Star formation and stellar evolution are such important drivers of galactic evolution that empirical laws to determine the star formation rate have been investigated for over 50 years. The results have never been very precise because star formation spans a wide range of scales, from cluster-forming cores to molecular clouds to the whole interstellar medium.

On the scale of a galaxy, the first idea was a proposed connection
between the total star formation rate and the mass of interstellar gas.
Schmidt (1959)
derived the star formation rate (SFR) over the history
of the Milky Way assuming a constant initial luminosity function for
stars,
(*M*_{V}),
a stellar lifetime function *T*(*M*_{V}), a
gas return per star equal to all of the stellar mass above
0.7 *M*_{},
and a star formation rate *f*(*t*) that scales with a
power *n* of the gas mass, *M*_{G}(*t*). Then
*f*(*t*)
_{MV}
(*M*_{V}) =
*C* [*M*_{G}(*t*)]^{n}, for a summation
_{MV} over
all stellar types.

Schmidt gave analytical solutions for *n* = 0,1,2. He noted that a
scale height for HI of 144 pc, a scale height for Cepheids of 80 pc,
and a scale height for clusters of 58 pc gave *n* = 2 to 3. The white
dwarf count gave *n* > 2, the He abundance suggested *n* =
2, the uniformity of HI suggested *n*
2, and the cluster mass
function gave *n* = 1 to 2. Schmidt also suggested that with
*n* = 2, dense galaxies
like ellipticals should now have less gas than low-density galaxies
like the LMC. His final comment was "It is hoped to study the
evolution of galaxies in more detail in the future." Following
Schmidt (1959),
many authors derived scaling relations between the
average surface density of star formation,
_{SFR}, and the
average surface density of gas.
Buat, Deharveng
& Donas (1989)
included molecular and
atomic gas and determined star formation rates from the UV flux
corrected for Milky Way and internal extinction. They assumed a
constant H_{2} / CO ratio and a
Scalo (1986)
IMF. The result was a good correlation between the average star
formation rate in a sample of 28 galaxies and the 1.65 ± 0.16 power
of the average total gas surface density. In the same year,
Kennicutt (1989)
used H for star
formation, and HI and CO for the gas
with a constant *H*_{2} / *CO* conversion factor, and
determined star formation rates both as a function of galactocentric
radius and averaged over whole galaxy disks. For whole galaxies, the
average H flux scaled
with the average gas surface density to a
power between 1 and 2; there was a lot of scatter in this relation and
the correlation was better for HI than *H*_{2}. More
interesting was
Kennicutt's (1989)
result that the star formation rate had an abrupt cutoff in radius where the
Toomre (1964)
stability condition indicated the onset of gravitationally stable
gas. Kennicutt derived a threshold gas column density for star formation,
_{crit} =
/ (3.36*G*) for
= 0.7; is the
velocity dispersion of the gas;
is the epicyclic
frequency, and *G* is the gravitational constant.

In a second study,
Kennicutt (1998)
examined the disk-average star formation rates using a larger sample of
galaxies with
H, HI, and CO. He found
that for normal galaxies, the slope of the
SFR-surface density relation ranged between 1.3 to 2.5, depending on
how the slope was measured; there was a lot of scatter. When starburst
galaxies with molecular surface densities in excess of 100
*M*_{}
were included, the overall slope became better defined and was around
1.4. This paper also found a good correlation with a star formation
rate that scaled directly with the average surface density of gas and
inversely with the rotation period of the disk. This second law
suggested that large-scale dynamical processes are involved.

Hunter et al. (1998) considered the same type of analysis for dwarf Irregulars and derived a critical surface density that was lower than the Kennicutt (1989) value by a factor of ~ 2. This meant that stars form in more stable gas in dwarf irregulars compared to spirals.

Boissiet et
al. (2003)
compared
_{SFR}
and _{gas}
versus radius in 16 resolved galaxies with three theoretical
expressions. The best fits were a SFR dependence on the gas surface
density as
_{SFR}
_{gas}^{2.06}, a more dynamical law from
Boissier &
Prantzos (1999)
which gave the fit
_{SFR}
_{gas}^{1.48}(*V* / *R*) for
rotation speed *V* and radius *R*, and a third type of law from
Dopita &
Ryder (1994),
which fit to
_{SFR}
_{gas}^{0.97} /
_{tot}^{0.61}.
Boissiet et
al. (2003)
assumed that H_{2} / CO varied with radius as the metallicity
(Boselli et
al. 2002).
Their conclusion was that the three laws are equally good, and that for
the pure gas law, *n* > 1.4.
Boissiet et
al. (2003)
also looked for a star formation threshold in the Milky Way. They
determined /
_{crit}
using both pure-gas for
_{crit}
and a gas+star
_{crit} from
Wang & Silk
(1994).
They found that the gas+star
_{crit} gave
the best threshold for determining where star formation occurs. The gas
alone was sub-threshold throughout the disk.

Zasov & Smirnova
(2005)
showed that a threshold like
_{crit} may be
used to determine the gas fraction in galaxies. If all galaxies have
(HI) approximately
at the critical
_{crit}
=
/
*G*, which is
proportional to *V* / *R* from
, then
*M*_{gas} =
_{R}
2 *R*
_{crit}
*dR*
*VR*. This was shown to be the case from observations. They also
considered that the total mass is *M*_{tot}
*V*^{2}*R*, in which
case *M*_{tot} / *M*_{gas}
*V*, the
rotation speed. This was also shown to be confirmed by observations. In
their interpretation, small galaxies are more gas-rich than large
galaxies because all galaxies have their gas column densities close to
the surface density threshold.

For the Milky Way,
Misiriotis et
al. (2006)
used COBE/DIRBE observations to
get both the gas and dust distributions and the SFR distribution. They
found a gas-law slope of 2.18 ± 0.20, which they claimed was similar
to Kennicutt's (1998)
bivariate fit slope *n* = 2.5 for normal galaxies.
Luna et al. (2006)
determined the Milky Way SFR from IRAS point sources and
the CO surface density from a southern hemisphere survey
(assuming constant H_{2} / CO). They found star formation
concentrated in low-shear spiral arms and suggested an additional
dependence on shear. Overall they derived
_{SFR} ~
_{gas}^{1.2±0.2}.
Vorobyov (2003)
also suggested a shear dependence for the SFR based on observations of
the Cartwheel galaxy, where there is an inner ring of star formation
with high shear that is too faint for the normal Kennicutt law, given
the gas column density.

A threshold for gravitational instabilities in rotating disks has been
derived for various ideal cases. For an infinitely thin disk of
isothermal gas, the dispersion relation for radial waves is
^{2} =
*k*^{2}
^{2} -
2 *G*
*k* +
^{2}. Solving
for the fastest growth rate
gives the wavenumber at
peak growth, *k* =
*G* /
^{2}, and the
wavelength, =
2^{2} / *G*
,
which is on the order of a kiloparsec in main galaxy disks. The
dominant unstable mass is
*M* ~ ( /
2)^{2}
= ^{4} /
*G*^{2}
~ 10^{7}
*M*_{} in local
spirals. The peak rate is given by

(1.1) |

which requires
*Q*
/
*G*
< 1 for
instability (i.e., when
_{peak}^{2}
< 0).

Disk thickness weakens the gravitational force in the in-plane
direction by an amount that depends on wavenumber, approximately as
1 / (1 + *kH*) for exponential scale height *H* (e.g.,
Elmegreen 1987,
Kim & Ostriker
2007).
Typically, *k* ~ 1/*H*, so this weakening can slow the
instability by a
factor of ~ 2, and it can make the disk slightly more stable by a
factor of 2 in *Q*. On the other hand, cooling during condensation
decreases the effective value of the velocity dispersion, which should
really be written
^{1/2}
for adiabatic index
that appears in the relation
*P*
^{} with
pressure *P* and density
. If *P*
is nearly constant for changes
in , as often
observed, then
~ 0.
Myers (1978)
found ~
0.25 for various thermal temperatures at interstellar
densities between 0.1 cm^{-3} and 100 cm^{-3}. Thus the
effects of disk thickness and a soft equation of state partially
compensate for each other.

There is also a *Q* threshold for the collapse of an expanding
shell of gas
(Elmegreen, Palous
& Ehlerova 2002).
Pressures from OB associations form giant shells of
gas and cause them to expand. Eventually they go unstable when the
accumulated gas is cold and massive enough, provided the induced
rotation and shear from Coriolis forces are small. Considering
thousands of initial conditions, these authors found that a sensitive
indicator of whether collapse occurs before the shell disperses is the
value of *Q* in the local galaxy disk, i.e., independent of the shell
itself. The fraction *f* of shells that collapsed scaled inversely with
*Q* as *f* ~ 0.5 - 0.4 log_{10} *Q*.

The Toomre *Q* parameter is also likely to play a role in the
occurrence of instabilities in turbulence-compressed gas on a galactic
scale
(Elmegreen 2002).
Isothermal compression has to include a mass
comparable to the ambient Jeans mass, *M*_{Jeans}, in order to
trigger instabilities. The turbulent outer scale in the galaxy is
comparable to the Jeans length, *L*_{Jeans}, which is about the
galactic gas scale height, *H*. If the compression distance exceeds the
epicyclic length, then Coriolis forces spin up the compressed gas,
leading to resistance from centrifugal forces. So instability needs
*L*_{Jeans}
*L*_{epicycle}, which
means *Q* 1, since
*L*_{Jeans} ~ *H* ~
^{2} /
*G*
. The epicyclic
length is *L*_{epicycle} ~
/*k*, so
*L*_{Jeans} / *L*_{epicycle} = *Q*.

The dimensionless parameter *Q* measures the ratio of the centrifugal
force from the Coriolis spin-up of a condensing gas perturbation to the
self-gravitational force, on the scale where gravity and pressure
forces are equal, which is the Jeans length. The derivation of *Q*
assumes that angular momentum is conserved, so the Coriolis force spins
up the gas to the maximum possible extent. When *Q* > 1, a
condensing
perturbation on the scale of the Jeans length spins up so fast that its
centrifugal force pulls it apart against self-gravity. Larger-scale
perturbations have the same self-gravitational acceleration (which
scales with ) and
stronger Coriolis acceleration (which scales
with ^{2} /
*k*); smaller-scale perturbations have stronger
accelerations from pressure. If angular momentum is not conserved, then
the disk can be unstable for a wider range of *Q* because there is less
spin up during condensation. For example, the Coriolis force can be
resisted by magnetic tension or viscosity and then the angular momentum
in a condensing cloud will get stripped away. This removes the *Q*
threshold completely
(Chandrasekhar
1954,
Stephenson
1961,
Lynden-Bell
1966,
Hunter & Horak
1983).
In the magnetic case, the result is the Magneto-Jeans instability,
which can dominate the gas condensation in low-shear environments like
spiral arms and some inner disks
(Elmegreen 1987,
Elmegreen 1991,
Elmegreen 1994,
Kim & Ostriker
2001,
Kim & Ostriker
2002,
Kim et al. 2002).
For the viscous case,
Gammie (1996)
showed that for *Q* close
to but larger than 1, i.e., in the otherwise stable regime, viscosity
can make the gas unstable with a growth rate equal to nearly one-third
of the full rate for a normally unstable (*Q* < 1) disk. A
dimensionless parameter for viscosity
is
^{3} /
*G*^{2}
^{2}, which is
~ 11 according to
Gammie (1996).
This is a large value indicating
that galaxy gas disks should be destabilized by viscosity. An important
dimensionless parameter for magnetic tension is *B*^{2} /
( *G*
^{2}) ~ 8,
which is also large enough to be important. Thus gas
disks should be generally unstable to form small spiral arms and
clouds, even with moderately stable *Q*, although the growth rate can
be low if *Q* is large.

**1.2. Modern Versions of the KS Law with ~ 1.5
slope**

Kennicutt et
al. (2007)
studied the local star formation law in M51 with 0.5-2 kpc
resolution using Pa-
and 24*µ* + H
lines for the SFR,
and a constant conversion factor for CO to H_{2}. There was
a correlation, mostly from the radial variation of both SFR and gas
surface density, with a slope of 1.56 ± 0.04. There was no
correlation with
(HI) alone, as this
atomic component had about constant column density ( ~ 10
*M*_{}). The
correlation with
molecules alone was about the same as the total gas correlation.

Leroy et
al. (2005)
studied dwarf galaxies and found that they have a
molecular KS index of 1.3 ± 0.1, indistinguishable from that of
spirals, except with a continuation to lower central *H*_{2}
column densities (i.e., down to ~ 10
*M*_{}
pc^{-2}).

Heyer et
al. (2004)
found a slope *n* = 1.36 for
_{SFR} versus
(*H*_{2})
in M33, where the molecular fraction,
*f*_{mol} is small. The correlation with the total gas was
much steeper. More recently,
Verley et
al. (2010)
studied M33 again and got
_{SFR}
_{H2}^{n} for *n* = 1 to 2, and
_{SFR}
_{total
gas}^{n} for *n* = 2 to 4. The steepening
for total gas is again because
_{HI} is about
constant, so the slope from HI alone is nearly infinite. This
correlation is dominated by the radial variations in both quantities,
as it is a point-by-point evaluation throughout the disk. Radial
changes in metallicity, spiral arm activation, tidal density, and so
on, are part of the total correlation.
Verley et
al. (2010)
also try other laws, such as
_{SFR}
(_{H2}
_{ISM}^{0.5})^{n}, for which
*n* = 1.16 ± 0.04, and
_{SFR}
_{ISM}^{n}, for which *n* = 1.07
± 0.02. These differ
by considering the conversion from column density to midplane density,
using a derivation of the gaseous scale height. The first of these
would have a slope of unity if the star formation rate per unit
molecular gas mass were proportional to the dynamical rate at the
average local (total) gas density. The second has the form of the
original Schmidt law, which depends only on density. To remove possible
effects of CO to H_{2} conversion, Verley et al. also
looked for a spatial correlation with the 160 *µ* opacity,
_{160}, which is a
measure of the total gas column density
independent of molecule formation. They found
_{SFR}
_{160}^{n}
for *n* = 1.13 ± 0.02, although the correlation
was not a single power law but a 2-component power law with a shallow
part (slope ~ 0.5) at low opacity
(_{160} <
10^{-4}) and a steep part (slope ~ 2) at high opacity.

**1.3. Explanations for the 1.5 slope**

Prior to around 2008, the popular form of the KS law had a slope of
around 1.5 when
_{SFR} was
plotted versus total gas column
density on a log-log scale. This follows from a dynamical model of star
formation in which the SFR per unit area equals the available gas mass
per unit area multiplied by the rate at which this gas mass gets
converted into stars, taken to be the dynamical rate,

(1.2) |

If the gas scale height is constant, then
_{gas}
_{gas}
and _{SFR}
_{gas}^{1.5}. In the model of star formation
where star-forming clouds
are made by large-scale gravitational instabilities, this 1.5 power law
would work only where the Toomre instability condition, *Q*
1.4, is
satisfied. Such a model accounts for the
Kennicutt (1989,
1998)
law with the *Q* < 1.4 threshold.

Several computer simulations have shown this dynamical effect.
Li et al. (2006)
did SPH simulations of galaxy disks with self-gravity
forming sink particles at densities larger than 10^{3}
cm^{-3}. They found a *Q* threshold for sink particle
formation, and had a nice fit to the KS law with a slope of ~ 1.5.
Tasker & Bryan
(2006)
ran ENZO, a 3D
adaptive mesh code, with star formation at various efficiencies,
various temperature floors in the cooling function, and various
threshold densities. Some models had a low efficiency with a low
threshold density and other models had a high efficiency with a high
threshold density. Some of their models had feedback from young stars.
They also got a KS slope of ~ 1.5 for both global and local star
formation, regardless of the details in the models.
Kravtsov (2003)
did cosmological simulations using N-body techniques in an Eulerian
adaptive mesh. He assumed a constant efficiency of star formation at
high gas density, and star formation only in the densest regions
(*n* > 50 cm^{-3}, the resolution limit), which are in the
tail of the density probability distribution function (pdf; cf.
Elmegreen 2002,
Krumholz & McKee
2005).
Kravtsov (2003)
got the KS law with a slope of 1.4 for total gas surface density.
Wada & Norman
(2007)
did a similar thing, using the
fraction of the mass at a density greater than a critical value from
the pdf
(_{crit}
= 10^{3} cm^{-3}) to determine the star
formation rate. Their analytical result had a slope of 1.5.
Harfst, Theis
& Hensler (2006)
had a code with a hierarchical tree for tracking
interacting star particles, SPH for the diffuse gas, and sticky
particles for the clouds. They included mass exchange by condensation
and evaporation, mass exchange from stars to clouds (via PNe) and from
stars to diffuse gas (SNe), and from clouds into stars during star
formation. New clouds were formed in expanding shells. Their KS slope
was 1.7 ± 0.1. They also got a drop in
_{SFR} at low
_{gas}, not
from a *Q* threshold but from an
inability of the gas to cool and form a thin disk (c.f.
Burkert et
al. 1992,
Elmegreen &
Parravano 1994).