Constraints on galactic wind models
Observational implications are derived for two standard models of supernovae-driven galactic winds: a freely expanding steady-state wind and a wind sourced by a self-similarly expanding superbubble including thermal heat conduction. It is shown that, for the steady-state wind, matching the measured correlation between the soft x-ray luminosity and star formation rate of starburst galaxies is equivalent to producing a scaled wind mass-loading factor relative to the star-formation rate of 0.5–3, in agreement with the amount inferred from metal absorption line measurements. The match requires the asymptotic wind velocity to scale with the star formation rate (in ) approximately as . The corresponding mass injection rate is close to the amount naturally provided by thermal evaporation from the wall of a superbubble in a galactic disc, suggesting thermal evaporation may be a major source of mass-loading. The predicted mass-loading factors from thermal evaporation within the galactic disc alone, however, are somewhat smaller, 0.2–2, so that a further contribution from cloud ablation or evaporation may be required. Both models may account for the 1.4 GHz luminosity of unresolved radio sources within starburst galaxies for plausible parameters describing the distribution of relativistic electrons. Further observational tests to distinguish the models are suggested.
keywords:galaxies: starburst – galaxies: star formation – X-rays: galaxies – X-rays: ISM – radio continuum: galaxies – radio continuum: ISM
Galactic winds have been known to be common features of star-forming galaxies for many years. While particularly spectacular winds, such as those of M82, NGC 1482 and NGC 253 (eg Martin, 1999; Strickland et al., 2004a), are exceptional, galaxies with modest winds are widespread (Veilleux et al., 2005), as revealed by extraplanar diffuse x-ray (Fabbiano et al., 1990; Armus et al., 1995; Strickland et al., 2004b) H (Heckman et al., 1990; Miller & Veilleux, 2003), and dust emission (Howk & Savage, 1999) in several galaxies.
Interest in the physical nature of the winds and their prevalence has increased since it has been recognised they appear to play key roles in galaxy formation and the distribution of metals in the Intergalactic Medium (IGM). Both photoionization and mechanical feedback from star-forming regions have long been expected to limit star-formation on small scales (McKee & Ostriker, 2007). Wind feedback has also been invoked to impede gas accretion and so limit the efficiency of star formation in galaxy formation models to account for disagreement between model predictions and observations (eg Dekel & Silk, 1986; Kereš et al., 2012; Keller et al., 2015; Oppenheimer & Davé, 2008; Weil et al., 1998). Winds extending over hundreds of kiloparsecs may account for intergalactic metal absorption systems (see the review by Meiksin, 2009).
It is widely believed galactic winds result from the collective impact of supernovae in compact star-forming regions on their surroundings (see the review by Veilleux et al., 2005). The details of the physical mechanism driving the winds, however, are still unknown. Most models are based on the injection of energy and matter by supernovae and stellar winds in a distributed region (Burke, 1968; Johnson & Axford, 1971; Mathews & Baker, 1971; Chevalier & Clegg, 1985). The most straightforward model of pressure-driven expanding gas expelled by the supernovae fails because it greatly over-predicts the gas temperature as inferred from x-ray spectra of the winds, and underpredicts the x-ray luminosities. Mass loading, in which interstellar gas is incorporated into the flow, is recognized as the most plausible explanation for the moderate temperatures and high x-ray luminosities detected (Tomisaka & Bregman, 1993; Suchkov et al., 1996). This possibility receives observational support from measurements of velocity-broadened metal absorption line systems (Heckman et al., 2015). Sources for the mass include pre-existing interstellar gas in the vicinity of the supernovae from stellar winds, hydrodynamic ablation of gas clouds entrained in the outflow and thermal evaporation from gas clouds (Suchkov et al., 1996; Strickland & Stevens, 2000; Pittard et al., 2001; Marcolini et al., 2005). Alternative wind models have also been considered, such as momentum-driven winds (Murray et al., 2005) and winds driven by cosmic ray streaming (Ipavich, 1975; Uhlig et al., 2012).
Based on the stellar wind model of Castor et al. (1975) and Weaver et al. (1977) for isolated stars, the galactic superbubble model of McCray & Kafatos (1987) and Mac Low & McCray (1988) naturally incorporates mass loading sourced by thermal evaporation off the wall of the wind cavity produced by supernovae, resulting in a self-similarly expanding supershell. Analytic modelling and numerical computations suggest the initially spherically expanding superbubble soon develops into a biconal outflow within the stratified interstellar medium of a disc galaxy (Schiano, 1985; Mac Low & McCray, 1988; Tomisaka & Ikeuchi, 1988). Asymptotically, at large distances from the galactic plane, the outflow may develop into a more spherical superwind.
A more sophisticated approach to modelling winds is to use numerical hydrodynamical computations to evolve a galactic wind from first principles, but uncertainties in the detailed structure of the interstellar medium limit the generality of the computations. The computational demands imposed by the spatial resolution necessary to capture all of the essential physics moreover precludes a fully self-consistent treatment using cosmological simulations with current resources. An element of ‘sub-grid physics’ is ultimately required (eg Springel & Hernquist, 2003; Dalla Vecchia & Schaye, 2012). Critically, all the numerical models suffer from one key deficiency: the unknown physical mechanism driving the wind. In particular, the roles of both clouds and thermal heat conduction, particularly as they affect the amount of mass loading, are unknown. While the interaction between winds and clouds has been investigated using numerical simulations (eg Cooper et al., 2008; Scannapieco & Brüggen, 2015), most simulations neglect thermal heat conduction (but see D’Ercole & Brighenti, 1999; Marcolini et al., 2005; Brüggen & Scannapieco, 2016). The superbubble model has recently been incorporated into cosmological simulations (Keller et al., 2014), producing galaxy properties in good agreement with observations (Keller et al., 2015).
A principal goal of this paper is to distinguish between models with and without thermal heat conduction. The implicit justification for neglecting heat conduction is that magnetic fields tangled by turbulence will suppress heat conductivity. On the other hand, winds will tend to comb out magnetic field lines, allowing some thermal heat conduction along the wind direction. Which of these effects dominates is not known.
The analytic modelling here is performed in the context of a homogeneous wind. The homogeneity of wind gas is currently not well constrained, although there is clear evidence for the presence of entrained gas clouds. Even if initially the gas is clumped into clouds, large mass-loading factors resulting from hydrodynamical ablation or thermal evaporation may render the hot gas interior to the wind bubble sufficiently smooth for the homogeneous models still to provide a good approximation to the wind structure. Observations suggest radiative losses, not included in the models, are generally too small to affect the energetics of the winds, although they may affect the metal column densities of embedded clouds (Heckman et al., 2001; Heckman et al., 2002). Gravity will play a secondary role in slowing the flow, an effect not readily incorporated into an analytic treatment (but see Bustard et al., 2016), while it may limit the outflow velocity of ram-pressure driven clouds (Heckman et al., 2015). The principal role of gravity is in stratifying the galactic disc gas, as will be discussed below.
The Chandra X-ray Observatory has enabled the development of observing campaigns to systematically investigate the x-ray properties of star-forming galaxies (Strickland et al., 2004b; Grimes et al., 2005; Mineo et al., 2012a, b; Mineo et al., 2014). The primary quantity focussed on in this paper is the radiative x-ray efficiency of the wind as quantified by the specific diffuse x-ray energy generated per solar mass of stars formed. The x-ray emission profiles extend to typical scales of several kiloparsecs (Strickland et al., 2004b; Mineo et al., 2012b), well beyond the active star-forming regions driving the outflow. Winds on these scales form bipolar cones through the stratified disc gas and become increasingly inhomogeneous due to the onset of Kelvin-Helmholtz and Rayleigh-Taylor instabilities, details not amenable to an analytic treatment. Since the diffuse x-ray profiles are strongly centrally peaked, however, the models considered here should still capture a fair fraction of the total x-ray luminosity, particularly for dwarf starbursts (Grimes et al., 2005). This paper seeks to quantify the radiative x-ray efficiency from the central region within the disc of the galaxies, allowing for a range in star-formation rates and wind properties. Although analytic models are approximate, they provide insight into the origin of observational trends in terms of the physical properties of the winds. They also provide invaluable guidance into the design and interpretation of numerical simulations.
In the next section, approximate analytic scaling relations are derived for the structure and some observational signatures of the winds. In Sec. 3, x-ray luminosity predictions are presented using full numerical integrations of the models. This is followed by predictions for radio luminosities in Sec. 4 and for metal absorption column densities in Sec. 5. The results are discussed in Sec. 6, followed by a summary of the key results in a conclusions section.
2 Analytic estimates of specific x-ray emission
The x-ray emission from a wind arises from both thermal free-free and line emission. To understand the dependence of the emission on the properties of the sources and the surrounding gas, it is helpful first to estimate the thermal free-free component analytically. In the following section, more accurate results from numerical computations are provided including line emission. Estimates use the model of Chevalier & Clegg (1985) for a steady-state wind and of McCray & Kafatos (1987) and Mac Low & McCray (1988) for superbubbles. Both assume gravitational acceleration is negligible, a good approximation in the central regions of a galaxy where the gas is much hotter than the galactic virial temperature. Gravity, however, produces stratification of the galactic disc gas, which results in biconal outflow. This is a limitation of both models: the steady-state wind assumes a homogeneous source, limited therefore to a region small compared with the scale-height of the disc. Comparison with numerical simulations, however, suggest the wind produced by a starburst in a cylindrical region in a disc may be rescaled to the spherically symmetric solution to high accuracy (Strickland & Heckman, 2009). The superbubble model assumes a homogeneous surrounding medium. Hydrodynamical simulations show this tends to limit the growth of the superbubble to within the disc as pressure-driven lobes emerge vertically (Mac Low et al., 1989). The analysis here concentrates on the structure of the superbubble when it reaches a size comparable to the scale-height of the disc.
2.1 Steady-state wind
The characteristic ejecta energy and mass of a core-collapse supernova are taken by Chevalier & Clegg (1985) to be and . Assuming a Salpeter stellar initial mass function (IMF), a lower progenitor mass limit of , for a core-collapse supernova gives a rate of about 1 core-collapse supernova per 100 solar masses of stars formed. (The value would rise by about 40 percent for a Kroupa IMF.) Characterizing the supernova rate as supernova per solar mass of stars formed, the energy and mass injection rates for a star-formation rate are and , where and allow for uncertainties in the mechanical energy and mass-loading, respectively. For expressed in , .
Most of the bolometric thermal free-free emission originates in the central source region at . The analytic estimate is based on emission from this region. Using the results in the Appendix for a gas, the central hydrogen density is
where is the star formation rate in units of and is the radius of the star forming region in units of 100 pc. The central temperature is
where is the mean mass per particle, and in the last expression the temperature is characterized by the asymptotic wind velocity at ,
In terms of , the central hydrogen density is .
Radiative cooling places a lower limit on the asymptotic wind velocity. The energy injection rate must exceed the cooling rate within the central region of the wind. The cooling rate is for electron and hydrogen number densities and , respectively, where with K and the metallicty relative to solar (Mac Low & McCray, 1988). Requiring imposes the robust restriction
corresponding to the limit on the mass-loading factor In the literature, a more commonly defined mass-loading factor is the ratio of mass outflow rate to star-formation rate. In terms of this ratio, designated here by , the cooling restriction imposes
A similar restriction is derived by Zhang et al. (2014). The mass injection rate is then limited to .
As shown in the Appendix, the core density and temperature are nearly uniform within . Approximating the density and temperature as constant within , the total thermal free-free luminosity is
for x-rays of energy in keV. Integrating Eq. (6) over the energy band keV, the x-ray energy produced per solar mass of stars formed is then
The model predicts a linear increase with the star formation rate. Taking and a typical mass-loading factor of , corresponding to the asymptotic wind velocity , gives
2.2 Superbubble with thermal heat conduction
Allowing for an ambient interstellar medium and equilibration of the temperature interior to the bubble cavity by thermal heat conduction, McCray & Kafatos (1987) and Mac Low & McCray (1988) model the superbubble as a self-similar expanding stellar wind. Normalized by the typical mechanical luminosity of an OB association, , and for an ambient hydrogen density outside the wind bubble , the bubble radius increases, assuming no radiative losses, like
where is the age of the bubble in units of yr. Adopting the thermal conductivity coefficient , including a possible conductivity suppression factor , the interior bubble temperature and ionized hydrogen number density are given in terms of the similarity variable for radius by
The bubble will cool primarily by line radiation at its surface. The characteristic radiative cooling time is
The factor 0.15 has been added to to account for hydrogen and helium cooling, where care is taken near the surface to ensure cooling is cut off below the recombination temperatures for helium and hydrogen for collisionally ionized gas, and was adopted for the surface layer, following Mac Low & McCray (1988). For high ambient hydrogen densities, cooling will limit the radius of the bubble to be smaller than the characteristic scale height of the stratified interstellar medium perpendicular to the disc. At lower densities, the bubble radius may reach the disc scale height. The wind will then evolve into a bipolar outflow perpendicular to the disc, and the expansion into the plane of the disc ceases, or may even reverse (Mac Low & McCray, 1988; Mac Low et al., 1989).
Expressing the temperature and density of the gas interior to the bubble as and , where , the thermal free-free emission emitted by a wind bubble of radius pc is
where is the Planck constant, is the Boltzmann constant, here is Euler’s constant, and the integral has been evaluated with its asymptotic leading order behaviour retained for the two limiting cases shown. A characteristic central temperature of K gives a transition energy between the two cases of about 3 keV.
At high densities, the growth of the wind bubble will be limited by cooling once the energy radiated matches the total mechanical energy deposited by the wind. This may be quantified as follows. At the cooling time , the central hydrogen density and gas temperature take on the values
independent of , and
respectively, where has been converted to the star formation rate () using (Sec. 2.1). The cooling radius may be expressed as
The corresponding bubble expansion velocity at this time is
The mass interior to the bubble is dominated by the evaporation off the bubble wall into the hot cavity at the rate (Castor et al., 1975), where is the thermal conductivity coefficient. The mass loading factor in the wind core referenced to the star formation rate becomes
From Eq. (13), the x-ray energy in the 0.5–2 keV band per solar mass of stars formed is then
independent of the star formation rate, the ambient hydrogen density and the rate of thermal heat conduction. It corresponds to 2 percent of the mechanical energy radiated as x-rays in this band. The wind will not immediately cease as the momentum of the outflow will continue to sweep up material, but at a reduced rate (Koo & McKee, 1992).
At lower densities, the bubble radius will be limited by the scale height of the gas perpendicular to the plane. The central hydrogen density and gas temperature when the bubble reaches a radius are
respectively. The corresponding wind velocity is
The mass loading factor in the bubble is
independent of the ambient gas density. The x-ray energy in the 0.5–2 keV band per solar mass of stars formed is then
decreasing weakly with increasing star formation rate.
The x-ray energy produced per solar mass of stars formed may then take on a wide range of values, depending on . For , the rate will be near , where it reaches a peak value independent of the ambient hydrogen density and the star formation rate once radiative cooling restricts the bubble growth.
It is instructive to compute the thermal heat conduction saturation parameter for these two limiting cases. Following Cowie & McKee (1977), a consideration of the ratio of the mean free path of the electrons to the temperature scale height for the wind, expressed as an ‘inverted cloud,’ shows that the surrounding density and temperature in the cloud case should be replaced by the central temperature and density of the wind. For a wind limited by radiative cooling, the saturation parameter becomes , where characterizes the uncertainty in the saturated heat flux. This is nearly identical to the value McKee & Cowie (1977) derive for interstellar clouds, below which clouds will cool and condense rather than evaporate. If the wind bubble is limited instead by the scale height of the disc to a radius , the saturation parameter becomes . Thermal heat conduction is thus close to being saturated () for typical values of the parameters. Only models with are considered here so that the classical heat conduction description applies.
In the following section, more precise numerical predictions are made for the models, including the contribution from metal emission lines. Comparisons with observations are also drawn.
3 Numerical evaluation of specific x-ray emission
3.1 Data and modelling
The high angular and spectral resolution of the Chandra X-ray Observatory have enabled quantification of the correlation between the soft x-ray diffuse emission associated with star forming regions within galaxies and the star formation rate. From measurements of 6 disc galaxies, Owen & Warwick (2009) find . Only the most luminous point sources were removed, so that their value may be conservatively viewed as an upper limit to the x-ray luminosity of a gas component. Li & Wang (2013) find a similar correlation between diffuse galactic coronal emission, corrected for observed or estimated stellar contributions, and the star formation rate of for 53 nearby disc galaxies. Based on star formation rate estimates from infra-red and UV measurements restricted to the same projected region as the diffuse x-ray emission, in a sample of galaxies cleaned of those showing evidence of an active nucleus and with detected or the estimated contribution of unresolved high mass x-ray binaries removed, Mineo et al. (2012b) obtained for a sample of 21 late-type galaxies. (A Salpeter stellar initial mass function was assumed.) On fitting a two-component thermal model to the spectra, they find a correlation between the gaseous contribution to the diffuse x-ray luminosity and the star formation rate of
or . For 9 galaxies, they find spectral evidence for substantial absorption internal to the galaxies. Using these systems, they estimate the intrinsic diffuse gaseous emission to be
Since the x-ray emission in the wind models peaks within the energy bands used to measure the emission, a more precise comparison between the models and the measurements requires numerical integration of the models. In addition to thermal free-free, x-ray line emission also contributes substantially to the overall x-ray budget. The CHIANTI rates (Landi et al., 2012) for collisionally ionized gas are adopted from Cloudy (13.03) (Ferland et al., 2013), and emission tables for solar and half-solar metallicity computed. Numerical integrations of the models interpolate on the tables. Comparisons with measurements are made separately below for the steady-state wind model and the superbubble model.
3.2 Steady-state wind
The specific x-ray emission in the bands keV and keV for the steady-state wind model is shown in Fig. 1 as a function of star formation rate. A source region of radius pc is adopted, with solar metallicity. Emission from outside the source region is included, although it diminishes rapidly with distance outward. The mass-loading factor is expressed in terms of the asymptotic wind velocity .
The specific emissivity is a decreasing function of . Expanding the source region to pc is found to decrease the emission in the keV band by about 30 percent, the same trend, but with a somewhat weaker dependence, as predicted by Eq. (7). The x-ray emission is diminished by 30–50 percent on going from solar to half-solar metallicity for .
In the broader energy band keV, the specific emissivity, shown in the right panel of Fig. 1, decreases with increasing volume of the source region, varying nearly as rapidly as , as in Eq. (7). The specific emissivity varies nearly linearly with metallicity, except for , for which the specific emissivity depends only weakly on the metallicity.
The predicted linearly increasing trend with star formation rate is not consistent with the observations. The data from Mineo et al. (2012b) suggests a constant amount of x-ray energy emitted per unit mass of star formed. In the energy band keV, this is matched by allowing a tight correlation between the asymptotic wind velocity and the star formation rate according to , corresponding to a central hydrogen density . This results in an increasing amount of mass loading for a decreasing star formation rate, a general requirement recognized by Zhang et al. (2014).
Results for galaxies corrected for internal absorption are shown in the right hand panel of Fig. 1 for the band keV. The near constancy of the specific emissivity with star formation rate persists in the data. Agreement with the data may again be achieved if the wind velocity were tightly correlated with the star formation rate according to , corresponding to a central hydrogen density . The scaling with star formation rate is bracketed by that expected from Eqs. (6) and (7), which give for a constant luminosity per rate of star formation the approximate analytic scaling with . The velocity correlations are close to the cooling restriction Eq. (4), suggesting a narrow range is allowed for viable winds (cf Zhang et al., 2014).
3.3 Superbubble with thermal heat conduction
Allowing for thermal evaporation from a surrounding medium results in a much narrower range in specific emissivity compared with the steady state model. Results assuming a maximum radius of 100 pc for the expanding bubble and for solar metallicity are shown in Fig. 2. The results shown are time-averaged over the duration of the spherical expansion of the wind, assumed to cease at the cooling time, Eq. (12), or when it reaches the maximum radius. X-ray emission only from within the maximum radius is computed. The emission will fall off rapidly away from the plane if the superbubble expands out of the disc, but emission from an extended region may be comparable to that from within the disc. A multi-dimensional model is required to estimate the full emission more accurately.
The specific x-ray emissivity increases with the ambient hydrogen density approximately as , in agreement with Eq. (24), except at the highest density and low star formation rate where the wind expansion is cooling limited. At high densities, the specific emissivity becomes nearly independent of the star formation rate and gas density, in agreement with Eq. (19).
For low star formation rates and low ambient hydrogen densities, the specific emissivity nearly halves on going from solar to half-solar metallicity. The x-ray emission is dominated by line emission. The difference is much more moderate at high star formation rates and high ambient densities, for which line emission no longer dominates. At low ambient hydrogen densities, the specific emissivity increases linearly with the maximum bubble radius, but less rapidly at higher densities as cooling becomes important, especially for low star formation rates, in accordance with Eqs. (19) and (24).
As shown in the left hand panel of Fig. 2, comparison with the measured specific emissivities using the data from Mineo et al. (2012b), assuming no internal absorption from the galaxies, shows good agreement for ambient gas densities of . For a fixed star formation rate, the required will scale like according to Eq. (24). Since the measured values likely exceed the emission from the inner region within the disc by a factor of a few, the implied hydrogen densities are likely somewhat smaller. Allowing for internal absorption, agreement with the data in the right hand panel shows values of are preferred. No other parameters need be adjusted: the model predicts the specific x-ray emissivity is only weakly dependent on the star formation rate, in agreement with the data.
4 Specific radio emission
4.1 Data and modelling
The radio continuum radiation emitted by star-forming galaxies scales with the star formation rate, at least for large radio luminosities (Condon, 1992; Bell, 2003). The physical origin of the emission is unknown, but it is suspected to arise both from shocks driven by stellar winds and supernovae and from cosmic rays in a large-scale magnetic field. Measurements suggest that 90 percent of the continuum emission at 1.4 GHz is synchrotron and 10 percent thermal free-free in nature, suggesting a component from H regions as well (Condon, 1992). Modelling all these effects is well beyond the scope of this paper. Here only the synchrotron and free-free radio emission from the wind regions are estimated. In comparing with radio data, it is unclear from which scale to take the emission. The correlation between the radio continuum and the star formation rate is based on extended regions that likely include emission from large-scale interstellar cosmic rays. A representative value is
(Condon et al., 2002). By contrast, the dominant emission from shocks within the wind region would be much more centrally concentrated.
The FIRST radio survey (Becker et al., 1995) includes data that matches the scale of the x-ray and star-forming regions, typically up to a few arcminutes, measured by Mineo et al. (2012b). We compare the models with two 1.4 MHz continuum measurements from the FIRST survey, the large-scale value centred on each galaxy and the brightest unresolved peak value in the nucleus of the galaxy, corresponding typically to a region within the central 100–500 pc of the galaxy for a source at a distance of 10–20 Mpc. As shown below, the large scale values agree well with Eq. (27), corresponding to .
The synchrotron and thermal free-free emission are computed for the models as in Meiksin & Whalen (2013). In brief, a power-law energy distribution is assumed for the relativistic electrons, with an energy density a fraction of the local thermal energy density. The magnetic field energy density is also taken to be for simplicity, corresponding to approximate equipartition. The relativistic electrons are allowed to cool by synchrotron and thermal free-free radiation and by excitation of plasmon waves following the passage of the wind-driven shock front into the interstellar gas. Thermal free-free and synchrotron self-absorption are included, although for the frequencies of interest these are generally negligible in the models considered. Observations of supernova remnant spectra suggest typical values for the relativistic electron energy index of (Chevalier, 1998; Weiler et al., 1986), while representative model values for the relativistic electron energy density fraction range over (Chevalier et al., 2006). The predictions of the wind models for radio emission are estimated to check they do not exceed the observed limits for plausible parameters. Virtually all the emission predicted by the models is synchrotron radiation; the thermal free-free component is two to three orders of magnitude smaller.
4.2 Steady-state wind
The x-ray measurements require models with mass-loading from a gas reservoir surrounding the supernovae. The wind will then drive a shock into the surroundings. Since the wind is in a steady state, however, the time since the shock passed a given radius is undetermined in the model. To allow an estimate of the synchrotron emission, the wind is arbitrarily assumed to have reached a distance of 10 kpc from the source region, corresponding to an age of .
For a characteristic wind age of , for low star formation rates the synchrotron emission is dominated by the source region , giving, for ,
For a typical value , .
At higher star formation rates, the density becomes sufficiently high that two additional effects become important: synchrotron and plasmon generation losses deplete the central region of relativistic electrons; for sufficiently strong synchrotron losses the critical frequency falls to several gigahertz or less. Emission then arises only from outside the source region. The criterion for plasmon and synchrotron losses not to deplete the population of relativistic electrons is , where is the gas pressure in the wind. Using the asymptotic limits for density and pressure, this requires emission to arise only from radii
resulting in a rate of radio energy generation at frequency Hz per solar mass of stars formed of
At even higher densities, once synchrotron losses lower the synchrotron critical frequency into the gigahertz range, emission above frequency occurs only at radii
resulting in a rate of radio energy generation per solar mass of stars formed
Results from numerically integrating the model are shown in Fig. 3. The 1.4 GHz emission is displayed as a function of the star-formation rate and for , and , . All three trends with the star formation rate are apparent: a rise (), followed by a decline () once synchrotron and plasmon losses pinch off the relativistic electron distribution within the wind source region, and finally a near constant level () once synchrotron losses restrict the generation of 1.4 GHz power to regions well outside the core. (The power at large star formation rates for the case differs from the trend with in Eq. (32) because the radius at which the critical frequency exceeds 1.4 GHz lies just outside the source region, where the asymptotic decrease of pressure with radius is no longer a good approximation.) For a high star formation rate, the radio spectrum for is found to steepen to with , rather than the expected , as is found for low star formation rates: the truncation of the emitting volume at high densities steepens the spectrum.
The predicted trend adopting the correlation is shown by the solid (black) curves. For , the fluxes for the unresolved radio sources are largely recovered or exceeded. The high flux values for the highest star-formation rates are matched only for , the limiting values inferred for supernova remnants. These high fluxes, however, are derived from distant galaxies within regions unresolved on the scales of 1–2 kpc, so the radio emission may be contaminated by emission from cosmic rays interacting with large-scale galactic magnetic fields. The predicted excess emission for unresolved sources with low flux values may indicate a reduced volume filling factor of emitting electrons, resulting in small volume-averaged values for the relativistic energy fraction. At high star formation rates, the radio flux also decreases inversely with the bubble size, according to Eq. (32). Alternatively, the relativistic electron energy distribution may be steeper. For , recovers the lower flux values of the unresolved sources.
4.3 Superbubble with thermal heat conduction
For the superbubble model, the characteristic radio emission for a scale-height limited superbubble from within is given approximately by
nearly independent of the star formation rate. The radio emission is independent of any suppression of thermal conductivity since the radio power is determined by the thermal energy density, not the gas density or temperature separately in the absence of significant attenuation. Results from numerical integration of the wind equations are shown in Fig. 4. The bubble region recovers the full range of measured large-scale radio power for and . The larger flux values, at high star formation rates, require a high ambient hydrogen density of , the upper value required by the x-ray data. The lower power from unresolved regions is over-predicted for the higher star formation rates, possibly indicating a reduced volume filling factor of emitting electrons. The steeper electron distribution case with requires increasing the relativistic electron energy fraction to , approaching the limit from supernova remnant modelling. A near constant specific radio power is found for a given ambient hydrogen density, only weakly dependent on the star formation rate, in agreement with Eq. (33).
5 Metal column densities
5.1 Data and modelling
An estimate of the column densities of metal ions within the winds may be made as follows. For solar metallicity, abundances by number of commonly detected metal atoms compared with hydrogen include: , , , , and (Asplund et al., 2009). Any given ionization state will dominate at a particular temperature where it contributes most to the column density of that ion, although for some species neighbouring ionization states share substantially in the ionization. Temperatures at which commonly measured ions peak include: C at K (), C at K (), N at K (), N at K (), O at K (), Si at K (), Si at K (), SIII at K () and S at K (), where the peak abundance fractions by number relative to hydrogen, , are indicated in parentheses. It is noted these values will be modified if the gas is not in collisional ionization equilibrium, as appears to be the case for some high velocity clouds in the Galactic halo (Shull et al., 2011).
For the steady-state wind model with and wind core radius pc, the temperatures at which these ion abundances peak are achieved only outside the core, where the gas density is rapidly declining. The column density of a typical ion like C is negligibly small, . Of the ions listed above, only O would achieve a measurable column density within the core, . It is noted, however, that at the interface of the wind shock and the interstellar medium in the galactic disc, detectable levels of absorption may arise (Dopita & Sutherland, 1996). Such systems could possibly be distinguished from those produced by superbubbles, discussed below, through their kinematics.
For , the wind temperature is . For a given ion , the radius and density at which are and The corresponding column density will be
The column densities decline very rapidly with . Representative values for some common ions, computed numerically by integrating along the full wind solution at , are shown in Fig. 5 (left panel). These represent the maximum column densities that would arise from the homogeneous wind for lines of site passing through the region . At larger distances, the column densities will rapidly decline.
In the superbubble model, the column densities are dominated by absorption from a very thin layer at the bubble interface with the interstellar medium. For the scale-height limited case with a bubble radius , the characteristic temperature scale height at the position where for a given ion is , corresponding to a hydrogen density . The column density of ion is then
nearly independent of the star formation rate and only weakly sensitive to the ambient gas density and size of the bubble. Representative values, computed numerically by integrating through the centre of the region , are shown in Fig. 5 (right panel). This is a minimal value that must arise in the thermal conductive interface with the interstellar medium. Substantially lower values would be evidence against the superbubble model. The column density for H is also a minimal value, and it will generally be small compared with the H column density through the surrounding disc.
6.1 X-ray constraints
Both the freely expanding steady-state wind model and the self-similarly expanding superbubble model with thermal heat conduction may account for the measured amount of diffuse soft x-ray energy generated per unit mass in stars formed. An additional assumption of a tight correlation between the asymptotic wind velocity and the star formation rate, however, is required for the steady state model. By contrast, thermal evaporation from the cavity walls in the superbubble model naturally accounts for the measured amount of difuse x-ray energy per unit mass in stars formed for characteristic interstellar gas densities , with the higher values favoured if much of the soft x-ray emission is absorbed internally to the galaxies.
In the steady state model, the required correlation between the asymptotic wind velocity and the star formation rate, particularly for the high x-ray luminosities when internal galactic absorption is allowed for, is close to the minimum wind velocity (Eq. ) for which a steady-state wind may be maintained against radiative cooling within the star-forming region, consistent with the narrow range in observed radiative efficiencies. But it does not provide a reason for the narrow range. One possibility is that the winds are driven by superbubbles. Once a superbubble expands to the scale-height of the galactic disc, its thermal pressure drives a vertical conical outflow rather than further expansion into the disc (Schiano, 1985; Mac Low & McCray, 1988). Simulations suggest the outflow is nearly adiabatic (Keller et al., 2015), so that it may be approximated by the steady state model with a superbubble as the source. The rate of mechanical energy injected by supernovae and the rate of evaporative mass loss from the disc may be used to define an effective asymptotic wind velocity for the superbubble:
close to the required relation found for the steady-state wind solution. This may reconcile the two models: thermal heat conduction sets the source terms that initiate the wind, which then ‘blows out’ vertically into a steady-state outflow (Mac Low et al., 1989; Mac Low & Ferrara, 1999; Fujita et al., 2003, 2004; Keller et al., 2014).
For a bubble to blow out, two criteria must be satisfied: the cooling radius must exceed the disc scale height and the bubble velocity must exceed the sound speed in the surrounding medium. Both of these may be expressed as a restriction on the average star formation rate per superbubble cross-sectional area, . The cooling criterion gives
The hydrodynamical computations of Mac Low & McCray (1988) suggest for the dynamical criterion that, in terms of their dynamical variable , the bubble velocity must exceed the disc sound speed by a factor , for . This gives
where is the temperature of the ambient disc gas in units of K. The dynamical criterion is similar to the estimate of Strickland et al. (2004a). For , these criteria give , comparable to the minimum observed star formation surface density in galaxies with winds (Veilleux et al., 2005). This raises the question: do the proxies for star formation actually probe the larger region of a superbubble, so that the minimum observed star formation surface density in galaxies with winds may be identified with the minimum required for blowout?
Differences in the hard x-ray luminosities of the wind cores are expected between the two models. In the steady-state model, the exponential sensitivity to the star formation rate results in a rapid decrease with decreasing star formation rate of the 2–10 keV luminosity when the correlation is imposed, as shown in Fig. 6. At