# Metal-Ion Absorption in Conductively Evaporating Clouds

###### Abstract

We present computations of the ionization structure and metal-absorption properties of thermally conductive interface layers that surround evaporating warm spherical clouds, embedded in a hot medium. We rely on the analytical steady-state formalism of Dalton and Balbus to calculate the temperature profile in the evaporating gas, and we explicitly solve the time-dependent ionization equations for H, He, C, N, O, Si, and S in the conductive interface. We include photoionization by an external field. We estimate how departures from equilibrium ionization affect the resonance-line cooling efficiencies in the evaporating gas, and determine the conditions for which radiative losses may be neglected in the solution for the evaporation dynamics and temperature profile. Our results indicate that non-equilibrium cooling significantly increases the value of the saturation parameter at which radiative losses begin to affect the flow dynamics. As applications we calculate the ion fractions and projected column densities arising in the evaporating layers surrounding dwarf-galaxy-scale objects that are also photoionized by metagalactic radiation. We compare our results to the UV metal-absorption column densities observed in local highly-ionized metal-absorbers, located in the Galactic corona or intergalactic medium. Conductive interfaces significantly enhance the formation of high-ions such as C, N, and O relative to purely photoionized clouds, especially for clouds embedded in a high-pressure corona. However, the enhanced columns are still too low to account for the O vi columns ( cm) observed in the local high-velocity metal-ion absorbers. We find that column densities larger than cm cannot be produced in evaporating clouds. Our results do support the conclusion of Savage and Lehner, that absorption due to evaporating O vi likely occurs in the local interstellar medium, with characteristic columns of cm.

###### Subject headings:

ISM:general – atomic processes – conduction – plasmas – quasars:absorption lines^{†}

^{†}slugcomment: Accepted to ApJ: June 3, 2010

## 1. Introduction

The physics of conductively evaporating clouds has applications to a variety of astrophysical environments. This includes the study of the Local Cloud and Local Bubble (Slavin 1989; Smith & Cox 2001, Slavin & Frisch 2002; Savage & Lehner 2006; Jenkins 2009), cloud evaporation in the interstellar medium (e.g. Bhringer & Hartquist 1987; Dalton & Balbus 1993; Nagashima et al. 2007; Slavin 2007; Vieser & Hensler 2007), supernovae remnants (e.g. Shelton 1998; Slavin & Cox 1992; Balsara et al. 2008), active galactic nuclei (e.g. McKee & Begelman 1990), and the galaxy formation process (Nipoti & Binney 2007).

In this paper, we present new computations of the non-equilibrium ionization states and metal absorption line signatures of thermally conductive interfaces that surround evaporating warm ( K) spherical gas clouds embedded in a hot ionized medium (HIM, K). In such interfaces, the warm clouds undergo steady evaporation, while heat from the hot ambient medium flows into the clouds (Cowie & McKee 1977; McKee & Cowie 1977). We compute the integrated metal-ion column densities through the conductive interfaces, for comparison to absorption line observations. We consider the effect of photoionization by the metagalactic radiation field on the non-equilibrium ionization states in the conduction fronts.

As matter flows from the warm cloud toward the hot ambient medium, its ionization state changes continuously. Non-equilibrium effects become significant when the ionization time-scale is long compared to the rate of temperature change. The gas then tends to remain “underionized” compared to gas in ionization equilibrium at the same temperature. Because the energy losses in the evaporating gas are dominated by atomic and ionic line emissions from many species, the cooling efficiencies are affected by the non-equilibrium abundances. For underionized gas, cooling is enhanced.

Our theoretical work is motivated by recent ultraviolet and X-ray absorption line spectroscopy of hot gas in the Galactic halo, around higher redshift galaxies, and in intergalactic environments (e.g. Sembach et al. 2003; Collins et al. 2005; Savage et al. 2005; Tumlinson et al. 2005; Fang et al. 2006; Stocke et al. 2006; Narayanan et al. 2010). We are also motivated by observations of discrete ionized “high-velocity clouds” (Sembach et al. 1999, 2000, 2002, 2003; Murphy et al. 2000; Wakker et al. 2003; Collins et al. 2003, 2004, 2007; Fox et al. 2005, 2006). These observations indicate that the ionized gas is not in equilibrium ionization (Sembach et al. 2002; Fox et al. 2004, 2005; Gnat & Sternberg 2004), and non-equilibrium processes such as time-dependent radiative cooling (e.g. Gnat & Sternberg 2007; Sutherland & Dopita 1993), shock ionization (e.g. Allen et al. 2008; Gnat & Sternberg 2009), or ionization in conductive interfaces (this paper) may be at play in these objects.

Theoretical models of cloud evaporation in a hot medium have been studied extensively. Cowie & McKee (1977) developed an analytical solution for the mass loss rate and temperature profile in static conductive interfaces around spherical clouds. In their solution, they included the possible effects of heat flow saturation that occurs when the electron mean free path is not small compared to the temperature scale height (see §2). The saturation in the flow can be measured by one global parameter, the “saturation parameter” . McKee & Cowie (1977) studied the effects of radiative losses on the interface. In particular, they demarcated the range of saturation parameters for which radiative losses may be neglected. The situation becomes much more complicated if the dynamics of the evaporation are affected by magnetic fields (e.g. Slavin 1989; Borkowski, Balbus & Fristrom 1990) or if the evaporation is time-dependent (e.g. Shelton 1998; Smith & Cox 2001; Vieser & Hensler 2007).

Non-equilibrium ionization in conductive interfaces was considered previously in several studies (Ballet, Arnaud & Rothenflug 1986; Bhringer & Hartquist 1987; Dalton & Balbus 1993; Slavin 1989; Slavin & Cox 1992; Shelton 1998; Smith & Cox 2001). However, most of these works explored only a limited range of parameters (with cm K and pc), appropriate for interstellar clouds. None of the previous computations included photoionization by an external radiation field.

Here we calculate the ion fractions created in the conductive interfaces surrounding evaporating gas clouds. We consider pressures between and cm K, cloud radii between pc and kpc, and ambient temperatures between K and K. We study how the metal absorption line properties depend on the interface parameters. We consider the effects of non-equilibrium cooling and reexamine the criterion derived by McKee & Cowie (1977) for the neglect of radiative losses in the evaporative flows.

As applications, we then focus on the example of the local “ionized high-velocity clouds”. Over the past decade, UV absorption line observations have revealed a population of local high-velocity metal-ion absorbers (Sembach et al. 1999, 2000, 2002, 2003; Murphy et al. 2000; Wakker et al. 2003; Collins et al. 2004; Fox et al. 2005, 2006). The observations, carried out with the Goddard High Resolution Spectrometer (GHRS) and Space Telescope Imaging Spectrograph (STIS) on board HST, and more recently with the Far Ultraviolet Spectroscopic Explorer (FUSE), indicate the presence of many high-ionization metal absorption lines, including Si iii , Si iv , and C iv with little or no associated H i (N cm). Of particular interest is O vi , for which absorbing column densities of order cm have been measured (Collins et al. 2004; Fox et al. 2006; Sembach et al. 2000, 2003).

Many of these ionized absorbers can be identified with the diffuse nearby H i high-velocity cloud complexes, such as the Magellanic Stream or Complex C (Sembach et al. 2003). However, some of these absorbers do not appear directly linked to such structures, and could be clouds at much larger distances, perhaps pervading the Local Group. The absorbers towards Mrk 509 and PKS 2155-304 are examples of such isolated high-velocity metal-ion absorbers (Sembach et al. 1999; Collins et al. 2004; Fox et al. 2005). As a guide to these observations, we list in Table 1 the data presented by Collins et al. (2004) for the metal-ion absorbers towards Mrk 509.

In Gnat & Sternberg (2004, hereafter GS04), we modeled the high-velocity
metal ion absorbers as photoionized gas clouds associated
with low-mass dark matter “minihalos” (see also Kepner et al. 1999).
For this purpose, we studied the metal photoionization properties of
hydrostatic gas clouds embedded in gravitationally dominant
dark-matter halos, that are photoionized by the present-day
cosmological metagalactic radiation field, and are
pressure-confined
by an external hot medium.
We considered minihalo models for dwarf-galaxy-scale objects
and for the lower mass compact H i high velocity
clouds (CHVCs), based on the properties derived by
Sternberg, McKee, & Wolfire^{1}^{1}1Sternberg, McKee, &
Wolfire (2002) constructed explicit minihalo models for Local Group
dwarf galaxies, based on the observed H I properties of Leo A
and Sag DIG. They then searched for minihalo models for the H I
CHVCs, that would simultaneously account for the range of observed
H I column densities, the existence of multiphased (cold/warm) cores,
and the total number of objects. (2002; see also Giovanelli et al. 2010).

For low mass objects ( M) embedded in the relatively high-pressure environment of the Galactic corona ( cm K), we found that the ionization parameter is too low to efficiently produce C iv and other high ions. However, for the more massive dwarf-galaxy-scale halos ( M, cm K) embedded in the low-pressure intergalactic medium (IGM), we found that their ionized envelopes are natural sites for the formation of high-ions. Photoionized envelopes of dwarf-galaxy-scale halos could be detectable as UV metal-line absorbers, with ionization states similar to those observed in the ionized high-velocity absorbers. O vi remains an important exception, as it is inefficiently produced by photoionization.

The strong O vi absorption in the high-velocity metal-ion absorbers strongly suggests that photoionization is not the only ionization mechanism at play in these objects (Sembach et al. 1999; Fox et al. 2004, 2005; GS04). The contribution of additional ionization processes may produce high-ions that are suppressed in the purely photoionized clouds.

The fact that minihalo clouds may be embedded in a hot plasma, such as the Galactic corona or the intergalactic medium at various redshifts, suggests the possibility of significant collisional interactions between the hot gas and the warm clouds. Here we use our evaporating cloud models to study the interface layers surrounding dwarf-galaxy-scale minihalos ( cm K, kpc, K, solar) for comparison with the observations of the high-velocity metal ion absorbers. As we describe below, we include the effect of photoionization by the metagalactic external radiation on the non-equilibrium ionization properties of the gas in the interface.

In §2 we describe the basic equations and numerical method. In §3 we present a set of computations of the ion fractions in conductive interfaces, and we discuss the resulting metal absorption column densities for the dwarf-galaxy-scale models in §4. We find that conductive interfaces significantly enhance the formation of high ions such as C, N and O relative to purely photoionized clouds, especially for clouds embedded in a high-pressure medium. However, the enhanced columns are still too low to account for the O vi columns ( cm) observed in the high-velocity metal-ion absorbers. We find that O vi column densities larger than cm cannot be produced in evaporating clouds. Our models do support the conclusion by Savage & Lehner (2006) that evaporating O vi absorption occurs in the local ISM, with characteristic columns of cm. We summarize in §5.

Ion | -300 km s | -240 km s |
---|---|---|

C ii | ||

N i | ||

Si ii | ||

S ii | ||

C iv | ||

N v | ||

O vi | ||

Si iii | ||

Si iv | ||

S iii |

Note. – Column densities for two velocity components toward Mrk 509 by Collins et al. (2004). The error estimates are the estimates, and the upper limits are limits.

## 2. Basic Equations and Processes

### 2.1. Evaporation

We are interested in studying the evolving non-equilibrium ionization states in thermally conductive gas that evaporates from warm purely photoionized clouds into a hot ambient medium. In our treatment of the ionization in the evaporating layers we use an analytic solution for the temperature profiles in the conductive interfaces surrounding steady-state, spherical, non-magnetic clouds. The temperature profiles are obtained by solving the equation of energy conservation in the flow, balancing the outward energy flux with the inward heat flux. Within the interface, the heat flux changes from a classical diffusive form,

(1) |

(Spitzer 1962) that applies when the electron mean free path is small compared to the temperature scale height (), to a saturated form,

(2) |

(Cowie & McKee 1977)
when the mean free path is comparable to, or greater than,
the temperature scale height.
In the above expressions, is the thermal
conductivity^{2}^{2}2 erg s cm K
where ,
is the electron density, and (Spitzer 1962).,
is the mass density, is the sound speed, and is a factor
of order unity
that allows for various uncertainties in the estimation of
.

The local saturation in the flow is defined by the ratio (Cowie & McKee 1977),

(3) |

and is proportional to the ratio of the mean free path to the temperature scale height.

For a classical diffusive interface, the rate of mass loss

(4) |

(Cowie & McKee 1977), where is the cloud radius, is the mean mass per particle, is the Boltzmann constant, and subscript “HIM” refers to the surrounding hot ionized medium. As the flow saturates, the heat flux is limited to a maximal rate given by equation (2), and the classical mass loss rate is reduced by a factor such that . The global level of saturation in the flow is parametrized by the saturation parameter,

(5) |

(Cowie & McKee 1977). In this expressions is the temperature, and is the gas pressure.

To obtain the steady state temperature profile in the conductive interface, the equation of energy conservation,

(6) |

must be solved, where is the cooling efficiency (erg cm s), and is the heating rate (erg s). The first term on the left hand side of equation (6) is a bulk kinetic energy term; the second is due to internal energy and work; and the next two terms account for radiative losses (cooling) and gains (heating). This outward energy flux is balanced by an inward heat flux represented by the right hand side of equation (6).

We rely on the formalism developed by Dalton & Balbus (1993, hereafter DB93) for determining the temperature profile in the interface. The DB93 formalism allows the thermal conductivity to change continuously from the classical diffusive form to a saturated form, by using an effective heat flux , defined as the harmonic mean of the local classical and saturated fluxes,

(7) |

DB93 then solve for the steady state temperature profile that develops in the interface layer between the warm cloud and the hot medium surrounding it.

When solving the energy equation, DB93 neglect the kinetic energy term, as well as the radiative losses (cooling) term. We discuss both assumptions in detail below. With these simplifying assumptions, the equation of energy conservation becomes,

(8) |

The temperature profile in the conductive interface^{3}^{3}3
DB93 write the solution for the scaled radius, , as
a function of the scaled temperature, , as
.
In this expression ,
are the modified Bessel functions of the
first kind, and are the modified Bessel functions of the
second kind.
then depends only
on the global saturation parameter, , which is a function
of the cloud radius, and the ambient pressure and temperature,
as given by equation (5).
Temperature profiles for various values of are
shown in Figure 1.

#### 2.1.1 On the Validity of the Subsonic Approximation

The kinetic energy term may be neglected when (see equation 6), or equivalently when . If the kinetic energy is small relative to the internal energy, the flow is isobaric, with a constant gas pressure .

In classical flows, is always less than unity, and the subsonic approximation holds. However, as the flow saturates, the maximal Mach number increases. In fully saturated flows, the heat flux is given by equation (2), and equation (6) reads

(9) |

This implies that in fully saturated flows

(10) |

(e.g. Cowie & McKee 1977). Therefore, for , the subsonic approximation holds throughout the flow.

As a consistency check, we also consider the local Mach number in the analytic profiles derived by Dalton & Balbus neglecting the kinetic energy term. In this case

(11) |

(DB93 equation B3).
Given a set of temperature profiles appropriate for
a range of saturation parameters (see Figure 1),
this equation shows that the maximal Mach number is an
increasing^{4}^{4}4DB93 argued, incorrectly, that the maximal Mach number is a
decreasing function of .
They used a specific set
of physical variables
( cm K,
K, pc) to derive the numerical
expression
(equation B5 in DB93).
They then used this expression to evaluate the Mach number
for a range of saturation parameters, and concluded that
the maximal Mach number is a decreasing function of .
However, the numerical coefficient in their expression was computed for
interface variables implying and is therefore
not valid for other saturation parameters. function of .
With the kinetic energy term neglected, the maximal
Mach number approaches in highly saturated flows^{5}^{5}5
As opposed to max resulting from equation
(10) which includes the kinetic energy term, for .,
implying that for the subsonic
approximation holds even for large .

Experimental evidence suggests that (see Balbus & McKee 1982). Throughout this paper we therefore assume , consistent with the use of subsonic approximation.

#### 2.1.2 On the Validity of the Non-Radiative Approximation

The fate of warm clouds that are embedded in a hot ambient medium depends of the delicate balance between the conductive heating and the radiative losses in the interface. Evaporation takes place if the radiative losses are small compared to conductive heating. If, on the other hand, cooling overcomes the conductive heating, the evaporation stops, and instead the ambient medium condenses onto the warm cloud.

McKee & Cowie (1977) used an approximate analytical method to determine the critical cloud radii at which radiative losses balance the conductive heating. They used fits to the equilibrium cooling functions of solar metallicity gas to derive these critical radii as a function of the external temperature and density. McKee & Cowie found that radiative losses are important only at low saturations, . Such small values of are generally out of the regime “occupied” by dwarf-galaxy scale halos (for which ).

Our models differ from those presented by McKee & Cowie. First, we explicitly follow the non-equilibrium ion fractions in the interface. As mentioned above, the outflowing gas may be underionized relative to gas in equilibrium at the same temperature. Underionized gas tends to radiate more efficiently (McCray 1987; Gnat & Sternberg 2007), thus increasing the contributions of radiative losses. Second, we include photoionization by an external background radiation field. This both increases the ionization level in the gas and provides heating, both effects acting to reduce the importance of radiative losses. Finally, we assume gas metallicity of solar, thus reducing the cooling efficiency. The relative importance of these three effects must be estimated numerically.

Using our numerical results we check, for each model, whether the emissivity associated with the heat flux is indeed everywhere larger than the radiative cooling. Given our non-equilibrium ion fractions (including photoionization, see section 2.2), we use the cooling functions included in Cloudy (version 07.02, Ferland et al. 1998) to compute the local cooling efficiencies. We then compare these values with .

Figure 2 displays our results.
Panel (a) shows the maximal ratio of the radiative to
conductive emissivity in each
model^{6}^{6}6We only display the maximum for
. As the temperature approaches the
ambient temperature, the heat flux approaches zero but
the cooling does not. as a function of the saturation.
Panel (b) shows the fraction of the interface temperature
over which the cooling is larger than the conductive
heating as a function of saturation.
The points (black and red) are for our
assumed heavy element abundances of times solar.
The gray circles show results for solar metallicity gas,
for a direct comparison with the critical derived
by McKee & Cowie (1977).

For times solar, both panels show that for (for ) radiative losses are significant and may not be neglected. For larger values of the importance of radiative cooling is generally smaller, and depends on the specific parameters of the evaporating cloud.

For , enhanced cooling and larger values of are associated with interfaces of higher gas densities, with larger neutral fractions at the cloud surface. We find that for our assumed K-photoionized clouds, the neutral fraction on the surface is significant when cm K. For example, for pressures of , , and cm K, the neutral fractions are , , and , respectively. Neutral gas cools very efficiently, and in underionized flows the lingering and enhanced contribution of H i Ly cooling significantly affects the energy balance.

To study the impact of the surface ionization on the flow
dynamics, we distinguish between “neutral cloud” models for
which cm K (red points),
and “ionized cloud” models with cm K (black
points)^{7}^{7}7 cm K corresponds to an ionization
parameter of on the cloud surface (see Section 3, equation 18)..
Figure 2 shows that for , radiative
losses are negligible for bounding pressures cm K.
However, in more neutral clouds cooling may be significant even at high saturations.

The fact that the limiting saturation parameter that we find is similar to the one derived by McKee & Cowie (1977, ) – despite the times lower metallicity that we use, and the inclusion of photoionization – implies that the cooling efficiencies in our non-equilibrium interfaces are considerably larger than their equilibrium counterparts. In fact, we find that Ly from underionized He dramatically increases the cooling efficiency in the interface. We discuss this point in detail in section 3.

For solar, we find that radiative losses are important up to a critical saturation parameter of (for ), and become less significant for larger values of . Since underionized He is one of the dominant coolants in the interface, increasing the metallicity from a tenth-solar to solar changes the critical saturation parameter by a factor smaller than . Figure 2 shows that this factor is .

For the tenth solar metallicity dwarf-galaxy-scale models that we use for comparison with the observations of the high-velocity metal-ion absorbers, the radiative losses are, in fact, everywhere smaller than the conductive heating.

Some examples are shown in Figure 3, in which we compare (solid curves) with the radiative losses (dashed curves) in three interface models. The left hand panels are for interfaces embedded in a high pressure corona, with cm K. We show results for a small, marginally saturated cloud ( kpc, ) in panel (a), and for a classical interface surrounding a larger cloud ( kpc, ) in panel (b). In panel (c) we present a highly saturated interface, with , surrounding a kpc cloud, embedded in a low pressure intergalactic medium ( cm K). In all cases, the radiative losses are everywhere smaller than the emissivity associated with the heat flux. These plots confirm that the significance of radiative losses is larger for lower saturation parameters, and becomes small at larger .

### 2.2. Ionization

Given the DB93 temperature profile, we solve
for the non-equilibrium abundances of the various
species in the interface.
We follow the time-dependent
ionization in the evaporating gas, given the initial photoionization
equilibrium state in the warm cloud.
We consider all ionization stages of the elements
H, He, C, N, O, Si, and S.
The temperature-dependent
ionization and recombination processes that we include are
collisional ionization by thermal electrons (Voronov 1997),
radiative recombination (Aldrovandi & Pequignot 1973;
Shull & van Steenberg 1982;
Landini & Monsignori Fossi 1990; Landini & Fossi 1991;
Pequignot, Petitjean, & Boisson 1991; Arnaud & Raymond 1992;
Verner et al. 1996),
dielectronic recombination (Aldrovandi & Pequignot 1973;
Arnaud & Raymond 1992;
Badnell et al. 2003, Badnell 2006;
Colgan et al. 2003, 2004, 2005;
Zatsarinny et al. 2003, 2004a, 2004b, 2005a, 2005b, 2006;
Altun et al. 2004, 2005, 2006;
Mitnik & Badnell 2004), and
neutralization and ionization by charge transfer reactions with
hydrogen and helium atoms and ions
(Kingdon & Ferland fits^{8}^{8}8See:
http://www-cfadc.phy.ornl.gov/astro/ps/data/cx
/hydrogen/rates/ct.html,
based on Kingdon & Ferland 1996,
Ferland et al. 1997, Clarke et al. 1998, Stancil et al. 1998;
Arnaud & Rothenflug 1985).

The time-dependent equations for the ion abundance fractions, , of element in ionization stage are,

(12) |

In this expression, is the gas velocity at radius in the evaporating flow. The parameters and are the rate coefficients for collisional ionization and recombination (radiative plus dielectronic), and , , , and are the rate coefficients for charge transfer reactions with hydrogen and helium that lead to ionization or neutralization. The quantities , , , , and are the particle densities (cm) for neutral hydrogen, ionized hydrogen, neutral helium, singly ionized helium, and electrons, respectively. are the photoionization rates of ions , due to externally incident radiation.

For the external radiation, we use the Sternberg, McKee & Wolfire (2002) fit for the present-day metagalactic field,

(13) |

to compute . In this expression erg s cm Hz sr, and is the Lyman limit frequency. This fit, which we plot in Figure 4, is based on observational constraints for the optical/UV and X-rays, (Bernstein et al. 2002; Martin et al. 1991; Chen et al. 1997) and on theoretical models (Haardt & Madau 1996) for the unobservable radiation from the Lyman limit to keV.

Recent observations and modeling have suggested that the spectral slope of the metagalactic radiation may be shallower near the Lyman limit, with varying as , and steepen gradually with increasing energy (e.g. Shull et al. 1999; Telfer et al. 2002; Scott et al. 2004; Shull et al. 2004; Zheng et al. 2004; Faucher-Giguere et al. 2009). In Appendix A we consider a radiation field (also plotted in Figure 4) that varies as from the Lyman limit to keV, and investigate the sensitivity of our results to the spectral slope. For reference, we also plot in Figure 4 the Faucher-Giguere et al. (2009) model computation of the present-day metagalactic field. In Appendix A we show that the column densities of the high ions (O vi and N v) are insensitive to the spectral slope at the Lyman limit, while the column densities of lower ions (e.g. Si iii) may be affected at a level of dex, mostly for low external bounding pressures.

The present-day metagalactic radiation field. The solid dark line is for the standard field as represented by Sternberg et al. (2002; see text and Equation 13), the gray line is for the Faucher-Giguere et al. (2009) field, and the dashed line is for the shallower ”maximal field” considered in Appendix A (Gnat & Sternberg 2004).

For each element , the ion fractions satisfy

(14) |

where is the density (cm) of ions in ionization stage of element , is total hydrogen density, and is the abundance of element relative to hydrogen. The sum is over all ionization stages of the element.

For the atomic elements that we include,
equations (12) are a set of coupled
ordinary differential equations (ODEs).
For the initial conditions, we assume that the cloud surface
is in photoionization equilibrium with the metagalactic filed,
and has a temperature of K. We advance the solutions
in small spatial steps , where is the current
distance from the cloud center (), which is associated with
a temperature through the analytical solutions
outlined by DB93.
We integrate equations (12) over the interval
using the same Livermore ODE solver^{9}^{9}9
See: http://www.netlib.org/odepack/
(Hindmarsh 1983) we used in Gnat & Sternberg (2007).
We verified that our code converges to the equilibrium abundances
given a constant temperature, by comparing our solutions to those
found by solving the set of algebraic equations for obtained
by setting in equation (12), as appropriate for steady
state.
We verified our time-dependent computations
by comparing with previous non-equilibrium models
(e.g. Sutherland & Dopita 1993), as discussed in detail in
Gnat & Sternberg (2007).

## 3. Metal Ions Fractions and Cooling in Evaporating Clouds

We computed the ionization states of the elements H, He, C, N, O, Si, and S in thermally conductive interfaces surrounding evaporating warm gas clouds. We calculate the non-equilibrium ionization states as a function of distance from the warm cloud surface. We consider a grid of models with pressures between and cm K, cloud radii between pc and kpc, and ambient (HIM) temperatures between and K. We assume heavy element abundance of times solar, as listed in Table 2. We begin with a brief discussion of our non-equilibrium conductive interfaces, and later focus on models for dwarf-galaxy-scale objects.

Element | Abundance |
---|---|

(X/H) | |

Carbon | |

Nitrogen | |

Oxygen | |

Silicon | |

Sulfur |

As the evaporating gas flows into the external, hotter, parts of the interface layer, its overall ionization state gradually increases. If the gas is heated faster than it is ionized, nonequilibrium effects become significant and the gas remains “underionized” throughout the flow. This happens when the heating time-scale, which is set by the flow time through the interface,

(15) |

becomes short compared to the ionization time scale for ion , .

In the absence of external photoionization, the ionization time scale is given by

(16) |

where is the collisional ionization coefficient for ion at temperature . The ionization state remains close to collisional ionization equilibrium when . Nonequilibrium effects become significant when . With no photoionization, the ratio of the flow to ionization time scale,

(17) |

is a function just of the saturation parameter and ambient temperature. The ion fractions and integrated metal-ion column densities then depend on just the two parameters and .

Here we also include photoionization by the external metagalactic radiation field, which increases the level of ionization in the gas, and provides heating. The ionization parameter at the surface of a photoionized, K cloud is given by,

(18) |

where is the Lyman limit frequency, is the speed of light, and is the hydrogen density (cm) at the cloud surface. Photoionization is more efficient in lower density gas. With photoionization included, the ionization states and integrated column densities depend on all three interface parameters, , , and . In general, we find that the ionization lag is so great that despite the inclusion of photoionization, the gas remains highly underionized, not only compared to photoionization equilibrium, but also compared to collisional ionization equilibrium (CIE).

Because the energy losses are dominated by atomic and ionic emission lines, the nonequilibrium cooling rates also differ (are enhanced for underionized gas) compared to equilibrium cooling. The lower ionization species that remain present at each temperature, most significantly He, are more efficiently excited by the thermal electrons

We demonstrate the effect of “under-ionization” in Figure 5, for a model with kpc, cm K, and K (). In the upper panel we show the nonequilibrium C ion fraction versus temperature (dashed curve), and the CIE ion fractions of C-C (solid curves, Gnat & Sternberg 2007). The nonequilibrium C abundance peaks at a temperature of K, whereas the CIE C abundance peaks at K . For CIE, the dominant ion at K is C.

The modified ionization states enhance the cooling efficiency in this interface. We computed the local cooling efficiencies using Cloudy (version 07.02, Ferland et al. 1998) given our nonequilibrium ion fractions. This is shown in panel (b). The CIE cooling efficiency is shown by the solid curve (Gnat & Sternberg 2007). The non-equilibrium cooling efficiency (dashed curve) is an order of magnitude larger in this model, over a significant temperature range. This is mostly due to He which persists to much higher temperatures in the nonequilibrium gas compared to CIE.

In Figure 6 we demonstrate the effect of non-equilibrium ionization for our complete set of models. In panel (a) we show the ionization lag for He. For each model, we plot the temperature to which He persists, with a fractional abundance larger than , versus its saturation parameter. These values are independent of the gas metallicity. The corresponding CIE value ( K) is shown by the dashed line. The shaded region indicates the range of saturation parameters for which no self-consistent evaporating solution exists (for solar). In this region, radiative cooling may not be neglected (see section 2.1.2), and results in condensation. For the evaporating models, departures from equilibrium ionization can clearly be seen. He persists to higher temperatures than in CIE, despite the contribution of external photoionization by the metagalactic radiation field. For highly saturated models, in which the temperature profile is extremely steep, He may persist to K.

The impact of these extended ion distributions on the cooling efficiencies is shown in panel (b). For each model we plot the maximal ratio of the non-equilibrium- to CIE-cooling efficiencies through the interface. The black points are for times solar, and the gray circles are for . For the evaporating models, cooling is indeed enhanced by factors to relative to the CIE cooling efficiency.

These enhanced cooling efficiencies explain the results presented in section 2.1.2 (Figure 2). We found that for solar and solar metallicity gas, radiative cooling has a small impact on the dynamics of the flow for saturation parameters and , respectively. Our results indicate that non-equilibrium effects significantly increase the value of at which radiative effects set in. For comparison, in CIE radiative losses in solar metallicity gas become significant for (for , McKee and Cowie 1977), a factor of lower than for nonequilibrium cooling. The reason for this is that the cooling efficiencies in the underionized gas are larger, due to the enhanced contributions of He and metal-ion resonance-line cooling.

Does non-equilibrium ionization modify the observational signatures of evaporating clouds? In Figure 7(a) we compare the equilibrium (dashed curves) and non-equilibrium (solid curves) ion distributions versus position in the evaporating layers, for a model with kpc, cm K, and K. We show the abundances of C, N, O, and Si including the effect of photoionization. The difference between the equilibrium and non-equilibrium curves is apparent: the non-equilibrium abundances reach their peaks further away from the cloud center (at higher temperatures) as the gas remains underionized, and they span considerably larger path-lengths than the equilibrium abundances. These larger path lengths result in enhanced column densities.

Finally, in panel (b) of Figure 7 we demonstrate the impact of photoionization by the metagalactic radiation field on the ionization states. Here we show a low-pressure model with cm K, K, and kpc. We compare the ionization state including photoionization by the metagalactic radiation field to the ionization in the absence of any external radiation. The presence of the external field increases the level of ionization both in the interface and in the surrounding HIM.

(K) | (kpc) | (cm K) | (g s) | max() | Gyr | |||
---|---|---|---|---|---|---|---|---|

Note. –

Results are shown for .

– The ionization parameter .

– The maximal Mach number in the interface, max.

– The evaporation time, Gyr,
where is the warm cloud gas mass in units of M,
and .

The highlighted lines correspond to the models considered in
sections 4.1 and 4.2 below.

### 3.1. Metal Ion Fractions in Dwarf-Galaxy-Scale Objects

We now focus on the ion fractions in conductive interfaces surrounding dwarf galaxy-scale halos for comparison with the observed metal-ion absorbers. As a guide to the physical properties of dwarf-scale halos, we rely on the results presented in GS04, where we considered median halos with a virial mass M. For the dwarf-scale halos, we assume ambient temperatures of K, K, and K, gas cloud radii of kpc, , and kpc, and bounding pressures in the range cm K. For bounding pressures in this range, the hydrogen in the K photoionized clouds is fully ionized at the surface, and the ionization parameter may be written as .

In Table 3 we list the input parameters for the dwarf-scale models that we consider. These include the ambient temperature , the cloud radius , and the gas pressure . We also list in Table 3 the ionization parameter , saturation parameter , the classical (diffusive) mass loss rate , the mass-loss suppression factor , the maximal Mach number in the flow, max, and the evaporation time scale, , for clouds with warm gas masses of M. All the values listed in Table 3 are computed assuming .

In Figure set 8, we present the ion fractions as a function of distance from the cloud surface for the grid of dwarf galaxy-scale models. The ambient temperature, , cloud radius, , and bounding pressure, are indicated in each panel. Figure 8 displays the ion fractions of the low ions C ii, N i, O i, Si ii, and S ii, and the high-ions C iv, N v, O vi, Si iii, Si iv, and S iii. These are the ions that were measured by Sembach et al. (1999, 2000), Collins et al. (2004), and Fox et al. (2005) in the high-velocity metal ion absorbers (see e.g. Table 1).

## 4. Metal-Absorption Column Densities

In this section we present computations of the integrated metal-ion column densities that are produced in the conductively evaporating layers.

The characteristic column density in an interface, , is given by (see equation 5),

(19) |

where is the mean mass per particle, is the HIM temperature in units of K, and . This equation shows that in general, models with lower saturation parameters produce larger columns. The column density of a given ionic species is related to this characteristic column by an “ionization correction”, that depends on the interface parameters.

For gas in collisional ionization equilibrium, the local ion fractions are a function of the temperature only. The ionization correction and implied metal-ion column densities therefore depend only on (through its control of the temperature profile, see Figure 1) and on the ambient temperature, . However, the functional dependence on and may be complicated.

For non-equilibrium conditions, departures from equilibrium ionization
and the ion fractions at a given temperature depend on the ratio of
the heating time-scale and the ionization time-scale.
Equation 17 shows that
in the absence of photoionization, the non-equilibrium abundances
and columns still depend only on and .
The inclusion of photoionization by the metagalactic radiation field
introduces an implicit dependence on the gas density (or ionization
parameter, see equation 18).
The non-equilibrium column densities are therefore
general functions of all three interface parameters^{10}^{10}10We find that the
column densities listed in Table 4 cannot be
expressed as simple power-law fits in , ,
and ..

Here we wish to find conditions that yield metal-ion column densities comparable to those observed in the high-velocity ionized absorbers, including the high observed O vi column ( cm). We focus on the column densities of the high ions C iv, N v, O vi, Si iii, Si iv, and S iii, as a function of the impact parameter from the cloud center, . In Table 4, we list the (two-sided) central () and maximal column densities for the different models considered in section 3. For the evaporating gas, the line-of-sight column density of ion of element observed at an impact parameter is,

(20) |

where is the abundance of element relative to hydrogen, is the position dependent hydrogen density, and are the position-dependent ion fractions, as presented in section 3. The maximal columns are evaluated for each ion independently, and occur at different impact parameters for each ion and model.

The results presented in Table 4 are computed for times solar. Since the fractional ion abundances are independent of metallicity, the integrated columns are proportional to . (However, as discussed in 2.1.2, the range of values for which a self-consistent evaporating solution can be found is a function of .)

Table 4 lists the central column densities produced in the interface layers. When observing evaporating clouds, the observed absorption line properties will generally depend not only on the conductive interface, but also on the ionization properties of the embedded photoionized clouds. In the text that follows we discuss, as illustrative examples, the total (cloud interface) column densities that arise in two specific models. First we consider a low-ionization cloud embedded in a high-pressure corona, with cm K (§4.1), and then we study a more highly-ionized cloud embedded a low-pressure medium, with cm K (in §4.2). The lines representing these two illustrative models are highlighted in Table 3.

### 4.1. A Low-Ionization Cloud Embedded in a High Pressure Medium

As an example for a low-ionization photoionized cloud embedded in a high-pressure corona, we rely on the physical properties of the “CHVC-scale” model presented in GS04 (“model A” in GS04). This model represents a M dark-matter halo that contains a warm gas mass of M, which is subjected to an external bounding pressure, cm K. The resulting photoionized cloud has a radius kpc. The ionization parameter on the cloud surface is low, , and high ions are thus inefficiently produces in this cloud. Here we assume that the cloud is surrounded by a hot Galactic corona, with a temperature K, and photoionized by the metagalactic radiation field.

These parameters imply that the cloud evaporates in a marginally-saturated flow, with . The mass loss rate is suppressed by a factor relative to that in a classical diffusive flow, and the maximal Mach number in the flow is .

Figure 8.1 displays the ion fractions that arise in the conductive interface in this model. In evaluating the absorption-line signatures that arise from the evaporating cloud, we use the abundances given in GS04 for “model A” (Figure 3 in GS04) for the ion fractions within the photoionized cloud, and the ion fractions computed here and presented in Figure 8.1 for the evaporating layers. We then evaluate the projected column densities as a function of impact parameter from the cloud center. These column densities include the contributions of both the photoionized cloud and the conductive interface surrounding it.

Figure 9 shows the projected column densities in our evaporating high-pressure model. The ionization in this conductive interface results in a maximal O vi column density of cm (compared to cm in the purely photoionized cloud), and a C iv column density of cm (compared to cm in the purely photoionized cloud). The evaporating layers increase the O vi column by orders of magnitude, and the C iv column by order of magnitude relative to the purely photoionized cloud discussed in GS04. However, for this model, the enhanced columns still do not bring the model into harmony with the observations of the high-velocity metal-ion absorbers (see e.g. Table 1).

For pressure supported photoionized clouds in high-pressure environments ( cm), the ionization parameter is generally too low to allow for efficient production of high ions such as C and O. The high-ion column densities are therefore dominated by ions produced in the evaporating layers, and are not significantly affected by the details of the cloud’s ionization state.

### 4.2. A Highly-ionized Cloud Embedded in a Low Pressure Medium

As an illustrative example for the physical properties of a more highly ionized photoionized cloud embedded in a low-pressure medium, we rely on the standard dwarf-scale model presented in GS04 (“model C” in GS04), based on the observed properties of Local Group dwarf galaxies Leo A and Sag DIG (Sternberg et al. 2002). This model represents a M dark-matter halo that contains a total warm gas mass of M, and is subjected to an external bounding pressure cm K. The resulting cloud has a radius kpc. The ionization parameter on the surface of this cloud is higher, , and high ions such as C and N are efficiently produced in the ionized envelopes of this cloud. Here we assume that the halo is surrounded by a hot intergalactic medium, with a temperature K, and is photoionized by the metagalactic radiation field.

T | R | P/k | C IV | N V | O VI | Si III | Si IV | S III | ||
---|---|---|---|---|---|---|---|---|---|---|

85.62 | central | |||||||||

maximal | ||||||||||

8.83 | central | |||||||||

maximal | ||||||||||

1.81 | central | |||||||||

maximal | ||||||||||

0.91 | central | |||||||||

maximal | ||||||||||

0.46 | central | |||||||||

maximal | ||||||||||

0.19 | central | |||||||||

maximal | ||||||||||

3.34 | central | |||||||||

maximal | ||||||||||

666.8 | central | |||||||||

maximal | ||||||||||

68.70 | central |