Choi & Nagamine (2010) compared three models for star formation in cosmological simulations, the Springel & Hernquist (2003) model, the Blitz & Rosolowsky (2006) model, and the Schaye & Dalla Vecchia (2008) model. These are instructive to review here so that the diversity of analytical models can be noted.

In the Springel & Hernquist (2003) model, there is cooling gas with a star formation rate

(9.1) |

where
equals the supernova gas return fraction,
_{c}
equals the density of cool clouds, *t*_{SFR} =
*t*_{0}^{*}(
/ _{th})^{-1/2} is the dynamical time,
where *t*_{0}^{*} = 2.1 Gyr gives the
local KS law. They also assumed
_{SFR} = 0
if <
_{th} for
threshold column density
_{th}, and
_{SFR} =
*A*(_{gas}
/ 1 *M*_{}
pc^{2})^{n} for *A* = 2.5 ± 0.7
*M*_{}
yr^{-1} kpc^{-2}, *n* = 1.4, and
_{th} = 10
*M*_{}
pc^{-2}. This equation for star formation rate is combined with
another equation for the rate of change of the cool cloud density,

(9.2) |

where *C* = *C*_{0}( /
_{th})^{-4/5}. This assumes that
supernovae evaporate and form cold clouds as in
McKee & Ostriker
(1977).

In the
Blitz &
Rosolowsky (2006)
model, the molecular fraction is given by
_{H2}
/ _{HI}
= (*P*_{ext} / *P*_{0})^{0.92}, where
*P*_{0} = 4.3 × 10^{4} k_{B} K
cm^{-3}. Then for a star formation rate, they assume

(9.3) |

which assumes
(a
constant scale height). This star formation rate
was applied only when *P*_{ext} <
*P*_{0}. For *P*_{ext} > *P*_{0},
Choi &
Nagamine (2010)
used the Springel & Hernquist law (i.e., the Kennicutt *n* =
1.4 law).

The third model was that of
Schaye &
Dalla Vecchia (2008).
These authors solved for the scale height using
_{gas} =
_{gas}
*L*_{Jeans} =
(
*f*_{g} *P*_{tot} / *G*)^{1/2},
where is
the adiabatic index: *P*_{tot} ~
_{gas}^{} (*P*_{tot} and
_{tot}
include stars), *f*_{g} equals the gas mass
fraction, and *f*_{th} equals the thermal pressure fraction
(*P* = *f*_{th} *P*_{tot}). They assumed
*f*_{g} = *f*_{th}, so if
_{SFR} =
*A* _{gas}^{n} =
_{gas} /
*t* which means *t* = _{gas}^{1-n} / *A*, then
*t*_{SFR} =
*A*^{-1}(*M*
pc^{2})^{n}( *P* / *G*)^{(1-n)/2}, and
finally *d*_{*} / *dt* =
_{gas}
/ *t*_{SFR}. Note that in this model,
_{SFR}
_{gas}
*P*^{0.2}. They also assumed a threshold density,
_{th} =
_{th}
/ *L*_{Jeans}, so
_{th} =
_{th}^{2} *G* /
*c*_{s}^{2} *f*_{g} for
*c*_{s} = 1.8 km s^{-1} (500K gas), and they
assumed *P* = *K*
^{4/3}.

Choi &
Nagamine (2010)
note that the Springel & Hernquist model forms too many stars at low
_{gas} and
this causes it to form stars too early in the Universe. The other models
have a pressure dependence for star formation which gives an acceptably
low rate in low pressure regions.

Genzel et al. (2010) reviewed the star formation and CO data for high-z galaxies in the context of the "main-sequence line" for star formation:

(9.4) |

(Bouche et al. 2010, Noeske et al. 2007, Daddi et al. 2007).

Genzel et al. noted that the gas depletion time depends weakly on *z*.
It is ~ 0.5 Gyr at *z* > 1, and 1.7 Gyr at *z* = 0, while
mergers have 2.5-7.5 × shorter depletion times than non-mergers. At
*z* > 1, the depletion time is comparable to the stellar age,
and it is always shorter than the Hubble time. This means there is a
continuous need for gas replenishment in galaxy disks.

Genzel et al. also found that the molecular star formation-column
density relation is in agreement with
Bigiel et
al. (2008).
There is no steepening at
_{gas} >
100 *M*_{}
pc^{-2} like there
appears to be in local starbursts. In general, they find no variation
in the empirical star formation law with redshift.

The dynamical version of the Kennicutt (1998) relation was examined by Genzel et al. too. The dynamical relation says that the star formation rate is proportional to the gas column density divided by the local orbit time. Even in this form, mergers were found to be more efficient at star formation than non-mergers. The star formation efficiency per unit dynamical time was about 1.7%.

These considerations led Genzel et al. to a fundamental plane of star formation, in which the total galactic star formation rate depends only on the dynamical time and the total molecular mass:

(9.5) |

all with a standard deviation of 0.47 dex. This is the same as

(9.6) |

Genzel et
al. (2010)
summarized the various star formation-density laws as
follows. The Kennicutt slope of ~ 1.4 includes HI, whereas the
Bigiel et
al. (2008)
slope of ~ 1 is just for CO (or H_{2}). Genzel et al. redid the
Kennicutt (1998)
slope of 1.4 with just H_{2} and got a slope
of ~ 1.33. They redid their own relation for
_{SFR}
versus _{gas}
including HI in addition to H_{2} and found
that it increases the slope from 1.17 to 1.28.
Kennicutt (1998)
used the same CO-H_{2} conversion factor everywhere. When Genzel
et al. redid the
Kennicutt (1998)
data with a variable CO-H_{2} conversion factor
including only H_{2}, the slope increased from 1.33 to
1.42. Thus the inclusion of HI and the constant CO-H_{2}
conversion factor somewhat cancel each other in the
Kennicutt (1998)
relation.
Genzel et
al. (2010)
also included new merger galaxies, which flatten the slope compared to that
in
Kennicutt (1998).
Writing the star formation rate as
_{mol} /
*t*_{dyn} works fairly well, including both mergers and normal
galaxies at all redshifts.

Daddi et
al. (2010b)
fitted Ultrahigh Luminosity Infrared Galaxies (ULIRGS)
and Submillimeter Wave Galaxies (SMGs) to the same star
formation-column density relation if the SF law is
_{SFR}
~ _{gas} /
*t*_{dyn} (*t*_{dyn} is the rotation
time at the outer disk radius). They suggested that the global star
formation rate is proportional so ~ (*M*_{total gas} /
*t*_{dyn})^{1.42}. High SFR galaxies consume their
gas faster than a rotation time at the outer radius. This suggests that
mergers are involved, or some other rapid accretion leading to
centralized star formation.