The statistical properties of the superbubble populations offer
another test of the standard evolutionary model for the shells.
Oey & Clarke (1997)
derived expressions for the differential size
distribution *N*(*R*) *dR* of superbubbles in a uniform
ISM, using the analytic expressions for adiabatic evolution (equation 1).
We considered a power-law mechanical luminosity function for the
parent OB associations,

(2) |

with
2, which
is robustly associated with the
H ii region
luminosity function (e.g.,
Kennicutt et al. 1989;
Oey & Clarke 1998a).
The superbubble growth is taken to be pressure-confined when the
interior pressure *P*_{i} = *P*_{0}. Star
formation is assumed to be coeval
within each OB association, with SNe therefore exploding over a period
*t*_{e} = 40 Myr, the lifetime of the lowest-mass SN
progenitors. For constant star-formation rate
and power-law
(*L*), we
found that:

(3) |

effectively yielding *N*(*R*)
*R*^{-3}
for = 2.
Oey & Clarke (1997)
also derive *N*(*R*) for
other combinations of and
(*L*).

This result agrees well with the
H i shell catalog for the
Small Magellanic Cloud (SMC) compiled by
Staveley-Smith et al. (1997).
This is by far the most complete sample of
H i shells obtained for any
galaxy, as evidenced by the fact that the relative number counts of
H ii regions and
H i shells are in excellent
agreement with their relative life expectancies. For shells having *R*
100 pc, the
fitted power-law slope
= 1-2
is 2.7 ± 0.6, in excellent agreement with the general prediction of
= 3.

We note that different models for ISM structure
yield different predictions for *N*(*R*). For example,
Stanimirovic et al. (1999)
suggest a possible fractal structure for the neutral ISM. From
the same H i dataset of the
SMC, they find a fractal dimension implying
a size distribution for H i
holes of = 3.5. It is
difficult to empirically differentiate this from our model, having
= 3; but it
is worth noting that the predictions are intrinsically different.

However, the superbubble size distribution presently is not a sensitive test in determining whether or not the objects evolve adiabatically. If all the internal energy is radiated away, the objects are predicted to follow the momentum-conserving law given by Steigman et al. (1975):

(4) |

The stall radius *R*_{f} in this case is only
1.3 times larger than for the adiabatic model, and the size
distribution follows the same law *N*(*R*)
*R*^{1-2}
(Oey & Clarke 1997).
The observations of hot gas are therefore vital confirmation that the
adiabatic model applies to a significant fraction of superbubbles.

We can also derive the distribution of expansion
velocities *N*(*v*) *dv*, which describes only the
growing objects
(Oey & Clarke 1998b):

(5) |

This again compares well with the SMC H i shell catalog: the fitted power-law slope is 2.9 ± 1.4. Thus, despite the crude assumptions in deriving the shell size and velocity distributions, the data suggest that the neutral ISM in the SMC is fully consistent with superbubble activity dominating the structure. Although most other available H i shell catalogs are highly incomplete, preliminary results for a few other galaxies also show agreement with our model for the size distribution (Kim et al. 1999; Mashchenko et al. 1999; Oey & Clarke 1997).