Density logn (cm-3)
Figure 2.5 (a) Theoretical prediction for the equilibrium temperature of interstellar gas, displayed as a function of the number density n. (b) Equilibrium pressure nT as a function of number density. The horizontal dashed line indicates the empirical nT -value for the interstellar medium.
a factor of two, by the corresponding product for the warm neutral medium (n = 0.5 cm~3; T = 8 x 103 K). The numbers suggest that HI clouds and the warm neutral medium might profitably be viewed as two phases of the interstellar medium that coexist in pressure equilibrium.
We may pursue this idea quantitatively as follows. The thermal energy content of any gas results from the balance of heating and cooling processes. As we will detail in Chapter 7, rates for these processes can be determined by both theoretical and empirical methods. Assuming this information to be at hand, imagine slowly compressing a parcel of interstellar gas to successively higher densities. At each step, the gas will assume an equilibrium temperature and pressure, which can be determined by equating the heating and cooling rates. We can then examine those states that have pressures matching the known interstellar value.
The results of such a calculation, for gas of solar composition, are displayed in Figure 2.5. Over the range of densities shown, heating is provided mostly by starlight, through its ejection of electrons from the surfaces of interstellar grains. Figure 2.5a shows that this heating yields, at lower densities, temperatures approaching 104 K, where internal electron levels of hydrogen are excited. The gas in this regime cools primarily through emission of the Lya line. At higher densities, hydrogen is in the ground electronic state and becomes an inefficient radiator. Atomic carbon, however, remains ionized by ambient ultraviolet photons, and the gas cools through the 158 pm transition of C II.
The solid curve in Figure 2.5b shows the predicted run of pressure (represented as nT = P/kB, where kB is Boltzmann's constant) as a function of density. A parcel of gas with pressure lying above this curve cools faster than it can be heated, while the reverse is true for points which lie below. The equilibrium curve crosses the mean empirical P/kB-value of 3 x 103 K cm~3 (dashed horizontal line) at three distinct points. Imagine first that the gas finds itself at point B. Suppose further that it is compressed slightly while maintaining pressure equilibrium with its surroundings. Since its representative point now crosses above the equilibrium curve, it must cool until it reaches point C. Conversely, any slight expansion of the same parcel causes it to heat up until it reaches point A. Point B, therefore, represents a thermally unstable state. Furthermore, the density and temperature at the stable point A (n = 0.4 cm~3; T = 7000 K), are seen to match those in the warm neutral medium. Conditions at point C
(n = 60 cm-3; T = 50 K), where the gas is said to be in the cold neutral medium, are just those in typical HI clouds.
From this analysis, we have strong reason to believe that any large mass of atomic hydrogen gas naturally divides into two components with very different properties. Of course, we still do not know how the actual separation occurs in detail. Nor does the foregoing reasoning tell us the relative fractional volumes of the two phases, much less the sizes and masses of individual HI clouds. At present, our knowledge of the atomic component is simply too rudimentary to address these issues from a theoretical perspective.
What of the molecular gas? Can it, too, be regarded as a phase of the interstellar medium? In any molecular cloud, densities are sufficiently high that self-gravity plays a dominant role in the cloud's mechanical equilibrium. In other words, the product nT deep inside can be much greater than the background value, since the interior pressure must also resist the weight of overlying gas. On the other hand, the more rarified material in the cloud's outer layers must still match the external pressure if this region is not to expand or contract. In Chapter 8, we will use this requirement to establish plausible surface conditions for molecular clouds.
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