### 5. ESTIMATES OF OUTFLOW RATES

While it is relatively straightforward to demonstrate that a superwind is present, it is more difficult to robustly calculate the rates at which mass, metals, and energy are being transported out by the wind. Several different types of data can be used, each with its own limitations and required set of assumptions.

X-Ray Emission: X-ray imaging spectroscopy yields the superwind's "emission integral". Presuming that the X-ray spectra are fit with the correct model for the hot gas it follows that the mass and energy of the X-ray gas scale as follows: MX (LXf )1/2 and EX (LXf )1/2TX(1 + 2). Here f is the volume-filling-factor of the X-ray gas and is its Mach number. Numerical hydrodynamical simulations of superwinds suggest that 2 = 2 to 3 (Strickland & Stevens 2000). The associated outflow rates ( and ) can then be estimated by dividing MX and EX by the crossing time of the observed region: t (R/cs), where cs is the speed-of-sound.

If the X-ray-emitting gas is assumed to be volume-filling (f unity), the resulting values for and are then very similar to the starburst's rates of kinetic energy deposition and star formation respectively. As described above, Chandra images (Fig. 1) show that the X-ray-emitting gas does not have unit volume filling factor. On morphological and physical grounds we have argued that f is of-order 10-1. This would mean that previous estimates of and are overestimated by a factor of 3.

Optical Emission: Optical data on the warm (T 104 K) ionized gas can be used to determine the outflow rates and in a way that is quite analogous to the X-ray data. In this case, the outflow velocities can be directly measured kinematically from spectroscopy. Martin (1999) found the implied values for are comparable to (and may even exceed) the star-formation rate.

In favorable cases, the densities and thermal pressures can be directly measured in the optical emission-line clouds using the appropriate ratios of emission lines. The thermal pressure in these clouds traces the ram-pressure in the faster outflowing wind that is accelerating them (hydrodynamical simulations suggest that Pram = Pcloud, where = 1 to 10). Thus, for a wind with a mass-flux that freely flows at a velocity v into a solid angle , we have

Based on observations and numerical models, the values v 103 km s-1, a few, and /4 a few tenths are reasonable. The radial pressure profiles Pcloud(r) measured in superwinds by Heckman, Armus, & Miley (1990) and Lehnert & Heckman (1996) then imply that is comparable to the star-formation rate and that is comparable to the starburst kinetic-energy injection rate (implying that radiative losses are not severe).

Interstellar Absorption-Lines: The use of interstellar absorption-lines to determine outflows rates offer several distinct advantages. First, since the gas is seen in absorption against the background starlight, there is no possible ambiguity as to the sign (inwards or outwards) of any radial flow that is detected, and the outflow speed can be measured directly (e.g. Fig. 2). Second, the strength of the absorption will be related to the column density of the gas. In contrast, the X-ray or optical surface-brightness of the emitting gas is proportional to the emission-measure. Thus, the absorption-lines more fully probe the whole range of gas densities in the outflow, rather than being strongly weighted in favor of the densest material (which may contain relatively little mass).

The biggest obstacle to estimating outflows rates is that the strong absorption-lines are usually saturated, so that their equivalent width is determined primarily by the velocity dispersion and covering factor, rather than by the ionic column density. In the cases where the rest-UV region can be probed with adequate signal-to-noise (Pettini et al. 2000; Heckman & Leitherer 1997), the total HI column in the outflow can be measured by fitting the damping wings of the Ly interstellar line, while ionic columns may be estimated from the weaker (less saturated) interstellar lines. In the Heckman et al. (2000) survey of the NaD line, we estimated NaI columns in the outflows based on the NaD doublet ratio (Spitzer 1968), and we then estimated the HI column assuming that the gas obeyed the same relation between NHI and NNaI as in the Milky Way. These HI columns agreed with columns estimated independently from the line-of-sight color excess E(B - V) toward the starburst, assuming a Galactic gas-to-dust ratio. From both the UV data and the NaD data, the typical inferred values for NHI are of-order 1021 cm-2.

We can then adopt a simple model of a superwind flowing into a solid angle at a velocity v from a minimum radius r* (taken to be the radius of the starburst within which the flow originates). This implies:

Based on this simple model, Heckman et al. (2000) estimated that the implied outflow rates of cool atomic gas are comparable to the star-formation rates (e.g. several tens of solar masses per year in powerful starbursts). The flux of kinetic energy carried by this material is substantial (of-order 10-1 of the kinetic energy supplied by the starburst). We also estimated that 1% of the mass in the outflow is the form of dust grains.

Summary: The various techniques for estimating the outflow rates in superwinds rely on simplifying assumptions (not all of which may be warranted). On the other hand, it is gratifying that the different techniques do seem to roughly agree: the outflows carry mass out of the starburst at a rate comparable to the star-formation rate and kinetic/thermal energy out at a rate comparable to the rate supplied by the starburst.