The theoretical astrophysics pertinent to superwinds has been extensively discussed in the literature. A comprehensive review of models of the closely related phenomenon of 'supershells' can be found in Tenario-Tagle & Bodenheimer (1988), while more specific theoretical aspects of the superwind phenomenon are treated by - for example - Chevalier & Clegg (1985 - hereafter CC), Tomisaka & Ikeuchi (1988), Norman & Ikeuchi (1989), Leitherer, Robert, & Drissent (1993 - hereafter LRD), and Balsara, Suchkov, & Heckman (1993).
Briefly, we expect a superwind to occur when the kinetic energy in the ejecta supplied by supernovae and winds from massive stars in a starburst is efficiently thermalized. This means that the collisions between stellar ejecta convert the kinetic energy of the ejecta into thermal energy via shocks, with little energy subsequently lost to radiation. This situation is expected to arise when the (suitably normalized) kinetic energy input rate is so high that most of the volume of the starburst's ISM is filled by hot, tenuous supernova-heated gas (e.g., McKee & Ostriker 1977; MM). The collective action of the supernovae and stellar winds then creates a 'bubble' of very hot (T up to 10^{8} K) gas with a pressure that is much larger than that of its surroundings.
As the over-pressured bubble expands and sweeps up ambient gas, it will rapidly enter the 'snowplow' or radiative phase, which occurs when the radiative cooling time of swept-up and shock-heated ambient material becomes shorter than the wind's expansion timescale. Should the bubble be expanding inside a disk-like ISM (as expected in a typical starburst), the bubble will expand most rapidly along the direction of the steepest pressure gradient (the disk's minor axis). It is likely that once the bubble has a diameter that is a few times the scale-height of the disk, it will evolve from the 'snowplow' phase into the 'blow-out' phase as the swept-up shell accelerates outward and fragments via Rayleigh-Taylor instabilities. The wind can then propagate out into the intergalactic medium at velocities of several thousand km s^{-1} (provided that radiative losses have been small - cf. CC).
2.1. Momentum & Energy Input in Starbursts
Superwinds will be driven by the kinetic energy and momentum supplied by the starburst. Supernovae (primarily types II and Ib), winds from massive stars, and radiation pressure will all contribute significantly in this regard.
In very useful recent paper, LRD have calculated the predicted return of mass, momentum, and kinetic energy from an ensemble of massive stars. These calculations have been made for a range of different metallicities and initial mass functions. The time-evolution of these injection rates has been followed for two extreme starburst 'histories': 1. an instantaneous burst with all stars formed at once, and 2. a constant star-formation rate lasting up to 5 × 10^{7} yr. LRD show that stellar winds are most important at early times, for the most top-heavy IMF's, and for the highest metallicities. Their results are relatively insensitive to the IMF and metallicity for starbursts older than about 10^{7} yr when supernovae start to dominate the injection rates. This is fortunate from the standpoint of comparisons of the models with the data, since most starbursts are believed to last 10^{7} to 10^{8} yr (e.g., Rieke et al. 1980; Bernlohr 1993a, b).
For concreteness and ease of comparison to the LRD models, we will assume a constant star-formation rate lasting for 5 × 10^{7} yr, a normal Salpeter IMF extending up to 100 M_{}, and solar metallicity. These seem reasonable approximations for the typical starbursts we will discuss later. Scaling of the corresponding LRD model for M 82 then leads to the following predicted relationships between the starburst bolometric luminosity and the various injection rates:
(1) | |
(2) | |
(3) |
where the bolometric luminosity is in units of 10^{11} L_{}.
Several comments are in order. First, for the IR-selected starbursts that will dominate most of the discussion in Sections 3 and 4 to follow, it is probably a reasonable approximation to equate the starburst bolometric luminosity to the total (1 to 1000µ) IR luminosity of the galaxy - as given (for example) by the two-dust-temperature fits to the IRAS data in Rice et al. (1988). Second, note that the momentum flux associated with the starburst's radiation pressure (L_{bol}/c) is about 25% as large as the momentum flux due to supernovae and stellar winds given in Equation 3. It is also worth noting that the mechanical energy output of the starburst is about 2% of the bolometric luminosity, and that the mass injection rate represents about 10% (25%) of the star-formation rate for a Salpeter IMF with a lower mass cut-off of 0.1 M_{} (1 M_{}) respectively.
The above equations are meant to represent 'typical' starbursts with durations of at least 10^{7} yr. Clearly, the relationships between such quantities as bolometric luminosity, Lyman continuum luminosity, and the input rates of mass, momentum, and energy could vary substantially with time for an instantaneous starburst. Since the kinetic energy input at times later than 10^{7} yr in a starburst is dominated by supernovae, since a normal IMF rises towards low mass, and since the kinetic energy per supernova is roughly independent of progenitor mass (e.g., Woosley & Weaver 1986), the rate at which an instantaneous starburst injects kinetic energy will actually peak at about the time the least massive type II supernova progenitors explode. A minimum supernova progenitor mass of 8 M_{} corresponds to an implied maximum kinetic energy injection rate at a time of about 5 × 10^{7} yr (cf. LRD). Thus, a starburst in which the duration of star-formation is much shorter than this time-scale (e.g., Bernlohr 1993a, b), will have a peak in the superwind's mechanical energy output at a time when no stars more massive than about 8 M_{} remain (i.e., at a main sequence turn-off at B4V).
Such a post-burst galaxy would have a significantly reduced bolometric (IR) luminosity. The instantaneous burst models of Charlot & Bruzual (1991) show that L_{bol} has dropped by a factor of about 40 at 5 × 10^{7} yr with respect the peak value. Even more importantly, the very steep dependence of the Lyman continuum luminosity with stellar mass means that this post-burst would have a truly negligible ionizing luminosity: the models of Charlot & Bruzual (1991) and those of De-Gioia-Eastwood (1985) together imply a drop in L_{LyC} by a factor of about 30,000 with respect to the peak value. Kornneef (1993) has dubbed such objects 'Postburst Infrared Galaxies' ('PIGs'), and has proposed that the superwind system NGC 4945 (HAM) represents the prototype. Other superwind galaxies like Arp 220, NGC 6240, and NGC 3079 in HAM which all have unusually small ratios of Br to bolometric luminosity (cf. Beck 1993) may be additional examples of PIGs. The dwarf galaxy NGC 1705 seems to be an example of a low-mass post-burst that is driving a global outflow (Meurer et al. 1992).
2.2. Simple Models of Superwind Hydrodynamics and Radiation
The simplest possible hydrodynamical model for a superwind would be spherically-symmetric and would ignore the effects of gravity, radiative cooling, and ambient gas (i.e., a 'free wind'). CC have discussed such a model in the context of M 82. They assume that there is a spherical region of radius r_{*} (the starburst) inside which mass and kinetic energy are injected at a constant rate per unit volume. The result is a region of hot gas which slowly expands through a sonic radius (located approximately at the radius of the starburst) and is then transformed into a supersonic outflow.
HAM have found that this model is a reasonable fit to the radial pressure profiles in a sample of well-studied starburst superwind systems (see Section 3.2 below). In such a model, the gas pressure inside the starburst (and hence inside the sonic radius of the wind) is just the static thermal pressure of the hot fluid that has been produced by thermalization (shock-heating) in high-speed collisions between material ejected in stellar winds and supernova explosions. As discussed by CC, this pressure is essentially determined by the size of the starburst and the rate at which mass and kinetic energy is injected. Since the sound-crossing time inside the starburst is much shorter than the outflow time (e.g., since the flow is subsonic inside the starburst itself), the pressure is roughly constant throughout this central region. Following CC, and using Equations 1 & 2 above to relate the injection rates to the starburst luminosity, we expect the central pressure to be given by:
(4) |
Thus, the smaller the starburst radius (r_{*}), the higher the central pressure for a given starburst luminosity.
For radii much larger than the starburst radius, the pressure of the outflow is dominated by the wind's ram pressure. This will essentially be set by the rate at which the starburst injects momentum into the flow, and will drop like the inverse square of distance from the starburst. Following CC and using Equation 3 above we get:
(5) |
where r is the distance from the starburst, assuming r >> r_{*}.
The temperature inside the starburst is just set by the mean amount of kinetic energy per gram in the material injected by supernovae and stellar winds, and the models of LRD then imply that T_{0} = 1.2 × 10^{8} K for the assumptions discussed in Section 2.1. As this hot fluid expands out though the sonic radius, it is adiabatically cooled (it is easy to show in the context of the CC models that adiabatic cooling dominates radiative cooling for the wind fluid). The CC models combined with the LRD models then imply that for r >> r_{*}:
(6) |
Finally, the terminal velocity of the outflow (velocity at r >> r_{*}) is just the square root of twice the kinetic energy per gram in the material injected by the supernovae and stellar winds (cf. CC). Equations 1 and 2 above imply that the terminal velocity will be about 2900 km s^{-1}.
The next level of complication is to retain spherical symmetry, but to put in a homogeneous ambient medium that the wind must push against. This is a well-understood problem first considered in the context of wind-blown bubbles inflated by individual stars (e.g., Castor, McCray, & Weaver 1975). Dyson (1989) has recently published an excellent review of the relevant physics.
Briefly, in this situation one has an 'onion-skin' structure of five concentric zones. From inside-out these are: 1. An innermost region inside which the mass and energy is injected (the starburst), 2. A region of supersonic outflow, 3. A region of hot gas (wind material that has passed through an internal shock that divides regions 2 and 3), 4. A thin, dense shell of ambient gas that has been swept-up, shocked, compressed, and then radiatively cooled as the 'piston' of hot gas in zone 3 expands into the ambient medium, and 5. The undisturbed ambient gas. Such an expanding bubble may be either energy-conserving or momentum-conserving (depending on whether radiative losses in zone 3 are negligible or not).
For concreteness, let us adopt an energy-conserving structure inflating in a medium of constant density n_{0} (cm^{-3}). Two of the most relevant properties of the gas in zone 3 (shocked wind material) are then given by (cf. MM):
(7) | |
(8) |
where t_{15} is the age of the bubble in units of 10^{15} s (about 32 Myr, and we have again used the relations from LRD in Equations 1 though 3 above to express T and L_{X} in terms of the starburst luminosity. We therefore expect zone thee to be a strong source of X-rays (though note that the X-ray luminosity is only about 10^{-4} of the starburst luminosity). It is also helpful to note that an ambient density of unity inside a sphere of radius r corresponds to a total gas mass of 10^{8} [r/kpc]^{3} M_{}.
For the same model the chief parameters of zone 4 (the expanding shell of shocked ambient gas) are given as (cf. Castor, McCray, & Weaver 1975):
(9) | |
(10) |
Thus, the shell expands far more slowly than would a freely expanding wind. The typical shock-speeds implied by Equation 10 yield post-shock temperatures in zone 4 of order few times 10^{5} K. This is near the peak of the cooling curve (e.g., Spitzer 1978), and cooling times in zone 4 should be much shorter than the bubble expansion time.
We would expect zone 4 to be a strong source of optical and UV line emission, with a predicted H luminosity (from shocks alone) of-order:
(11) |
Thus, the H luminosity of zone 4 is a few times larger than the X-ray luminosity of zone 3, and is a few times 10^{-4} of the starburst luminosity. Since the H line typically represents 1 to 2% of the total emission-line luminosity in a shock (e.g., Shull & McKee 1979), the total line luminosity due to shock-heating of zone 4 will be of-order 1 to 2% of the starburst bolometric luminosity. Since the total wind kinetic energy flux is about 2% of the starburst bolometric luminosity (see Equation 1 above), this may be regarded as an upper bound on the luminosity of gas that has been shock-heated by the wind.
For the more general case in which an energy-conserving bubble is inflated into an ambient medium in which the density declines with radius like n r^{}, the expression corresponding to Equation 10 above has v t^{-(+2/+5)} - cf. Dyson (1989). For values of < -2 the shell therefore accelerates outward. Thus, it is expected that Rayleigh-Taylor instabilities will cause the shell to fragment, allowing the wind fluid in zones 2 and 3 to then freely escape from the ruptured bubble. Such a situation is referred to as 'blow-out' in the jargon of the trade.
As a more geometrically realistic model, we can consider a bubble being inflated in a plane-parallel stratified atmosphere (e.g., the gaseous disk of a starburst). The structure and evolution of the bubble is very similar to the spherical model above. However, in the disk case, the bubble will expand most rapidly along the direction of the steepest pressure gradient (e.g., normal to the plane of the disk). This process has been studied by numerous authors (e.g., Schiano 1985; MM; Tomisaka & Ikeuchi 1988; Balsara, Suchkov, & Heckman 1993; and see the review by Tenario-Tagle & Bodenheimer 1988). For an exponential density law in the vertical direction, the shell of shocked ambient gas will accelerate and fragment (e.g., blow-out will occur) when the bubble's radius reaches a few disk scale heights. Thereafter, the hot gas injected inside the starburst could vent directly out into the galactic halo (or beyond).
A schematic version of the evolution of this process is shown in Figure 1a, b. Figure 1a shows the radiative phase before 'blow-out'. The molecular disk fueling the starburst is labeled 'MD', and the associated supernovae and stellar winds are indicated by stylized stars. The starburst has partially evacuated a central cavity (labeled 'C') that is filled with the hot (T ~ 10^{8} K) thermalized supernova and stellar wind ejecta. This hot fluid passes through a sonic radius and then expands outward at several thousand km s^{-1} as a free wind (arrows labeled 'FW'). It passes though an internal wind shock (dashed line labeled 'WS') and is then reheated back to T ~ 10^{8} K. This shocked-wind material ('SW') can be an important source of thermal X-ray emission. The pressure of the wind inflates a 'bubble' which is elongated along the vertical axis, the direction of the maximum pressure gradient in the gaseous halo ('GH'). The solid contours are isodensity contours of the gaseous halo, which has a vertical scale-height labeled 'SH'. The shock driven by the expanding hot bubble will sweep up the ambient gas, accelerating it to velocities of a few hundred km s^{-1} (with a velocity field indicated by the short arrows). The shocked gas then cools radiatively into a thin shell ('S'). This shock-heated shell will be an important source of optical line-emission.
Figure 1b shows the situation after the bubble expands more than a couple of vertical scale-heights it can 'blow-out' of the halo. The bulk of the interior of the region evacuated by the wind is now occupied by the freely expanding wind fluid, with a small shell of shocked-wind fluid at the leading edge of the outflow separated from the ambient halo gas by a contact discontinuity (dot-dashed line labeled 'CD'). Bow shocks in the wind ('BS') will form around density inhomogeneities in the galaxy halo ('SC' for 'shocked clouds'). These clouds will be accelerated outward (short arrows), and may radiate optical/UV emission-lines and soft X-rays depending on the speed of the shocks driven into them.
Figure 1. Schematic diagrams showing the essential features predicted for two evolutionary phases of a superwind expanding into the gaseous halo of a disk galaxy. |
2.3. Conditions for Driving Superwinds
In this section, we will argue that there are two basic conditions to be met if a starburst is to drive a superwind. First, the energy injection rate inside the starburst must be high enough to create a cavity of hot gas whose radiative cooling time is much greater than its expansion time, and second, the wind must be sufficiently powerful to ultimately blow out of the ISM of the starburst (i.e., not remain trapped as a hot bubble inside the galaxy). We will show that these conditions are in fact met in typical starbursts (see HAM for details). Norman & Ikeuchi (1989) have reached the same conclusion using related arguments.
In order to drive a superwind, the kinetic energy supplied by supernovae and stellar winds must be efficiently converted into the bulk kinetic energy of the outflow - that is, very little of the energy should be lost in the form of radiation from the hot gas at the base of the outflow. This condition will be met if the energy injection rate is high enough to create a hot, tenuous gas phase that dominates the ISM of the starburst (i.e., has a volume-filling factor near unity). In this case, supernova and stellar wind ejecta will expand into a low-density medium and the radiative cooling time of the ejecta will exceed the wind outflow time. This criterion can be expressed in terms of the well-known McKee - Ostriker (1977) 'porosity' parameter. The volume filling factor of the hot, tenuous phase is given by:
(12) |
where
(13) |
In Equation 13, E_{51} is the kinetic energy per supernova in units of 10^{51} ergs, S_{-8} is the supernova rate in units of 10^{-8} SNe per year per pc^{3}, n_{0,100} is the mean gas density in starburst in units of 100 cm^{-3}, and P_{7} is the pressure in the starburst in units of 10^{7} K cm^{-3}. For typical parameters appropriate to starbursts, HAM concluded that Q_{SNR} will be of-order ten, so that ff_{HOT} is near unity.
An alternative way of evaluating whether radiative losses from the hot gas are important enough to stall the superwind is to use the expression derived by MM for the ratio of the radiative cooling time to the bubble expansion time for a supernova-driven bubble expanding in an disk with an exponential vertical density law with scale-height H and mid-plane density
(14) |
where we have used Equations 1 - 3 from the LRD models to write Equation 14 in terms of the starburst luminosity. Again, HAM showed that t_{cool} >> t_{exp} for typical starburst parameters.
Finally, for the same 'bubble in an exponential disk' model, MM conclude that blow-out will occur if the dimensionless wind power > 100. Using our Equations 1 - 3 to express this power in terms of the starburst bolometric luminosity, and following MM, we find:
(15) |
Typical parameters appropriate to starbursts then imply that blow-out is likely (see HAM for details).
So far, we have considered the theory of superwinds only in the context of a wind blowing into a homogeneous (single-phase) medium. In reality, the gaseous halos of star-forming galaxies are clearly multi-phase 'cloudy' systems (cf. Dettmar 1993; Bloemen 1991; and the review of QSO absorption-line systems by Steidel and Lanzetta in this volume). Such clouds could have a profound impact on the thermal and dynamical evolution of a superwind, and on the strength and character of the radiation they produce.
Let us then imagine a case in which clouds occupy only a small fraction of the volume swept by the superwind, but contribute a significant fraction of the gas mass within that volume. The evolution of the wind-blown cavity will still be qualitatively similar to the scenarios discussed above. The wind will propagate though and shock-heat the tenuous intercloud medium, inflating an expanding cavity. The clouds engulfed by the wind will suddenly find themselves highly over-pressured (and hence shock-heated) as they are subjected to the thermal pressure of the hot intercloud medium and/or the ram pressure of the wind itself (cf. Fig 1). The clouds will then presumably be destroyed by either hydrodynamical processes (cf. Hartquist & Dyson 1988; Stone & Norman 1992), by evaporation due to conductive heating (e.g., Cowie & McKee 1977), or by a combination of these processes in the 'mixing layers' at the cloud-intercloud interface (e.g., Begelman & Fabian 1990; Slavin, Shull, & Begelman 1993).
This 'mass-loading' by clouds of the intercloud medium will have several important consequences. First, the clouds will provide a source of cool, dense material which (when mixed with the intercloud medium) could significantly decrease the cooling time inside the wind-blown cavity. This could turn an energy-conserving outflow into a momentum-conserving one, and could even rob the bubble's interior of so much thermal energy that the bubble collapses before blow-out occurs (cf. Hartquist & Dyson 1988; White & Long 1991; McKee, Van Buren, & Lazareff 1984; Berman & Suchkov 1991). Second, the increased radiative cooling inside the superwind cavity should make the superwind's emission more readily detectable.
Since this review is primarily about the observational manifestations of superwinds, we will address this second point in a bit more detail. It is likely that engulfed clouds may in fact be the dominant sources of the soft X-rays and optical emission-lines observed from superwinds (see Sections 3.1 and 3.2 below). For example, Begelman & Fabian (1990) and Slavin, Shull, & Begelman (1993) have proposed that hydrodynamical processes at the interface between a cloud and the intercloud medium through which it moves will lead to the development of a 'mixing layer': a region containing gas from both media whose temperature will be roughly equal to the logarithmic average of the temperatures of the two media. These mixing layers and the clouds they irradiate can be a copious source of optical, EUV, and soft X-ray emission.
Clouds heated by shocks driven into them by the superwind will also be sources of radiation. Balancing momentum across the shock implies that the speed of the shock driven into the cloud will be related to the superwind velocity by the inverse of the square root of the cloud-wind density contrast. Using Equation 2 above for the mass outflow rate in the wind, and adopting a wind speed of 2900 km s^{-1} for consistency (Section 2.2 above), implies:
(16) |
for a cloud with a particle density n_{cloud} (cm^{-3}) located a distance r (in kpc) from the starburst.
Such a shock will heat the cloud to a temperature
(17) |
and we therefore see that such shocked clouds would produce soft X-rays, ionizing radiation, and optical emission-lines (cf. Shull & McKee 1979; Binette, Dopita, & Tuohy 1985).
The wind will also accelerate these clouds outward. Following CC, we can estimate the terminal velocity reached by a cloud with a column density typical of interstellar clouds in the Milky Way (N_{21} in units of 10^{21} cm^{-2}) accelerated outward from an initial location r_{kpc} (well outside the sonic radius of the wind):
(18) |
Here we have used Equations 1 and 2 above to relate the mass and energy flux in the wind to the starburst luminosity. We therefore expect the shocked clouds responsible for optical emission-lines to have typical outflow speeds of a few hundred km s^{-1}.