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–1– 5. Thermal Properties and Line Diagnostics for HII Regions 5.1

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–1– 5. Thermal Properties and Line Diagnostics for HII Regions 5.1 { 1 { 5. Thermal Properties and Line Diagnostics for HII Regions 5.1. Heating and Cooling The temperature of the ISM is governed by a number of heating and cooling processes. These include both adiabatic changes in volume, which do not alter the entropy of the gas, and collisional and radiative processes that do. The First Law of Thermodynamics can be expressed as de = dq ¡ pd½¡1; (1) where dq is the heat added to the system per unit mass, ½¡1 the speci¯c volume (i.e., volume per unit mass), 1 p e = + e (2) γ ¡ 1 ½ I the speci¯c energy, γ the ratio of speci¯c heats, and eI the speci¯c chemical binding plus ionization 5 energy. For a monatomic gas, γ = 3 , whereas a gas of diatomic molecules at about room temperture 7 has two additional degrees of freedom and γ = 5 . At temperatures low enough that the rotational 5 degrees of freedom cannot be excited (as is the case for H2 in cold molecular clouds), γ = 3 . Here we shall assume that γ = const, noting that the results here must be modi¯ed if one wishes to treat the case in which molecular gas is heated from low to moderate temperatures. The chemical/ionization ¡1 energy term is eI = ½ n®I®, where ® labels the ion species and I® is the chemical/ionization energy per particle. For convenience, we de¯ne the zero point for e to consist of neutral atoms P I (H I). It follows that IH¤ = 13:6 eV and IH2 = ¡4:48 eV. Let ¹ be the mass per particle so that ½ = n¹, where n is the number density of particles, and p = nkT = ½kT=¹. Converting equation (1) to a rate equation and factoring out p=½ = kT=¹, we ¯nd dq p 1 d p 1 de = ln + d ln + I ; (3) dt ½ γ ¡ 1 dt ½ ½ dt · µ ¶ µ ¶¸ kT d p de = ln + I ; (4) ¹(γ ¡ 1) dt ½γ dt µ ¶ where d=dt = @=@t + v ¢ 5 is the comoving, or Lagrangian, derivative. The quantity ½dq is the total amount of heat added to the gas per unit volume. If dq = 0, the process is said to be adiabatic; for γ deI = 0, this implies p / ½ . Since heating processes are generally per particle, we de¯ne the heating rate per unit volume 1 as nH¡. Cooling processes generally involve collisions between two particles, so they scale as the 1 The mass density ½ can also be written in terms of the number density of hydrogen nuclei as ½ = nH¹H, where ¹H ¡24 is the mass per hydrogen nucleus. For a He abundance that is 10% that of hydrogen, ¹H = 1:4mH = 2:34 £ 10 g. Expressing the mass density in terms of nH is convenient if the gas is being ionized, for example, since then nH is constant at ¯xed volume, whereas n changes. Equivalently, ¹H is a constant in an ionizing gas, but ¹ changes. { 2 { 2 2 square of the density; we therefore write the cooling rate per unit volume as nH¤. The total rate at which heat is added to the gas per unit volume is then dq p d p de ½ = n ¡ ¡ n2 ¤ = ln + ½ I : (5) dt H H γ ¡ 1 dt ½γ dt µ ¶ 5 For γ = 3 and constant composition, this simpli¯es to d 3 dn n kT ¡ kT = n ¡ ¡ n2 ¤; (6) dt 2 dt H H which is the form given in Spitzer (eq. 6.1). If the gas is in a steady state, equation (5) becomes ¡ = nH¤ (steady state). (7) It is sometimes convenient to describe the evolution of the thermal state of the gas in terms of the speci¯c entropy s. For reversible processes, T ds = dq; for irreversible ones (like shocks), T ds ¸ dq, which is a statement of the Second Law of Thermodynamics. The speci¯c entropy is a function of the state variables of the system, s = s[T; n(H0); n(H+); n(He0); :::]. For reversible processes with constant composition and internal energy (¹; eI const), equation (4) can be integrated to give the speci¯c entropy of the gas, k T 1=(γ¡1) s = ln + const. (8) ¹ " n # 5 For γ = 3 the argument of the logarithm is proportional to the volume in phase space, as expected, since T 3=2=n / momentum3 £ volume. 5.1.1. Heating and Cooling Processes Here we shall give a brief summary of the heating and cooling processes that occur throughout the ISM, not just in H II regions. The heating processes fall into several categories: Radiative heating processes, nH¡ = 4¼·J: - Photoionization (Lec. 4). This dominates the heating of H II regions and of the warm ionized medium (WIM) in the ISM. - X-ray photoionization: This requires special treatment for two reasons. First, much of the heating is to due photoionization of the metals, in which the X-ray knocks out a K- shell electron. Second, the electron that is ionized has a high energy and can collisionally ionize and excite other atoms and heat ambient electrons (see Wol¯re et al 1995). 2 Note that some people rede¯ne ¡ and ¤ so as to include the deI =dt term. For example, if a 1 eV electron recombines to the ground state of hydrogen, the energy loss would be taken to be 1 eV instead of 14.6 eV, which is the energy emitted. { 3 { - Photoelectric heating by dust: When FUV photons are absorbed by dust grains, a photoelectron is ejected, heating the ambient gas. This is believed to dominate the heating of the H I in the ISM. - Photodissociation heating: This is analogous to photoionization heating, except that it occurs when H2 molecules are photodissociated. - Collisional heating (or cooling) by dust: As we shall see in the next two lectures, dust is coupled very strongly to optical and UV radiation, so it is possible for the dust to be at a di®erent temperature than the gas. Near strong sources of radiation, such as early-type stars, the dust can be hotter than the gas, and the gas will be heated by collisions with the dust grains. Since this is a collisional process that varies as n2, it is important only at high densities. Dynamical heating - Shock heating: Shocks are hydrodynamic discontinuities in that the equations of hy- drodynamics, which are based on small mean free paths, break down in the thin shock transition layer. The shock jump conditions, to be discussed later in the course, deter- mine the temperature and density behind a shock; the entropy always increases across a shock. Shocks are believed to dominate the heating of the hot component of the ISM (the hot ionized medium, or HIM). - Turbulent heating: The ISM is observed to have substantial bulk motions that are believed to be turbulent. If the velocity on a scale ` is v(`), then the characteristic damping time for these motions is the dynamical time `=v(`). The heating rate per unit mass is then ¡ v(`)2=2 v(`)3 » = : (9) ¹H `=v(`) 2` The actual heating mechanism is that the motions cascade to smaller and smaller scales until they are dissipated by viscosity. It follows that the heating rate is independent of scale ` so that v(`) / `1=3, as ¯rst shown by Kolmogorov. Turbulence in the ISM is complicated by the fact that it is magnetized and can be supersonic, but the Kolmogorov relation remains approximately valid. The importance of turbulent heating in the ISM is under active investigation. Cosmic ray heating: - High energy particles permeate the ISM, and we can observe those that are able to penetrate through the solar wind to reach the Earth or spacecraft in the solar system. These particles ionize atoms, and the ejected electrons heat the gas. We shall discuss this further when we talk about cosmic rays in more detail. Magnetic heating { 4 { - Reconnection: The magnetic energy in the ISM is comparable to or greater than the thermal energy. If this energy can be dissipated, it can be a signi¯cant heat source. In ionized components of the ISM, the ¯eld is \frozen" to the plasma and cannot directly heat it. However, on su±ciently small scales, flux freezing breaks down and the topology of the ¯eld can be altered by magnetic reconnection. Reconnection is observed in the Sun and in the Earth's magnetosphere. Energy goes both into heat and into bulk motions, but the process is not well understood. - Ambipolar di®usion: The magnetic ¯eld is directly coupled to the ionized component of the gas, which in turn is coupled to the neutral component only through collisions. If the drift velocity between the neutral and ionized components is signi¯cant, as it can be in the weakly ionized gas in molecular clouds, then this process can lead to signi¯cant heating. Next we consider the cooling processes, which are only three in number: Collisional excitation (bound-bound): - Particle collisions (electrons, atoms or molecules impinging on atoms or molecules) can excite internal degrees of freedom which can then decay by radiative emission. This is the dominant cooling process in the ISM. Recombination cooling (bound-free). - When an electron recombines with an ion, an energy ee; rec + I, where I is the ionization energy, is radiated. In Case A, the radiation escapes from the gas, and the recombination (1) coe±cient is written as ®A or ® , indicating that recombinations to all states are e®ective. In Case B, photons arising from recombinations to the ground state photoionize other atoms and do not escape; as a result, only recombinations to excited states result in a net recombination.
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