuse gas, the pressure will be too high, while the
star formation rate will be too low. In this situation, the cooling
would exceed heating, and mass would “drop out” of the diffuse
component to produce additional star-forming gas. With addi-
tional star formation, the FUV intensity would raise the heating
rate in the diffuse gas until it matches the cooling.
6
Dopita (1985) previously showed that assuming the pressure to be
proportional to the star formation rate yields scaling properties similar to
observed relationships.

No. 2, 2010 REGULATION OF STAR FORMATION RATES IN MULTIPHASE GALACTIC DISKS 977
In the remainder of this section, we formalize these ideas
mathematically, first defining terms (Section 2.2), then con-
sidering the requirements of dynamical balance (Section 2.3)
and thermal balance (Section 2.4), and finally combining these
to obtain an expression for the star formation rate when both
equilibria are satisfied (Section 2.5). We then discuss, from
a physical point of view, how the various feedback processes
might act to adjust the system over time, steering it toward
the equilibrium we have identified (Section 2.6). While evolv-
ing to an equilibrium of this kind is plausible, we emphasize
that this is an assumption of the present model, which must be
tested by detailed time-dependent simulations.
7
A worked ex-
ample applying the model to an idealized galaxy is presented in
Section 2.7. In developing the present model, we have adopted a
number of simplifications that a more refined treatment should
address; we enumerate several of these issues in Section 2.8.
2.2. Gas Components
In this model, we divide the neutral ISM into two components.
One component consists of the gas that is collected into
gravitationally bound clouds (GBCs) localized near the galactic
midplane, with mean surface density (averaged over ∼ kpc
scales) of Σ
GBC
. The other component consists of gas that
is diffuse (i.e., not gravitationally bound), with mean surface
density Σ
diff
. Here, we use the term “diffuse” in the sense of
being widely dispersed or scattered throughout the volume; the
diffuse component may include both tenuous, volume-filling
gas and small, dense cloudlets (see below). All star formation
is assumed to take place within the GBC component. In normal
galaxies, the GBC component is identified with the population
of giant molecular clouds (GMCs). Note that while observed
GMCs in the Milky Way consist primarily of molecular gas,
they also contain atomic gas in shielding layers. More generally,
as we shall discuss further below, the relative proportions of
molecular and dense atomic gas in GBCs depends on the cloud
column and metallicity, and GBCs could even be primarily
atomic if the metallicity is sufficiently low.
The diffuse component is identified (in normal galaxies)
with the atomic ISM. We treat the diffuse gas as a two-phase
cloud–intercloud medium in thermal pressure equilibrium, with
turbulent vertical velocity dispersion v
2
z
assumedtobethe
same for warm and cold phases. Although the cold cloudlets
within the diffuse component have much higher internal density
than the warm intercloud gas, they are (by definition) each of
sufficiently low mass that they are non-self-gravitating, such
that their thermal pressure (approximately) matches that of
their surroundings. The pressure in the interior of GBCs is
considerably higher than the pressure of the surrounding diffuse
gas (cf. Koyama & Ostriker 2009b).
In reality, the diffuse gas would not have a single unique
pressure even if the radiative heating rate is constant because of
time-dependent dynamical effects: turbulent compressions and
rarefactions heat and cool the gas, altering what would otherwise
be a balance between radiative heating and cooling processes.
Nevertheless, simulations of turbulent gas with atomic-ISM
heating and cooling indicate that the majority of the gas has
pressure within ∼50% of the mean value (Piontek & Ostriker
2005, 2007), although the breadth of the pressure peak depends
on the timescale of turbulent forcing ∼L
turb
/v
turb
compared to
the cooling time (Audit & Hennebelle 2005, 2010; Hennebelle
7
Very recent numerical studies provide support for the quasi-equilibrium
assumption—see C.-G. Kim et al. (2010, in preparation).
& Audit 2007; Gazol et al. 2005, 2009; Joung & Mac Low 2006;
Joung et al. 2009). Observations indicate a range of pressures
in the cold atomic gas in the solar neighborhood, with a small
fraction of the gas at very high pressures, and ∼50% of the gas
at pressures within ∼50% of the mean value (Jenkins & Tripp
2001, 2007).
In general, the volume-weighted mean thermal pressure at the
midplane is given by
P
th

vol
=

P
th
d
3
x

d
3
x
=

(P
th
/ρ)ρd
3
x

d
3
x
=

ρd
3
x

d
3
x

v
2
th
dm

dm
= ρ
0

v
2
th

mass
, (1)
where ρ
0
is the volume-weighted mean midplane density of
diffuse gas. The quantity v
2
th

mass
is the mass-weighted mean
thermal velocity dispersion; for a medium with warm and cold
gas with respective mass fractions (in the diffuse component)
f
w
and f
c
= 1 − f
w
and temperatures T
w
and T
c
,

v
2
th

mass
c
2
w
= f
w
+
T
c
T
w
(1 − f
w
) ≡
˜
f
w
. (2)
Here, c
w
≡ (P
w
/ρ
w
)
1/2
= (kT
w
/μ)
1/2
is the thermal speed of
warm gas. Since the ratio T
w
/T
c
is typically ∼100,
˜
f
w
≈ f
w
unless f
w
is extremely small.
If the thermal pressures in the warm and cold diffuse-gas
phases are the same, P
th

vol
= P
w
= ρ
w
c
2
w
, so that from
Equations (1) and (2),
ρ
w
ρ
0
=

v
2
th

mass
c
2
w
=
˜
f
w
. (3)
This result still holds approximately even if the warm and
cold medium pressures differ somewhat, since the warm gas
fills most of the volume, P
th

vol
≈ P
w
. Note that one can
also write ρ
w
/ρ
0
= f
w
(V
tot
/V
w
)forV
tot
and V
w
the total and
warm-medium volumes, so that
˜
f
w
≈ f
w
provided the warm
medium fills most of the volume. If the medium is all cold gas,
˜
f
w
= T
c
/T
w
. Henceforth, we shall assume the warm and cold
gas pressures are equal at the midplane so that P
th

vol
→ P
th
;
for convenience, we shall also omit the subscript on v
2
th

mass
.
2.3. Vertical Dynamical Equilibrium of Diffuse Gas
By averaging the momentum equation of the diffuse com-
ponent horizontally and in time, and integrating outward from
the midplane, it is straightforward to show that the difference
in the total vertical momentum flux across the disk thickness
(i.e., between midplane and z
diff,max
) must be equal to the total
weight of the diffuse gas (e.g., Boulares & Cox 1990; Piontek
&Ostriker2007; Koyama & Ostriker 2009b). This total weight
has three terms. The first term is the weight of the diffuse gas in
its own gravitational field,

z
diff,max
0
ρ
dΦ
diff
dz
dz =
1
8πG

z
diff,max
0
d

dΦ
diff
dz

2
dz
dz =
πGΣ
2
diff
2
,
(4)
where we have used |dΦ
diff
/dz|
z
diff,max
= 2πGΣ
diff
for a slab.
The second term is the weight of the diffuse gas in the mean
gravitational field associated with the GBCs,

z
diff,max
0
ρ
dΦ
GBC
dz
dz ≈ πGΣ
GBC
Σ
diff
, (5)

978 OSTRIKER, MCKEE, & LEROY Vol. 721
where we have assumed that the scale height of the GBC
distribution is much smaller than that of the diffuse gas so that
|dΦ
GBC
/dz|≈2πGΣ
GBC
over most of the integral. Note that
Equation (5) gives an upper bound on this term in the weight,
with a lower bound given by πGΣ
GBC
Σ
diff
/2, corresponding to
the case in which the vertical distributions of the diffuse and
gravitationally bound components are the same. The third term
is the weight in the gravitational field associated with the disk
stars plus dark matter,

z
diff,max
0
ρ

dΦ
s
dz
+
dΦ
dm
dz

dz ≡ 2πζ
d
G
ρ
sd
Σ
2
diff
ρ
0
. (6)
Here, ρ
sd
= ρ
s
+ ρ
dm
is the midplane density of the stellar
disk plus that of the dark matter halo; we have assumed
a flat rotation curve V
c
= const for the dark halo so that
ρ
dm
= (V
c
/R)
2
/(4πG).
8
The stellar disk’s scale height is
assumed to be larger than that of the diffuse gas, so that
g
z
≈ 4πGρ
sd
z within the diffuse-gas layer. The numerical
value of ζ
d
depends, but not sensitively, on the exact vertical
distribution of the gas, which in turn depends on whether self-
or external gravity dominates; ζ
d
≈ 0.33 within 5% for a range
of cases between zero external gravity and zero self gravity.
Allowing for a gradient in the vertical stellar density within the
gas distribution, the stellar contribution to the weight would
be reduced by a factor ∼1 − (2/3)(H
g
/H
s
)
2
, where H
g
and
H
s
are the gaseous and stellar scale heights. In the (unlikely)
circumstance that the diffuse-gas scale height is much larger
than that of the stars, g
z
≈ 2πGΣ
s
would be substituted for
the gravity of the stellar component, yielding a contribution
analogous to that in Equation (5) with Σ
GBC
→ Σ
s
.
Including both thermal and kinetic terms, and taking ρ → 0
at the top of the diffuse-gas layer, the difference in the gaseous
vertical momentum flux between z = 0 and z
diff,max
is given by
P
th
+ ρ
0
v
2
z
.Thetermv
2
z
is formally a mass-weighted quantity
(analogous to v
2
th

mass
), but we assume a similar turbulent
velocity dispersion for the diffuse warm and cold atomic gas
(Heiles & Troland 2003). If the magnetic field is significant,
a term equal to the difference between B
2
/(8π) − B
2
z
/(4π)at
z = 0 and z
diff,max
is added (Boulares & Cox 1990; Piontek &
Ostriker 2007). Like other pressures, these magnetic terms are
volume weighted; both observations (Heiles & Troland 2005)
and numerical simulations (Piontek & Ostriker 2005) indicate
that field strengths in the warm and cold atomic medium are
similar. If the scale height of the magnetic field is larger than
that of the diffuse gas (as some observations indicate; see, e.g.,
Ferri
`
ere 2001), then this term will be small, while it will provide
an appreciable effect if B → 0 where ρ → 0. In any case, the
magnetic term in the vertical momentum flux may be accounted
for by taking ρ
0
v
2
z
→ ρ
0
v
2
z
+(ΔB
2
/ 2 − ΔB
2
z
)/ 4π ≡ ρ
0
v
2
t
,
where Δ indicates the difference between values of the squared
magnetic field at z = 0 and z
diff,max
. Cosmic rays have a much
8
If the vertical stellar distribution in the disk varies as ρ
s
∝ sech
2
(z/H
s
)
with H
s
= v
2
z,s
/(πGΣ
s
), then the midplane stellar density is
ρ
s
= Σ
s
/(2H
s
) = πGΣ
2
s
/(2v
2
z,s
). Existing photometric and kinematic
observations suggest that H
s
∼ const and v
z,s
∝ Σ
1/2
s
(van der Kruit & Searle
1982; Bottema 1993), but these are sensitive primarily to the central parts of
the disk. Note that if the Toomre parameter for the stellar disk and the
vertical-to-horizontal velocity dispersion ratio are both constant with radius,
then ρ
s
∝ ρ
dm
∝ (V
c
/R)
2
. In this case, the ratio of the gas-to-stellar scale
height is ∼v
z,g
/v
z,s
; since the gas can dissipate turbulence and cool to
maintain constant v
z,g
while the stellar velocity dispersion secularly increases
over time, the gas layer will tend to be thinner than the stellar layer even if
both components flare in the outer parts of galaxies.
larger scale height than that of the diffuse (neutral) gas, such that
difference in the cosmic-ray pressure between z = 0 and z
diff,max
may be neglected. We have also neglected the contribution from
diffuse warm ionized gas, which has a low mean density and
large scale height compared to that of the neutral gas (e.g.,
Gaensler et al. 2008).
Equating the momentum flux difference with the total weight,
we have
P
th

1+
v
2
t
c
2
w
˜
f
w

=
πG
2
Σ
2
diff
+ πGΣ
GBC
Σ
diff
+2πζ
d
Gc
2
w
˜
f
w
ρ
sd
Σ
2
diff
P
th
. (7)
Here, we have used Equations (1) and (2) to substitute
˜
f
w
c
2
w
/P
th
for ρ
−1
0
on the right-hand side. As noted above, the second term
on the right-hand side could be reduced by up to a factor of two,
if the scale height of the GBC distribution approaches that of
the diffuse gas. It is convenient to define
α ≡ 1+
v
2
t
c
2
w
˜
f
w
=

v
2
th

+ v
2
t

v
2
th

=
P
th
+ ρ
0
v
2
z
+ Δ

B
2
/2 − B
2
z

(4π)
P
th
, (8)
which represents the midplane ratio of total effective pressure to
thermal pressure. If the magnetic contribution is small (which
wouldbetrueifΔB
2
 B
2
, even if magnetic and thermal
pressures are comparable at the midplane), α is the total
observed velocity dispersion σ
2
z
divided by the mean thermal
value. We shall treat v
t
, c
w
, and
˜
f
w
as parameters that do not
vary strongly within a galaxy or from one galaxy to another (see
below), and Σ
diff
, Σ
GBC
, and P
th
as (interdependent) variables.
At any location in a galaxy, we shall consider ρ
sd
(and the total
gas surface density Σ = Σ
diff
+ Σ
GBC
) as a given environmental
conditions.
Equation (7) is a quadratic in both Σ
diff
and P
th
.Thus,ifP
th
and Σ
GBC
are known, we may solve to obtain the surface density
of diffuse gas:
Σ
diff
=
2αP
th
πGΣ
GBC
+

(πGΣ
GBC
)
2
+2πGα

P
th
+4ζ
d
c
2
w
˜
f
w
ρ
sd

1/2
.
(9)
Scaling the variables to astronomical units, the result in
Equation (9) can also be expressed as
Σ
diff
=

9.5 M

pc
−2
α

P
th
/k
3000 K cm
−3

×

0.11

Σ
GBC
1 M

pc
−2

+

0.011

Σ
GBC
1 M

pc
−2

2
+ α

P
th
/k
3000 K cm
−3

+10α
˜
f
w

ρ
sd
0.1 M

pc
−3

1/2

−1
.
(10)
What are appropriate parameter values to use? Since thermal
balance in the warm medium is controlled by line cooling
(Wolfire et al. 1995, 2003), the warm-medium temperature is
relatively insensitive to local conditions in a galaxy; we shall
adopt c
w
= 8kms
−1
. Numerical simulations in multiphase

No. 2, 2010 REGULATION OF STAR FORMATION RATES IN MULTIPHASE GALACTIC DISKS 979
gas have shown that the magnetic field is amplified by the
magnetorotational instability to a level B
2
/(8π) = (1 − 2)P
th
,
independent of the mass fractions of cold and warm gas and the
vertical gravitational field strength (Piontek & Ostriker 2005,
2007), while |B
z
/B
φ
|1. This is consistent with observed
magnetic field strengths measured in the Milky Way and in
external galaxies (Heiles & Troland 2005; Beck 2008). Large-
scale turbulent velocity dispersions observed in local H i gas
(both warm and cold) are ∼7kms
−1
, comparable to c
w
(Heiles
& Troland 2003; Mohan et al. 2004). Total vertical velocity
dispersions in H i gas in external galaxies are also observed
to be in the range 5–15 km s
−1
, decreasing outward from the
center (Tamburro et al. 2009).
The most uncertain parameter is
˜
f
w
≈ f
w
, the fraction of
the diffuse mass in the warm phase. In the solar neighborhood,
this is ∼0.6 (Heiles & Troland 2003), and in external galaxies
the presence of both narrower and broader components of
21 cm emission suggests that both warm and cold gas are
present (de Blok & Walter 2006), with some indication based
on “universality” in line profile shapes that the warm-to-cold
mass ratio does not strongly vary with position (Petric & Rupen
2007). In the outer Milky Way, the ratio of H i emission to
absorption appears nearly constant out to ∼25 kpc, indicating
that the warm-to-cold ratio does not vary significantly (Dickey
et al. 2009). In dwarf galaxies as well, observations indicate that
both a cold and warm H i component is present (Young & Lo
1996). While uncertain, it is likely that
˜
f
w
∼ 0.5–1, at least in
outer galaxies.
Thus, allowing for the full range of observed variation,
α ∼ 2–10, α
˜
f
w
∼ 1–5, and α/
˜
f
w
∼ 2–20; we shall adopt
α = 5 and
˜
f
w
= 0.5 as typical for mid-to-outer-disk conditions.
For these fiducial parameters, and taking midplane thermal
pressure P
th,0
/k ∼ 3000 K cm
−3
(see Jenkins & Tripp 2001
and Wolfire et al. 2003), ρ
sd
= 0.05 M

pc
−3
(Holmberg &
Flynn 2000), and Σ
GBC
 2 M

pc
−2
(Dame et al. 1987, 2001;
Bronfman et al. 1988; Luna et al. 2006; Nakanishi & Sofue
2006) near the Sun, the result from Equation (10) is consistent
with the observed total surface density estimate ∼10 M

pc
−2
of atomic gas in the solar neighborhood (Dickey & Lockman
1990; Kalberla & Kerp 2009).
Equation (7) may also be solved to obtain the thermal pressure
in terms of Σ
diff
, Σ
GBC
, ρ
sd
and the diffuse-gas parameters α
and
˜
f
w
:
P
th
=
πGΣ
2
diff
4α

1+2
Σ
GBC
Σ
diff
+


1+2
Σ
GBC
Σ
diff

2
+
32ζ
d
c
2
w
˜
f
w
α
πG
ρ
sd
Σ
2
diff

1/2
⎫
⎬
⎭
. (11)
Over most of the disk in normal galaxies, the term in
Equation (11) that is proportional to ρ
sd
(arising from the weight
in the stellar-plus-dark-matter gravitational field) dominates;
this yields
P
th
∼ Σ
diff
(
2Gρ
sd
)
1/2

πζ
d
˜
f
w
α

1/2
c
w
. (12)
For given ρ
sd
and Σ, the thermal pressure therefore increases
approximately proportional to the fraction of gas in the diffuse
phase, f
diff
≡ Σ
diff
/Σ. With
(
πζ
d
)
1/2
≈ 1 and
˜
f
w
≈ f
w
,
P
th
∼ Σ
w
(
2Gρ
sd
)
1/2
c
2
w
/(v
2
th
 + v
2
t
)
1/2
in this limit; i.e., it is
the surface density of the volume-filling warm medium that sets
the thermal pressure. Multiplying Equation (12)byα and using
α
˜
f
w
= (v
2
th
 + v
2
t
)/c
2
w
yields
P
tot
∼ Σ
diff
(
2Gρ
sd
)
1/2

v
2
th

+ v
2
t

1/2
. (13)
This is the same as the formula for midplane pressure adopted
by Blitz & Rosolowsky (2004, 2006), except that instead of
Σ
diff
their expression contains the total gas surface density Σ,
instead of (v
2
th
+ v
2
t
)
1/2
they use the thermal+turbulent vertical
velocity dispersion (these are equal if vertical magnetic support
is negligible), and they omit the dark matter contribution to
ρ
sd
.Usingρ
s
= πGΣ
2
s
/(2v
2
z,s
) and taking Σ
GBC
, ρ
dm
→ 0,
Equation (11) yields
P
tot
∼
πGΣ
2
2

1+

v
2
th

+ v
2
t

1/2

1/2
Σ
s
v
z,s
Σ

, (14)
recovering the result of Elmegreen (1989; except that he includes
the total B
2
, rather than ΔB
2
,inv
2
t
and P
tot
).
2.4. Thermal Equilibrium of Diffuse Gas
As expressed by Equation (11), the thermal pressure in the
diffuse gas must respond to the dynamical constraint imposed
by vertical momentum conservation in the disk. In addition,
the thermal pressure is also regulated by the microphysics of
heating and cooling. Namely, if the atomic gas is in the two-
phase regime (as is expected in a star-forming region of a
galaxy; see Section 2.5), then the thermal pressure must lie
between the minimum value for which a cold phase is possible,
P
min,cold
, and the maximum value for which a warm phase is
possible, P
max,warm
. Wolfire et al. (2003) found, based on detailed
modeling of heating and cooling in the solar neighborhood, that
geometric mean of these two equilibrium extrema, P
two-phase
≡
(P
min,cold
P
max,warm
)
1/2
, is comparable to the local empirically
estimated thermal pressure, and that two phases are expected to
be present in the Milky Way out to ∼ 18 kpc. Based on turbulent
numerical simulations with a bistable cooling curve, Piontek &
Ostriker (2005, 2007) found that the mean midplane pressure
evolves to a value near the geometric mean pressure P
two-phase
,
for a wide range of vertical gravitational fields and warm-to-
cold mass fractions. Thus, we expect the midplane thermal
pressure in the diffuse gas to be comparable to the two-phase
value defined by the thermal equilibrium curve, P
th
≈ P
two-phase
.
Since P
max,warm
/P
min,cold
∼ 2–5 (Wolfire et al. 2003), even if
P
th
= P
two-phase
does not hold precisely, the midplane pressure
P
th
will be within a factor ∼ 2ofP
two-phase
provided the diffuse
gas is in the two-phase regime.
For the fiducial solar-neighborhood model of Wolfire et al.
(2003), the geometric mean thermal pressure is P
two-phase
/k ∼
P
th,0
/k ∼ 3000 K cm
−3
. For other environments, the values of
P
min,cold
and P
max,warm
depend on the heating of the gas: en-
hanced heating pushes the transition pressures upward (Wolfire
et al. 1995). Because the dominant heating is provided by the
photoelectric effect on small grains, P
two-phase
increases ap-
proximately linearly with the FUV intensity. Assuming that
P
two-phase
scales with P
min,cold
, we adapt the expression given in
Wolfire et al. (2003) and normalize P
two-phase
using the solar-
neighborhood value:
P
two-phase
k
= 12,000 K cm
−3
G

0
Z

d
/Z

g
1+3.1(G

0
Z

d
/ζ

t
)
0.365
. (15)